数值流形法中基于适合分析T样条的局部网格加密算法
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摘要
数值流形方法中一般采用有限元网格或规则网格作为其数学覆盖系统,而规则的网格突出的优点是不需要适应求解域的边界和各种不连续面。采用规则的矩形网格作为数值流形方法中的数学网格,并借助适合分析的T样条实现了数值流形方法中的局部加密。适合分析的T样条定义在一个限制的T网格上,其基函数具有线性无关、单位分解、局部加密等许多重要性质,使得其非常适合用于工程设计及分析。当对一个适合分析的T网格加密后,所产生的新的网格往往不再是适合分析的T网格。基于此,提出了一种简单的数学网格加密算法,该算法能保证局部加密后的数学网格仍然是适合分析的。算例结果表明:在应力集中区和裂纹尖端等应力梯度较大区域,该算法均具有较强的适用性。
Generally, a finite element mesh or a regular mesh is used as the mathematical covering system in the numerical manifold method(NMM). The advantage of the regular mesh is that it has no requirement to conform to the boundary of the solution domain and various discontinuities. In this paper, the regular rectangular mesh was used as the mathematical mesh in NMM, and the analysis-suitable T-spline was introduced into NMM to realize the local refinement. As the analysis-suitable T-spline was defined over a mildly restricted T-mesh, it presents many important mathematical properties, such as linear independence, partition of unity and highly localized refinement capability. However, after an analysis-suitable T-mesh was locally refined, the generated new T-spline was not analysis-suitable. Therefore, a simple local refinement algorithm was developed in this paper to make sure that the refined mathematical mesh was still analysis-suitable. Moreover, the results from numerical examples show that the algorithm has strong applicability in large-stress gradient areas such as stress concentration area and crack tips.
Generally, a finite element mesh or a regular mesh is used as the mathematical covering system in the numerical manifold method(NMM). The advantage of the regular mesh is that it has no requirement to conform to the boundary of the solution domain and various discontinuities. In this paper, the regular rectangular mesh was used as the mathematical mesh in NMM, and the analysis-suitable T-spline was introduced into NMM to realize the local refinement. As the analysis-suitable T-spline was defined over a mildly restricted T-mesh, it presents many important mathematical properties, such as linear independence, partition of unity and highly localized refinement capability. However, after an analysis-suitable T-mesh was locally refined, the generated new T-spline was not analysis-suitable. Therefore, a simple local refinement algorithm was developed in this paper to make sure that the refined mathematical mesh was still analysis-suitable. Moreover, the results from numerical examples show that the algorithm has strong applicability in large-stress gradient areas such as stress concentration area and crack tips.
引文
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[10]YANG Y T,ZHENG H.A three-node triangular element fitted to numerical manifold method with continuous nodal stress for crack analysis[J].Engineering Fracture Mechanics,2016,162:51-75.
[11]HE J,LIU Q,WU Z.Creep crack analysis of viscoelastic material by numerical manifold method[J].Engineering Analysis With Boundary Elements,2017,80:72-86.
[12]JIANG Q H,DENG S S,ZHOU C B,et al.Modeling unconfined seepage flow using three-dimensional numerical manifold method[J].Journal of Hydrodynamics,2010,22(4):554-561.
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[14]YANG S K,MA G W,REN X H,et al.Cover refinement of numerical manifold for crack propagation simulation[J].Engineering Analysis with Boundary Elements,2014,43:37-49.
[15]苏海东,祁勇峰.部分重叠覆盖流形法的覆盖加密方法[J].长江科学院院报,2013,30(7):95-100.SU Hai-dong,QI Yong-feng.Cover refinement for numerical manifold method with partially overlapping covers[J].Journal of Yangtze River Scientific Research Institute,2013,30(7):95-100.
[16]苏海东.固定网格的数值流形方法研究[J].力学学报,2011,43(1):169-178.SU Hai-dong.Study on numerical manifold method with fixed meshs[J].Chinese Journal of Theoretical and Applied Mechanics,2011,43(1):169-178.
[17]刘治军.二维结构化网格上的数值流形方法[D].武汉:中国科学院武汉岩土力学研究所,2015.LIU Zhi-jun.Two dimensional numerical manifold method based on structured mesh[D].Wuhan:Institute of Rock and Soil Mechanics,Chinese Academy of Science,2015.
[18]刘治军,郑宏,董苇,等.基于加密物理片的数值流形法中局部网格加密[J].岩土力学,2017,38(4):1211-1217.LIU Zhi-jun,ZHENG Hong,DONG Wei,et al.Local mesh refinement in numerical manifold method based on refined physical patches[J].Rock and Soil Mechanics,2017,38(4):1211-1217.
[19]ZHANG Y L,LIU D X,TAN F.Numerical manifold method based on isogeometric analysis[J].Science China Technological Sciences,2015,58:1520-1532.
[20]张友良,刘登学,刘高敏.数值流形法的T样条局部加密算法[J].岩土力学,2016,37(8):2404-2410.ZHANG You-liang,LIU Deng-xue,LIU Gao-min.T-splines local refinement algorithm for numerical manifold element method[J].Rock and Soil Mechanics,2016,37(8):2404-2410.
[21]SCOTT M A,LI X,SEDERBERG T W,et al.Local refinement of Analysis-suitable T-splines[J].Computer Methods in Applied Mechanics and Engineering,2012,213-216(1-3):206-222.
[22]SEDERBERG T W,ZHENG J,BAKENOV A,et al.T-splines and T-nurccs[J].ACM Transactions on Graphic,2003,22:477-484.
[23]王平.基于层次T网格上样条的等几何分析及其应用[D].合肥:中国科学技术大学,2015.WANG Ping.Isogeometric analysis based on polynomial splines over hierarchical T-meshes and its application[D].Hefei:University of Science and Technology of China,2015.
[24]LI X,ZHENG J,SEDERBERG T W,et al.On linear independence of T-splines blending functions[J].Computer Aided Geometric Design,2012,29(1):63-76.
[25]刘登学,张友良,刘高敏.基于适合分析T样条的高阶数值流形方法[J].力学学报,2017,49(1):212-222.LIU Deng-xue,ZHANG You-liang,LIU Gao-min.Higher-order numerical manifold method based on analysis-suitable T-spline[J].Chinese Journal of Theoretical and Applied Mechanics,2017,49(1):212-222.
[26]COX M G.The numerical evaluation of B-Splines[J].IMA Journal of Applied Mathematics,1972,10:134-149.
[27]CARL D B.On calculating with B-Splines[J].Journal of Approximation Theory,1972,6:50-62.
[28]CHENG Y M,ZHANG Y H.Formulation of a three-dimensional manifold method with tetrahedron elements[J].Rock Mechanics and Rock Engineering,2008,41(4):601-628.
[29]TADA H,PARIS P C,IRWIN G R.The stress analysis of cracks handbook[M].[S.l.]:[s.n.],1985:614.
[30]SUKUMAR N,HUANG Z Y,PREVOST J H,et al.Partition of unity enrichment for bimaterial interface cracks[J].International Journal for Numerical Methods in Engineering,2004,59(8):1075-1102.
[2]石根华,裴觉民.数值流形方法与非连续变形分析[M].北京:清华大学出版社,1997:29-56.SHI Gen-hua,PEI Jue-min.Numerical manifold method and discontinue deformation analysis[M].Beijing:Tsinghua University Press,1997:29-56.
[3]朱爱军.数值流形方法研究及其在岩土工程中的应用[D].重庆:重庆大学,2005.ZHU Ai-jun.Study on numerical manifold method and application in geotechnical engineering[D].Chongqing:Chongqing University,2005.
[4]MA G W,AN X M,ZHANG H H,et al.Modelling complex crack problems using the numerical manifold method[J].International Journal of Fracture,2009,156(1):21-35.
[5]曹文贵,速宝玉.岩体锚固支护的数值流形方法模拟及其应用[J].岩土工程学报,2001,23(5):581-583.CAO Wen-gui,SU Bao-yu.Simulation of numerical manifold method for supports by reinforced bolt and its application[J].Chinese Journal of Geotechnical Engineering,2001,23(5):581-583.
[6]董志宏,邬爱清,丁秀丽.数值流形方法中的锚固支护模拟及初步应用[J].岩石力学与工程学报,2005,24(20):3754-3760.DONG Zhi-hong,WU Ai-qing,DING Xiu-li.Rockbolts simulation by numerical manifold method and its preliminary application[J].Chinese Journal of Rock Mechanics and Engineering,2005,24(20):3754-3760.
[7]姜清辉,王书法.锚固岩体的三维数值流形方法[J].岩石力学与工程学报,2006,25(3):528-532.JIANG Qing-hui,WANG Shu-fa.Three dimensional numerical manifold method simulation of anchor bolt supported rock mass[J].Chinese Journal of Rock Mechanics and Engineering,2006,25(3):528-532.
[8]WU Z J,WONG L N Y.Friction crack initiation and propagation analysis using the numerical manifold method[J].Computers and Geotechnics,2012,39:38-53.
[9]ZHENG H,XU D D.New strategies for some issues of numerical manifold method in simulation of crack propagation[J].International Journal for Numerical Methods in Engineering,2014,97(13):986-1010.
[10]YANG Y T,ZHENG H.A three-node triangular element fitted to numerical manifold method with continuous nodal stress for crack analysis[J].Engineering Fracture Mechanics,2016,162:51-75.
[11]HE J,LIU Q,WU Z.Creep crack analysis of viscoelastic material by numerical manifold method[J].Engineering Analysis With Boundary Elements,2017,80:72-86.
[12]JIANG Q H,DENG S S,ZHOU C B,et al.Modeling unconfined seepage flow using three-dimensional numerical manifold method[J].Journal of Hydrodynamics,2010,22(4):554-561.
[13]WANG Y,HU M S,ZHOU Q L,et al A new second-order numerical manifold method model with an efficient scheme for analyzing free surface flow with inner drains[J].Applied Mathematical Modelling,2016,40(2):1427-1445.
[14]YANG S K,MA G W,REN X H,et al.Cover refinement of numerical manifold for crack propagation simulation[J].Engineering Analysis with Boundary Elements,2014,43:37-49.
[15]苏海东,祁勇峰.部分重叠覆盖流形法的覆盖加密方法[J].长江科学院院报,2013,30(7):95-100.SU Hai-dong,QI Yong-feng.Cover refinement for numerical manifold method with partially overlapping covers[J].Journal of Yangtze River Scientific Research Institute,2013,30(7):95-100.
[16]苏海东.固定网格的数值流形方法研究[J].力学学报,2011,43(1):169-178.SU Hai-dong.Study on numerical manifold method with fixed meshs[J].Chinese Journal of Theoretical and Applied Mechanics,2011,43(1):169-178.
[17]刘治军.二维结构化网格上的数值流形方法[D].武汉:中国科学院武汉岩土力学研究所,2015.LIU Zhi-jun.Two dimensional numerical manifold method based on structured mesh[D].Wuhan:Institute of Rock and Soil Mechanics,Chinese Academy of Science,2015.
[18]刘治军,郑宏,董苇,等.基于加密物理片的数值流形法中局部网格加密[J].岩土力学,2017,38(4):1211-1217.LIU Zhi-jun,ZHENG Hong,DONG Wei,et al.Local mesh refinement in numerical manifold method based on refined physical patches[J].Rock and Soil Mechanics,2017,38(4):1211-1217.
[19]ZHANG Y L,LIU D X,TAN F.Numerical manifold method based on isogeometric analysis[J].Science China Technological Sciences,2015,58:1520-1532.
[20]张友良,刘登学,刘高敏.数值流形法的T样条局部加密算法[J].岩土力学,2016,37(8):2404-2410.ZHANG You-liang,LIU Deng-xue,LIU Gao-min.T-splines local refinement algorithm for numerical manifold element method[J].Rock and Soil Mechanics,2016,37(8):2404-2410.
[21]SCOTT M A,LI X,SEDERBERG T W,et al.Local refinement of Analysis-suitable T-splines[J].Computer Methods in Applied Mechanics and Engineering,2012,213-216(1-3):206-222.
[22]SEDERBERG T W,ZHENG J,BAKENOV A,et al.T-splines and T-nurccs[J].ACM Transactions on Graphic,2003,22:477-484.
[23]王平.基于层次T网格上样条的等几何分析及其应用[D].合肥:中国科学技术大学,2015.WANG Ping.Isogeometric analysis based on polynomial splines over hierarchical T-meshes and its application[D].Hefei:University of Science and Technology of China,2015.
[24]LI X,ZHENG J,SEDERBERG T W,et al.On linear independence of T-splines blending functions[J].Computer Aided Geometric Design,2012,29(1):63-76.
[25]刘登学,张友良,刘高敏.基于适合分析T样条的高阶数值流形方法[J].力学学报,2017,49(1):212-222.LIU Deng-xue,ZHANG You-liang,LIU Gao-min.Higher-order numerical manifold method based on analysis-suitable T-spline[J].Chinese Journal of Theoretical and Applied Mechanics,2017,49(1):212-222.
[26]COX M G.The numerical evaluation of B-Splines[J].IMA Journal of Applied Mathematics,1972,10:134-149.
[27]CARL D B.On calculating with B-Splines[J].Journal of Approximation Theory,1972,6:50-62.
[28]CHENG Y M,ZHANG Y H.Formulation of a three-dimensional manifold method with tetrahedron elements[J].Rock Mechanics and Rock Engineering,2008,41(4):601-628.
[29]TADA H,PARIS P C,IRWIN G R.The stress analysis of cracks handbook[M].[S.l.]:[s.n.],1985:614.
[30]SUKUMAR N,HUANG Z Y,PREVOST J H,et al.Partition of unity enrichment for bimaterial interface cracks[J].International Journal for Numerical Methods in Engineering,2004,59(8):1075-1102.
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