An accurate improved complex variable element-free method for numerical solutions of elastodynamic problems

Abstract In this paper, an improved complex variable element-free method (ICVEFM) is proposed and implemented for elastodynamic problems. The modeling process involves element-free approximation based on an improved complex variable moving least-squares (ICVMLS) and employs the Galerkin procedure to derive the discretized equation system. Since the best mean-square approximation for both real and imaginary parts is simultaneously achieved, the ICVEFM facilitates a higher computing efficiency. The Newmark-β method for time integration and penalty method for essential boundary conditions are introduced to derive the corresponding formulae. Several numerical examples including free vibration and forced vibration with time history analyses are presented to demonstrate the accuracy of the proposed method, vis-a-vis the analytical and numerical solutions.

[1]  Wang Jian-fei,et al.  A new complex variable element-free Galerkin method for two-dimensional potential problems , 2012 .

[2]  Yumin Cheng,et al.  The complex variable element-free Galerkin (CVEFG) method for elasto-plasticity problems , 2011 .

[3]  K. Liew,et al.  Analyzing elastoplastic large deformation problems with the complex variable element-free Galerkin method , 2014 .

[4]  Cheng Yu-min,et al.  Reproducing kernel particle method with complex variables for elasticity , 2008 .

[5]  Zhang Wu,et al.  An improved boundary element-free method (IBEFM) for two-dimensional potential problems , 2009 .

[6]  M. Aliabadi,et al.  Elastodynamic problems by meshless local integral method: Analytical formulation , 2013 .

[7]  Jianfeng Wang,et al.  An improved complex variable element-free Galerkin method for two-dimensional elasticity problems , 2012 .

[8]  Shucai Li,et al.  Enriched meshless manifold method for two-dimensional crack modeling , 2005 .

[9]  K. M. Liew,et al.  Large deflection geometrically nonlinear analysis of carbon nanotube-reinforced functionally graded cylindrical panels , 2014 .

[10]  Jianfeng Wang,et al.  The Complex Variable Element-Free Galerkin (cvefg) Method for Two-Dimensional Elastodynamics Problems , 2012 .

[11]  K. Liew,et al.  A novel complex variable element-free Galerkin method for two-dimensional large deformation problems , 2012 .

[12]  Roy R. Craig,et al.  Structural Dynamics: An Introduction to Computer Methods , 1981 .

[13]  L. W. Zhang,et al.  Modeling of biological population problems using the element-free kp-Ritz method , 2014, Appl. Math. Comput..

[14]  Baodong Dai,et al.  AN IMPROVED LOCAL BOUNDARY INTEGRAL EQUATION METHOD FOR TWO-DIMENSIONAL POTENTIAL PROBLEMS , 2010 .

[15]  L. W. Zhang,et al.  An improved moving least-squares Ritz method for two-dimensional elasticity problems , 2014, Appl. Math. Comput..

[16]  K. M. Liew,et al.  Complex variable boundary element-free method for two-dimensional elastodynamic problems , 2009 .

[17]  Yumin Cheng,et al.  Complex variable element-free Galerkin method for viscoelasticity problems , 2012 .

[18]  Glen V. Berg.  Elements of structural dynamics , 1989 .

[19]  W. Mansur,et al.  Two-dimensional elastodynamics by the time-domain boundary element method: Lagrange interpolation strategy in time integration , 2012 .

[20]  Yumin Cheng,et al.  A COMPLEX VARIABLE MESHLESS MANIFOLD METHOD FOR FRACTURE PROBLEMS , 2010 .

[21]  Yumin Cheng,et al.  The complex variable reproducing kernel particle method for two-dimensional elastodynamics , 2010 .

[22]  Zan Zhang,et al.  The complex variable reproducing kernel particle method for two-dimensional inverse heat conduction problems , 2014 .

[23]  A. Idesman.  A new exact, closed‐form a priori global error estimator for second‐ and higher‐order time‐integration methods for linear elastodynamics , 2011 .

[24]  Cheng Yu-Min,et al.  Error estimates of element-free Galerkin method for potential problems , 2008 .

[25]  Satya N. Atluri,et al.  A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method , 1998 .

[26]  K. M. Liew,et al.  Geometrically nonlinear thermomechanical analysis of moderately thick functionally graded plates using a local Petrov–Galerkin approach with moving Kriging interpolation , 2014 .

[27]  Yumin Cheng,et al.  Analyzing three-dimensional viscoelasticity problems via the improved element-free Galerkin (IEFG) method , 2014 .

[28]  K. Liew,et al.  A higher-order gradient theory for modeling of the vibration behavior of single-wall carbon nanocones , 2014 .

[29]  Yumin Cheng,et al.  A boundary element-free method (BEFM) for two-dimensional potential problems , 2009 .

[30]  Cheng,et al.  AN INVESTIGATION ON A NEW INTEGRAL TRANSFORM-BOUNDARY ELEMENT METHOD FOR ELASTODYNAMICS , 1997 .

[31]  A. Sequeira,et al.  A new LBIE method for solving elastodynamic problems , 2011 .

[32]  K. M. Liew,et al.  Static and dynamic of carbon nanotube reinforced functionally graded cylindrical panels , 2014 .

[33]  L. W. Zhang,et al.  Numerical analysis of generalized regularized long wave equation using the element-free kp-Ritz method , 2014, Appl. Math. Comput..

[34]  Cheng Yu-min,et al.  Boundary element-free method for elastodynamics , 2005 .

[35]  K. M. Liew,et al.  Postbuckling of carbon nanotube-reinforced functionally graded cylindrical panels under axial compression using a meshless approach , 2014 .

[36]  K. Liew,et al.  The improved element-free Galerkin method for two-dimensional elastodynamics problems , 2013 .

[37]  J. Sládek,et al.  Modified meshless local Petrov–Galerkin formulations for elastodynamics , 2012 .

[38]  Cheng Yu-Min,et al.  A meshless method with complex variables for elasticity , 2005 .

[39]  K. Liew,et al.  An improved element-free Galerkin method for numerical modeling of the biological population problems , 2014 .

[40]  Yumin Cheng,et al.  A complex variable meshless method for fracture problems , 2006 .

[41]  K. M. Liew,et al.  Thermal buckling of functionally graded plates using a local Kriging meshless method , 2014 .

[42]  Pei Liu,et al.  THE COMPLEX VARIABLE ELEMENT-FREE GALERKIN (CVEFG) METHOD FOR TWO-DIMENSIONAL ELASTICITY PROBLEMS , 2009 .

[43]  A linear θ time-marching algorithm in 3D BEM formulation for elastodynamics , 1999 .

[44]  K. M. Liew,et al.  Dynamic stability analysis of carbon nanotube-reinforced functionally graded cylindrical panels using the element-free kp-Ritz method , 2014 .

[45]  G. Manolis,et al.  Three-dimensional BEM for transient elastodynamics based on the velocity reciprocal theorem , 2011 .

[46]  Wang Jian-fei,et al.  A new complex variable meshless method for transient heat conduction problems , 2012 .

[47]  Wang Jian-fei,et al.  New complex variable meshless method for advection—diffusion problems , 2013 .