The material point method (MPM) is often applied to large deformation problems that involve sharp gradients in the solution field. Representative examples in geomechanics are interactions between soils and various “structures” such as foundations, penetrometers, and machines, where the displacement fields exhibit sharp gradients around the soil‐structure interfaces. Such sharp gradients should be captured properly in the MPM discretization to ensure that the numerical solution is sufficiently accurate. In the MPM literature, several types of locally refined discretizations have been developed and used for this purpose. However, these local refinement schemes are not only quite complicated but also restricted to certain types of basis functions or update schemes. In this work, we propose a new MPM formulation, called the mapped MPM, that can efficiently capture sharp gradients with a uniform background grid compatible with every standard MPM basis function and scheme. The mapped MPM is built on the method of auxiliary mapping that reparameterizes the given problem in a different domain whereby sharp gradients become much smoother. Because the reparameterized problem is free of undesirably sharp gradients, it can be well solved with the standard MPM ingredients including a uniform background grid. We verify and demonstrate the mapped MPM through several numerical examples, with particular attention to soil‐structure interaction problems.

中文翻译：

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锐梯度大变形问题的映射质点法及其在土-结构相互作用中的应用

质点法 (MPM) 通常应用于解场中涉及尖锐梯度的大变形问题。地质力学中的代表性例子是土壤与各种“结构”（例如地基、贯入仪和机器）之间的相互作用，其中位移场在土壤-结构界面周围表现出尖锐的梯度。这种尖锐的梯度应该在 MPM 离散化中正确捕获，以确保数值解足够准确。在 MPM 文献中，已经为此目的开发并使用了几种类型的局部细化离散化。然而，这些局部细化方案不仅相当复杂，而且仅限于某些类型的基函数或更新方案。在这项工作中，我们提出了一种新的 MPM 公式，称为映射 MPM，它可以有效地捕获具有与每个标准 MPM 基函数和方案兼容的均匀背景网格的尖锐梯度。映射 MPM 建立在辅助映射方法的基础上，该方法在不同的域中重新参数化给定问题，从而使尖锐的梯度变得更加平滑。由于重新参数化的问题不存在不需要的尖锐梯度，因此可以使用包括均匀背景网格的标准 MPM 成分很好地解决该问题。我们通过几个数值例子验证和演示了映射的 MPM，特别关注土壤-结构相互作用问题。