一类求解含静态裂缝线弹性问题的预条件扩展有限元方法

doi:10.19596/j.cnki.1001-246x.8683

摘要/Abstract

摘要：

基于几何型扩展有限元离散方法, 研究含静态裂缝线弹性问题的高效区域分解预条件算法。为了构造Schwarz型预条件算法, 采用一种特殊的裂尖型区域分解策略, 将计算区域分解为包含所有分支增强自由度的裂尖子区域和仅包含标准有限元自由度与Heaviside增强自由度的常规子区域。基于该区域分解策略, 推导一类高效的乘性和限制型乘性Schwarz区域分解预条件子, 对裂尖子问题进行精确求解, 而对常规子问题则非精确求解。数值实验验证了算法的有效性。

关键词: 扩展有限元算法, 区域分解, 预条件子, 静态裂缝问题

Abstract:

This paper mainly discusses some effective domain decomposition preconditioners for static elastic crack problems modeled by geometrical extended finite element method. To construct the Schwarz type preconditioners, we adopt a special crack-tip domain decomposition strategy. The finite element mesh is decomposed into "crack tip" subdomains, which contain all the degrees of freedom (DOFs) of the branch enrichment functions, and "regular" subdomains, which contain the standard DOFs and the DOFs of the Heaviside enrichment functions. Based on the crack-tip domain decomposition strategy, an effective class of multiplicative and restrict multiplicative Schwarz preconditioners are derived. In the preconditioners, the crack tip subproblems are solved exactly and the regular subproblems are solved by some inexact solvers. Numerical experiments demonstrate the effectiveness of the preconditioners.

Key words: extended finite element method, domain decomposition, preconditioners, static crack problem

中图分类号:

O357.41

范鹤潇, 陈星玎. 一类求解含静态裂缝线弹性问题的预条件扩展有限元方法[J]. 计算物理, 2024, 41(2): 151-160.

Hexiao FAN, Xingding CHEN. A Class of Preconditioners for Static Elastic Crack Problems Modeled by Extended Finite Element Method[J]. Chinese Journal of Computational Physics, 2024, 41(2): 151-160.

图/表 14

图1 含裂缝的线弹性体(Ω代表线弹性体区域，Γ代表线弹性体边界，Γc代表裂缝。)

Fig.1 Linear elastic body with a crack (The domain Ω with boundary Γ and a crack Γc.)

图2 结构化网格中的裂缝(圆点和方点分属裂缝增强节点集合SH和裂尖增强节点集合SC。)

Fig.2 Crack located on a structured mesh (Nodes in sets SH and SC are denoted by circles and squares, respectively.)

图3 裂缝型区域分解(红线代表裂缝，蓝色子区域代表裂缝子区域，其余部分为好子区域。)

Fig.3 Schematic representation of two subdomains in the "crack line" domain decomposition (The red line denotes cracks, the blue subdomain denotes the "cracked" domain, the other is the "healthy" domain.)

图4 裂尖型区域分解(蓝色和黑色子区域分别代表裂尖子区域和常规子区域。)

Fig.4 Schematic representation of subdomains in the "crack tip" domain decomposition(The blue subdomain denotes the "crack tip" domain, the black subdomains denote "regular" domains.)

图5 单轴拉伸边界裂缝问题的初始几何图形

Fig.5 The initial geometry for the uniaxial tensile boundary crack problem

图6 几何型XFEM示意图(圆点和方点分属裂缝增强节点集合SH和裂尖增强节点集合SG，红线代表裂缝线。) (a) 网格剖分为19×19；(b) 网格剖分为39×39

Fig.6 The geometric XFEM (The red line denotes the crack, nodes in sets SH and SG are denoted by circles and squares, respectively.) (a) the mesh scale is 19×19; (b) the mesh scale is 39×39

图7 两子区域问题示意图(红线代表裂缝线，蓝色区域为裂尖子区域。)

Fig.7 Schematic representation of two subdomains in the domain decomposition (The red line denotes cracks, the blue subdomain denotes the "crack tip" domain.)

表1 两子区域问题的总刚度矩阵条件数(δ=2，CondNPre、CondRAS、CondRMS分别代表无预条件子，采用RAS预条件和采用RMS预条件后总刚度矩阵的条件数。)

Table 1 The condition number of the stiffness matrix for the two subdomains preconditioner (δ=2, CondNPre, CondRAS, CondRMS denotes condition number of the stiffness matrix with no preconditioner, with RAS preconditioner and with RMS preconditioner respectively.)

网格剖分	Cond_NPre	Cond_RAS	Cond_RMS
19×19	3.08×10⁶	7.40×10³	8.92×10³
39×39	1.62×10⁹	3.10×10⁶	3.76×10⁶
59×59	3.05×10¹⁰	4.22×10⁷	5.05×10⁷
79×79	2.13×10¹¹	2.19×10⁸	2.66×10⁸
99×99	1.43×10¹²	6.46×10⁸	1.20×10⁹
119×119	4.83×10¹²	3.10×10⁹	3.74×10⁹

表2 两子区域问题迭代步数(NonPre代表无预条件子，E代表在常规子区域精确求解。)

Table 2 Number of iterations for two subdomains preconditioner (NonPre denotes no preconditioner, E denotes exact solution in the "regular" domain.)

网格剖分	NonPre	EAS	ERAS	EMS	ERMS
19×19	594	8	8	4	4
39×39	2 437	10	11	5	6
59×59	2 657	13	14	7	7
79×79	3 106	15	15	8	8
99×99	3 651	16	17	8	9
119×119	3 659	18	18	9	10

表3 两子区域问题迭代步数(在常规子区域采用ILU非精确求解，调降容差是10-3。)

Table 3 Number of iterations for two subdomains preconditioner (Inexact solution by using ILU decomposition with a 10-3 drop tolerance in the "regular" domain.)

网格剖分	AS	RAS	MS	RMS
19×19	13	11	7	8
39×39	19	19	13	14
59×59	25	25	19	20
79×79	37	38	26	26
99×99	55	56	45	47
119×119	84	86	57	60

表4 采用不同预条件子的迭代步数(常规子区域为4×4个，精确求解，δ=1。)

Table 4 Number of iterations for different preconditioners with a 4×4 patition, δ=1

网格剖分	NonPre	EAS	ERAS	EMS	ERMS
39×39	2 437	31	26	27	24
59×59	2 657	38	33	34	29
79×79	3 106	43	37	38	35
99×99	3 651	51	41	42	38
119×119	3 659	56	46	48	42
139×139	3 599	63	50	53	46
159×159	3 622	69	54	58	49

表5 采用不同预条件子的迭代步数(常规子区域为4×4个，精确求解，δ=2。)

Table 5 Number of iterations for different preconditioners with a 4×4 patition, δ=2

网格剖分	EAS	ERAS	EMS	ERMS
39×39	28	22	24	20
59×59	30	26	26	24
79×79	35	29	30	27
99×99	39	33	34	30
119×119	42	36	37	34
139×139	47	39	39	35
159×159	52	41	43	38

表6 不同区域分解下不同预条件的迭代步数(δ=2，常规子区域精确求解。)

Table 6 Number of iterations for different preconditioners with different patitions (δ=2, exact solutions in "regular" domains.)

网格剖分	4×4			6×6			8×8
网格剖分	ERAS	EMS	ERMS	ERAS	EMS	ERMS	ERAS	EMS	ERMS
39×39	22	24	20	26	29	23	30	29	29
59×59	26	26	24	32	35	28	37	41	34
79×79	29	30	27	36	41	35	41	47	37
99×99	33	34	30	41	50	38	47	57	42
119×119	36	37	34	44	48	41	53	63	48
139×139	39	39	35	48	55	45	58	71	54
159×159	41	43	38	55	59	51	63	72	58

表7 不同区域分解下预条件GMRES迭代步数(δ=2，常规子区域采用ILU非精确求解。)

Table 7 Number of iterations for different preconditioners with different patitions (δ=2, inexact solutions using ILU decomposition with a 10-3 drop tolerance in the "regular" domain.)

网格剖分	4×4			6×6			8×8
网格剖分	RAS	MS	RMS	RAS	MS	RMS	RAS	MS	RMS
39×39	24	26	22	27	29	24	32	29	26
59×59	34	36	30	40	39	34	41	42	38
79×79	47	47	43	49	48	45	50	52	46
99×99	57	59	54	56	62	54	58	64	55
119×119	77	75	72	74	73	69	75	75	70
139×139	85	84	82	86	83	81	86	87	82
159×159	109	109	106	115	117	110	104	107	101

参考文献 22

1	高英俊, 罗志荣, 邓芊芊, 等. 韧性材料的微裂纹扩展与分叉的晶体相场模拟[J]. 计算物理, 2014, 31 (4): 471- 478.
2	黄朝琴, 姚军, 王月英, 等. 基于离散裂缝模型的裂缝性油藏注水开发数值模拟[J]. 计算物理, 2011, 28 (1): 41- 49.
3	李开泰, 黄艾香, 黄庆怀. 有限元方法及其应用[M]. 北京: 科学出版社, 2005: 1- 160.
4	BELYTSCHKO T , BLACK T . Elastic crack growth in finite elements with minimal remeshing[J]. International Journal for Numerical Methods in Engineering, 1999, 45 (5): 601- 620. DOI
5	庄茁, 柳占立, 成斌斌, 等. 扩展有限单元法[M]. 北京: 清华大学出版社, 2012: 35- 47.
6	杨璞, 牛红攀, 肖世富. 平面应变问题广义有限元法[J]. 计算物理, 2016, 33 (3): 358- 366.
7	BABUŠKA I , BANERJEE U . Stable generalized finite element method (SGFEM)[J]. Computer Methods in Applied Mechanics & Engineering, 2011, 201 (1): 91- 111.
8	GUPTA V , DUARTE C A , BABUŠKA I , et al. A stable and optimally convergent generalized FEM (SGFEM) for linear elastic fracture mechanics[J]. Computer Methods in Applied Mechanics & Engineering, 2013, 266 (1): 23- 39.
9	ZHANG Q H , BABUŠKA I , BANERJEE U . Robustness in stable generalized finite element methods (SGFEM) applied to Poisson problems with crack singularities[J]. Computer Methods in Applied Mechanics & Engineering, 2016, 311 (1): 476- 502.
10	田荣, 文龙飞. 改进型XFEM进展[J]. 计算力学学报, 2016, 33 (4): 469- 477.
11	ZHANG Q H . DOF-gathering stable generalized finite element methods for crack problems[J]. Numerical Methods for Partial Differential Equations, 2020, 36 (6): 1209- 1233. DOI
12	MENK A , BORDAS S P A . A robust preconditioning technique for the extended finite element method[J]. International Journal for Numerical Methods in Engineering, 2011, 85 (13): 1609- 1632. DOI
13	LANG C , MAKHIJA D , DOOSTAN A , et al. A simple and efficient preconditioning scheme for heaviside enriched XFEM[J]. Computational Mechanics: Solids, Fluids, Fracture Transport Phenomena and Variational Methods, 2014, 54 (5): 1357- 1374.
14	GILBERT S . Computational science and engineering[M]. Wellesley, MA: Wellesley-Cambridge Press, 2007: 551- 597.
15	WAISMAN H , BERGER-VERGIAT L . An adaptive domain decomposition preconditioner for crack propagation problems modeled by XFEM[J]. International Journal for Multiscale Computational Engineering, 2013, 11 (6): 633- 654. DOI
16	BERGER-VERGIAT L , WAISMAN H , HIRIYUR B , et al. Inexact Schwarz-algebraic multigrid preconditioners for crack problems modeled by extended finite element methods[J]. International Journal for Numerical Methods in Engineering, 2012, 90 (3): 311- 328. DOI
17	TIAN R, WEN L F. An extra dof-free and well-conditioned XFEM[C]//Proceedings of the 5th International Conference on Computational Methods, 2014: 784-793.
18	TIAN W , HUANG J J , JIANG Y , et al. A parallel scalable domain decomposition preconditioner for elastic crack simulation using XFEM[J]. International Journal for Numerical Methods in Engineering, 2022, 123 (15): 3393- 3417. DOI
19	GUPTA V , DUARTE C A , BABUŠKA I , et al. Stable GFEM (SGFEM): Improved conditioning and accuracy of GFEM/XFEM for three-dimensional fracture mechanics[J]. Computer Methods in Applied Mechanics and Engineering, 2015, 289, 355- 386. DOI
20	GUPTA V , DUARTE C A , BABUŠKA I , et al. A stable and optimally convergent generalized FEM (SGFEM) for linear elastic fracture mechanics[J]. Computer Methods in Applied Mechanics and Engineering, 2013, 266, 23- 39. DOI
21	SAAD Y . Iterative methods for sparse linear systems[M]. Philadelphia: Society for Industrial and Applied Mathematics, 2000: 157- 218.
22	CAI X C , CHEN X D . Effective two-level domain decomposition preconditioners for elastic crack problems modeled by extended finite element method[J]. Communications in Computational Physics, 2020, 28 (4): 1561- 1584. DOI

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