An Adaptive and Efficient Boundary Approach for Density-Based Topology Optimization

Abstract

This paper presents an adaptive nodal boundary condition scheme to systematically enhance the computational efficiency and circumvent numerical instabilities of the finite element analysis in density-based topology optimization problems. The approach revisits the idea originally proposed by Bruns and Tortorelli to eliminate the contribution of void elements from the finite element model and extends this idea to modern projection methods to stabilize the implementation, facilitate reintroduction of material, and consider additional physics. The computational domain is discretized on a fixed finite element mesh and a threshold density is used to determine if an element is sufficiently low relative density to be “removed” from the finite element analysis. By eliminating low-density elements from the design domain, the number of free Degrees-Of-Freedom (DOFs) is reduced, thereby reducing the solution cost of the finite element equations. Perhaps more importantly, it circumvents numerical instabilities such as element distortion when considering large deformations. Unlike traditional solids-only modeling approaches, a key feature of the projection-based scheme is that the design and finite element spaces are separate, allowing the design variable sensitivities in a region to remain active (and potentially non-zero) even if the corresponding analysis elements are removed from the finite element model. This ultimately means material reintroduction is systematic and driven by the design sensitivities. The Solid Isotropic Material with Penalization (SIMP) approach is used to interpolate material properties and the Heaviside Projection Method (HPM) is used to regularize the optimization problem and facilitate material reintroduction through the gradient-based optimizer. Several benchmark examples in areas of linear and nonlinear structural mechanics are presented to demonstrate the performance of the proposed approach. The resulting optimized designs are consistent with literature and results reveal the performance and efficiency of the developed method in reducing computational costs without numerical instabilities known to be due to modeling near-void elements.