A paradigm for higher-order polygonal elements in finite elasticity using a gradient correction scheme
A paradigm for higher-order polygonal elements in finite elasticity using a gradient correction scheme
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DOI：
10.1016/j.cma.2015.12.025
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发表时间：
2016-07
影响因子：
7.2
通讯作者：
Heng Chi;Cameron Talischi;O. Lopez-Pamies;G. Paulino
中科院分区：
文献类型：
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作者：
Heng Chi;Cameron Talischi;O. Lopez-Pamies;G. Paulino
近年来的研究表明，多边形单元在研究有限变形下的非线性弹性材料中具有巨大的潜力。一方面，这些元素非常适合模拟复杂的微观结构（例如颗粒微观结构和涉及不同长度尺度的微观结构），并结合周期性的边界条件。另一方面，多边形元素被发现是更宽容的大的局部变形比标准的有限元，并产生更准确的结果，在弯曲和剪切。与混合配方，低阶混合多边形元素也被证明是数值稳定的Voronoi型网格上没有任何额外的稳定化处理。然而，多边形单元通常遭受持续的一致性误差下网格加密与常用的数值积分方案。因此，通常会出现非收敛的有限元结果，这严重限制了它们的应用。在这项工作中，一般的梯度校正方案，通过添加一个最小的扰动位移场的梯度恢复多项式的一致性。通过对有限弹性力学中几个边值问题的数值研究，证实了修正格式可以恢复线性和二次位移插值的位移基和混合格式解的最优收敛性.此外，对于混合多边形单元，讨论了压力场近似的各种选择，并数值研究了它们在稳定性和精度方面的性能。我们目前的应用程序中的这些元素的物理为基础的例子，包括填充弹性体的研究与interphasial效果和定性比较与纤维增强弹性体的空化实验。
Recent studies have demonstrated that polygonal elements possess great potential in the study of nonlinear elastic materials under finite deformations. On the one hand, these elements are well suited to model complex microstructures (e.g. particulate microstructures and microstructures involving different length scales) and incorporating periodic boundary conditions. On the other hand, polygonal elements are found to be more tolerant to large localized deformations than the standard finite elements, and to produce more accurate results in bending and shear. With mixed formulations, lower order mixed polygonal elements are also shown to be numerically stable on Voronoi-type meshes without any additional stabilization treatment. However, polygonal elements generally suffer from persistent consistency errors under mesh refinement with the commonly used numerical integration schemes. As a result, non-convergent finite element results typically occur, which severely limit their applications. In this work, a general gradient correction scheme is adopted that restores the polynomial consistency by adding a minimal perturbation to the gradient of the displacement field. With the correction scheme, the recovery of optimal convergence for solutions of displacement-based and mixed formulations with both linear and quadratic displacement interpolants is confirmed by numerical studies of several boundary value problems in finite elasticity. In addition, for mixed polygonal elements, the various choices of the pressure field approximations are discussed, and their performance on stability and accuracy are numerically investigated. We present applications of those elements in physically-based examples including a study of filled elastomers with interphasial effect and a qualitative comparison with cavitation experiments for fiber reinforced elastomers.