* in the current (spatial) configuration through the nonlinear map
* @f[
* \mathbf{x}
- * := \boldsymbol{\varphi} \left( \mathbf{X} \right)
+ * \dealcoloneq \boldsymbol{\varphi} \left( \mathbf{X} \right)
* = \mathbf{X} + \mathbf{u}(\mathbf{X}) \, ,
* @f]
* where the $\mathbf{u}(\mathbf{X})$ represents the displacement vector.
* From this we can compute the deformation gradient tensor as
* @f[
- * \mathbf{F} := \mathbf{I} + \nabla_{0}\mathbf{u} \, ,
+ * \mathbf{F} \dealcoloneq \mathbf{I} + \nabla_{0}\mathbf{u} \, ,
* @f]
* wherein the differential operator $\nabla_{0}$ is defined as
* $\frac{\partial}{\partial \mathbf{X}}$ and $\mathbf{I}$ is the identity
*
* We then define the symmetric and skew-symmetric fourth-order unit tensors by
* @f[
- * \mathcal{S} := \dfrac{1}{2}[\mathcal{I} + \overline{\mathcal{I}}]
- * \qquad \text{and} \qquad
- * \mathcal{W} := \dfrac{1}{2}[\mathcal{I} - \overline{\mathcal{I}}] \, ,
+ * \mathcal{S} \dealcoloneq
+ * \dfrac{1}{2}[\mathcal{I} + \overline{\mathcal{I}}]
+ * \qquad \text{and} \qquad
+ * \mathcal{W} \dealcoloneq
+ * \dfrac{1}{2}[\mathcal{I} - \overline{\mathcal{I}}] \, ,
* @f]
* such that
* @f[
}
s/\$projectname// unless (m/<title>/);
+
+# Finally, define some extra commands for MathJax in every file. These are in an
+# undisplayed div so that the \newcommand text does not pop up and then
+# disappear while MathJax works.
+if (eof)
+{
+ CORE::say '<!--Extra macros for MathJax:-->';
+ CORE::say '<div style="display:none">';
+ CORE::say '\(\newcommand{\dealcoloneq}{\mathrel{\vcenter{:}}=}\)';
+ CORE::say '</div>';
+}
vector field, $u$ the (scalar) solution
function, $g$ a boundary value function,
@f[
-\Gamma_-:=\{{\bf x}\in\Gamma, {\mathbf \beta}({\bf x})\cdot{\bf n}({\bf x})<0\}
+\Gamma_- \dealcoloneq \{{\bf x}\in\Gamma, {\mathbf \beta}({\bf x})\cdot{\bf n}({\bf x})<0\}
@f]
the inflow part of the boundary of the domain and ${\bf n}$ denotes
the unit outward normal to the boundary $\Gamma$. This equation is the
In particular, we solve the advection equation on
$\Omega=[0,1]^2$ with ${\mathbf \beta}=\frac{1}{|x|}(-x_2, x_1)$
representing a circular counterclockwise flow field, and $g=1$
-on ${\bf x}\in\Gamma_-^1:=[0,0.5]\times\{0\}$ and $g=0$ on ${\bf x}\in
+on ${\bf x}\in\Gamma_-^1 \dealcoloneq [0,0.5]\times\{0\}$ and $g=0$ on ${\bf x}\in
\Gamma_-\setminus \Gamma_-^1$.
We apply the well-known upwind discontinuous Galerkin method. To this
Using that for the normal vectors on adjacent cells we have $n'=-n$, we define the jump of the
normal derivative by
@f[
- [\partial_n u_h] := \partial_n u_h|_K + \partial_{n'} u_h|_{K'}
+ [\partial_n u_h] \dealcoloneq \partial_n u_h|_K + \partial_{n'} u_h|_{K'}
=
\partial_n u_h|_K - \partial_n u_h|_{K'},
@f]
@f}
with
@f[
- F(u):= -\nabla \cdot \left( \frac{1}{\sqrt{1+|\nabla u|^{2}}}\nabla u \right)
+ F(u) \dealcoloneq -\nabla \cdot \left( \frac{1}{\sqrt{1+|\nabla u|^{2}}}\nabla u \right)
@f]
and $F'(u,\delta u)$ the derivative of F in direction of $\delta u$:
@f[
\delta u^{n}=\sum_{j=0}^{N-1} \delta U_{j} \varphi_{j}.
@f]
-Using the basis functions as test functions and defining $a_{n}:=\frac{1}
+Using the basis functions as test functions and defining $a_{n} \dealcoloneq \frac{1}
{\sqrt{1+|\nabla u^{n}|^{2}}}$, we can rewrite the weak formulation:
@f[
where the entries of the matrix $A^{n}$ are given by:
@f[
- A^{n}_{ij}:= \left( \nabla \varphi_{i} , a_{n} \nabla \varphi_{j} \right) -
+ A^{n}_{ij} \dealcoloneq \left( \nabla \varphi_{i} , a_{n} \nabla \varphi_{j} \right) -
\left(\nabla u^{n}\cdot \nabla \varphi_{i} , a_{n}^{3} \nabla u^{n} \cdot \nabla
\varphi_{j} \right),
@f]
and the right hand side $b^{n}$ is given by:
@f[
- b^{n}_{i}:=-\left( \nabla \varphi_{i} , a_{n} \nabla u^{n}\right).
+ b^{n}_{i} \dealcoloneq -\left( \nabla \varphi_{i} , a_{n} \nabla u^{n}\right).
@f]
The testcase we solve is chosen as follows: We seek to find the solution of
minimal surface over the unit disk $\Omega=\{\mathbf x: \|\mathbf
x\|<1\}\subset {\mathbb R}^2$ where the surface attains the values
-$u(x,y)|{\partial\Omega} = g(x,y):=\sin(2 \pi (x+y))$ along the boundary.
+$u(x,y)|{\partial\Omega} = g(x,y) \dealcoloneq \sin(2 \pi (x+y))$ along the
+boundary.
+
\left(F(S^n_-) (\mathbf n \cdot \mathbf{u}^{n+1}_-), \sigma\right)_{\partial K_-},
@f}
-where $\partial K_{-}:= \{x\in \partial K, \mathbf{u}(x) \cdot \mathbf{n}<0\}$
-denotes the inflow boundary and $\partial K_{+}:= \{\partial K \setminus
+where $\partial K_{-} \dealcoloneq \{x\in \partial K, \mathbf{u}(x) \cdot \mathbf{n}<0\}$
+denotes the inflow boundary and $\partial K_{+} \dealcoloneq \{\partial K \setminus
\partial K_{-}\}$ is the outflow part of the boundary.
The quantities $S_+,\mathbf{u}_+$ then correspond to the values of these
variables on the present cell, whereas $S_-,\mathbf{u}_-$ (needed on the
$\theta=\frac{1}{2}$, which gives the second-order accurate
Crank-Nicolson scheme. Henceforth, a superscript $n$ denotes the
values of the variables at the $n^{\mathrm{th}}$ time step, i.e. at
-$t=t_n:= n k$, where $k$ is the (fixed) time step size. Thus,
+$t=t_n \dealcoloneq n k$, where $k$ is the (fixed) time step size. Thus,
the split formulation of the time-discretized sine-Gordon equation becomes
\f{eqnarray*}
\frac{u^n - u^{n-1}}{k} - \left[\theta v^n + (1-\theta) v^{n-1}\right] &=& 0,\\
\Gamma$, where the "boundary" at infinity is defined as
\f[
-\Gamma_\infty := \lim_{r\to\infty} \partial B_r(0).
+\Gamma_\infty \dealcoloneq \lim_{r\to\infty} \partial B_r(0).
\f]
In our program the normals are defined as <i>outer</i> to the domain
$\alpha(\mathbf{x})$ by which the point $\mathbf{x}$ sees the domain
$\Omega$ can be defined using the double layer potential itself:
\f[
-\alpha(\mathbf{x}) := 1 -
+\alpha(\mathbf{x}) \dealcoloneq 1 -
\frac{1}{2(n-1)\pi}\int_{\partial \Omega} \frac{ (\mathbf{y}-\mathbf{x})\cdot\mathbf{n}_y }
{ |\mathbf{y}-\mathbf{x}|^{n} }\phi(\mathbf{y})\,ds_y = 1+
\int_{\partial \Omega} \frac{ \partial G(\mathbf{y}-\mathbf{x}) }{\partial \mathbf{n}_y} \, ds_y.
We define the finite dimensional space $V_h$ as
\f[
\label{eq:definition-Vh}
- V_h := \{ v \in C^0(\Gamma) \text{ s.t. } v|_{K_i} \in \mathcal{Q}^1(K_i),
+ V_h \dealcoloneq \{ v \in C^0(\Gamma) \text{ s.t. } v|_{K_i} \in \mathcal{Q}^1(K_i),
\forall i\},
\f]
with basis functions $\psi_i(\mathbf{x})$ for which we will use the usual FE_Q
$\phi_i$, that is:
\f[
\label{eq:definition-of-element}
- \phi_h(\mathbf{x}) := \phi_i \psi_i(\mathbf{x}), \qquad
- \boldsymbol{\phi} := \{ \phi_i \},
+ \phi_h(\mathbf{x}) \dealcoloneq \phi_i \psi_i(\mathbf{x}), \qquad
+ \boldsymbol{\phi} \dealcoloneq \{ \phi_i \},
\f]
where summation is implied over repeated indexes. Note that we could use
discontinuous elements here — in fact, there is no real reason to use
simple domain, i.e., we assume that each element $K_i$ of
$\mathcal{T}_h$ can be expressed as a linear (in two dimensions) or
bi-linear (in three dimensions) transformation of the reference
-boundary element $\hat K := [0,1]^{n-1}$, and we perform the integrations after a
+boundary element $\hat K \dealcoloneq [0,1]^{n-1}$, and we perform the integrations after a
change of variables from the real element $K_i$ to the reference
element $\hat K$.
i.e. each point $\hat{\mathbf x}\in\hat S$ induces a point ${\mathbf
x}_S(\hat{\mathbf x}) \in S$. Then let
@f[
-G_S:= (D \mathbf{x}_S)^T \ D \mathbf{x}_S
+G_S\dealcoloneq (D \mathbf{x}_S)^T \ D \mathbf{x}_S
@f]
denotes the corresponding first fundamental form, where $D
\mathbf{x}_S=\left(\frac{\partial x_{S,i}(\hat{\mathbf x})}{\partial \hat x_j}\right)_{ij}$ is the
We are now in position to define the tangential gradient of a function $v : S \rightarrow \mathbb
R$ by
@f[
-(\nabla_S v)\circ \mathbf x_S := D \mathbf x_S \ G_S^{-1} \ \nabla (v \circ \mathbf x_S).
+(\nabla_S v)\circ \mathbf x_S \dealcoloneq D \mathbf x_S \ G_S^{-1} \ \nabla (v \circ \mathbf x_S).
@f]
The surface Laplacian (also called the Laplace-Beltrami operator) is then
-defined as $\Delta_S:= \nabla_S \cdot \nabla_S$.
-Note that an alternate way to compute the surface gradient on smooth surfaces $\Gamma$ is
+defined as $\Delta_S \dealcoloneq \nabla_S \cdot \nabla_S$.
+Note that an alternate way to compute the surface gradient on smooth surfaces $\Gamma$ is
@f[
\nabla_S v = \nabla \tilde v - \mathbf n (\mathbf n \cdot \nabla \tilde v),
@f]
{\mathbb T}} \int_K f \ v \qquad \forall v \in H^1_0(\Gamma).
@f]
Moreover, each integral in the above expression is computed in the reference
-element $\hat K:= [0,1]^2$
+element $\hat K \dealcoloneq [0,1]^2$
so that
@f{align*}
\int_{K} \nabla_{K} u \cdot \nabla_{K} v
An obvious way to obtain the variational formulation of the obstacle problem is to consider the total potential energy:
@f{equation*}
- E(u):=\dfrac{1}{2}\int\limits_{\Omega} \nabla u \cdot \nabla u - \int\limits_{\Omega} fu.
+ E(u) \dealcoloneq \dfrac{1}{2}\int\limits_{\Omega} \nabla u \cdot \nabla u - \int\limits_{\Omega} fu.
@f}
We have to find a solution $u\in G$ of the following minimization problem:
@f{equation*}
@f}
with the convex set of admissible displacements:
@f{equation*}
- G:=\lbrace v\in V: v\geq g \text{ a.e. in } \Omega\rbrace,\quad V:=H^1_0(\Omega).
+ G \dealcoloneq \lbrace v\in V: v\geq g \text{ a.e. in } \Omega\rbrace,\quad V\dealcoloneq H^1_0(\Omega).
@f}
This set takes care of the third and fifth conditions above (the boundary
values and the complementarity condition).
Consider now the minimizer $u\in G$ of $E$ and any other function $v\in
G$. Then the function
@f{equation*}
- F(\varepsilon) := E(u+\varepsilon(v-u)),\quad\varepsilon\in\left[0,1\right],
+ F(\varepsilon) \dealcoloneq E(u+\varepsilon(v-u)),\quad\varepsilon\in\left[0,1\right],
@f}
takes its minimum at $\varepsilon = 0$ (because $u$ is a minimizer of the
energy functional $E(\cdot)$), so that $F'(0)\geq 0$ for any choice
The variational inequality above is awkward to work with. We would therefore
like to reformulate it as an equivalent saddle point problem. We introduce a
Lagrange multiplier $\lambda$ and the convex cone $K\subset V'$, $V'$
-dual space of $V$, $K:=\{\mu\in V': \langle\mu,v\rangle\geq 0,\quad \forall
+dual space of $V$, $K \dealcoloneq \{\mu\in V': \langle\mu,v\rangle\geq 0,\quad \forall
v\in V, v \le 0 \}$ of
Lagrange multipliers, where $\langle\cdot,\cdot\rangle$ denotes the duality
pairing between $V'$ and $V$. Intuitively, $K$ is the cone of all "non-positive
@f}
<i>with</i>
@f{align*}
- a(u,v) &:= \left(\nabla u, \nabla v\right),\quad &&u,v\in V\\
- b(u,\mu) &:= \langle u,\mu\rangle,\quad &&u\in V,\quad\mu\in V'.
+ a(u,v) &\dealcoloneq \left(\nabla u, \nabla v\right),\quad &&u,v\in V\\
+ b(u,\mu) &\dealcoloneq \langle u,\mu\rangle,\quad &&u\in V,\quad\mu\in V'.
@f}
In other words, we can consider $\lambda$ as the negative of the additional, positive force that the
obstacle exerts on the membrane. The inequality in the second line of the
Now we define for each degree of freedom $i$ the function
@f{equation*}
- C([BU]_i,\Lambda_i):=-\Lambda_i + \min\lbrace 0, \Lambda_i + c([BU]_i - G_i) \rbrace,
+ C([BU]_i,\Lambda_i) \dealcoloneq -\Lambda_i + \min\lbrace 0, \Lambda_i + c([BU]_i - G_i) \rbrace,
@f}
with some $c>0$. (In this program we choose $c = 100$. It is a kind of a
penalty parameter which depends on the problem itself and needs to be chosen
3. Define the new active and inactive sets by
@f{equation*}
\begin{split}
- \mathcal{A}_{k+1}:=\lbrace i\in\mathcal{S}:\Lambda^k_i + c([BU^k]_i - G_i)< 0\rbrace,\\
- \mathcal{F}_{k+1}:=\lbrace i\in\mathcal{S}:\Lambda^k_i + c([BU^k]_i - G_i)\geq 0\rbrace.
+ \mathcal{A}_{k+1} \dealcoloneq \lbrace i\in\mathcal{S}:\Lambda^k_i + c([BU^k]_i - G_i)< 0\rbrace,\\
+ \mathcal{F}_{k+1} \dealcoloneq \lbrace i\in\mathcal{S}:\Lambda^k_i + c([BU^k]_i - G_i)\geq 0\rbrace.
\end{split}
@f}
4. If $\mathcal{A}_{k+1}=\mathcal{A}_k$ (and then, obviously, also
${\mathcal{A}_k}$ or the inactive set ${\mathcal{F}_k}$.
Rather than solving for updates $\delta U, \delta \Lambda$, we can also solve
-for the variables we are interested in right away by setting $\delta U^k :=
-U^{k+1} - U^k$ and $\delta \Lambda^k := \Lambda^{k+1} - \Lambda^k$ and
+for the variables we are interested in right away by setting $\delta U^k \dealcoloneq
+U^{k+1} - U^k$ and $\delta \Lambda^k \dealcoloneq \Lambda^{k+1} - \Lambda^k$ and
bringing all known terms to the right hand side. This yields
@f{equation*}
\begin{pmatrix}
@f}
where the projector $P_\Pi$ is defined as
@f{align*}
- P_{\Pi}(\tau):=\begin{cases}
+ P_{\Pi}(\tau) \dealcoloneq \begin{cases}
\tau, & \text{if }\vert\tau^D\vert \leq \sigma_0,\\
\left[
\dfrac{\gamma^{\text{iso}}}{2\mu + \gamma^{\text{iso}}} +
\mathcal{F}_i = \emptyset$ and set $i = 1$. Here, $\mathcal{S}$ is the set of
all degrees of freedom located at the surface of the domain where contact
may happen.
- The start value $\hat U^0 :=
+ The start value $\hat U^0 \dealcoloneq
P_{\mathcal{A}_k}(0)$ fulfills our obstacle condition, i.e., we project an
initial zero displacement onto the set of feasible displacements.
- <li> Assemble the Newton matrix $A_{pq} := a'(
+ <li> Assemble the Newton matrix $A_{pq} \dealcoloneq a'(
U^{i-1};\varphi_p,\varphi_q)$ and the right-hand-side $F(\hat U^{i-1})$.
These correspond to the linearized Newton step, ignoring for the moment
the contact inequality.
<li> Damp the Newton iteration for $i>2$ by applying a line search and
calculating a linear combination of $U^{i-1}$ and $\tilde U^i$. This
requires finding an
- $\alpha^i_l:=2^{-l},(l=0,\ldots,10)$ so that
- @f{gather*}U^i := \alpha^i_l\bar U^i +
+ $\alpha^i_l \dealcoloneq 2^{-l},(l=0,\ldots,10)$ so that
+ @f{gather*}U^i \dealcoloneq \alpha^i_l\bar U^i +
(1-\alpha^i_l)U^{i-1}@f}
satisfies
@f{gather*}
and (ii) elements that correspond to hanging nodes, which we eliminate in the usual manner.
<li> Define the new active and inactive sets by
- @f{gather*}\mathcal{A}_{i+1}:=\lbrace p\in\mathcal{S}:\Lambda^i_p +
+ @f{gather*}\mathcal{A}_{i+1} \dealcoloneq \lbrace p\in\mathcal{S}:\Lambda^i_p +
c\left(\left[B^TU^i\right]_p - G_p\right) > 0\rbrace,@f}
- @f{gather*}\mathcal{F}_{i+1}:=\lbrace p\in\mathcal{S}:\Lambda^i_p +
+ @f{gather*}\mathcal{F}_{i+1} \dealcoloneq \lbrace p\in\mathcal{S}:\Lambda^i_p +
c\left(\left[B^TU^i\right]_p - G_p\right) \leq 0\rbrace.@f}
<li>Project $U^i$ so that it satisfies the contact inequality,
- @f{gather*}\hat U^i := P_{\mathcal{A}_{i+1}}(U^i).@f}
+ @f{gather*}\hat U^i \dealcoloneq P_{\mathcal{A}_{i+1}}(U^i).@f}
Here,
$P_{\mathcal{A}}(U)$ is the projection of the active
components in $\mathcal{A}$ to the gap
- @f{gather*}P_{\mathcal{A}}(U)_p:=\begin{cases}
+ @f{gather*}P_{\mathcal{A}}(U)_p \dealcoloneq \begin{cases}
U_p, & \textrm{if}\quad p\notin\mathcal{A}\\
g_{h,p}, & \textrm{if}\quad
p\in\mathcal{A},
Note $\mathcal{I} \neq \overline{\mathcal{I}}^T$.
Furthermore, we define the symmetric and skew-symmetric fourth-order unit tensors by
@f[
- \mathcal{S} := \dfrac{1}{2}[\mathcal{I} + \overline{\mathcal{I}}]
+ \mathcal{S} \dealcoloneq \dfrac{1}{2}[\mathcal{I} + \overline{\mathcal{I}}]
\qquad \text{and} \qquad
- \mathcal{W} := \dfrac{1}{2}[\mathcal{I} - \overline{\mathcal{I}}] \, ,
+ \mathcal{W} \dealcoloneq \dfrac{1}{2}[\mathcal{I} - \overline{\mathcal{I}}] \, ,
@f]
such that
@f[
The deformation gradient $\mathbf{F}$ is defined as the material gradient of the motion:
@f[
\mathbf{F}(\mathbf{X},t)
- := \dfrac{\partial \boldsymbol{\varphi}(\mathbf{X},t)}{\partial \mathbf{X}}
+ \dealcoloneq \dfrac{\partial \boldsymbol{\varphi}(\mathbf{X},t)}{\partial \mathbf{X}}
= \textrm{Grad}\ \mathbf{x}(\mathbf{X},t)
= \mathbf{I} + \textrm{Grad}\ \mathbf{U} \, .
@f]
The determinant of the of the deformation gradient
-$J(\mathbf{X},t):= \textrm{det}\ \mathbf{F}(\mathbf{X},t) > 0$
+$J(\mathbf{X},t) \dealcoloneq \textrm{det}\ \mathbf{F}(\mathbf{X},t) > 0$
maps corresponding volume elements in the reference and current configurations, denoted
$\textrm{d}V$ and $\textrm{d}v$,
respectively, as
\textrm{d}v = J(\mathbf{X},t)\; \textrm{d}V \, .
@f]
-Two important measures of the deformation in terms of the spatial and material coordinates are the left and right Cauchy-Green tensors, respectively,
-and denoted $\mathbf{b} := \mathbf{F}\mathbf{F}^T$ and $\mathbf{C} := \mathbf{F}^T\mathbf{F}$.
+Two important measures of the deformation in terms of the spatial and material coordinates are the left and right Cauchy-Green tensors, respectively,
+and denoted $\mathbf{b} \dealcoloneq \mathbf{F}\mathbf{F}^T$ and $\mathbf{C} \dealcoloneq \mathbf{F}^T\mathbf{F}$.
They are both symmetric and positive definite.
The Green-Lagrange strain tensor is defined by
@f[
- \mathbf{E}:= \frac{1}{2}[\mathbf{C} - \mathbf{I} ]
+ \mathbf{E} \dealcoloneq \frac{1}{2}[\mathbf{C} - \mathbf{I} ]
= \underbrace{\frac{1}{2}[\textrm{Grad}^T \mathbf{U} + \textrm{Grad}\mathbf{U}]}_{\boldsymbol{\varepsilon}}
+ \frac{1}{2}[\textrm{Grad}^T\ \mathbf{U}][\textrm{Grad}\ \mathbf{U}] \, .
@f]
The derivative of the spatial velocity field with respect to the spatial coordinates gives the spatial velocity gradient $\mathbf{l}(\mathbf{x},t)$, that is
@f[
\mathbf{l}(\mathbf{x},t)
- := \dfrac{\partial \mathbf{v}(\mathbf{x},t)}{\partial \mathbf{x}}
+ \dealcoloneq \dfrac{\partial \mathbf{v}(\mathbf{x},t)}{\partial \mathbf{x}}
= \textrm{grad}\ \mathbf{v}(\mathbf{x},t) \, ,
@f]
where $\textrm{grad} \{\bullet \}
The push-forward and-pull back operations for second-order covariant tensors $(\bullet)^{\text{cov}}$ are respectively given by:
@f[
- \chi_{*}(\bullet)^{\text{cov}}:= \mathbf{F}^{-T} (\bullet)^{\text{cov}} \mathbf{F}^{-1}
+ \chi_{*}(\bullet)^{\text{cov}} \dealcoloneq \mathbf{F}^{-T} (\bullet)^{\text{cov}} \mathbf{F}^{-1}
\qquad \text{and} \qquad
- \chi^{-1}_{*}(\bullet)^{\text{cov}}:= \mathbf{F}^{T} (\bullet)^{\text{cov}} \mathbf{F} \, .
+ \chi^{-1}_{*}(\bullet)^{\text{cov}} \dealcoloneq \mathbf{F}^{T} (\bullet)^{\text{cov}} \mathbf{F} \, .
@f]
The push-forward and pull back operations for second-order contravariant tensors $(\bullet)^{\text{con}}$ are respectively given by:
@f[
- \chi_{*}(\bullet)^{\text{con}}:= \mathbf{F} (\bullet)^{\text{con}} \mathbf{F}^T
+ \chi_{*}(\bullet)^{\text{con}} \dealcoloneq \mathbf{F} (\bullet)^{\text{con}} \mathbf{F}^T
\qquad \text{and} \qquad
- \chi^{-1}_{*}(\bullet)^{\text{con}}:= \mathbf{F}^{-1} (\bullet)^{\text{con}} \mathbf{F}^{-T} \, .
+ \chi^{-1}_{*}(\bullet)^{\text{con}} \dealcoloneq \mathbf{F}^{-1} (\bullet)^{\text{con}} \mathbf{F}^{-T} \, .
@f]
For example $\boldsymbol{\tau} = \chi_{*}(\mathbf{S})$.
&= \underbrace{( \mathcal{I} - \dfrac{1}{3} \mathbf{I} \otimes \mathbf{I})}_{\mathbb{P}} : \overline{\boldsymbol{\tau}} \, ,
@f}
where
-$p := \dfrac{\partial \Psi_{\text{vol}}(J)}{\partial J}$ is the pressure response.
-$\mathbb{P}$ is the projection tensor which provides the deviatoric operator in the Eulerian setting.
+$p \dealcoloneq \dfrac{\partial \Psi_{\text{vol}}(J)}{\partial J}$ is the pressure response.
+$\mathbb{P}$ is the projection tensor which provides the deviatoric operator in the Eulerian setting.
The fictitious Kirchhoff stress tensor $\overline{\boldsymbol{\tau}}$ is defined by
@f[
\overline{\boldsymbol{\tau}}
- := 2 \overline{\mathbf{b}} \dfrac{\partial \Psi_{\textrm{iso}}(\overline{\mathbf{b}})}{\partial \overline{\mathbf{b}}} \, .
+ \dealcoloneq 2 \overline{\mathbf{b}} \dfrac{\partial \Psi_{\textrm{iso}}(\overline{\mathbf{b}})}{\partial \overline{\mathbf{b}}} \, .
@f]
\underbrace{\kappa [ \mathcal{G}(J) ] }_{\Psi_{\textrm{vol}}(J)}
+ \underbrace{\bigl[c_1 [ \overline{I}_1 - 3] \bigr]}_{\Psi_{\text{iso}}(\overline{\mathbf{b}})} \, ,
@f]
-where $\kappa := \lambda + 2/3 \mu$ is the bulk modulus ($\lambda$ and $\mu$ are the Lame parameters)
-and $\overline{I}_1 := \textrm{tr}\ \overline{\mathbf{b}}$.
-The function $\mathcal{G}(J)$ is required to be strictly convex and satisfy the condition $\mathcal{G}(1) = 0$,
+where $\kappa \dealcoloneq \lambda + 2/3 \mu$ is the bulk modulus ($\lambda$ and $\mu$ are the Lame parameters)
+and $\overline{I}_1 \dealcoloneq \textrm{tr}\ \overline{\mathbf{b}}$.
+The function $\mathcal{G}(J)$ is required to be strictly convex and satisfy the condition $\mathcal{G}(1) = 0$,
among others, see Holzapfel (2001) for further details.
-In this work $\mathcal{G}:=\frac{1}{4} [ J^2 - 1 - 2\textrm{ln}J ]$.
+In this work $\mathcal{G} \dealcoloneq \frac{1}{4} [ J^2 - 1 - 2\textrm{ln}J ]$.
Incompressibility imposes the isochoric constraint that $J=1$ for all motions $\boldsymbol{\varphi}$.
The Helmholtz free energy corresponding to an incompressible neo-Hookean material is given by
\Psi \equiv
\underbrace{\bigl[ c_1 [ I_1 - 3] \bigr] }_{\Psi_{\textrm{iso}}(\mathbf{b})} \, ,
@f]
-where $ I_1 := \textrm{tr}\mathbf{b} $.
+where $ I_1 \dealcoloneq \textrm{tr}\mathbf{b} $.
Thus, the incompressible response is obtained by removing the volumetric component from the compressible free energy and enforcing $J=1$.
\\
&= J[\widehat{p}\, \mathbf{I} \otimes \mathbf{I} - 2p \mathcal{I}]
\qquad \text{where} \qquad
- \widehat{p} := p + \dfrac{\textrm{d} p}{\textrm{d}J} \, ,
+ \widehat{p} \dealcoloneq p + \dfrac{\textrm{d} p}{\textrm{d}J} \, ,
\\
J \mathfrak{c}_{\text{iso}}
&= 4 \mathbf{b} \dfrac{\partial^2 \Psi_{\text{iso}}(\overline{\mathbf{b}})} {\partial \mathbf{b} \partial \mathbf{b}} \mathbf{b}
As mentioned above, we adopt a three-field formulation.
We denote the set of primary unknowns by
-$\mathbf{\Xi}:= \{ \mathbf{u}, \widetilde{p}, \widetilde{J} \}$.
+$\mathbf{\Xi} \dealcoloneq \{ \mathbf{u}, \widetilde{p}, \widetilde{J} \}$.
The independent kinematic variable $\widetilde{J}$ enters the formulation as a constraint on $J$ enforced by the Lagrange multiplier $\widetilde{p}$ (the pressure, as we shall see).
The three-field variational principle used here is given by
@f[
- \Pi(\mathbf{\Xi}) := \int_\Omega \bigl[
+ \Pi(\mathbf{\Xi}) \dealcoloneq \int_\Omega \bigl[
\Psi_{\textrm{vol}}(\widetilde{J})
+ \widetilde{p}\,[J(\mathbf{u}) - \widetilde{J}]
+ \Psi_{\textrm{iso}}(\overline{\mathbf{b}}(\mathbf{u}))
The value of a quantity at the current iteration $\textrm{i}$ is denoted
${ \{ \bullet \} }^{\textrm{n}}_{\textrm{i}} = { \{ \bullet \} }_{\textrm{i}}$.
The incremental change between iterations $\textrm{i}$ and $\textrm{i}+1$ is denoted
-$d \{ \bullet \} := \{ \bullet \}_{\textrm{i}+1} - \{ \bullet \}_{\textrm{i}}$.
+$d \{ \bullet \} \dealcoloneq \{ \bullet \}_{\textrm{i}+1} - \{ \bullet \}_{\textrm{i}}$.
Assume that the state of the system is known for some iteration $\textrm{i}$.
The linearised approximation to nonlinear governing equations to be solved using the Newton-Raphson method is:
@f]
where
@f[
- \overline{\overline{\mathbf{\mathsf{K}}}} :=
+ \overline{\overline{\mathbf{\mathsf{K}}}} \dealcoloneq
\mathbf{\mathsf{K}}_{u\widetilde{p}} \overline{\mathbf{\mathsf{K}}} \mathbf{\mathsf{K}}_{\widetilde{p}u} \, .
@f]
Note that due to the choice of $\widetilde{p}$ and $\widetilde{J}$ as discontinuous at the element level, all matrices that need to be inverted are defined at the element level.
That is
@f[
\mathbf{\mathsf{K}}_{\textrm{store}}
-:=
+\dealcoloneq
\begin{bmatrix}
\mathbf{\mathsf{K}}_{\textrm{con}} & \mathbf{\mathsf{K}}_{u\widetilde{p}} & \mathbf{0}
\\
Ideally this class would derive from a class HyperelasticMaterial which would derive from the base class Material.
The three-field nature of the formulation used here also complicates the matter.
-The Helmholtz free energy function for the three field formulation is $\Psi = \Psi_\text{vol}(\widetilde{J}) + \Psi_\text{iso}(\overline{\mathbf{b}})$.
-The isochoric part of the Kirchhoff stress ${\boldsymbol{\tau}}_{\text{iso}}(\overline{\mathbf{b}})$ is identical to that obtained using a one-field formulation for a hyperelastic material.
-However, the volumetric part of the free energy is now a function of the primary variable $\widetilde{J}$.
-Thus, for a three field formulation the constitutive response for the volumetric part of the Kirchhoff stress ${\boldsymbol{\tau}}_{\text{vol}}$ (and the tangent) is not given by the hyperelastic constitutive law as in a one-field formulation.
+The Helmholtz free energy function for the three field formulation is $\Psi = \Psi_\text{vol}(\widetilde{J}) + \Psi_\text{iso}(\overline{\mathbf{b}})$.
+The isochoric part of the Kirchhoff stress ${\boldsymbol{\tau}}_{\text{iso}}(\overline{\mathbf{b}})$ is identical to that obtained using a one-field formulation for a hyperelastic material.
+However, the volumetric part of the free energy is now a function of the primary variable $\widetilde{J}$.
+Thus, for a three field formulation the constitutive response for the volumetric part of the Kirchhoff stress ${\boldsymbol{\tau}}_{\text{vol}}$ (and the tangent) is not given by the hyperelastic constitutive law as in a one-field formulation.
One can label the term
$\boldsymbol{\tau}_{\textrm{vol}} \equiv \widetilde{p} J \mathbf{I}$
as the volumetric Kirchhoff stress, but the pressure $\widetilde{p}$ is not derived from the free energy; it is a primary field.
// \Psi_{\text{iso}}(\overline{\mathbf{b}}) = c_{1} [\overline{I}_{1} - 3] $
// where $ c_{1} = \frac{\mu}{2} $ and $\overline{I}_{1}$ is the first
// invariant of the left- or right-isochoric Cauchy-Green deformation tensors.
- // That is $\overline{I}_1 :=\textrm{tr}(\overline{\mathbf{b}})$. In this
- // example the SEF that governs the volumetric response is defined as $
- // \Psi_{\text{vol}}(\widetilde{J}) = \kappa \frac{1}{4} [ \widetilde{J}^2 - 1
- // - 2\textrm{ln}\; \widetilde{J} ]$, where $\kappa:= \lambda + 2/3 \mu$ is
- // the <a href="http://en.wikipedia.org/wiki/Bulk_modulus">bulk modulus</a>
- // and $\lambda$ is <a
+ // That is $\overline{I}_1 \dealcoloneq \textrm{tr}(\overline{\mathbf{b}})$.
+ // In this example the SEF that governs the volumetric response is defined as
+ // $ \Psi_{\text{vol}}(\widetilde{J}) = \kappa \frac{1}{4} [ \widetilde{J}^2 -
+ // 1 - 2\textrm{ln}\; \widetilde{J} ]$, where $\kappa \dealcoloneq \lambda +
+ // 2/3 \mu$ is the <a href="http://en.wikipedia.org/wiki/Bulk_modulus">bulk
+ //
modulus</a> and $\lambda$ is <a
// href="http://en.wikipedia.org/wiki/Lam%C3%A9_parameters">Lame's first
// parameter</a>.
//
time.increment();
// We then declare the incremental solution update $\varDelta
- // \mathbf{\Xi}:= \{\varDelta \mathbf{u},\varDelta \widetilde{p},
+ // \mathbf{\Xi} \dealcoloneq \{\varDelta \mathbf{u},\varDelta \widetilde{p},
// \varDelta \widetilde{J} \}$ and start the loop over the time domain.
//
// At the beginning, we reset the solution update for this time step...
return vol_current;
}
- // Calculate how well the dilatation $\widetilde{J}$ agrees with $J :=
- // \textrm{det}\ \mathbf{F}$ from the $L^2$ error $ \bigl[ \int_{\Omega_0} {[
- // J
- // - \widetilde{J}]}^{2}\textrm{d}V \bigr]^{1/2}$.
+ // Calculate how well the dilatation $\widetilde{J}$ agrees with $J
+ // \dealcoloneq \textrm{det}\ \mathbf{F}$ from the $L^2$ error $ \bigl[
+ // \int_{\Omega_0} {[ J - \widetilde{J}]}^{2}\textrm{d}V \bigr]^{1/2}$.
// We also return the ratio of the current volume of the
// domain to the reference volume. This is of interest for incompressible
// media where we want to check how well the isochoric constraint has been
// with $\mathsf{\mathbf{k}}_{\textrm{con}} = \bigl[
// \mathsf{\mathbf{k}}_{uu} +\overline{\overline{\mathsf{\mathbf{k}}}}~
// \bigr]$ where $ \overline{\overline{\mathsf{\mathbf{k}}}}
- // := \mathsf{\mathbf{k}}_{u\widetilde{p}} \overline{\mathsf{\mathbf{k}}}
- // \mathsf{\mathbf{k}}_{\widetilde{p}u}
+ // \dealcoloneq \mathsf{\mathbf{k}}_{u\widetilde{p}}
+ // \overline{\mathsf{\mathbf{k}}} \mathsf{\mathbf{k}}_{\widetilde{p}u}
// $
// and
// $
// the tangent matrix. For the following, recall that
// @f{align*}
// \mathsf{\mathbf{K}}_{\textrm{store}}
- //:=
+ //\dealcoloneq
// \begin{bmatrix}
// \mathsf{\mathbf{K}}_{\textrm{con}} &
// \mathsf{\mathbf{K}}_{u\widetilde{p}} & \mathbf{0}
// @f]
// where
// @f[
- // \overline{\overline{\mathsf{\mathbf{K}}}} :=
+ // \overline{\overline{\mathsf{\mathbf{K}}}} \dealcoloneq
// \mathsf{\mathbf{K}}_{u\widetilde{p}}
// \overline{\mathsf{\mathbf{K}}}
// \mathsf{\mathbf{K}}_{\widetilde{p}u} \, .
-\textrm{div}\; \textbf{u}&=&0,\\
\textbf{u}|_{\Gamma_1}&=&{\bf 0},
@f}
-where the boundary $\Gamma_1$ is defined as $\Gamma_1:=\{x\in \partial\Omega: \|x\|\in\{0.5,1\}\}$.
+where the boundary $\Gamma_1$ is defined as $\Gamma_1 \dealcoloneq \{x\in \partial\Omega: \|x\|\in\{0.5,1\}\}$.
For the remaining parts of the boundary we are going to use periodic boundary conditions, i.e.
@f{align*}
u_x(0,\nu)&=-u_y(\nu,0)&\nu&\in[0,1]\\
<h3> Problem specific data </h3>
For this tutorial program, we consider almost the same test case as in
-step-7. The computational domain is $\Omega := [-1,1]^d$ and the exact
+step-7. The computational domain is $\Omega \dealcoloneq [-1,1]^d$ and the exact
solution corresponds to the one in step-7, except for a scaling. We use the
following source centers $x_i$ for the exponentials
<ul>
where
@f{eqnarray*}{
-K_{ij} &:=& (\nabla v_j, \nabla v_i)_\Omega \qquad i,j=1,\dots,n \\
-C_{\alpha j} &:=& (v_j, q_\alpha)_\Gamma \qquad j=1,\dots,n, \alpha = 1,\dots, m \\\\
-G_{\alpha} &:=& (g, q_\alpha)_\Gamma \qquad \alpha = 1,\dots, m.
+K_{ij} &\dealcoloneq& (\nabla v_j, \nabla v_i)_\Omega \qquad i,j=1,\dots,n \\
+C_{\alpha j} &\dealcoloneq& (v_j, q_\alpha)_\Gamma \qquad j=1,\dots,n, \alpha = 1,\dots, m \\\\
+G_{\alpha} &\dealcoloneq& (g, q_\alpha)_\Gamma \qquad \alpha = 1,\dots, m.
@f}
While the matrix $K$ is the standard stiffness matrix for the Poisson problem on
and compute the integral on $\hat K$ using a quadrature formula:
\f[
-C_{\alpha j} := (v_j, q_\alpha)_\Gamma = \sum_{K\in \Gamma} \int_{\hat K}
+C_{\alpha j} \dealcoloneq (v_j, q_\alpha)_\Gamma = \sum_{K\in \Gamma} \int_{\hat K}
\hat q_\alpha(\hat x) (v_j \circ F_{K}) (\hat x) J_K (\hat x) \mathrm{d} \hat x =
\sum_{K\in \Gamma} \sum_{i=1}^{n_q} \big(\hat q_\alpha(\hat x_i) (v_j \circ F_{K}) (\hat x_i) J_K (\hat x_i) w_i \big)
\f]
To evaluate $(v_j \circ F_{K}) (\hat x_i)$ the following steps needs to be
taken (as shown in the picture below):
-- For a given cell $K$ in $\Gamma$ compute the real point $y_i := F_{K} (\hat
+- For a given cell $K$ in $\Gamma$ compute the real point $y_i \dealcoloneq F_{K} (\hat
x_i)$, where $x_i$ is one of the quadrature points used for the integral on $K
\subseteq \Gamma$.
* where the matrix $B$ is given by $B_{ij} = v[j][i]-v[0][i]$.
*
* The weights are scaled with the absolute value of the determinant of $B$,
- * that is $J := |\text{det}(B)|$. If $J$ is zero, an empty quadrature is
- * returned. This may happen, in two dimensions, if the three vertices are
- * aligned, or in three dimensions if the four vertices are on the same
- * plane.
+ * that is $J \dealcoloneq |\text{det}(B)|$. If $J$ is zero, an empty
+ * quadrature is returned. This may happen, in two dimensions, if the three
+ * vertices are aligned, or in three dimensions if the four vertices are on
+ * the same plane.
*
* @param[in] vertices The vertices of the simplex you wish to integrate on
* @return A quadrature object that can be used to integrate on the simplex
* \frac{\hat x}{\sin(\theta)+\cos(\theta)} cos(\theta) \\
* \frac{\hat x}{\sin(\theta)+\cos(\theta)} sin(\theta)
* \end{pmatrix}
- * \qquad \theta := \frac\pi 2 \hat y
+ * \qquad \theta \dealcoloneq \frac\pi 2 \hat y
* \f]
*
* @author Luca Heltai, 2017
* The adjugate of a tensor $\left(\bullet\right)$ is defined as
* @f[
* \textrm{adj}\left(\bullet\right)
- * := \textrm{det}\left(\bullet\right) \; \left(\bullet\right)^{-1} \; .
+ * \dealcoloneq \textrm{det}\left(\bullet\right) \; \left(\bullet\right)^{-1}
+ * \; .
* @f]
*
* @note This requires that the tensor is invertible.
* The cofactor of a tensor $\left(\bullet\right)$ is defined as
* @f[
* \textrm{cof}\left(\bullet\right)
- * := \textrm{det}\left(\bullet\right) \; \left(\bullet\right)^{-T}
+ * \dealcoloneq \textrm{det}\left(\bullet\right) \; \left(\bullet\right)^{-T}
* = \left[ \textrm{adj}\left(\bullet\right) \right]^{T} \; .
* @f]
*
* For 1d this function does not make any sense. Thus it is not
* implemented for <code>spacedim=1</code>. In 2d the curl is defined as
* @f{equation*}{
- * \operatorname{curl}(u):=\frac{du_2}{dx} -\frac{du_1}{dy},
+ * \operatorname{curl}(u) \dealcoloneq \frac{du_2}{dx} -\frac{du_1}{dy},
* @f}
* whereas in 3d it is given by
* @f{equation*}{
- * \operatorname{curl}(u):=\left( \begin{array}{c}
+ * \operatorname{curl}(u) \dealcoloneq \left( \begin{array}{c}
* \frac{du_3}{dy}-\frac{du_2}{dz}\\ \frac{du_1}{dz}-\frac{du_3}{dx}\\
* \frac{du_2}{dx}-\frac{du_1}{dy} \end{array} \right).
* @f}
// A_{ij} = A_{ji} and there is only one (if diagonal) or two non-zero
// entries in the tensorial representation. define the
// divergence as:
- // b_i := \dfrac{\partial phi_{ij}}{\partial x_j}.
+ // b_i \dealcoloneq \dfrac{\partial phi_{ij}}{\partial x_j}.
// (which is incidentally also
- // b_j := \dfrac{\partial phi_{ij}}{\partial x_i}).
+ // b_j \dealcoloneq \dfrac{\partial phi_{ij}}{\partial x_i}).
// In both cases, a sum is implied.
//
// Now, we know the nonzero component in unrolled form: it is indicated
// given the form of the divergence above, if ii=jj there is only a
// single nonzero component of the full tensor and the gradient
// equals
- // b_ii := \dfrac{\partial phi_{ii,ii}}{\partial x_ii}.
+ // b_ii \dealcoloneq \dfrac{\partial phi_{ii,ii}}{\partial x_ii}.
// all other entries of 'b' are zero
//
// on the other hand, if ii!=jj, then there are two nonzero entries in
// the full tensor and
- // b_ii := \dfrac{\partial phi_{ii,jj}}{\partial x_ii}.
- // b_jj := \dfrac{\partial phi_{ii,jj}}{\partial x_jj}.
+ // b_ii \dealcoloneq \dfrac{\partial phi_{ii,jj}}{\partial x_ii}.
+ // b_jj \dealcoloneq \dfrac{\partial phi_{ii,jj}}{\partial x_jj}.
// again, all other entries of 'b' are zero
const dealii::Tensor<1, spacedim> &phi_grad =
fe_values->finite_element_output.shape_gradients[snc][q_point];
* @relatesalso LinearOperator
*
* Addition of two linear operators @p first_op and @p second_op given by
- * $(\text{first\_op}+\text{second\_op})x := \text{first\_op}(x) +
+ * $(\text{first\_op}+\text{second\_op})x \dealcoloneq \text{first\_op}(x) +
* \text{second\_op}(x)$
*
* @ingroup LAOperators
* @relatesalso LinearOperator
*
* Subtraction of two linear operators @p first_op and @p second_op given by
- * $(\text{first\_op}-\text{second\_op})x := \text{first\_op}(x) -
+ * $(\text{first\_op}-\text{second\_op})x \dealcoloneq \text{first\_op}(x) -
* \text{second\_op}(x)$
*
* @ingroup LAOperators
* @relatesalso LinearOperator
*
* Composition of two linear operators @p first_op and @p second_op given by
- * $(\text{first\_op}*\text{second\_op})x :=
+ * $(\text{first\_op}*\text{second\_op})x \dealcoloneq
* \text{first\_op}(\text{second\_op}(x))$
*
* @ingroup LAOperators
/**
* Perform one of the matrix-matrix operations:
- * sub( C ) := alpha*op( sub( A ) )*op( sub( B ) ) + beta*sub( C ),
+ * @f{align*}
+ * \mathrm{sub}(C) &\dealcoloneq \alpha op(\mathrm{sub}(A))op(\mathrm{sub}(B))
+ * + \beta \mathrm{sub}(C), \\
+ * \mathrm{sub}(C) &\dealcoloneq \alpha op(\mathrm{sub}(A))op(\mathrm{sub}(B))
+ * + beta sub(C),
+ * @f
* where
- * sub( C ) denotes C(IC:IC+M-1,JC:JC+N-1), and, op( X ) is one of
- * op( X ) = X or op( X ) = X'.
+ * $\mathrm{sub}(C)$ denotes C(IC:IC+M-1,JC:JC+N-1), and, $op(X)$ is one of
+ * $op(X) = X$ or $op(X) = X^T$.
*/
void
pdgemm_(const char * transa,
int * info);
/**
- * Copy all or a part of a distributed matrix A to another
- * distributed matrix B. No communication is performed, pdlacpy
- * performs a local copy sub(A) := sub(B), where sub(A) denotes
- * A(ia:ia+m-1,ja:ja+n-1) and sub(B) denotes B(ib:ib+m-1,jb:jb+n-1)
+ * Copy all or a part of a distributed matrix A to another distributed matrix
+ * B. No communication is performed, pdlacpy performs a local copy
+ * $\mathrm{sub}(A) \dealcoloneq \mathrm{sub}(B)$, where $\mathrm{sub}(A)$
+ * denotes $A(ia:ia+m-1, ja:ja+n-1)$ and $\mathrm{sub}(B)$ denotes
+ * $B(ib:ib+m-1, jb:jb+n-1)$.
*/
void
pdlacpy_(const char * uplo,
/*
* Perform matrix sum:
- * C := beta*C + alpha*op(A),
- * where op(A) denotes either op(A)=A or op(A)=A^T
+ * @f{equation*}
+ * C \dealcoloneq \beta C + \alpha op(A),
+ * @f
+ * where $op(A)$ denotes either $op(A) = A$ or $op(A)=A^T$.
*/
void
pdgeadd_(const char * transa,
/**
* Enables the failure check. Solving is stopped with @p ReturnState @p
* failure if <tt>residual>failure_residual</tt> with
- * <tt>failure_residual:=rel_failure_residual*first_residual</tt>.
+ * <tt>failure_residual := rel_failure_residual*first_residual</tt>.
*/
void
set_failure_criterion(const double rel_failure_residual);
* \text{span}\{w_j\}_{j=0}^m$, compute the sparsity pattern that would be
* necessary to assemble the matrix
* \f[
- * M_{ij} := \int_{B} v_i(x) w_j(x) dx, \quad i \in [0,n), j \in [0,m),
+ * M_{ij} \dealcoloneq \int_{B} v_i(x) w_j(x) dx,
+ * \quad i \in [0,n), j \in [0,m),
* \f]
* where $V(\Omega)$ is the finite element space associated with the
* `space_dh` passed to this function (or part of it, if specified in
* $V(\Omega) = \text{span}\{v_i\}_{i=0}^n$ and $Q(B) =
* \text{span}\{w_j\}_{j=0}^m$, compute the coupling matrix
* \f[
- * M_{ij} := \int_{B} v_i(x) w_j(x) dx, \quad i \in [0,n), j \in [0,m),
+ * M_{ij} \dealcoloneq \int_{B} v_i(x) w_j(x) dx,
+ * \quad i \in [0,n), j \in [0,m),
* \f]
* where $V(\Omega)$ is the finite element space associated with the
* `space_dh` passed to this function (or part of it, if specified in
* also the normalized normal for each quadrature point. This can be viewed
* as storing a discrete surface element,
* @f[
- * \Delta \hat{S}_q := w_q \hat{n}_q \approx d\hat{S}(\hat{x}_q),
+ * \Delta \hat{S}_q \dealcoloneq w_q \hat{n}_q \approx d\hat{S}(\hat{x}_q),
* @f]
* for each quadrature point. The surface integral in real space would then be
* approximated as
* The result is expressed as
* @f[
* \mathbf{F}
- * := \nabla_{0} \boldsymbol{\varphi} \left( \mathbf{X} \right)
+ * \dealcoloneq \nabla_{0} \boldsymbol{\varphi}
+ * \left( \mathbf{X} \right)
* =\mathbf{I} + \nabla_{0}\mathbf{u}
* @f]
* where $\mathbf{u} = \mathbf{u}\left(\mathbf{X}\right)$ is the
* tensor @p F .
* The result is expressed as
* @f[
- * \mathbf{F}^{\text{iso}} := J^{-1/\textrm{dim}} \mathbf{F}
+ * \mathbf{F}^{\text{iso}} \dealcoloneq J^{-1/\textrm{dim}} \mathbf{F}
* @f]
* where $J = \text{det}\left(\mathbf{F}\right)$.
*
* tensor @p F .
* The result is expressed as
* @f[
- * \mathbf{F}^{\text{vol}} := J^{1/\textrm{dim}} \mathbf{I}
+ * \mathbf{F}^{\text{vol}} \dealcoloneq J^{1/\textrm{dim}} \mathbf{I}
* @f]
* where $J = \text{det}\left(\mathbf{F}\right)$.
*
* as constructed from the deformation gradient tensor @p F.
* The result is expressed as
* @f[
- * \mathbf{C} := \mathbf{F}^{T}\cdot\mathbf{F} \, .
+ * \mathbf{C} \dealcoloneq \mathbf{F}^{T}\cdot\mathbf{F} \, .
* @f]
*
* @dealiiWriggersA{23,3.15}
* as constructed from the deformation gradient tensor @p F.
* The result is expressed as
* @f[
- * \mathbf{b} := \mathbf{F}\cdot\mathbf{F}^{T} \, .
+ * \mathbf{b} \dealcoloneq \mathbf{F}\cdot\mathbf{F}^{T} \, .
* @f]
*
* @dealiiWriggersA{28,3.25}
* as constructed from the deformation gradient tensor @p F.
* The result is expressed as
* @f[
- * \mathbf{E} := \frac{1}{2}[\mathbf{F}^{T}\cdot\mathbf{F} - \mathbf{I}]
- * \, .
+ * \mathbf{E} \dealcoloneq \frac{1}{2}
+ * \left[ \mathbf{F}^{T}\cdot\mathbf{F} - \mathbf{I} \right] \, .
* @f]
*
* @dealiiWriggersA{23,3.15}
* as constructed from the displacement gradient tensor @p Grad_u.
* The result is expressed as
* @f[
- * \boldsymbol{\varepsilon} := \frac{1}{2} \left[ \nabla_{0}\mathbf{u}
- * + [\nabla_{0}\mathbf{u}]^{T} \right] \, .
+ * \boldsymbol{\varepsilon} \dealcoloneq \frac{1}{2}
+ * \left[ \nabla_{0}\mathbf{u} + [\nabla_{0}\mathbf{u}]^{T} \right] \, .
* @f]
* where $\mathbf{u} = \mathbf{u}(\mathbf{X})$ is the displacement at
* position
* as constructed from the deformation gradient tensor @p F.
* The result is expressed as
* @f[
- * \mathbf{e} := \frac{1}{2} \left[ \mathbf{I}
+ * \mathbf{e} \dealcoloneq \frac{1}{2} \left[ \mathbf{I}
* - \mathbf{F}^{-T}\cdot\mathbf{F}^{-1} \right] \, .
* @f]
*
* gradient).
* The result is expressed as
* @f[
- * \mathbf{l} := \dot{\mathbf{F}}\cdot\mathbf{F}^{-1} \, .
+ * \mathbf{l} \dealcoloneq \dot{\mathbf{F}}\cdot\mathbf{F}^{-1} \, .
* @f]
*
* @dealiiWriggersA{32,3.47}
* gradient).
* The result is expressed as
* @f[
- * \mathbf{d} := \frac{1}{2} \left[ \mathbf{l} + \mathbf{l}^{T} \right]
+ * \mathbf{d} \dealcoloneq \frac{1}{2}
+ * \left[ \mathbf{l} + \mathbf{l}^{T} \right]
* @f]
* where
* @f[
* gradient).
* The result is expressed as
* @f[
- * \mathbf{w} := \frac{1}{2} \left[ \mathbf{l} - \mathbf{l}^{T} \right]
+ * \mathbf{w} \dealcoloneq \frac{1}{2}
+ * \left[ \mathbf{l} - \mathbf{l}^{T} \right]
* @f]
* where
* @f[
* \}$ the following holds:
* @f[
* \mathcal{S} : \{ \hat{\bullet} \}
- * := \dfrac{1}{2}[\{ \hat{\bullet} \} + \{ \hat{\bullet} \}^T] \, .
+ * \dealcoloneq \dfrac{1}{2}
+ * \left[ \{ \hat{\bullet} \} + \{ \hat{\bullet} \}^T \right] \, .
* @f]
*
* As a corollary to this, for any second-order symmetric tensor $\{
* This is defined as
* @f[
* \mathcal{P}
- * := \mathcal{S} - \frac{1}{\textrm{dim}} \mathbf{I} \otimes
- * \mathbf{I}
+ * \dealcoloneq \mathcal{S} - \frac{1}{\textrm{dim}} \mathbf{I}
+ * \otimes \mathbf{I}
* @f]
* where $\mathcal{S}$ is the fourth-order unit symmetric tensor and
* $\mathbf{I}$ is the second-order identity tensor.
* For any second-order (spatial) symmetric tensor the following holds:
* @f[
* \mathcal{P} : \{ \bullet \}
- * := \{ \bullet \} - \frac{1}{\textrm{dim}} \left[ \{ \bullet \} :
- * \mathbf{I} \right]\mathbf{I} = \mathcal{P}^{T} : \{ \bullet \} =
- * \texttt{dev\_P} \left( \{ \bullet \} \right)
+ * \dealcoloneq \{ \bullet \} - \frac{1}{\textrm{dim}}
+ * \left[ \{ \bullet \} : \mathbf{I} \right]\mathbf{I}
+ * = \mathcal{P}^{T} : \{ \bullet \}
+ * = \texttt{dev\_P} \left( \{ \bullet \} \right)
* @f]
* and, therefore,
* @f[
* This referential isochoric projection tensor is defined as
* @f[
* \hat{\mathcal{P}}
- * := \frac{\partial \bar{\mathbf{C}}}{\partial \mathbf{C}}
+ * \dealcoloneq \frac{\partial \bar{\mathbf{C}}}{\partial \mathbf{C}}
* @f]
* with
* @f[
- * \bar{\mathbf{C}} := J^{-2/\textrm{dim}} \mathbf{C}
+ * \bar{\mathbf{C}} \dealcoloneq J^{-2/\textrm{dim}} \mathbf{C}
* \qquad \text{,} \qquad
* \mathbf{C} = \mathbf{F}^{T}\cdot\mathbf{F}
* \qquad \text{and} \qquad
* the following holds:
* @f[
* \{ \bullet \} : \hat{\mathcal{P}}
- * := J^{-2/\textrm{dim}} \left[ \{ \bullet \} -
+ * \dealcoloneq J^{-2/\textrm{dim}} \left[ \{ \bullet \} -
* \frac{1}{\textrm{dim}}\left[\mathbf{C} : \{ \bullet \}\right]
* \mathbf{C}^{-1} \right] = \texttt{Dev\_P} \left( \{ \bullet \} \right)
* \, .
* @f[
* \left[ \frac{\partial \mathbf{C}^{-1}}{\partial \mathbf{C}}
* \right]_{IJKL}
- * := -\frac{1}{2}[ C^{-1}_{IK}C^{-1}_{JL}
+ * \dealcoloneq -\frac{1}{2}[ C^{-1}_{IK}C^{-1}_{JL}
* + C^{-1}_{IL}C^{-1}_{JK} ]
* @f]
*
* a rank-2 symmetric tensor $\mathbf{S}$ we enumerate its tensor
* components
* @f[
- * \mathbf{S} := \left[ \begin{array}{ccc}
+ * \mathbf{S} \dealcoloneq \left[ \begin{array}{ccc}
* S_{00} & S_{01} & S_{02} \\
* S_{10} = S_{01} & S_{11} & S_{12} \\
* S_{20} = S_{02} & S_{21} = S_{12} & S_{22}
* where $n$ denotes the Kelvin index for the tensor component,
* while for a general rank-2 tensor $\mathbf{T}$
* @f[
- * \mathbf{T} := \left[ \begin{array}{ccc}
+ * \mathbf{T} \dealcoloneq \left[ \begin{array}{ccc}
* T_{00} & T_{01} & T_{02} \\
* T_{10} & T_{11} & T_{12} \\
* T_{20} & T_{21} & T_{22}
* @f]
* and for a rank-1 tensor $\mathbf{v}$
* @f[
- * \mathbf{v} := \left[ \begin{array}{c}
+ * \mathbf{v} \dealcoloneq \left[ \begin{array}{c}
* v_{0} \\ v_{1} \\ v_{2}
* \end{array}\right]
* \quad \Rightarrow \quad
/**
* Return the rotation matrix for 2-d Euclidean space, namely
* @f[
- * \mathbf{R} := \left[ \begin{array}{cc}
+ * \mathbf{R} \dealcoloneq \left[ \begin{array}{cc}
* cos(\theta) & sin(\theta) \\
* -sin(\theta) & cos(\theta)
* \end{array}\right]
* stated using the Rodrigues' rotation formula, this function returns
* the equivalent of
* @f[
- * \mathbf{R} := cos(\theta)\mathbf{I} + sin(\theta)\mathbf{W}
+ * \mathbf{R} \dealcoloneq cos(\theta)\mathbf{I} + sin(\theta)\mathbf{W}
* + (1-cos(\theta))\mathbf{u}\otimes\mathbf{u}
* @f]
* where $\mathbf{u}$ is the axial vector (an axial vector) and $\theta$
* contravariant vector, i.e.
* @f[
* \chi\left(\bullet\right)^{\sharp}
- * := \mathbf{F} \cdot \left(\bullet\right)^{\sharp}
+ * \dealcoloneq \mathbf{F} \cdot \left(\bullet\right)^{\sharp}
* @f]
*
* @param[in] V The (referential) vector to be operated on
* contravariant tensor, i.e.
* @f[
* \chi\left(\bullet\right)^{\sharp}
- * := \mathbf{F} \cdot \left(\bullet\right)^{\sharp} \cdot
+ * \dealcoloneq \mathbf{F} \cdot \left(\bullet\right)^{\sharp} \cdot
* \mathbf{F}^{T}
* @f]
*
* contravariant symmetric tensor, i.e.
* @f[
* \chi\left(\bullet\right)^{\sharp}
- * := \mathbf{F} \cdot \left(\bullet\right)^{\sharp} \cdot
+ * \dealcoloneq \mathbf{F} \cdot \left(\bullet\right)^{\sharp} \cdot
* \mathbf{F}^{T}
* @f]
*
* contravariant tensor, i.e. (in index notation)
* @f[
* \left[ \chi\left(\bullet\right)^{\sharp} \right]_{ijkl}
- * := F_{iI} F_{jJ} \left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL}
+ * \dealcoloneq F_{iI} F_{jJ}
+ * \left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL}
* @f]
*
* @param[in] H The (referential) rank-4 tensor to be operated on
* contravariant symmetric tensor, i.e. (in index notation)
* @f[
* \left[ \chi\left(\bullet\right)^{\sharp} \right]_{ijkl}
- * := F_{iI} F_{jJ} \left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL}
+ * \dealcoloneq F_{iI} F_{jJ}
+ * \left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL}
* @f]
*
* @param[in] H The (referential) rank-4 symmetric tensor to be operated
* vector, i.e.
* @f[
* \chi^{-1}\left(\bullet\right)^{\sharp}
- * := \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp}
+ * \dealcoloneq \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp}
* @f]
*
* @param[in] v The (spatial) vector to be operated on
* contravariant tensor, i.e.
* @f[
* \chi^{-1}\left(\bullet\right)^{\sharp}
- * := \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp} \cdot
- * \mathbf{F}^{-T}
+ * \dealcoloneq \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp}
+ * \cdot \mathbf{F}^{-T}
* @f]
*
* @param[in] t The (spatial) tensor to be operated on
* contravariant symmetric tensor, i.e.
* @f[
* \chi^{-1}\left(\bullet\right)^{\sharp}
- * := \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp} \cdot
- * \mathbf{F}^{-T}
+ * \dealcoloneq \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp}
+ * \cdot \mathbf{F}^{-T}
* @f]
*
* @param[in] t The (spatial) symmetric tensor to be operated on
* contravariant tensor, i.e. (in index notation)
* @f[
* \left[ \chi^{-1}\left(\bullet\right)^{\sharp} \right]_{IJKL}
- * := F^{-1}_{Ii} F^{-1}_{Jj} \left(\bullet\right)^{\sharp}_{ijkl}
- * F^{-1}_{Kk} F^{-1}_{Ll}
+ * \dealcoloneq F^{-1}_{Ii} F^{-1}_{Jj}
+ * \left(\bullet\right)^{\sharp}_{ijkl} F^{-1}_{Kk} F^{-1}_{Ll}
* @f]
*
* @param[in] h The (spatial) tensor to be operated on
* contravariant symmetric tensor, i.e. (in index notation)
* @f[
* \left[ \chi^{-1}\left(\bullet\right)^{\sharp} \right]_{IJKL}
- * := F^{-1}_{Ii} F^{-1}_{Jj} \left(\bullet\right)^{\sharp}_{ijkl}
- * F^{-1}_{Kk} F^{-1}_{Ll}
+ * \dealcoloneq F^{-1}_{Ii} F^{-1}_{Jj}
+ * \left(\bullet\right)^{\sharp}_{ijkl} F^{-1}_{Kk} F^{-1}_{Ll}
* @f]
*
* @param[in] h The (spatial) symmetric tensor to be operated on
* vector, i.e.
* @f[
* \chi\left(\bullet\right)^{\flat}
- * := \mathbf{F}^{-T} \cdot \left(\bullet\right)^{\flat}
+ * \dealcoloneq \mathbf{F}^{-T} \cdot \left(\bullet\right)^{\flat}
* @f]
*
* @param[in] V The (referential) vector to be operated on
* covariant tensor, i.e.
* @f[
* \chi\left(\bullet\right)^{\flat}
- * := \mathbf{F}^{-T} \cdot \left(\bullet\right)^{\flat} \cdot
- * \mathbf{F}^{-1}
+ * \dealcoloneq \mathbf{F}^{-T} \cdot \left(\bullet\right)^{\flat}
+ * \cdot \mathbf{F}^{-1}
* @f]
*
* @param[in] T The (referential) rank-2 tensor to be operated on
* covariant symmetric tensor, i.e.
* @f[
* \chi\left(\bullet\right)^{\flat}
- * := \mathbf{F}^{-T} \cdot \left(\bullet\right)^{\flat} \cdot
- * \mathbf{F}^{-1}
+ * \dealcoloneq \mathbf{F}^{-T} \cdot \left(\bullet\right)^{\flat}
+ * \cdot \mathbf{F}^{-1}
* @f]
*
* @param[in] T The (referential) rank-2 symmetric tensor to be operated
* covariant tensor, i.e. (in index notation)
* @f[
* \left[ \chi\left(\bullet\right)^{\flat} \right]_{ijkl}
- * := F^{-T}_{iI} F^{-T}_{jJ} \left(\bullet\right)^{\flat}_{IJKL}
- * F^{-T}_{kK} F^{-T}_{lL}
+ * \dealcoloneq F^{-T}_{iI} F^{-T}_{jJ}
+ * \left(\bullet\right)^{\flat}_{IJKL} F^{-T}_{kK} F^{-T}_{lL}
* @f]
*
* @param[in] H The (referential) rank-4 tensor to be operated on
* covariant symmetric tensor, i.e. (in index notation)
* @f[
* \left[ \chi\left(\bullet\right)^{\flat} \right]_{ijkl}
- * := F^{-T}_{iI} F^{-T}_{jJ} \left(\bullet\right)^{\flat}_{IJKL}
- * F^{-T}_{kK} F^{-T}_{lL}
+ * \dealcoloneq F^{-T}_{iI} F^{-T}_{jJ}
+ * \left(\bullet\right)^{\flat}_{IJKL} F^{-T}_{kK} F^{-T}_{lL}
* @f]
*
* @param[in] H The (referential) rank-4 symmetric tensor to be operated
* vector, i.e.
* @f[
* \chi^{-1}\left(\bullet\right)^{\flat}
- * := \mathbf{F}^{T} \cdot \left(\bullet\right)^{\flat}
+ * \dealcoloneq \mathbf{F}^{T} \cdot \left(\bullet\right)^{\flat}
* @f]
*
* @param[in] v The (spatial) vector to be operated on
* covariant tensor, i.e.
* @f[
* \chi^{-1}\left(\bullet\right)^{\flat}
- * := \mathbf{F}^{T} \cdot \left(\bullet\right)^{\flat} \cdot
+ * \dealcoloneq \mathbf{F}^{T} \cdot \left(\bullet\right)^{\flat} \cdot
* \mathbf{F}
* @f]
*
* covariant symmetric tensor, i.e.
* @f[
* \chi^{-1}\left(\bullet\right)^{\flat}
- * := \mathbf{F}^{T} \cdot \left(\bullet\right)^{\flat} \cdot
- * \mathbf{F}
+ * \dealcoloneq \mathbf{F}^{T} \cdot \left(\bullet\right)^{\flat}
+ * \cdot \mathbf{F}
* @f]
*
* @param[in] t The (spatial) symmetric tensor to be operated on
* contravariant tensor, i.e. (in index notation)
* @f[
* \left[ \chi^{-1}\left(\bullet\right)^{\flat} \right]_{IJKL}
- * := F^{T}_{Ii} F^{T}_{Jj} \left(\bullet\right)^{\flat}_{ijkl}
- * F^{T}_{Kk} F^{T}_{Ll}
+ * \dealcoloneq F^{T}_{Ii} F^{T}_{Jj}
+ * \left(\bullet\right)^{\flat}_{ijkl} F^{T}_{Kk} F^{T}_{Ll}
* @f]
*
* @param[in] h The (spatial) tensor to be operated on
* contravariant symmetric tensor, i.e. (in index notation)
* @f[
* \left[ \chi^{-1}\left(\bullet\right)^{\flat} \right]_{IJKL}
- * := F^{T}_{Ii} F^{T}_{Jj} \left(\bullet\right)^{\flat}_{ijkl}
- * F^{T}_{Kk} F^{T}_{Ll}
+ * \dealcoloneq F^{T}_{Ii} F^{T}_{Jj}
+ * \left(\bullet\right)^{\flat}_{ijkl} F^{T}_{Kk} F^{T}_{Ll}
* @f]
*
* @param[in] h The (spatial) symmetric tensor to be operated on
* contravariant vector, i.e.
* @f[
* \textrm{det} \mathbf{F}^{-1} \; \chi\left(\bullet\right)^{\sharp}
- * := \frac{1}{\textrm{det} \mathbf{F}} \; \mathbf{F} \cdot
- * \left(\bullet\right)^{\sharp}
+ * \dealcoloneq \frac{1}{\textrm{det} \mathbf{F}} \; \mathbf{F} \cdot
+ * \left(\bullet\right)^{\sharp}
* @f]
*
* @param[in] V The (referential) vector to be operated on
* contravariant tensor, i.e.
* @f[
* \textrm{det} \mathbf{F}^{-1} \; \chi\left(\bullet\right)^{\sharp}
- * := \frac{1}{\textrm{det} \mathbf{F}} \; \mathbf{F} \cdot
+ * \dealcoloneq \frac{1}{\textrm{det} \mathbf{F}} \; \mathbf{F} \cdot
* \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{T}
* @f]
*
* contravariant symmetric tensor, i.e.
* @f[
* \textrm{det} \mathbf{F}^{-1} \; \chi\left(\bullet\right)^{\sharp}
- * := \frac{1}{\textrm{det} \mathbf{F}} \; \mathbf{F} \cdot
+ * \dealcoloneq \frac{1}{\textrm{det} \mathbf{F}} \; \mathbf{F} \cdot
* \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{T}
* @f]
*
* @f[
* \textrm{det} \mathbf{F}^{-1} \; \left[
* \chi\left(\bullet\right)^{\sharp} \right]_{ijkl}
- * := \frac{1}{\textrm{det} \mathbf{F}} \; F_{iI} F_{jJ}
+ * \dealcoloneq \frac{1}{\textrm{det} \mathbf{F}} \; F_{iI} F_{jJ}
* \left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL}
* @f]
*
* @f[
* \textrm{det} \mathbf{F}^{-1} \; \left[
* \chi\left(\bullet\right)^{\sharp} \right]_{ijkl}
- * := \frac{1}{\textrm{det} \mathbf{F}} \; F_{iI} F_{jJ}
+ * \dealcoloneq \frac{1}{\textrm{det} \mathbf{F}} \; F_{iI} F_{jJ}
* \left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL}
* @f]
*
* vector, i.e.
* @f[
* \textrm{det} \mathbf{F} \; \chi^{-1}\left(\bullet\right)^{\sharp}
- * := \textrm{det} \mathbf{F} \; \mathbf{F}^{-1} \cdot
+ * \dealcoloneq \textrm{det} \mathbf{F} \; \mathbf{F}^{-1} \cdot
* \left(\bullet\right)^{\sharp}
* @f]
*
* contravariant tensor, i.e.
* @f[
* \textrm{det} \mathbf{F} \; \chi^{-1}\left(\bullet\right)^{\sharp}
- * := \textrm{det} \mathbf{F} \; \mathbf{F}^{-1} \cdot
+ * \dealcoloneq \textrm{det} \mathbf{F} \; \mathbf{F}^{-1} \cdot
* \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{-T}
* @f]
*
* contravariant symmetric tensor, i.e.
* @f[
* \textrm{det} \mathbf{F} \; \chi^{-1}\left(\bullet\right)^{\sharp}
- * := \textrm{det} \mathbf{F} \; \mathbf{F}^{-1} \cdot
+ * \dealcoloneq \textrm{det} \mathbf{F} \; \mathbf{F}^{-1} \cdot
* \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{-T}
* @f]
*
* @f[
* \textrm{det} \mathbf{F} \; \left[
* \chi^{-1}\left(\bullet\right)^{\sharp} \right]_{IJKL}
- * := \textrm{det} \mathbf{F} \; F^{-1}_{Ii} F^{-1}_{Jj}
+ * \dealcoloneq \textrm{det} \mathbf{F} \; F^{-1}_{Ii} F^{-1}_{Jj}
* \left(\bullet\right)^{\sharp}_{ijkl} F^{-1}_{Kk} F^{-1}_{Ll}
* @f]
*
* @f[
* \textrm{det} \mathbf{F} \; \left[
* \chi^{-1}\left(\bullet\right)^{\sharp} \right]_{IJKL}
- * := \textrm{det} \mathbf{F} \; F^{-1}_{Ii} F^{-1}_{Jj}
+ * \dealcoloneq \textrm{det} \mathbf{F} \; F^{-1}_{Ii} F^{-1}_{Jj}
* \left(\bullet\right)^{\sharp}_{ijkl} F^{-1}_{Kk} F^{-1}_{Ll}
* @f]
*
* between the reference and spatial surface elements, i.e.
* @f[
* \mathbf{n} \frac{da}{dA}
- * := \textrm{det} \mathbf{F} \, \mathbf{F}^{-T} \cdot \mathbf{N}
- * = \textrm{cof} \mathbf{F} \cdot \mathbf{N} \, .
+ * \dealcoloneq \textrm{det} \mathbf{F} \, \mathbf{F}^{-T} \cdot \mathbf{N}
+ * = \textrm{cof} \mathbf{F} \cdot \mathbf{N} \, .
* @f]
*
* @param[in] N The referential normal unit vector $\mathbf{N}$
*
* For both DIRK and ARK methods, an implicit system of the form
* \f[
- * G(z_i) := M z_i − h_n A^I_{i,i} f_I (t^I_{n,i}, z_i) − a_i = 0
+ * G(z_i) \dealcoloneq M z_i − h_n A^I_{i,i} f_I (t^I_{n,i}, z_i) − a_i = 0
* \f]
* must be solved for each stage $z_i , i = 1, \ldot, s$, where
* we have the data
* \f[
- * a_i := M y_{n−1} + h_n \sum_{j=1}^{i−1} [ A^E_{i,j} f_E(t^E_{n,j}, z_j)
+ * a_i \dealcoloneq
+ * M y_{n−1} + h_n \sum_{j=1}^{i−1} [ A^E_{i,j} f_E(t^E_{n,j}, z_j)
* + A^I_{i,j} f_I (t^I_{n,j}, z_j)]
* \f]
* for the ARK methods, or
* \f[
- * a_i := M y_{n−1} + h_n \sum_{j=1}^{i−1} A^I_{i,j} f_I (t^I_{n,j}, z_j)
+ * a_i \dealcoloneq
+ * M y_{n−1} + h_n \sum_{j=1}^{i−1} A^I_{i,j} f_I (t^I_{n,j}, z_j)
* \f]
* for the DIRK methods. Here $A^I_{i,j}$ and $A^E_{i,j}$ are the Butcher's
* tables for the chosen solver.
* \f]
* where
* \f[
- * N := M - \gamma J, \quad J := \frac{\partial f_I}{\partial y},
- * \qquad \gamma:= h_n A^I_{i,i}.
+ * N \dealcoloneq M - \gamma J, \quad J
+ * \dealcoloneq \frac{\partial f_I}{\partial y},
+ * \qquad \gamma\dealcoloneq h_n A^I_{i,i}.
* \f]
*
* As an alternate to Newton’s method, ARKode may solve for each stage $z_i ,i
* That is $y' = A y$
* where
* \f[
- * A:=
+ * A \dealcoloneq
* \begin{matrix}
* 0 & 1 \\
* -k^2 &0
ScaLAPACKMatrix<NumberType>::operator=(const FullMatrix<NumberType> &matrix)
{
// FIXME: another way to copy is to use pdgeadd_ PBLAS routine.
- // This routine computes the sum of two matrices B:=a*A+b*B.
+ // This routine computes the sum of two matrices B := a*A + b*B.
// Matrices can have different distribution,in particular matrix A can
// be owned by only one process, so we can set a=1 and b=0 to copy
// non-distributed matrix A into distributed matrix B.
projects / dealii.git / commitdiff