A regularization scheme for explicit level-set XFEM topology optimization

Abstract

Regularization of the level-set (LS) field is a critical part of LS-based topology optimization (TO) approaches. Traditionally this is achieved by advancing the LS field through the solution of a Hamilton-Jacobi equation combined with a reinitialization scheme. This approach, however, may limit the maximum step size and introduces discontinuities in the design process. Alternatively, energy functionals and intermediate LS value penalizations have been proposed. This paper introduces a novel LS regularization approach based on a signed distance field (SDF) which is applicable to explicit LSbased TO. The SDF is obtained using the heat method (HM) and is reconstructed for every design in the optimization process. The governing equations of the HM, as well as the ones describing the physical response of the system of interest, are discretized by the extended finite element method (XFEM). Numerical examples for problems modeled by linear elasticity, nonlinear hyperelasticity and the incompressible Navier-Stokes equations in two and three dimensions are presented to show the applicability of the proposed scheme to a broad range of design optimization problems.

Level set band method: A combination of density-based and level set methods for the topology optimization of continuums

“…The mesh and boundary are difficult to be conformed, and the elements around the boundary are usually cut through by the zero level set. An accurate way to calculate the contribution of the stiffness of "half" elements is using the extended FEM [48][49][50]; another simpler but with lower accuracy way is to use an approximate density to represent the stiffness contribution of the element around the boundary [43]. The latter can be considered a means to interpolate the density of elements falling into the band or cut by the boundaries, as illustrated in Fig.…”

A stabilized parametric level‐set XFEM topology optimization method for thermal‐fluid problem

“…However, the double-well distance potential method and the reinitialization method still suffer from it due to the lack of sufficient control on the global level set field. 54 Based on the level set field shown in Figure 20C,D, it can be observed that the double-well distance potential function and reinitialization procedure do have some control on the level set function. However, the regularization effect seems to be random because some hollows with |∇Φ| = 1 come into being inside the fluid area as shown in Figure 20 and they lead to the reappearance of small holes and islands in the optimized patterns.…”

“…The level set field in Figure 20A clearly shows that the constraint |∇Φ| = 1 is enforced around the structural boundaries while the property |∇Φ| = 0 is obtained in the rest of design area. Other methods like formulating the penalized gradient function, 38 adopting the heat method to construct the signed distance field 54 and enforcing geometric constraints 31 can remedy this issue to some extent. The proposed method inherits the merits of distance potential function and extends its use from the Ersatz material model to the immersed boundary method.…”

Hole seeding in level set topology optimization via density fields

“…To avoid spurious oscillations in the LSF, the regularization scheme of Geiss et al (2019a) is adopted. This approach promotes a uniform spatial gradient of the LSF near the solid-void interface while converging to either a positive or negative target value away from the interface.…”

“…where φBnd is the difference between the upper, φup , and lower, φlow , bounds in the target LSF. In Geiss et al (2019a), the weights w φ and w ∇φ were kept constant in the entire design domain. In the current work, to balance the influence of the regularization components in the vicinity and away from the material interface, these weights are defined as:…”