Introduction to PDEs and Numerical Methods Tutorial 10. Finite Element Analysis

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1 Patzhater für Bid, Bid auf Titefoie hinter das Logo einsetzen Introduction to PDEs and Numerica Methods Tutoria. Finite Eement Anaysis Dr. Noemi Friedman, 3..2

2 FROM STRONG FORM TO WEAK FORM inhomogeneous Dirichet boundary conditions Strong form: Exampe: LL(x) = f(x) Δu x = f u = g u n = h.) Mutipy by test function φ and integrate convert to homogeneous probem: u = ω + v v known function, u = g on Γ D ω:new function that we ook for Δu x φ x d f x φ x d = 2.) Reduce order of LL, φ by using divergence theoreem Δu x φ x d = u x x d u φ x dγ n Dr. Noemi Friedman PDE tutoria Seite 2

3 FROM STRONG FORM TO WEAK FORM inhomogeneous Dirichet boundary conditions Δu x φ x d = u x x d 3.) Appy boundary conditions φ x dγ = φ x dγ + Γ D h u φ x dγ n φ x dγ = hφ x dγ u x x d = f x φ x d + hφ x dγ ω x + v x x d = f x φ x d + hφ x dγ ω x x d = f x φ x d + hφ x dγ v x x d Dr. Noemi Friedman PDE tutoria Seite 3

4 Discretisation Gaerkin method - FEM Approximate soution to the weak equation from a given finite-dimansiona subspace ω x N c i Φ i (x) φ i x = Φ i (x) Pugging it in the weak form: ω x φ j x d c i Φ i (x) Φ j x d N = f x φ x d + hφ x dγ v x x d φ j = f x Φ j x d + hφ j x dγ v x Φ j x d N c i Φ i (x) φ j x d = f x φ j x d + + hφ j x dγ v x Φ j x d K ij f j KK = f Dr. Noemi Friedman PDE tutoria Seite

5 D Exampe with inear noda basis p(x) /5 /5 /5 /5 /5 p x = aa Strong form: u = u = Discretisation of the weak form: u x u i φ i (x) Weak form: EE = p x φ x Not efficient to cacuate a the eements of the stiffness matrix one by one! u j EE φ i(x) φ j (x) dx K ij = p(x)φ j (x)dx f j Cacuate eement stiffness matrices and assembe Dr. Noemi Friedman PDE tutoria Seite 5

6 D Exampe with inear noda basis p(x) /5 2 /5 3 /5 /5 5 /5 6 instead: Compute stiffness matrix eementwisey and then assembe Goba stiffness matrix K e, K e,2 K e 2, K e 2,2 φ 2 = EE φ (x) φ (x) φ 3 = EE φ (x) φ 5 (x) = EE φ 5(x) φ (x) = EE φ 5(x) φ 5 (x) φ φ K e 2, K e K e 2 (,2) (2,2) K e 3, K e K e K e 2 (2,) 2 (2,2) 3 (,2) K e (,) K e (,2) K e 3 (2,) K e 3 (2,2) K e (2,) K e (2,2) K 5 e (,) u u 2 u 3 u u 5 = f 2 f 3 f f 5 K e = K e (,) K e (,2) 6 u 6 5 K e (2,) K e (2,2) Dr. Noemi Friedman PDE tutoria Seite 6 K u f

7 D Exampe with inear noda basis p(x) instead: Compute stiffness matrix eementwisey and then assembe /5 2 /5 3 /5 /5 5 / φ 2 φ 3 φ φ 5 u = f e = p(x)φ (x) 2 K e 2, K e K e 2 (,2) (2,2) u 2 f e 2 f e 2 f e 2 = p(x)φ 5 (x) 3 K e 3, K e 2 (2,) K e 2 (2,2) K 3 e (,2) u 3 f e 2 2 f e 3 f e = e p(x ) p(x 5 ) f e = f e 5 6 K e (,) K e (,2) K e 3 (2,) K e 3 (2,2) K e (2,) K e (2,2) K 5 e (,) u u 5 u 6 f e 3 2 f e f e 2 f e 5 5 f e Dr. Noemi Friedman PDE tutoria Seite 7 K u f

8 Loca/goba coordinate system D Idea: coordinate transformation to have unit ength eements eement stiffnes matrix is the same for each eement ξ = [,] ξ = e K e i, j = EE φ i(x) φ j (x) x x K e i, j = EE φ i(ξ) ξ φ j (ξ) ξ ξ ξ = EE e 2 φ i(ξ) φ j (ξ) ξ ξ e e K e i, j = EE e 2 φ i(ξ) φ j (ξ) ξ ξ dξ dξ = EE e φ i(ξ) φ j (ξ) dξ ξ ξ e Dr. Noemi Friedman PDE tutoria Seite 8

9 Loca/ coordinate system, isoparametric mapping D Basis functions: coordinate transformation using the ansatzfunctions isoparametric mapping functions of ower order: subparametric functions of higher order: superparametric Transformation from oca to goba coordinates: x gggg ξ = x i N ξ + x i+ N 2 ξ = N ξ Stiffness matrix with isoparametric eements: N 2 ξ x i x i+ oca coordinate ξ = ξ = ξ = goba coordinate x i x Dr. Noemi Friedman PDE tutoria Seite 9

10 Loca/ coordinate system, isoparametric mapping 2D quadriatera eements Basis functions: (,) 2 η 3 (,) ξ (, ) (, ) Transformation from oca to goba coordinates: x gggg ξ, η y gggg ξ, η Stiffness matrix: = N ξ, η N 2 ξ, η N ξ, η N 3 ξ, η N 2 ξ, η N 3 ξ, η N ξ, η N ξ, η x y x 2 y 2 x 3 y 3 x y Dr. Noemi Friedman PDE tutoria Seite

Loca/ coordinate system, isoparametric mapping 2D quadriatera eements Stiffness matrix: 2 η 3 ξ Stiffness matrix with oca coordinates: substitution rue

11 Loca/ coordinate system, isoparametric mapping 2D quadriatera eements Stiffness matrix: 2 η 3 ξ Stiffness matrix with oca coordinates: substitution rue determinant shoud not be negative or zero! where: J = N i ξ, η ξ N i ξ, η ξ x i y i N i ξ, η η N i ξ, η η x i y i Dr. Noemi Friedman PDE tutoria Seite

Stiffness matrix: = N ξ, η N 2 ξ, η N ξ, η formuation ike in quadriatera case N

12 Loca/ coordinate system, isoparametric mapping 2D trianguar eements Basis functions: Transformation from oca to goba coordinates: x gggg ξ, η y gggg ξ, η Stiffness matrix: = N ξ, η N 2 ξ, η N ξ, η formuation ike in quadriatera case N 3 ξ, η N 2 ξ, η N 3 ξ, η x y x 2 y 2 x 3 y Dr. Noemi Friedman PDE tutoria Seite 2

Numerical methods for PDEs FEM - abstract formulation, the Galerkin method

Numerical methods for PDEs FEM - abstract formulation, the Galerkin method Patzhater für Bid, Bid auf Titefoie hinter das Logo einsetzen Numerica methods for PDEs FEM - abstract formuation, the Gaerkin method Dr. Noemi Friedman Contents of the course Fundamentas of functiona