PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Tytuł artykułu

Recent developments in stabilized Galerkin and collocation meshfree methods

Autorzy
Wybrane pełne teksty z tego czasopisma
Identyfikatory
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Meshfree methods have been developed based on Galerkin type weak formulation and strong formulationwith collocation. Galerkin type formulation in conjunction with the compactly supported approximation functions and polynomial reproducibility yields algebraic convergence, while strong form collocationmethod with nonlocal approximation such as radial basis functions offers exponential convergence. In thiswork, we discuss rank instability resulting from the nodal integration of Galerkin type meshfree methodas well as the ill-conditioning type instability in the radial basis collocation method. We present the recentadvances in resolving these diffculties in meshfree methods, and demonstrate how meshfree methods can be applied to problems di?cult to be modeled by the conventional ?nite element methods due to their intrinsic regularity constraints.
Rocznik
Strony
3--21
Opis fizyczny
Bibliogr. 53 poz., rys., tab., wykr.
Twórcy
autor
autor
autor
  • Department of Civil & Environmental Engineering University of California Los Angeles, CA 90095, USA, huhy@ thu.edu.tw
Bibliografia
  • [1] S. Beissel, T. Belytschko. Nodal integration of the element-free Galerkin method. Computer Methods in Applied Mechanics and Engineering, 139: 49–74, 1996.
  • [2] T. Belytschko, Y.Y. Lu, L. Gu. Element-free Galerkin methods. International Journal for Numerical Methods in Engineering, 37: 229–256, 1994.
  • [3] T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, P. Krysl. Meshless Methods: An Overview and Recent Development. Comput. Meth. Appl. Mech. Engng., 139: 3–49, 1996.
  • [4] J. Bonet, S. Kulasegaram. Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulation. International Journal for Numerical Methods in Engineering, 47: 1189–1214, 1999.
  • [5] J.S. Chen, C. Pan, C.T. Wu, W.K. Liu. Reproducing Kernel Particle Methods for Large Deformation Analysis of Nonlinear Structures. Computer Methods in Applied Mechanics and Engineering, 139: 195–227, 1996.
  • [6] J.S. Chen, C. Pan, C.T. Wu. Large Deformation Analysis of Rubber Based on a Reproducing Kernel Particle Method, Computational Mechanics, 19: 211–227, 1997.
  • [7] J.S. Chen, C. Pan, C.M.O.L. Roque, H.P. Wang. A Lagrangian Reproducing Kernel Particle Method for Metal Forming Analysis. Computational Mechanics, 22: 289-307, 1998.
  • [8] J.S. Chen, H.P. Wang. New boundary condition treatments for meshless computation of contact problems. Computer Methods in Applied Mechanics and Engineering, 187: 441–468, 2000.
  • [9] J.S. Chen, H.P. Wang. Some Recent Improvements in Meshfree Methods for Incompressible Finite Elasticity Boundary Value Problems with Contact. Computational Mechanics, 25: 137–156, 2000.
  • [10] J.S. Chen, S. Yoon, H.P. Wang, W.K. Liu. An Improved Reproducing Kernel Particle Method for Nearly Incompressible Hyperelastic Solids. Computer Methods in Applied Mechanics and Engineering, 181: 117–145, 2000.
  • [11] J.S. Chen, H.P. Wang. New Boundary Condition Treatments for Meshless Computation of Contact Problems. Computer Methods in Applied Mechanics and Engineering, 187: 441–468, 2000.
  • [12] J.S. Chen, C.T. Wu, T. Belytschko. Regularization of Material Instabilities by Meshfree Approximation with Intrinsic Length Scales. International Journal for Numerical Methods in Engineering, 47: 1303–1322, 2000.
  • [13] J.S. Chen, C.T.Wu, S. Yoon, Y. You. A Stabilized Conforming Nodal Integration for Galerkin Meshfree Methods.International Journal for Numerical Methods in Engineering, 50: 435–466, 2001.
  • [14] J.S. Chen, S. Yoon, C.T. Wu. Nonlinear Version of Stabilized Conforming Nodal Integration for Galerkin Meshfree Methods. International Journal for Numerical Methods in Engineering, 53: 2587–2615, 2002.
  • [15] J.S. Chen, W. Han, Y.You, X. Meng. A reproducing kernel method with nodal interpolation property. International Journal for Numerical Methods in Engineering, 56: 935–960, 2003.
  • [16] J.S. Chen, X. Zhang, T. Belytschko. An Implicit Gradient Model by a Reproducing Kernel Strain Regularization in Strain Localization Problems. Computer Methods in Applied Mechanics and Engineering, 193: 2827–2844, 2004.
  • [17] J.S. Chen, D.D. Wang. A Constrained Reproducing Kernel Particle Formulation for Shear Deformable Shell in Cartesian Coordinate. International Journal for Numerical Methods in Engineering, 68: 151–172, 2006.
  • [18] J.S. Chen, W. Hu, M. Puso. Orbital HP-Cloud for Schr¨odinger Equation in Quantum Mechanics. Computer Methods in Applied Mechanics and Engineering, 196: 3693–3705, 2007.
  • [19] J.S. Chen, Y.Wu. Stability in Lagrangian and Semi-Lagrangian Reproducing Kernel Discretizations Using Nodal Integration in Nonlinear Solid Mechanics. In: V.M.A. Leitao, C.J.S. Alves, C.A. Duarte [Eds.], Computational Methods in Applied Sciences, 55–77. Springer, O?nate, Eugenio, 2007.
  • [20] J.S. Chen,W. Hu, H.Y. Hu. Reproducing Kernel Enhanced Local Radial Basis Collocation Method. International Journal for Numerical Methods in Engineering, 75: 600–627, 2008.
  • [21] J.S. Chen, L. Wang, H.Y. Hu, S.W. Chi. Subdomain Radial Basis Collocation Method for Heterogeneous Media. International Journal for Numerical Methods in Engineering, 80: 163–190, 2009.
  • [22] J.S. Chen, Y. Wu, P.C. Guan, H. Teng, J. Gaidos, K. Hofstetter, M. Alsaleh. A Semi-Lagrangian Reproducing Kernel Formulation for Modeling Earth Moving Operations. Mechanics of Materials, 41: 670-683, 2009.
  • [23] J. Dolbow, T. Belytschko. Numerical integration of Galerkin weak form in meshfree methods. Computational Mechanics, 23: 219–230, 1999.
  • [24] P.C. Guan, S.W. Chi, J.S. Chen, T.R. Slawson, M.J. Roth. Semi-Lagrangian Reproducing Kernel Particle Method for Fragment-Impact Problems, accepted. International Journal of Impact Engineering, 2011.
  • [25] W. Han, G.J. Wagner, W.K. Liu. Convergence analysis of a hierarchical enrichment of dirichlet boundary conditions in a meshfree method. International Journal for Numerical Methods in Engineering, 53: 1323–1336, 2002.
  • [26] H.Y. Hu, Z.C. Li. Collocation methods for Poisson’s equation. Computer Methods in Applied Mechanics and Engineering, 195: 4139–4160, 2006.
  • [27] H.Y. Hu, J.S. Chen, W. Hu. Weighted Radial Basis Collocation Method for Boundary Value Problems. International Journal for Numerical Methods in Engineering, 69: 2736–2757, 2007.
  • [28] H.Y. Hu., J.S. Chen,W. Hu. Error Analysis of Collocation Method Based on Reproducing Kernel Approximation. Numerical Methods for Partial Differential Equations, 27: 554–580, 2011.
  • [29] H.Y. Hu, J.S. Chen. Perturbation and Stability Analysis of Strong Form Collocation with Reproducing Kernel Approximation. International Journal for Numerical Methods in Engineering, 88: 157–179, 2011.
  • [30] A. Huerta, S. Fernandez-Mendez. Enrichment and coupling of the finite element and meshless methods. International Journal for Numerical Methods in Engineering, 48: 1615–1636, 2000.
  • [31] I. Kaljevic, S. Saigal. An improved element free Galerkin formulation. International Journal for Numerical Methods in Engineering, 40: 2953–2974, 1997.
  • [32] E.J. Kansa. Multiquadrics – A scattered data approximation scheme with applications to computational fluiddynamics – I Surface approximations and partial derivatives. Computers and Mathematics with Applications, 19: 127–145, 1992.
  • [33] E.J. Kansa. Multiquadrics – A scattered data approximation scheme with applications to computational fluiddynamics – II Solutions to parabolic, hyperbolic and elliptic partial differential equations. Computers and Mathematics with Applications, 19: 147–161, 1992.
  • [34] N.H. Kim, K.K. Choi, J.S. Chen. Shape Design Sensitivity Analysis and Optimization of Elasto-Plasticity with Frictional Contact. AIAA, 38: 1742–1753, 2000.
  • [35] N.H. Kim, K.K. Choi, J.S. Chen. Die Shape Design of Sheet Metal Stamping Process Using Meshfree Method: Design Sensitivity Analysis. International Journal for Numerical Methods in Engineering, 51: 1385–1405, 2001.
  • [36] Y. Krongauz, T. Belytschko. Enforcement of essential boundary conditions in meshless approximations using finite elements. Computer Methods in Applied Mechanics and Engineering, 131: 133–145, 1996.
  • [37] W.K. Liu, S. Jun, Y.F. Zhang. Reproducing kernel particle methods. International Journal for Numerical Methods in Fluids, 20: 1081–1106, 1995.
  • [38] W. R. Madych and S. A. Nelson. Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation. Journal of Approximation Theory, 70: 94–114, 1992.
  • [39] J. Nitsche. ¨Uber ein Variationsprinzip zur L¨osung von Dirichlet-Problemen bei Verwendung von Teilr¨aumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg, 36: 9–15, 1971.
  • [40] M. Puso, E. Zywicz, J.S. Chen. A New Stabilized Nodal Integration Approach. Lecture Notes in Computational Science and Engineering, 57: 207–218, 2006.
  • [41] M. Puso, J.S. Chen, E. Zywick,W. Elmer. Meshfree and Finite Element Nodal IntegrationMethods. International Journal for Numerical Methods in Engineering, 74: 416–446, 2008.
  • [42] P.W. Randles, L.D. Libersky, A.G. Petschek. On neighbors, derivatives, and viscosity in particle codes. Proceeding of ECCM Conference, Munich, Germany, 31 August – 3 September, 1999.
  • [43] X. Ren, J.S. Chen, J. Li. Micro-cracks Informed Damage Models for Brittle Solids. accepted. International Journal of Solids and Structures, 48: 1560–1571, 2011.
  • [44] M.J. Roth, J.S. Chen, T.R. Slawson, R.N. Boone, X. Ren, S.W. Chi, C.H. Lee, P.C. Guan. Multiscale RKPM Formulation for Modeling Penetration of an Ultra High-Strength Concrete Material. Proceeding, Third International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, May 26–28, 2011, Corfu, Greece.
  • [45] R. Schaback, H. Wendland. Using compactly supported radial basis functions to solve partial differential equations. Boundary Element Technology XIII, 1999.
  • [46] S.P. Timoshenko, J.N. Goodier, Theory of Elasticity. McGraw-Hill. New York, USA, 1970.
  • [47] D. Wang, J.S. Chen. Locking Free Stabilized Conforming Nodal Integration for Meshfree Mindlin-Reissner Plate Formulation. Computer Methods in Applied Mechanics and Engineering, 193: 1065–1083, 2004.
  • [48] D. Wang, J.S. Chen. A Locking-free Meshfree Curved Beam Formulation with the Stabilized Conforming Nodal Integration. Computational Mechanics, 39: 83–90, 2006.
  • [49] D. Wang, J.S. Chen. A Hermite Reproducing Kernel Approximation for Thin Plate Analysis with Sub-domain Stabilized Conforming Integration. International Journal for Numerical Methods in Engineering, 74: 368–390, 2008.
  • [50] L. Wang, J.S. Chen, H.Y. Hu. Subdomain Radial Basis Collocation Method for Fracture Mechanics. Int. J. Numer. Meth. Engng., 83: 851–876, 2010.
  • [51] Y. You, J.S. Chen, H. Lu. Filter, Reproducing Kernel, and Adaptive Meshfree Methods. Computational Mechanics, 31: 316-326, 2003.
  • [52] T. Zhu, S.N. Atluri. A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method. Computational Mechanics, 21: 211–222, 1998.
  • [53] W. Han, X. Meng. Error analysis of the reproducing kernel particle method. Computer Methods in Applied Mechanics and Engineering, 190: 6157-–6181, 2001.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-BPB2-0069-0001
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.