Locking materials and the topology of optimal shapes

Abstract: We study a problem of structural optimization using the fictitious material approach.  This is connected with the equilibrium of locking materials, which can be approximated by strongly non linear materials. A finite elements simulation allows us to experiment some conjectures about the topology of the optimal solutions.
 

Keywords: shape optimization, locking material, topological optimization, fictitious material, finite elements.

Introduction:

Structural optimization is a major objective in the conception of industrial systems. In partonception of industrial systems. In particular, for a given desired performance, engineers may need to minimize the mass of a structure by using an adapted geometry. First methods consist of progressively tuning some geometric parameters of a postulated initial shape. The optimal design obtained in this way depends strongly
on the initial one : the optimization process cannot change the fundamental topological characteristics of the shape. Methods which allow such modifications (Allaire et al., 1997), (Bendsoe, 1995) are called topological optimization.  Here we consider a structure submitted to a given load and we try to get a structure with a smallest volume for a desired compliance (i.e.  global stiffness, or stored elastic energy for the given load). It is now well known that this problem may admit no solution in the classical sense:  the optimal "shape" may consist of an intricate mixture of material
and holes.  The optimal solution has to be understood in the framework of homogenization theory and the optimization problem has to be set at the very beginning in a relaxed form. However, as the set of all possible effective stiffness tensors for a given volume fraction of the material is rather intricate (Allaire and Khon, 1993), (Aubry, 1999), (Bendsoe, 1995), it seems difficult to find analytic solutions or mathematical properties for optimal designs.  That is why we consider a very similar but much simpler problem : w simpler problem : we assume that the stiffness tensor depends linearly on the volume fraction (or the mass density) of the material. In 3-D this assumption is not realistic as the stiffness of a composite material with a given volume fraction is weaker than the stiffness obtained assuming a linear dependence: the problem we consider is then called "fictitious material optimization". In 2-D, the problem corresponds to the optimization of the thickness of a plate submitted to a plane load (plane stress).
The optimal designs obtained using homogenization theory or fictitious approach are qualitatively similar.  Then we hope that our results could be extended to the optimal structures obtained by the homogenization method.
We show that the problem of fictitious material optimization is equivalent to the equilibrium problem of a perfect locking material (i.e. a material whose strain tensor must belong to a given bounded set, in that case its internal energy density vanishes). This enables us to give non-trivial analytical solutions (section 4.1). We show that a regularization of this last problem leads to the resolution of a simple non-linear elasticity problem (section 3). Some 2-D numerical solutions are presented (section 5) and compared to analytical solutions. The numerical solutions enable us to test some conjectures about the topology of optimal solutions: it seems that the support of the optimal design is a simply-connegn is a simply-connected bounded set. In particular, we can prove that circular holes
cannot be present in an optimal design.

This conjecture, if confirmed, is remarkable : the topological complexity is not introduced by the optimization process, but only by a later penalization process (such a penalization is generally used to get out of the relaxed formulation and to obtain a classical solution (Allaire et al., 1997).  Only this penalization process justifies the name of ``topological optimization" for the whole process.