Constrained optimization framework for interface-aware sub-scale dynamics closure model for multimaterial cells in Lagrangian and arbitrary Lagrangian–Eulerian hydrodynamics

Abstract

A systematic description of the new interface-aware sub-scale-dynamics (IA-SSD) closure model for the Lagrangian stage of multimaterial arbitrary Lagrangian–Eulerian methods is presented. The IA-SSD closure model consists of two stages. During the first, bulk, stage, the well known equal compressibility model is used. During the second stage, sub-scale interactions of the materials inside the multimaterial cell are taken into account. At this stage, information about the topology of the materials inside the multimaterial cell is utilized, allowing the orientations of internal interfaces to be included in the model. Each material interacts in a pair-wise fashion with the materials with which it has a common boundary. The interactions are based on the solution of the acoustic Riemann problem between each pair of materials and is limited using physically justified constraints: positivity of volume, positivity of internal energy and controlled rate of pressure relaxation. To determine the values of the limiter coefficients, a constrained-optimization framework is employed using a quadratic objective function with linear constraints. The algorithm guarantees the positivity of the material volume and internal energy as well as the smooth relaxation of the pressure – this allows a significant increase in the robustness of the overall algorithm.The results of comprehensive testing of the new model have been presented for one- and two-dimensional multimaterial Lagrangian hydrodynamics along with representative results for 2D multimaterial arbitrary Lagrangian–Eulerian (ALE) calculations. The numerical tests have shown that in most cases the new IA-SSD closure model produces better results compared to the well known Tipton's closure model.