In this paper, we modify the sensitivity analysis of a structural optimization process by using a coupled finite element–element-free Galerkin method. The aim is to improve the sensitivity analysis and to avoid a mesh degradation that can occur when the design variables are perturbed while using classical finite element method. The idea is to replace the finite element mesh by EFGM nodes in areas where the sensitivities of the structural responses have to be computed. The proposed methodology is illustrated by a realistic numerical example for which the finite element method cannot give correct results.
Over the last few years, numerical simulation has become a tool systematically used in the design phase of industrial parts. The design engineers must analyze complex structures in order to meet ever more drastic specifications. Furthermore, the industrial needs now require optimized mechanical designs.
The optimization of structures has been the subject of specific developments, which allow the engineer to choose the design variables (geometric inputs) and the design constraints (stress or displacement value upper limit…) while performing a finite element simulation.
However, the finite element method is marked by some shortcomings such as discontinuous stress field, need of remeshing in case of severe distortion of the mesh, etc. These shortcomings are even more acute when dealing with shape design optimization problems. In this kind of problems, it is essential to obtain accurate structural responses and their derivatives. In a FEM-based iterative process for improving the design, the need of remeshing to avoid loss of accuracy due to distortion often becomes a necessity and represents a burden in terms of computational time.
Design sensitivity analysis is concerned with finding the variation of a structural response due to a variation of some design parameters, describing the geometry of the domain. The sensitivities are needed in a gradient-based optimization process in order to provide the gradients of the objective function (for example, the area or the mass of a structural part) and of the constraint functions (for example, the admissible stress in the structure) [2]. A review paper in this domain is Haftka and Grandhi [1].
Bobaru et al. developed shape sensitivity analysis in the EFG method context [3]. The derivative of the weak form are computed before discretization. This enables to avoid differentiating the EFG shape functions with respect to the design variables.
Moving least-squares approximation (MLSA) is computationally expensive, since for each integration point, a linear system has to be solved. Moreover, a dense integration pattern is necessary to get accurate values since the EFG method makes use of functions that are non rational. Therefore, from the viewpoint of computational time, it is more convenient to use EFG only on the part of the domain where we want to achieve a better approximation of the solution, and to use the FE method for the remaining part of the domain. The coupling between FE and EFG methods has been studied by Belytschko et al. [8] and Hegen [11].
Our research about structural optimization has revealed a main difficulty with the geometry perturbation. Indeed, in order to compute the sensitivity of a structural response with respect to a design variable, the geometry is modified and a new mesh is generated. For most of optimization algorithms, this perturbed mesh must have the same topology as the initial one. This restriction cannot be satisfied if the mesh is generated with the aim of controlling its accuracy. Therefore, we propose to take advantage of the meshless methods. Based on formulations similar to finite element methods, they present the advantage of not requiring a mesh. Hence, they will allow us an easy geometry modification and a more accurate sensitivity analysis. We have demonstrated the applicability of such an approach, first for completely meshless simulations, and then, with the aim of reducing the computational time, for simulations in which the mesh has been deleted in areas where the sensitivities are computed, leaving the finite element mesh elsewhere.
The outline of this paper is as follows: The sensitivity analysis and the usefulness of meshless methods is described in Section 2. Section 3 summarizes the element-free Galerkin method formulation while Section 4 describes a coupled formulation finite element–element-free Galerkin method. A numerical test illustrates our work in Section 5.
The EFG method has been applied where FE method was no longer adequate. We must emphasize the fact that the problem was due to mesh degradation linked to the necessity to keep topology whilst perturbing the design parameters. Indeed, the sensitivity analysis, when carried out with the FE method, is only possible if the original mesh keeps the same topology after perturbing the geometry. Therefore, the user is required to use a transfinite mesher, what often leads to unacceptable mesh distortion. Alternatively, we could have generated free meshes using the mesh perturbator, especially conceived to insure the conservation of the topology after perturbation. The clumsiness of this tool has driven us to study possibilities to exploit meshless methods to carry out the sensitivity analysis.
The main advantage of these methods is to give stress fields with high continuity degree, which allows to compute structural responses for every point in the structure.
