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<li class="md-nav__item">
|
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<a href="../fdtd/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
fdtd
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
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|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item md-nav__item--active">
|
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<input class="md-nav__toggle md-toggle" type="checkbox" id="__toc">
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|
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|
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<label class="md-nav__link md-nav__link--active" for="__toc">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
fdmath
|
||
|
||
|
||
|
||
</span>
|
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|
||
|
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|
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<span class="md-nav__icon md-icon"></span>
|
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</label>
|
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|
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<a href="./" class="md-nav__link md-nav__link--active">
|
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|
||
|
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|
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<span class="md-ellipsis">
|
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|
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|
||
fdmath
|
||
|
||
|
||
|
||
</span>
|
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|
||
|
||
|
||
</a>
|
||
|
||
|
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|
||
<nav class="md-nav md-nav--secondary" aria-label="Table of contents">
|
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|
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|
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|
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|
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|
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<label class="md-nav__title" for="__toc">
|
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<span class="md-nav__icon md-icon"></span>
|
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Table of contents
|
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</label>
|
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
|
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|
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<li class="md-nav__item">
|
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<a href="#meanas.fdmath" class="md-nav__link">
|
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<span class="md-ellipsis">
|
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|
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<code class="doc-symbol doc-symbol-toc doc-symbol-module"></code> fdmath
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|
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</span>
|
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</a>
|
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|
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<nav class="md-nav" aria-label=" fdmath">
|
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<ul class="md-nav__list">
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|
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<a href="#meanas.fdmath--fields-functions-and-operators" class="md-nav__link">
|
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<span class="md-ellipsis">
|
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|
||
Fields, Functions, and Operators
|
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</span>
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</a>
|
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|
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</li>
|
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|
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<li class="md-nav__item">
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<a href="#meanas.fdmath--discrete-calculus" class="md-nav__link">
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<span class="md-ellipsis">
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Discrete calculus
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|
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<nav class="md-nav" aria-label="Discrete calculus">
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|
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<span class="md-ellipsis">
|
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|
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Scalar derivatives and cell shifts
|
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|
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</span>
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||
</a>
|
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|
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</li>
|
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|
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<li class="md-nav__item">
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<a href="#meanas.fdmath--gradients-and-fore-vectors" class="md-nav__link">
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<span class="md-ellipsis">
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Gradients and fore-vectors
|
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</span>
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</a>
|
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</li>
|
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|
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<li class="md-nav__item">
|
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<a href="#meanas.fdmath--divergences" class="md-nav__link">
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<span class="md-ellipsis">
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Divergences
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</span>
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</a>
|
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|
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</li>
|
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|
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<li class="md-nav__item">
|
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<a href="#meanas.fdmath--curls" class="md-nav__link">
|
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<span class="md-ellipsis">
|
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|
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Curls
|
||
|
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</span>
|
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</a>
|
||
|
||
</li>
|
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|
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</ul>
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</nav>
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</li>
|
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<li class="md-nav__item">
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<a href="#meanas.fdmath--maxwells-equations" class="md-nav__link">
|
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<span class="md-ellipsis">
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Maxwell's Equations
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|
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</span>
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</a>
|
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|
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<nav class="md-nav" aria-label="Maxwell's Equations">
|
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<ul class="md-nav__list">
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<li class="md-nav__item">
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<a href="#meanas.fdmath--wave-equation" class="md-nav__link">
|
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<span class="md-ellipsis">
|
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|
||
Wave equation
|
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|
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</span>
|
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</a>
|
||
|
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</li>
|
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|
||
<li class="md-nav__item">
|
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<a href="#meanas.fdmath--frequency-domain" class="md-nav__link">
|
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<span class="md-ellipsis">
|
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|
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Frequency domain
|
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|
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</span>
|
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</a>
|
||
|
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</li>
|
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|
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<li class="md-nav__item">
|
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<a href="#meanas.fdmath--plane-waves-and-dispersion-relation" class="md-nav__link">
|
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<span class="md-ellipsis">
|
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|
||
Plane waves and Dispersion relation
|
||
|
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</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
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</ul>
|
||
</nav>
|
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|
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</li>
|
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|
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<li class="md-nav__item">
|
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<a href="#meanas.fdmath--grid-description" class="md-nav__link">
|
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<span class="md-ellipsis">
|
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|
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Grid description
|
||
|
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</span>
|
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</a>
|
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|
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<nav class="md-nav" aria-label="Grid description">
|
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<ul class="md-nav__list">
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<li class="md-nav__item">
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<a href="#meanas.fdmath--datastructure-dx_lists_t" class="md-nav__link">
|
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<span class="md-ellipsis">
|
||
|
||
Datastructure: dx_lists_t
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
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|
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</ul>
|
||
</nav>
|
||
|
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</li>
|
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|
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<li class="md-nav__item">
|
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<a href="#meanas.fdmath--permittivity-and-permeability" class="md-nav__link">
|
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<span class="md-ellipsis">
|
||
|
||
Permittivity and Permeability
|
||
|
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</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
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</ul>
|
||
</nav>
|
||
|
||
</li>
|
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|
||
<li class="md-nav__item">
|
||
<a href="#functional-and-sparse-operators" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
Functional and sparse operators
|
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|
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</span>
|
||
</a>
|
||
|
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</li>
|
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|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.functional" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-module"></code> functional
|
||
|
||
</span>
|
||
</a>
|
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|
||
<nav class="md-nav" aria-label=" functional">
|
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<ul class="md-nav__list">
|
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|
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<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.functional.deriv_forward" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> deriv_forward
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
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|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.functional.deriv_back" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> deriv_back
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.functional.curl_forward" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> curl_forward
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.functional.curl_back" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> curl_back
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
</ul>
|
||
</nav>
|
||
|
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</li>
|
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|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.operators" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-module"></code> operators
|
||
|
||
</span>
|
||
</a>
|
||
|
||
<nav class="md-nav" aria-label=" operators">
|
||
<ul class="md-nav__list">
|
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|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.operators.shift_circ" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> shift_circ
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.operators.shift_with_mirror" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> shift_with_mirror
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.operators.deriv_forward" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> deriv_forward
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.operators.deriv_back" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> deriv_back
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.operators.cross" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> cross
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.operators.vec_cross" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> vec_cross
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.operators.avg_forward" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> avg_forward
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.operators.avg_back" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> avg_back
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.operators.curl_forward" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> curl_forward
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.operators.curl_back" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> curl_back
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
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|
||
<li class="md-nav__item">
|
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<a href="#meanas.fdmath.vectorization" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-module"></code> vectorization
|
||
|
||
</span>
|
||
</a>
|
||
|
||
<nav class="md-nav" aria-label=" vectorization">
|
||
<ul class="md-nav__list">
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.vectorization.vec" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> vec
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.vectorization.unvec" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> unvec
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
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</ul>
|
||
</nav>
|
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|
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</li>
|
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|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-module"></code> types
|
||
|
||
</span>
|
||
</a>
|
||
|
||
<nav class="md-nav" aria-label=" types">
|
||
<ul class="md-nav__list">
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types.dx_lists_t" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-attribute"></code> dx_lists_t
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
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|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types.dx_lists2_t" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-attribute"></code> dx_lists2_t
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types.dx_lists_mut" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-attribute"></code> dx_lists_mut
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types.dx_lists2_mut" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-attribute"></code> dx_lists2_mut
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types.fdfield_updater_t" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-attribute"></code> fdfield_updater_t
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types.cfdfield_updater_t" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-attribute"></code> cfdfield_updater_t
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types.fdfield" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> fdfield
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types.vfdfield" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> vfdfield
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types.cfdfield" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> cfdfield
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types.vcfdfield" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> vcfdfield
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types.fdslice" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> fdslice
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types.vfdslice" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> vfdslice
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types.cfdslice" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> cfdslice
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types.vcfdslice" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> vcfdslice
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
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|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types.fdfield2" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> fdfield2
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types.vfdfield2" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
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|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> vfdfield2
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
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|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types.cfdfield2" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> cfdfield2
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
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|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types.vcfdfield2" class="md-nav__link">
|
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<span class="md-ellipsis">
|
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|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> vcfdfield2
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
</ul>
|
||
|
||
</nav>
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||
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
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</div>
|
||
</div>
|
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</div>
|
||
|
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|
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|
||
<div class="md-sidebar md-sidebar--secondary" data-md-component="sidebar" data-md-type="toc" >
|
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<div class="md-sidebar__scrollwrap">
|
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<div class="md-sidebar__inner">
|
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|
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|
||
<nav class="md-nav md-nav--secondary" aria-label="Table of contents">
|
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|
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|
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|
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|
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|
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|
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<label class="md-nav__title" for="__toc">
|
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<span class="md-nav__icon md-icon"></span>
|
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Table of contents
|
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</label>
|
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
|
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|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-module"></code> fdmath
|
||
|
||
</span>
|
||
</a>
|
||
|
||
<nav class="md-nav" aria-label=" fdmath">
|
||
<ul class="md-nav__list">
|
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|
||
<li class="md-nav__item">
|
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<a href="#meanas.fdmath--fields-functions-and-operators" class="md-nav__link">
|
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<span class="md-ellipsis">
|
||
|
||
Fields, Functions, and Operators
|
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|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
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|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath--discrete-calculus" class="md-nav__link">
|
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<span class="md-ellipsis">
|
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|
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Discrete calculus
|
||
|
||
</span>
|
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</a>
|
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|
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<nav class="md-nav" aria-label="Discrete calculus">
|
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<ul class="md-nav__list">
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|
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|
||
<span class="md-ellipsis">
|
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|
||
Scalar derivatives and cell shifts
|
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|
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</span>
|
||
</a>
|
||
|
||
</li>
|
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|
||
<li class="md-nav__item">
|
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<a href="#meanas.fdmath--gradients-and-fore-vectors" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
Gradients and fore-vectors
|
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|
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</span>
|
||
</a>
|
||
|
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</li>
|
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|
||
<li class="md-nav__item">
|
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<a href="#meanas.fdmath--divergences" class="md-nav__link">
|
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<span class="md-ellipsis">
|
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|
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Divergences
|
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|
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</span>
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</a>
|
||
|
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</li>
|
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|
||
<li class="md-nav__item">
|
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<a href="#meanas.fdmath--curls" class="md-nav__link">
|
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<span class="md-ellipsis">
|
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|
||
Curls
|
||
|
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</span>
|
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</a>
|
||
|
||
</li>
|
||
|
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</ul>
|
||
</nav>
|
||
|
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</li>
|
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|
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<li class="md-nav__item">
|
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<a href="#meanas.fdmath--maxwells-equations" class="md-nav__link">
|
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<span class="md-ellipsis">
|
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|
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Maxwell's Equations
|
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|
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</span>
|
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</a>
|
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|
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<nav class="md-nav" aria-label="Maxwell's Equations">
|
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<ul class="md-nav__list">
|
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|
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<li class="md-nav__item">
|
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<a href="#meanas.fdmath--wave-equation" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
Wave equation
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
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|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath--frequency-domain" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
Frequency domain
|
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|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
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|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath--plane-waves-and-dispersion-relation" class="md-nav__link">
|
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<span class="md-ellipsis">
|
||
|
||
Plane waves and Dispersion relation
|
||
|
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</span>
|
||
</a>
|
||
|
||
</li>
|
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|
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</ul>
|
||
</nav>
|
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|
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</li>
|
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|
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<li class="md-nav__item">
|
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<a href="#meanas.fdmath--grid-description" class="md-nav__link">
|
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<span class="md-ellipsis">
|
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|
||
Grid description
|
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|
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</span>
|
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</a>
|
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|
||
<nav class="md-nav" aria-label="Grid description">
|
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<ul class="md-nav__list">
|
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|
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<li class="md-nav__item">
|
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<a href="#meanas.fdmath--datastructure-dx_lists_t" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
Datastructure: dx_lists_t
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
</ul>
|
||
</nav>
|
||
|
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</li>
|
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|
||
<li class="md-nav__item">
|
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<a href="#meanas.fdmath--permittivity-and-permeability" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
Permittivity and Permeability
|
||
|
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</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
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<li class="md-nav__item">
|
||
<a href="#functional-and-sparse-operators" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
Functional and sparse operators
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.functional" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-module"></code> functional
|
||
|
||
</span>
|
||
</a>
|
||
|
||
<nav class="md-nav" aria-label=" functional">
|
||
<ul class="md-nav__list">
|
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|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.functional.deriv_forward" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> deriv_forward
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.functional.deriv_back" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> deriv_back
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.functional.curl_forward" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> curl_forward
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.functional.curl_back" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> curl_back
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.operators" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-module"></code> operators
|
||
|
||
</span>
|
||
</a>
|
||
|
||
<nav class="md-nav" aria-label=" operators">
|
||
<ul class="md-nav__list">
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.operators.shift_circ" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> shift_circ
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.operators.shift_with_mirror" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> shift_with_mirror
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.operators.deriv_forward" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> deriv_forward
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.operators.deriv_back" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> deriv_back
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.operators.cross" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> cross
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.operators.vec_cross" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> vec_cross
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.operators.avg_forward" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> avg_forward
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.operators.avg_back" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> avg_back
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.operators.curl_forward" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> curl_forward
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.operators.curl_back" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> curl_back
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.vectorization" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-module"></code> vectorization
|
||
|
||
</span>
|
||
</a>
|
||
|
||
<nav class="md-nav" aria-label=" vectorization">
|
||
<ul class="md-nav__list">
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.vectorization.vec" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> vec
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.vectorization.unvec" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> unvec
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
</ul>
|
||
</nav>
|
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|
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</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-module"></code> types
|
||
|
||
</span>
|
||
</a>
|
||
|
||
<nav class="md-nav" aria-label=" types">
|
||
<ul class="md-nav__list">
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types.dx_lists_t" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-attribute"></code> dx_lists_t
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types.dx_lists2_t" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-attribute"></code> dx_lists2_t
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types.dx_lists_mut" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-attribute"></code> dx_lists_mut
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#meanas.fdmath.types.dx_lists2_mut" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-attribute"></code> dx_lists2_mut
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
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|
||
<a href="#meanas.fdmath.types.fdfield_updater_t" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-attribute"></code> fdfield_updater_t
|
||
|
||
</span>
|
||
</a>
|
||
|
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|
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|
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<a href="#meanas.fdmath.types.cfdfield_updater_t" class="md-nav__link">
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<span class="md-ellipsis">
|
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|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-attribute"></code> cfdfield_updater_t
|
||
|
||
</span>
|
||
</a>
|
||
|
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<a href="#meanas.fdmath.types.fdfield" class="md-nav__link">
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<span class="md-ellipsis">
|
||
|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> fdfield
|
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|
||
</span>
|
||
</a>
|
||
|
||
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|
||
|
||
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|
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<a href="#meanas.fdmath.types.vfdfield" class="md-nav__link">
|
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<span class="md-ellipsis">
|
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|
||
<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> vfdfield
|
||
|
||
</span>
|
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</a>
|
||
|
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|
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<a href="#meanas.fdmath.types.cfdfield" class="md-nav__link">
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<span class="md-ellipsis">
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<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> cfdfield
|
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|
||
</span>
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</a>
|
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|
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</li>
|
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<li class="md-nav__item">
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<a href="#meanas.fdmath.types.vcfdfield" class="md-nav__link">
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<span class="md-ellipsis">
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<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> vcfdfield
|
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|
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</span>
|
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</a>
|
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</li>
|
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<li class="md-nav__item">
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<a href="#meanas.fdmath.types.fdslice" class="md-nav__link">
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<span class="md-ellipsis">
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|
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<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> fdslice
|
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|
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</span>
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</a>
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|
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</li>
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<li class="md-nav__item">
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<a href="#meanas.fdmath.types.vfdslice" class="md-nav__link">
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<span class="md-ellipsis">
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<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> vfdslice
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|
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</span>
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</a>
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</li>
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<a href="#meanas.fdmath.types.cfdslice" class="md-nav__link">
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<span class="md-ellipsis">
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<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> cfdslice
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#meanas.fdmath.types.vcfdslice" class="md-nav__link">
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<span class="md-ellipsis">
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<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> vcfdslice
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</span>
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</a>
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</li>
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<a href="#meanas.fdmath.types.fdfield2" class="md-nav__link">
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<span class="md-ellipsis">
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<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> fdfield2
|
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|
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#meanas.fdmath.types.vfdfield2" class="md-nav__link">
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<span class="md-ellipsis">
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<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> vfdfield2
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#meanas.fdmath.types.cfdfield2" class="md-nav__link">
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<span class="md-ellipsis">
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<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> cfdfield2
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</span>
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</a>
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<li class="md-nav__item">
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<a href="#meanas.fdmath.types.vcfdfield2" class="md-nav__link">
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<span class="md-ellipsis">
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<code class="doc-symbol doc-symbol-toc doc-symbol-type_alias"></code> vcfdfield2
|
||
|
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</span>
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</a>
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</ul>
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</nav>
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</nav>
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</div>
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<div class="md-content" data-md-component="content">
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||
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<article class="md-content__inner md-typeset">
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||
|
||
|
||
|
||
|
||
|
||
<h1 id="fdmath">fdmath<a class="headerlink" href="#fdmath" title="Permanent link">¶</a></h1>
|
||
|
||
|
||
<div class="doc doc-object doc-module">
|
||
|
||
|
||
|
||
<h2 id="meanas.fdmath" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-module"></code> <span class="doc doc-object-name doc-module-name">meanas.fdmath</span>
|
||
|
||
|
||
<a href="#meanas.fdmath" class="headerlink" title="Permanent link">¶</a></h2>
|
||
|
||
<div class="doc doc-contents first">
|
||
|
||
<p>Basic discrete calculus for finite difference (fd) simulations.</p>
|
||
<h3 id="meanas.fdmath--fields-functions-and-operators">Fields, Functions, and Operators<a class="headerlink" href="#meanas.fdmath--fields-functions-and-operators" title="Permanent link">¶</a></h3>
|
||
<p>Discrete fields are stored in one of two forms:</p>
|
||
<ul>
|
||
<li>The <code>fdfield_t</code> form is a multidimensional <code>numpy.NDArray</code><ul>
|
||
<li>For a scalar field, this is just <code>U[m, n, p]</code>, where <code>m</code>, <code>n</code>, and <code>p</code> are
|
||
discrete indices referring to positions on the x, y, and z axes respectively.</li>
|
||
<li>For a vector field, the first index specifies which vector component is accessed:
|
||
<code>E[:, m, n, p] = [Ex[m, n, p], Ey[m, n, p], Ez[m, n, p]]</code>.</li>
|
||
</ul>
|
||
</li>
|
||
<li>The <code>vfdfield_t</code> form is simply a vectorzied (i.e. 1D) version of the <code>fdfield_t</code>,
|
||
as obtained by <code>meanas.fdmath.vectorization.vec</code> (effectively just <code>numpy.ravel</code>)</li>
|
||
</ul>
|
||
|
||
|
||
<details class="operators-which-act-on-fields-also-come-in-two-forms" open>
|
||
<summary>Operators which act on fields also come in two forms</summary>
|
||
<ul>
|
||
<li>Python functions, created by the functions in <code>meanas.fdmath.functional</code>.
|
||
The generated functions act on fields in the <code>fdfield_t</code> form.</li>
|
||
<li>Linear operators, usually 2D sparse matrices using <code>scipy.sparse</code>, created
|
||
by <code>meanas.fdmath.operators</code>. These operators act on vectorized fields in the
|
||
<code>vfdfield_t</code> form.</li>
|
||
</ul>
|
||
</details> <p>The operations performed should be equivalent: <code>functional.op(*args)(E)</code> should be
|
||
equivalent to <code>unvec(operators.op(*args) @ vec(E), E.shape[1:])</code>.</p>
|
||
<p>Generally speaking the <code>field_t</code> form is easier to work with, but can be harder or less
|
||
efficient to compose (e.g. it is easy to generate a single matrix by multiplying a
|
||
series of other matrices).</p>
|
||
<h3 id="meanas.fdmath--discrete-calculus">Discrete calculus<a class="headerlink" href="#meanas.fdmath--discrete-calculus" title="Permanent link">¶</a></h3>
|
||
<p>This documentation and approach is roughly based on W.C. Chew's excellent
|
||
"Electromagnetic Theory on a Lattice" (doi:10.1063/1.355770),
|
||
which covers a superset of this material with similar notation and more detail.</p>
|
||
<h4 id="meanas.fdmath--scalar-derivatives-and-cell-shifts">Scalar derivatives and cell shifts<a class="headerlink" href="#meanas.fdmath--scalar-derivatives-and-cell-shifts" title="Permanent link">¶</a></h4>
|
||
<p>Define the discrete forward derivative as
|
||
|
||
<div class="arithmatex">\[ [\tilde{\partial}_x f]_{m + \frac{1}{2}} = \frac{1}{\Delta_{x, m}} (f_{m + 1} - f_m) \]</div></p>
|
||
<p>where <span class="arithmatex">\(f\)</span> is a function defined at discrete locations on the x-axis (labeled using <span class="arithmatex">\(m\)</span>).
|
||
The value at <span class="arithmatex">\(m\)</span> occupies a length <span class="arithmatex">\(\Delta_{x, m}\)</span> along the x-axis. Note that <span class="arithmatex">\(m\)</span>
|
||
is an index along the x-axis, <em>not</em> necessarily an x-coordinate, since each length
|
||
<span class="arithmatex">\(\Delta_{x, m}, \Delta_{x, m+1}, ...\)</span> is independently chosen.</p>
|
||
<p>If we treat <code>f</code> as a 1D array of values, with the <code>i</code>-th value <code>f[i]</code> taking up a length <code>dx[i]</code>
|
||
along the x-axis, the forward derivative is</p>
|
||
<div class="highlight"><pre><span></span><code>deriv_forward(f)[i] = (f[i + 1] - f[i]) / dx[i]
|
||
</code></pre></div>
|
||
<p>Likewise, discrete reverse derivative is
|
||
|
||
<div class="arithmatex">\[ [\hat{\partial}_x f ]_{m - \frac{1}{2}} = \frac{1}{\Delta_{x, m}} (f_{m} - f_{m - 1}) \]</div></p>
|
||
<p>or</p>
|
||
<div class="highlight"><pre><span></span><code>deriv_back(f)[i] = (f[i] - f[i - 1]) / dx[i]
|
||
</code></pre></div>
|
||
<p>The derivatives' values are shifted by a half-cell relative to the original function, and
|
||
will have different cell widths if all the <code>dx[i]</code> ( <span class="arithmatex">\(\Delta_{x, m}\)</span> ) are not
|
||
identical:</p>
|
||
<div class="highlight"><pre><span></span><code>[figure: derivatives and cell sizes]
|
||
dx0 dx1 dx2 dx3 cell sizes for function
|
||
----- ----- ----------- -----
|
||
______________________________
|
||
| | | |
|
||
f0 | f1 | f2 | f3 | function
|
||
_____|_____|___________|_____|
|
||
| | | |
|
||
| Df0 | Df1 | Df2 | Df3 forward derivative (periodic boundary)
|
||
__|_____|________|________|___
|
||
|
||
dx'3] dx'0 dx'1 dx'2 [dx'3 cell sizes for forward derivative
|
||
-- ----- -------- -------- ---
|
||
dx'0] dx'1 dx'2 dx'3 [dx'0 cell sizes for reverse derivative
|
||
______________________________
|
||
| | | |
|
||
| df1 | df2 | df3 | df0 reverse derivative (periodic boundary)
|
||
__|_____|________|________|___
|
||
|
||
Periodic boundaries are used here and elsewhere unless otherwise noted.
|
||
</code></pre></div>
|
||
<p>In the above figure,
|
||
<code>f0 =</code> <span class="arithmatex">\(f_0\)</span>, <code>f1 =</code> <span class="arithmatex">\(f_1\)</span>
|
||
<code>Df0 =</code> <span class="arithmatex">\([\tilde{\partial}f]_{0 + \frac{1}{2}}\)</span>
|
||
<code>Df1 =</code> <span class="arithmatex">\([\tilde{\partial}f]_{1 + \frac{1}{2}}\)</span>
|
||
<code>df0 =</code> <span class="arithmatex">\([\hat{\partial}f]_{0 - \frac{1}{2}}\)</span>
|
||
etc.</p>
|
||
<p>The fractional subscript <span class="arithmatex">\(m + \frac{1}{2}\)</span> is used to indicate values defined
|
||
at shifted locations relative to the original <span class="arithmatex">\(m\)</span>, with corresponding lengths
|
||
|
||
<div class="arithmatex">\[ \Delta_{x, m + \frac{1}{2}} = \frac{1}{2} * (\Delta_{x, m} + \Delta_{x, m + 1}) \]</div>
|
||
</p>
|
||
<p>Just as <span class="arithmatex">\(m\)</span> is not itself an x-coordinate, neither is <span class="arithmatex">\(m + \frac{1}{2}\)</span>;
|
||
carefully note the positions of the various cells in the above figure vs their labels.
|
||
If the positions labeled with <span class="arithmatex">\(m\)</span> are considered the "base" or "original" grid,
|
||
the positions labeled with <span class="arithmatex">\(m + \frac{1}{2}\)</span> are said to lie on a "dual" or
|
||
"derived" grid.</p>
|
||
<p>For the remainder of the <code>Discrete calculus</code> section, all figures will show
|
||
constant-length cells in order to focus on the vector derivatives themselves.
|
||
See the <code>Grid description</code> section below for additional information on this topic
|
||
and generalization to three dimensions.</p>
|
||
<h4 id="meanas.fdmath--gradients-and-fore-vectors">Gradients and fore-vectors<a class="headerlink" href="#meanas.fdmath--gradients-and-fore-vectors" title="Permanent link">¶</a></h4>
|
||
<p>Expanding to three dimensions, we can define two gradients
|
||
<br />
|
||
<div class="arithmatex">\[
|
||
[\tilde{\nabla} f]_{m,n,p} = \vec{x} [\tilde{\partial}_x f]_{m + \frac{1}{2},n,p} +
|
||
\vec{y} [\tilde{\partial}_y f]_{m,n + \frac{1}{2},p} +
|
||
\vec{z} [\tilde{\partial}_z f]_{m,n,p + \frac{1}{2}}
|
||
\]</div></p>
|
||
<div class="arithmatex">\[
|
||
[\hat{\nabla} f]_{m,n,p} = \vec{x} [\hat{\partial}_x f]_{m + \frac{1}{2},n,p} +
|
||
\vec{y} [\hat{\partial}_y f]_{m,n + \frac{1}{2},p} +
|
||
\vec{z} [\hat{\partial}_z f]_{m,n,p + \frac{1}{2}}
|
||
\]</div>
|
||
|
||
<p>or</p>
|
||
<div class="highlight"><pre><span></span><code>[code: gradients]
|
||
grad_forward(f)[i,j,k] = [Dx_forward(f)[i, j, k],
|
||
Dy_forward(f)[i, j, k],
|
||
Dz_forward(f)[i, j, k]]
|
||
= [(f[i + 1, j, k] - f[i, j, k]) / dx[i],
|
||
(f[i, j + 1, k] - f[i, j, k]) / dy[i],
|
||
(f[i, j, k + 1] - f[i, j, k]) / dz[i]]
|
||
|
||
grad_back(f)[i,j,k] = [Dx_back(f)[i, j, k],
|
||
Dy_back(f)[i, j, k],
|
||
Dz_back(f)[i, j, k]]
|
||
= [(f[i, j, k] - f[i - 1, j, k]) / dx[i],
|
||
(f[i, j, k] - f[i, j - 1, k]) / dy[i],
|
||
(f[i, j, k] - f[i, j, k - 1]) / dz[i]]
|
||
</code></pre></div>
|
||
<p>The three derivatives in the gradient cause shifts in different
|
||
directions, so the x/y/z components of the resulting "vector" are defined
|
||
at different points: the x-component is shifted in the x-direction,
|
||
y in y, and z in z.</p>
|
||
<p>We call the resulting object a "fore-vector" or "back-vector", depending
|
||
on the direction of the shift. We write it as
|
||
<br />
|
||
<div class="arithmatex">\[
|
||
\tilde{g}_{m,n,p} = \vec{x} g^x_{m + \frac{1}{2},n,p} +
|
||
\vec{y} g^y_{m,n + \frac{1}{2},p} +
|
||
\vec{z} g^z_{m,n,p + \frac{1}{2}}
|
||
\]</div></p>
|
||
<div class="arithmatex">\[
|
||
\hat{g}_{m,n,p} = \vec{x} g^x_{m - \frac{1}{2},n,p} +
|
||
\vec{y} g^y_{m,n - \frac{1}{2},p} +
|
||
\vec{z} g^z_{m,n,p - \frac{1}{2}}
|
||
\]</div>
|
||
|
||
<div class="highlight"><pre><span></span><code>[figure: gradient / fore-vector]
|
||
(m, n+1, p+1) ______________ (m+1, n+1, p+1)
|
||
/: /|
|
||
/ : / |
|
||
/ : / |
|
||
(m, n, p+1)/_____________/ | The forward derivatives are defined
|
||
| : | | at the Dx, Dy, Dz points,
|
||
| :.........|...| but the forward-gradient fore-vector
|
||
z y Dz / | / is the set of all three
|
||
|/_x | Dy | / and is said to be "located" at (m,n,p)
|
||
|/ |/
|
||
(m, n, p)|_____Dx______| (m+1, n, p)
|
||
</code></pre></div>
|
||
<h4 id="meanas.fdmath--divergences">Divergences<a class="headerlink" href="#meanas.fdmath--divergences" title="Permanent link">¶</a></h4>
|
||
<p>There are also two divergences,</p>
|
||
<div class="arithmatex">\[
|
||
d_{n,m,p} = [\tilde{\nabla} \cdot \hat{g}]_{n,m,p}
|
||
= [\tilde{\partial}_x g^x]_{m,n,p} +
|
||
[\tilde{\partial}_y g^y]_{m,n,p} +
|
||
[\tilde{\partial}_z g^z]_{m,n,p}
|
||
\]</div>
|
||
|
||
<div class="arithmatex">\[
|
||
d_{n,m,p} = [\hat{\nabla} \cdot \tilde{g}]_{n,m,p}
|
||
= [\hat{\partial}_x g^x]_{m,n,p} +
|
||
[\hat{\partial}_y g^y]_{m,n,p} +
|
||
[\hat{\partial}_z g^z]_{m,n,p}
|
||
\]</div>
|
||
|
||
<p>or</p>
|
||
<div class="highlight"><pre><span></span><code>[code: divergences]
|
||
div_forward(g)[i,j,k] = Dx_forward(gx)[i, j, k] +
|
||
Dy_forward(gy)[i, j, k] +
|
||
Dz_forward(gz)[i, j, k]
|
||
= (gx[i + 1, j, k] - gx[i, j, k]) / dx[i] +
|
||
(gy[i, j + 1, k] - gy[i, j, k]) / dy[i] +
|
||
(gz[i, j, k + 1] - gz[i, j, k]) / dz[i]
|
||
|
||
div_back(g)[i,j,k] = Dx_back(gx)[i, j, k] +
|
||
Dy_back(gy)[i, j, k] +
|
||
Dz_back(gz)[i, j, k]
|
||
= (gx[i, j, k] - gx[i - 1, j, k]) / dx[i] +
|
||
(gy[i, j, k] - gy[i, j - 1, k]) / dy[i] +
|
||
(gz[i, j, k] - gz[i, j, k - 1]) / dz[i]
|
||
</code></pre></div>
|
||
<p>where <code>g = [gx, gy, gz]</code> is a fore- or back-vector field.</p>
|
||
<p>Since we applied the forward divergence to the back-vector (and vice-versa), the resulting scalar value
|
||
is defined at the back-vector's (fore-vector's) location <span class="arithmatex">\((m,n,p)\)</span> and not at the locations of its components
|
||
<span class="arithmatex">\((m \pm \frac{1}{2},n,p)\)</span> etc.</p>
|
||
<div class="highlight"><pre><span></span><code>[figure: divergence]
|
||
^^
|
||
(m-1/2, n+1/2, p+1/2) _____||_______ (m+1/2, n+1/2, p+1/2)
|
||
/: || ,, /|
|
||
/ : || // / | The divergence at (m, n, p) (the center
|
||
/ : // / | of this cube) of a fore-vector field
|
||
(m-1/2, n-1/2, p+1/2)/_____________/ | is the sum of the outward-pointing
|
||
| : | | fore-vector components, which are
|
||
z y <==|== :.........|.====> located at the face centers.
|
||
|/_x | / | /
|
||
| / // | / Note that in a nonuniform grid, each
|
||
|/ // || |/ dimension is normalized by the cell width.
|
||
(m-1/2, n-1/2, p-1/2)|____//_______| (m+1/2, n-1/2, p-1/2)
|
||
'' ||
|
||
VV
|
||
</code></pre></div>
|
||
<h4 id="meanas.fdmath--curls">Curls<a class="headerlink" href="#meanas.fdmath--curls" title="Permanent link">¶</a></h4>
|
||
<p>The two curls are then</p>
|
||
<div class="arithmatex">\[
|
||
\begin{aligned}
|
||
\hat{h}_{m + \frac{1}{2}, n + \frac{1}{2}, p + \frac{1}{2}} &= \\
|
||
[\tilde{\nabla} \times \tilde{g}]_{m + \frac{1}{2}, n + \frac{1}{2}, p + \frac{1}{2}} &=
|
||
\vec{x} (\tilde{\partial}_y g^z_{m,n,p + \frac{1}{2}} - \tilde{\partial}_z g^y_{m,n + \frac{1}{2},p}) \\
|
||
&+ \vec{y} (\tilde{\partial}_z g^x_{m + \frac{1}{2},n,p} - \tilde{\partial}_x g^z_{m,n,p + \frac{1}{2}}) \\
|
||
&+ \vec{z} (\tilde{\partial}_x g^y_{m,n + \frac{1}{2},p} - \tilde{\partial}_y g^z_{m + \frac{1}{2},n,p})
|
||
\end{aligned}
|
||
\]</div>
|
||
|
||
<p>and</p>
|
||
<div class="arithmatex">\[
|
||
\tilde{h}_{m - \frac{1}{2}, n - \frac{1}{2}, p - \frac{1}{2}} =
|
||
[\hat{\nabla} \times \hat{g}]_{m - \frac{1}{2}, n - \frac{1}{2}, p - \frac{1}{2}}
|
||
\]</div>
|
||
|
||
<p>where <span class="arithmatex">\(\hat{g}\)</span> and <span class="arithmatex">\(\tilde{g}\)</span> are located at <span class="arithmatex">\((m,n,p)\)</span>
|
||
with components at <span class="arithmatex">\((m \pm \frac{1}{2},n,p)\)</span> etc.,
|
||
while <span class="arithmatex">\(\hat{h}\)</span> and <span class="arithmatex">\(\tilde{h}\)</span> are located at <span class="arithmatex">\((m \pm \frac{1}{2}, n \pm \frac{1}{2}, p \pm \frac{1}{2})\)</span>
|
||
with components at <span class="arithmatex">\((m, n \pm \frac{1}{2}, p \pm \frac{1}{2})\)</span> etc.</p>
|
||
<div class="highlight"><pre><span></span><code>[code: curls]
|
||
curl_forward(g)[i,j,k] = [Dy_forward(gz)[i, j, k] - Dz_forward(gy)[i, j, k],
|
||
Dz_forward(gx)[i, j, k] - Dx_forward(gz)[i, j, k],
|
||
Dx_forward(gy)[i, j, k] - Dy_forward(gx)[i, j, k]]
|
||
|
||
curl_back(g)[i,j,k] = [Dy_back(gz)[i, j, k] - Dz_back(gy)[i, j, k],
|
||
Dz_back(gx)[i, j, k] - Dx_back(gz)[i, j, k],
|
||
Dx_back(gy)[i, j, k] - Dy_back(gx)[i, j, k]]
|
||
</code></pre></div>
|
||
<p>For example, consider the forward curl, at (m, n, p), of a back-vector field <code>g</code>, defined
|
||
on a grid containing (m + 1/2, n + 1/2, p + 1/2).
|
||
The curl will be a fore-vector, so its z-component will be defined at (m, n, p + 1/2).
|
||
Take the nearest x- and y-components of <code>g</code> in the xy plane where the curl's z-component
|
||
is located; these are</p>
|
||
<div class="highlight"><pre><span></span><code>[curl components]
|
||
(m, n + 1/2, p + 1/2) : x-component of back-vector at (m + 1/2, n + 1/2, p + 1/2)
|
||
(m + 1, n + 1/2, p + 1/2) : x-component of back-vector at (m + 3/2, n + 1/2, p + 1/2)
|
||
(m + 1/2, n , p + 1/2) : y-component of back-vector at (m + 1/2, n + 1/2, p + 1/2)
|
||
(m + 1/2, n + 1 , p + 1/2) : y-component of back-vector at (m + 1/2, n + 3/2, p + 1/2)
|
||
</code></pre></div>
|
||
<p>These four xy-components can be used to form a loop around the curl's z-component; its magnitude and sign
|
||
is set by their loop-oriented sum (i.e. two have their signs flipped to complete the loop).</p>
|
||
<div class="highlight"><pre><span></span><code>[figure: z-component of curl]
|
||
: |
|
||
z y : ^^ |
|
||
|/_x :....||.<.....| (m+1, n+1, p+1/2)
|
||
/ || /
|
||
| v || | ^
|
||
|/ |/
|
||
(m, n, p+1/2) |_____>______| (m+1, n, p+1/2)
|
||
</code></pre></div>
|
||
<h3 id="meanas.fdmath--maxwells-equations">Maxwell's Equations<a class="headerlink" href="#meanas.fdmath--maxwells-equations" title="Permanent link">¶</a></h3>
|
||
<p>If we discretize both space (m,n,p) and time (l), Maxwell's equations become</p>
|
||
<div class="arithmatex">\[
|
||
\begin{aligned}
|
||
\tilde{\nabla} \times \tilde{E}_{l,\vec{r}} &= -\tilde{\partial}_t \hat{B}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}}
|
||
- \hat{M}_{l, \vec{r} + \frac{1}{2}} \\
|
||
\hat{\nabla} \times \hat{H}_{l-\frac{1}{2},\vec{r} + \frac{1}{2}} &= \hat{\partial}_t \tilde{D}_{l, \vec{r}}
|
||
+ \tilde{J}_{l-\frac{1}{2},\vec{r}} \\
|
||
\tilde{\nabla} \cdot \hat{B}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}} &= 0 \\
|
||
\hat{\nabla} \cdot \tilde{D}_{l,\vec{r}} &= \rho_{l,\vec{r}}
|
||
\end{aligned}
|
||
\]</div>
|
||
|
||
<p>with</p>
|
||
<div class="arithmatex">\[
|
||
\begin{aligned}
|
||
\hat{B}_{\vec{r}} &= \mu_{\vec{r} + \frac{1}{2}} \cdot \hat{H}_{\vec{r} + \frac{1}{2}} \\
|
||
\tilde{D}_{\vec{r}} &= \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}}
|
||
\end{aligned}
|
||
\]</div>
|
||
|
||
<p>where the spatial subscripts are abbreviated as <span class="arithmatex">\(\vec{r} = (m, n, p)\)</span> and
|
||
<span class="arithmatex">\(\vec{r} + \frac{1}{2} = (m + \frac{1}{2}, n + \frac{1}{2}, p + \frac{1}{2})\)</span>,
|
||
<span class="arithmatex">\(\tilde{E}\)</span> and <span class="arithmatex">\(\hat{H}\)</span> are the electric and magnetic fields,
|
||
<span class="arithmatex">\(\tilde{J}\)</span> and <span class="arithmatex">\(\hat{M}\)</span> are the electric and magnetic current distributions,
|
||
and <span class="arithmatex">\(\epsilon\)</span> and <span class="arithmatex">\(\mu\)</span> are the dielectric permittivity and magnetic permeability.</p>
|
||
<p>The above is Yee's algorithm, written in a form analogous to Maxwell's equations.
|
||
The time derivatives can be expanded to form the update equations:</p>
|
||
<div class="highlight"><pre><span></span><code>[code: Maxwell's equations updates]
|
||
H[i, j, k] -= dt * (curl_forward(E)[i, j, k] + M[t, i, j, k]) / mu[i, j, k]
|
||
E[i, j, k] += dt * (curl_back( H)[i, j, k] + J[t, i, j, k]) / epsilon[i, j, k]
|
||
</code></pre></div>
|
||
<p>Note that the E-field fore-vector and H-field back-vector are offset by a half-cell, resulting
|
||
in distinct locations for all six E- and H-field components:</p>
|
||
<div class="highlight"><pre><span></span><code>[figure: Field components]
|
||
|
||
(m - 1/2,=> ____________Hx__________[H] <= r + 1/2 = (m + 1/2,
|
||
n + 1/2, /: /: /| n + 1/2,
|
||
z y p + 1/2) / : / : / | p + 1/2)
|
||
|/_x / : / : / |
|
||
/ : Ez__________Hy | Locations of the E- and
|
||
/ : : : /| | H-field components for the
|
||
(m - 1/2, / : : Ey...../.|..Hz [E] fore-vector at r = (m,n,p)
|
||
n - 1/2, =>/________________________/ | /| (the large cube's center)
|
||
p + 1/2) | : : / | | / | and [H] back-vector at r + 1/2
|
||
| : :/ | |/ | (the top right corner)
|
||
| : [E].......|.Ex |
|
||
| :.................|......| <= (m + 1/2, n + 1/2, p + 1/2)
|
||
| / | /
|
||
| / | /
|
||
| / | / This is the Yee discretization
|
||
| / | / scheme ("Yee cell").
|
||
r - 1/2 = | / | /
|
||
(m - 1/2, |/ |/
|
||
n - 1/2,=> |________________________| <= (m + 1/2, n - 1/2, p - 1/2)
|
||
p - 1/2)
|
||
</code></pre></div>
|
||
<p>Each component forms its own grid, offset from the others:</p>
|
||
<div class="highlight"><pre><span></span><code>[figure: E-fields for adjacent cells]
|
||
|
||
H1__________Hx0_________H0
|
||
z y /: /|
|
||
|/_x / : / | This figure shows H back-vector locations
|
||
/ : / | H0, H1, etc. and their associated components
|
||
Hy1 : Hy0 | H0 = (Hx0, Hy0, Hz0) etc.
|
||
/ : / |
|
||
/ Hz1 / Hz0
|
||
H2___________Hx3_________H3 | The equivalent drawing for E would have
|
||
| : | | fore-vectors located at the cube's
|
||
| : | | center (and the centers of adjacent cubes),
|
||
| : | | with components on the cube's faces.
|
||
| H5..........Hx4...|......H4
|
||
| / | /
|
||
Hz2 / Hz2 /
|
||
| / | /
|
||
| Hy6 | Hy4
|
||
| / | /
|
||
|/ |/
|
||
H6__________Hx7__________H7
|
||
</code></pre></div>
|
||
<p>The divergence equations can be derived by taking the divergence of the curl equations
|
||
and combining them with charge continuity,
|
||
|
||
<div class="arithmatex">\[ \hat{\nabla} \cdot \tilde{J} + \hat{\partial}_t \rho = 0 \]</div></p>
|
||
<p>implying that the discrete Maxwell's equations do not produce spurious charges.</p>
|
||
<h4 id="meanas.fdmath--wave-equation">Wave equation<a class="headerlink" href="#meanas.fdmath--wave-equation" title="Permanent link">¶</a></h4>
|
||
<p>Taking the backward curl of the <span class="arithmatex">\(\tilde{\nabla} \times \tilde{E}\)</span> equation and
|
||
replacing the resulting <span class="arithmatex">\(\hat{\nabla} \times \hat{H}\)</span> term using its respective equation,
|
||
and setting <span class="arithmatex">\(\hat{M}\)</span> to zero, we can form the discrete wave equation:</p>
|
||
<div class="arithmatex">\[
|
||
\begin{aligned}
|
||
\tilde{\nabla} \times \tilde{E}_{l,\vec{r}} &=
|
||
-\tilde{\partial}_t \hat{B}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}}
|
||
- \hat{M}_{l-1, \vec{r} + \frac{1}{2}} \\
|
||
\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}} &=
|
||
-\tilde{\partial}_t \hat{H}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}} \\
|
||
\hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}}) &=
|
||
\hat{\nabla} \times (-\tilde{\partial}_t \hat{H}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}}) \\
|
||
\hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}}) &=
|
||
-\tilde{\partial}_t \hat{\nabla} \times \hat{H}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}} \\
|
||
\hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}}) &=
|
||
-\tilde{\partial}_t \hat{\partial}_t \epsilon_{\vec{r}} \tilde{E}_{l, \vec{r}} + \hat{\partial}_t \tilde{J}_{l-\frac{1}{2},\vec{r}} \\
|
||
\hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}})
|
||
+ \tilde{\partial}_t \hat{\partial}_t \epsilon_{\vec{r}} \cdot \tilde{E}_{l, \vec{r}}
|
||
&= \tilde{\partial}_t \tilde{J}_{l - \frac{1}{2}, \vec{r}}
|
||
\end{aligned}
|
||
\]</div>
|
||
|
||
<h4 id="meanas.fdmath--frequency-domain">Frequency domain<a class="headerlink" href="#meanas.fdmath--frequency-domain" title="Permanent link">¶</a></h4>
|
||
<p>We can substitute in a time-harmonic fields</p>
|
||
<div class="arithmatex">\[
|
||
\begin{aligned}
|
||
\tilde{E}_{l, \vec{r}} &= \tilde{E}_{\vec{r}} e^{-\imath \omega l \Delta_t} \\
|
||
\tilde{J}_{l, \vec{r}} &= \tilde{J}_{\vec{r}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t}
|
||
\end{aligned}
|
||
\]</div>
|
||
|
||
<p>resulting in</p>
|
||
<div class="arithmatex">\[
|
||
\begin{aligned}
|
||
\tilde{\partial}_t &\Rightarrow (e^{ \imath \omega \Delta_t} - 1) / \Delta_t = \frac{-2 \imath}{\Delta_t} \sin(\omega \Delta_t / 2) e^{-\imath \omega \Delta_t / 2} = -\imath \Omega e^{-\imath \omega \Delta_t / 2}\\
|
||
\hat{\partial}_t &\Rightarrow (1 - e^{-\imath \omega \Delta_t}) / \Delta_t = \frac{-2 \imath}{\Delta_t} \sin(\omega \Delta_t / 2) e^{ \imath \omega \Delta_t / 2} = -\imath \Omega e^{ \imath \omega \Delta_t / 2}\\
|
||
\Omega &= 2 \sin(\omega \Delta_t / 2) / \Delta_t
|
||
\end{aligned}
|
||
\]</div>
|
||
|
||
<p>This gives the frequency-domain wave equation,</p>
|
||
<div class="arithmatex">\[
|
||
\hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{\vec{r}})
|
||
-\Omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} = -\imath \Omega \tilde{J}_{\vec{r}} e^{\imath \omega \Delta_t / 2} \\
|
||
\]</div>
|
||
|
||
<h4 id="meanas.fdmath--plane-waves-and-dispersion-relation">Plane waves and Dispersion relation<a class="headerlink" href="#meanas.fdmath--plane-waves-and-dispersion-relation" title="Permanent link">¶</a></h4>
|
||
<p>With uniform material distribution and no sources</p>
|
||
<div class="arithmatex">\[
|
||
\begin{aligned}
|
||
\mu_{\vec{r} + \frac{1}{2}} &= \mu \\
|
||
\epsilon_{\vec{r}} &= \epsilon \\
|
||
\tilde{J}_{\vec{r}} &= 0 \\
|
||
\end{aligned}
|
||
\]</div>
|
||
|
||
<p>the frequency domain wave equation simplifies to</p>
|
||
<div class="arithmatex">\[ \hat{\nabla} \times \tilde{\nabla} \times \tilde{E}_{\vec{r}} - \Omega^2 \epsilon \mu \tilde{E}_{\vec{r}} = 0 \]</div>
|
||
|
||
<p>Since <span class="arithmatex">\(\hat{\nabla} \cdot \tilde{E}_{\vec{r}} = 0\)</span>, we can simplify</p>
|
||
<div class="arithmatex">\[
|
||
\begin{aligned}
|
||
\hat{\nabla} \times \tilde{\nabla} \times \tilde{E}_{\vec{r}}
|
||
&= \tilde{\nabla}(\hat{\nabla} \cdot \tilde{E}_{\vec{r}}) - \hat{\nabla} \cdot \tilde{\nabla} \tilde{E}_{\vec{r}} \\
|
||
&= - \hat{\nabla} \cdot \tilde{\nabla} \tilde{E}_{\vec{r}} \\
|
||
&= - \tilde{\nabla}^2 \tilde{E}_{\vec{r}}
|
||
\end{aligned}
|
||
\]</div>
|
||
|
||
<p>and we get</p>
|
||
<div class="arithmatex">\[ \tilde{\nabla}^2 \tilde{E}_{\vec{r}} + \Omega^2 \epsilon \mu \tilde{E}_{\vec{r}} = 0 \]</div>
|
||
|
||
<p>We can convert this to three scalar-wave equations of the form</p>
|
||
<div class="arithmatex">\[ (\tilde{\nabla}^2 + K^2) \phi_{\vec{r}} = 0 \]</div>
|
||
|
||
<p>with <span class="arithmatex">\(K^2 = \Omega^2 \mu \epsilon\)</span>. Now we let</p>
|
||
<div class="arithmatex">\[ \phi_{\vec{r}} = A e^{\imath (k_x m \Delta_x + k_y n \Delta_y + k_z p \Delta_z)} \]</div>
|
||
|
||
<p>resulting in</p>
|
||
<div class="arithmatex">\[
|
||
\begin{aligned}
|
||
\tilde{\partial}_x &\Rightarrow (e^{ \imath k_x \Delta_x} - 1) / \Delta_t = \frac{-2 \imath}{\Delta_x} \sin(k_x \Delta_x / 2) e^{ \imath k_x \Delta_x / 2} = \imath K_x e^{ \imath k_x \Delta_x / 2}\\
|
||
\hat{\partial}_x &\Rightarrow (1 - e^{-\imath k_x \Delta_x}) / \Delta_t = \frac{-2 \imath}{\Delta_x} \sin(k_x \Delta_x / 2) e^{-\imath k_x \Delta_x / 2} = \imath K_x e^{-\imath k_x \Delta_x / 2}\\
|
||
K_x &= 2 \sin(k_x \Delta_x / 2) / \Delta_x \\
|
||
\end{aligned}
|
||
\]</div>
|
||
|
||
<p>with similar expressions for the y and z dimnsions (and <span class="arithmatex">\(K_y, K_z\)</span>).</p>
|
||
<p>This implies</p>
|
||
<div class="arithmatex">\[
|
||
\tilde{\nabla}^2 = -(K_x^2 + K_y^2 + K_z^2) \phi_{\vec{r}} \\
|
||
K_x^2 + K_y^2 + K_z^2 = \Omega^2 \mu \epsilon = \Omega^2 / c^2
|
||
\]</div>
|
||
|
||
<p>where <span class="arithmatex">\(c = \sqrt{\mu \epsilon}\)</span>.</p>
|
||
<p>Assuming real <span class="arithmatex">\((k_x, k_y, k_z), \omega\)</span> will be real only if</p>
|
||
<div class="arithmatex">\[ c^2 \Delta_t^2 = \frac{\Delta_t^2}{\mu \epsilon} < 1/(\frac{1}{\Delta_x^2} + \frac{1}{\Delta_y^2} + \frac{1}{\Delta_z^2}) \]</div>
|
||
|
||
<p>If <span class="arithmatex">\(\Delta_x = \Delta_y = \Delta_z\)</span>, this simplifies to <span class="arithmatex">\(c \Delta_t < \Delta_x / \sqrt{3}\)</span>.
|
||
This last form can be interpreted as enforcing causality; the distance that light
|
||
travels in one timestep (i.e., <span class="arithmatex">\(c \Delta_t\)</span>) must be less than the diagonal
|
||
of the smallest cell ( <span class="arithmatex">\(\Delta_x / \sqrt{3}\)</span> when on a uniform cubic grid).</p>
|
||
<h3 id="meanas.fdmath--grid-description">Grid description<a class="headerlink" href="#meanas.fdmath--grid-description" title="Permanent link">¶</a></h3>
|
||
<p>As described in the section on scalar discrete derivatives above, cell widths
|
||
(<code>dx[i]</code>, <code>dy[j]</code>, <code>dz[k]</code>) along each axis can be arbitrary and independently
|
||
defined. Moreover, all field components are actually defined at "derived" or "dual"
|
||
positions, in-between the "base" grid points on one or more axes.</p>
|
||
<p>To get a better sense of how this works, let's start by drawing a grid with uniform
|
||
<code>dy</code> and <code>dz</code> and nonuniform <code>dx</code>. We will only draw one cell in the y and z dimensions
|
||
to make the illustration simpler; we need at least two cells in the x dimension to
|
||
demonstrate how nonuniform <code>dx</code> affects the various components.</p>
|
||
<p>Place the E fore-vectors at integer indices <span class="arithmatex">\(r = (m, n, p)\)</span> and the H back-vectors
|
||
at fractional indices <span class="arithmatex">\(r + \frac{1}{2} = (m + \frac{1}{2}, n + \frac{1}{2},
|
||
p + \frac{1}{2})\)</span>. Remember that these are indices and not coordinates; they can
|
||
correspond to arbitrary (monotonically increasing) coordinates depending on the cell widths.</p>
|
||
<p>Draw lines to denote the planes on which the H components and back-vectors are defined.
|
||
For simplicity, don't draw the equivalent planes for the E components and fore-vectors,
|
||
except as necessary to show their locations -- it's easiest to just connect them to their
|
||
associated H-equivalents.</p>
|
||
<p>The result looks something like this:</p>
|
||
<div class="highlight"><pre><span></span><code>[figure: Component centers]
|
||
p=
|
||
[H]__________Hx___________[H]_____Hx______[H] __ +1/2
|
||
z y /: /: /: /: /| | |
|
||
|/_x / : / : / : / : / | | |
|
||
/ : / : / : / : / | | |
|
||
Hy : Ez...........Hy : Ez......Hy | | |
|
||
/: : : : /: : : : /| | | |
|
||
/ : Hz : Ey....../.:..Hz : Ey./.|..Hz __ 0 | dz[0]
|
||
/ : /: : / / : /: : / / | /| | |
|
||
/_________________________/_______________/ | / | | |
|
||
| :/ : :/ | :/ : :/ | |/ | | |
|
||
| Ex : [E].......|..Ex : [E]..|..Ex | | |
|
||
| : | : | | | |
|
||
| [H]..........Hx....|......[H].....H|x.....[H] __ --------- (n=+1/2, p=-1/2)
|
||
| / | / | / / /
|
||
Hz / Hz / Hz / / /
|
||
| / | / | / / /
|
||
| Hy | Hy | Hy __ 0 / dy[0]
|
||
| / | / | / / /
|
||
| / | / | / / /
|
||
|/ |/ |/ / /
|
||
[H]__________Hx___________[H]_____Hx______[H] __ -1/2 /
|
||
=n
|
||
|------------|------------|-------|-------|
|
||
-1/2 0 +1/2 +1 +3/2 = m
|
||
|
||
------------------------- ----------------
|
||
dx[0] dx[1]
|
||
|
||
Part of a nonuniform "base grid", with labels specifying
|
||
positions of the various field components. [E] fore-vectors
|
||
are at the cell centers, and [H] back-vectors are at the
|
||
vertices. H components along the near (-y) top (+z) edge
|
||
have been omitted to make the insides of the cubes easier
|
||
to visualize.
|
||
</code></pre></div>
|
||
<p>The above figure shows where all the components are located; however, it is also useful to show
|
||
what volumes those components correspond to. Consider the Ex component at <code>m = +1/2</code>: it is
|
||
shifted in the x-direction by a half-cell from the E fore-vector at <code>m = 0</code> (labeled <code>[E]</code>
|
||
in the figure). It corresponds to a volume between <code>m = 0</code> and <code>m = +1</code> (the other
|
||
dimensions are not shifted, i.e. they are still bounded by <code>n, p = +-1/2</code>). (See figure
|
||
below). Since <code>m</code> is an index and not an x-coordinate, the Ex component is not necessarily
|
||
at the center of the volume it represents, and the x-length of its volume is the derived
|
||
quantity <code>dx'[0] = (dx[0] + dx[1]) / 2</code> rather than the base <code>dx</code>.
|
||
(See also <code>Scalar derivatives and cell shifts</code>).</p>
|
||
<div class="highlight"><pre><span></span><code>[figure: Ex volumes]
|
||
p=
|
||
<_________________________________________> __ +1/2
|
||
z y << /: / /: >> | |
|
||
|/_x < < / : / / : > > | |
|
||
< < / : / / : > > | |
|
||
< < / : / / : > > | |
|
||
<: < / : : / : >: > | |
|
||
< : < / : : / : > : > __ 0 | dz[0]
|
||
< : < / : : / :> : > | |
|
||
<____________/____________________/_______> : > | |
|
||
< : < | : : | > : > | |
|
||
< Ex < | : Ex | > Ex > | |
|
||
< : < | : : | > : > | |
|
||
< : <....|.......:........:...|.......>...:...> __ --------- (n=+1/2, p=-1/2)
|
||
< : < | / : /| /> : > / /
|
||
< : < | / : / | / > : > / /
|
||
< :< | / :/ | / > :> / /
|
||
< < | / : | / > > _ 0 / dy[0]
|
||
< < | / | / > > / /
|
||
< < | / | / > > / /
|
||
<< |/ |/ >> / /
|
||
<____________|____________________|_______> __ -1/2 /
|
||
=n
|
||
|------------|------------|-------|-------|
|
||
-1/2 0 +1/2 +1 +3/2 = m
|
||
|
||
~------------ -------------------- -------~
|
||
dx'[-1] dx'[0] dx'[1]
|
||
|
||
The Ex values are positioned on the x-faces of the base
|
||
grid. They represent the Ex field in volumes shifted by
|
||
a half-cell in the x-dimension, as shown here. Only the
|
||
center cell (with width dx'[0]) is fully shown; the
|
||
other two are truncated (shown using >< markers).
|
||
|
||
Note that the Ex positions are the in the same positions
|
||
as the previous figure; only the cell boundaries have moved.
|
||
Also note that the points at which Ex is defined are not
|
||
necessarily centered in the volumes they represent; non-
|
||
uniform cell sizes result in off-center volumes like the
|
||
center cell here.
|
||
</code></pre></div>
|
||
<p>The next figure shows the volumes corresponding to the Hy components, which
|
||
are shifted in two dimensions (x and z) compared to the base grid.</p>
|
||
<div class="highlight"><pre><span></span><code>[figure: Hy volumes]
|
||
p=
|
||
z y mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm __ +1/2 s
|
||
|/_x << m: m: >> | |
|
||
< < m : m : > > | | dz'[1]
|
||
< < m : m : > > | |
|
||
Hy........... m........Hy...........m......Hy > | |
|
||
< < m : m : > > | |
|
||
< ______ m_____:_______________m_____:_>______ __ 0
|
||
< < m /: m / > > | |
|
||
mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm > | |
|
||
< < | / : | / > > | | dz'[0]
|
||
< < | / : | / > > | |
|
||
< < | / : | / > > | |
|
||
< wwwww|w/wwwwwwwwwwwwwwwwwww|w/wwwww>wwwwwwww __ s
|
||
< < |/ w |/ w> > / /
|
||
_____________|_____________________|________ > / /
|
||
< < | w | w > > / /
|
||
< Hy........|...w........Hy.......|...w...>..Hy _ 0 / dy[0]
|
||
< < | w | w > > / /
|
||
<< | w | w > > / /
|
||
< |w |w >> / /
|
||
wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww __ -1/2 /
|
||
|
||
|------------|------------|--------|-------|
|
||
-1/2 0 +1/2 +1 +3/2 = m
|
||
|
||
~------------ --------------------- -------~
|
||
dx'[-1] dx'[0] dx'[1]
|
||
|
||
The Hy values are positioned on the y-edges of the base
|
||
grid. Again here, the 'Hy' labels represent the same points
|
||
as in the basic grid figure above; the edges have shifted
|
||
by a half-cell along the x- and z-axes.
|
||
|
||
The grid lines _|:/ are edges of the area represented by
|
||
each Hy value, and the lines drawn using <m>.w represent
|
||
edges where a cell's faces extend beyond the drawn area
|
||
(i.e. where the drawing is truncated in the x- or z-
|
||
directions).
|
||
</code></pre></div>
|
||
<h4 id="meanas.fdmath--datastructure-dx_lists_t">Datastructure: dx_lists_t<a class="headerlink" href="#meanas.fdmath--datastructure-dx_lists_t" title="Permanent link">¶</a></h4>
|
||
<p>In this documentation, the E fore-vectors are placed on the base grid. An
|
||
equivalent formulation could place the H back-vectors on the base grid instead.
|
||
However, in the case of a non-uniform grid, the operation to get from the "base"
|
||
cell widths to "derived" ones is not its own inverse.</p>
|
||
<p>The base grid's cell sizes could be fully described by a list of three 1D arrays,
|
||
specifying the cell widths along all three axes:</p>
|
||
<div class="highlight"><pre><span></span><code>[dx, dy, dz] = [[dx[0], dx[1], ...], [dy[0], ...], [dz[0], ...]]
|
||
</code></pre></div>
|
||
<p>Note that this is a list-of-arrays rather than a 2D array, as the simulation domain
|
||
may have a different number of cells along each axis.</p>
|
||
<p>Knowing the base grid's cell widths and the boundary conditions (periodic unless
|
||
otherwise noted) is enough information to calculate the cell widths <code>dx'</code>, <code>dy'</code>,
|
||
and <code>dz'</code> for the derived grids.</p>
|
||
<p>However, since most operations are trivially generalized to allow either E or H
|
||
to be defined on the base grid, they are written to take the a full set of base
|
||
and derived cell widths, distinguished by which field they apply to rather than
|
||
their "base" or "derived" status. This removes the need for each function to
|
||
generate the derived widths, and makes the "base" vs "derived" distinction
|
||
unnecessary in the code.</p>
|
||
<p>The resulting data structure containing all the cell widths takes the form of a
|
||
list-of-lists-of-arrays. The first list-of-arrays provides the cell widths for
|
||
the E-field fore-vectors, while the second list-of-arrays does the same for the
|
||
H-field back-vectors:</p>
|
||
<div class="highlight"><pre><span></span><code> [[[dx_e[0], dx_e[1], ...], [dy_e[0], ...], [dz_e[0], ...]],
|
||
[[dx_h[0], dx_h[1], ...], [dy_h[0], ...], [dz_h[0], ...]]]
|
||
</code></pre></div>
|
||
<p>where <code>dx_e[0]</code> is the x-width of the <code>m=0</code> cells, as used when calculating dE/dx,
|
||
and <code>dy_h[0]</code> is the y-width of the <code>n=0</code> cells, as used when calculating dH/dy, etc.</p>
|
||
<h3 id="meanas.fdmath--permittivity-and-permeability">Permittivity and Permeability<a class="headerlink" href="#meanas.fdmath--permittivity-and-permeability" title="Permanent link">¶</a></h3>
|
||
<p>Since each vector component of E and H is defined in a different location and represents
|
||
a different volume, the value of the spatially-discrete <code>epsilon</code> and <code>mu</code> can also be
|
||
different for all three field components, even when representing a simple planar interface
|
||
between two isotropic materials.</p>
|
||
<p>As a result, <code>epsilon</code> and <code>mu</code> are taken to have the same dimensions as the field, and
|
||
composed of the three diagonal tensor components:</p>
|
||
<div class="highlight"><pre><span></span><code>[equations: epsilon_and_mu]
|
||
epsilon = [epsilon_xx, epsilon_yy, epsilon_zz]
|
||
mu = [mu_xx, mu_yy, mu_zz]
|
||
</code></pre></div>
|
||
<p>or</p>
|
||
<div class="arithmatex">\[
|
||
\epsilon = \begin{bmatrix} \epsilon_{xx} & 0 & 0 \\
|
||
0 & \epsilon_{yy} & 0 \\
|
||
0 & 0 & \epsilon_{zz} \end{bmatrix}
|
||
\]</div>
|
||
<div class="arithmatex">\[
|
||
\mu = \begin{bmatrix} \mu_{xx} & 0 & 0 \\
|
||
0 & \mu_{yy} & 0 \\
|
||
0 & 0 & \mu_{zz} \end{bmatrix}
|
||
\]</div>
|
||
|
||
<p>where the off-diagonal terms (e.g. <code>epsilon_xy</code>) are assumed to be zero.</p>
|
||
<p>High-accuracy volumetric integration of shapes on multiple grids can be performed
|
||
by the <a href="https://mpxd.net/code/jan/gridlock">gridlock</a> module.</p>
|
||
<p>The values of the vacuum permittivity and permability effectively become scaling
|
||
factors that appear in several locations (e.g. between the E and H fields). In
|
||
order to limit floating-point inaccuracy and simplify calculations, they are often
|
||
set to 1 and relative permittivities and permeabilities are used in their places;
|
||
the true values can be multiplied back in after the simulation is complete if non-
|
||
normalized results are needed.</p>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="doc doc-children">
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
</div><h2 id="functional-and-sparse-operators">Functional and sparse operators<a class="headerlink" href="#functional-and-sparse-operators" title="Permanent link">¶</a></h2>
|
||
|
||
|
||
<div class="doc doc-object doc-module">
|
||
|
||
|
||
|
||
<h2 id="meanas.fdmath.functional" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-module"></code> <span class="doc doc-object-name doc-module-name">meanas.fdmath.functional</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.functional" class="headerlink" title="Permanent link">¶</a></h2>
|
||
|
||
<div class="doc doc-contents first">
|
||
|
||
<p>Math functions for finite difference simulations</p>
|
||
<p>Basic discrete calculus etc.</p>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="doc doc-children">
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="doc doc-object doc-function">
|
||
|
||
|
||
<h3 id="meanas.fdmath.functional.deriv_forward" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-function"></code> <span class="doc doc-object-name doc-function-name">deriv_forward</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.functional.deriv_forward" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">deriv_forward</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">dx_e</span><span class="p">:</span> <span class="n">Sequence</span><span class="p">[</span><span class="n">NDArray</span><span class="p">[</span><span class="n">floating</span> <span class="o">|</span> <span class="n">complexfloating</span><span class="p">]]</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span><span class="p">,</span>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="p">)</span> <span class="o">-></span> <span class="nb">tuple</span><span class="p">[</span>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> <span class="n">fdfield_updater_t</span><span class="p">,</span> <span class="n">fdfield_updater_t</span><span class="p">,</span> <span class="n">fdfield_updater_t</span>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Utility operators for taking discretized derivatives (backward variant).</p>
|
||
|
||
|
||
<p><span class="doc-section-title">Parameters:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Name</th>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
<th>Default</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>dx_e</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="collections.abc.Sequence">Sequence</span>[<span title="numpy.typing.NDArray">NDArray</span>[<span title="numpy.floating">floating</span> | <span title="numpy.complexfloating">complexfloating</span>]] | None</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Lists of cell sizes for all axes
|
||
<code>[[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...]</code>.</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<code>None</code>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<p><span class="doc-section-title">Returns:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code><span title="tuple">tuple</span>[<a class="autorefs autorefs-internal" title=" fdfield_updater_t
|
||
|
||
|
||
|
||
module-attribute
|
||
(meanas.fdmath.types.fdfield_updater_t)" href="#meanas.fdmath.types.fdfield_updater_t">fdfield_updater_t</a>, <a class="autorefs autorefs-internal" title=" fdfield_updater_t
|
||
|
||
|
||
|
||
module-attribute
|
||
(meanas.fdmath.types.fdfield_updater_t)" href="#meanas.fdmath.types.fdfield_updater_t">fdfield_updater_t</a>, <a class="autorefs autorefs-internal" title=" fdfield_updater_t
|
||
|
||
|
||
|
||
module-attribute
|
||
(meanas.fdmath.types.fdfield_updater_t)" href="#meanas.fdmath.types.fdfield_updater_t">fdfield_updater_t</a>]</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>List of functions for taking forward derivatives along each axis.</p>
|
||
</div>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-function">
|
||
|
||
|
||
<h3 id="meanas.fdmath.functional.deriv_back" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-function"></code> <span class="doc doc-object-name doc-function-name">deriv_back</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.functional.deriv_back" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">deriv_back</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">dx_h</span><span class="p">:</span> <span class="n">Sequence</span><span class="p">[</span><span class="n">NDArray</span><span class="p">[</span><span class="n">floating</span> <span class="o">|</span> <span class="n">complexfloating</span><span class="p">]]</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span><span class="p">,</span>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="p">)</span> <span class="o">-></span> <span class="nb">tuple</span><span class="p">[</span>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> <span class="n">fdfield_updater_t</span><span class="p">,</span> <span class="n">fdfield_updater_t</span><span class="p">,</span> <span class="n">fdfield_updater_t</span>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Utility operators for taking discretized derivatives (forward variant).</p>
|
||
|
||
|
||
<p><span class="doc-section-title">Parameters:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Name</th>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
<th>Default</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>dx_h</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="collections.abc.Sequence">Sequence</span>[<span title="numpy.typing.NDArray">NDArray</span>[<span title="numpy.floating">floating</span> | <span title="numpy.complexfloating">complexfloating</span>]] | None</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Lists of cell sizes for all axes
|
||
<code>[[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...]</code>.</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<code>None</code>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<p><span class="doc-section-title">Returns:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code><span title="tuple">tuple</span>[<a class="autorefs autorefs-internal" title=" fdfield_updater_t
|
||
|
||
|
||
|
||
module-attribute
|
||
(meanas.fdmath.types.fdfield_updater_t)" href="#meanas.fdmath.types.fdfield_updater_t">fdfield_updater_t</a>, <a class="autorefs autorefs-internal" title=" fdfield_updater_t
|
||
|
||
|
||
|
||
module-attribute
|
||
(meanas.fdmath.types.fdfield_updater_t)" href="#meanas.fdmath.types.fdfield_updater_t">fdfield_updater_t</a>, <a class="autorefs autorefs-internal" title=" fdfield_updater_t
|
||
|
||
|
||
|
||
module-attribute
|
||
(meanas.fdmath.types.fdfield_updater_t)" href="#meanas.fdmath.types.fdfield_updater_t">fdfield_updater_t</a>]</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>List of functions for taking forward derivatives along each axis.</p>
|
||
</div>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-function">
|
||
|
||
|
||
<h3 id="meanas.fdmath.functional.curl_forward" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-function"></code> <span class="doc doc-object-name doc-function-name">curl_forward</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.functional.curl_forward" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">curl_forward</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">dx_e</span><span class="p">:</span> <span class="n">Sequence</span><span class="p">[</span><span class="n">NDArray</span><span class="p">[</span><span class="n">floating</span> <span class="o">|</span> <span class="n">complexfloating</span><span class="p">]]</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span><span class="p">,</span>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="p">)</span> <span class="o">-></span> <span class="n">Callable</span><span class="p">[[</span><span class="n">TT</span><span class="p">],</span> <span class="n">TT</span><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Curl operator for use with the E field.</p>
|
||
|
||
|
||
<p><span class="doc-section-title">Parameters:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Name</th>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
<th>Default</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>dx_e</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="collections.abc.Sequence">Sequence</span>[<span title="numpy.typing.NDArray">NDArray</span>[<span title="numpy.floating">floating</span> | <span title="numpy.complexfloating">complexfloating</span>]] | None</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Lists of cell sizes for all axes
|
||
<code>[[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...]</code>.</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<code>None</code>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<p><span class="doc-section-title">Returns:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code><span title="collections.abc.Callable">Callable</span>[[<span title="meanas.fdmath.functional.TT">TT</span>], <span title="meanas.fdmath.functional.TT">TT</span>]</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Function <code>f</code> for taking the discrete forward curl of a field,</p>
|
||
</div>
|
||
</td>
|
||
</tr>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code><span title="collections.abc.Callable">Callable</span>[[<span title="meanas.fdmath.functional.TT">TT</span>], <span title="meanas.fdmath.functional.TT">TT</span>]</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p><code>f(E)</code> -> curlE <span class="arithmatex">\(= \nabla_f \times E\)</span></p>
|
||
</div>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-function">
|
||
|
||
|
||
<h3 id="meanas.fdmath.functional.curl_back" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-function"></code> <span class="doc doc-object-name doc-function-name">curl_back</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.functional.curl_back" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">curl_back</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">dx_h</span><span class="p">:</span> <span class="n">Sequence</span><span class="p">[</span><span class="n">NDArray</span><span class="p">[</span><span class="n">floating</span> <span class="o">|</span> <span class="n">complexfloating</span><span class="p">]]</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span><span class="p">,</span>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="p">)</span> <span class="o">-></span> <span class="n">Callable</span><span class="p">[[</span><span class="n">TT</span><span class="p">],</span> <span class="n">TT</span><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Create a function which takes the backward curl of a field.</p>
|
||
|
||
|
||
<p><span class="doc-section-title">Parameters:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Name</th>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
<th>Default</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>dx_h</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="collections.abc.Sequence">Sequence</span>[<span title="numpy.typing.NDArray">NDArray</span>[<span title="numpy.floating">floating</span> | <span title="numpy.complexfloating">complexfloating</span>]] | None</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Lists of cell sizes for all axes
|
||
<code>[[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...]</code>.</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<code>None</code>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<p><span class="doc-section-title">Returns:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code><span title="collections.abc.Callable">Callable</span>[[<span title="meanas.fdmath.functional.TT">TT</span>], <span title="meanas.fdmath.functional.TT">TT</span>]</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Function <code>f</code> for taking the discrete backward curl of a field,</p>
|
||
</div>
|
||
</td>
|
||
</tr>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code><span title="collections.abc.Callable">Callable</span>[[<span title="meanas.fdmath.functional.TT">TT</span>], <span title="meanas.fdmath.functional.TT">TT</span>]</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p><code>f(H)</code> -> curlH <span class="arithmatex">\(= \nabla_b \times H\)</span></p>
|
||
</div>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-module">
|
||
|
||
|
||
|
||
<h2 id="meanas.fdmath.operators" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-module"></code> <span class="doc doc-object-name doc-module-name">meanas.fdmath.operators</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.operators" class="headerlink" title="Permanent link">¶</a></h2>
|
||
|
||
<div class="doc doc-contents first">
|
||
|
||
<p>Matrix operators for finite difference simulations</p>
|
||
<p>Basic discrete calculus etc.</p>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="doc doc-children">
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="doc doc-object doc-function">
|
||
|
||
|
||
<h3 id="meanas.fdmath.operators.shift_circ" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-function"></code> <span class="doc doc-object-name doc-function-name">shift_circ</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.operators.shift_circ" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">shift_circ</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">axis</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">shape</span><span class="p">:</span> <span class="n">Sequence</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">shift_distance</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">1</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="p">)</span> <span class="o">-></span> <span class="n">sparse</span><span class="o">.</span><span class="n">sparray</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Utility operator for performing a circular shift along a specified axis by a
|
||
specified number of elements.</p>
|
||
|
||
|
||
<p><span class="doc-section-title">Parameters:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Name</th>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
<th>Default</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>axis</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="int">int</span></code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Axis to shift along. x=0, y=1, z=2</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<em>required</em>
|
||
</td>
|
||
</tr>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>shape</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="collections.abc.Sequence">Sequence</span>[<span title="int">int</span>]</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Shape of the grid being shifted</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<em>required</em>
|
||
</td>
|
||
</tr>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>shift_distance</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="int">int</span></code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Number of cells to shift by. May be negative. Default 1.</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<code>1</code>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<p><span class="doc-section-title">Returns:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code><span title="scipy.sparse.sparray">sparray</span></code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Sparse matrix for performing the circular shift.</p>
|
||
</div>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-function">
|
||
|
||
|
||
<h3 id="meanas.fdmath.operators.shift_with_mirror" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-function"></code> <span class="doc doc-object-name doc-function-name">shift_with_mirror</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.operators.shift_with_mirror" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">shift_with_mirror</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">axis</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">shape</span><span class="p">:</span> <span class="n">Sequence</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">shift_distance</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">1</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="p">)</span> <span class="o">-></span> <span class="n">sparse</span><span class="o">.</span><span class="n">sparray</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Utility operator for performing an n-element shift along a specified axis, with mirror
|
||
boundary conditions applied to the cells beyond the receding edge.</p>
|
||
|
||
|
||
<p><span class="doc-section-title">Parameters:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Name</th>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
<th>Default</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>axis</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="int">int</span></code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Axis to shift along. x=0, y=1, z=2</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<em>required</em>
|
||
</td>
|
||
</tr>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>shape</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="collections.abc.Sequence">Sequence</span>[<span title="int">int</span>]</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Shape of the grid being shifted</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<em>required</em>
|
||
</td>
|
||
</tr>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>shift_distance</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="int">int</span></code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Number of cells to shift by. May be negative. Default 1.</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<code>1</code>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<p><span class="doc-section-title">Returns:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code><span title="scipy.sparse.sparray">sparray</span></code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Sparse matrix for performing the shift-with-mirror.</p>
|
||
</div>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-function">
|
||
|
||
|
||
<h3 id="meanas.fdmath.operators.deriv_forward" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-function"></code> <span class="doc doc-object-name doc-function-name">deriv_forward</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.operators.deriv_forward" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">deriv_forward</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">dx_e</span><span class="p">:</span> <span class="n">Sequence</span><span class="p">[</span><span class="n">NDArray</span><span class="p">[</span><span class="n">floating</span> <span class="o">|</span> <span class="n">complexfloating</span><span class="p">]],</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="p">)</span> <span class="o">-></span> <span class="nb">list</span><span class="p">[</span><span class="n">sparse</span><span class="o">.</span><span class="n">sparray</span><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Utility operators for taking discretized derivatives (forward variant).</p>
|
||
|
||
|
||
<p><span class="doc-section-title">Parameters:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Name</th>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
<th>Default</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>dx_e</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="collections.abc.Sequence">Sequence</span>[<span title="numpy.typing.NDArray">NDArray</span>[<span title="numpy.floating">floating</span> | <span title="numpy.complexfloating">complexfloating</span>]]</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Lists of cell sizes for all axes
|
||
<code>[[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...]</code>.</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<em>required</em>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<p><span class="doc-section-title">Returns:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code><span title="list">list</span>[<span title="scipy.sparse.sparray">sparray</span>]</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>List of operators for taking forward derivatives along each axis.</p>
|
||
</div>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-function">
|
||
|
||
|
||
<h3 id="meanas.fdmath.operators.deriv_back" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-function"></code> <span class="doc doc-object-name doc-function-name">deriv_back</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.operators.deriv_back" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">deriv_back</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">dx_h</span><span class="p">:</span> <span class="n">Sequence</span><span class="p">[</span><span class="n">NDArray</span><span class="p">[</span><span class="n">floating</span> <span class="o">|</span> <span class="n">complexfloating</span><span class="p">]],</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="p">)</span> <span class="o">-></span> <span class="nb">list</span><span class="p">[</span><span class="n">sparse</span><span class="o">.</span><span class="n">sparray</span><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Utility operators for taking discretized derivatives (backward variant).</p>
|
||
|
||
|
||
<p><span class="doc-section-title">Parameters:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Name</th>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
<th>Default</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>dx_h</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="collections.abc.Sequence">Sequence</span>[<span title="numpy.typing.NDArray">NDArray</span>[<span title="numpy.floating">floating</span> | <span title="numpy.complexfloating">complexfloating</span>]]</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Lists of cell sizes for all axes
|
||
<code>[[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...]</code>.</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<em>required</em>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<p><span class="doc-section-title">Returns:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code><span title="list">list</span>[<span title="scipy.sparse.sparray">sparray</span>]</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>List of operators for taking forward derivatives along each axis.</p>
|
||
</div>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-function">
|
||
|
||
|
||
<h3 id="meanas.fdmath.operators.cross" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-function"></code> <span class="doc doc-object-name doc-function-name">cross</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.operators.cross" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">cross</span><span class="p">(</span><span class="n">B</span><span class="p">:</span> <span class="n">Sequence</span><span class="p">[</span><span class="n">sparray</span><span class="p">])</span> <span class="o">-></span> <span class="n">sparse</span><span class="o">.</span><span class="n">sparray</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Cross product operator</p>
|
||
|
||
|
||
<p><span class="doc-section-title">Parameters:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Name</th>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
<th>Default</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>B</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="collections.abc.Sequence">Sequence</span>[<span title="scipy.sparse.sparray">sparray</span>]</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>List <code>[Bx, By, Bz]</code> of sparse matrices corresponding to the x, y, z
|
||
portions of the operator on the left side of the cross product.</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<em>required</em>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<p><span class="doc-section-title">Returns:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code><span title="scipy.sparse.sparray">sparray</span></code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Sparse matrix corresponding to (B x), where x is the cross product.</p>
|
||
</div>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-function">
|
||
|
||
|
||
<h3 id="meanas.fdmath.operators.vec_cross" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-function"></code> <span class="doc doc-object-name doc-function-name">vec_cross</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.operators.vec_cross" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">vec_cross</span><span class="p">(</span><span class="n">b</span><span class="p">:</span> <span class="n">vfdfield</span><span class="p">)</span> <span class="o">-></span> <span class="n">sparse</span><span class="o">.</span><span class="n">sparray</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Vector cross product operator</p>
|
||
|
||
|
||
<p><span class="doc-section-title">Parameters:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Name</th>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
<th>Default</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>b</code>
|
||
</td>
|
||
<td>
|
||
<code><a class="autorefs autorefs-internal" title=" vfdfield (meanas.fdmath.types.vfdfield)" href="#meanas.fdmath.types.vfdfield">vfdfield</a></code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Vector on the left side of the cross product.</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<em>required</em>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<p>Returns:</p>
|
||
<div class="highlight"><pre><span></span><code>Sparse matrix corresponding to (b x), where x is the cross product.
|
||
</code></pre></div>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-function">
|
||
|
||
|
||
<h3 id="meanas.fdmath.operators.avg_forward" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-function"></code> <span class="doc doc-object-name doc-function-name">avg_forward</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.operators.avg_forward" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">avg_forward</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">axis</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">shape</span><span class="p">:</span> <span class="n">Sequence</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="p">)</span> <span class="o">-></span> <span class="n">sparse</span><span class="o">.</span><span class="n">sparray</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Forward average operator <code>(x4 = (x4 + x5) / 2)</code></p>
|
||
|
||
|
||
<p><span class="doc-section-title">Parameters:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Name</th>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
<th>Default</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>axis</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="int">int</span></code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Axis to average along (x=0, y=1, z=2)</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<em>required</em>
|
||
</td>
|
||
</tr>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>shape</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="collections.abc.Sequence">Sequence</span>[<span title="int">int</span>]</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Shape of the grid to average</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<em>required</em>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<p><span class="doc-section-title">Returns:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code><span title="scipy.sparse.sparray">sparray</span></code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Sparse matrix for forward average operation.</p>
|
||
</div>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-function">
|
||
|
||
|
||
<h3 id="meanas.fdmath.operators.avg_back" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-function"></code> <span class="doc doc-object-name doc-function-name">avg_back</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.operators.avg_back" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">avg_back</span><span class="p">(</span><span class="n">axis</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">shape</span><span class="p">:</span> <span class="n">Sequence</span><span class="p">[</span><span class="nb">int</span><span class="p">])</span> <span class="o">-></span> <span class="n">sparse</span><span class="o">.</span><span class="n">sparray</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Backward average operator <code>(x4 = (x4 + x3) / 2)</code></p>
|
||
|
||
|
||
<p><span class="doc-section-title">Parameters:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Name</th>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
<th>Default</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>axis</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="int">int</span></code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Axis to average along (x=0, y=1, z=2)</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<em>required</em>
|
||
</td>
|
||
</tr>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>shape</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="collections.abc.Sequence">Sequence</span>[<span title="int">int</span>]</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Shape of the grid to average</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<em>required</em>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<p><span class="doc-section-title">Returns:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code><span title="scipy.sparse.sparray">sparray</span></code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Sparse matrix for backward average operation.</p>
|
||
</div>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-function">
|
||
|
||
|
||
<h3 id="meanas.fdmath.operators.curl_forward" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-function"></code> <span class="doc doc-object-name doc-function-name">curl_forward</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.operators.curl_forward" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">curl_forward</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">dx_e</span><span class="p">:</span> <span class="n">Sequence</span><span class="p">[</span><span class="n">NDArray</span><span class="p">[</span><span class="n">floating</span> <span class="o">|</span> <span class="n">complexfloating</span><span class="p">]],</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="p">)</span> <span class="o">-></span> <span class="n">sparse</span><span class="o">.</span><span class="n">sparray</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Curl operator for use with the E field.</p>
|
||
|
||
|
||
<p><span class="doc-section-title">Parameters:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Name</th>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
<th>Default</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>dx_e</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="collections.abc.Sequence">Sequence</span>[<span title="numpy.typing.NDArray">NDArray</span>[<span title="numpy.floating">floating</span> | <span title="numpy.complexfloating">complexfloating</span>]]</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Lists of cell sizes for all axes
|
||
<code>[[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...]</code>.</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<em>required</em>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<p><span class="doc-section-title">Returns:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code><span title="scipy.sparse.sparray">sparray</span></code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Sparse matrix for taking the discretized curl of the E-field</p>
|
||
</div>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-function">
|
||
|
||
|
||
<h3 id="meanas.fdmath.operators.curl_back" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-function"></code> <span class="doc doc-object-name doc-function-name">curl_back</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.operators.curl_back" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">curl_back</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">dx_h</span><span class="p">:</span> <span class="n">Sequence</span><span class="p">[</span><span class="n">NDArray</span><span class="p">[</span><span class="n">floating</span> <span class="o">|</span> <span class="n">complexfloating</span><span class="p">]],</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="p">)</span> <span class="o">-></span> <span class="n">sparse</span><span class="o">.</span><span class="n">sparray</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Curl operator for use with the H field.</p>
|
||
|
||
|
||
<p><span class="doc-section-title">Parameters:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Name</th>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
<th>Default</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>dx_h</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="collections.abc.Sequence">Sequence</span>[<span title="numpy.typing.NDArray">NDArray</span>[<span title="numpy.floating">floating</span> | <span title="numpy.complexfloating">complexfloating</span>]]</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Lists of cell sizes for all axes
|
||
<code>[[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...]</code>.</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<em>required</em>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<p><span class="doc-section-title">Returns:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code><span title="scipy.sparse.sparray">sparray</span></code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Sparse matrix for taking the discretized curl of the H-field</p>
|
||
</div>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-module">
|
||
|
||
|
||
|
||
<h2 id="meanas.fdmath.vectorization" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-module"></code> <span class="doc doc-object-name doc-module-name">meanas.fdmath.vectorization</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.vectorization" class="headerlink" title="Permanent link">¶</a></h2>
|
||
|
||
<div class="doc doc-contents first">
|
||
|
||
<p>Functions for moving between a vector field (list of 3 ndarrays, <code>[f_x, f_y, f_z]</code>)
|
||
and a 1D array representation of that field <code>[f_x0, f_x1, f_x2,... f_y0,... f_z0,...]</code>.
|
||
Vectorized versions of the field use row-major (ie., C-style) ordering.</p>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="doc doc-children">
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="doc doc-object doc-function">
|
||
|
||
|
||
<h3 id="meanas.fdmath.vectorization.vec" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-function"></code> <span class="doc doc-object-name doc-function-name">vec</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.vectorization.vec" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-overloads">
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">vec</span><span class="p">(</span><span class="n">f</span><span class="p">:</span> <span class="kc">None</span><span class="p">)</span> <span class="o">-></span> <span class="kc">None</span>
|
||
</code></pre></div><div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">vec</span><span class="p">(</span><span class="n">f</span><span class="p">:</span> <span class="n">fdfield_t</span><span class="p">)</span> <span class="o">-></span> <span class="n">vfdfield_t</span>
|
||
</code></pre></div><div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">vec</span><span class="p">(</span><span class="n">f</span><span class="p">:</span> <span class="n">cfdfield_t</span><span class="p">)</span> <span class="o">-></span> <span class="n">vcfdfield_t</span>
|
||
</code></pre></div><div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">vec</span><span class="p">(</span><span class="n">f</span><span class="p">:</span> <span class="n">fdfield2_t</span><span class="p">)</span> <span class="o">-></span> <span class="n">vfdfield2_t</span>
|
||
</code></pre></div><div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">vec</span><span class="p">(</span><span class="n">f</span><span class="p">:</span> <span class="n">cfdfield2_t</span><span class="p">)</span> <span class="o">-></span> <span class="n">vcfdfield2_t</span>
|
||
</code></pre></div><div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">vec</span><span class="p">(</span><span class="n">f</span><span class="p">:</span> <span class="n">fdslice_t</span><span class="p">)</span> <span class="o">-></span> <span class="n">vfdslice_t</span>
|
||
</code></pre></div><div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">vec</span><span class="p">(</span><span class="n">f</span><span class="p">:</span> <span class="n">cfdslice_t</span><span class="p">)</span> <span class="o">-></span> <span class="n">vcfdslice_t</span>
|
||
</code></pre></div><div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">vec</span><span class="p">(</span><span class="n">f</span><span class="p">:</span> <span class="n">ArrayLike</span><span class="p">)</span> <span class="o">-></span> <span class="n">NDArray</span>
|
||
</code></pre></div> </div>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">vec</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">f</span><span class="p">:</span> <span class="n">fdfield_t</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a> <span class="o">|</span> <span class="n">cfdfield_t</span>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a> <span class="o">|</span> <span class="n">fdfield2_t</span>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> <span class="o">|</span> <span class="n">cfdfield2_t</span>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a> <span class="o">|</span> <span class="n">fdslice_t</span>
|
||
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a> <span class="o">|</span> <span class="n">cfdslice_t</span>
|
||
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a> <span class="o">|</span> <span class="n">ArrayLike</span>
|
||
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a> <span class="o">|</span> <span class="kc">None</span><span class="p">,</span>
|
||
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="p">)</span> <span class="o">-></span> <span class="p">(</span>
|
||
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a> <span class="n">vfdfield_t</span>
|
||
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a> <span class="o">|</span> <span class="n">vcfdfield_t</span>
|
||
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a> <span class="o">|</span> <span class="n">vfdfield2_t</span>
|
||
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a> <span class="o">|</span> <span class="n">vcfdfield2_t</span>
|
||
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a> <span class="o">|</span> <span class="n">vfdslice_t</span>
|
||
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a> <span class="o">|</span> <span class="n">vcfdslice_t</span>
|
||
<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a> <span class="o">|</span> <span class="n">NDArray</span>
|
||
<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a> <span class="o">|</span> <span class="kc">None</span>
|
||
<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a><span class="p">)</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Create a 1D ndarray from a vector field which spans a 1-3D region.</p>
|
||
<p>Returns <code>None</code> if called with <code>f=None</code>.</p>
|
||
|
||
|
||
<p><span class="doc-section-title">Parameters:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Name</th>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
<th>Default</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>f</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="meanas.fdmath.types.fdfield_t">fdfield_t</span> | <span title="meanas.fdmath.types.cfdfield_t">cfdfield_t</span> | <span title="meanas.fdmath.types.fdfield2_t">fdfield2_t</span> | <span title="meanas.fdmath.types.cfdfield2_t">cfdfield2_t</span> | <span title="meanas.fdmath.types.fdslice_t">fdslice_t</span> | <span title="meanas.fdmath.types.cfdslice_t">cfdslice_t</span> | <span title="numpy.typing.ArrayLike">ArrayLike</span> | None</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>A vector field, e.g. <code>[f_x, f_y, f_z]</code> where each <code>f_</code> component is a 1- to
|
||
3-D ndarray (<code>f_*</code> should all be the same size). Doesn't fail with <code>f=None</code>.</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<em>required</em>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<p><span class="doc-section-title">Returns:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code><span title="meanas.fdmath.types.vfdfield_t">vfdfield_t</span> | <span title="meanas.fdmath.types.vcfdfield_t">vcfdfield_t</span> | <span title="meanas.fdmath.types.vfdfield2_t">vfdfield2_t</span> | <span title="meanas.fdmath.types.vcfdfield2_t">vcfdfield2_t</span> | <span title="meanas.fdmath.types.vfdslice_t">vfdslice_t</span> | <span title="meanas.fdmath.types.vcfdslice_t">vcfdslice_t</span> | <span title="numpy.typing.NDArray">NDArray</span> | None</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>1D ndarray containing the linearized field (or <code>None</code>)</p>
|
||
</div>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-function">
|
||
|
||
|
||
<h3 id="meanas.fdmath.vectorization.unvec" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-function"></code> <span class="doc doc-object-name doc-function-name">unvec</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.vectorization.unvec" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-overloads">
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">unvec</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">v</span><span class="p">:</span> <span class="kc">None</span><span class="p">,</span> <span class="n">shape</span><span class="p">:</span> <span class="n">Sequence</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">nvdim</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">3</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="p">)</span> <span class="o">-></span> <span class="kc">None</span>
|
||
</code></pre></div><div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">unvec</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">v</span><span class="p">:</span> <span class="n">vfdfield_t</span><span class="p">,</span> <span class="n">shape</span><span class="p">:</span> <span class="n">Sequence</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">nvdim</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">3</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="p">)</span> <span class="o">-></span> <span class="n">fdfield_t</span>
|
||
</code></pre></div><div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">unvec</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">v</span><span class="p">:</span> <span class="n">vcfdfield_t</span><span class="p">,</span> <span class="n">shape</span><span class="p">:</span> <span class="n">Sequence</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">nvdim</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">3</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="p">)</span> <span class="o">-></span> <span class="n">cfdfield_t</span>
|
||
</code></pre></div><div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">unvec</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">v</span><span class="p">:</span> <span class="n">vfdfield2_t</span><span class="p">,</span> <span class="n">shape</span><span class="p">:</span> <span class="n">Sequence</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">nvdim</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">3</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="p">)</span> <span class="o">-></span> <span class="n">fdfield2_t</span>
|
||
</code></pre></div><div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">unvec</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">v</span><span class="p">:</span> <span class="n">vcfdfield2_t</span><span class="p">,</span> <span class="n">shape</span><span class="p">:</span> <span class="n">Sequence</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">nvdim</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">3</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="p">)</span> <span class="o">-></span> <span class="n">cfdfield2_t</span>
|
||
</code></pre></div><div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">unvec</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">v</span><span class="p">:</span> <span class="n">vfdslice_t</span><span class="p">,</span> <span class="n">shape</span><span class="p">:</span> <span class="n">Sequence</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">nvdim</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">3</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="p">)</span> <span class="o">-></span> <span class="n">fdslice_t</span>
|
||
</code></pre></div><div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">unvec</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">v</span><span class="p">:</span> <span class="n">vcfdslice_t</span><span class="p">,</span> <span class="n">shape</span><span class="p">:</span> <span class="n">Sequence</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">nvdim</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">3</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="p">)</span> <span class="o">-></span> <span class="n">cfdslice_t</span>
|
||
</code></pre></div><div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">unvec</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">v</span><span class="p">:</span> <span class="n">ArrayLike</span><span class="p">,</span> <span class="n">shape</span><span class="p">:</span> <span class="n">Sequence</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">nvdim</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">3</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="p">)</span> <span class="o">-></span> <span class="n">NDArray</span>
|
||
</code></pre></div> </div>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">unvec</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">v</span><span class="p">:</span> <span class="n">vfdfield_t</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a> <span class="o">|</span> <span class="n">vcfdfield_t</span>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a> <span class="o">|</span> <span class="n">vfdfield2_t</span>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> <span class="o">|</span> <span class="n">vcfdfield2_t</span>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a> <span class="o">|</span> <span class="n">vfdslice_t</span>
|
||
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a> <span class="o">|</span> <span class="n">vcfdslice_t</span>
|
||
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a> <span class="o">|</span> <span class="n">ArrayLike</span>
|
||
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a> <span class="o">|</span> <span class="kc">None</span><span class="p">,</span>
|
||
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a> <span class="n">shape</span><span class="p">:</span> <span class="n">Sequence</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span>
|
||
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a> <span class="n">nvdim</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span>
|
||
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="p">)</span> <span class="o">-></span> <span class="p">(</span>
|
||
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a> <span class="n">fdfield_t</span>
|
||
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a> <span class="o">|</span> <span class="n">cfdfield_t</span>
|
||
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a> <span class="o">|</span> <span class="n">fdfield2_t</span>
|
||
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a> <span class="o">|</span> <span class="n">cfdfield2_t</span>
|
||
<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a> <span class="o">|</span> <span class="n">fdslice_t</span>
|
||
<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a> <span class="o">|</span> <span class="n">cfdslice_t</span>
|
||
<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a> <span class="o">|</span> <span class="n">NDArray</span>
|
||
<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a> <span class="o">|</span> <span class="kc">None</span>
|
||
<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="p">)</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Perform the inverse of vec(): take a 1D ndarray and output an <code>nvdim</code>-component field
|
||
of form e.g. <code>[f_x, f_y, f_z]</code> (<code>nvdim=3</code>) where each of <code>f_*</code> is a len(shape)-dimensional
|
||
ndarray.</p>
|
||
<p>Returns <code>None</code> if called with <code>v=None</code>.</p>
|
||
|
||
|
||
<p><span class="doc-section-title">Parameters:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Name</th>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
<th>Default</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>v</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="meanas.fdmath.types.vfdfield_t">vfdfield_t</span> | <span title="meanas.fdmath.types.vcfdfield_t">vcfdfield_t</span> | <span title="meanas.fdmath.types.vfdfield2_t">vfdfield2_t</span> | <span title="meanas.fdmath.types.vcfdfield2_t">vcfdfield2_t</span> | <span title="meanas.fdmath.types.vfdslice_t">vfdslice_t</span> | <span title="meanas.fdmath.types.vcfdslice_t">vcfdslice_t</span> | <span title="numpy.typing.ArrayLike">ArrayLike</span> | None</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>1D ndarray representing a vector field of shape shape (or None)</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<em>required</em>
|
||
</td>
|
||
</tr>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>shape</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="collections.abc.Sequence">Sequence</span>[<span title="int">int</span>]</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>shape of the vector field</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<em>required</em>
|
||
</td>
|
||
</tr>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>nvdim</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="int">int</span></code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Number of components in each vector</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<code>3</code>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<p><span class="doc-section-title">Returns:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code><span title="meanas.fdmath.types.fdfield_t">fdfield_t</span> | <span title="meanas.fdmath.types.cfdfield_t">cfdfield_t</span> | <span title="meanas.fdmath.types.fdfield2_t">fdfield2_t</span> | <span title="meanas.fdmath.types.cfdfield2_t">cfdfield2_t</span> | <span title="meanas.fdmath.types.fdslice_t">fdslice_t</span> | <span title="meanas.fdmath.types.cfdslice_t">cfdslice_t</span> | <span title="numpy.typing.NDArray">NDArray</span> | None</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p><code>[f_x, f_y, f_z]</code> where each <code>f_</code> is a <code>len(shape)</code> dimensional ndarray (or <code>None</code>)</p>
|
||
</div>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-module">
|
||
|
||
|
||
|
||
<h2 id="meanas.fdmath.types" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-module"></code> <span class="doc doc-object-name doc-module-name">meanas.fdmath.types</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.types" class="headerlink" title="Permanent link">¶</a></h2>
|
||
|
||
<div class="doc doc-contents first">
|
||
|
||
<p>Types shared across multiple submodules</p>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="doc doc-children">
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="doc doc-object doc-attribute">
|
||
|
||
|
||
|
||
<h3 id="meanas.fdmath.types.dx_lists_t" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-attribute"></code> <span class="doc doc-object-name doc-attribute-name">dx_lists_t</span>
|
||
|
||
|
||
<span class="doc doc-labels">
|
||
<small class="doc doc-label doc-label-module-attribute"><code>module-attribute</code></small>
|
||
</span>
|
||
|
||
<a href="#meanas.fdmath.types.dx_lists_t" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="n">dx_lists_t</span> <span class="o">=</span> <span class="n">Sequence</span><span class="p">[</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">Sequence</span><span class="p">[</span><span class="n">NDArray</span><span class="p">[</span><span class="n">floating</span> <span class="o">|</span> <span class="n">complexfloating</span><span class="p">]]</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>'dxes' datastructure which contains grid cell width information in the following format:</p>
|
||
<div class="highlight"><pre><span></span><code>[[[dx_e[0], dx_e[1], ...], [dy_e[0], ...], [dz_e[0], ...]],
|
||
[[dx_h[0], dx_h[1], ...], [dy_h[0], ...], [dz_h[0], ...]]]
|
||
</code></pre></div>
|
||
<p>where <code>dx_e[0]</code> is the x-width of the <code>x=0</code> cells, as used when calculating dE/dx,
|
||
and <code>dy_h[0]</code> is the y-width of the <code>y=0</code> cells, as used when calculating dH/dy, etc.</p>
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-attribute">
|
||
|
||
|
||
|
||
<h3 id="meanas.fdmath.types.dx_lists2_t" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-attribute"></code> <span class="doc doc-object-name doc-attribute-name">dx_lists2_t</span>
|
||
|
||
|
||
<span class="doc doc-labels">
|
||
<small class="doc doc-label doc-label-module-attribute"><code>module-attribute</code></small>
|
||
</span>
|
||
|
||
<a href="#meanas.fdmath.types.dx_lists2_t" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="n">dx_lists2_t</span> <span class="o">=</span> <span class="n">Sequence</span><span class="p">[</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">Sequence</span><span class="p">[</span><span class="n">NDArray</span><span class="p">[</span><span class="n">floating</span> <span class="o">|</span> <span class="n">complexfloating</span><span class="p">]]</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>2D 'dxes' datastructure which contains grid cell width information in the following format:</p>
|
||
<div class="highlight"><pre><span></span><code>[[[dx_e[0], dx_e[1], ...], [dy_e[0], ...]],
|
||
[[dx_h[0], dx_h[1], ...], [dy_h[0], ...]]]
|
||
</code></pre></div>
|
||
<p>where <code>dx_e[0]</code> is the x-width of the <code>x=0</code> cells, as used when calculating dE/dx,
|
||
and <code>dy_h[0]</code> is the y-width of the <code>y=0</code> cells, as used when calculating dH/dy, etc.</p>
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-attribute">
|
||
|
||
|
||
|
||
<h3 id="meanas.fdmath.types.dx_lists_mut" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-attribute"></code> <span class="doc doc-object-name doc-attribute-name">dx_lists_mut</span>
|
||
|
||
|
||
<span class="doc doc-labels">
|
||
<small class="doc doc-label doc-label-module-attribute"><code>module-attribute</code></small>
|
||
</span>
|
||
|
||
<a href="#meanas.fdmath.types.dx_lists_mut" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="n">dx_lists_mut</span> <span class="o">=</span> <span class="n">MutableSequence</span><span class="p">[</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">MutableSequence</span><span class="p">[</span><span class="n">NDArray</span><span class="p">[</span><span class="n">floating</span> <span class="o">|</span> <span class="n">complexfloating</span><span class="p">]]</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Mutable version of <code>dx_lists_t</code></p>
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-attribute">
|
||
|
||
|
||
|
||
<h3 id="meanas.fdmath.types.dx_lists2_mut" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-attribute"></code> <span class="doc doc-object-name doc-attribute-name">dx_lists2_mut</span>
|
||
|
||
|
||
<span class="doc doc-labels">
|
||
<small class="doc doc-label doc-label-module-attribute"><code>module-attribute</code></small>
|
||
</span>
|
||
|
||
<a href="#meanas.fdmath.types.dx_lists2_mut" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="n">dx_lists2_mut</span> <span class="o">=</span> <span class="n">MutableSequence</span><span class="p">[</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">MutableSequence</span><span class="p">[</span><span class="n">NDArray</span><span class="p">[</span><span class="n">floating</span> <span class="o">|</span> <span class="n">complexfloating</span><span class="p">]]</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Mutable version of <code>dx_lists2_t</code></p>
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-attribute">
|
||
|
||
|
||
|
||
<h3 id="meanas.fdmath.types.fdfield_updater_t" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-attribute"></code> <span class="doc doc-object-name doc-attribute-name">fdfield_updater_t</span>
|
||
|
||
|
||
<span class="doc doc-labels">
|
||
<small class="doc doc-label doc-label-module-attribute"><code>module-attribute</code></small>
|
||
</span>
|
||
|
||
<a href="#meanas.fdmath.types.fdfield_updater_t" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="n">fdfield_updater_t</span> <span class="o">=</span> <span class="n">Callable</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">fdfield</span><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Convenience type for functions which take and return a real <code>fdfield</code></p>
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-attribute">
|
||
|
||
|
||
|
||
<h3 id="meanas.fdmath.types.cfdfield_updater_t" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-attribute"></code> <span class="doc doc-object-name doc-attribute-name">cfdfield_updater_t</span>
|
||
|
||
|
||
<span class="doc doc-labels">
|
||
<small class="doc doc-label doc-label-module-attribute"><code>module-attribute</code></small>
|
||
</span>
|
||
|
||
<a href="#meanas.fdmath.types.cfdfield_updater_t" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="n">cfdfield_updater_t</span> <span class="o">=</span> <span class="n">Callable</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">cfdfield</span><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Convenience type for functions which take and return a complex <code>cfdfield</code></p>
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
|
||
<div class="doc doc-object doc-type_alias">
|
||
|
||
|
||
|
||
<h3 id="meanas.fdmath.types.fdfield" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-type_alias"></code> <span class="doc doc-object-name doc-type_alias-name">fdfield</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.types.fdfield" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nc">fdfield</span> <span class="o">=</span> <span class="n">fdfield_t</span> <span class="o">|</span> <span class="n">NDArray</span><span class="p">[</span><span class="n">floating</span><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Vector field with shape (3, X, Y, Z) (e.g. <code>[E_x, E_y, E_z]</code>)</p>
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-type_alias">
|
||
|
||
|
||
|
||
<h3 id="meanas.fdmath.types.vfdfield" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-type_alias"></code> <span class="doc doc-object-name doc-type_alias-name">vfdfield</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.types.vfdfield" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nc">vfdfield</span> <span class="o">=</span> <span class="n">vfdfield_t</span> <span class="o">|</span> <span class="n">NDArray</span><span class="p">[</span><span class="n">floating</span><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Linearized vector field (single vector of length 3<em>X</em>Y*Z)</p>
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-type_alias">
|
||
|
||
|
||
|
||
<h3 id="meanas.fdmath.types.cfdfield" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-type_alias"></code> <span class="doc doc-object-name doc-type_alias-name">cfdfield</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.types.cfdfield" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nc">cfdfield</span> <span class="o">=</span> <span class="n">cfdfield_t</span> <span class="o">|</span> <span class="n">NDArray</span><span class="p">[</span><span class="n">complexfloating</span><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Complex vector field with shape (3, X, Y, Z) (e.g. <code>[E_x, E_y, E_z]</code>)</p>
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-type_alias">
|
||
|
||
|
||
|
||
<h3 id="meanas.fdmath.types.vcfdfield" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-type_alias"></code> <span class="doc doc-object-name doc-type_alias-name">vcfdfield</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.types.vcfdfield" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nc">vcfdfield</span> <span class="o">=</span> <span class="n">vcfdfield_t</span> <span class="o">|</span> <span class="n">NDArray</span><span class="p">[</span><span class="n">complexfloating</span><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Linearized complex vector field (single vector of length 3<em>X</em>Y*Z)</p>
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-type_alias">
|
||
|
||
|
||
|
||
<h3 id="meanas.fdmath.types.fdslice" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-type_alias"></code> <span class="doc doc-object-name doc-type_alias-name">fdslice</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.types.fdslice" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nc">fdslice</span> <span class="o">=</span> <span class="n">fdslice_t</span> <span class="o">|</span> <span class="n">NDArray</span><span class="p">[</span><span class="n">floating</span><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Vector field slice with shape (3, X, Y) (e.g. <code>[E_x, E_y, E_z]</code> at a single Z position)</p>
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-type_alias">
|
||
|
||
|
||
|
||
<h3 id="meanas.fdmath.types.vfdslice" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-type_alias"></code> <span class="doc doc-object-name doc-type_alias-name">vfdslice</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.types.vfdslice" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nc">vfdslice</span> <span class="o">=</span> <span class="n">vfdslice_t</span> <span class="o">|</span> <span class="n">NDArray</span><span class="p">[</span><span class="n">floating</span><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Linearized vector field slice (single vector of length 3<em>X</em>Y)</p>
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-type_alias">
|
||
|
||
|
||
|
||
<h3 id="meanas.fdmath.types.cfdslice" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-type_alias"></code> <span class="doc doc-object-name doc-type_alias-name">cfdslice</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.types.cfdslice" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nc">cfdslice</span> <span class="o">=</span> <span class="n">cfdslice_t</span> <span class="o">|</span> <span class="n">NDArray</span><span class="p">[</span><span class="n">complexfloating</span><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Complex vector field slice with shape (3, X, Y) (e.g. <code>[E_x, E_y, E_z]</code> at a single Z position)</p>
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-type_alias">
|
||
|
||
|
||
|
||
<h3 id="meanas.fdmath.types.vcfdslice" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-type_alias"></code> <span class="doc doc-object-name doc-type_alias-name">vcfdslice</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.types.vcfdslice" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nc">vcfdslice</span> <span class="o">=</span> <span class="n">vcfdslice_t</span> <span class="o">|</span> <span class="n">NDArray</span><span class="p">[</span><span class="n">complexfloating</span><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Linearized complex vector field slice (single vector of length 3<em>X</em>Y)</p>
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-type_alias">
|
||
|
||
|
||
|
||
<h3 id="meanas.fdmath.types.fdfield2" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-type_alias"></code> <span class="doc doc-object-name doc-type_alias-name">fdfield2</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.types.fdfield2" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nc">fdfield2</span> <span class="o">=</span> <span class="n">fdfield2_t</span> <span class="o">|</span> <span class="n">NDArray</span><span class="p">[</span><span class="n">floating</span><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>2D Vector field with shape (2, X, Y) (e.g. <code>[E_x, E_y]</code>)</p>
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-type_alias">
|
||
|
||
|
||
|
||
<h3 id="meanas.fdmath.types.vfdfield2" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-type_alias"></code> <span class="doc doc-object-name doc-type_alias-name">vfdfield2</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.types.vfdfield2" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nc">vfdfield2</span> <span class="o">=</span> <span class="n">vfdfield2_t</span> <span class="o">|</span> <span class="n">NDArray</span><span class="p">[</span><span class="n">floating</span><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>2D Linearized vector field (single vector of length 2<em>X</em>Y)</p>
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-type_alias">
|
||
|
||
|
||
|
||
<h3 id="meanas.fdmath.types.cfdfield2" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-type_alias"></code> <span class="doc doc-object-name doc-type_alias-name">cfdfield2</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.types.cfdfield2" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nc">cfdfield2</span> <span class="o">=</span> <span class="n">cfdfield2_t</span> <span class="o">|</span> <span class="n">NDArray</span><span class="p">[</span><span class="n">complexfloating</span><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>2D Complex vector field with shape (2, X, Y) (e.g. <code>[E_x, E_y]</code>)</p>
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-type_alias">
|
||
|
||
|
||
|
||
<h3 id="meanas.fdmath.types.vcfdfield2" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-type_alias"></code> <span class="doc doc-object-name doc-type_alias-name">vcfdfield2</span>
|
||
|
||
|
||
<a href="#meanas.fdmath.types.vcfdfield2" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nc">vcfdfield2</span> <span class="o">=</span> <span class="n">vcfdfield2_t</span> <span class="o">|</span> <span class="n">NDArray</span><span class="p">[</span><span class="n">complexfloating</span><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>2D Linearized complex vector field (single vector of length 2<em>X</em>Y)</p>
|
||
</div>
|
||
|
||
</div>
|
||
|
||
|
||
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
</article>
|
||
</div>
|
||
|
||
|
||
<script>var target=document.getElementById(location.hash.slice(1));target&&target.name&&(target.checked=target.name.startsWith("__tabbed_"))</script>
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