光滑有限元方法研究进展
摘要
光滑有限元法作为一种新型数值方法,由于其良好的性能,受到了国内外学者的广泛关注与研究,但对其梳理总结的较少。文章对光滑有限元方法的类型进行了介绍、归纳,总结了光滑有限元法在各个方面应用的研究现状,并对光滑有限元方法的难点与发展趋势进行了展望:为后续光滑有限元研究工作的推进提供参考。
引文
[1]张永刚.有限元法发展及其应用[J].科技情报开发与经济,2007,17(11):178-179.
[2] 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.
[3] MO?S N,DOLBOW J,BELYTSCHKO T. A finite element method for crack growth without remeshing[J]. International Journal for Numerical Methods in Engineering,1999,46(1):131-150.
[4] CUETO E,SUKUMAR N,CALVO B,et al. Overview and recent advances in natural neighbour Galerkin methods[J]. Archives of Computational Methods in Engineering,2003,10(4):307-384.
[5] LIU G R,ZHANG G Y,DAI K Y,et al. A linearly conforming point interpolation method(lc-pim)for 2d solid mechanics problems[J]. International Journal of Computational Methods,2005,2(4):645-665.
[6] LIU G R,DAI K Y,NGUYEN T T. A Smoothed Finite Element Method for Mechanics Problems[J]. Computational Mechanics,2007,39(6):859-877.
[7] DAI K Y,LIU G R,NGUYEN T T. An n-sided polygonal smoothed finite element method(nSFEM)for solid mechanics[J]. Finite Elem Anal Des,2007,43(11):847–860.
[8] LIU G R,NGUYEN T T,DAI K Y,et al. Theoretical aspects of the smoothed finite element method(SFEM)[J]. International Journal for Numerical Methods in Engineering,2007,71(8):902–930.
[9] DAI K Y,LIU G R. Free and forced vibration analysis using the smoothed finite element method(SFEM)[J]. Journal of Sound&Vibration 2007,301(3):803–820.
[10] NGUYEN T T,LIU G R,DAI K Y,et al. Selective smoothed finite element method[J]. Tsinghua Science and Technology,2007,12(5):497–508.
[11] LIU G R,NGUYEN T T,LAM K Y. A novel alpha finite element method(aFEM)for exact solution to mechanics problems using triangular and tetrahedral elements[J]. Computer Methods in Applied Mechanics&Engineering,2008,197(45):3883–3897.
[12] LIU G R. A generalized Gradient smoothing technique and the smoothed bilinear form for Galerkin formulation of a wide class of computational methods[J]. International Journal of Computational Methods,2008,5(2):199–236.
[13] LIU G R,NGUYEN T T,NGUYEN X H,et al. A node-based smoothed finite element method(NS-FEM)for upper bound solutions to solid mechanics problems[J]. Computers and Structures,2009,87(1):14–26.
[14] LIU G R,NGUYEN T T,LAM K Y. An edge-based smoothed finite element method(ES-FEM)for static,free and forced vibration analyses of solids[J]. Journal of Sound and Vibration,2009,320(4):1100–1130.
[15] NGUYEN T T,LIU G R,LAM K Y,et al. A face-based smoothed finite element method(FS-FEM)for 3D linear and geometrically non-linear solid mechanics problems using 4-node tetrahedral elements[J]. International Journal for Numerical Methods in Engineering,2009,78(3):324-353.
[16] CUI X Y,LIU G R,LI G Y,et al. A smoothed finite element method(SFEM)for linear and geometrically nonlinear analysis of plates and shells[J]. CMES:Computer Modeling in Engineering and Sciences,2008,28(2):109–125.
[17] HE Z C,LIU G R,ZHONG Z H,et al. A coupled edge-/face-based smoothed finite element method for structural acoustic problems[J]. Applied Acoustics,2010,71(10):955–964.
[18] ZHANG Z Q,YAO J,LIU G R. An immersed smoothed finite element method for fluid–structure interaction problems[J]. International Journal of Computational Methods,2011,8(4):747–757.
[19] VU B N,NGUYEN X H,CHEN L,et al. A node-based smoothed XFEM for fracture mechanics[J]. CMES:Computer Modeling in Engineering and Sciences,2011,73(0):331–356.
[20] CHEN L,RABCZUK T,BORDAS SPA,et al. Extended finite element method with edge-based strain smoothing(ESmXFEM)for linear elastic crack growth[J]. Computer Methods in Applied Mechanics and Engineering,2012(209):250–265.
[21] ZENG W,LIU G R,KITAMURA Y,et al. A threedimensional ES-FEM for fracture mechanics problems in elastic solids[J].Engineering Fracture Mechanics,2013,114(114):127–150.
[22] NGUYEN X H,LIU G R. An edge-based smoothed finite element method softened with a bubble function(bES-FEM)for solid mechanics problems[J]. Computers and Structures,2013,128(9):14–30.
[23] JIANG C,ZHANG Z Q,LIU G R,et al. An edgebased/node-based selective smoothed finite element method using tetrahedrons for cardiovascular tissues[J]. Engineering Analysis with Boundary Elements,2015(59):62–77.
[24] ZENG W,LIU G R,LI D,et al. A smoothing technique based beta finite element method(bFEM)for crystal plasticity modeling[J].Computers and structures,2016,162(C):48-67.
[25]张旭明,孙建国.光滑有限单元法及其应用[J].河海大学学报(自然科学版),2012,40(5):582-585.
[26] ZENG W,LIU G R. Smoothed Finite Element Methods(S-FEM):An Overview and Recent Developments[J]. Archives of computational Methods in Engineering,2018,25(2):397-435.
[27]李静.有限元方法在电磁场分析中的应用[D].重庆:重庆师范大学,2011.
[28]李永利.边光滑有限元法在汽车电磁兼容计算中的研究及应用[D].长沙:湖南大学,2016.
[29] LI S,CUI X Y,LI G Y. Multi-physics analysis of electromagnetic forming process using an edge-based smoothed finite element method[J]. International Journal of Mechanical Sciences,2017,13(12)244-252.
[30]何智成,李光耀,成艾国,等.基于边光滑有限元的声固耦合研究[J].机械工程学报,2014,50(4):113-119.
[31] HE Z C,LI G Y,ZHONG Z H,et al. An ES-FEM for accurate analysis of 3D mid-frequency acoustics using tetrahedron mesh[J]. Computers and Structures,2012,106/107(5):125-134.
[32] HE Z C,LIU G R,ZHONG Z H,et al. Cheng. An edge-based smoothed finite element method(ES-FEM)for analyzing three-dimensional acoustic problems[J]. Computer Methods in Applied Mechanics and Engineering,2009,199(1):20-33.
[33]何智成.汽车中频NVH高效高精度计算理论与方法[D].长沙:湖南大学,2014.
[34]聂昕,李永利,何智成.基于一种新型数值算法的电磁场仿真研究[J].机械强度,2018,40(2):378-383.
[35] LI E,HE Z C,XU X,et al. Hybrid smoothed finite element method for acoustic problems[J]. Computer Methods in Applied Mechanics and Engineering,2015(283):664-688.
[36] WANG G,CUI X Y,FENG H,et al. A stable node-based smoothed finite element method for acoustic problems[J]. Computer Methods in Applied Mechanics and Engineering,2015(297):348-370.
[37] CHAI Y B,LI W,LI T Y,ea tl. Analysis of underwater acoustic scattering problems using stable node-based smoothed finite element method[J]. Engineering Analysis with Boundary Elements,2016(72)27-41.
[38]常舒.两步型Taylor-Galerkin光滑有限元法在波动和大变形问题中的应用研究[D].长沙:湖南大学,2015.
[39]周立明,孟广伟,王晖,等.含裂纹压电材料的Cell-Based光滑有限元法[J].东北大学学报(自然科学版),2015,36(11):1581-1585.
[40]周立明,蔡斌,孟广伟,等.含裂纹压电材料的Cell-Based光滑扩展有限元法[J].复合材料学报,2016,33(4):929-938.
[41] NGUYEN X H,LOC V,TRAN T,et al. Analysis of functionally graded plates using an edge-based smoothed finite element method[J]. Composite Structures,2011,93(11):3019-3039.
[42]顾帅.基于非均匀光滑有限元法的功能梯度压电梁的动力学分析[D].长春:吉林大学,2017.
[43]游翔宇,郑文成,李威,等.基于边光滑有限元法的加筋板静力和自由振动分析[J].计算力学学报,2018,35(1):28-34.
[44]吴飞.汽车车身振动噪音控制混合数值方法研究[D].长沙:湖南大学,2015.
[45]李威,李诤,柴应彬,等.基于边光滑有限元法的结构振动与稳定性研究[J].噪声与振动控制,2014,34(2):1-4.
[46]李明,李宁,李诤,等.基于边光滑有限元方法的三角形单元在复合材料反对称铺设角层合板的自由振动分析中的应用[J].船海工程,2013,42(3):21-24.
[47]李诤.基于光滑有限元方法的复合结构力学性能分析与振动试验测试[D].武汉:华中科技大学,2013.
[48] WU F,ZENG W,YAO L Y,et al. A generalized probabilistic edge-based smoothed finite element method for elastostatic analysis of Reissner–Mindlin plates[J]. Applied Mathematical Modelling,2018,(53):333-352.
[49] HU X B,CUI X Y,FENG H,et al. Stochastic analysis using the generalized perturbation stable node-based smoothed finite element method[J]. Engineering Analysis with Boundary Elements,2016,70(9):40-55.
[50] LIU G R,ZENG W,NGUYEN X H. Generalized stochastic cell-based smoothed finite element method(GS_CS-FEM)for solid mechanics[J]. Finite Elements in Analysis&Design,2013,63(1):51-61.
[51] KIM J,IM S. Polygonal type variable-node elements by means of the smoothed finite element method for analysis of two-dimensional fluid-solid interaction problems in viscous incompressible flows[J]. Computers&Structures,2017(182):475-490.
[52] HE T. Towards straightforward use of cell-based smoothed finite element method in fluid–structure interaction[J]. Ocean Engineering,2018(157):350-363.
[53] YUE J H,LIU G R,LI M,et al. A Cell-Based Smoothed Finite Element Method for Multi-body Contact Analysis using Linear Complementarity Formulation[J]. International Journal of Solids&Structures,2018(141/142):110-126.
[54] SELLAM M,NATARAJAN S,KANNAN K. Smoothed polygonal finite element method for generalized elastic solids subjected to torsion[J]. Computers and Structures,2017(188):32-44.
[55] WAN D T,HU D,YANG G,et al. A fully smoothed finite element method for analysis of axisymmetric problems[J]. Engineering Analysis with Boundary Elements,2016(72):78-88.
[56] CHEN H D,WANG Q S,LIU G R,et al. Simulation of thermoelastic crack problems using singular edge-based smoothed finite element method[J]. International Journal of Mechanical Sciences,2016(115/116):123-134.
[57] LI E,HE Z C,XU X. An edge-based smoothed tetrahedron finite element method(ES-T-FEM)for thermomechanical problems[J]. International Journal of Heat and Mass Transfer,2013,66(11):723-732.
[58] FENG S Z,CUI X Y,LI G Y. Analysis of transient thermo-elastic problems using edge-based smoothed finite element method[J]. International Journal of Thermal Sciences,2013(65):127-135.
[59] RODRIGUES J D,NATARAJAN S,A J M,et al. Analysis of composite plates through cell-based smoothed finite element and 4-noded mixed interpolation of tensorial components techniques[J]. Computers and Structures,2014,135(8):83-87.
[60]樊现行.光滑有限元法理论及算法研究[D].济南:山东大学,2013.
[61]张刚.基于光滑有限元法消除体积锁定和剪切锁定问题研究[D].济南:山东大学,2012.
[62]魏云鹏.应变重构光滑有限元法及其在反问题中的应用[D].长春:吉林大学,2017.
[63] ZHANG W,DAI B B,LIU Z,et al. Modeling discontinuous rock mass based on smoothed finite element method[J]. Computers and Geotechnics,2016(79):22-30.
[64] BIE Y H,CUI X Y,LI Z C. A coupling approach of state-based peridynamics with node-based smoothed finite element method[J]. Computer Methods in Applied Mechanics and Engineering,2018(331):675-700.
[65]万德涛.二维光滑扩展有限元法研究[D].长沙:湖南大学,2014.
[66] FENG H,CUI X Y,LI G Y,et al. A t-emporal stable node-based smoothed finite element method for three-dimensional elasticity problems[J]. Computational Mechanics,2014,53(5):859-876.
[2] 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.
[3] MO?S N,DOLBOW J,BELYTSCHKO T. A finite element method for crack growth without remeshing[J]. International Journal for Numerical Methods in Engineering,1999,46(1):131-150.
[4] CUETO E,SUKUMAR N,CALVO B,et al. Overview and recent advances in natural neighbour Galerkin methods[J]. Archives of Computational Methods in Engineering,2003,10(4):307-384.
[5] LIU G R,ZHANG G Y,DAI K Y,et al. A linearly conforming point interpolation method(lc-pim)for 2d solid mechanics problems[J]. International Journal of Computational Methods,2005,2(4):645-665.
[6] LIU G R,DAI K Y,NGUYEN T T. A Smoothed Finite Element Method for Mechanics Problems[J]. Computational Mechanics,2007,39(6):859-877.
[7] DAI K Y,LIU G R,NGUYEN T T. An n-sided polygonal smoothed finite element method(nSFEM)for solid mechanics[J]. Finite Elem Anal Des,2007,43(11):847–860.
[8] LIU G R,NGUYEN T T,DAI K Y,et al. Theoretical aspects of the smoothed finite element method(SFEM)[J]. International Journal for Numerical Methods in Engineering,2007,71(8):902–930.
[9] DAI K Y,LIU G R. Free and forced vibration analysis using the smoothed finite element method(SFEM)[J]. Journal of Sound&Vibration 2007,301(3):803–820.
[10] NGUYEN T T,LIU G R,DAI K Y,et al. Selective smoothed finite element method[J]. Tsinghua Science and Technology,2007,12(5):497–508.
[11] LIU G R,NGUYEN T T,LAM K Y. A novel alpha finite element method(aFEM)for exact solution to mechanics problems using triangular and tetrahedral elements[J]. Computer Methods in Applied Mechanics&Engineering,2008,197(45):3883–3897.
[12] LIU G R. A generalized Gradient smoothing technique and the smoothed bilinear form for Galerkin formulation of a wide class of computational methods[J]. International Journal of Computational Methods,2008,5(2):199–236.
[13] LIU G R,NGUYEN T T,NGUYEN X H,et al. A node-based smoothed finite element method(NS-FEM)for upper bound solutions to solid mechanics problems[J]. Computers and Structures,2009,87(1):14–26.
[14] LIU G R,NGUYEN T T,LAM K Y. An edge-based smoothed finite element method(ES-FEM)for static,free and forced vibration analyses of solids[J]. Journal of Sound and Vibration,2009,320(4):1100–1130.
[15] NGUYEN T T,LIU G R,LAM K Y,et al. A face-based smoothed finite element method(FS-FEM)for 3D linear and geometrically non-linear solid mechanics problems using 4-node tetrahedral elements[J]. International Journal for Numerical Methods in Engineering,2009,78(3):324-353.
[16] CUI X Y,LIU G R,LI G Y,et al. A smoothed finite element method(SFEM)for linear and geometrically nonlinear analysis of plates and shells[J]. CMES:Computer Modeling in Engineering and Sciences,2008,28(2):109–125.
[17] HE Z C,LIU G R,ZHONG Z H,et al. A coupled edge-/face-based smoothed finite element method for structural acoustic problems[J]. Applied Acoustics,2010,71(10):955–964.
[18] ZHANG Z Q,YAO J,LIU G R. An immersed smoothed finite element method for fluid–structure interaction problems[J]. International Journal of Computational Methods,2011,8(4):747–757.
[19] VU B N,NGUYEN X H,CHEN L,et al. A node-based smoothed XFEM for fracture mechanics[J]. CMES:Computer Modeling in Engineering and Sciences,2011,73(0):331–356.
[20] CHEN L,RABCZUK T,BORDAS SPA,et al. Extended finite element method with edge-based strain smoothing(ESmXFEM)for linear elastic crack growth[J]. Computer Methods in Applied Mechanics and Engineering,2012(209):250–265.
[21] ZENG W,LIU G R,KITAMURA Y,et al. A threedimensional ES-FEM for fracture mechanics problems in elastic solids[J].Engineering Fracture Mechanics,2013,114(114):127–150.
[22] NGUYEN X H,LIU G R. An edge-based smoothed finite element method softened with a bubble function(bES-FEM)for solid mechanics problems[J]. Computers and Structures,2013,128(9):14–30.
[23] JIANG C,ZHANG Z Q,LIU G R,et al. An edgebased/node-based selective smoothed finite element method using tetrahedrons for cardiovascular tissues[J]. Engineering Analysis with Boundary Elements,2015(59):62–77.
[24] ZENG W,LIU G R,LI D,et al. A smoothing technique based beta finite element method(bFEM)for crystal plasticity modeling[J].Computers and structures,2016,162(C):48-67.
[25]张旭明,孙建国.光滑有限单元法及其应用[J].河海大学学报(自然科学版),2012,40(5):582-585.
[26] ZENG W,LIU G R. Smoothed Finite Element Methods(S-FEM):An Overview and Recent Developments[J]. Archives of computational Methods in Engineering,2018,25(2):397-435.
[27]李静.有限元方法在电磁场分析中的应用[D].重庆:重庆师范大学,2011.
[28]李永利.边光滑有限元法在汽车电磁兼容计算中的研究及应用[D].长沙:湖南大学,2016.
[29] LI S,CUI X Y,LI G Y. Multi-physics analysis of electromagnetic forming process using an edge-based smoothed finite element method[J]. International Journal of Mechanical Sciences,2017,13(12)244-252.
[30]何智成,李光耀,成艾国,等.基于边光滑有限元的声固耦合研究[J].机械工程学报,2014,50(4):113-119.
[31] HE Z C,LI G Y,ZHONG Z H,et al. An ES-FEM for accurate analysis of 3D mid-frequency acoustics using tetrahedron mesh[J]. Computers and Structures,2012,106/107(5):125-134.
[32] HE Z C,LIU G R,ZHONG Z H,et al. Cheng. An edge-based smoothed finite element method(ES-FEM)for analyzing three-dimensional acoustic problems[J]. Computer Methods in Applied Mechanics and Engineering,2009,199(1):20-33.
[33]何智成.汽车中频NVH高效高精度计算理论与方法[D].长沙:湖南大学,2014.
[34]聂昕,李永利,何智成.基于一种新型数值算法的电磁场仿真研究[J].机械强度,2018,40(2):378-383.
[35] LI E,HE Z C,XU X,et al. Hybrid smoothed finite element method for acoustic problems[J]. Computer Methods in Applied Mechanics and Engineering,2015(283):664-688.
[36] WANG G,CUI X Y,FENG H,et al. A stable node-based smoothed finite element method for acoustic problems[J]. Computer Methods in Applied Mechanics and Engineering,2015(297):348-370.
[37] CHAI Y B,LI W,LI T Y,ea tl. Analysis of underwater acoustic scattering problems using stable node-based smoothed finite element method[J]. Engineering Analysis with Boundary Elements,2016(72)27-41.
[38]常舒.两步型Taylor-Galerkin光滑有限元法在波动和大变形问题中的应用研究[D].长沙:湖南大学,2015.
[39]周立明,孟广伟,王晖,等.含裂纹压电材料的Cell-Based光滑有限元法[J].东北大学学报(自然科学版),2015,36(11):1581-1585.
[40]周立明,蔡斌,孟广伟,等.含裂纹压电材料的Cell-Based光滑扩展有限元法[J].复合材料学报,2016,33(4):929-938.
[41] NGUYEN X H,LOC V,TRAN T,et al. Analysis of functionally graded plates using an edge-based smoothed finite element method[J]. Composite Structures,2011,93(11):3019-3039.
[42]顾帅.基于非均匀光滑有限元法的功能梯度压电梁的动力学分析[D].长春:吉林大学,2017.
[43]游翔宇,郑文成,李威,等.基于边光滑有限元法的加筋板静力和自由振动分析[J].计算力学学报,2018,35(1):28-34.
[44]吴飞.汽车车身振动噪音控制混合数值方法研究[D].长沙:湖南大学,2015.
[45]李威,李诤,柴应彬,等.基于边光滑有限元法的结构振动与稳定性研究[J].噪声与振动控制,2014,34(2):1-4.
[46]李明,李宁,李诤,等.基于边光滑有限元方法的三角形单元在复合材料反对称铺设角层合板的自由振动分析中的应用[J].船海工程,2013,42(3):21-24.
[47]李诤.基于光滑有限元方法的复合结构力学性能分析与振动试验测试[D].武汉:华中科技大学,2013.
[48] WU F,ZENG W,YAO L Y,et al. A generalized probabilistic edge-based smoothed finite element method for elastostatic analysis of Reissner–Mindlin plates[J]. Applied Mathematical Modelling,2018,(53):333-352.
[49] HU X B,CUI X Y,FENG H,et al. Stochastic analysis using the generalized perturbation stable node-based smoothed finite element method[J]. Engineering Analysis with Boundary Elements,2016,70(9):40-55.
[50] LIU G R,ZENG W,NGUYEN X H. Generalized stochastic cell-based smoothed finite element method(GS_CS-FEM)for solid mechanics[J]. Finite Elements in Analysis&Design,2013,63(1):51-61.
[51] KIM J,IM S. Polygonal type variable-node elements by means of the smoothed finite element method for analysis of two-dimensional fluid-solid interaction problems in viscous incompressible flows[J]. Computers&Structures,2017(182):475-490.
[52] HE T. Towards straightforward use of cell-based smoothed finite element method in fluid–structure interaction[J]. Ocean Engineering,2018(157):350-363.
[53] YUE J H,LIU G R,LI M,et al. A Cell-Based Smoothed Finite Element Method for Multi-body Contact Analysis using Linear Complementarity Formulation[J]. International Journal of Solids&Structures,2018(141/142):110-126.
[54] SELLAM M,NATARAJAN S,KANNAN K. Smoothed polygonal finite element method for generalized elastic solids subjected to torsion[J]. Computers and Structures,2017(188):32-44.
[55] WAN D T,HU D,YANG G,et al. A fully smoothed finite element method for analysis of axisymmetric problems[J]. Engineering Analysis with Boundary Elements,2016(72):78-88.
[56] CHEN H D,WANG Q S,LIU G R,et al. Simulation of thermoelastic crack problems using singular edge-based smoothed finite element method[J]. International Journal of Mechanical Sciences,2016(115/116):123-134.
[57] LI E,HE Z C,XU X. An edge-based smoothed tetrahedron finite element method(ES-T-FEM)for thermomechanical problems[J]. International Journal of Heat and Mass Transfer,2013,66(11):723-732.
[58] FENG S Z,CUI X Y,LI G Y. Analysis of transient thermo-elastic problems using edge-based smoothed finite element method[J]. International Journal of Thermal Sciences,2013(65):127-135.
[59] RODRIGUES J D,NATARAJAN S,A J M,et al. Analysis of composite plates through cell-based smoothed finite element and 4-noded mixed interpolation of tensorial components techniques[J]. Computers and Structures,2014,135(8):83-87.
[60]樊现行.光滑有限元法理论及算法研究[D].济南:山东大学,2013.
[61]张刚.基于光滑有限元法消除体积锁定和剪切锁定问题研究[D].济南:山东大学,2012.
[62]魏云鹏.应变重构光滑有限元法及其在反问题中的应用[D].长春:吉林大学,2017.
[63] ZHANG W,DAI B B,LIU Z,et al. Modeling discontinuous rock mass based on smoothed finite element method[J]. Computers and Geotechnics,2016(79):22-30.
[64] BIE Y H,CUI X Y,LI Z C. A coupling approach of state-based peridynamics with node-based smoothed finite element method[J]. Computer Methods in Applied Mechanics and Engineering,2018(331):675-700.
[65]万德涛.二维光滑扩展有限元法研究[D].长沙:湖南大学,2014.
[66] FENG H,CUI X Y,LI G Y,et al. A t-emporal stable node-based smoothed finite element method for three-dimensional elasticity problems[J]. Computational Mechanics,2014,53(5):859-876.