強沖擊問題的物質點有限元法

張雄 廉艷平 劉巖

摘 要：材料和結構在強沖擊載荷作用下表現出很強的非線性，出現超大變形、斷裂、破碎，甚至出現相變、熔化、氣化等現象，對數值模擬分析提出了巨大挑戰，有限元法等基于網格的傳統數值分析方法不易有效地求解此類問題。物質點法中使用一組拉格朗日質點和一套歐拉背景網格，質點用以離散物質區域，攜帶質量、速度、應力、能量等物理量，背景網格用以計算空間導數及求解動量方程。物質點法兼具拉格朗日格式和歐拉格式的優點，不會出現網格畸變問題，易于跟蹤歷史變量和物質界面，非常適合求解強沖擊問題。但物質點法在小變形問題中的精度和效率均低于顯式有限元法，鋼筋混凝土沖擊等問題中各組成部分的特征尺寸差別較大，也對物質點離散提出了挑戰。該報告介紹了研究組針對上述物質點法的不足，所發展的針對強沖擊問題的物質點法與有限元法的貫通結合方法。該報告首先扼要介紹了物質點法的基本理論，指出物質點法與有限元法在理論上的相似性，給出兩者相互結合的理論基礎。之后依次闡述了耦合物質點有限元法、自適應物質點有限元法和雜交物質點有限元法，詳細論述了這些方法的基本思想和理論，給出了這些方法用于侵徹等強沖擊問題的實例，顯示出其相對于標準物質點法的精度與效率優勢。耦合物質點有限元法的基本思想是采用有限元法離散小變形物體、用物質點法離散大變形物體，不同離散區域之間通過接觸算法相互耦合。報告詳細介紹了耦合過程的處理，主要包括接觸探測、接觸法線計算、接觸力計算以及考慮兩者接觸情況下的時間積分。自適應物質點有限元法的基本思想是初始時采用有限元離散全部區域，在計算過程中將可能發生畸變或破壞的單元自動轉化為物質點求解。報告詳細介紹了實現單元到質點的自動轉化的轉化算法，包括轉化判據和轉化方案，以及不同離散區域間的相互作用與相互接觸的處理。雜交物質點有限元法主要針對鋼筋混凝土沖擊等問題，其基本思想是采用桿單元離散鋼筋，采用質點離散混凝土。報告詳細介紹了如何通過背景網格實現不同離散格式相互作用和變形協調，討論了鋼筋失效的模擬方案。上述方法充分發揮了物質點法和有限元法各自的優勢，克服了其不足，較之標準物質點法和傳統有限元法能夠更有效地模擬強沖擊載荷問題。通過這些方法的研究，建立了物質點有限元貫通框架，從理論上推動了物質點法和有限元法的深化研究，為相關設計分析工作提供了強有力的數值手段，對工程問題的高效解決具有重要應用價值。

關鍵詞：強沖擊載荷 物質點法 有限元法 耦合 自適應轉化 鋼筋混凝土

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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