SIMP‐ALL: A generalized SIMP method based on the topological derivative concept

Abstract

Summary Topology optimization has emerged in the last years as a promising research field with a wide range of applications. One of the most successful approaches, the SIMP method, is based on regularizing the problem and proposing a penalization interpolation function. In this work, we propose an alternative interpolation function, the SIMP‐ALL method that is based on the topological derivative concept. First, we show the strong relation in plane linear elasticity between the Hashin‐Shtrikman (H‐S) bounds and the topological derivative, providing a new interpretation of the last one. Then, we show that the SIMP‐ALL interpolation remains always in between the H‐S bounds regardless the materials to be interpolated. This result allows us to interpret intermediate values as real microstructures. Finally, we verify numerically this result and we show the convenience of the proposed SIMP‐ALL interpolation for obtaining auto‐penalized optimal design in a wider range of cases. A MATLAB code of the SIMP‐ALL interpolation function is also provided.

Multifrequency Topological Derivative Approach to Inverse Scattering Problems in Attenuating Media

“…The formulas obtained in this paper could be combined with other strategies to define iterative methods. For instance, the reconstructions obtained by the first computation of the topological derivative in R d − could be used as initial guesses for other types of iterative methods, such as Newton-type algorithms [57], level set methods [14,15] or other topological derivative-based approaches [58][59][60][61][62]. Iterative methods that alternate topological derivative computations with regularized Gauss-Newton iterations [63] could also be explored.…”

Inverse homogenization using the topological derivative

“…We recall that in work Ferrer (2019), the proposed SIMP-ALL interpolation for the shear and the bulk modulus were proved to remain in between the HS bounds for any pair of materials at any value of the density. This is μ UB ðρÞ ≥ μ SA ðρÞ ≥ μ LB ðρÞ and κ UB ðρÞ ≥ κ SA ðρÞ ≥ κ LB ðρÞ ∀ρ ∈ ½0; 1:…”

“…For macroscopic problems, a similar interpolation was firstly studied in Amstutz (2011b) and generalized for any Poisson ratio and proved to be in between the HS bound in Ferrer (2019). For all these reasons, we use the SIMP-ALL interpolation method in this work, which is briefly presented in the following.…”

“…Recently, in Ferrer (2019), the topological derivative has succeeded to solve topology optimization problems through a relaxed formulation. This approach proposed on an isotropic interpolation, such as SIMP methodologies, such that the derivative in the void and material part of the domain coincides with the topological derivative.…”

“…This interpolation function is called SIMP-ALL. In addition, the SIMP-ALL interpolation has been proved in Ferrer (2019) to remain in between the Hashin-Shtrikman (HS) bounds ensuring that the intermediate values that may appear can always be interpreted as the homogenization of some micro-structures. The main property of the approach is that the sensitivity of the shapefunction when a topological change is considered now becomes a continuous change and therefore continuous optimization algorithms become available.…”