The simulation of incompressible materials suffers from locking when using the standard finite element method (FEM) and coarse linear tetrahedral meshes. Locking increases as the Poisson ratio ν gets close to 0.5 and often lower Poisson ratio values are used to reduce locking, affecting volume preservation. We propose a novel mixed FEM approach to simulating incompressible solids that alleviates the locking problem for tetrahedra. Our method uses linear shape functions for both displacements and pressure, and adds one scalar per node. It can accommodate nonlinear isotropic materials described by a Young’s modulus and any Poisson ratio value by enforcing a volumetric constitutive law. The most realistic such material is Neo-Hookean, and we focus on adapting it to our method. For ν = 0.5, we can obtain full volume preservation up to any desired numerical accuracy. We show that standard Neo-Hookean simulations using tetrahedra are often locking, which, in turn, affects accuracy. We show that our method gives better results and that our Newton solver is more robust. As an alternative, we propose a dual ascent solver that is simple and has a good convergence rate. We validate these results using numerical experiments and quantitative analysis.

中文翻译：

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防锁定四面体

当使用标准有限元法 (FEM) 和粗线性四面体网格时，不可压缩材料的模拟会出现锁定问题。随着泊松比 ν 接近 0.5，锁定增加，并且通常使用较低的泊松比值来减少锁定，从而影响体积保存。我们提出了一种新的混合有限元方法来模拟不可压缩的固体，以减轻四面体的锁定问题。我们的方法对位移和压力都使用线性形状函数，并为每个节点添加一个标量。它可以通过执行体积本构法来适应由杨氏模量和任何泊松比值描述的非线性各向同性材料。最真实的此类材料是 Neo-Hookean，我们专注于使其适应我们的方法。对于 ν = 0.5，我们可以获得完整的体积保存，达到任何所需的数值精度。我们表明，使用四面体的标准 Neo-Hookean 模拟通常是锁定的，这反过来又会影响准确性。我们展示了我们的方法提供了更好的结果，并且我们的牛顿求解器更加稳健。作为替代方案，我们提出了一种简单且具有良好收敛速度的双上升求解器。我们使用数值实验和定量分析验证了这些结果。