## SIMP vs Level Set Topology Optimization

Topology optimization is a mathematical technique that produces an optimized shape and material distribution for the given design domain, loading, boundary conditions and performance targets (mass, displacement, stress etc.) of the structure. The finite element method is typically used to discretize the structure. The goal is to obtain a binary material distribution in the finite element mesh i.e., an element in the design domain is either a void or fully dense at the end of the optimization. So, topology optimization basically involves the determination of the number, shape and location of the voids/holes and how the fully dense material is connected in the presence of the voids/holes. There are two main approaches to determining the material distribution namely, the material approach and the geometry approach.

In the material approach, the design variable used is an artificial density of the element which alters the stiffness of the element to achieve the goal of either a void or fully dense element. SIMP (Solid Isotropic Material with Penalization) is one of the most widely used material approaches. In practice. some intermediate density elements can be present in the optimized design as explained later.

In the geometry approach, the design variables used are certain scalar functions that implicitly define/update the geometry of the elements to achieve the goal of a void or a fully dense element. The Level Set method is one of the most widely used geometry approaches. When using this approach, a transition zone of intermediate density elements can exist between the solid material and voids. But in general, the Discreteness Index (DI) of the final design can be much higher than that obtained using the material approach.

The two approaches are described briefly in the subsequent sections as a detailed theoretical discussion is beyond the technical scope of this document. In the author’s earlier studies KB0115720 & KB0120099 , stress constrained topology optimization was demonstrated using the SIMP method employing the stress-NORM response and ALM (Augmented Lagrange Method) respectively. Two examples used in the author’s earlier studies namely the  L-Bracket and the Control Arm are revisited in this study using the Level Set method with the stress-NORM response and the results are compared with those obtained using the SIMP method with the stress-NORM response. OptiStruct version 2023 is used for both.

Details are given in the PDF file that is separately attached. Since the animation is not included in the PDF file, it is embedded in this article body with the same title as that used in the article.

Animation of Iso-contour (threshold=0.5) of the element density (slide 9)