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.
In the first research thrust, we study nonlocal problems that are related to peridynamics and nonlocal diffusion and the impact of nonlocality on numerical methods. The addressed problems are new and the proposed solutions are totally novel. The nonlocal research we conducted is beyond the state-of-the-art in the field of numerical mathematics and has a great transformative potential both in mathematical and engineering fields. Peridynamics is a nonlocal extension of continuum mechanics. It has been shown to be an effective modeling paradigm in capturing cracks and failure in structural mechanics applications. Nonlocal diffusion has been used in a wide array of areas which include biological models, image processing, particle systems, coagulation, and phase transition. In the involved formulations, integral operators are used instead of derivatives. For instance, the success of peridynamics is largely due to this integral formulation. A crack is inherently discontinuous, hence, its description with an integral operator is more natural compared to a (weak) derivative based operator. Domain Decomposition Methods (DDMs) play a critical role in modeling and simulation because they allow dividing the problem into smaller pieces. That way, small pieces of a large, realistic simulation can be computed in a significantly shorter time by parallel computing. The basic performance indicator of a DDM, or in general a numerical method, is a single quantity called the condition number. When the condition number is large, the method becomes unstable, the underlying equations are referred as ill-conditioned. The mechanism, usually in the form of numerical methods, designed to rectify ill-conditioning of the underlying equation is called preconditioning, the PI's area of expertise. The bigger goal of this project is to construct effective numerical methods, especially robust and scalable preconditioners, to solve nonlocal problems. We put great emphasis on the basis of designing and constructing such preconditioners. Quantifying the condition number in terms of the involved parameters is a crucial step towards acquiring a fundamental understanding of the underlying equation. Preconditioners are constructed under the guidance of quantifications provided by the conditioning analysis. If the quantifications on which the preconditioner is constructed are not sharp, the preconditioner may have an incorrect design which may lead to an ineffective numerical method. Obtaining sharpness is a challenging open problem we solve. Hence, our study lays the foundation of preconditioner research for nonlocal problems. A more specialized goal is the extension of conventional (local) DDMs to nonlocal ones. Constructing a nonlocal method from a local standpoint is a demanding task because the nature of locality and nonlocality seem inherently different and they often do not reconcile. Construction of conventional DDMs heavily rely on utilizing local boundary conditions, and hence, transferring them to nonlocal theories is not obvious. We go beyond standard numerical mathematics, by using operator theory, we discover a natural way to incorporate local boundary conditions into nonlocal theories. This is a major breakthrough which should enable us to construct nonlocal DDMs. Our natural approach to treating boundary conditions will also attract many engineering groups and will enable them to freely use classical boundary conditions. That way, they will have a more diverse set of experiments to study and compare with classical continuum mechanics which has an established history. In addition, engineering groups would be able to solve bigger and more realistic application problems. Peridynamics is being used in material science, nanotechnology, and aerospace. Hence, our numerical methods will be utilized by these disciplines. In the second research thrust, we develop preconditioners that are robust with respect to contrast size (physical quantities such as viscosity and permeability) and the mesh resolution simultaneously. Addressing contrast size and mesh resolution is vital for realistic simulations as they call for rather large contrasts and small mesh size. We can now solve the high-contrast Stokes equation, for instance, used in computational geomechanics for the study of earth's mantle dynamics. In addition, we can solve the high-contrast biharmonic plate equation used in the modeling of composite materials. For instance, composites are now being used in modern aircrafts by Airbus and Boeing. Since the modeling and simulation capability of composites increase, we address the imminent need for robust preconditioning technology in the computational material science community. Consequently, this allows us to solve a wide family elliptic partial differential equations with variants of the preconditioner. Furthermore, the same preconditioner is used for different discretizations. Therefore, we have accomplished a desirable preconditioner design goal.
