From: Jean-Paul Pelteret Date: Thu, 6 May 2021 16:07:37 +0000 (+0200) Subject: Fix some typos in doc for step-71 X-Git-Tag: v9.3.0-rc1~121^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=304e6efdd8cc4eef1a08f70edc49bc25a0080393;p=dealii.git Fix some typos in doc for step-71 --- diff --git a/examples/step-71/doc/intro.dox b/examples/step-71/doc/intro.dox index 1ad1cfa8b9..b27d7d77d5 100644 --- a/examples/step-71/doc/intro.dox +++ b/examples/step-71/doc/intro.dox @@ -331,7 +331,7 @@ multi-variate function returns the derivative of that function with respect to one of those variables while holding the others constant: @f[ \frac{\partial f\left(x, y\right)}{\partial x} - = \frac{d f\left(x, y\right)}{d x} \vert_{y} . + = \frac{d f\left(x, y\right)}{d x} \Big\vert_{y} . @f] More specific to what's encoded in the dissipation inequality (with the very general free energy density function $\psi_{0}$ with its parameterization yet to be formalized), @@ -341,10 +341,10 @@ derivative would imply judicious use of the chain rule. This can be better understood by comparing the following two statements: @f{align*} \frac{\partial f\left(x, y\left(x\right)\right)}{\partial x} - &= \frac{d f\left(x, y\left(x\right)\right)}{d x} \vert_{y} \\ + &= \frac{d f\left(x, y\left(x\right)\right)}{d x} \Big\vert_{y} \\ \frac{d f\left(x, y\left(x\right)\right)}{d x} - &= \frac{d f\left(x, y\left(x\right)\right)}{d x} \vert_{y} - + \frac{d f\left(x, y\left(x\right)\right)}{d y} \vert_{x} \frac{d y\left(x\right)}{x} . + &= \frac{d f\left(x, y\left(x\right)\right)}{d x} \Big\vert_{y} + + \frac{d f\left(x, y\left(x\right)\right)}{d y} \Big\vert_{x} \frac{d y\left(x\right)}{x} . @f} Returning to the thermodynamics of the problem, we next exploit the arbitrariness @@ -406,18 +406,18 @@ these quantities may be computed by For the case of rate-dependent materials, this expands to @f{align*}{ \mathcal{H}^{\text{tot}} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right) - &= 2 \frac{d^{2} \psi_{0} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right)}{d \mathbf{C} \otimes \partial \mathbf{C}} , \\ + &= 4 \frac{d^{2} \psi_{0} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right)}{\partial \mathbf{C} \otimes d \mathbf{C}} , \\ \mathbb{D} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right) - &= -\frac{d^{2} \psi_{0} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right)}{d \boldsymbol{\mathbb{H}} \otimes \partial \boldsymbol{\mathbb{H}}} , \\ + &= -\frac{d^{2} \psi_{0} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right)}{\partial \boldsymbol{\mathbb{H}} \otimes d \boldsymbol{\mathbb{H}}} , \\ \mathfrak{P}^{\text{tot}} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right) - &= - 2 \frac{d^{2} \psi_{0} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right)}{d \boldsymbol{\mathbb{H}} \otimes \partial \mathbf{C}} , \\ + &= - 2 \frac{d^{2} \psi_{0} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right)}{\partial \boldsymbol{\mathbb{H}} \otimes d \mathbf{C}} , \\ \left[ \mathfrak{P}^{\text{tot}} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right) \right]^{T} - &= - 2 \frac{d^{2} \psi_{0} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right)}{d \mathbf{C} \otimes \partial \boldsymbol{\mathbb{H}}} , + &= - 2 \frac{d^{2} \psi_{0} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right)}{\partial \mathbf{C} \otimes d \boldsymbol{\mathbb{H}}} , @f} while for rate-independent materials the linearizations are @f{align*}{ \mathcal{H}^{\text{tot}} \left( \mathbf{C}, \boldsymbol{\mathbb{H}} \right) - &= 2 \frac{d^{2} \psi_{0} \left( \mathbf{C}, \boldsymbol{\mathbb{H}} \right)}{d \mathbf{C} \otimes d \mathbf{C}} , \\ + &= 4 \frac{d^{2} \psi_{0} \left( \mathbf{C}, \boldsymbol{\mathbb{H}} \right)}{d \mathbf{C} \otimes d \mathbf{C}} , \\ \mathbb{D} \left( \mathbf{C}, \boldsymbol{\mathbb{H}} \right) &= -\frac{d^{2} \psi_{0} \left( \mathbf{C}, \boldsymbol{\mathbb{H}} \right)}{d \boldsymbol{\mathbb{H}} \otimes d \boldsymbol{\mathbb{H}}} , \\ \mathfrak{P}^{\text{tot}} \left( \mathbf{C}, \boldsymbol{\mathbb{H}} \right) diff --git a/examples/step-71/step-71.cc b/examples/step-71/step-71.cc index bc9e23043f..95d668d757 100644 --- a/examples/step-71/step-71.cc +++ b/examples/step-71/step-71.cc @@ -1691,7 +1691,7 @@ namespace Step71 // @cite Pelteret2018a, equation 29). Thereafter we can compute the // dissipative component of the energy density function; its expression // is stated in @cite Pelteret2018a (equation 28), which is a - // straight-forward extension of an energy density function formulated by + // straight-forward extension of an energy density function formulated in // @cite Linder2011a (equation 46). const Differentiation::SD::Expression f_mu_v_sd = 1.0 + @@ -1718,11 +1718,11 @@ namespace Step71 // This means that deriving any function $f = f(\mathbf{C}, \mathbf{Q})$ // with respect to $\mathbf{C}$ will return partial derivatives // $\frac{\partial f(\mathbf{C}, \mathbf{Q})}{\partial \mathbf{C}} - // \Big\vert_{\mathbf{C}_{v}}$ as opposed to the total derivative + // \Big\vert_{\mathbf{Q}}$ as opposed to the total derivative // $\frac{d f(\mathbf{C}, \mathbf{Q}(\mathbf{C}))}{d \mathbf{C}} = // \frac{\partial f(\mathbf{C}, \mathbf{Q}(\mathbf{C}))}{\partial - // \mathbf{C}} \Big\vert_{\mathbf{C}_{v}} + \frac{\partial f(\mathbf{C}, - // \mathbf{Q}(\mathbf{C}))}{\partial \mathbf{C}_{v}} + // \mathbf{C}} \Big\vert_{\mathbf{Q}} + \frac{\partial f(\mathbf{C}, + // \mathbf{Q}(\mathbf{C}))}{\partial \mathbf{Q}} // \Big\vert_{\mathbf{C}} : \frac{d \mathbf{Q}(\mathbf{C}))}{d // \mathbf{C}}$. //