Nonlocal interactions raise numerical challenges such as high computational cost and geometric complexity. In the context of continuum mechanics, this thesis studies numerical methods for two nonlocal problems: weak form peridynamics and nanoscale strain engineering.

Unlike the classical local theory, the weak formulation of peridynamics involves a double integral and the additional integral operator needs an efficient quadrature rule. For this reason, the thesis investigates convergence behaviours of a promising quadrature rule based on Generalized Moving Least Squares (GMLS) when applied to the double integral. For uniform discretizations, second-order convergence is observed with a mesh extension for global symmetrical inner quadrature. For non-uniform discretizations, a proposed strategy for symmetrically placing inner quadrature points shows decaying second-order convergence, while increasing the number of outer quadrature points leads to a more persistent convergence behaviour. Numerical tests in 1D demonstrate the above properties and 2D tests show consistent behaviours.

Nanoscale strain engineering aims at tuning the electronic properties of a semiconductor by modulating its nanoscale stain field, and the nonlocal interaction through Van der Waals forces is a possible mechanism for the modulation. To better understand the interaction process, based on the Lennard-Jones (LJ) model, the thesis builds a continuum model to simulate nonlocal interactions between a monolayer MoS2 and a multihole Si3N4 substrate. A low-dimensional model is first built as a proof of concept before considering the real problem. The monolayer MoS2 is then modeled by a Kirchhoff–Love shell while the integration of LJ potential over the substrate is approximated by a Riemann sum in a finite range and optimized using octrees. An alternative approach based on a semi-infinite integral is proposed for the integration over curved substrates, as a preliminary study for future work.