An effective, robust and parallel implementation of an interior point algorithm for limit state optimization

A robust and effective finite element based implementation of lower bound limit state analysis applying an interior point formulation is presented in this paper. The lower bound formulation results in a convex optimization problem consisting of a number of linear constraints from the equilibrium equations and a number of convex non-linear constraints from the yield criteria. The computational robustness has been improved by eliminating a large number of the equilibrium equations a priori leaving only the statical redundant variables as free optimization variables. The elimination of equilibrium equations is based on a optimized numbering of elements and stress variables based on the frontal method approach used in the standard finite element method. The optimized numbering secures sparsity in the formulation. The convex non-linear yield criteria are treated directly in the interior point formulation and calculation of the search gradients takes into account the curvature of the yield criteria. Contrary to the cone based optimization methods the present implementation allows for fully general yield criteria. The optimized numbering secures an effective calculation of the Hessian matrix used in the determination of the search direction in each iteration step, and the formualtion also allows for parallel computation. The implementation has been used in load optimization of reinforced concrete slabs but is fully general. Different examples are treated to benchmark the algorithm with previous work in the field of lower bound optimization problems