Structural rigidity optimization with frictionless unilateral contact

The problem of maximization the global rigidity (measured by the compliance) of an elastic structure with frictionless unilateral contact is considered in the framework of topology optimization. The frictionless unilateral contact is introduced in the continuous formulation of the elastic problem (under the assumption of small strains and small displacements) in the regularized form of an interface with an asymmetric behavior law relating the normal component of the stress vector transmitted through the contact surface to the normal displacement (in the case of contact with a rigid foundation) or the jump of normal displacement (in the case of internal contact of two surfaces of the elastic medium). Using the concept of homogeneous thermodynamical potentials, we extend a convergent and numerically efficient optimization algorithm introduced in the framework of linear elasticity to this nonlinear case of an elastic structure with unilateral contact. Numerical examples in two-dimensional elasticity are presented.