This work considers an efficient approach for length scale control in high-resolution three-dimensional level set–based topology optimization. A contrast parameter–based cut element method is employed to ensure a crisp interface representation while working on fixed background hexahedral meshes. This enables the use of efficient multigrid preconditioned Krylov methods in massively parallelized computations. The minimum length scale is controlled using a projection filter without the need for β-continuation. The capabilities of the proposed approach is demonstrated on several numerical examples using more than 62 million hexahedral elements.

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Variable thickness sheet and homogenization-based topology optimization often result in spread-out, non-well-defined solutions that are difficult to interpret or de-homogenize to sensible final designs. By extensive numerical investigations, we demonstrate that such solutions are due to non-uniqueness of solutions or at least very flat minima. Much clearer and better-defined solutions may be obtained by adding a measure of non-void space to the objective function with little if any increase in structural compliance. We discuss various alternatives for cleaning up solutions and propose two efficient approaches which both introduce an auxiliary field to control non-void space: one approach based on a cut element based auxiliary field (hybrid approach) and another approach based on an auxiliary element based field (density approach). At the end, we demonstrate significant qualitative and quantitative improvements in variable thickness sheet and de-homogenization designs resulting from the proposed cleaning schemes.

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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.