A moving superimposed finite element method for structural topology optimization

Abstract

SUMMARYLevel set methods are becoming an attractive design tool in shape and topology optimization for obtaining efficient and lighter structures. In this paper, a dynamic implicit boundary-based moving superimposed finite element method (s-version FEM or S-FEM) is developed for structural topology optimization using the level set methods, in which the variational interior and exterior boundaries are represented by the zero level set. Both a global mesh and an overlaying local mesh are integrated into the moving S-FEM analysis model. A relatively coarse fixed Eulerian mesh consisting of bilinear rectangular elements is used as a global mesh. The local mesh consisting of flexible linear triangular elements is constructed to match the dynamic implicit boundary captured from nodal values of the implicit level set function. In numerical integration using the Gauss quadrature rule, the practical difficulty due to the discontinuities is overcome by the coincidence of the global and local meshes. A double mapping technique is developed to perform the numerical integration for the global and coupling matrices of the overlapped elements with two different co-ordinate systems. An element killing strategy is presented to reduce the total number of degrees of freedom to improve the computational efficiency. A simple constraint handling approach is proposed to perform minimum compliance design with a volume constraint. A physically meaningful and numerically efficient velocity extension method is developed to avoid the complicated PDE solving procedure. The proposed moving S-FEM is applied to structural topology optimization using the level set methods as an effective tool for the numerical analysis of the linear elasticity topology optimization problems. For the classical elasticity problems in the literature, the present S-FEM can achieve numerical results in good agreement with those from the theoretical solutions and/or numerical results from the standard FEM. For the minimum compliance topology optimization problems in structural optimization, the present approach significantly outperforms the well-recognized 'ersatz material' approach as expected in the accuracy of the strain field, numerical stability, and representation fidelity at the expense of increased computational time. It is also shown that the present approach is able to produce structures near the theoretical optimum. It is suggested that the present S-FEM can be a promising tool for shape and topology optimization using the level set methods.

A boundary element and level set based topology optimisation using sensitivity analysis

“…In order to evolve the structural geometry, normal velocities at the boundary need to be calculated using Equation (12). However, a necessary condition for the solution of (12) requires the Lagrange multiplier ℓ to be known in advance.…”

Design parameterization for topology optimization by intersection of an implicit function

“…There are several approaches in the literature to generate finite element (FE) matrices from an implicit function, such as the ersatz material (or density) method [23], smoothed Heaviside function [24], eXtended finite element method [25] or re-meshing to generate a boundary fitted mesh [26]. In this paper the inner-loop optimization problem is solved using a mathematical programming method, thus it is desirable to use a method that enables an explicit link between the design variables and objective and constraint functions, so that analytical gradients can be computed.…”

An isoparametric approach to level set topology optimization using a body-fitted finite-element mesh

“…(3) corresponds to moving the material boundary by a distance of vdt in the outward normal direction from the interface. The time step dt is typically chosen so that the Courant-Friedrichs-Lewy (CFL) condition is satisfied [16]. When the advection velocity is given by the shape sensitivities of the objective function, each HamiltonJacobi update is equivalent to a steepest descent step.…”

“…Compliance minimization is commonly used among researchers of the level set method [3,1,16] to demonstrate new concepts, largely because of the ease with which it can be implemented. As shown by Allaire et al [3], for the compliance objective function (i.e.…”

“…(16). The expression can be simplified by introducing the strain-displacement matrix, B, which is comprised of the set of first derivatives of the element shape functions (Eq.…”