[论文解读] A stabilized finite element formulation of non-smooth contact

Ghadir Haikal|ArXiv.org|Jun 2, 2009

Contact Mechanics and Variational Inequalities参考文献 86被引用 3

一句话总结

本文提出了一种基于节点到面间隙函数和基于间断伽辽金（DG）的稳定化方法的稳定有限元格式，用于非光滑接触问题，以确保界面间压力传递的准确性。该方法能有效处理尖锐拐角和非线性表面，通过拉格朗日乘子法保证能量守恒，并实现超弹性与接触问题中非协调网格的稳健耦合。

ABSTRACT

The computational modeling of many engineering problems using the Finite Element method involves the modeling of two or more bodies that meet through an interface. The interface can be physical, as in multi-physics and contact problems, or purely numerical, as in the coupling of non-conforming meshes. The most critical part of the modeling process is to ensure geometric compatibility and a complete transfer of surface tractions between the different components at the connecting interfaces. Popular contact modeling techniques rely on geometric projections to detect and resolve overlapping or mass interpenetration between two or more contacting bodies. Such approaches have been shown to have two major drawbacks: they are not suitable for contact at highly nonlinear surfaces and sharp corners where smooth normal projections are not feasible, and they fail to guarantee a complete and accurate transfer of pressure across the interface. This dissertation presents a novel formulation for the modeling of contact problems that possesses the ability to resolve complicated contact scenarios effectively, while being simpler to implement and more widely applicable than currently available methods. We show that the formulation boils down to a node-to-surface gap function that works effectively for non-smooth contact. The numerical implementation using the midpoint rule shows the need to guarantee the conservation of the total energy during impact, for which a Lagrange multiplier method is used. We propose a local enrichment of the interface and a simple stabilization procedure based on the discontinuous Galerkin method to guarantee an accurate transfer of the pressure field. The result is a robust interface formulation for contact problems and the coupling of non-conforming meshes.

研究动机与目标

• 解决传统接触方法依赖光滑法向投影而失效于尖锐拐角和高度非线性表面的问题。
• 开发一种鲁棒的界面格式，能够准确地在非光滑和非协调界面上传递牵引力。
• 通过在时间积分中采用拉格朗日乘子法，确保在碰撞过程中能量守恒。
• 为超弹性问题中非协调网格的耦合提供稳定且一致的格式。
• 实现复杂接触场景（包括滑动和分离）的有效模拟，提升数值稳定性和精度。

提出的方法

• 基于节点到面间隙函数，采用有向体积方法表述接触约束，避免依赖光滑法向投影。
• 在时间积分中采用中点法，并结合拉格朗日乘子法以在碰撞事件中实现能量守恒。
• 引入界面的局部增强，以提高接触边界处的逼近质量。
• 应用基于间断伽辽金（DG）的稳定化过程，确保界面间压力传递的一致性和准确性。
• 利用从接触约束函数推导出的雅可比矩阵和海森矩阵，实现牛顿-拉夫森求解方法中的一致线性化。
• 采用多点约束（MPC）方法实现非协调网格的耦合，并通过片体试验和收敛性研究进行验证。

实验结果

研究问题

• RQ1 如何在传统法向投影失效的尖锐拐角和高度非线性表面处，有效表述接触约束？
• RQ2 何种稳定化技术可确保在非协调或非光滑界面间实现准确且一致的表面牵引力传递？
• RQ3 在动态接触模拟中，如何保证碰撞过程中的能量守恒？
• RQ4 节点到面间隙函数能否与DG稳定化有效结合，以提升接触和耦合问题的鲁棒性？
• RQ5 该格式在超弹性材料中非协调网格耦合的收敛性和一致性方面表现如何？

主要发现

• 所提出的节点到面间隙函数在非光滑几何形状（包括尖锐拐角）处成功实现了接触，而传统基于投影的方法在此类情况下失效。
• 基于DG的稳定化确保了界面间压力传递的准确性和一致性，避免了虚假振荡并提升了解的质量。
• 通过在中点法时间积分格式中引入拉格朗日乘子法，实现了碰撞过程中的能量守恒。
• 该格式通过了片体试验，并在数值算例中表现出最优收敛率，证实了其一致性和稳定性。
• 该方法在滑动接触和复杂三维接触场景（如双悬臂梁试验）中表现出色，具有鲁棒的收敛性和准确的应力传递。
• 接触约束的线性化产生对称的海森矩阵，支持高效且鲁棒的牛顿-拉夫森求解过程。