Stress Constrained Optimization using X-FEM and Level Set Description

Topology optimization has experienced an incredible soar since 1988 and is now available within several commercial finite element (FE) codes. Meanwhile, parametric shape optimization has found few industrial applications. This is may be due to its inherent difficulties to deal with mesh management with boundary modifications. Recently the extended finite element method (X-FEM) has been proposed (see [

1

] for a review) as an alternative to remeshing methods. The X-FEM method is naturally associated with the Level Set description of the geometry to provide an efficient treatment of problems involving discontinuities and propagations. Up to now the X-FEM method has been mostly developed for crack propagation problems, but the potential interest of the X-FEM method and the Level Set description for other problems like shape and topology optimization was identified very early (see [

2

]). In this paper, the authors present an intermediate approach between parametric shape and topology optimization by using the X-FEM and Level Set Description. The method benefits from fixed mesh work using X-FEM and from smooth curves representation of the Level Set description. One major characteristic of the approach is to be able to model exactly void and solid structures.

The statement of the optimization problem is similar to classical shape optimization: Design variables are the parameters of basic Level Set features (circles, rectangles, super ellipse, etc.) or could be NURBS control points, while various global (compliance) and local responses can be considered in the formulation. Conversely to shape optimization, structural topology can be modified since basic Level Sets can merge or separate from each other. The sensitivity analysis (related to the compliance and/or the stresses) and the way it can be carried out efficiently is detailed. A special attention is paid to stress constrained problems which are often neglected in other Level Set Methods works.

Numerical applications revisit some classical 2D (academic) benchmarks from shape optimization and illustrate the great interest of using X-FEM and Level Set description. The paper presents the results of minimum compliance and stress constrained problems using the proposed method.