A Class of Preconditioners for Static Elastic Crack Problems Modeled by Extended Finite Element Method

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

Abstract

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

CLC Number:

O357.41

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.

Figures/Tables 14

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

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

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.)

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.)

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

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

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

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

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

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

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

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

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

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

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