We introduce an immersed meshfree formulation for modeling heterogeneous materials with flexible non-body-fitted discretizations, approximations, and quadrature rules. The interfacial compatibility condition is imposed by a volumetric constraint, which avoids a tedious contour integral for complex material geometry. The proposed immersed approach is formulated under a variational multiscale based formulation, termed the variational multiscale immersed method (VMIM). Under this framework, the solution approximation on either the foreground or the background can be decoupled into coarse-scale and fine-scale in the variational equations, where the fine-scale approximation represents a correction to the residual of the coarse-scale equations. The resulting fine-scale solution leads to a residual-based stabilization in the VMIM discrete equations. The employment of reproducing kernel (RK) approximation for the coarse- and fine-scale variables allows arbitrary order of continuity in the approximation, which is particularly advantageous for modeling heterogeneous materials. The effectiveness of VMIM is demonstrated with several numerical examples, showing accuracy, stability, and discretization efficiency of the proposed method.

我们引入了一种无网格沉浸式公式，用于通过灵活的非人体离散，近似和正交规则对异构材料进行建模。界面相容性条件是由体积约束强加的，它避免了复杂材料几何形状的繁琐轮廓轮廓。拟议的沉浸方法是在基于变分多尺度沉浸法的基础上制定的，称为变分多尺度沉浸法（VMIM）。在此框架下，前景或背景上的解近似值可以在变分方程中解耦为粗尺度和精细尺度，其中精细尺度近似表示对粗尺度方程残差的校正。所得的精细解决方案导致VMIM离散方程中基于残差的稳定化。对粗略和精细尺度变量使用再生核（RK）近似值可以使近似值具有任意连续性，这对于建模异质材料特别有利。通过几个数值示例证明了VMIM的有效性，显示了所提方法的准确性，稳定性和离散化效率。