2005 2006.2.22-1 -
1 Fig. 1 2005 2006.2.22-2 -
Element-Free Galerkin Method (EFGM) Meshless Local Petrov-Galerkin Method (MLPGM) 2005 2006.2.22-3 -
2 MLS u h (x) 1 p T (x) = [1, x, y]. (1) φ(x) 0.5 φ(x) N Σ (1) J[a] = w i (x) p T (x i )a(x) u i 2 i=1 0-0.5 0 0.5 x Fig. 2 1 φ i (x) N : w(r) : u i u(x i ) 2005 2006.2.22-4 -
w(r) MLS Gauss : : Table 1 MLS 1 c = h R = 4h ( 5 ) Gauss h : 2005 2006.2.22-5 -
3 Element-Free Galerkin Method 2 Poisson (2) (3) (4) 2 Poisson : Fig. 3 Ω Ω (5) λ(s) : Lagrange ( Ω s ) (5) (6) 2 Poisson (6) 2005 2006.2.22-6 -
Table 2 Ω = ( 1, 1) ( 1, 1) Dirichlet u = sinπx cosπy p = 2π 2 sinπx cosπy 1 1024 256 Fig. 4 2005 2006.2.22-7 -
CPU 10 3 CPU Time [s] 10 1 10-1 Gauss ICCG 10-3 10 1 10 2 10 3 10 4 Total Number of Nodes, N Fig. 5(6) CPU 2005 2006.2.22-8 -
4 Meshless Local Petrov-Galerkin Method 2 Poisson 2 Poisson 2 Poisson : 2 Poisson : Fig. 6 (7) α : (7) Ku = f. (8) 2 Poisson (8) 2005 2006.2.22-9 -
Table 3 Ω = ( 1, 1) ( 1, 1) 1 Dirichlet u = sinπx cosπy p = 2π 2 sinπx cosπy 3200 Fig. 7 2005 2006.2.22-10 -
(a) r s = 0.5h (b) r s = h (c) r s = h Fig. 8 2005 2006.2.22-11 -
10 1 Relative Error, ε 10-1 10-3 10 2 10 3 10 4 Total Number of Nodes, N Fig. 9 (α = 10 6 ) : r s = 0.5h : r s = : r s = 2h : r s = 4h : r s = 10h 2005 2006.2.22-12 - 2 2 h : r s = h
10 1 Relative Error, ε 10-1 10-3 10 2 10 3 10 4 Total Number of Nodes, N Fig. 10 (r s = 4h) : α = 1.0 10 2 : α = 1.0 10 4 : α = 1.0 10 6 : α = 1.0 10 8 : α = 1.0 10 10 : α = 1.0 10 12 2005 2006.2.22-13 -
5 EFGM MLPGM Table 4 EFGM MLPGM u δu EFGM Lagrange (Galerkin ) MLPGM (Petrov-Galerkin ) 2005 2006.2.22-14 -
EFG MLPG EFG EFG MLPG MLPG Table 5 Lagrange Lagrange 2005 2006.2.22-15 -
Table 6 w σ φ ψ Gauss Shepard (0 MLS ) Gauss MLS MLS Table 7 Galerkin(w) w w Galerkin(σ) σ σ Galerkin(φ) φ φ Petrov-Galerkin(φ, w) φ w Petrov-Galerkin(φ, σ) φ σ Petrov-Galerkin(φ, ψ) φ ψ 2005 2006.2.22-16 -
Lagrange EFG 10 1 10 1 10 0 10 0 Relative Error, ε 10-1 10-2 Relative Error, ε 10-1 10-2 10-3 10 1 10 2 10 3 10 4 Total Number of Nodes, N 10-3 10 1 10 2 10 3 10 4 Total Number of Nodes, N (a) Galerkin (b) Petrov-Galerkin Fig. 11 : Galerkin(w) : Galerkin(σ) : Galerkin(φ) : Petrov-Galerkin(φ, w) : Petrov-Galerkin(φ, σ) : Petrov- Galerkin(φ, ψ) 2005 2006.2.22-17 -
EFG 10 1 10 1 10 0 10 0 Relative Error, ε 10-1 10-2 Relative Error, ε 10-1 10-2 10-3 10 1 10 2 10 3 10 4 10-3 10 1 10 2 10 3 10 4 Total Number of Nodes, N Total Number of Nodes, N (a) Galerkin (b) Petrov-Galerkin Fig. 12 : Galerkin(w) : Galerkin(σ) : Galerkin(φ) : Petrov-Galerkin(φ, w) : Petrov-Galerkin(φ, σ) : Petrov- Galerkin(φ, ψ) 2005 2006.2.22-18 -
Lagrange MLPG 10 0 10-1 Relative Error, ε 10-2 10-3 10-4 10 2 10 3 10 4 Total Number of Nodes, N Fig. 13 : Galerkin(w) : Galerkin(φ) : Petrov- Galerkin(φ, w) 2005 2006.2.22-19 -
MLPG 10 1 10 0 Relative Error, ε 10-1 10-2 10-3 10 2 10 3 10 4 Total Number of Nodes, N Fig. 14 : Galerkin(w) : Galerkin(φ) : Petrov- Galerkin(φ, w) 2005 2006.2.22-20 -
6 EFGM ICCG Gauss MLPGM Lagrange MLPG Petrov-Galerkin(φ, w) Petrov-Galerkin(φ, w) 2005 2006.2.22-21 -