maximum stress placed on a structure by selecting a region $E$ where material is
placed. In other words,
@f[
- \text{Minimize}\| \boldsymbol{\sigma} (\mathbf{u}) \|_\infty
+ \text{minimize}\| \boldsymbol{\sigma} (\mathbf{u}) \|_\infty
@f]
@f[
\text{subject to } |E|\leq V_{\max},
also works as a measure of total deformation over the structure.
@f[
- \text{Minimize } \int_E \frac{1}{2}\boldsymbol{\sigma} : \boldsymbol{\varepsilon} dV
+ \text{minimize } \int_E \frac{1}{2}\boldsymbol{\sigma} : \boldsymbol{\varepsilon} dV
@f]
@f[
\text{subject to } \|E\| \leq V_{\max}
following:
@f[
- \text{Minimize } \int_\Omega \frac{1}{2}\boldsymbol{\sigma}(\rho) : \boldsymbol{\varepsilon}(\rho) d_\Omega
+ \text{minimize } \int_\Omega \frac{1}{2}\boldsymbol{\sigma}(\rho) : \boldsymbol{\varepsilon}(\rho) d\Omega
@f]
@f[
- \text{subject to } \int_\Omega \rho(x) d_\Omega= V_{\max},
+ \text{subject to } \int_\Omega \rho(x) d\Omega= V_{\max},
@f]
@f[
0<\rho_{\min}\leq \rho(x) \leq 1,
\nabla \cdot \boldsymbol{\sigma}(\rho) + \mathbf{F} = 0 \quad \text{on } \Omega
@f]
-The final constraint, the balance of linear momentum (sometimes referred to as the elasticity equation),
+The final constraint, the balance of linear momentum (which we will refer to as the elasticity equation),
gives a method for finding $\boldsymbol{\sigma}$ and $\boldsymbol{\varepsilon}$ given the density $\rho$.
<h3>Elasticity Equation</h3>
formulations).
Furthermore, we will make the assumption that the material is linear isotropic,
-in which case the stress-strain tensor can be expressed in terms of the Lam\'{e}
+in which case the stress-strain tensor can be expressed in terms of the Lamé
parameters $\lambda,\mu$ such that
@f{align}
\boldsymbol{\sigma} &= \rho^p (\lambda \text{tr}(\boldsymbol{\varepsilon}) \mathbf{I} + 2 \mu \boldsymbol{\varepsilon}) , \\
Lagrange multipliers. Specifically, we will use the following symbols for the
Lagrange multipliers for the various constraints:
<ol>
- <li> $\mathbf{y}_1 $ - a Lagrange multiplier corresponding to the
- elasticity constraint </li>
- <li> $y_2$ - a Lagrange multiplier corresponding to the convolution
- filter constraint </li>
- <li> $z_1$ - a Lagrange multiplier corresponding to the lower slack variable </li>
- <li> $z_2$ - a Lagrange multiplier corresponding to the upper slack variable. </li>
+ <li> $\mathbf{y}_1 $: a Lagrange multiplier corresponding to the
+ elasticity constraint, </li>
+ <li> $y_2$: a Lagrange multiplier corresponding to the convolution
+ filter constraint, </li>
+ <li> $z_1$: a Lagrange multiplier corresponding to the lower slack variable, and </li>
+ <li> $z_2$: a Lagrange multiplier corresponding to the upper slack variable. </li>
</ol>
With these variables, the Lagrangian function reads as follows:
@f{align}{
\mathcal{L} =& \int_{\partial\Omega} \mathbf{u} \cdot \mathbf{t} d\partial\Omega
- - \alpha \int_\Omega \left(\log(s_1) + \log(s_2)\right) d\Omega- \left(\int_\Omega
+ - \alpha \int_\Omega \left(\log(s_1) + \log(s_2)\right) d\Omega- \int_\Omega
\rho^p \left(\frac{\mu}{2}\left(\boldsymbol{\varepsilon}(\mathbf{y}_1):\boldsymbol{\varepsilon}(\mathbf{u}))
\right) + \lambda \left( \nabla \cdot \mathbf{u} \nabla \cdot \mathbf{y}_1
- \right) d\Omega \right)- \int_{\partial \Omega} \mathbf{y}_1 \cdot \mathbf{t} d\partial\Omega \right) \\
+ \right)\right) d\Omega - \int_{\partial \Omega} \mathbf{y}_1 \cdot \mathbf{t} d\partial\Omega \\
& -\int_\Omega y_2 (\rho - H(\varrho)) d\Omega - \int_\Omega z_1 (\varrho-s_1) d\Omega
- \int_\Omega z_2 (1 - s_2 -\varrho) d\Omega
@f}
Newton method to compute search directions, and come back to how to deal with
the inequalities below when talking about step length procedures.
-Newton's method applied to the equations above results in the following system.
-Here, variational derivatives with respect to the $\{\bullet\}$ variable are taken
- in the $c_{\{\bullet\}}$ direction. This gives
+Newton's method applied to the equations above results in the system of equations
+listed below.
+Therein, variational derivatives with respect to the $\{\bullet\}$ variable are
+taken in the $c_{\{\bullet\}}$ direction.
<ol>
-<li> Stationarity - these equations ensure we are at a critical point of the
-objective function when constrained
+<li> Stationarity: These equations ensure we are at a critical point of the
+objective function when constrained.
Equation 1
@f{align}{
@f]
</li>
-<li> Primal Feasibility - these equations ensure the equality constraints
+<li> Primal Feasibility: These equations ensure the equality constraints
are met.
Equation 4
@f]
</li>
-<li>Complementary Slackness - these equations essentially ensure the barrier
-is met - in the final solution, we need $s^T z = 0$
+<li>Complementary Slackness: These equations essentially ensure the barrier
+is met - in the final solution, we need $s^T z = 0$.
Equation 8
@f[
@f]
</li>
-<li>Dual Feasibility - Multiplier on slacks and slack variables must be kept
+<li>Dual Feasibility: The Lagrange multiplier on slacks and slack variables must be kept
greater than 0. (This is the only part not implemented in the
-SANDTopOpt::assemble_system() function)
+`SANDTopOpt::assemble_system()` function.)
@f[
s,z \geq 0
@f]
factors that are independent of the current solution. That said, such methods
are also more complicated and we will not do this here.
-<h3>Merit Function</h3>
+<h3>Merit %Function</h3>
The algorithm outlined above makes use of a "merit function". Merit functions
are used to determine whether a step from $x_k$ to a proposed point $x_{k+1}$ is
minima at the same location as the original problem. This penalty parameter is
updated (by recommendation of Nocedal and Wright @cite Benson2002) as follows:
@f[
- p > \frac{\frac{1}{2} \mathbf{x}^T \cdot \mathbf{H} \cdot \mathbf{x} - \mathbf{x}^T \cdot \nabla f}{\|c_i\|_{l_\infty}, i \in \mathcal{E}} ,
+ p > \frac{\frac{1}{2} \mathbf{x}^T \cdot \mathbf{H} \cdot \mathbf{x} - \mathbf{x}^T \cdot \nabla f}{\|c_i\|_{l_\infty}}
+ \quad , i \in \mathcal{E},
@f]
where $\mathbf{H}$ is the Hessian of the objective function, $\mathbf{x}$ is a vector of our
decision (primal) variables, $f$ is the objective function, and $c_i$ is the error on a
// The use of the symbolic names defined in namespace
// `SolutionComponents` helps understand what each of the
// following terms corresponds to, but it also makes the
- // expressions lengthy and unwieldy: An term such as
+ // expressions lengthy and unwieldy: A term such as
// `coupling[SolutionComponents::density_upper_slack_multiplier<dim>][SolutionComponents::density<dim>]`
// just doesn't read very well, and would either have to be
// split over several lines or run off the right edge of
// us how far off our filtered density is from the filter
// applied to the unfiltered density. That is because while at
// the solution of the nonlinear problem, we have
- // $\rho=H\sigma$, but at intermediate iterations, we in
- // general have $\rho^k\neq H\sigma^k$ and the "residual"
- // $\rho^k-H\sigma^k$ will then appear as the right hand side
+ // $\rho=H\varrho$, but at intermediate iterations, we in
+ // general have $\rho^k\neq H\varrho^k$ and the "residual"
+ // $\rho^k-H\varrho^k$ will then appear as the right hand side
// of one of the Newton update equations that we compute
// below.
BlockVector<double> filtered_unfiltered_density_solution =
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