强冲击问题的物质点有限元法

张雄 廉艳平 刘岩

摘 要：材料和结构在强冲击载荷作用下表现出很强的非线性，出现超大变形、断裂、破碎，甚至出现相变、熔化、气化等现象，对数值模拟分析提出了巨大挑战，有限元法等基于网格的传统数值分析方法不易有效地求解此类问题。物质点法中使用一组拉格朗日质点和一套欧拉背景网格，质点用以离散物质区域，携带质量、速度、应力、能量等物理量，背景网格用以计算空间导数及求解动量方程。物质点法兼具拉格朗日格式和欧拉格式的优点，不会出现网格畸变问题，易于跟踪历史变量和物质界面，非常适合求解强冲击问题。但物质点法在小变形问题中的精度和效率均低于显式有限元法，钢筋混凝土冲击等问题中各组成部分的特征尺寸差别较大，也对物质点离散提出了挑战。该报告介绍了研究组针对上述物质点法的不足，所发展的针对强冲击问题的物质点法与有限元法的贯通结合方法。该报告首先扼要介绍了物质点法的基本理论，指出物质点法与有限元法在理论上的相似性，给出两者相互结合的理论基础。之后依次阐述了耦合物质点有限元法、自适应物质点有限元法和杂交物质点有限元法，详细论述了这些方法的基本思想和理论，给出了这些方法用于侵彻等强冲击问题的实例，显示出其相对于标准物质点法的精度与效率优势。耦合物质点有限元法的基本思想是采用有限元法离散小变形物体、用物质点法离散大变形物体，不同离散区域之间通过接触算法相互耦合。报告详细介绍了耦合过程的处理，主要包括接触探测、接触法线计算、接触力计算以及考虑两者接触情况下的时间积分。自适应物质点有限元法的基本思想是初始时采用有限元离散全部区域，在计算过程中将可能发生畸变或破坏的单元自动转化为物质点求解。报告详细介绍了实现单元到质点的自动转化的转化算法，包括转化判据和转化方案，以及不同离散区域间的相互作用与相互接触的处理。杂交物质点有限元法主要针对钢筋混凝土冲击等问题，其基本思想是采用杆单元离散钢筋，采用质点离散混凝土。报告详细介绍了如何通过背景网格实现不同离散格式相互作用和变形协调，讨论了钢筋失效的模拟方案。上述方法充分发挥了物质点法和有限元法各自的优势，克服了其不足，较之标准物质点法和传统有限元法能够更有效地模拟强冲击载荷问题。通过这些方法的研究，建立了物质点有限元贯通框架，从理论上推动了物质点法和有限元法的深化研究，为相关设计分析工作提供了强有力的数值手段，对工程问题的高效解决具有重要应用价值。

关键词：强冲击载荷 物质点法 有限元法 耦合 自适应转化 钢筋混凝土

Finite Element Material Point Method for Intensive Impact Loading

Zhang Xiong Lian Yanping Liu Yan

（Tsinghua University）

Abstract：Materials and structures show strong nonlinearities under intensive impact loading， which pose great challenges on numerical analysis. It is not an easy task for mesh-based methods such as finite element method （FEM） to solve such problems effectively. The material point method （MPM） has both the advantages of Lagrangian method and Eulerian method. No mesh distortion exists， and history variables and the material interface can be easily traced in MPM. So MPM is very appropriate for intensive impact problems. But the accuracy and the efficiency of MPM for small deformation problems are lower than those of explicit FEM. The characteristic lengths of different components in the reinforced concrete （RC） problems are very different， which also poses challenges on MPM. The combined finite element material point method， which is proposed by the research group， is introduced in this report. MPM theory is briefly investigated， and the similarity between MPM and FEM is the foundation of the combination methods. The coupled finite element material point method （CFEMP）， the adaptive finite element material point method （AFEMP）， and the hybrid finite element material point method （HFEMP） are introduced sequentially. The basic ideas and the theories of the above methods are elaborated， and numerical examples of intensive impact problems such as perforation are given. The results show advantages in accuracy and efficiency over standard MPM. CFEMP employs FEM for small deformation objects and MPM for large deformation objects. Different discretization regions are coupled through contact algorithms. The coupling process is explained in detail. AFEMP employs FEM for the whole domain initially and automatically converts the elements before distortion or failure to material points. The conversion algorithm is introduced thoroughly， and interactions and contacts between different discretization regions are stated as well. HFEMP focuses on RC problems. The bar elements are used for the reinforcement and the material points are used for the concrete. The interaction between different discretization and the deformation consistency are realized through the background mesh. The above methods have both the advantages of FEM and MPM and overcome their shortcomings， so they are more effective in simulating intensive impact problems. A unified framework of FEM and MPM is established. The above methods are significant to design and analysis of practical problems.

Key Words：Intensive impact loading； Material point method； Finite element method； Coupling； Adaptive conversion； Reinforced concrete

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