Complementarity Problems in Engineering Mechanics: Models and Solution

A large class of problems in engineering mechanics involves a so-called “complementarity” relationship representing the orthogonality of two sign-constrained vectors. Typical instances are plasticity laws and contact-like conditions. For state problems, the formulation leads to a mixed complementarity problem (MCP) whereas in synthesis (e.g. minimum weight design) or identification problems, a mathematical program with equilibrium constraints (MPEC) is obtained. The aim of this paper is two-fold. Firstly, it describes, through two typical models, how some important engineering mechanics problems can be formulated elegantly and naturally as either an MCP or an MPEC. Secondly, it describes a powerful computer-oriented environment for constructing and solving these mathematical programming problems, with features such as sparsity and automatic differentiation facilities being transparently accessible. This involves the use of the modeling language GAMS (an acronym for General Algebraic Modeling System) and its associated mathematical programming solvers (e.g. the industry standard MCP solver PATH). A simple generic model suitable for solving the state problem for trusses is used to clarify the syntax of GAMS models and to illustrate the ease with which they can be built and solved.