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Copy file name to clipboardExpand all lines: examples/ceed/index.rst
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@@ -50,37 +50,17 @@ Similarly to :ref:`Ex1-Volume`, it computes:
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I = \int_{\partial\Omega} \mathbf{1} \, dS .
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:label: eq-ex2-surface
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but this time by solving a Laplace's equation for a harmonic function
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:math:`u(\mathbf{x})`. We write the Laplace's equation
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but this time by applying the divergence theorem using a Laplacian.
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In particular, we select :math:`u(\bm x) = x_0 + x_1 + x_2`, for which :math:`\nabla u = [1, 1, 1]^T`, and thus :math:`\nabla u \cdot\hat{\bm n} = 1`.
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.. math::
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\nabla\cdot\nabla u = 0, \textrm{ for } \mathbf{x} \in\Omega .
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:label: eq-laplace
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We can rewrite this via the bilinear form :math:`a(\cdot, \cdot)` and the linear form
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:math:`\langle\cdot, \cdot\rangle` as
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Given Laplace's equation,
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.. math::
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a(u,v) = \langle, v,f \rangle
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-\nabla\cdot\nabla u = 0, \textrm{ for } \mathbf{x} \in\Omega
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where :math:`v` is the test function, and for which :math:`\langle, v,f \rangle=0` in
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this case. We
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obtain
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multiply by a test function :math:`v` and integrate by parts to obtain
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.. math::
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a(u,v) = \int_\Omega v \nabla\cdot\nabla u \, dV = \int_{\partial\Omega} v \nabla u \cdot\mathbf{n}\, dS - \int_\Omega\nabla v \cdot\nabla u \, dV = 0 ,
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where we have used integration by parts.
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\int_\Omega\nabla v \cdot\nabla u \, dV - \int_{\partial\Omega} v \nabla u \cdot\hat{\bm n}\, dS = 0 .
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:math:`a(u,v) = 0` because we have chosen :math:`u(\mathbf{x})` to be harmonic, so we
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can write
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.. math::
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\int_{\partial\Omega} v \nabla u \cdot\mathbf{n}\, dS = \int_\Omega\nabla v \cdot\nabla u \, dV
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:label: eq-laplace-by-parts
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and use the :ref:`CeedOperator` for Laplace's operator to compute the right-hand side of
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equation :math:numref:`eq-laplace-by-parts`. This way, the left-hand side of equation
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:math:numref:`eq-laplace-by-parts` (which gives :math:numref:`eq-ex2-surface` because
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we have chosen :math:`u(\mathbf{x}) = (x + y + z)` such that
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:math:`\nabla u \cdot\mathbf{n} = 1`) is readily found.
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Since we have chosen :math:`u` such that the boundary integrand is :math:`v 1`, we may evaluate the surface integral by applying the volumetric Laplacian and summing the result.
Copy file name to clipboardExpand all lines: examples/solids/index.rst
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@@ -458,7 +458,7 @@ That is, given the linearization point :math:`\bm F` and solution increment :mat
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#. conclude by :math:numref:`eq-diff-P`, where :math:`\bm S` is either stored or recomputed from its definition exactly as in the nonlinear residual evaluation.
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.. note::
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The decision of whether to recompute or store functions of the current state :math:`\bm F` depends on a roofline analysis :cite:`williams2009roofline,Brown:2010` of the computation and the cost of the constitutive model.
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The decision of whether to recompute or store functions of the current state :math:`\bm F` depends on a roofline analysis :cite:`williams2009roofline,brown2010` of the computation and the cost of the constitutive model.
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For low-order elements where flops tend to be in surplus relative to memory bandwidth, recomputation is likely to be preferable, where as the opposite may be true for high-order elements.
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Similarly, analysis with a simple constitutive model may see better performance while storing little or nothing while an expensive model such as Arruda-Boyce :cite:`arruda1993largestretch`, which contains many special functions, may be faster when using more storage to avoid recomputation.
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In the case where complete linearization is preferred, note the symmetry :math:`\mathsf C_{IJKL} = \mathsf C_{KLIJ}` evident in :math:numref:`eq-neo-hookean-incremental-stress-index`, thus :math:`\mathsf C` can be stored as a symmetric :math:`6\times6` matrix, which has 21 unique entries.
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