廉艳平. 自适应物质点有限元法及其在冲击侵彻问题中的应用. 清华大学博士学位论文，2012.5

开展冲击侵彻问题的数值计算方法研究是计算力学领域的一项重要课题,在 国防军事和公共安全中有广泛而重要的应用。由于涉及应力波传播、高应变率、 大变形、摩擦和磨损等多种物理现象,冲击侵彻问题往往非常复杂。相比于拉格 朗日和欧拉算法,近年来发展的无网格法在处理该类问题时更具有优势。物质点 法(MPM)是一种无网格法,采用质点均匀离散材料区域,用背景网格计算动量方 程和空间导数,解决了有限元法的网格畸变问题,也避免了欧拉法的缺点。本文 主要针对冲击侵彻问题,基于MPM,研究高效稳定的数值计算方法。
针对MPM在模拟钢筋混凝土结构时离散规模庞大的问题,本文提出了杂交物 质点有限元法(HFEMP)。在 HFEMP中,将有限元法的杆单元引入物质点法中, 构造了离散钢筋的杆单元,很好地解决了这一问题。 HFEMP采用质点离散混凝 土,用杆单元离散钢筋,避免了在钢筋直径方向上的离散,降低了MPM的离散规 模。采用该算法模拟了弹体侵彻钢筋混凝土的问题,结果表明钢筋可以提高混凝 土靶体的抗侵彻性能。
在冲击侵彻问题中,大部分材料区域始终处于小变形阶段,而物质点法在计 算材料小变形时的精度和效率均低于有限元法。针对该问题,本文提出了耦合 物质点有限元法(CFEMP)。在CFEMP中,采用有限元离散仅涉及材料小变形的物 体,用质点离散涉及材料特大变形的物体,通过接触算法实现两个离散体之间的 相互作用。数值计算表明该算法的计算精度和效率高于物质点法,适合模拟冲击 侵彻和流固耦合等问题。
考虑到材料从小变形到特大变形的时间历程,为了充分发挥有限元法计算材 料小变形时效率和精度的优势以及物质点法模拟材料特大变形的优势,本文进一 步提出了自适应物质点有限元法(AFEMP)。在 AFEMP中,采用有限元离散所有 物体,在计算过程中将发生畸变或破坏的单元自动转化为质点,通过背景网格实 现同一个物体内不同离散区域之间的相互作用,采用CFEMP计算各物体之间的接 触问题,实现了有限元法到物质点法求解的自适应转化。数值计算表明,在计算 精度一致的情况下,AFEMP的计算效率显著高于物质点法。
最终,本文为冲击侵彻及其他涉及材料特大变形的问题建立了一类高效稳定 的自适应物质点有限元法。

The numerical method for impact and penetration problems is an important research project in computational mechanics, which has broad and significant applications in the national defense, military and public security affairs. However, the impact and pene- tration problems are often very complicated, due to covering a broad range of physical phenomena, such as stress wave propagation, high strain rate, large strain, friction and abrasion. Compared with the Lagrangian and Eulerian mesh methods, meshless meth- ods developed in recent decades have some advantages to solve such problems. Material point method (MPM) is a meshless method, which discretizes material domain by a set of uniform particles, and uses background grid to solve the momentum equations and calculate spatial derivatives. Therefore, MPM not only eliminates the mesh distortion difficulties in finite element method (FEM), but also avoids the shortcomings of Eulerian methods. In this study, the new efficient and stable numerical methods are researched based on MPM with the aim of solving impact and penetration problems.
When apply MPM for reinforced concrete (RC) structures simulation, the discretiza- tion model is large-scale. To solve this problem, a hybrid finite element material point (HFEMP) method is developed by introducing the bar element into MPM for the steel bars in RC. In HFEMP, the concrete is discretiezed by MPM particles, while the steel bars by bar elements in the length direction without discretization in the diameter direc- tion, which can decrease the scale of the RC discretization model significantly. The RC slab subjected to projectile penetration is studied by HFEMP, which indicates that steel bars can increase the capability of concrete resisting penetration.
In fact, the most part of the material domain is still with the mild deformation in the impact and penetration problems. But for material with mild deformation, both the efficiency and accuracy of MPM are lower than those of FEM. A coupled finite element material point (CFEMP) method is proposed for such problems in this study. CFEMP uses FEM for body with mild deformations and MPM for body with extreme deforma- tions, the interaction between two different discretization bodies is handled by the contact method. The numerical results indicate that CFEMP is more accurate and efficient than MPM, and suitable for impact/penetration and fluid solid interaction problems.
Considering the mild deformation always before extreme deformation, an adaptive finite element material point (AFEMP) method is proposed based on CFEMP to take full advantages of FEM for material mild deformation and MPM for material extreme defor- mation. In AFEMP, all of the bodies are firstly discretized by finite elements, then the distorted elements are automatically converted into MPM particles during the calcula- tion process, the interaction between finite elements domain and MPM particles domain within the same body is implemented based on the background grid, and the contact be- tween bodies is handled by CFEMP. So the conversion from FEM solver to MPM solver is adaptive by conversion scheme in AFEMP. Numerical results indicate that the efficiency of AFEMP is significantly higher than that of MPM.
Finally, a class of adaptive material point finite element methods is developed for impact/penetration problems and problems involving material extreme deformation in this study.