Numerical methods for PDEs FEM implementation: element stiffness matrix, isoparametric mapping, assembling global stiffness matrix

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1 Platzhaltr für Bild, Bild auf Titlfoli hintr das Logo instzn Numrical mthods for PDEs FEM implmntation: lmnt stiffnss matrix, isoparamtric mapping, assmbling global stiffnss matrix Dr. Nomi Fridman

2 Contnts of th cours Fundamntals of functional analysis Abstract formulation FEM Application to conrt formulations Convrgnc, rgularity Variational crims Implmntation Mixd formulations (.g. Stoks) Stabilisation for flow problms Error indicators/stimation Adaptivity Implmntation Dr. Nomi Fridman PDE2 tutorial Sit 2

3 Implmntation Picwis polynomials and th FEM Sparsity of th stiffnss matrix Calculating th stiffns matrix lmntwisly Isoparamtric mapping Condition numbr of th stiffnss matrix Sparsity of th stiffnss matrix Numrical intgration Implmntation Dr. Nomi Fridman PDE2 tutorial Sit

4 D Exampl with linar nodal basis p(x) l l/5 l/5 l/5 l/5 l/5 p x = aa Strong form: u = u l = Discrtisation of th wak form: u x u i ψ i (x) i= Wak form: l EE dψ l = p x ψ x Not fficint to calculat all th lmnts of th stiffnss matrix on by on! u j EE ψ i(x) ψ j (x) dx i= l K ij = p(x)ψ j (x)dx l f j Calculat lmnt stiffnss matrics and assmbl Implmntation Dr. Nomi Fridman PDE2 tutorial Sit

5 D Exampl with linar nodal basis l p(x) l/5 2 l/5 l/5 l/5 5 l/5 6 instad: Comput stiffnss matrix lmntwisly and thn assmbl Global stiffnss matrix K, K,2 K 2, K 2,2 ψ 2 = EE ψ (x) ψ (x) Ω ψ = EE ψ (x) ψ 5 (x) Ω = EE ψ 5(x) ψ (x) Ω = EE ψ 5(x) Ω ψ 5 (x) ψ ψ K 2, K K 2 (,2) (2,2) K, K K K 2 (2,) 2 (2,2) (,2) K (,) K (,2) K (2,) K (2,2) K (2,) K (2,2) K 5 (,) u u 2 u u u 5 = f 2 f f f 5 K = K (,) K (,2) 6 u 6 5 K (2,) K (2,2) K u Implmntation Dr. Nomi Fridman PDE2 tutorial Sit 5 f

6 D Exampl with linar nodal basis l p(x) instad: Comput stiffnss matrix lmntwisly and thn assmbl l/5 2 l/5 l/5 l/5 5 l/ ψ 2 ψ ψ ψ 5 u = f = p(x)ψ (x) Ω 2 K 2, K K 2 (,2) (2,2) u 2 f 2 f 2 f 2 = p(x)ψ 5 (x) Ω K, K 2 (2,) K 2 (2,2) K (,2) u f 2 2 f K (,) K (,2) K (2,) K (2,2) u f 2 f f = f 5 6 K (2,) K (2,2) K 5 (,) u 5 u 6 f 2 f 5 5 f 2 K u Implmntation Dr. Nomi Fridman PDE2 tutorial Sit 6 f

7 Local/global coordinat systm D Ida: coordinat transformation to hav unit lngth lmnts lmnt stiffns matrix is th sam for ach lmnt ξ = [,] = l K k, l = EE ψ (x) ψ 5 (x) x x i, j [,5] Ω k, l [,2] K k, l = EE N k(ξ) N l (ξ) = EE Ω l 2 N k(ξ) l l Ω N l (ξ) K k, l = EE l 2 N k(ξ) N l (ξ) (ξ) = EE l N k(ξ) N l (ξ) l Implmntation Dr. Nomi Fridman PDE2 tutorial Sit 7

8 Local/global coordinat systm D f l = p(x)ψ (x) Ω p(x)ψ 5 (x) Ω = p(ξ)n l (ξ) (ξ) = l p(ξ)n l (ξ) l [,2] Implmntation Dr. Nomi Fridman PDE2 tutorial Sit 8

9 Local/ coordinat systm, isoparamtric mapping D x Shap functions: ξ = [,] coordinat transformation using th ansatzfunctions N ξ = ξ N 2 ξ = ξ Transformation from local to global coordinats: x ξ = x i N ξ + x i+ N 2 ξ = N ξ Stiffnss matrix with isoparamtric lmnts: local coordinat global coordinat x i x isoparamtric mapping 2 N 2 ξ ξ = ξ = x i x i+ K k, l = EE ψ i(x) ψ j (x) = EE N k(ξ) N l (ξ) x x K k, l = EE N k(ξ) functions of lowr ordr: subparamtric functions of highr ordr: suprparamtric Ω Ω k, l [,2] Nl (ξ) (ξ) Implmntation Dr. Nomi Fridman PDE2 tutorial Sit 9 = x dn ξ i = ξ + x i+ 2 ξ 2 ξ x i x i+ i, j [,5]

ξ, η Stiffnss matrix: = N ξ, η N 2 ξ, η N ξ, η N ξ, η N 2 ξ, η N ξ,

10 Isoparamtric linar mapping 2D triangular lmnts Basis functions: Transformation from local to global coordinats: x gggg ξ, η y gggg ξ, η Stiffnss matrix: = N ξ, η N 2 ξ, η N ξ, η N ξ, η N 2 ξ, η N ξ, η x y x 2 y 2 x y i, j [,2,] Implmntation Dr. Nomi Fridman PDE2 tutorial Sit

11 Isoparamtric linar mapping 2D triangular lmnts Stiffnss matrix: i, j [,2,] Stiffnss matrix with local coordinats: N j N i η K ii = J T J N T j N i η η whr: J ξξ substitution rul dtrminant should not b ngativ or zro! i, j [,2,] J = N i ξ, η i= N i ξ, η i= x i y i N i ξ, η η i= N i ξ, η η i= x i y i Implmntation Dr. Nomi Fridman PDE2 tutorial Sit

12 Isoparamtric linar mapping 2D triangular lmnts, xampl 5 6 Transformation from local to global coordinats (isoparamtric mapping): x gggg ξ, η y gggg ξ, η x ξ, η y ξ, η = = N ξ, η N 2 ξ, η N ξ, η ξ η ξ ξ η N ξ, η N 2 ξ, η N ξ, η η ξ η Implmntation Dr. Nomi Fridman PDE2 tutorial Sit 2 x y x 2 y 2 x y

13 Local/ coordinat systm, isoparamtric mapping 2D triangular lmnts, xampl Stiffnss matrix with local coordinats: K ii = η J T N j N j η J T N i N i η J ξξ i, j [,2,] Implmntation Dr. Nomi Fridman PDE2 tutorial Sit

14 Local/ coordinat systm, isoparamtric mapping 2D triangular lmnts, xampl J T 5 7 = J = J T = J = 2 K ii = K ii = η J T η N j N j η J T N j N j η N i N i η J ξξ N i N i η ξξ J T = 2 J T 7 5 K K 2 K K 2 K 22 K 2 K K 2 K K 2 = η ξξ == η ξξ K 2 ==.8 η ξξ =.8 2 =.559 Implmntation Dr. Nomi Fridman PDE2 tutorial Sit

15 Local/ coordinat systm, isoparamtric mapping 2D triangular lmnts, xampl K K 2 K K 2 K 22 K 2 K K 2 K u u 2 u = f f 2 f J T = J = f = p(x)n (x, y) Ω p(x)n 2 (x, y) Ω p(x)n (x, y) Ω = η p(ξ)n (ξ, η) η η p(ξ)n 2 (ξ, η) p(ξ)n (ξ, η) J dη J dη J dη Implmntation Dr. Nomi Fridman PDE2 tutorial Sit 5

16 Local/ coordinat systm, isoparamtric mapping 2D triangular lmnts, xampl u = K 22 K 2 K 2 K 2 K K K 2 K K u 2 u u u 5 u 6 f 2 f f local 2 global K K 2 K K 2 K 22 K 2 K K 2 K u u 2 u = f f 2 f 6 Implmntation Dr. Nomi Fridman PDE2 tutorial Sit 6

17 Condition numbr of th stiffnss matrix [Chaptr..] Condition numbr What happns with th roundoff rrors in K = LL = K + δδ K Ku = f K + δk u = f + δf u u u λ mmm λ mmm δf f λ mmm λ mmm = κ(k) Condition numbr of K with nodal bass with 2D triangular msh: For th Poisson quation N c i i= Ψ i (x) Ψ j x dω Ω = f x Ψ j x dω Ω turns il-conditiond for rfind msh!!! K ij Implmntation Dr. Nomi Fridman PDE2 tutorial Sit 7

Numerical methods for PDEs FEM convergence, error estimates, piecewise polynomials

Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Numerical methods for PDEs FEM convergence, error estimates, piecewise polynomials Dr. Noemi Friedman Contents of the course Fundamentals