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無網格法 介紹 Introduction to Meshless Methods

無網格法 介紹 Introduction to Meshless Methods. Reporter : 蔡文瀚 Wen-Han, Tsai Date : 2013/08/15. Content. Introduction Radial B asis Functions (RBFs) Types of Meshless Methods Examples. Introduction - Why use meshless methods ?.

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無網格法 介紹 Introduction to Meshless Methods

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  1. 無網格法介紹Introduction to Meshless Methods Reporter : 蔡文瀚Wen-Han, Tsai Date : 2013/08/15

  2. Content • Introduction • Radial Basis Functions (RBFs) • Types of Meshless Methods • Examples

  3. Introduction-Why use meshless methods? • Save the great deal of time cost by meshing and remeshing in traditional mesh-dependent numerical methods in groundwater flow simulation.

  4. Introduction-Applications(1) • Scattered data interpolation 3D Printing

  5. Introduction-Applications(2) • Solve DEs or PDEs -Steady-state equations : (1) Lapalce equation ( (2) Poisson equation ( -Transient equations : (1) Heat equation (2) Wave equation (3) Telegraph equation

  6. Introduction-Literature review • Hardy (1971) developed the algorithm for scattered data interpolation by MQ-RBF. • Kansa (1990) first applied MQ-RBF in solving partial differential equations. • Young et al. (2006) used MFS to solve seepage problem. • Meenal and Eldho (2012) applied MQ-LRBFCM in 1D and 2D groundwater flow simulation.

  7. Radial Basis Functions (RBFs) : unknown , : undetermined coefficient , : radial basis function (RBF) : distance between ith point of interest and jth node. • Types of common used RBF - , Multiquardrics (MQ) - , Inverse Multiquardrics - , Gaussian (EXP) - , Thin plate splines (TPS)

  8. Types of meshless methods • Common used meshless method (1) Domain-type -(Global) Radial basis collocation method (RBFCM) -Localized radial basis function collocation method (LRBFCM) (2) Boundary-type -Method of fundamental solution (MFS) -Method of particular solution (MPS) -Trefftz method (TM) Node-distribution for RBFCM (Hu et al. 2007) Node distribution for MFS (Young et al. 2006)

  9. Examples-Algorithm of Domain Type Meshless Method (RBFCM) 1 2 3 • , , Let , For G.E. For B.C.s x=0 x=L R

  10. Examples-Algorithm of Domain Type Meshless Method (LRBFCM) • , , Let , From G.E. For Group 1. For Group 2. Global Assembly Impose B.C.s 1 2 1 2 3 x=0 x=L

  11. Examples-Algorithm of Boundary Type Meshless Method (MFS) • , , Let , For B.C.s 1 3 2 x=0 x=L

  12. Thank you for your attention~

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