Topology Optimization using CutFEM

Abstract

The demand for better performing structural designs gives rise to the interest in topology optimization. An accurate geometry description is fundamental in the optimization process. The Cut Finite Element Method (CutFEM) can describe the object surface on a fixed grid with an immersed boundary. To this level set method, multiple techniques from the density method can be added. In this thesis project, the performance of 3D topology optimization using CutFEM is tested. A topology optimization model using CutFEM was developed. For the elements on the boundary (Cut Elements), a cut is made to split the element in a solid and fluid part. The Gauss points that represent each part are calculated in order to find the stiffness of the Cut Elements. The performance has been tested by performing finite element analysis with CutFEM. In order to perform topology optimization, the sensitivities are calculated with the adjoint method, a filter was used with Heaviside projection and mapping, and the Method of Moving Asymptotes (MMA) is used in order to find the design of the next iteration. In order to increase the length scale, an optional robust design method was implemented which creates an eroded and dilated design. The performance of topology optimization using CutFEM was tested by optimizing a structure for minimum compliance for a set of loading conditions. This was compared to topology optimization with classical Solid Isotropic Material with Penalization (SIMP) method. Firstly, it was found that Cut Elements are an accurate method to perform a finite element analysis. Next, it was found that topology optimization using CutFEM is able to obtain a better objective function than topology optimization using the SIMP method. The computational costs of the CutFEM method are substancially higher. In 3D topology optimization using CutFEM, the design changes can happen everywhere on the boundary, so that the initial structural design is of less importance than for 2D. Next, it was found that the robust design method works well in increasing the length scale, but the objective function is decreased and the initial design is more important. It is thought that CutFEM could best be used to perform optimization with the initial design computed by the SIMP method. Finally, the CutFEM method has been used in a Navier Stokes fluid solver with Brinkman penalization implementation. It was found that this does not work, and a fluid solver with a hard boundary method is required. It is recommended to implement the CutFEM method in a fluid solver with Nitsches method.