TOTAL LAGRANGIAN MATERIAL POINT METHOD FOR THE DYNAMIC ANALYSIS OF NEARLY INCOMPRESSIBLE SOFT MATERIALS
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摘要: 物质点法(MPM)在模拟非线性动力问题时具有很好的效果, 其已被广泛应用于许多大变形动力问题的分析中. 然而传统的MPM在模拟不可压或近似不可压材料的动力学行为时会产生体积自锁, 极大地影响模拟精度和收敛性. 本文针对近似不可压软材料的大变形动力学行为, 提出一种混合格式的显式完全拉格朗日物质点法(TLMPM). 首先基于近似不可压软材料的体积部分应变能密度, 引入关于静水压力的方程; 之后将该方程与动量方程基于显式物质点法框架进行离散, 并采用完全拉格朗日格式消除物质点跨网格产生的误差, 提升大变形问题的模拟精度; 对位移和压强场采用不同阶次的B样条插值函数并通过引入针对体积变形的重映射技术改进了算法, 提升算法的准确性. 此外, 算法通过实施一种交错求解格式在每个时间步对位移场和压强场依次进行求解. 最后, 给出几个典型数值算例来验证本文所提出的混合格式TLMPM的有效性和准确性, 计算结果表明该方法可以有效处理体积自锁, 准确地模拟近似不可压软材料的大变形动力学行为.
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关键词:
- 完全拉格朗日物质点法 /
- 软材料 /
- 体积自锁 /
- 大变形 /
- 显式动力学
Abstract: The material point method (MPM) shows good performance in modeling nonlinear dynamic problems and has been widely used to simulate various types of large deformation dynamic problems. However, the classical MPM may suffer from the volumetric locking when modeling the dynamic responses of the incompressible or nearly incompressible materials, which reduces the computational accuracy and affects the convergence behavior greatly. In this work, a displacement-pressure mixed total Lagrangian material point method (TLMPM) with explicit time integration is proposed for the large deformation dynamic behavior of nearly incompressible soft materials. In this method, an equation about the hydrostatic pressure is introduced based on the volumetric part of the strain energy density of nearly incompressible soft materials. Then the introduced equation as well as the momentum equation is discretized within the framework of the explicit MPM and the total Lagrangian formulation is implemented to overcome the cell-crossing noise, which increases the computational accuracy for the problems involving large deformation. Furthermore, the B-spline interpolation functions with different orders are applied for displacement and pressure fields respectively and the mixed TLMPM is improved to increase the accuracy by introducing a remapping technique for the volumetric deformation. In addition, the staggered solving scheme is adopted and the displacement and the pressure are required to be solved sequentially in a single time step. Finally, several typical numerical examples are simulated by the mixed TLMPM and the convergence and accuracy are analyzed. The results demonstrate that the proposed mixed TLMPM is able to deal with the volumetric locking effectively and simulate the dynamic behavior of nearly incompressible soft materials involving large deformation accurately. -
图 2 库克膜: 采用不同数量物质点离散的t = 3 s时右上角竖向位移模拟结果
Figure 2. Cook’s membrane: simulation results of the vertical displacement of the top right corner at t = 3 s using different numbers of material points
图 3 库克膜: t = 3 s时的静水压力分布
Figure 3. Cook’s membrane: hydrostatic pressure distribution at t = 3 s
图 5 二维软梁的弯曲: A点x方向位移变化曲线
Figure 5. Bending of 2D soft beam: history curves of x-displacement of point A
图 6 二维软梁的弯曲: 不同时刻的静水压力分布. (a)~(b) 标准格式线性插值; (c)~(d) 标准格式二次插值; (e)~(f) 线性−常数混合格式; (g)~(h) 二次−线性混合
Figure 6. Bending of 2D soft beam: hydrostatic distribution at different times. (a)~(b) Standard linear interpolation; (c)~(d) standard quadratic interpolation; (e)~(f) linear-constant mixed formulation and (g)~(h) quadratic-linear mixed formulation
图 7 三维柱体扭转: 构型及静水压力分布. (a) 几何模型; (b) t = 0.1 s, 标准TLMPM; (c) t = 0.3 s, 标准TLMPM; (d) t = 0.1 s, 混合 TLMPM; (e) t = 0.3 s, 混合 TLMPM
Figure 7. Twisting of a 3D column: configurations and distribution of hydrostatic pressure. (a) Geometry; (b) t = 0.1 s, standard TLMPM; (c) t = 0.3 s, standard TLMPM; (d) t = 0.1 s, mixed TLMPM and (e) t = 0.3 s, mixed TLMPM
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