# Simultaneous Material and Topology Optimization Based on Topological Derivatives

## Abstract

We use an asymptotic expansion of the compliance cost functional in linear elasticity to find the optimal material inside elliptic inclusions. We extend the proposed method to material optimization on the whole domain and compare the global quality of the solutions for different inclusion sizes. Specifically, we use an adjusted free material optimization problem, that can be solved globally, as a global lower material optimization bound. Finally, the asymptotic expansion is used as a topological derivative in a simultaneous material and topology optimization problem.

## Keywords

Material optimization Topology optimization Material orientation Asymptotic expansions Discrete material optimization## 1 Introduction

We investigate material and
topology optimization of compliance problems in two dimensions. To this end, we first present an asymptotic formula^{1} of the compliance functional for the insertion of a number of ellipsoidal bodies in an elastic domain, originally derived in [1]. Later, we study numerically the feasibility of replacing all material using the same asymptotic expansion as for the ellipses and finally make use of the formula as topological derivative.

The problem described above is by far not new. There are many publications dealing with similar types of problems. For the rotational optimization considered later, in [2] an analytical formula for the strain energy is derived, to directly compute the optimal material orientation. In [3], this has been embedded into a structural optimization algorithm for compliance minimization. A similar approach is discussed in [4] for a plate model. The method proposed in this article, however, can be used for a broader spectrum in material optimization as well, such as discrete material optimization. The algorithm for simultaneous material and topology optimization presented at the end of this article is very similar to topology gradient methods, see e.g. [5]. For more references, see [6].

## 2 Material and Topology Optimization

We consider a domain \(\varOmega \subset \mathbb {R}^2\) of isotropic elastic material. The domain is subject to exterior traction and other boundary conditions (e.g. homogeneous Dirichlet conditions). The objective is to find the optimal material \(C^0\) in a set of admissible materials \(\mathcal {C}\) to insert into an inclusion, for which the compliance as defined in (1) is minimized.

*Hooke’s law*

### 2.1 Optimal Material in Elliptic Inclusions

### 2.2 Admissible Material Choices

### 2.3 From Elliptic Inclusions to Material Optimization

While the asymptotic model (2) rigorously holds only for elliptic inclusions of small size, in the following we will also numerically investigate the behavior when replacing the material inside squared patches of finite elements. Choosing the elements properly, the material in the whole domain can be replaced this way with the FE patches still being disjoint. Using a large number of patches, the size of the inclusion stays small compared to the domain size. Thus, we will study increasingly bigger ellipses and compare the compliance values of the different parametrizations to global lower material optimization bounds computed with an FMO solver.

**Validation Methods.** For the numerical evaluation, we discretize the domain \(\varOmega \) using rectangular finite elements. This discretization is necessary to compute the displacements used in the asymptotic expansion. The elliptic inclusions are approximated by those finite elements, for which the coordinates of their center point are contained in the inclusion \(\omega _i\). In order to obtain the actual compliance value for the optimization result, the material used in those elements is then replaced by the optimal value of \(C_i^0\). When replacing all material, we use equally sized squared FE patches that are uniformly distributed, disjoint and cover the whole domain.

**Numerical Results.**We consider the example from Fig. 1 with \(10\times 10\) ellipses and discretize the domain \(\varOmega \) using \(100\times 100\) finite elements. We compare the different admissible material sets as defined in Sect. 2.2. For \(\mathcal {C}_{\text {FMO}}\) we choose \(\underline{\tau }=1\) as lower eigenvalue bound and \(\overline{\rho }=17\) as upper trace bound both in the asymptotic material optimization as in the FMO solver. The results are shown in Table 1 and a visualization in Fig. 2. Although the error compared to the exact FMO result increases heavily with the size of the inclusions, this is largely due to the decrease of the overall compliance value. The absolute value does not increase much from the largest ellipses to the squared FE patches.

Compliance values and FMO comparison for increasing ellipse size.

\(a=b\): | 0.02 | 0.04 | 0.05 | squared patch | ||||
---|---|---|---|---|---|---|---|---|

\(\mathcal {C}_{\theta ,s=0}\) | 15.420 | 1.0 % | 11.053 | 6.15 % | 8.3173 | 14.2 % | 4.3730 | 42.3 % |

\(\mathcal {C}_{\theta ,s}\) | 15.332 | 0.42 % | 10.738 | 3.12 % | 7.7140 | 5.93 % | 3.5668 | 16.1 % |

\(\mathcal {C}_\text {Eng}\) | 15.328 | 0.39 % | 10.647 | 2.25 % | 7.6135 | 4.55 % | 3.5014 | 13.9 % |

\(\mathcal {C}_\text {FMO}\) | 15.289 | 0.14 % | 10.551 | 1.33 % | 7.4840 | 2.77 % | 3.2912 | 7.10 % |

FMO | 15.268 | 10.413 | 7.2821 | 3.0730 |

Compliance values and FMO comparison for 50x50 squared FE patches.

\(\mathcal {C}_{\theta ,s=0}\) | 3.6850 | 38.89 % |

\(\mathcal {C}_{\theta ,s}\) | 2.9963 | 12.93 % |

\(\mathcal {C}_\text {Eng}\) | 3.0535 | 15.08 % |

\(\mathcal {C}_\text {FMO}\) | 2.8964 | 9.16 % |

FMO | 2.6533 |

For the squared FE patches, we furthermore study the different parametrizations separating the domain into \(50\times 50\) patches. The results are found in Table 2 and Fig. 3(a). We can see, that for the parametrization with nonlinear subproblems \(\mathcal {C}_\text {Eng}\), apparently a local minimum was found, which stresses the importance of a global solution of the local material optimization problems.

### 2.4 Material and Topology Optimization

In Fig. 3(b), the result of the algorithm for the admissible material set \(\mathcal {C}_{\theta ,s}\) and \(50\times 50\) squared FE patches is shown. In this experiment, we removed in the iteration \(k\) a total of \(200*0.83^k\) FE patches until \(3\) FE patches or less where removed, which lead to a final volume fraction of \(0.542\). The strategy for the removal of ellipses can be varied, however a decreasing volume fraction should be removed in order to obtain a smoother convergence.

## 3 Conclusion

We proposed an efficient algorithm for material optimization on multiple elliptic inclusions. The numerical evidence suggests that the accuracy of the proposed method decreases only slightly, when replacing all material instead of just the material in elliptic inclusions. A major advantage of this algorithm is the possibility of avoiding local minima, however at the cost of only having an approximate solution. The total error in the studied example for the finer resolution was about 10 %. From experience in practice, the proposed algorithm for simultaneous material and topology optimization seems to work well in large parts, however oftentimes small bars are left over and if a hole happens to be drilled in a “bad” position the algorithm struggles, as material is not reintroduced.

Nevertheless, both algorithms allow for a very efficient solution of usually quite complicated problems, such as discrete material optimization and rotational optimization. When the accuracy provided in this method does not suffice or the optimized topology appears flawed, the optimization result may still be used as a high quality initial design for other solution schemes, such as fully parametrized approaches.

## Footnotes

- 1.
We note that the asymptotic formulae are also available for the three-dimensional case.

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