The computation of bounds for linear-functional outputs of weak solutions to the two-dimensional elasticity equations

We present a method for the computation of upper and lower bounds for linear-functional outputs of the exact solutions to the two dimensional elasticity equations. The method can be regarded as a generalization of the well known complementary energy principle. The desired output is cast as the supremum of a quadratic-linear convex functional over an infinite dimensional domain. Using duality the computation of an upper bound for the output of interest is reduced to a feasibility problem for the complementary, or dual, problem. In order to make the problem tractable from a computational perspective two additional relaxations that preserve the bounding properties are introduced. First, the domain is triangulated and a domain decomposition strategy is used to generate a sequence of independent problems to be solved over each triangle. The Lagrange multipliers enforcing continuity are approximated using piecewise linear functions over the edges of the triangulation. Second, the solution of the adjoint problem is approximated over the triangulation using a standard Galerkin finite element approach. A lower bound for the output of interest is computed by repeating the process for the negative of the output. Reversing the sign of the computed upper bound for the negative of the output yields a lower bound for the actual output. The method can be easily generalized to three dimensions. However, a constructive proof for the existence of feasible solutions for the outputs of interest is only given in two dimensions. The computed bound gaps are found to converge optimally, that is, at the same rate as the finite element approximation. An attractive feature of the proposed approach is that it allows for a data set to be generated that can be used to certify and document the computed bounds. Using this data set and a simple algorithm, the correctness of the computed bounds can be established without recourse to the original code used to compute them. In the present paper, only computational domains whose boundary is made up of straight sided segments and polynomially varying loads are considered. Two examples are given to illustrate the proposed methodology.