Variational Multiscale Element-Free Galerkin Method and Precise Time Step Integration Method for Convection–Diffusion Problems

REFERENCES

• H. Lin and S. N. Atluri , Meshless Local Petrov–Galerkin (MLPG) Method for Convection-Diffusion Problems , Comput. Model. Eng. Sci. , vol. 1 , pp. 45 – 60 , 2000 .
   Web of Science ®Google Scholar
• X. H. Wu , Y. J. Dai , and W. Q. Tao , MLPG/SUPG Method for Convection-Dominated Problems , Numer. Heat Transfer , vol. 61 , pp. 36 – 51 , 2012 .
   Web of Science ®Google Scholar
• X. H. Zhang , J. Ouyang , and J. Y. Wang , Stabilization Meshless Method for Convection Dominated Problems , Appl. Math. Mech. , vol. 29 , pp. 1067 – 1075 , 2008 .
   Web of Science ®Google Scholar
• E. Oñate , S. Idelsohn , O. C. Zienkiewicz , R. L. Taylor , and C. Sacco , A Point Method in Computational Mechanics: Application to Convections to Transport and Fluid Flow , Comput. Methods Appl. Mech. Eng. , vol. 39 , pp. 3839 – 3866 , 1996 .
   Google Scholar
• X. Q. Qin , X. B. Duan , and W. Guo , The Characteristic Radial Basis Meshless Method for Convection-Dominated Diffusion Equations , Appl. Mech. Mater. , vol. 182–183 , pp. 1756 – 1760 , 2012 .
   Google Scholar
• C. M. Fan , C. S. Chien , H. F. Chan , and C. L. Chiu , The Local RBF Collocation Method for Solving the Double-Diffusive Natural Convection in Fluid-Saturated Porous Media , Int. J. Heat Mass Transfer , vol. 57 , pp. 500 – 503 , 2013 .
   Web of Science ®Google Scholar
• W. K. Liu , S. Jun , D. T. Sihling , Y. Chen , and W. Hao , Multiresolution Reproducing Kerner Particle Method for Computational Fluid Dynamics , Int. J. Numer. Methods Fluids , vol. 24 , pp. 1391 – 1415 , 1997 .
   Web of Science ®Google Scholar
• G. R. Liu and Y. T. Gu , An Introduction to Meshfree Methods and their Programming , Springer , Dordrecht , 2005 .
   Google Scholar
• E. Oñate , S. Idelsohn , O. C. Zienkiewicz , R. L. Taylor , C. Sacco , and R. L. Taylor , A Stabilized Finite Point Method for Analysis of Fluid Mechanics Problems , Comput. Methods Appl. Mech. Eng. , vol. 139 , pp. 315 – 346 , 1996 .
   Web of Science ®Google Scholar
• E. Oñate and S. Idelsohn , A Mesh-Free Finite Point Method for Advective-Diffusive Transport and Fluid Flow Problems , Comput. Mech. , vol. 21 , pp. 283 – 292 , 1998 .
   Web of Science ®Google Scholar
• L. Zhang , J. Ouyang , X. X. Wang , and X. H. Zhang , Variational Multiscale Element-Free Galerkin Method for 2D Burgers’ Equation , J. Comput. Phys. , vol. 229 , pp. 7147 – 7161 , 2010 .
   Web of Science ®Google Scholar
• L. Zhang , J. Ouyang , T. Jiang , and Chunlei Ruan , Variational Multiscale Element Free Galerkin Method for the Water Wave Problems, J. Comput. Phys. , vol. 230, pp. 5045–5060, 2011.
   Web of Science ®Google Scholar
• L. Zhang , J. Ouyang , X. H. Zhang , and Wenbin Zhang , On a Multi-Scale Element-Free Galerkin Method for the Stokes Problem , Appl. Math. Comput. , vol. 203 , pp. 745 – 753 , 2008 .
   Web of Science ®Google Scholar
• L. Zhang , J. Ouyang , and X. H. Zhang , The Variational Multiscale Element Free Galerkin Method for MHD Flows at High Hartmann Numbers , Comput. Phys. Commun. , vol. 184 , pp. 1106 – 1118 , 2013 .
   Web of Science ®Google Scholar
• T. J. R. Hughes , Multiscale Phenomena: Green's Function, the Dirichlet-to-Neumann Formulation, Subgrid Scale Models, Bubbles and the Origins of Stabilized Methods , Comput. Methods. Appl. Mech. Eng. , vol. 127 , pp. 387 – 401 , 1995 .
   Web of Science ®Google Scholar
• T. J. R. Hughes , G. R. Feijóo , L. Mazzei , and J. Quincy , The Variational Multiscale Method: A Paradigm for Computational Mechanics , Comput. Methods. Appl. Mech. Eng. , vol. 166 , pp. 3 – 24 , 1998 .
   Web of Science ®Google Scholar
• X. H. Zhang and P. Zhang , The Research on Fast Solution Algorithm Based on Meshless Method for Unsteady Heat Conduction Problems , Chin. J. Appl. Mech. , vol. 30 , pp. 262 – 267 , 2013 .
   Google Scholar
• W. X. Zhong and F. W. Williams , A Precise Time Step Integration Method , J. Mech. Eng. Sci. , vol. 208 , pp. 427 – 430 , 1994 .
   Web of Science ®Google Scholar
• M. F. Wang and F. T. K. Au , Assessment and Improvement of Precise Time Step Integration Method , Comput. Struct. , vol. 84 , pp. 779 – 786 , 2006 .
   Web of Science ®Google Scholar
• P. Smolinski and T. Palmer , Procedures for Multi-Time Step Integration of Element-Free Galerkin Methods for Diffusion Problems , Comput. Struct. , vol. 77 , pp. 171 – 183 , 2000 .
   Web of Science ®Google Scholar
• Q. H. Li , S. S. Chen , and G. X. Kou , Transient Heat Conduction Analysis Using the MLPG Method and Modified Precise Time Step Integration Method , J. Comput. Phys. , vol. 230 , pp. 2736 – 2750 , 2011 .
   Web of Science ®Google Scholar
• S. N. Atluri and T. Zhu , A New Meshless Local Petrov–Galerkin (MLPG) Approach in Computational Mechanics , Comput. Mech. , vol. 22 , pp. 117 – 127 , 1998 .
   Web of Science ®Google Scholar
• V. P. Nguyen , T. Rabczuk , S. Bordas , and M. Duflot , Meshless Methods: A Review and Computer Implementation Aspects , Math. Comput. Simul. , vol. 79 , pp. 763 – 813 , 2008 .
   Web of Science ®Google Scholar
• L. J. Ding , Numerical Computation Method , Beijing Institute of Technology Press , Beijing , 1997 .
   Google Scholar
• Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/unht.
   Google Scholar