This paper studies the characteristics and applicability of the CutFEM approach [1] as the core of a robust topology optimization framework for 3D laminar incompressible flow and species transport problems at low Reynolds number (Re < 200). CutFEM is a methodology for discretizing partial differential equations on complex geometries by immersed boundary techniques. In this study, the geometry of the fluid domain is described by an explicit level set method, where the parameters of a level set function are defined as functions of the optimization variables. The fluid behavior is modeled by the incompressible Navier-Stokes equations. Species transport is modeled by an advection-diffusion equation. The governing equations are discretized in space by a generalized extended finite element method. Face-oriented ghost-penalty terms are added for stability reasons and to improve the conditioning of the system. The boundary conditions are enforced weakly via Nitsche's method. The emergence of isolated volumes of fluid surrounded by solid during the optimization process leads to a singular analysis problem. An auxiliary indicator field is modeled to identify these volumes and to impose a constraint on the average pressure. Numerical results for 3D, steady-state and transient problems demonstrate that the CutFEM analyses are sufficiently accurate, and the optimized designs agree well with results from prior studies solved in 2D or by density approaches.

As the capabilities of additive manufacturing techniques increase, topology optimization provides a promising approach to design geometrically sophisticated structures which can be directly manufactured. Traditional topology optimization methods aim at finding the conceptual design but often lack a sufficient resolution of the geometry and structural response, needed to directly use the optimized design for manufacturing. To overcome these limitations, this paper studies the viability and characteristics of the eXtended Finite Element Method (XFEM) in combination with the LevelSet Method (LSM) for topology optimization of three dimensional structural design problems. The LSM describes the geometry by defining the nodal level set values via explicit functions of the optimization variables. The structural response is predicted by a generalized version of the XFEM. The LSM-XFEM approach is compared against results from a traditional Solid Isotropic Material with Penalization (SIMP) method for two-phase "solid-void" and "solid-solid" problems. The numerical results demonstrate that the LSM-XFEM approach can describe crisply the geometry and predict the structural response of complex three-dimensional structures with acceptable accuracy even on coarse meshes. However, the LSM-XFEM studied here lacks a robust and intuitive formulation to control the minimum feature size, and the optimization results may depend on the initial design.

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