Comparison of high order finite element and discontinuous Galerkin methods for phase field equations: Application to structural damage
Comparison of high order finite element and discontinuous Galerkin methods for phase field equations: Application to structural damage
复制标题
DOI：
10.1016/j.camwa.2017.05.003
复制
发表时间：
2017-10-01
影响因子：
2.9
通讯作者：
Bittencourt, M. L.
中科院分区：
文献类型：
--
作者：
Chiarelli, L. R.;Fumes, F. G.;Bittencourt, M. L.
相场方程用于模拟各种多相问题，如流体分离、凝固、粘指、断裂和疲劳等。在文献中可以找到各种各样的数值求解相场方程的方法。特别是，当需要提高精度时，高阶方法是一种有效的选择。本文第一部分分析了高阶有限元法（FEM）和不连续伽辽金法（DG）在二阶Allen-Cahn （AC）和四阶Cahn-Hilliard （CH）方程求解中的精度和计算效率。对这些方程采用了几种时间积分格式。显式格式是Gottlieb et al.（2011）中描述的正演欧拉、经典四阶龙格-库塔（RK4）和强稳定性保持十阶龙格-库塔（rkssp -10,4）。在全隐式和半隐式格式中采用了后向欧拉和梯形隐式方法，由Eyre（未发表）提出。为了评估误差和比较不同的数值方法，使用了一维问题的人造解。在第二部分中，通过对第一部分工作的先前分析得出的AC方程选择适当的离散化，我们提出了一种数值半隐式方案来求解Boldrini et al.（2016）中描述的损伤和断裂模型。该程序采用有限元法进行空间离散化，采用Newmark法对运动学方程进行时间积分，采用后向欧拉法进行损伤相场演化。最后，给出了断裂相场模型的二维基准试验结果，验证了损伤相场层伽玛在小宽度下收敛为尖锐裂纹。(C) 2017 Elsevier Ltd.版权所有。
Phase field equations are used to model a wide range of multiphase problems such as separation of fluids, solidification, viscous fingering, fracture and fatigue. A wide variety of methods to numerically solve phase field equations can be found in the literature. In particular, high order methods are an effective option when accuracy improvement is desired. In the first part of this work, we analyze the accuracy and computational efficiency of the high order finite element method (FEM) and discontinuous Galerkin (DG) method applied to the second-order Allen-Cahn (AC) and fourth-order Cahn-Hilliard (CH) equations. Several schemes for time integration are used for these equations. The explicit schemes are the forward Euler, classical fourth-order Runge-Kutta (RK4) and the strong stability preserving ten stages fourth-order Runge-Kutta (RKSSP-10,4) described in Gottlieb et al. (2011). The backward Euler and trapezoidal implicit methods are adopted in the full and semi implicit schemes, as proposed in Eyre (unpublished). Manufactured solutions for one dimensional problems are used in order to evaluate the errors and to compare the different numerical methods. By choosing an adequate discretization for AC equations resulting from the previous analysis of the first part of the work, in the second part, we propose a numerical semi implicit scheme to solve the damage and fracture model described in Boldrini et al. (2016). This procedure employs the FEM for spatial discretization, the Newmark method for time integration of the kinematics equation and the backward Euler for the damage phase field evolution. Finally, results for 2D benchmark tests are presented for the fracture phase field model and the convergence to a sharp crack for a small width of the damage phase field layer gamma is verified. (C) 2017 Elsevier Ltd. All rights reserved.