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 $[0,t_1]$. 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.

本文关注基于无网格的降阶模型 (ROM) 到具有非均匀体积约束的时间相关非局部模型的实现。通常，当对非局部模型使用 ROM 时，需要在每个时间步计算非局部体积约束的投影以处理非局部边界条件。到目前为止，只有有限元方法 (FEM) 才能很好地为非局部模型构建 ROM，因为 FEM 基函数的插值特性可以很容易地获得这样的投影。但是如果尝试基于现有的非局部模型的无网格方法开发ROM，每个时间步的投影将导致全阶离散系统并且非常耗时，因为这些方法的基函数不满足插值性质. 为克服上述困难，我们引入了具有节点插值特性的混合再生核 (RK) 近似来开发非局部模型的无网格搭配方法，并用它来构建 ROM。由于节点插值特性，可以明确获得非局部边界条件的投影。该 ROM 是使用数值结果作为全阶模型在小时间间隔内的快照开发的$[0,{吨}_1]$. 通过POD（适当正交分解）-Galerkin方法构建的代理模型导致离散系统的自由度远低于原始无网格方法。提出了包括非局部扩散和近场动力学在内的非局部问题的数值实验，以表明与全阶无网格方法相比，我们的方法以非常小的计算成本满足几乎相同的精度。