弹性力学问题的插值型无单元伽辽金比例边界法
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摘要
比例边界法是一种半解析数值方法,在处理应力奇异性问题和无限域问题时十分有效.在改进的插值型移动最小二乘法的框架下将无单元伽辽金法与比例边界法结合,本文首次提出插值型无单元伽辽金比例边界法求解弹性力学问题.该方法在径向具有解析性质,只需计算域边界上用节点进行离散,并且环向上形函数的高阶连续性可以进一步提高计算精度和收敛速度.运用插值型无单元伽辽金比例边界法进行计算时,不需要基本解,也不存在奇异积分问题.改进的插值型移动最小二乘法形函数具有Kronecker delta函数的性质,可以直接施加本质边界条件.此外,改进的插值型移动最小二乘法不仅克服了Lancaster和Salkauskas的插值型移动最小二乘法采用奇异权函数的缺点,而且计算形函数时待定系数比传统的移动最小二乘法少一个.最后给出了数值算例,并验证了所提分析方法的有效性和正确性.
As a newly-developed semi-analytical method,the scaled boundary finite element method is very powerful to deal with singular and unbounded problems.By combining the element-free Galerkin(EFG) method with the scaled boundary method in the frame of improved interpolating moving least-squares(IIMLS) method,an interpolating element-free Galerkin scaled boundary method(IEFG-SBM) is firstly proposed to solve elasticity problems in this paper.In the IEFG-SBM,the solution in the radial direction is obtained analytically and only nodes are required to discretize the boundaries of the computational domain.In addition,higher accuracy and faster convergence are obtained due to the higher continuity of the shape functions in the circumferential direction.The IEFG-SBM does not need the fundamental solution and thus no singular integrations are involved.The shape function of the IIMLS method satisfies the Kronecker delta function property and thus the essential boundary conditions can be imposed directly as in the traditional finite element method.In comparison with the interpolating moving least-squares(IMLS) method proposed by Lancaster and Salkauskas,the key advantage of the IIMLS method is that it does not require singular weight function and thus any weight function used in the MLS approximation can also be applied in the IIMLS method.In addition,there are less unknown coefficients in the IIMLS method than in the conventional moving least-squares(MLS) approximation.Thus fewer nodes are required in the local influence domain and a higher computational accuracy can be reached in the IIMLS-based meshless method.At last,several numerical examples are presented to verify the effectiveness and accuracy of the developed method for the elasticity problem.
As a newly-developed semi-analytical method,the scaled boundary finite element method is very powerful to deal with singular and unbounded problems.By combining the element-free Galerkin(EFG) method with the scaled boundary method in the frame of improved interpolating moving least-squares(IIMLS) method,an interpolating element-free Galerkin scaled boundary method(IEFG-SBM) is firstly proposed to solve elasticity problems in this paper.In the IEFG-SBM,the solution in the radial direction is obtained analytically and only nodes are required to discretize the boundaries of the computational domain.In addition,higher accuracy and faster convergence are obtained due to the higher continuity of the shape functions in the circumferential direction.The IEFG-SBM does not need the fundamental solution and thus no singular integrations are involved.The shape function of the IIMLS method satisfies the Kronecker delta function property and thus the essential boundary conditions can be imposed directly as in the traditional finite element method.In comparison with the interpolating moving least-squares(IMLS) method proposed by Lancaster and Salkauskas,the key advantage of the IIMLS method is that it does not require singular weight function and thus any weight function used in the MLS approximation can also be applied in the IIMLS method.In addition,there are less unknown coefficients in the IIMLS method than in the conventional moving least-squares(MLS) approximation.Thus fewer nodes are required in the local influence domain and a higher computational accuracy can be reached in the IIMLS-based meshless method.At last,several numerical examples are presented to verify the effectiveness and accuracy of the developed method for the elasticity problem.
引文
1 Wolf J P,Song C.The scaled boundary finite-element method-a fundamental solution-less boundary-element method.Comp Methods Appl Mech Eng,2001,190:5551-5568
2 Song C.A matrix function solution for the scaled boundary finite-element equation in statics.Comp Methods Appl Mech Eng,2004,193:2325-2356
3 Zhu C L,Li JB,Lin G.J integral method for calculation of fracture energy of mixed mode inclined cracks based on SBFEM(in Chinese).China Civil Eng J,2011,44:16-22[朱朝磊,李建波,林皋.基于SBFEM任意角度混合型裂纹断裂能计算的J积分方法研究.土木工程学报,2011,44:16-22]
4 Jiang C,Long X Y,Han X,et al.A crack propagation path analysis method of random cracked structure(in Chinese).Chin J Solid Mech,2014,35:30-38[姜潮,龙湘云,韩旭,等.一种随机裂纹结构的裂纹扩展路径分析方法.固体力学学报,2014,35:30-38]
5 Deeks A J,Wolf J P.Semi-analytical solution of Laplace's equation in non-equilibrating unbounded problems.Comp Struct,2003,81:1525-1537
6 Belytschko T,Krongauz Y,Organ D,et al.Meshless methods:An overview and recent developments.Comp Methods Appl Mech Eng,1996,139:3-47
7 Cheng Y M,Peng M J.Boundary element free method for elastodynamics(in Chinese).Sci Sin-Phys Mech Astron,2005,35:435-448[程玉民,彭妙娟.弹性动力学的边界无单元法.中国科学G辑:物理学力学天文学,2005,35:435-448]
8 Belytschko T,Lu Y Y,Gu L.Element-free Galerkin methods.Int J Numer Meth Engng,1994,37:229-256
9 Atluri S N,Zhu T.A new Meshless Local Petrov-Galerkin(MLPG)approach in computational mechanics.Comp Mech,1998,22:117-127
10 He Y,Yang H,Deeks A J.An Element-free Galerkin(EFG)scaled boundary method.Finite Elements Anal Des,2012,62:28-36
11 He Y,Yang H,Deeks A J.An Element-Free Galerkin scaled boundary method for steady-state heat transfer problems.Numer Heat Trans Part B-Fundam,2013,64:199-217
12 Lancaster P,Salkauskas K.Surfaces generated by moving least squares methods.Math Comp,1981,37:141
13 Ren H P,Cheng Y M,Zhang W.An interpolating boundary element free method(IBEFM)for the elasticity problem(in Chinese).Sci Sin-Phys Mech Astron,2010,40:361-369[任红萍,程玉民,张武.弹性力学的插值型边界无单元法.中国科学:物理学力学天文学,2010,40:361-369]
14 Ren H P,Cheng Y M,Zhang W.Researches on the improved interpolating moving least-squares method(in Chinese).Chin J Eng Math,2010,27:1021-1029[任红萍,程玉民,张武.改进的移动最小二乘插值法研究.工程数学学报,2010,27:1021-1029]
15 Wang J F,Sun F X,Cheng Y M.An improved interpolating element-free Galerkin method with a nonsingular weight function for two-dimensional potential problems.Chin Phys B,2012,21:090204
16 Wang J,Wang J,Sun F,et al.An interpolating boundary element-free method with nonsingular weight function for two-dimensional potential problems.Int J Comput Methods,2013,10:1350043
17 Timoshenko S P,Goodier J N.Theory of Elasticity.3th ed.New York:McGraw-Hill Inc.,1970.
18 Civelek M B,Erdogan F.Crack problems for a rectangular plate and an infinite strip.Int J Fract,1982,19:139-159
19 Kitagawa H,Yuuki R.Analysis of arbitrarily shaped crack in a finite plate using conformal mapping.First report:Construction of analysis procedure and its applicability.Trans Jpn Soc Mech Eng,1977,43:4354-4362
20 Tsang D KL,Oyadiji S O.Super singular element method for two-dimensional crack analysis.Proc R Soc A-Math Phys Eng Sci,2008,464:2629-2648
2 Song C.A matrix function solution for the scaled boundary finite-element equation in statics.Comp Methods Appl Mech Eng,2004,193:2325-2356
3 Zhu C L,Li JB,Lin G.J integral method for calculation of fracture energy of mixed mode inclined cracks based on SBFEM(in Chinese).China Civil Eng J,2011,44:16-22[朱朝磊,李建波,林皋.基于SBFEM任意角度混合型裂纹断裂能计算的J积分方法研究.土木工程学报,2011,44:16-22]
4 Jiang C,Long X Y,Han X,et al.A crack propagation path analysis method of random cracked structure(in Chinese).Chin J Solid Mech,2014,35:30-38[姜潮,龙湘云,韩旭,等.一种随机裂纹结构的裂纹扩展路径分析方法.固体力学学报,2014,35:30-38]
5 Deeks A J,Wolf J P.Semi-analytical solution of Laplace's equation in non-equilibrating unbounded problems.Comp Struct,2003,81:1525-1537
6 Belytschko T,Krongauz Y,Organ D,et al.Meshless methods:An overview and recent developments.Comp Methods Appl Mech Eng,1996,139:3-47
7 Cheng Y M,Peng M J.Boundary element free method for elastodynamics(in Chinese).Sci Sin-Phys Mech Astron,2005,35:435-448[程玉民,彭妙娟.弹性动力学的边界无单元法.中国科学G辑:物理学力学天文学,2005,35:435-448]
8 Belytschko T,Lu Y Y,Gu L.Element-free Galerkin methods.Int J Numer Meth Engng,1994,37:229-256
9 Atluri S N,Zhu T.A new Meshless Local Petrov-Galerkin(MLPG)approach in computational mechanics.Comp Mech,1998,22:117-127
10 He Y,Yang H,Deeks A J.An Element-free Galerkin(EFG)scaled boundary method.Finite Elements Anal Des,2012,62:28-36
11 He Y,Yang H,Deeks A J.An Element-Free Galerkin scaled boundary method for steady-state heat transfer problems.Numer Heat Trans Part B-Fundam,2013,64:199-217
12 Lancaster P,Salkauskas K.Surfaces generated by moving least squares methods.Math Comp,1981,37:141
13 Ren H P,Cheng Y M,Zhang W.An interpolating boundary element free method(IBEFM)for the elasticity problem(in Chinese).Sci Sin-Phys Mech Astron,2010,40:361-369[任红萍,程玉民,张武.弹性力学的插值型边界无单元法.中国科学:物理学力学天文学,2010,40:361-369]
14 Ren H P,Cheng Y M,Zhang W.Researches on the improved interpolating moving least-squares method(in Chinese).Chin J Eng Math,2010,27:1021-1029[任红萍,程玉民,张武.改进的移动最小二乘插值法研究.工程数学学报,2010,27:1021-1029]
15 Wang J F,Sun F X,Cheng Y M.An improved interpolating element-free Galerkin method with a nonsingular weight function for two-dimensional potential problems.Chin Phys B,2012,21:090204
16 Wang J,Wang J,Sun F,et al.An interpolating boundary element-free method with nonsingular weight function for two-dimensional potential problems.Int J Comput Methods,2013,10:1350043
17 Timoshenko S P,Goodier J N.Theory of Elasticity.3th ed.New York:McGraw-Hill Inc.,1970.
18 Civelek M B,Erdogan F.Crack problems for a rectangular plate and an infinite strip.Int J Fract,1982,19:139-159
19 Kitagawa H,Yuuki R.Analysis of arbitrarily shaped crack in a finite plate using conformal mapping.First report:Construction of analysis procedure and its applicability.Trans Jpn Soc Mech Eng,1977,43:4354-4362
20 Tsang D KL,Oyadiji S O.Super singular element method for two-dimensional crack analysis.Proc R Soc A-Math Phys Eng Sci,2008,464:2629-2648