Principle of cluster minimum complementary energy of FEM-cluster-based reduced order method: fast updating the interaction matrix and predicting effective nonlinear properties of heterogeneous material

The present paper studies the efficient prediction of effective mechanical properties of heterogeneous material by the FCA (FEM-Cluster based reduced order model Analysis) method proposed in Cheng et al. (Comput Methods Appl Mech Eng 348:157–168, 2019). The principle of minimum complementary energy and its cluster form for the RUC subjected to applied uniform eigenstrains and the PHBCs (Periodic Homogeneous Boundary Conditions) are developed. By using the known interaction matrix, an alternative form of the principle of cluster minimum complementary energy is constructed and proved very efficient for updating the interaction matrix and the effective elastic modulus when the material properties of clusters change. Moreover, the proposed principle of cluster minimum complementary energy is applied for the incremental nonlinear analysis of the cluster reduced order model, and thus greatly improves the prediction of nonlinear effective properties of the RUC in online stage computed in Cheng et al. (Comput Methods Appl Mech Eng 348:157–168, 2019). A number of numerical examples illustrate the effectiveness and efficiency of the FCA approach with the proposed principle of cluster minimum complementary energy.