CutFEM topology optimization of 3D laminar incompressible flow problems

Abstract

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.

Acoustic shape optimization using cut finite elements

“…20 More recent studies have concerned multiple materials and inclusions. [21][22][23] There are also publications covering other fields of engineering, such as fluid flow, 24,25 electrostatics, 26 and magnetic actuators. 27 However, to the authors' best knowledge, there does not seem to be any previous study on level-set-based optimization using finite element methods on fixed meshes for wave propagation problems.…”

Cut topology optimization for linear elasticity with coupling to parametric nondesign domain regions

“…In shape and topology optimization CutFEM has previously been applied to problems where the geometry is represented using a level-set that is updated either using an optimization approach or an evolution equation, see [8,13,14,37]. The present work is however the first contribution where CutFEM is combined with a density based topology optimization approach.…”

Shape morphing and topology optimization of fluid channels by explicit boundary tracking

“…Although, in general, providing reliable results, such blurry boundary conditions may, in some cases, have trouble capturing refined flow and boundary effects, and hence, a number of level set–based papers have considered explicit conformal structural meshing for an accurate evaluation of the signed‐distance function and for mechanical analysis . As an alternative to full remeshing, Villanueva and Maute have just recently suggested accurate boundary modeling for fluid problems using a CutFEM approach. The DSC approach for topology and shape optimization of fluid problems suggested in this paper represents an alternative way of imposing accurate boundary conditions, a possibility for topological changes through merging and splitting of holes and including procedures for ensuring well‐formed boundary conforming meshes.…”

“…Ignoring its ability to perform topological changes, the method closely follows and yields similar results to classical shape optimization papers from the literature. However, together with the CutFEM approach, the presented methodology is thus far unique in its ability to perform simultaneous shape and topology optimization of fluid problems.…”

“…In addition, the level set-based topology optimization approach 26 is another powerful tool for shape and topology optimization that has been applied to fluid problems. [27][28][29][30][31] Common for most density-based and level set-based topology optimization approaches based on fixed background meshes is that resulting boundaries between fluid and solid are described by jagged edges and/or elements taking intermediate fluid properties through density interpolations or ersatz material properties. Although, in general, providing reliable results, such blurry boundary conditions may, in some cases, have trouble capturing refined flow and boundary effects, and hence, a number of level set-based papers have considered explicit conformal structural meshing for an accurate evaluation of the signed-distance function 32 and for mechanical analysis.…”