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Nodal arrangement for boundary treatment in EFGM

Nodal arrangement for boundary treatment in EFGM. R. Tian A. Nishida N. Nakajima G. Yagawa Center for promotion of Computational Science and Engineering (CCSE), JAERI. 1. Background. u, u h. x. Difficulties in boundary treatment in EFGM:

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Nodal arrangement for boundary treatment in EFGM

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  1. Nodal arrangement for boundary treatment in EFGM R. Tian A. Nishida N. Nakajima G. Yagawa Center for promotion of Computational Science and Engineering (CCSE), JAERI R. Tian CCSE, JAERI

  2. 1. Background u, uh x Difficulties in boundary treatment in EFGM: (1) Losing delta properties => approximation does not pass nodes; (2) No longer presenting a linear variation between two neighbor nodes. Current solution: (1)Lagrange multiplier method (2)Coupling with finite elements; (3)Penalty method; (4)… • Substitutes: • Nodal Interpolation Method; • Kriging interpolation; • Radial basis function methods; • Others R. Tian CCSE, JAERI

  3. 2. Does this difficulty exist in a 1-D EFGM? exact numer The function y=sin(x) is approximated using MLSs with three different nodal radii. 3 pts 2 pts (1)A usual MLS approximation (2)A C0 MLS interpolation on the whole of an domain Reason: 2-point MLS(linear basis) = C0 linear interpolation In 1-D, the difficulty is trivial and can be easily overcome by only adjusting the nodal influence at the node of interest. 2 pts R. Tian CCSE, JAERI (c)C0 MLS interpolation only at nodes of interest

  4. An Introduction to Programming the Meshless Element Free Galerkin Method by J. Dolbow and T. Belytschko Archives in Computational Mechanics, Vol 5, No 3, pp. 207-241 (1998)(from John Dolbow's homepage) In this paper, for 1-D test, since influence radius was set to r=2h, the MLS approximation in this case naturally interpolates at essential boundary, Lagrange multiplier method for boundary treatment has no effect. R. Tian CCSE, JAERI

  5. 3. A 2-D case A 3-point MLS approximation (linear basis) = C0 linear interpolation regardless of weight functions. R. Tian CCSE, JAERI

  6. 4. Nodal arrangement for boundary treatment in EFGM a b e d a b c e c d a e b d c Triangular NA: Square NA: R. Tian CCSE, JAERI

  7. 5. Examples: 1-D A loaded string or a one-dimensional heat transfer problem: The exact solution is where E is the young’s modulus or the conductivity. The nodal arrangement (NA) scheme is compared with Lagrange multiplier (LM) and penalty method (PN) treatment. Energy norm for convergence study: R. Tian CCSE, JAERI

  8. 5. Examples: 1-D NA: nodal arrangement LM: Lagrange multiplier PN: penalty method In case a, the Dirichlet boundary is satisfied naturally, a special treatment is in fact not necessary! In other cases, the nodal arrangement is quite simple yet effective. R. Tian CCSE, JAERI

  9. 6. Examples: a 2-D cantilever problem E=1000, v=0.3, t=1, L=10, H=1 B (1) Problem: (2) NA: A (4) Shape functions: (3) Nodal connectivity at the root: MLS shape function at A A B MLS shape function at B R. Tian CCSE, JAERI

  10. 6. Examples: a 2-D cantilever problem NA: nodal arrangement; PN: penalty method (1)Accuracy comparison between methods PN and NA. (2)Convergence studies of methods PN and NA. (a) Usual nodal arangement (b) (c) • (3)Vonmises stresses of the beam obtained by • penalty method; (b) nodal arrangement; (c) the exact solution. Special nodal arrangement R. Tian CCSE, JAERI

  11. 7. NAs for different boundaries R. Tian CCSE, JAERI

  12. 7. NAs for different boundaries R. Tian CCSE, JAERI

  13. 8. Discussions Merits: (1) very simple a): only nodal radius customization; b): boundary treatment follows standard FEM procedures; c): no addition programming. (2) Effective a): better accuracy and convergence rate compared to penalty method. Demerit: Dependent on geometries of boundaries. C0 interface Additionally possible application: seamlessly coupling with finite elements. R. Tian CCSE, JAERI

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