Solving 2D/3D Heat Conduction Problems by Combining Topology Optimization and Anisotropic Mesh Adaptation

Abstract

Topology optimization was recently combined with anisotropic mesh adaptation to solve 3D minimum compliance problems in a fast and robust way. This paper demonstrates that the methodology is also applicable to 2D/3D heat conduction problems. Nodal design variables are used and the objective function is chosen such that the problem is self-adjoint. There is no way around the book keeping associated with mesh adaptation, so the whole 5527 line MATLAB code is published (https://github.com/kristianE86/trullekrul). The design variables as well as the sensitivities have to be interpolated between meshes, but MATLAB does not support interpolation on simplex meshes and it is thus handled as part of the local operations in the mesh adaptation. This functionality is available for nodal as well as element-wise design variables, but we have found the former to be superior. Results are shown for various discretizations demonstrating that the objective function converges, but comparison to optimizations with fixed meshes indicate that the use of mesh adaptation results in worse objective functions, particularly in 3D. Out of the 5018 statements only 100 is used for the actual optimization loop, 100 for 2D/3D geometry/mesh setup and 50 for the forward problem. It is thus feasible to use the script as a platform for solving other problems or for investigating the details of the methodology itself.

8 Appendix

Equation (1) can be reformulated by multiplying with a Lagrange multiplier, $\lambda$,

$$\begin{aligned} 0= & {} \int _\Omega \lambda \left[ \nabla \cdot (-\kappa \varvec{\nabla }T) - Q\right] d\Omega \nonumber \\= & {} \int _\Omega \varvec{\nabla }\lambda \cdot \kappa \varvec{\nabla }T d\Omega - \int _\Omega Q \lambda d\Omega + \int _{\partial \Omega } \lambda (\overbrace{\kappa \varvec{\nabla }T \cdot \mathbf {\hat{n}}}^{\mathbf {q}_\mathrm {bnd}}) ds. \end{aligned}$$

(11)

We now consider a variation in the design variable, $\delta \rho$, which gives rise to a variation in the filtered design variable, $\delta \tilde{\rho }$ and thus a variation in Eq. (11). Defining $\lambda$ to be orthogonal to this variation gives

$$\begin{aligned} 0= & {} \int _\Omega \varvec{\nabla }\lambda \cdot \left[ \frac{\partial \kappa }{\partial \tilde{\rho }} \varvec{\nabla }T + \kappa \frac{\partial \varvec{\nabla }T}{\partial \tilde{\rho }}\right] \delta \tilde{\rho } d\Omega + \int _{\partial \Omega } \lambda \left( \frac{\partial \mathbf {q}_\mathrm {bnd}}{\partial \tilde{\rho }} \cdot \mathbf {\hat{n}}\right) \delta \tilde{\rho } ds. \end{aligned}$$

(12)

By subtracting Eq. (12) from the variation of Eq. (6), we arrive at the variation of the objective function,

$$\begin{aligned} \delta \phi= & {} \int _\Omega \left[ \frac{\partial \kappa }{\partial \tilde{\rho }} (\varvec{\nabla }T)^2 + 2 \kappa \varvec{\nabla }T \cdot \frac{\partial \varvec{\nabla }T}{\partial \tilde{\rho }} \right] \delta \tilde{\rho } d\Omega - \int _\Omega \varvec{\nabla }\lambda \cdot \left[ \frac{\partial \kappa }{\partial \tilde{\rho }} \varvec{\nabla }T + \kappa \frac{\partial \varvec{\nabla }T}{\partial \tilde{\rho }}\right] \delta \tilde{\rho } d\Omega \nonumber \\- & {} \int _{\partial \Omega } \lambda \left( \frac{\partial \mathbf {q}_\mathrm {bnd}}{\partial \tilde{\rho }} \cdot \mathbf {\hat{n}}\right) \delta \tilde{\rho } ds \nonumber \\= & {} \int _\Omega \frac{\partial \varvec{\nabla }T}{\partial \tilde{\rho }} \left[ 2\kappa \varvec{\nabla }T - \kappa \varvec{\nabla }\lambda \right] \delta \tilde{\rho } d\Omega + \int _\Omega \frac{\partial \kappa }{\partial \tilde{\rho }} \varvec{\nabla }T \left[ \varvec{\nabla }T - \varvec{\nabla }\lambda \right] \delta \tilde{\rho } d\Omega \nonumber \\- & {} \int _{\partial \Omega } \lambda \left( \frac{\partial \mathbf {q}_\mathrm {bnd}}{\partial \tilde{\rho }} \cdot \mathbf {\hat{n}}\right) \delta \tilde{\rho } ds, \end{aligned}$$

(13)

so if we set

$$\begin{aligned} \lambda= & {} 2T , \end{aligned}$$

the first integral in Eq. (13) drops out. The last term also drops out, because $\lambda =0$ holds at the Dirichlet boundary and otherwise we have $\mathbf {q}_\mathrm {bnd}=0$. We thus arrive at

$$\begin{aligned} \delta \phi= & {} - \int _\Omega \frac{\partial \kappa }{\partial \tilde{\rho }} (\varvec{\nabla }T)^2 \delta \tilde{\rho } d\Omega , \end{aligned}$$

which we can easily relate to the variation in the design variable using the chain-rule

$$\begin{aligned} \delta \phi= & {} - \int _\Omega \frac{\partial \kappa }{\partial \tilde{\rho }} (\varvec{\nabla }T)^2 \frac{\partial \tilde{\rho }}{\partial \rho } \delta \rho d\Omega \end{aligned}$$

(14)

We now introduce another Lagrange multiplier, $\nu$, and multiply it with Eq. (4)

$$\begin{aligned} 0= & {} \int _\Omega \nu \left( L_\mathrm {min}^2 \nabla ^2\tilde{\rho } + \rho - \tilde{\rho }\right) d\Omega \nonumber \\= & {} - \int _\Omega \varvec{\nabla }\nu \cdot L_\mathrm {min}^2 \varvec{\nabla }\tilde{\rho } d\Omega + \int _\Omega \nu (\rho -\tilde{\rho }) d\Omega + \int _{\partial \Omega } \nu (\overbrace{\varvec{\nabla }\tilde{\rho } \cdot \mathbf {\hat{n}}}^{0}) ds \nonumber \\= & {} \int _\Omega \tilde{\rho } (L_\mathrm {min}^2 \nabla ^2 \nu ) d\Omega + \int _\Omega \nu (\rho -\tilde{\rho }) d\Omega + \int _{\partial \Omega } \tilde{\rho } (\varvec{\nabla }\nu \cdot \mathbf {\hat{n}}) ds. \end{aligned}$$

(15)

Defining $\nu$ to be orthogonal with respect to variations in the design variable yields

$$\begin{aligned} 0= & {} \int _\Omega \frac{\partial \tilde{\rho }}{\partial \rho } \left( L_\mathrm {min}^2 \nabla ^2 \nu - \nu \right) \delta \rho d\Omega + \int _\Omega \nu \delta \rho d\Omega + \int _{\partial \Omega } \frac{\partial \tilde{\rho }}{\partial \rho } (\varvec{\nabla }\nu \cdot \mathbf {\hat{n}}) \delta \rho ds \end{aligned}$$

(16)

Adding Eqs. (14) and (16) gives

$$\begin{aligned} \delta \phi= & {} - \int _\Omega \frac{\partial \kappa }{\partial \tilde{\rho }} (\varvec{\nabla }T)^2 \frac{\partial \tilde{\rho }}{\partial \rho } \delta \rho d\Omega + \int _\Omega \frac{\partial \tilde{\rho }}{\partial \rho } \left( L_\mathrm {min}^2 \nabla ^2 \nu - \nu \right) \partial \rho d\Omega + \int _\Omega \nu \delta \rho d\Omega \\+ & {} \int _{\partial \Omega } \frac{\partial \tilde{\rho }}{\partial \rho } (\varvec{\nabla }\nu \cdot \mathbf {\hat{n}}) \delta \rho ds \\= & {} \int _\Omega \frac{\partial \tilde{\rho }}{\partial \rho } \left[ L_\mathrm {min}^2 \nabla ^2 \nu -\frac{\partial \kappa }{\partial \tilde{\rho }} (\varvec{\nabla }T)^2 - \nu \right] \delta \rho d\Omega + \int _\Omega \nu \delta \rho d\Omega + \int _{\partial \Omega } \frac{\partial \tilde{\rho }}{\partial \rho } (\varvec{\nabla }\nu \cdot \mathbf {\hat{n}}) \delta \rho ds, \end{aligned}$$

so the variation in the objective function becomes

$$\begin{aligned} \delta \phi= & {} \int _\Omega \nu \delta \rho d\Omega , \quad \mathrm {where} \\ \nu= & {} L_\mathrm {min}^2 \nabla ^2 \nu -\frac{\partial \kappa }{\partial \tilde{\rho }} (\varvec{\nabla }T)^2 \quad \mathrm {and} \quad \mathbf {\hat{n}} \cdot \varvec{\nabla }\nu = 0 \quad \mathrm {on} \, \partial \Omega . \nonumber \end{aligned}$$

(17)