A FEM cluster-based basis reduction method for shakedown analysis of heterogeneous materials

Shakedown analysis with Melan’s theorem is an important approach to predicting the ultimate load-bearing capacity of heterogeneous materials under varying loads. However, this approach entails dealing with a large-scale nonlinear mathematical programming problem with numerous element-wise yielding constraints and unknown time-independent beneficial residual stress variables, resulting in a substantial computational burden. The well-known basis reduction method expresses the unknown time-independent beneficial residual stress as a linear combination of a set of self-equilibrium stress (SES) bases, and the corresponding coefficients are the unknowns. This method is effective only if the set of SES basis vectors is small and easily available. Based on the representative volume element (RVE) and FEM-cluster based analysis (FCA) method, this paper proposes a FEM cluster-based basis reduction method to fast predict the shakedown domain of heterogeneous materials. The novel data-driven clustering method is introduced to divide the RVE into several clusters. The SES basis is constructed by applying the cluster eigenstrain to RVE under periodic boundary conditions. Numerical experiments show that the unknown time-independent beneficial residual stress can be well represented with this small set of SES basis vectors. In this way, the unknown variables are reduced dramatically. In addition, to further reduce the number of nonlinear constraints, a constraint reduction strategy based on the reduced-order model of FCA is implemented to remove the element-wise yielding constraints for the elements far from yielding. Several numerical examples demonstrate its efficiency and accuracy.