Partial differential equation_1 Finite element method

偏微分方程式入門の第一回目として、有限要素法の解説用資料の公開版です。
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https://note.mu/lib_arts/n/n45f1fc24bafb

Transcript

1. / FreeFEM Python FreeFEM Python 1 1 clock@LiberalArtsCommunity clock@LiberalArtsCommunity 2019/10/19 2019/10/19 1 Copyright © Liberal Arts Community. All Rights Reserved.
2. / 2 . 1 Copyright © Liberal Arts Community. All Rights Reserved.
3. / 2 ( ) (Poisson ) ( ) ( ) = α∇ u ∂t ∂u 2 −∇ u = f(r) 2 = ∇ u c2 1 ∂t2 ∂ u 2 2 ∇ = 2 + ∂x2 ∂2 ∂y2 ∂2 2 . 2 Copyright © Liberal Arts Community. All Rights Reserved.
4. / (field): ( ) ( ) 2 . 3 Copyright © Liberal Arts Community. All Rights Reserved.
5. / 1. ( ) n n 2 . 4 Copyright © Liberal Arts Community. All Rights Reserved.
6. / 2. ( ) Fourier etc ( ) etc 2 . 5 Copyright © Liberal Arts Community. All Rights Reserved.
7. / 3 . 1 Copyright © Liberal Arts Community. All Rights Reserved.
8. / FreeFEM FreeFEM 3 . 2 Copyright © Liberal Arts Community. All Rights Reserved.
9. / Dirichlet Poisson ( ) u −∇ u = f(r), u = 0 on ∂Ω. 2 3 . 3 Copyright © Liberal Arts Community. All Rights Reserved.
10. / ( ) ( ) v v = ∣ ∂Ω 0 u ∇u ⋅ ∇v dV = fv dV ∫ Ω ∫ Ω 3 . 4 Copyright © Liberal Arts Community. All Rights Reserved.
11. / 2 FreeFEM 3 . 5 Copyright © Liberal Arts Community. All Rights Reserved.
12. / 1. 2. 3. 3 . 6 Copyright © Liberal Arts Community. All Rights Reserved.
13. / 1. 2. 1 P1 1 , P2 2 3. : int n = 100; border C(t=0,1){x=cos(2*pi*t); y=sin(2*pi*t);}; mesh Th = buildmesh(C(n)); fespace Vh(Th,P1); ∇u ⋅ ∫ Ω ∇v dV = fv dV ∫ Ω Vh u,v; problem Poisson(u,v) = int2d(Th)( dx(u)*dx(v)+dy(u)*dy(v) ) - int2d(Th)( f*v ); Poisson; 3 . 7 Copyright © Liberal Arts Community. All Rights Reserved.
14. / FreeFEM FreeFEM 4 . 1 Copyright © Liberal Arts Community. All Rights Reserved.
15. / Poisson Poisson 2 (Dirichlet ) Poisson ( ) −T∇ u = −ρg 2 g T 4 . 2 Copyright © Liberal Arts Community. All Rights Reserved.
16. / 4 . 3 Copyright © Liberal Arts Community. All Rights Reserved.
17. / : ( ): : = α∇ u ∂t ∂u 2 ∼ ∂t ∂u Δt u −u i i−1 v dV + α∇u ⋅ ∇v dV = 0 ∫ Ω ( Δt u − u i i−1 ) ∫ Ω i 4 . 4 Copyright © Liberal Arts Community. All Rights Reserved.
18. / 2 (i) 0 ( ) (ii) ( ) u = ∣ ∂Ω 0 ∇u = ∣ ∂Ω 0 4 . 5 Copyright © Liberal Arts Community. All Rights Reserved.
19. / Neumann Dirichlet 4 . 6 Copyright © Liberal Arts Community. All Rights Reserved.
20. / Dirichlet : Neumann : t → ∞ 4 . 7 ( ǟ Copyright © Liberal Arts Community. All Rights Reserved.