* }
* @endcode
*
- * ### Exploitation of the chain-rule
+ * @subsection Exploitation of the chain-rule
*
* In the most practical sense, any of the above categories exploit the chain-rule to compute the total
* derivative of a composite function. To perform this action, they typically use one of two mechanisms to
DEAL_II_ADOLC_WITH_ADVANCED_BRANCHING=1 \
DEAL_II_ADOLC_WITH_ATRIG_ERF=1 \
DEAL_II_ADOLC_WITH_TAPELESS_REFCOUNTING=1 \
+ DEAL_II_CXX20_REQUIRES(x)= \
DEAL_II_DEPRECATED_EARLY= \
DEAL_II_DEPRECATED_EARLY_WITH_COMMENT(x)= \
DEAL_II_WITH_ARBORX=1 \
functions, and get
@f{eqnarray*}{
M\bar{p}^{n}-k \theta M v^n & = & M\bar{p}^{n-1}+k (1-\theta)Mv^{n-1},\\
-
(-c_0^2k \theta A-c_0 B)\bar{p}^n-Mv^{n} & = &
(c_0^2k(1-\theta)A-c_0B)\bar{p}^{n-1}-Mv^{n-1}+c_0^2k(\theta F^{n}+(1-\theta)F^{n-1}).
@f}
// $\mathbf{z}_i$ is the $i$th vector valued test function.
// Furthermore, the scalar product
// $\left(\mathbf{F}(\mathbf{w}), \nabla\mathbf{z}_i\right)_K$ is
- // understood as $\int_K \sum_{c=1}^{\text{n_components}}
+ // understood as $\int_K \sum_{c=1}^{\text{n\_components}}
// \sum_{d=1}^{\text{dim}} \mathbf{F}(\mathbf{w})_{cd}
// \frac{\partial z^c_i}{x_d}$ where $z^c_i$ is the $c$th component of
// the $i$th test function.
// @f{eqnarray*}{
// R_i &=&
// \left(\frac{(\mathbf{w}_{n+1} -
- // \mathbf{w}_n)_{\text{component_i}}}{\delta
- // t},(\mathbf{z}_i)_{\text{component_i}}\right)_K
+ // \mathbf{w}_n)_{\text{component\_i}}}{\delta
+ // t},(\mathbf{z}_i)_{\text{component\_i}}\right)_K
// \\ &-& \sum_{d=1}^{\text{dim}} \left( \theta \mathbf{F}
- // ({\mathbf{w}^k_{n+1}})_{\text{component_i},d} + (1-\theta)
- // \mathbf{F} ({\mathbf{w}_{n}})_{\text{component_i},d} ,
- // \frac{\partial(\mathbf{z}_i)_{\text{component_i}}} {\partial
+ // ({\mathbf{w}^k_{n+1}})_{\text{component\_i},d} + (1-\theta)
+ // \mathbf{F} ({\mathbf{w}_{n}})_{\text{component\_i},d} ,
+ // \frac{\partial(\mathbf{z}_i)_{\text{component\_i}}} {\partial
// x_d}\right)_K
// \\ &+& \sum_{d=1}^{\text{dim}} h^{\eta} \left( \theta \frac{\partial
- // (\mathbf{w}^k_{n+1})_{\text{component_i}}}{\partial x_d} + (1-\theta)
- // \frac{\partial (\mathbf{w}_n)_{\text{component_i}}}{\partial x_d} ,
- // \frac{\partial (\mathbf{z}_i)_{\text{component_i}}}{\partial x_d}
+ // (\mathbf{w}^k_{n+1})_{\text{component\_i}}}{\partial x_d} + (1-\theta)
+ // \frac{\partial (\mathbf{w}_n)_{\text{component\_i}}}{\partial x_d} ,
+ // \frac{\partial (\mathbf{z}_i)_{\text{component\_i}}}{\partial x_d}
// \right)_K
- // \\ &-& \left( \theta\mathbf{G}({\mathbf{w}^k_n+1} )_{\text{component_i}}
- // + (1-\theta)\mathbf{G}({\mathbf{w}_n})_{\text{component_i}} ,
- // (\mathbf{z}_i)_{\text{component_i}} \right)_K ,
+ // \\ &-& \left( \theta\mathbf{G}({\mathbf{w}^k_n+1} )_{\text{component\_i}}
+ // + (1-\theta)\mathbf{G}({\mathbf{w}_n})_{\text{component\_i}} ,
+ // (\mathbf{z}_i)_{\text{component\_i}} \right)_K ,
// @f}
// where integrals are
// understood to be evaluated through summation over quadrature points.
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 \dealcoloneq \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},
set strategy for non-linear multibody contact problems, Comput. Methods Appl. Mech. Engrg.
194, 2005, pp. 3147-3166).
The vector $G$ is defined by a suitable approximation $g_h$ of the gap $g$
-@f{gather*}G_p = \begin{cases}{
+@f{gather*}G_p = \begin{cases}
g_{h,p}, & \text{if}\quad p\in\mathcal{S}\\
0, & \text{if}\quad p\notin\mathcal{S}.
\end{cases}@f}
, solver_type(solver_type)
, triangulation(Triangulation<dim>::maximum_smoothing)
,
- // Finite element for the velocity only -- we choose the
- // $Q_{\text{pressure_degree}}^d$ element:
+ // Finite element for the velocity only:
velocity_fe(FE_Q<dim>(pressure_degree + 1) ^ dim)
,
// Finite element for the whole system:
0<\rho_{\min}\leq \rho(x) \leq 1,
@f]
@f[
-
\nabla \cdot \boldsymbol{\sigma}(\rho) + \mathbf{F} = 0 \quad \text{on } \Omega
@f]
The final constraint, the balance of linear momentum (which we will refer to as the elasticity equation),
@f[
\int_\Omega -d_\varrho z_1 + d_\varrho z_2 + H(d_\varrho)y_2 d\Omega= 0\;\;\forall
d_\varrho
-
@f]
</li>
<li> Primal Feasibility:
We use the free-space wave number $k_0 = \omega\sqrt{\varepsilon_0\mu_0}$ and
the dipole strength, $J_0$ to arrive at the following rescaling of the vector
fields and coordinates:
-@f[
-\begin{align*}
+@f{align*}{
\hat{x} = k_0x, &\qquad
\hat{\nabla} = \frac{1}{k_0}\nabla,\\
\hat{\mathbf{H}} = \frac{k_0}{J_0}\mu^{-1}\mathbf{H},&\qquad
\hat{\mathbf{E}} = \frac{k_0^2}{\omega\mu_0 J_0}\mathbf{E},\\
\hat{\mathbf{J}}_a = \frac{1}{J_0}\mathbf{J}_a,&\qquad
\hat{\mathbf{M}}_a = \frac{k_0}{\omega\mu_0 J_0}\mathbf{M}_a.
-\end{align*}
-@f]
+@f}
Finally, the interface conductivity is rescaled as
@f[
Assume that $\sigma_r^{\Sigma} \in L^{\infty}(\Sigma)^{2\times 2}$ is matrix-valued
and symmetric, and has a semidefinite real and complex part. Let $\varepsilon_r$
-be a smooth scalar function with $–\text{Im}(\varepsilon_r) = 0$, or
+be a smooth scalar function with $-\text{Im}(\varepsilon_r) = 0$, or
$\text{Im}(\varepsilon_r)\ge c > 0$ in $\Omega$. $\mu_r^{-1}$ is a smooth scalar
such that $\sqrt{\mu_r^{-1}\varepsilon_r}$ is real valued and strictly positive
in $\partial\Omega$.
evaluation
@f[
u(\boldsymbol{x}_q) = \sum_{i} \hat{N}^K_i(\hat{\boldsymbol{x}}_q) u_i^K
-\quad\text{with}\quad i\in[0,n_{\text{dofs_per_cell}}),
+\quad\text{with}\quad i\in[0,n_{\text{dofs\_per\_cell}}),
@f]
with $\hat{N}^K_i$ being the shape functions defined on the reference cell and
$u_i^{K}$ the solution coefficients
// =
// \sum_i\text{tr}\left({\nabla \boldsymbol{N}_i (\boldsymbol{x}_q)
// \boldsymbol n_i}\right)
- // \;\text{with}\; i\in[0,n_{\text{dofs_per_cell}}),
+ // \;\text{with}\; i\in[0,n_{\text{dofs\_per\_cell}}),
// @f]
// which we can apply since the immersed mesh is consistently
// orientated. The surface tension coefficient is set to 1 for the
*
* \f[
* \phi_{1,i}(x) = \begin{cases} v_i(x) & \text{ if } i \in [0,n_l) \\
- * 0 & \text{ if ) i \in [n_1, n_1+n_2] \end{cases},\quad \phi_{1,i}(x) =
+ * 0 & \text{ if } i \in [n_1, n_1+n_2] \end{cases},\quad \phi_{1,i}(x) =
* \begin{cases} 0(x) & \text{ if } i \in [0,n_1) \\
- * w_{i-n_1}(x) & \text{ if ) i \in [n_1, n_1+n_2] \end{cases},
+ * w_{i-n_1}(x) & \text{ if } i \in [n_1, n_1+n_2] \end{cases},
* \f]
*
- * where $phi_{1,i}$ is the first basis function with index $i$ and $n_{1,2}$
+ * where $\phi_{1,i}$ is the first basis function with index $i$ and $n_{1,2}$
* are the number of local dofs on the first and second FEValuesBase objects.
*
* This enum is used in the constructor of FECouplingValues to specify how to
* \partial_1\partial_2 u_2 - \partial_2^2 u_1 \\
* \partial_1\partial_2 u_1 - \partial_1^2 u_2
* \end{pmatrix}
- *
+ * @f]
+ * and
+ * @f[
* \nabla\times\nabla\times \mathbf u = \begin{pmatrix}
* \partial_1\partial_2 u_2 + \partial_1\partial_3 u_3
* - (\partial_2^2+\partial_3^2) u_1 \\
* - (\partial_3^2+\partial_1^2) u_2 \\
* \partial_3\partial_1 u_1 + \partial_3\partial_2 u_2
* - (\partial_1^2+\partial_2^2) u_3
- * \end{pmatrix}
+ * \end{pmatrix}.
* @f]
*
* @note The third tensor argument is not used in two dimensions and can
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