Numerical methods for heterogeneity and nonlocality

Louisiana State University, Baton Rouge LA

Investigators

Abstract

The PI proposes two main research thrusts. The first is to develop scalable solvers, in particular, iterative substructuring methods, for integral equation based nonlocal (NL) problems such as peridynamics (PD). As a component in the first thrust, heterogeneity is incorporated to study composite materials which is of utmost importance to numerous applications in material science and structural mechanics. Within the first thrust, robustness of the solvers with respect to heterogeneity and multiscale finite element discretizations are the subsequent directions to pursue. The second research thrust concentrates on preconditioning for partial differential equation (PDE) based (local) problems with rough coefficients. The robustness aspect strongly connects the first research thrust to the second. Since the impact of nonlocality on solvers has never been studied before, the first research thrust is unique, transformative, and has great potential to create a solver subfield: nonlocal domain decomposition methods (DDM). Solver research has the potential to reveal multiscale implications associated to NL modeling. The PI proved fundamental results indicating that the weak formulation of PD gives rise to conditioning bounds that are independent of the mesh size, meaning that the length scale is carried by the horizon instead of the mesh size. The study of composite materials requires solvers that are robust with respect to heterogeneity as well as discretizations supporting multiscale features and nonlinearities. For robustness, the PI will capitalize on his existing preconditioning technology for (local) PDE based problems. The second research thrust calls for a qualitative understanding of the PDE operators and their dependence on the coefficients because such understanding is essential for designing preconditioners. This process draws heavily upon effective utilization of theoretical tools such as methods in operator theory. The resulting control of the behaviour of the operators should allow the detection of the main features that provide a basis for the construction of robust preconditioners. After discretization, singular perturbation analysis (SPA) is used to detect and exploit algebraic features such as low-rank perturbations and decoupling of solution parts to construct computationally more feasible preconditioners. With the insights provided by operator theory and SPA, one acquires control of the effectiveness and computational feasibility simultaneously. Scalable and robust solver technologies will create a great impact on modeling and simulation capabilities in material science and structural mechanics, the two vital fields that would maintain the nation's leadership in the aerospace industry. There is imminent need for effective numerical methods in these fields as composite materials have become industry standard. For instance, Airbus and Boeing heavily use light weight composite materials in modern aircrafts. NL models, especially PD, have become increasingly useful for multiscale material modeling as well. The effectiveness of PD has been established in sophisticated nanoscience applications such as fracture and failure of composites, nanofiber networks, and polycrystal fracture. In addition, the prediction of crack paths has been successfully modeled by PD. Furthermore, NL modeling has been used in abundant applications which include fracture of solids, stress fields at dislocation cores and cracks tips, microscale heat transfer, and fluid flow in microscale channels. There are other fields important to national interest where NL models are critically needed for the effective modeling and simulation of complex phenomena. Examples include evolution equations for species population densities, image processing, porous media flow, and turbulence.