A reduced-order fast reproducing kernel collocation method for nonlocal models with inhomogeneous volume constraints

This paper is concerned with the implementations of the meshfree-based reduced-order model (ROM) to time-dependent nonlocal models with inhomogeneous volume constraints. Generally, when using ROM for nonlocal models, the projection of nonlocal volume constraints needs to be computed in every time step to handle the nonlocal boundary conditions. Up to now, only finite element methods (FEM) can work well in constructing ROM for nonlocal models, since the interpolation property of the FEM basis functions makes it easy to obtain such a projection. But if one tries to develop ROM based on existing meshfree methods for nonlocal models, the projection in every time step will lead to a full-order discrete system and is highly time-consuming, since the basis functions of these methods do not meet interpolation property. To overcome the above difficulties, we introduce a mixed reproducing kernel (RK) approximation with nodal interpolation property to develop a meshfree collocation method for nonlocal models and use it to construct ROM. Thanks to the nodal interpolation property, the projection of nonlocal boundary conditions can be obtained explicitly. This ROM is developed using numerical results as snapshots by a full-order model in a small time interval [0,t₁]. The surrogate model, which is constructed by POD (proper orthogonal decomposition)-Galerkin approach, leads to a discrete system with far fewer degrees of freedom than the original meshfree method. Numerical experiments for nonlocal problems including nonlocal diffusion and peridynamics are presented to show that our method meets almost the same accuracy with a very small computational cost compared with the full-order meshfree approach.