We propose to develop scalable solvers for integral equation based nonlocal (NL) problems such as peridynamics (PD). Heterogeneity will also be studied due to utmost importance of composite materials to numerous applications in material science and structural mechanics. Robustness of the solvers with respect to heterogeneity and multiscale finite element discretizations are the subsequent directions to pursue.
Since the impact of nonlocality on solvers has never been studied before, this research initiative is unique, transformative, and has great potential to create a solver subfield: nonlocal domain decomposition methods (DDM). We propose to study both the algorithmic and theoretical aspects of DDM. The solver research has the potential to reveal multiscale implications associated to NL modeling. We recently proved fundamental conditioning results indicating that the weak formulation of PD can be bounded independently of the mesh size, meaning that one can increase the resolution without increasing the condition number.
Scalable and robust solver technologies will create a great impact on simulation capabilities of nonlocal problems at large. In particular, PD will be used for more complex and realistic NL applications because scalable solvers will directly impact the modeling and simulation capability. There is also imminent need for robust preconditioning in the computational material science community as composite materials become industry standard. For instance, Airbus heavily uses light weight composite materials in modern aircrafts.
Fields of science
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