A unified contact overlap, termed the Minkowski overlap, between any two shapes is proposed in this article. This overlap is based on the concept of the Minkowski difference of two shapes, and particularly on the equivalence between the contact state of the two shapes and the location of the origin relative to their Minkowski difference. The Minkowski contact features of a contact, including the overlap, normal direction, and contact points, are also defined for convex shapes. In particular, an important property of the Minkowski overlap is introduced which lays the solid theoretical foundation for proposing a Minkowski overlap based energy-conserving contact model in the current work. The energy-conserving property for cases where the contact normal direction and point may be subject to discrete changes is also rigorously proved. For convex particles, the computational procedures combining both GJK and EPA algorithms are outlined, and uniqueness and ambiguity issues associated with some special cases are clarified and resolved. The elastic energy conservation of the proposed contact model for convex shapes in elastic impact is further verified using two numerical examples, and two more examples involving more convex particles with different sizes and shapes are also conducted to demonstrate the robustness and applicability of the proposed Minkowski overlap contact model and the computational procedures.

中文翻译：

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用于凸非球形粒子离散元建模的 Minkowski 重叠和能量守恒接触模型

本文提出了任何两个形状之间的统一接触重叠，称为 Minkowski 重叠。这种重叠基于两个形状的 Minkowski 差异的概念，特别是基于两个形状的接触状态和原点相对于它们的 Minkowski 差异的位置之间的等价性。接触的 Minkowski 接触特征，包括重叠、法线方向和接触点，也定义为凸形。特别是，介绍了闵可夫斯基重叠的一个重要性质，为在当前工作中提出基于闵可夫斯基重叠的节能接触模型奠定了坚实的理论基础。对于接触法线方向和点可能受到离散变化的情况的能量守恒性质也得到了严格的证明。对于凸粒子，概述了结合 GJK 和 EPA 算法的计算过程，并澄清和解决了与一些特殊情况相关的唯一性和二义性问题。使用两个数值例子进一步验证了弹性冲击中凸形接触模型的弹性能量守恒，另外两个例子还涉及更多不同尺寸和形状的凸粒子，以证明所提出的 Minkowski 重叠的鲁棒性和适用性接触模型和计算程序。