Generalized Shape Optimization using XFEM and Level Set Description

CAD based shape optimization aims at finding the shapes of internal and external boundaries
of a structural components. The method is able to improve the design of structures against var-
ious criteria such as restricted displacements, stress criteria, eigenfrequencies, etc. However, this
technique has been quite unsuccessful in industrial applications because of the mesh management
problems coming from the large shape modifications. The main technical problems stems from
the sensitivity analysis requiring the calculation of the so-called velocity field related to mesh
modifications. If 2D problems are quite well mastered, 3D solid and shell problems are difficult
to handle in the most general way. It turns out that shape optimization remains generally quite
fragile and delicate to use in industrial context. To circumvent the technical difficulties of the
moving mesh problems, a couple of methods have been proposed such as the fictitious domain
approach, the fixed grid finite elements and the projection methods.
The present work relies on the application of the extended finite element method (X-FEM) to
handle parametric shape optimization. The X-FEM method is naturally associated with the
Level Set description of the geometry to provide an efficient and flexible treatment of problems
involving moving boundaries or discontinuities. On the one hand, the method proposed benefits
from the fixed mesh approach using X-FEM to prevent from mesh management difficulties. On
the other hand, the Level Set description provides a smooth curves representation while being
able to treat topology modifications naturally.
In this thesis, we focus on the material-void and bi-material X-FEM elements for mechanical
structures. The representation of the geometry is realized with a Level Set description. Basic
shapes can be modeled from simple Level Set such as plane, circle, ... NURBS curves and surfaces
that can be combined together using a Constructive Solid Geometry approach to represent com-
plex geometries. The design variables of the optimization problem are the parameters of basic
Level Set features or the NURBS control points. Classical global (compliance, eigenfrequencies,
volume) and local responses (such as stress constraint) can be considered in the optimization
problem that is solved using a mathematical programming approach with the CONLIN optimizer.
The problem of the computation of the shape sensitivity analysis with X-FEM is carefully ad-
dressed and investigated using several original methods based on semi-analytical and analytical
approaches that are developed. Academic examples are first considered to illustrate that the
proposed method is able to tackle accurately shape optimization problems. Then, real life struc-
tures including 2D and 3D complex geometries illustrate the advantages and the drawbacks of
using X-FEM and Level Set description for generalized shape optimization.