Automated formulation and resolution of limit analysis problems

Abstract

The present manuscript presents a framework for automating the formulation and resolution of limit analysis problems in a very general manner. This framework relies on FEniCS domain-specific language and the representation of material strength criteria and their corresponding support function in the conic programming setting. Various choices of finite element discretization, including discontinuous Galerkin interpolations, are offered by FEniCS, enabling to formulate lower bound equilibrium elements or upper bound elements including discontinuities for instance. The numerical resolution of the corresponding optimization problem is carried out by the interior-point solver Mosek which takes advantage of the conic representation for yield criteria. Through various illustrative examples ranging from classical continuum limit analysis problems to generalized mechanical models such as plates, shells, strain gradient or Cosserat continua, we show that limit analysis problems can be formulated using only a few lines of code, discretized in a very simple manner and solved extremely efficiently. This paper is accompanied by a FEniCS toolbox implementing the above-mentioned framework.