机械结构分析中的新型低阶高精度单元理论研究
|
摘要
在计算金属体积成型或车身覆盖件冲压和碰撞等牵涉到动态大变形的实际机械类工程问题时,有限元方法存在精度较低或计算能力不足的问题；无网格方法能很好地用于大变形问题,却存在计算复杂、计算效率过低的问题。针对这些现有数值算法的不足,本文旨在形成一套新型的单元构造体系,提出几种有效的低阶高精度板壳单元,使其能够有效地用于工程计算,为工程实际问题的求解提供有效的数值手段和理论基础。基于这一目的,本文从弱-弱形式和广义梯度光滑技术出发,提出几种有效的低阶高精度数值计算方法和板壳单元,并对其精度、收敛特性等进行了分析,建立了相应的单元算法构造体系。具体工作为：
1.基于弱-弱形式的新型数值算法理论
提出子区域应变光滑的概念和区域光滑径向点插值法,将传统无网格所用的高斯积分转化为可任意控制积分点数目的子区域光滑积分,并给出基于背景网格的支撑点选点方案,将无网格插值方式和有网格选点进行有机结合,既保证了新算法处理网格畸变的能力,又提高了计算效率和精度。
给出了边光滑弹塑性有限单元法,并从理论上证明边光滑技术能够降低系统的模型刚度,有效地解决有限元存在的模型过刚问题。通过求得边光滑域所对应的等效弹性模量和等效泊松比求解弹塑性问题,有效获得材料发生弹塑性变形而产生硬化后的材料参数。
2.建立几种新型Mindlin板壳单元
给出基于区域光滑径向点插值法的Mindlin-Reissner板构造过程,提出消除剪切自锁的方案,获得了具有高应力精度的中厚板公式。
构造了基于光滑有限单元法的四边形板壳单元,并提出平均形函数的概念,有效消除了剪切自锁问题。直接在笛卡尔坐标系下构造形函数,不需要进行等参变换,并将传统的面积积分转化为线积分,既能够提高计算效率,又能够有效地处理网格畸变问题,是一种在工程上较为实用的优质四边形板壳元。
提出了边光滑三角形板壳单元构造理论和边局部坐标系的概念,给出了非同面内的梯度光滑实现方案。在现有低精度三角形板壳单元基础上只需做简单改动便能够获得高精度,受网格畸变影响较小并能有效地用于解决工程问题的三角形板壳元,是一种简单高效实用的三角形板壳单元构造方式。
3.建立基于线性插值的薄梁薄壳元构造理论
提出采用线性插值求解四阶微分方程思想,首次构建相关理论框架,并通过构造线性插值薄梁单元,证明采用线性插值求解四阶微分方程问题的可行性。
提出一种连续性二次松弛技术,进一步形成了全新的薄板单元构造新理论。通过散度定理将域积分转换为边界积分对积分域内的曲率进行平均处理,能够有效地降低试函数的连续性要求,对于2后阶偏微分方程,k-1阶连续的试函数便能够有效地离散数值模型。对于需要试函数具有C1连续性的四阶偏微分方程的薄板问题能够很容易地采用C0连续的试函数进行离散求解,并能够有效地进行求解。
基于径向点插值法和三角形背景网格,给出了不同的积分域形成方案,并构造了相应的连续性二次松弛薄板单元,讨论不同的积分域形成方案对计算结果的影响,证实了采用连续性二次松弛技术构造薄板单元的有效性与精确性。
提出了曲率重构理论,基于线性插值的挠度场构造兴趣点处的转角,并利用构造的转角重构积分域内的曲率场,从而获得相应的离散方程。给出了几种曲率场重构方案,构造了薄板公式,讨论了不同方案的计算精度,并选择了合适的方案用于构造薄壳单元。
4.构建基于新型单元理论的动态大变形显式求解格式
将基于弱-弱形式的数值计算方法用于求解包含接触、几何和材料三大非线性的金属成形问题,给出了相关的接触算法,并提出了光滑变形梯度和光滑应力应变张量的概念,为金属成形分析模拟提供了新而有效的数值工具。
将前面提出的两种表现较好的三角形壳单元用于求解显式动态问题,给出了所提出的三角形壳单元显式格式,并在光滑迦辽金弱形式下推导光滑应变率形式的节点内力、节点外力以及节点惯性力。
The finite element method meets low accuracy or can not work for some engineering practice problems, such as metal forming、vehicle body panel forming and crashing, containing dynamic large deformation analysis. Meshless methods can work perfectly for large deformation analysis, but they encounter the major technical barrier-low computation efficiency. To solve these shortages of existing numerical method, this thesis propose a novel numerical theory system, which are very effective for engineering analysis. To pursue this goal, this thesis present several effective numerical method and plate and shell elements based on weakend weak form and generalized gradient smoothing technique. Some important properties, such as accuracy and convergence, are proved, and a numerical theory system is constructed. Main work includes the following aspects:
1. Novel numerical methods based on weakend weak form
The subdomain gradient smoothing technique and a cell-based smoothing radial point interpolation method are proposed in this thesis. Conventional gauss integration becomes line integration along smoothing subdomain edges. The supporting node selection for shape function construction is based on the background cells, which incorporating the meshless interpotation can deal with the mesh distortion effectively, and also has a high accuracy and efficiency.
An edge-based smoothed finite element method (ES-FEM) for elastic-plastic analysis is formulated. Theorem-proof of the results demonstrate that the edge-based smoothing operation reduces the stiffness of the discretized system, and compensates nicely the "over-stiffness" of the FEM model. The effective elastic modulus and Possion's ratio in each smoothing domain are obtained for computing the elastic-plastic problems, and the effective material parameters are the real material parameters of the hardening material.
2. Proposed several novel Mindin plate and shell elements
A formulation of Mindlin-Reissner plates is proposed using the cell-based smoothed radial point interpolation method with sub-domain smoothing operations. Effective treatment for shear-locking are given and a plate formulation with high accuracy of stress is obtained.
A novel quadrilateral plate and shell element is presented based on the smoothed finite element method. An average shape function is proposed, and shear locking phenomenon is avoided by using only one smoothing cell. Area integration over each smoothing cells is recast into line integration along its edges, and the shape functions are obtained directly in the Cartesian coordinate, hence no mapping is needed and extremely distorted elements can be used. This quadrilateral plate and shell element provides very stable and accurate results with the low computational effort compared, which is very effective for engineering analysis.
A novel edge-based smoothed triangular plate and shell element theory is proposed, and an edge local coordinate system is introduced for performing strain smoothing operations for the elements consisting the smoothing domain not in a plane. The present method can be easily implemented with little changes to the esxisting triangular shell elements. The new element is very high accuracy and not sensitive to mesh distortion, which is very useful to engineering analysis.
3. Novel thin plate and shell element theory using linear interpolation
This thesis presents a way to solve the4th order boundary value problems using simple linear point interpolation method, and a rotation-free Euler-Bernoulli beam element is proposed. A novel numerical theory is formed, and the feasibility of this theory is proved.
A novel thin plate element theory is proposed based on a continuity re-relaxed technique to overcome the high continuity requirement for thin plate formulation which need the high computation cost. The curvatures over integration domains are restructured through the divergence theorem, and the continuity requirement of the trial deflection function for thin plate problems can be re-relaxed. Using the proposed continuity re-relaxed technique, the weak form of a2k order PDE can be modeled using shape functions of k-1consistence. The C1continuity thin plate problem can be easily formulated and stable results can be obtained only using Co interpolating functions.
Based on the radial point interpolation method and triangular background cells, several types of smoothing domains are constructed, and different thin plate formulation using continuity re-relaxed technique. The properties of the different smoothing domains constructing scheme are discussed, the efficiency and accuracy of the novel thin plate element theory are confirmed.
A curvature-constructed method (CCM) for thin plate problems is proposed using three-node triangular cells and assumed piecewisely linear deflection field. The slopes at special points are obtained using the gradient smoothing techniques (GST) over different smoothing domains. The curvature in each cell can be constructed using these slopes, and then used to create the discretized system equations. Some schemes are devised using the present CCM and the numerical results are presented, to test the efficiency and accuracy of these schemes.Outstanding scheme is choosed for constructing shell formulation.
4. Dynamic explicit formulation of proposed numerical method and shell element for large deformation analysis
An edge-based gradient smoothing based on weakened weak form is proposed for the simulation of metal forming processes, which involves geometric, material, and contact non-linear. A simple but general contact searching algorithm is used to treat the contact interface and an algorithm for the contact force is presented. The smoothed deformation gradient and the smoothed stress tensor are derived at the reference (or initial) configuration. The integration domains are different from the interpolation domain, and the method can work for extremely distorted meshes. Solutions to upsetting, extrusion of metals with large material distortions are given to show the effectiveness and efficiency of the proposed method.
The explicit dynamic formulations of two outstanding triangular shell elements are extended for large deformation analysis. The nodel internal force, external force and inertial force are obtained using the weakened-weak form. Various numerical tests demonstrate the accuracy and efficiency of the proposed triangular shell elements.
1.基于弱-弱形式的新型数值算法理论
提出子区域应变光滑的概念和区域光滑径向点插值法,将传统无网格所用的高斯积分转化为可任意控制积分点数目的子区域光滑积分,并给出基于背景网格的支撑点选点方案,将无网格插值方式和有网格选点进行有机结合,既保证了新算法处理网格畸变的能力,又提高了计算效率和精度。
给出了边光滑弹塑性有限单元法,并从理论上证明边光滑技术能够降低系统的模型刚度,有效地解决有限元存在的模型过刚问题。通过求得边光滑域所对应的等效弹性模量和等效泊松比求解弹塑性问题,有效获得材料发生弹塑性变形而产生硬化后的材料参数。
2.建立几种新型Mindlin板壳单元
给出基于区域光滑径向点插值法的Mindlin-Reissner板构造过程,提出消除剪切自锁的方案,获得了具有高应力精度的中厚板公式。
构造了基于光滑有限单元法的四边形板壳单元,并提出平均形函数的概念,有效消除了剪切自锁问题。直接在笛卡尔坐标系下构造形函数,不需要进行等参变换,并将传统的面积积分转化为线积分,既能够提高计算效率,又能够有效地处理网格畸变问题,是一种在工程上较为实用的优质四边形板壳元。
提出了边光滑三角形板壳单元构造理论和边局部坐标系的概念,给出了非同面内的梯度光滑实现方案。在现有低精度三角形板壳单元基础上只需做简单改动便能够获得高精度,受网格畸变影响较小并能有效地用于解决工程问题的三角形板壳元,是一种简单高效实用的三角形板壳单元构造方式。
3.建立基于线性插值的薄梁薄壳元构造理论
提出采用线性插值求解四阶微分方程思想,首次构建相关理论框架,并通过构造线性插值薄梁单元,证明采用线性插值求解四阶微分方程问题的可行性。
提出一种连续性二次松弛技术,进一步形成了全新的薄板单元构造新理论。通过散度定理将域积分转换为边界积分对积分域内的曲率进行平均处理,能够有效地降低试函数的连续性要求,对于2后阶偏微分方程,k-1阶连续的试函数便能够有效地离散数值模型。对于需要试函数具有C1连续性的四阶偏微分方程的薄板问题能够很容易地采用C0连续的试函数进行离散求解,并能够有效地进行求解。
基于径向点插值法和三角形背景网格,给出了不同的积分域形成方案,并构造了相应的连续性二次松弛薄板单元,讨论不同的积分域形成方案对计算结果的影响,证实了采用连续性二次松弛技术构造薄板单元的有效性与精确性。
提出了曲率重构理论,基于线性插值的挠度场构造兴趣点处的转角,并利用构造的转角重构积分域内的曲率场,从而获得相应的离散方程。给出了几种曲率场重构方案,构造了薄板公式,讨论了不同方案的计算精度,并选择了合适的方案用于构造薄壳单元。
4.构建基于新型单元理论的动态大变形显式求解格式
将基于弱-弱形式的数值计算方法用于求解包含接触、几何和材料三大非线性的金属成形问题,给出了相关的接触算法,并提出了光滑变形梯度和光滑应力应变张量的概念,为金属成形分析模拟提供了新而有效的数值工具。
将前面提出的两种表现较好的三角形壳单元用于求解显式动态问题,给出了所提出的三角形壳单元显式格式,并在光滑迦辽金弱形式下推导光滑应变率形式的节点内力、节点外力以及节点惯性力。
The finite element method meets low accuracy or can not work for some engineering practice problems, such as metal forming、vehicle body panel forming and crashing, containing dynamic large deformation analysis. Meshless methods can work perfectly for large deformation analysis, but they encounter the major technical barrier-low computation efficiency. To solve these shortages of existing numerical method, this thesis propose a novel numerical theory system, which are very effective for engineering analysis. To pursue this goal, this thesis present several effective numerical method and plate and shell elements based on weakend weak form and generalized gradient smoothing technique. Some important properties, such as accuracy and convergence, are proved, and a numerical theory system is constructed. Main work includes the following aspects:
1. Novel numerical methods based on weakend weak form
The subdomain gradient smoothing technique and a cell-based smoothing radial point interpolation method are proposed in this thesis. Conventional gauss integration becomes line integration along smoothing subdomain edges. The supporting node selection for shape function construction is based on the background cells, which incorporating the meshless interpotation can deal with the mesh distortion effectively, and also has a high accuracy and efficiency.
An edge-based smoothed finite element method (ES-FEM) for elastic-plastic analysis is formulated. Theorem-proof of the results demonstrate that the edge-based smoothing operation reduces the stiffness of the discretized system, and compensates nicely the "over-stiffness" of the FEM model. The effective elastic modulus and Possion's ratio in each smoothing domain are obtained for computing the elastic-plastic problems, and the effective material parameters are the real material parameters of the hardening material.
2. Proposed several novel Mindin plate and shell elements
A formulation of Mindlin-Reissner plates is proposed using the cell-based smoothed radial point interpolation method with sub-domain smoothing operations. Effective treatment for shear-locking are given and a plate formulation with high accuracy of stress is obtained.
A novel quadrilateral plate and shell element is presented based on the smoothed finite element method. An average shape function is proposed, and shear locking phenomenon is avoided by using only one smoothing cell. Area integration over each smoothing cells is recast into line integration along its edges, and the shape functions are obtained directly in the Cartesian coordinate, hence no mapping is needed and extremely distorted elements can be used. This quadrilateral plate and shell element provides very stable and accurate results with the low computational effort compared, which is very effective for engineering analysis.
A novel edge-based smoothed triangular plate and shell element theory is proposed, and an edge local coordinate system is introduced for performing strain smoothing operations for the elements consisting the smoothing domain not in a plane. The present method can be easily implemented with little changes to the esxisting triangular shell elements. The new element is very high accuracy and not sensitive to mesh distortion, which is very useful to engineering analysis.
3. Novel thin plate and shell element theory using linear interpolation
This thesis presents a way to solve the4th order boundary value problems using simple linear point interpolation method, and a rotation-free Euler-Bernoulli beam element is proposed. A novel numerical theory is formed, and the feasibility of this theory is proved.
A novel thin plate element theory is proposed based on a continuity re-relaxed technique to overcome the high continuity requirement for thin plate formulation which need the high computation cost. The curvatures over integration domains are restructured through the divergence theorem, and the continuity requirement of the trial deflection function for thin plate problems can be re-relaxed. Using the proposed continuity re-relaxed technique, the weak form of a2k order PDE can be modeled using shape functions of k-1consistence. The C1continuity thin plate problem can be easily formulated and stable results can be obtained only using Co interpolating functions.
Based on the radial point interpolation method and triangular background cells, several types of smoothing domains are constructed, and different thin plate formulation using continuity re-relaxed technique. The properties of the different smoothing domains constructing scheme are discussed, the efficiency and accuracy of the novel thin plate element theory are confirmed.
A curvature-constructed method (CCM) for thin plate problems is proposed using three-node triangular cells and assumed piecewisely linear deflection field. The slopes at special points are obtained using the gradient smoothing techniques (GST) over different smoothing domains. The curvature in each cell can be constructed using these slopes, and then used to create the discretized system equations. Some schemes are devised using the present CCM and the numerical results are presented, to test the efficiency and accuracy of these schemes.Outstanding scheme is choosed for constructing shell formulation.
4. Dynamic explicit formulation of proposed numerical method and shell element for large deformation analysis
An edge-based gradient smoothing based on weakened weak form is proposed for the simulation of metal forming processes, which involves geometric, material, and contact non-linear. A simple but general contact searching algorithm is used to treat the contact interface and an algorithm for the contact force is presented. The smoothed deformation gradient and the smoothed stress tensor are derived at the reference (or initial) configuration. The integration domains are different from the interpolation domain, and the method can work for extremely distorted meshes. Solutions to upsetting, extrusion of metals with large material distortions are given to show the effectiveness and efficiency of the proposed method.
The explicit dynamic formulations of two outstanding triangular shell elements are extended for large deformation analysis. The nodel internal force, external force and inertial force are obtained using the weakened-weak form. Various numerical tests demonstrate the accuracy and efficiency of the proposed triangular shell elements.
引文
[1]王勖成.有限单元法.清华大学出版社.2003.
[2]Zienkiewicz O.C., Taylor R.L. The Finite Element Method. (6th ed.). New York: Elsevier Butterworth-Heinemann,2005.
[3]Liu G.R. Meshfree Methods:Moving Beyond the Finite Element Method. Boca Raton:CRC Press,2002.
[4]Liu G.R. Meshfree methods:Moving Beyond the Finite Element Method,2nd edition. Boca Raton:CRC Press,2009.
[5]Atluri S.N., Shen S.P. The Meshless Local Petrov-Galerkin (MLPG) Method. Netherlands:Springer Press,2005.
[6]Liu G.R., Gu Y.T. An Introduction to Meshfree Methods and Their Programming. Netherlands:Springer Press,2005.
[7]Belytschko T., Krongauz Y., Organ D. Meshless method:an overview and recent development. Computer methods in Applied Mechanics and Engineering, 1996,139(1-4):3-47.
[8]张雄,刘岩.无网格法.北京：清华大学出版社,2004.
[9]刘更,刘天祥,谢琴.无网格法及其应用.西安：西北工业大学出版社,2005.
[10]Lucy L.B. A numerical approach to testing of the fission hypothesis. The Astronomical Journal,1977,8(12):1013-1024.
[11]Gingold R.A., Monaghan J.J. Smoothed Particle hydrodynamics:theory and applications to non-spherical stars. Royal Astronomical Society, Monthly Notic, 1977,181:375-389.
[12]Monaghan J.J. An introduction to SPH. Computer Physics Communications, 1998,48(1):89-96.
[13]张锁春.光滑质点流体动力学(SPH)方法(综述).计算物理,1996,13(4)：385-397.
[14]Liu G.R., Liu M.B. Smoothed Particle Hydrodynamics-A Meshfree Particle Method. Singapore:World Scientific Publishing Company,2003.
[15]Liu G.R., Liu M.B著(韩旭,杨刚,强洪夫(译)).光滑粒子流体动力学—种无网格粒子法.湖南大学出版社,2005.
[16]Swegle J.W., Attaway S.W. On the feasibility of using smoothed particle hydrodynamics for underwater explosion calculations. Computational Mechanics,1995,17(3):151-168.
[17]Liu M.B., Liu G.R., Lam K.Y. et al. Smoothed particle hydrodynamics for numerical simulation of underwater explosion. Computational Mechanics,2003, 30(2):106-118.
[18]Cleary P.W., Prakash M., Ha J. Novel applications of smoothed particle hydrodynamics (SPH) in metal forming. Journal of Materials Processing Technology,2006,177(1-3):41-48.
[19]Jeong J.H., Jhon M.S., Halow J.S. et al. Smoothed particle hydrodynamics: Applications to heat conduction. Computer Physics Communications,2003, 153(1):71-84.
[20]Monaghan J.J., Huppert H.E., Worster M.G. Solidification using smoothed particle hydrodynamics. Journal of Computational Physics,2005,206(2): 684-705.
[21]Zhang M.Y., Zhang H., Zheng L.L. Application of smoothed particle hydrodynamics method to free surface and solidification problems. Numerical Heat Transfer A-Applications,2007,52(4):299-314.
[22]Zhang M.Y., Zhang H., Zheng L.L. Simulation of droplet spreading, splashing and solidification using smoothed particle hydrodynamics method. International Journal of Heat and Mass Transfer,2008,51(13-14):3410-3419.
[23]Lancaster P., Salkauskas K. Surfaces generated by moving least square methods. Mathematics of Computation,1981,37(155):141-158.
[24]Nayroles B., Touzot G., Villon P. Generalized the finite element method:diffuse approximation and diffuse elements. Computational Mechanics,1992,10(5): 307-318.
[25]Belytschko T., Lu Y.Y., Gu L. Element-free Galerkin method. International Journal for Numerical Methods in Engineering,1994,37(2):229-256.
[26]Belytschko T., Organ D., Krongauz Y. A coupled finite element-element-free Galerkin method. Computational Mechanics,1995,17:186-195.
[27]Gu Y.T., Liu G.R. A coupled element free Galerkin/boundary element method for stress analysis of two-dimensional solids. Computer Methods in Applied Mechanics and Engineering,2001,190:4405-4419.
[28]Ponthot J.P., Belytsehko T. Arbitrary Lagrangian-Eulerian formulation for element-free Galerkin method. Computer Methods in Applied Mechanics and Engineering,1998,152:19-46.
[29]Liu G.R., Tu Z.H. An adaptive procedure based on background cells for meshless method. Computer Methods in Applied Mechanics and Engineering, 2002,191:1923-1943.
[30]Chung H.J., Belytsehko T. An error estimate in the EFG method. Computational Mechanics,1998,21(2):91-100.
[31]Gavete L., Cuesta J.L., Ruiz A. A procedure for approximation of the error in the EFG method. International Journal for Numerical Methods in Engineering, 2002,53(3):677-690.
[32]Gavete L., Gavete M.L., Alonso B., Martin A.J. A posteriori error approximation in EFG method. International Journal for Numerical Methods in Engineering,2003,58:2239-2263.
[33]Krysl P., Belytschko T. Analysis of thin plates by the element-free Galerkin method. Computational Mechanics,1995,17(1):26-35.
[34]Krysl P., Belytschko T. Analysis of thin shells by the element-free Galerkin method. International Journal of Solids and Structures,1996,33:3057-3080.
[35]Belytschko T., Lu Y.Y., Gu L. Fracture and crack growth by element-free Galerkin method. Modeling and Simulation in Material Science and Engineering,1994,2:519-534.
[36]Belytschko T., Tabbara M. Dynamic fracture using element-free Galerkin method. International Journal for Numerical Methods in Engineering,1996,39: 141-158.
[37]Kargarnovin M.H., Toussi H.E., Fariborz S.J. Elasto-Plastic element-free Galerkin method. Computational Mechanics,2004,33(3):206-214.
[38]Ren J., Liew K.M., Meguid S.A. Modelling and simulation of the superelastic behaviour of shape memory alloys using the element-free Galerkin method. International Journal of Mechanical Science,2002,44(12):393-2413.
[39]Belytschko T., Krysl P., Krongauz Y. A three-dimensional explicit element-free Galerkin method. International Journal for Numerical Methods in Fluids,1997, 24(12):1253-1270.
[40]Li G.Y., Belytscko T. Element-free Galerkin method for contact problems in metal forming analysis. Engineering Computation,2001,18(1):62-78.
[41]Guan Y.J., Wu X., Zhao G.Q., Lu P. A nonlinear numerical analysis for metal-forming process using the rigid-(visco) plastic element-free Galerkin method. International Journal of Advanced Manufacturing Technology,2009, 42:83-92.
[42]Liu W.K., Jun S., Zhang Y.F. Reproducing kernel particle methods. International Journal for Numerical Methods in Fluids,1995,20:1081-1106.
[43]Liu W.K., Chen Y. Wavelet and multiple scale reprodueing kernel method. International Journal for Numerical Methods in Fluids,1995,21:901-931.
[44]Liu W.K., Jun S., Li S.F. Reprodueing kernel particle methods for structural dynamics. International Journal for Numerical Methods in Engineering.1995, 38:1655-1679.
[45]Liu W.K., Jun S., Sihling D.T. et al. Multiresolution reproducing kernel particle method for computational fluid dynamics. International Journal for Numerical Methods in Fluids,1997,24(12):1391-1415.
[46]Chen J.S., Pan C., Wu C.T. Large deformation analysis of rubber based on a reproducing kernel particle method. Computational Mechanics,1997,19(3): 211-227.
[47]Liew K.M., Ng T.Y., Wu Y.C. Meshfree method for large deformation analysis-a reproducing kernel particle approach. Engineering Structures,2002, 24(5):543-551.
[48]Li G.Y., Sidibe K., Liu G.R. Meshfree method for 3D forming analysis with lower order integration scheme. Engineering Analysis with Boundary Elements, 2004,28(10):1283-1292.
[49]Park Y.H. Rigid-plastic analysis for metal forming processes using a reproducing kernel particle method. Journal of Materials Processing Technology,2007,183(2-3):256-263.
[50]Atluri S.N., Zhu T. A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Computational Mechanics,1998,22(2):117-127.
[51]Zhu T., Zhang J.D., Atluri S.N. A local boundary integral equation (LBIE) method in computational mechanics and a meshless diseretization approach. Computational Mechanics,1998,21(3):223-235.
[52]Sladek J., Sladek V., Zhang C. et al. Analysis of orthotropic thick plates by meshless local Petrov-Galerkin method (MLPG) method. International Journal for Numerical Methods in Engineering,2006,67(13):1830-1850.
[53]Qian L.F., Batra R.C. Three-dimensional transient heat conduction in a functionally graded thick plate with a higher-order plate theory and a meshless local Petrov-Galerkin method. Computational Mechanics,2005,35(3): 214-226.
[54]Sladek J., Sladek V., Wen P. et al. Meshless local Petrov-Galerkin method for shear deformable shells analysis. CMES-Computer Modeling in Engineering and Science,2006,13(2):103-117.
[55]Li D., Lin Z.Q., Li S.H. Numerical analysis of Mindlin shell by meshless local Petrov-Galerkin method. Acta Mechanica Solida Sinica,2008,21(2):160-169.
[56]Xiong Y.B., Long S.Y., Hu D.A. et al. A meshless local Petrov-Galerkin method for geometrically nonlinear analysis. Acta Mechanica Solida Sinica,2005, 18(4):348-356.
[57]Hu D.A., Long S.Y., Han X. et al. A meshless Petrov-Galerkin method for large deformation contact analysis of elastomers. Engineering Analysis with Boundary Elements,2007,31(7):61-74.
[58]张希.无网格局部彼得洛夫伽辽金法在大变形问题中的应用.清华大学博士论文,2006.
[59]Arefmanesh A., Najafi M., Abdi H. A meshless local Petrov-Galerkin method for fluid dynamics and heat transfer applications. Journal of Fluids Engineering-Transactions of the ASME,2005,127(4):647-655.
[60]Sladek J., Sladek V., Hellmich C. et al. Heat conduction analysis of 3D axisymmetric and anisotropic FGM bodies by meshless local Petrov-Galerkin method. Computational Mechnics,2007,39(3):323-333.
[61]Sladek J., Sladek V., Zhang C. et al. Meshless local Petrov-Galerkin method for linear coupled thermoelastic analysis. CMES-Computer Modeling in Engineering and Science,2007,16(1):57-68.
[62]Liu K.Y., Long S.Y., Li G.Y. A simple and less-costly meshless local Petrov-Galerkin method for the dynamic fracture problem. Engineering Analysis with Boundary Elements,2006,30(1):72-76.
[63]Long S.Y., Liu K.Y., Li G.Y. An analysis for the elasto-plastic fracture problem by the meshless local Petrov-Galerkin method. CMES-Computer Modeling in Engineering and Science,2008,28(3):203-216.
[64]Zhu T., Atluri N.. Modified collocation method and a penalty function for enforcing essential boundary conditions in the element free Galerkin method. Computational Mechanics,1998,21:211-222.
[65]Lu Y.Y., Belytschko T., Gu L. A new implementation of the element-free Galerkin method. Computer Methods in Applied Mechanics and Engineering, 1994,113:397-414.
[66]Kaljevic I., Saigal S. An improved element free Galerkin formulation. International Journal for Numerical Methods in Engineering,1997,40: 2953-2974.
[67]Krongauz Y., Belytschko T. Enforcement of essential boundary conditions in meshless approximations using finite elements. Computer Methods in Applied Mechanics and Engineering,1996,131:133-145.
[68]Most T. A natural neighbour-based moving least-squares approach for the element-free Galerkin method. International Journal for Numerical Methods in Engineering,2007,71:224-252.
[69]Liu G.R., Gu Y.T. A point interpolation method for two-dimensional solids. International Journal for Numerical Methods in Engineering,2001,50:937-951.
[70]Gu Y.T., Liu G.R. A boundary Point interpolation method for stress analysis of solids. Computational Mechanics,2002,28(1):47-54.
[71]Liu G.R., Gu Y.T. A local point interpolation method for stress analysis of two-dimensional solids. International Journal of Structural Engineering and Mechanics,2001,11:221-236.
[72]Gu Y.T., Liu G.R. A local point interpolation method for static and dynamic analysis of thin beams. Computer Methods in Applied Mechanics and Engineering,2001,190:5515-5528.
[73]Gu Y.T., Liu G.R. A local radial point interpolation method (LRPIM) for free vibration analyses of 2-d solids, Journal of Sound and Vibration,2001,246(1): 29-46.
[74]Wang J.G., Liu G.R. A point interpolation meshless method based on radial basis functions. International Journal for Numerical Methods in Engineering, 2002,54:1623-1648.
[75]Braun J. A numerical method for solving Partial differential equations on highly irregular evolving grids. Nature,1995,76:655-660.
[76]Sukumar N., Moran B., Belytsehko T. The natural element method in solid mechanics. International Journal for Numerical Methods in Engineering,1998, 43(5):839-887.
[77]Sukumar N., Moran B., Semenov A.Y., et al. Natural neighbour Galerkin methods. International Journal for Numerical Methods in Engineering,2001, 50(1):1-27.
[78]Duarte C.A., Oden J.T. Hp clouds-hp meshless method. Numerical Methods for Partial Differential Equations,1996,12(6):673-705.
[79]Duarte C.A, Oden J.T. An hp Adaptive Method Using Clouds, Computer Methods in Applied Mechanics and Engineering,1996,139:237-262.
[80]Liszka T.J., Duarte C.A., Tworzydlo W.W. Hp-Meshless Cloud Method, Computer Methods in Applied Mechanics and Engineering,1996,139: 263-288.
[81]Oden J.T., Duarte C. A., Zienkiewicz, O.C., A New Cloud-Based hp Finite Element Method, Computer Methods in Applied Mechanics and Engineering, 1998,153:117-126.
[82]Babuska I., Melenk J.M. The partition of unity methods. International Journal for Numerical Methods in Engineering,1998,40(4):727-758.
[83]Melenk J.M., Babuska I. The partition of unity finite element method:Basic theory and applications. Computer Methods in Applied Mechanics and Engineering,1996,139:289-314.
[84]Onate E., Idelsohn S., Zienkiewicz O.C. et al. A finite point method in computational mechanics:applications to convective transport and fluid flow. International Journal for Numerical Methods in Engineering,1998,39(22): 3839-3866.
[85]Onate E., Idelsohn S., Zienkiewicz O.C. et al. A stabilized finite point method for analysis of fluid mechanics problems. Computer Methods in Applied Mechanics and Engineering,1996,139:315-346.
[86]Mukherjee Y.X., Mukherjee S. The boundary node method for potential problems. International Journal for Numerical Methods in Engineering,1998, 40(5):795-815.
[87]张见明,姚振汉,李宏.二维势问题的杂交边界点法.重庆建筑大学学报,2000,22(6):105-107.
[88]Zhang J.M., Yao Z.H., Li H. A hybrid boundary node method. International Journal for Numerical Methods in Engineering,2002,53(4):751-763.
[89]张雄,胡炜,潘小飞等.加权最小二乘无网格法.力学学报,2003,35(4)：425-431.
[90]Beissel S., Belytschko T. Nodal integration of the element-free Galerkin method. Computer Methods in Applied Mechanics and Engineering,1996,139: 49-74.
[91]Liu G.R., Zhang G.Y., Wang Y.Y. A nodal integration technique for meshfree radial point interpolation method (NI-RPIM). International Journal of Solids and Structures,2007,44:3840-3860.
[92]Chen J.S., Wu C.T., Yoon S. et al. A stabilized conforming nodal integration for Galerkin meshfree methods. International Journal for Numerical Methods in Engineering,2001,50(2):435-466.
[93]Chen J.S., Yoon S., Wu C.T. Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods. International Journal for Numerical Methods in Engineering,2002,53:2587-2615.
[94]Yoo J., Moran B., Chen J.S. Stabilized conforming nodal integration in the natural element method. International Journal for Numerical Methods in Engineering,2004,60:861-890.
[95]Liu G.R., Zhang G.Y., Dai K.Y. et al. A linearly conforming point interpolation method (LC-PIM) for 2D solid mechanics problems. International Journal of Computational Methods,2005,2(4):645-665.
[96]Liu G.R, Li Y., Dai K.Y. A linearly conforming radial point interpolation method for solid mechanics problems. International Journal of Computational Methods,2006,3(4):401-428.
[97]Zhang G.Y., Liu G.R., Wang Y.Y. et al. A linearly conforming point interpolation method (LC-PIM) for three-dimensional elasticity problems. International Journal for Numerical Methods in Engineering,2007,72(13): 1524-1543.
[98]Liu G.R., Zhang G.Y. Upper bound solution to elasticity problems:A unique property of the linearly conforming point interpolation method (LC-PIM). International Journal for Numerical Methods in Engineering,2008,74(7): 1128-1161.
[99]Liu G.R., Dai K.Y, Nguyen, T.T. A smoothed finite element method for mechanics problems. Computational Mechanics,2007,39:859-877.
[100]Liu G.R., Nguyen T.T., Dai K.Y., Lam, K.Y. Theoretical aspects of the smoothed finite element method (SFEM). International Journal for Numerical Methods in Engineering,2007,71:902-930.
[101]Dai K.Y., Liu G.R. Free and forced vibration analysis using the smoothed finite element method (SFEM). Journal of Sound and Vibration,2007,301:803-820.
[102]Nguyen-Van H., Mai-Duy N., Tran-Cong, T. A smoothed four-node piezoelectric element analysis of two-dimensional smart structures. CMES: Computer Modeling in Engineering & Science,2008,23:209-222.
[103]Nguyen-Xuan H., Rabczuk T., Bordas S. et al. A smoothed finite element method for plate analysis. Computer Methods in Applied Mechanics and Engineering,2008,197:1184-1203.
[104]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. CMES- Computer Modeling in Engineering & Sciences,2008,28(2):109-126.
[105]Cui X.Y., Liu G.R., Li G.Y. An explicit smoothed finite element method (SFEM) for dynamic problems of solids. International Journal of Computational Methods, (Submitted)
[106]Liu G.R. A generalized gradient smoothing technique and smoothed bilinear form for Galerkin formulation of a wide class of computational methods. International Journal of Computational Methods,2008,5(2):199-236.
[107]Liu G.R. On a G space theory. Journal of Computational Methods,2009,6(2): 257-289.
[108]Liu G.R. A G space theory and weakened weak (W2) form for a unified formulation of compatible and incompatible methods, Part Ⅰ:Theory and Part Ⅱ: Applications to solid mechanics problems. International Journal for Numerical Methods in Engineering,2009,81:1093-1156.
[109]Liu G.R., Nguyen T.T., Nguyen H.X. et al. A node-based smoothed finite element method for upper bound solution to solid problems (NS-FEM). Computers and Structures,2008,87(1-2):14-26.
[110]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. Computer Methods in Applied Mechanics and Engineering,2008,197: 3883-3897.
[111]Liu G.R., Nguyen T.T., Lam K.Y. A novel FEM by scaling the gradient of strains with factor a (aFEM). Computational Mechanics,2008,411:457-472.
[112]Liu G.R., Nguyen-Xuan H., Nguyen-Thoi T., Xu X. A novel Galerkin-like weakform and a superconvergent alpha finite element method (SaFEM) for mechanics problems using triangular meshes. Journal of Computational Physics, 2009,228:4055-4087.
[113]Liu G.R., Xu X., Zhang G.Y. et al. An extended Galerkin weak form and a point interpolation method with continuous strain field and superconvergence using triangular mesh. Computational Mechanics,2009,43:369-391.
[114]Liu G.R, Nguyen T.T, Dai K.Y, Lam K.Y. An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analysis. Journal of Sound and Vibration,2008,320:1110-1130.
[115]Cui X.Y, Liu G.R, Li G.Y et al. Elasto-plasticity analysis using edge-based smoothed finite element method (ES-FEM). International Journal of Pressure Vessels and Piping,2009,86:711-718.
[116]Nguyen-Thoil T., Liu G.R., Lam K.Y., Zhang G.Y. A face-based smoothed finite element method (FS-FEM) for 3D linear and geometrically non-linear solid mechanics problems using 4-node tetrahedral elements. International Journal for Numerical Methods in Engineering,2009,78:324-353.
[117]Liu G.R., Zhang G.Y. Edge-based smoothed point interpolation methods. International Journal of Computational Methods,2008,5(4):621-646.
[118]Liu G.R., Zhang G.Y. A normed G space and weakened weak (W2) formulation of a cell-based Smoothed Point Interpolation Method. International Journal of Computational Methods,2009,6(1):147-179.
[119]Liu G.R., Zhang G.Y. A novel scheme of strain-constructed point interpolation method for static and dynamic mechanics problems. International Journal of Applied Mechanics,2009,1:233-258.
[120]Xu X. Liu G.R., Zhang G.Y. A point interpolation method with least square strain field (PIM-LSS) for solution bounds and ultra-accurate solutions using triangular mesh. Computer Methods in Applied Mechanics and Engineering, 2009,198:1486-1499.
[121]Cui X.Y., Liu G.R., Li G.Y. A cell-based smoothed radial point interpolation method (CS-RPIM) for static and free vibration of solids. Engineering Analysis with Boundary Elements,2010,34:144-157.
[122]Li Y., Liu G.R., Luan M.T. et al. Contact analysis for solids based on linearly conforming radial point interpolation method. Computational Mechanics,2007, 39(4):537-554.
[123]Cheng J., Chang X.L., Wu S.C., Zhang G.Y. Certified solutions for hydraulic structures using the node-based smoothed point interpolation method (NS-PIM). International journal for numerical and analytical methods in geomechanics, 2010,34(15):1560-1585.
[124]Zhang G.Y., Liu G.R., Li Y. An efficient adaptive analysis procedure for certified solutions with exact bounds of strain energy for elasticity problems. Finite Elements in Analysis and Design,2008,44(14):831-841.
[125]Chen L., Zhang G.Y., Zhang J. et al. An adaptive edge-based smoothed point interpolation method for mechanics problems. International Journal of Computer Mathematics, DOI:10.1080/00207160.2010.539682. (In press)
[126]Tang Q., Liu G.R., Zhong Z.H. et al. An efficient adaptive analysis procedure for node-based smoothed point interpolation method (NS-PIM), Applied Mathematics and Computation, doi:10.1016/j.amc.2011.03.036. (In press)
[127]Wu S.C., Liu G.R., Zhang H.O. et al. A node-based smoothed point interpolation method (NS-PIM) for three-dimensional thermoelastic problems. Numerical Heat Transfer A-Applications,2008,54(12):1121-1147.
[128]Wu S.C., Liu G.R., Zhang H.O. et al. A node-based smoothed point interpolation method (NS-PIM) for thermoelastic problems with solution bounds. International Journal of Heat and Mass Transfer,2009,52:1464-1471.
[129]Wu S.C., Liu G.R., Zhang H.O. et al. A node-based smoothed point interpolation method (NS-PIM) for three-dimensional heat transfer problems. International Journal of Thermal Sciences,2009,48(6):1154-1165.
[130]Wu S.C., Liu G.R., Cui X.Y. An Edge-based Smoothed Point Interpolation Method (ES-PIM) for Heat Transfer Analysis of Rapid Manufacturing System. International Journal of Heat and Mass Transfer,2010,53:1938-1950.
[131]Cui X.Y., Liu G.R., Li G.Y., Zheng G. A rotation free formulation for static and free vibration analysis of thin beams using gradient smoothing technique. CMES-Computer Modeling in Engineering & Sciences,2008,38(3):217-229.
[132]Cui X.Y., Liu G.R., Li G.Y. A curvature-constructed method for bending analysis of thin plates using 3-node triangular cells. International Journal of Computational methods,2010. (Accepted)
[133]Cui X.Y., Li G.Y. Nodal integration thin plate formulation using linear interpolation and triangular cells. International Journal of Computational methods,2010. (Accepted)
[134]Cui X.Y., Li G.Y., Zheng G. Novel thin plate element theory based on a continuity re-relaxed technique. Science China:Technological Sciences,2010, 53:2450-2457.
[135]Cui X.Y., Liu G.R., Li G.Y., Zhang G.Y. A thin plate formulation without rotation DOFs based on the radial point interpolation method and triangular cells. International Journal for Numerical Methods in Engineering,2011,85(8): 958-986.
[136]Kirchhoff G.R. Uber das gleichgewichi und die bewegung einer elastishem scheibe. J. Fuer die Reine und Angewandte Mathematik 1850; 40:51-88.
[137]Rissner E. The effect of transverse shear deformation on the bending of elastic plates. Jounral of Applied Mechnaics,1945,12:69-76.
[138]Mindlin R.D. Influence of rotatory inertia and shear in flexural motions of isotropic elastic plates. Jounral of Applied Mechnaics,1951,37:1031-1036.
[139]Bazeley G.P., Cheung Y.K., Irons B.M., Zienkiewicz O.C. Triangular elements in bending conforming and nonconforming solution. Proceedings of the Conference on Matrix Methods in Structural Mechanics,1965,547-576.
[140]Clough R.W., Tocher J.L. A conforming triangular finite element plate bending solution. International Journal for Numerical Methods in Engineering,1970,2, 259-264.
[141]Morley L.S.D. The constant-moment plate-bending element. Journal of strain analysis,1971,6:20-24.
[142]Alwood R.J., Cornes G.M. A polygonal finite element for plate bending problems using the assumed stress approach. International Journal for Numerical Methods in Engineering,1969,1:135-149.
[143]Batoz J.L., Bathe K.J., Ho L.W. A study of three-node triangular plate bending elements, International Journal for Numerical Methods in Engineering,1980, 15:1771-1812.
[144]Hughes T.J.R, Cohen M., Haroun M. Reduced and selective integration techniques in finite element analysis of plates. Nuclear Engineering Design, 1978,46:203-222.
[145]Belytschko T., Stolarski H. Stress projection for membrane and shear locking in shell elements. Computer Methods in Applied Mechanics and Engineering, 1985,51:221-258.
[146]Liu W.K., Law E.S., Lam D., Belytschko T. Resultant-stress Degenerated-shell Element. Computer Methods in Applied Mechanics and Engineering,1986, 55:259-300.
[147]Bathe K.J, Dvorkin E.N. A formulation of general shell elements-the use of mixed interpolation of tensorial components. International Journal for Numerical Methods in Engineering,1986,22:697-722.
[148]Belytschko, T., Lin, J.I., Tsay, C. Explicit Algorithms for the Nonlinear Dynamics of Shells. Computer Methods in Applied Mechanics and Engineering, 1984,42:225-251.
[149]Belytschko T., Stolarski H., Carpenter N. A Co triangular plate element with one-point quadrature. International Journal for Numerical Methods in Engineering,1984,20:787-802.
[150]Bathe K.J., Brezzi F. The MITC7 and MITC9 plate bending element. Computers & Structures,1989,32:797-814.
[151]Bathe K.J., Iosilevich A., Chapelle D. An evaluation of the MITC shell elements. Computers & Structures,2000,75:1-30.
[152]Lee P.S., Bathe K.J. Development of MITC isotropic triangular shell finite elements. Computers & Structures,2004,82:945-962.
[153]Bletzinger K.U., Bischoff M., Ramm E. A unified approach for shear-locking-free triangular and rectangular shell finite elements, Computers & Structures,2000,75:321-334.
[154]Kim J.H., Kim Y.H. Three-node macro triangular shell element based on the assumed natural strains. Computational Mechanics,2002,29:441-458.
[155]Chen W.J., Cheung Y.K. Refined 9-Dof triangular Mindin plate elements. International Journal for Numerical Methods in Engineering,2001,51: 1259-1281.
[156]Chen W.J. Refined 15-DOF triangular discrete degenerated shell element, International Journal for Numerical Methods in Engineering,2004,60: 1817-1846.
[157]岑松,龙志飞.一种新型厚薄板通用三角形广义协调元.工程力学.1998,15：10-22.
[158]陈永亮,岑松,姚振汉,龙驭球.厚薄通用三角形三结点平板壳元TSLT18.清华大学学报(自然科学版).2003,48：1069-1073.
[159]Dhatt G. Numerical analysis of thin shells by curved triangular elements based on discrete Kirchhoff hypothesis. Proceedings of the ASCE Symposium on Applications of FEM in Civil Engineering,1969,255-278.
[160]Dhatt G. An efficient triangular shell element (Kirchhoff triangular shell element design via linear shell theory), AIAA Journal,1970,8:2100-2102.
[161]Ayad R., Dhatt G., Batoz J.L. A new hybrid-mixed variational approach for Reissner-Mindlin plates, The MiSP model. International Journal for Numerical Methods in Engineering,1998,42:1149-1479.
[162]Wu S., Li G.Y., Belytschko T. A DKT shell element for dynamic large deformation analysis. Communications in numerical methods in engineering, 2005,21:651-674.
[163]Cui X.Y., Liu G.R., Li G.Y., Zhang G.Y., Zheng G. Analysis of plates and shells using an edge-based smoothed finite element method. Computational Mechanics,2010,45:141-156.
[164]Zheng G., Cui X.Y., Li G.Y., Wu S.Z. An edge-based smoothed triangle element for non-linear explicit dynamic analysis of shells. Computational Mechanics, 2011,48(1):65-80.
[165]Liu G.R., Chen X.L. A mesh-free Galerkin method for static and free vibration analyses of thin plates of complicated shape. Journal of Sound and Vibration, 2001,241(5):839-855.
[166]Liu L., Liu G.R., Tan V.B.C. Element free method for static and free vibration analysis of spatial thin shell structures. Computer Methods in Applied Mechanics and Engineering,2002,191:5923-5942.
[167]Gu Y.T., Liu G.R. A meshless local Petrov-Galerkin (MLPG) formulation for static and free vibration analyses of thin plates. Computer Modeling in Engineering & Sciences,2001,2(4):463-476.
[168]Long S.R., Atluri S.N. A meshless local Petrov-Galerkin (MLPG) method for solving the bending problem of a thin plate. Computer Modeling in Engineering & Sciences,2002,3:53-63.
[169]Wang D.D., Chen J.S. A Hermite reproducing kernel approximation for thin-plate analysis with sub-domain stabilized conforming integration. International Journal for Numerical Methods in Engineering,2008,74(3): 368-390.
[170]Cui X.Y., Liu G.R., Li G.Y. A smoothed Hermite radial point interpolation method for thin plate analysis using gradient smoothing operation. Archive of Applied Mechanics,2011,81(1):1-18.
[171]Donning B.M., Liu W.K. Meshless methods for shear-deformable beams and plates. Computer Methods in Applied Mechanics and Engineering,1998,152: 47-71.
[172]Noguchi H., Kawashima T., Miyamura T. Element free analyses of shell and spatial structures. International Journal for Numerical Methods in Engineering, 2000,47(6):1215-1240.
[173]Kanok-Nukulchai W., Barry W., Saran-Yasoontorn K, Bouillard PH. On elimination of shear locking in the element-free Galerkin method. International Journal for Numerical Methods in Engineering,2001,52(7):705-725.
[174]Soric J., Li Q., Jarak T., Atluri S.N. Meshless local Petrov-Galerkin (MLPG) formulation for analysis of thick plates. Computer Modeling in Engineering and Sciences,2004,6(4):349-357.
[175]Sladek J., Sladek V., Mang H.A. Meshless formulations for simply supported and clamped plate problems. International Journal for Numerical Methods in Engineering,2002,55(3):359-375.
[176]Liu L., Chua L.P., Ghista D.N. Conforming radial point interpolation method for spatial shell structures on the stress-resultant shell theory. Archive of Applied Mechanics,2006,75(4-5):248-267.
[177]Liu L., Chua L.P., Ghista D.N. Applications of point interpolation method for spatial general shells structures. Computer Methods in Applied Mechanics and Engineering,2007,196(9-12):1633-1647.
[178]Wang D.D., Chen J.S. Locking-free stabilized conforming nodal integration for meshfree Mindlin-Reissener plate formulation. Computer Methods in Applied Mechanics and Engineering,2004,193:1065-1083.
[179]Chen J.S., Wang D.D. A constrained reproducing kernel particle formulation for shear deformable shell in Cartesian coordinates. International Journal for Numerical Methods in Engineering,2006,68(2):151-172.
[180]Zhang Z., Noguchi H., Chen J.S. Moving least-squares approximation with discontinuous derivative basis functions for shell structures with slope discontinuities. International Journal for Numerical Methods in Engineering, 2008,76(8):1202-1230.
[181]Liu G.R., Zhao X., Dai K.Y., et al. Static and free vibration analysis of laminated composite plates using the conforming radial point interpolation method. Composites Science and technology,2008,68:354-366.
[182]Zhao X., Liu G.R., Dai K.Y., et al. Free-vibration analysis of shells via a linearly conforming radial point interpolation method (LC-RPIM). Finite Elements in Analysis and Design,2009,45:917-924.
[183]Zhao X., Liu G.R., Dai K.Y., et al. A linearly conforming radial point interpolation method (LC-RPIM) for shells. Computational Mechanics,2009, 43:403-413.
[184]Zhao X., Liu G.R, Dai K.Y, et al. Geometric nonlinear analysis of plates and cylindrical shells via a linearly conforming radial point interpolation method. Computational Mechanics,2009,42:133-144.
[185]Hughes T.J.R. The finite element method, Prentice-Hall, London,1987.
[186]Golberg M.A., Chen C.S. Bowman H. Some recent results and proposals for the use of radial basis functions in the BEM. Engineering Analysis with Boundary Elements,1999,23:285-296.
[187]Timoshenko S.P., Goodier J.N. Theory of Elasticity (3rd edn). McGraw-Hill: New York,1970.
[188]Nagashima T. Node-by-node meshless approach and its application to structural analysis. International Journal for Numerical Methods in Engineering,1999, 46:341-385.
[189]Jahed H., Sethuraman R., Dubey R.N. A variable material property approach for solving elastic-plastic problems, International Journal of Pressure Vessels and Piping,1997,71:285-291.
[190]Desikan V., Sethuraman R. Analysis of material nonlinear problems using pseudo-elastic finite element method, Journal of pressure vessel technology-transactions of the ASME,2000,122:457-461.
[191]Zienkiewicz O.C., Taylor R.L. The Finite Element Method. The fifth edition, Vol 2:Solid Mechanics, Butterworth-Heinemann, Oxford,2000.
[192]Fung Y.C. Foundations of solid mechanics. London:Prentice Hall,1965.
[193]Ugural A.C. Stresses in plates and shells.McGraw-Hill, New York,1981.
[194]Gorman D.J. Free vibration analysis of rectangular plates. New York:Elsevier, 1982.
[195]Abbassian F., Dawswell D.J., Knowles N.C. Free vibration benchmarks. Atkins Engineering Sciences:Glasgow,1987.
[196]Crisfield M.A. Non-Linear Finite Element Analysis of Solids and Structures. Vol.1. Wiley:Chichester,1997.
[197]Malkus D.S., Hughes T.J.R. Mixed finite element methods-reduced and selective integration techniques:A unification of concepts. Computer Methods in Applied Mechanics and Engineering,1978,15:63-81.
[198]Reismann H. Elastic Plates:Theory and Applications. Wiley, New York,1988.
[199]Bathe K.J., Dvorkin E.H. A four-node plate bending element based on Mindlin-Reissner plate theory and mixed interpolation. International Journal for Numerical Methods in Engineering,1985,21:367-383.
[200]Batoz J.L., Tahar M.B. Evaluation of a new quadrilateral thin plate bending element. International Journal for Numerical Methods in Engineering,1982,18: 1655-1677.
[201]Chia C.Y. Nonlinear Analysis of Plate. McGraw-Hill, New York,1980.
[202]Hughes T.J.R., Liu W.K. Nonlinear finite element analysis shells:Part Ⅰ, Three-dimensional shells. Computer Methods in Applied Mechanics and Engineering,1981,26:331-362.
[203]Engelmann B.E., Whirley R.G. A new elasto-plastic shell element formulation for DYNA3D, Lawrence Livermore National Laboratory, Report UCRL-JC-104826,1990.
[204]Dvorkin E.N., Bathe K.J. A continuum mechanics based four-node shell element for general nonlinear analysis, Engineering Computations,1984,1: 77-88.
[205]Belytschko T., Leviathan I. Physical stabilization of the 4-node shell element with one point quadrature. Computer Methods in Applied Mechanics and Engineering,1994,113:321-350.
[206]Horrigmoe G., Bergan P.G. Non-linear analysis of free-form shells by flat finite elements. Computer Methods in Applied Mechanics and Engineering,1978:16: 11-35.
[207]Dhatt G.S. Instability of thin shells by the finite element method, IASS Symposium for Folded Plates and Prismatic Structures, Vienna,1970.
[208]Sabir A.B., Lock A.C. The application of finite elements to the large deflection geometrically nonlinear behaviour of cylindrical shells. International Conference on Variational Methods in Engineering, Southampton, England; United Kingdom,1973.
[209]Batoz J.L. An explicit formulation for an efficient triangular plate-bending element. International Journal for Numerical Methods in Engineering,1982,18: 1077-1089.
[210]Timoshenko S., Woinowsky-Krieger S. Theory of Plates and Shells. New York: McGraw-Hill,1940.
[211]Sze K.Y, Zhu D. A quadratic assumed natural strain shell curved triangular element. Computer Methods in Applied Mechanics and Engineering,1999,174: 57-71.
[212]Jirousek J, Lan Guex. The hybrid-trefftz finite element model and its application to plate bending. International Journal for Numerical Methods in Engineering,1986,23:651-693.
[213]Scordelis A.C., Lo K.S. Computer Analysis of Cylindrical Shells. Journal of the American Concrete Institute,1969,61:539-561.
[214]Simo J.C., Hughes T.J.R. Computational Inelasticity, New York, Springer Verlag,1998.
[215]Zhong Z.H. Finite Element Procedures for Contact-Impact Problems, Oxford University Press, New York,1993.
[216]钟志华,李光耀.薄板冲压成形过程的计算机仿真与应用,第1版.北京：北京理工大学出版社,1998,1-7
[217]Male A.T., Cockcroft M.G. A Method for the Determination of the Coefficient of Friction of Metals under Conditons of Bulk Plastic Deformation, Journal of the Institute of Metals,1965,93:38-46.
[218]Morino L., Leech J.W., Witmer E.A. An improved numerical calculation technique for large elastic-plastic transient deformations of thin shells:Part 2-evaluation and applications. Journal of Applied Mechanics,1971,38:429-435.
[2]Zienkiewicz O.C., Taylor R.L. The Finite Element Method. (6th ed.). New York: Elsevier Butterworth-Heinemann,2005.
[3]Liu G.R. Meshfree Methods:Moving Beyond the Finite Element Method. Boca Raton:CRC Press,2002.
[4]Liu G.R. Meshfree methods:Moving Beyond the Finite Element Method,2nd edition. Boca Raton:CRC Press,2009.
[5]Atluri S.N., Shen S.P. The Meshless Local Petrov-Galerkin (MLPG) Method. Netherlands:Springer Press,2005.
[6]Liu G.R., Gu Y.T. An Introduction to Meshfree Methods and Their Programming. Netherlands:Springer Press,2005.
[7]Belytschko T., Krongauz Y., Organ D. Meshless method:an overview and recent development. Computer methods in Applied Mechanics and Engineering, 1996,139(1-4):3-47.
[8]张雄,刘岩.无网格法.北京：清华大学出版社,2004.
[9]刘更,刘天祥,谢琴.无网格法及其应用.西安：西北工业大学出版社,2005.
[10]Lucy L.B. A numerical approach to testing of the fission hypothesis. The Astronomical Journal,1977,8(12):1013-1024.
[11]Gingold R.A., Monaghan J.J. Smoothed Particle hydrodynamics:theory and applications to non-spherical stars. Royal Astronomical Society, Monthly Notic, 1977,181:375-389.
[12]Monaghan J.J. An introduction to SPH. Computer Physics Communications, 1998,48(1):89-96.
[13]张锁春.光滑质点流体动力学(SPH)方法(综述).计算物理,1996,13(4)：385-397.
[14]Liu G.R., Liu M.B. Smoothed Particle Hydrodynamics-A Meshfree Particle Method. Singapore:World Scientific Publishing Company,2003.
[15]Liu G.R., Liu M.B著(韩旭,杨刚,强洪夫(译)).光滑粒子流体动力学—种无网格粒子法.湖南大学出版社,2005.
[16]Swegle J.W., Attaway S.W. On the feasibility of using smoothed particle hydrodynamics for underwater explosion calculations. Computational Mechanics,1995,17(3):151-168.
[17]Liu M.B., Liu G.R., Lam K.Y. et al. Smoothed particle hydrodynamics for numerical simulation of underwater explosion. Computational Mechanics,2003, 30(2):106-118.
[18]Cleary P.W., Prakash M., Ha J. Novel applications of smoothed particle hydrodynamics (SPH) in metal forming. Journal of Materials Processing Technology,2006,177(1-3):41-48.
[19]Jeong J.H., Jhon M.S., Halow J.S. et al. Smoothed particle hydrodynamics: Applications to heat conduction. Computer Physics Communications,2003, 153(1):71-84.
[20]Monaghan J.J., Huppert H.E., Worster M.G. Solidification using smoothed particle hydrodynamics. Journal of Computational Physics,2005,206(2): 684-705.
[21]Zhang M.Y., Zhang H., Zheng L.L. Application of smoothed particle hydrodynamics method to free surface and solidification problems. Numerical Heat Transfer A-Applications,2007,52(4):299-314.
[22]Zhang M.Y., Zhang H., Zheng L.L. Simulation of droplet spreading, splashing and solidification using smoothed particle hydrodynamics method. International Journal of Heat and Mass Transfer,2008,51(13-14):3410-3419.
[23]Lancaster P., Salkauskas K. Surfaces generated by moving least square methods. Mathematics of Computation,1981,37(155):141-158.
[24]Nayroles B., Touzot G., Villon P. Generalized the finite element method:diffuse approximation and diffuse elements. Computational Mechanics,1992,10(5): 307-318.
[25]Belytschko T., Lu Y.Y., Gu L. Element-free Galerkin method. International Journal for Numerical Methods in Engineering,1994,37(2):229-256.
[26]Belytschko T., Organ D., Krongauz Y. A coupled finite element-element-free Galerkin method. Computational Mechanics,1995,17:186-195.
[27]Gu Y.T., Liu G.R. A coupled element free Galerkin/boundary element method for stress analysis of two-dimensional solids. Computer Methods in Applied Mechanics and Engineering,2001,190:4405-4419.
[28]Ponthot J.P., Belytsehko T. Arbitrary Lagrangian-Eulerian formulation for element-free Galerkin method. Computer Methods in Applied Mechanics and Engineering,1998,152:19-46.
[29]Liu G.R., Tu Z.H. An adaptive procedure based on background cells for meshless method. Computer Methods in Applied Mechanics and Engineering, 2002,191:1923-1943.
[30]Chung H.J., Belytsehko T. An error estimate in the EFG method. Computational Mechanics,1998,21(2):91-100.
[31]Gavete L., Cuesta J.L., Ruiz A. A procedure for approximation of the error in the EFG method. International Journal for Numerical Methods in Engineering, 2002,53(3):677-690.
[32]Gavete L., Gavete M.L., Alonso B., Martin A.J. A posteriori error approximation in EFG method. International Journal for Numerical Methods in Engineering,2003,58:2239-2263.
[33]Krysl P., Belytschko T. Analysis of thin plates by the element-free Galerkin method. Computational Mechanics,1995,17(1):26-35.
[34]Krysl P., Belytschko T. Analysis of thin shells by the element-free Galerkin method. International Journal of Solids and Structures,1996,33:3057-3080.
[35]Belytschko T., Lu Y.Y., Gu L. Fracture and crack growth by element-free Galerkin method. Modeling and Simulation in Material Science and Engineering,1994,2:519-534.
[36]Belytschko T., Tabbara M. Dynamic fracture using element-free Galerkin method. International Journal for Numerical Methods in Engineering,1996,39: 141-158.
[37]Kargarnovin M.H., Toussi H.E., Fariborz S.J. Elasto-Plastic element-free Galerkin method. Computational Mechanics,2004,33(3):206-214.
[38]Ren J., Liew K.M., Meguid S.A. Modelling and simulation of the superelastic behaviour of shape memory alloys using the element-free Galerkin method. International Journal of Mechanical Science,2002,44(12):393-2413.
[39]Belytschko T., Krysl P., Krongauz Y. A three-dimensional explicit element-free Galerkin method. International Journal for Numerical Methods in Fluids,1997, 24(12):1253-1270.
[40]Li G.Y., Belytscko T. Element-free Galerkin method for contact problems in metal forming analysis. Engineering Computation,2001,18(1):62-78.
[41]Guan Y.J., Wu X., Zhao G.Q., Lu P. A nonlinear numerical analysis for metal-forming process using the rigid-(visco) plastic element-free Galerkin method. International Journal of Advanced Manufacturing Technology,2009, 42:83-92.
[42]Liu W.K., Jun S., Zhang Y.F. Reproducing kernel particle methods. International Journal for Numerical Methods in Fluids,1995,20:1081-1106.
[43]Liu W.K., Chen Y. Wavelet and multiple scale reprodueing kernel method. International Journal for Numerical Methods in Fluids,1995,21:901-931.
[44]Liu W.K., Jun S., Li S.F. Reprodueing kernel particle methods for structural dynamics. International Journal for Numerical Methods in Engineering.1995, 38:1655-1679.
[45]Liu W.K., Jun S., Sihling D.T. et al. Multiresolution reproducing kernel particle method for computational fluid dynamics. International Journal for Numerical Methods in Fluids,1997,24(12):1391-1415.
[46]Chen J.S., Pan C., Wu C.T. Large deformation analysis of rubber based on a reproducing kernel particle method. Computational Mechanics,1997,19(3): 211-227.
[47]Liew K.M., Ng T.Y., Wu Y.C. Meshfree method for large deformation analysis-a reproducing kernel particle approach. Engineering Structures,2002, 24(5):543-551.
[48]Li G.Y., Sidibe K., Liu G.R. Meshfree method for 3D forming analysis with lower order integration scheme. Engineering Analysis with Boundary Elements, 2004,28(10):1283-1292.
[49]Park Y.H. Rigid-plastic analysis for metal forming processes using a reproducing kernel particle method. Journal of Materials Processing Technology,2007,183(2-3):256-263.
[50]Atluri S.N., Zhu T. A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Computational Mechanics,1998,22(2):117-127.
[51]Zhu T., Zhang J.D., Atluri S.N. A local boundary integral equation (LBIE) method in computational mechanics and a meshless diseretization approach. Computational Mechanics,1998,21(3):223-235.
[52]Sladek J., Sladek V., Zhang C. et al. Analysis of orthotropic thick plates by meshless local Petrov-Galerkin method (MLPG) method. International Journal for Numerical Methods in Engineering,2006,67(13):1830-1850.
[53]Qian L.F., Batra R.C. Three-dimensional transient heat conduction in a functionally graded thick plate with a higher-order plate theory and a meshless local Petrov-Galerkin method. Computational Mechanics,2005,35(3): 214-226.
[54]Sladek J., Sladek V., Wen P. et al. Meshless local Petrov-Galerkin method for shear deformable shells analysis. CMES-Computer Modeling in Engineering and Science,2006,13(2):103-117.
[55]Li D., Lin Z.Q., Li S.H. Numerical analysis of Mindlin shell by meshless local Petrov-Galerkin method. Acta Mechanica Solida Sinica,2008,21(2):160-169.
[56]Xiong Y.B., Long S.Y., Hu D.A. et al. A meshless local Petrov-Galerkin method for geometrically nonlinear analysis. Acta Mechanica Solida Sinica,2005, 18(4):348-356.
[57]Hu D.A., Long S.Y., Han X. et al. A meshless Petrov-Galerkin method for large deformation contact analysis of elastomers. Engineering Analysis with Boundary Elements,2007,31(7):61-74.
[58]张希.无网格局部彼得洛夫伽辽金法在大变形问题中的应用.清华大学博士论文,2006.
[59]Arefmanesh A., Najafi M., Abdi H. A meshless local Petrov-Galerkin method for fluid dynamics and heat transfer applications. Journal of Fluids Engineering-Transactions of the ASME,2005,127(4):647-655.
[60]Sladek J., Sladek V., Hellmich C. et al. Heat conduction analysis of 3D axisymmetric and anisotropic FGM bodies by meshless local Petrov-Galerkin method. Computational Mechnics,2007,39(3):323-333.
[61]Sladek J., Sladek V., Zhang C. et al. Meshless local Petrov-Galerkin method for linear coupled thermoelastic analysis. CMES-Computer Modeling in Engineering and Science,2007,16(1):57-68.
[62]Liu K.Y., Long S.Y., Li G.Y. A simple and less-costly meshless local Petrov-Galerkin method for the dynamic fracture problem. Engineering Analysis with Boundary Elements,2006,30(1):72-76.
[63]Long S.Y., Liu K.Y., Li G.Y. An analysis for the elasto-plastic fracture problem by the meshless local Petrov-Galerkin method. CMES-Computer Modeling in Engineering and Science,2008,28(3):203-216.
[64]Zhu T., Atluri N.. Modified collocation method and a penalty function for enforcing essential boundary conditions in the element free Galerkin method. Computational Mechanics,1998,21:211-222.
[65]Lu Y.Y., Belytschko T., Gu L. A new implementation of the element-free Galerkin method. Computer Methods in Applied Mechanics and Engineering, 1994,113:397-414.
[66]Kaljevic I., Saigal S. An improved element free Galerkin formulation. International Journal for Numerical Methods in Engineering,1997,40: 2953-2974.
[67]Krongauz Y., Belytschko T. Enforcement of essential boundary conditions in meshless approximations using finite elements. Computer Methods in Applied Mechanics and Engineering,1996,131:133-145.
[68]Most T. A natural neighbour-based moving least-squares approach for the element-free Galerkin method. International Journal for Numerical Methods in Engineering,2007,71:224-252.
[69]Liu G.R., Gu Y.T. A point interpolation method for two-dimensional solids. International Journal for Numerical Methods in Engineering,2001,50:937-951.
[70]Gu Y.T., Liu G.R. A boundary Point interpolation method for stress analysis of solids. Computational Mechanics,2002,28(1):47-54.
[71]Liu G.R., Gu Y.T. A local point interpolation method for stress analysis of two-dimensional solids. International Journal of Structural Engineering and Mechanics,2001,11:221-236.
[72]Gu Y.T., Liu G.R. A local point interpolation method for static and dynamic analysis of thin beams. Computer Methods in Applied Mechanics and Engineering,2001,190:5515-5528.
[73]Gu Y.T., Liu G.R. A local radial point interpolation method (LRPIM) for free vibration analyses of 2-d solids, Journal of Sound and Vibration,2001,246(1): 29-46.
[74]Wang J.G., Liu G.R. A point interpolation meshless method based on radial basis functions. International Journal for Numerical Methods in Engineering, 2002,54:1623-1648.
[75]Braun J. A numerical method for solving Partial differential equations on highly irregular evolving grids. Nature,1995,76:655-660.
[76]Sukumar N., Moran B., Belytsehko T. The natural element method in solid mechanics. International Journal for Numerical Methods in Engineering,1998, 43(5):839-887.
[77]Sukumar N., Moran B., Semenov A.Y., et al. Natural neighbour Galerkin methods. International Journal for Numerical Methods in Engineering,2001, 50(1):1-27.
[78]Duarte C.A., Oden J.T. Hp clouds-hp meshless method. Numerical Methods for Partial Differential Equations,1996,12(6):673-705.
[79]Duarte C.A, Oden J.T. An hp Adaptive Method Using Clouds, Computer Methods in Applied Mechanics and Engineering,1996,139:237-262.
[80]Liszka T.J., Duarte C.A., Tworzydlo W.W. Hp-Meshless Cloud Method, Computer Methods in Applied Mechanics and Engineering,1996,139: 263-288.
[81]Oden J.T., Duarte C. A., Zienkiewicz, O.C., A New Cloud-Based hp Finite Element Method, Computer Methods in Applied Mechanics and Engineering, 1998,153:117-126.
[82]Babuska I., Melenk J.M. The partition of unity methods. International Journal for Numerical Methods in Engineering,1998,40(4):727-758.
[83]Melenk J.M., Babuska I. The partition of unity finite element method:Basic theory and applications. Computer Methods in Applied Mechanics and Engineering,1996,139:289-314.
[84]Onate E., Idelsohn S., Zienkiewicz O.C. et al. A finite point method in computational mechanics:applications to convective transport and fluid flow. International Journal for Numerical Methods in Engineering,1998,39(22): 3839-3866.
[85]Onate E., Idelsohn S., Zienkiewicz O.C. et al. A stabilized finite point method for analysis of fluid mechanics problems. Computer Methods in Applied Mechanics and Engineering,1996,139:315-346.
[86]Mukherjee Y.X., Mukherjee S. The boundary node method for potential problems. International Journal for Numerical Methods in Engineering,1998, 40(5):795-815.
[87]张见明,姚振汉,李宏.二维势问题的杂交边界点法.重庆建筑大学学报,2000,22(6):105-107.
[88]Zhang J.M., Yao Z.H., Li H. A hybrid boundary node method. International Journal for Numerical Methods in Engineering,2002,53(4):751-763.
[89]张雄,胡炜,潘小飞等.加权最小二乘无网格法.力学学报,2003,35(4)：425-431.
[90]Beissel S., Belytschko T. Nodal integration of the element-free Galerkin method. Computer Methods in Applied Mechanics and Engineering,1996,139: 49-74.
[91]Liu G.R., Zhang G.Y., Wang Y.Y. A nodal integration technique for meshfree radial point interpolation method (NI-RPIM). International Journal of Solids and Structures,2007,44:3840-3860.
[92]Chen J.S., Wu C.T., Yoon S. et al. A stabilized conforming nodal integration for Galerkin meshfree methods. International Journal for Numerical Methods in Engineering,2001,50(2):435-466.
[93]Chen J.S., Yoon S., Wu C.T. Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods. International Journal for Numerical Methods in Engineering,2002,53:2587-2615.
[94]Yoo J., Moran B., Chen J.S. Stabilized conforming nodal integration in the natural element method. International Journal for Numerical Methods in Engineering,2004,60:861-890.
[95]Liu G.R., Zhang G.Y., Dai K.Y. et al. A linearly conforming point interpolation method (LC-PIM) for 2D solid mechanics problems. International Journal of Computational Methods,2005,2(4):645-665.
[96]Liu G.R, Li Y., Dai K.Y. A linearly conforming radial point interpolation method for solid mechanics problems. International Journal of Computational Methods,2006,3(4):401-428.
[97]Zhang G.Y., Liu G.R., Wang Y.Y. et al. A linearly conforming point interpolation method (LC-PIM) for three-dimensional elasticity problems. International Journal for Numerical Methods in Engineering,2007,72(13): 1524-1543.
[98]Liu G.R., Zhang G.Y. Upper bound solution to elasticity problems:A unique property of the linearly conforming point interpolation method (LC-PIM). International Journal for Numerical Methods in Engineering,2008,74(7): 1128-1161.
[99]Liu G.R., Dai K.Y, Nguyen, T.T. A smoothed finite element method for mechanics problems. Computational Mechanics,2007,39:859-877.
[100]Liu G.R., Nguyen T.T., Dai K.Y., Lam, K.Y. Theoretical aspects of the smoothed finite element method (SFEM). International Journal for Numerical Methods in Engineering,2007,71:902-930.
[101]Dai K.Y., Liu G.R. Free and forced vibration analysis using the smoothed finite element method (SFEM). Journal of Sound and Vibration,2007,301:803-820.
[102]Nguyen-Van H., Mai-Duy N., Tran-Cong, T. A smoothed four-node piezoelectric element analysis of two-dimensional smart structures. CMES: Computer Modeling in Engineering & Science,2008,23:209-222.
[103]Nguyen-Xuan H., Rabczuk T., Bordas S. et al. A smoothed finite element method for plate analysis. Computer Methods in Applied Mechanics and Engineering,2008,197:1184-1203.
[104]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. CMES- Computer Modeling in Engineering & Sciences,2008,28(2):109-126.
[105]Cui X.Y., Liu G.R., Li G.Y. An explicit smoothed finite element method (SFEM) for dynamic problems of solids. International Journal of Computational Methods, (Submitted)
[106]Liu G.R. A generalized gradient smoothing technique and smoothed bilinear form for Galerkin formulation of a wide class of computational methods. International Journal of Computational Methods,2008,5(2):199-236.
[107]Liu G.R. On a G space theory. Journal of Computational Methods,2009,6(2): 257-289.
[108]Liu G.R. A G space theory and weakened weak (W2) form for a unified formulation of compatible and incompatible methods, Part Ⅰ:Theory and Part Ⅱ: Applications to solid mechanics problems. International Journal for Numerical Methods in Engineering,2009,81:1093-1156.
[109]Liu G.R., Nguyen T.T., Nguyen H.X. et al. A node-based smoothed finite element method for upper bound solution to solid problems (NS-FEM). Computers and Structures,2008,87(1-2):14-26.
[110]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. Computer Methods in Applied Mechanics and Engineering,2008,197: 3883-3897.
[111]Liu G.R., Nguyen T.T., Lam K.Y. A novel FEM by scaling the gradient of strains with factor a (aFEM). Computational Mechanics,2008,411:457-472.
[112]Liu G.R., Nguyen-Xuan H., Nguyen-Thoi T., Xu X. A novel Galerkin-like weakform and a superconvergent alpha finite element method (SaFEM) for mechanics problems using triangular meshes. Journal of Computational Physics, 2009,228:4055-4087.
[113]Liu G.R., Xu X., Zhang G.Y. et al. An extended Galerkin weak form and a point interpolation method with continuous strain field and superconvergence using triangular mesh. Computational Mechanics,2009,43:369-391.
[114]Liu G.R, Nguyen T.T, Dai K.Y, Lam K.Y. An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analysis. Journal of Sound and Vibration,2008,320:1110-1130.
[115]Cui X.Y, Liu G.R, Li G.Y et al. Elasto-plasticity analysis using edge-based smoothed finite element method (ES-FEM). International Journal of Pressure Vessels and Piping,2009,86:711-718.
[116]Nguyen-Thoil T., Liu G.R., Lam K.Y., Zhang G.Y. A face-based smoothed finite element method (FS-FEM) for 3D linear and geometrically non-linear solid mechanics problems using 4-node tetrahedral elements. International Journal for Numerical Methods in Engineering,2009,78:324-353.
[117]Liu G.R., Zhang G.Y. Edge-based smoothed point interpolation methods. International Journal of Computational Methods,2008,5(4):621-646.
[118]Liu G.R., Zhang G.Y. A normed G space and weakened weak (W2) formulation of a cell-based Smoothed Point Interpolation Method. International Journal of Computational Methods,2009,6(1):147-179.
[119]Liu G.R., Zhang G.Y. A novel scheme of strain-constructed point interpolation method for static and dynamic mechanics problems. International Journal of Applied Mechanics,2009,1:233-258.
[120]Xu X. Liu G.R., Zhang G.Y. A point interpolation method with least square strain field (PIM-LSS) for solution bounds and ultra-accurate solutions using triangular mesh. Computer Methods in Applied Mechanics and Engineering, 2009,198:1486-1499.
[121]Cui X.Y., Liu G.R., Li G.Y. A cell-based smoothed radial point interpolation method (CS-RPIM) for static and free vibration of solids. Engineering Analysis with Boundary Elements,2010,34:144-157.
[122]Li Y., Liu G.R., Luan M.T. et al. Contact analysis for solids based on linearly conforming radial point interpolation method. Computational Mechanics,2007, 39(4):537-554.
[123]Cheng J., Chang X.L., Wu S.C., Zhang G.Y. Certified solutions for hydraulic structures using the node-based smoothed point interpolation method (NS-PIM). International journal for numerical and analytical methods in geomechanics, 2010,34(15):1560-1585.
[124]Zhang G.Y., Liu G.R., Li Y. An efficient adaptive analysis procedure for certified solutions with exact bounds of strain energy for elasticity problems. Finite Elements in Analysis and Design,2008,44(14):831-841.
[125]Chen L., Zhang G.Y., Zhang J. et al. An adaptive edge-based smoothed point interpolation method for mechanics problems. International Journal of Computer Mathematics, DOI:10.1080/00207160.2010.539682. (In press)
[126]Tang Q., Liu G.R., Zhong Z.H. et al. An efficient adaptive analysis procedure for node-based smoothed point interpolation method (NS-PIM), Applied Mathematics and Computation, doi:10.1016/j.amc.2011.03.036. (In press)
[127]Wu S.C., Liu G.R., Zhang H.O. et al. A node-based smoothed point interpolation method (NS-PIM) for three-dimensional thermoelastic problems. Numerical Heat Transfer A-Applications,2008,54(12):1121-1147.
[128]Wu S.C., Liu G.R., Zhang H.O. et al. A node-based smoothed point interpolation method (NS-PIM) for thermoelastic problems with solution bounds. International Journal of Heat and Mass Transfer,2009,52:1464-1471.
[129]Wu S.C., Liu G.R., Zhang H.O. et al. A node-based smoothed point interpolation method (NS-PIM) for three-dimensional heat transfer problems. International Journal of Thermal Sciences,2009,48(6):1154-1165.
[130]Wu S.C., Liu G.R., Cui X.Y. An Edge-based Smoothed Point Interpolation Method (ES-PIM) for Heat Transfer Analysis of Rapid Manufacturing System. International Journal of Heat and Mass Transfer,2010,53:1938-1950.
[131]Cui X.Y., Liu G.R., Li G.Y., Zheng G. A rotation free formulation for static and free vibration analysis of thin beams using gradient smoothing technique. CMES-Computer Modeling in Engineering & Sciences,2008,38(3):217-229.
[132]Cui X.Y., Liu G.R., Li G.Y. A curvature-constructed method for bending analysis of thin plates using 3-node triangular cells. International Journal of Computational methods,2010. (Accepted)
[133]Cui X.Y., Li G.Y. Nodal integration thin plate formulation using linear interpolation and triangular cells. International Journal of Computational methods,2010. (Accepted)
[134]Cui X.Y., Li G.Y., Zheng G. Novel thin plate element theory based on a continuity re-relaxed technique. Science China:Technological Sciences,2010, 53:2450-2457.
[135]Cui X.Y., Liu G.R., Li G.Y., Zhang G.Y. A thin plate formulation without rotation DOFs based on the radial point interpolation method and triangular cells. International Journal for Numerical Methods in Engineering,2011,85(8): 958-986.
[136]Kirchhoff G.R. Uber das gleichgewichi und die bewegung einer elastishem scheibe. J. Fuer die Reine und Angewandte Mathematik 1850; 40:51-88.
[137]Rissner E. The effect of transverse shear deformation on the bending of elastic plates. Jounral of Applied Mechnaics,1945,12:69-76.
[138]Mindlin R.D. Influence of rotatory inertia and shear in flexural motions of isotropic elastic plates. Jounral of Applied Mechnaics,1951,37:1031-1036.
[139]Bazeley G.P., Cheung Y.K., Irons B.M., Zienkiewicz O.C. Triangular elements in bending conforming and nonconforming solution. Proceedings of the Conference on Matrix Methods in Structural Mechanics,1965,547-576.
[140]Clough R.W., Tocher J.L. A conforming triangular finite element plate bending solution. International Journal for Numerical Methods in Engineering,1970,2, 259-264.
[141]Morley L.S.D. The constant-moment plate-bending element. Journal of strain analysis,1971,6:20-24.
[142]Alwood R.J., Cornes G.M. A polygonal finite element for plate bending problems using the assumed stress approach. International Journal for Numerical Methods in Engineering,1969,1:135-149.
[143]Batoz J.L., Bathe K.J., Ho L.W. A study of three-node triangular plate bending elements, International Journal for Numerical Methods in Engineering,1980, 15:1771-1812.
[144]Hughes T.J.R, Cohen M., Haroun M. Reduced and selective integration techniques in finite element analysis of plates. Nuclear Engineering Design, 1978,46:203-222.
[145]Belytschko T., Stolarski H. Stress projection for membrane and shear locking in shell elements. Computer Methods in Applied Mechanics and Engineering, 1985,51:221-258.
[146]Liu W.K., Law E.S., Lam D., Belytschko T. Resultant-stress Degenerated-shell Element. Computer Methods in Applied Mechanics and Engineering,1986, 55:259-300.
[147]Bathe K.J, Dvorkin E.N. A formulation of general shell elements-the use of mixed interpolation of tensorial components. International Journal for Numerical Methods in Engineering,1986,22:697-722.
[148]Belytschko, T., Lin, J.I., Tsay, C. Explicit Algorithms for the Nonlinear Dynamics of Shells. Computer Methods in Applied Mechanics and Engineering, 1984,42:225-251.
[149]Belytschko T., Stolarski H., Carpenter N. A Co triangular plate element with one-point quadrature. International Journal for Numerical Methods in Engineering,1984,20:787-802.
[150]Bathe K.J., Brezzi F. The MITC7 and MITC9 plate bending element. Computers & Structures,1989,32:797-814.
[151]Bathe K.J., Iosilevich A., Chapelle D. An evaluation of the MITC shell elements. Computers & Structures,2000,75:1-30.
[152]Lee P.S., Bathe K.J. Development of MITC isotropic triangular shell finite elements. Computers & Structures,2004,82:945-962.
[153]Bletzinger K.U., Bischoff M., Ramm E. A unified approach for shear-locking-free triangular and rectangular shell finite elements, Computers & Structures,2000,75:321-334.
[154]Kim J.H., Kim Y.H. Three-node macro triangular shell element based on the assumed natural strains. Computational Mechanics,2002,29:441-458.
[155]Chen W.J., Cheung Y.K. Refined 9-Dof triangular Mindin plate elements. International Journal for Numerical Methods in Engineering,2001,51: 1259-1281.
[156]Chen W.J. Refined 15-DOF triangular discrete degenerated shell element, International Journal for Numerical Methods in Engineering,2004,60: 1817-1846.
[157]岑松,龙志飞.一种新型厚薄板通用三角形广义协调元.工程力学.1998,15：10-22.
[158]陈永亮,岑松,姚振汉,龙驭球.厚薄通用三角形三结点平板壳元TSLT18.清华大学学报(自然科学版).2003,48：1069-1073.
[159]Dhatt G. Numerical analysis of thin shells by curved triangular elements based on discrete Kirchhoff hypothesis. Proceedings of the ASCE Symposium on Applications of FEM in Civil Engineering,1969,255-278.
[160]Dhatt G. An efficient triangular shell element (Kirchhoff triangular shell element design via linear shell theory), AIAA Journal,1970,8:2100-2102.
[161]Ayad R., Dhatt G., Batoz J.L. A new hybrid-mixed variational approach for Reissner-Mindlin plates, The MiSP model. International Journal for Numerical Methods in Engineering,1998,42:1149-1479.
[162]Wu S., Li G.Y., Belytschko T. A DKT shell element for dynamic large deformation analysis. Communications in numerical methods in engineering, 2005,21:651-674.
[163]Cui X.Y., Liu G.R., Li G.Y., Zhang G.Y., Zheng G. Analysis of plates and shells using an edge-based smoothed finite element method. Computational Mechanics,2010,45:141-156.
[164]Zheng G., Cui X.Y., Li G.Y., Wu S.Z. An edge-based smoothed triangle element for non-linear explicit dynamic analysis of shells. Computational Mechanics, 2011,48(1):65-80.
[165]Liu G.R., Chen X.L. A mesh-free Galerkin method for static and free vibration analyses of thin plates of complicated shape. Journal of Sound and Vibration, 2001,241(5):839-855.
[166]Liu L., Liu G.R., Tan V.B.C. Element free method for static and free vibration analysis of spatial thin shell structures. Computer Methods in Applied Mechanics and Engineering,2002,191:5923-5942.
[167]Gu Y.T., Liu G.R. A meshless local Petrov-Galerkin (MLPG) formulation for static and free vibration analyses of thin plates. Computer Modeling in Engineering & Sciences,2001,2(4):463-476.
[168]Long S.R., Atluri S.N. A meshless local Petrov-Galerkin (MLPG) method for solving the bending problem of a thin plate. Computer Modeling in Engineering & Sciences,2002,3:53-63.
[169]Wang D.D., Chen J.S. A Hermite reproducing kernel approximation for thin-plate analysis with sub-domain stabilized conforming integration. International Journal for Numerical Methods in Engineering,2008,74(3): 368-390.
[170]Cui X.Y., Liu G.R., Li G.Y. A smoothed Hermite radial point interpolation method for thin plate analysis using gradient smoothing operation. Archive of Applied Mechanics,2011,81(1):1-18.
[171]Donning B.M., Liu W.K. Meshless methods for shear-deformable beams and plates. Computer Methods in Applied Mechanics and Engineering,1998,152: 47-71.
[172]Noguchi H., Kawashima T., Miyamura T. Element free analyses of shell and spatial structures. International Journal for Numerical Methods in Engineering, 2000,47(6):1215-1240.
[173]Kanok-Nukulchai W., Barry W., Saran-Yasoontorn K, Bouillard PH. On elimination of shear locking in the element-free Galerkin method. International Journal for Numerical Methods in Engineering,2001,52(7):705-725.
[174]Soric J., Li Q., Jarak T., Atluri S.N. Meshless local Petrov-Galerkin (MLPG) formulation for analysis of thick plates. Computer Modeling in Engineering and Sciences,2004,6(4):349-357.
[175]Sladek J., Sladek V., Mang H.A. Meshless formulations for simply supported and clamped plate problems. International Journal for Numerical Methods in Engineering,2002,55(3):359-375.
[176]Liu L., Chua L.P., Ghista D.N. Conforming radial point interpolation method for spatial shell structures on the stress-resultant shell theory. Archive of Applied Mechanics,2006,75(4-5):248-267.
[177]Liu L., Chua L.P., Ghista D.N. Applications of point interpolation method for spatial general shells structures. Computer Methods in Applied Mechanics and Engineering,2007,196(9-12):1633-1647.
[178]Wang D.D., Chen J.S. Locking-free stabilized conforming nodal integration for meshfree Mindlin-Reissener plate formulation. Computer Methods in Applied Mechanics and Engineering,2004,193:1065-1083.
[179]Chen J.S., Wang D.D. A constrained reproducing kernel particle formulation for shear deformable shell in Cartesian coordinates. International Journal for Numerical Methods in Engineering,2006,68(2):151-172.
[180]Zhang Z., Noguchi H., Chen J.S. Moving least-squares approximation with discontinuous derivative basis functions for shell structures with slope discontinuities. International Journal for Numerical Methods in Engineering, 2008,76(8):1202-1230.
[181]Liu G.R., Zhao X., Dai K.Y., et al. Static and free vibration analysis of laminated composite plates using the conforming radial point interpolation method. Composites Science and technology,2008,68:354-366.
[182]Zhao X., Liu G.R., Dai K.Y., et al. Free-vibration analysis of shells via a linearly conforming radial point interpolation method (LC-RPIM). Finite Elements in Analysis and Design,2009,45:917-924.
[183]Zhao X., Liu G.R., Dai K.Y., et al. A linearly conforming radial point interpolation method (LC-RPIM) for shells. Computational Mechanics,2009, 43:403-413.
[184]Zhao X., Liu G.R, Dai K.Y, et al. Geometric nonlinear analysis of plates and cylindrical shells via a linearly conforming radial point interpolation method. Computational Mechanics,2009,42:133-144.
[185]Hughes T.J.R. The finite element method, Prentice-Hall, London,1987.
[186]Golberg M.A., Chen C.S. Bowman H. Some recent results and proposals for the use of radial basis functions in the BEM. Engineering Analysis with Boundary Elements,1999,23:285-296.
[187]Timoshenko S.P., Goodier J.N. Theory of Elasticity (3rd edn). McGraw-Hill: New York,1970.
[188]Nagashima T. Node-by-node meshless approach and its application to structural analysis. International Journal for Numerical Methods in Engineering,1999, 46:341-385.
[189]Jahed H., Sethuraman R., Dubey R.N. A variable material property approach for solving elastic-plastic problems, International Journal of Pressure Vessels and Piping,1997,71:285-291.
[190]Desikan V., Sethuraman R. Analysis of material nonlinear problems using pseudo-elastic finite element method, Journal of pressure vessel technology-transactions of the ASME,2000,122:457-461.
[191]Zienkiewicz O.C., Taylor R.L. The Finite Element Method. The fifth edition, Vol 2:Solid Mechanics, Butterworth-Heinemann, Oxford,2000.
[192]Fung Y.C. Foundations of solid mechanics. London:Prentice Hall,1965.
[193]Ugural A.C. Stresses in plates and shells.McGraw-Hill, New York,1981.
[194]Gorman D.J. Free vibration analysis of rectangular plates. New York:Elsevier, 1982.
[195]Abbassian F., Dawswell D.J., Knowles N.C. Free vibration benchmarks. Atkins Engineering Sciences:Glasgow,1987.
[196]Crisfield M.A. Non-Linear Finite Element Analysis of Solids and Structures. Vol.1. Wiley:Chichester,1997.
[197]Malkus D.S., Hughes T.J.R. Mixed finite element methods-reduced and selective integration techniques:A unification of concepts. Computer Methods in Applied Mechanics and Engineering,1978,15:63-81.
[198]Reismann H. Elastic Plates:Theory and Applications. Wiley, New York,1988.
[199]Bathe K.J., Dvorkin E.H. A four-node plate bending element based on Mindlin-Reissner plate theory and mixed interpolation. International Journal for Numerical Methods in Engineering,1985,21:367-383.
[200]Batoz J.L., Tahar M.B. Evaluation of a new quadrilateral thin plate bending element. International Journal for Numerical Methods in Engineering,1982,18: 1655-1677.
[201]Chia C.Y. Nonlinear Analysis of Plate. McGraw-Hill, New York,1980.
[202]Hughes T.J.R., Liu W.K. Nonlinear finite element analysis shells:Part Ⅰ, Three-dimensional shells. Computer Methods in Applied Mechanics and Engineering,1981,26:331-362.
[203]Engelmann B.E., Whirley R.G. A new elasto-plastic shell element formulation for DYNA3D, Lawrence Livermore National Laboratory, Report UCRL-JC-104826,1990.
[204]Dvorkin E.N., Bathe K.J. A continuum mechanics based four-node shell element for general nonlinear analysis, Engineering Computations,1984,1: 77-88.
[205]Belytschko T., Leviathan I. Physical stabilization of the 4-node shell element with one point quadrature. Computer Methods in Applied Mechanics and Engineering,1994,113:321-350.
[206]Horrigmoe G., Bergan P.G. Non-linear analysis of free-form shells by flat finite elements. Computer Methods in Applied Mechanics and Engineering,1978:16: 11-35.
[207]Dhatt G.S. Instability of thin shells by the finite element method, IASS Symposium for Folded Plates and Prismatic Structures, Vienna,1970.
[208]Sabir A.B., Lock A.C. The application of finite elements to the large deflection geometrically nonlinear behaviour of cylindrical shells. International Conference on Variational Methods in Engineering, Southampton, England; United Kingdom,1973.
[209]Batoz J.L. An explicit formulation for an efficient triangular plate-bending element. International Journal for Numerical Methods in Engineering,1982,18: 1077-1089.
[210]Timoshenko S., Woinowsky-Krieger S. Theory of Plates and Shells. New York: McGraw-Hill,1940.
[211]Sze K.Y, Zhu D. A quadratic assumed natural strain shell curved triangular element. Computer Methods in Applied Mechanics and Engineering,1999,174: 57-71.
[212]Jirousek J, Lan Guex. The hybrid-trefftz finite element model and its application to plate bending. International Journal for Numerical Methods in Engineering,1986,23:651-693.
[213]Scordelis A.C., Lo K.S. Computer Analysis of Cylindrical Shells. Journal of the American Concrete Institute,1969,61:539-561.
[214]Simo J.C., Hughes T.J.R. Computational Inelasticity, New York, Springer Verlag,1998.
[215]Zhong Z.H. Finite Element Procedures for Contact-Impact Problems, Oxford University Press, New York,1993.
[216]钟志华,李光耀.薄板冲压成形过程的计算机仿真与应用,第1版.北京：北京理工大学出版社,1998,1-7
[217]Male A.T., Cockcroft M.G. A Method for the Determination of the Coefficient of Friction of Metals under Conditons of Bulk Plastic Deformation, Journal of the Institute of Metals,1965,93:38-46.
[218]Morino L., Leech J.W., Witmer E.A. An improved numerical calculation technique for large elastic-plastic transient deformations of thin shells:Part 2-evaluation and applications. Journal of Applied Mechanics,1971,38:429-435.