Differential Complexes and Mixed Finite Elements for Elasticity

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1 Differential Complexes an Mixe Finite Elements for Elasticity Douglas N. Arnol Institute for Mathematics an its Applications Richar Falk Rutgers University Ragnar Winther Centre of Mathematics for Applications, University of Oslo Institute for Mathematics an its Applications

2 Outline 1. Mixe finite elements for elasticity 2. Exterior calculus 3. Discrete exterior calculus 4. Construction of the elasticity complex Construction of the elements 1

3 Outline 1. Mixe finite elements for elasticity 2. Exterior calculus 3. Discrete exterior calculus 4. Construction of the elasticity complex 2

4 Linear elasticity isplacement u : Ω V := R n Aσ = ɛ u := [ u + ( u) T ]/2 stress σ : Ω S:= R n n sym iv σ = f σ H(iv, Ω; S), u L 2 (Ω; V) satisfy Ω Aσ : τ x + iv τ u x = 0 τ H(iv, Ω; S) Ω iv σ v x = f v v L 2 (Ω; V) Ω (σ, u) H(iv, Ω; S) L 2 (Ω; V) sale point of L(τ, v) = Ω Ω (1 2 Aτ : τ + iv τ v f v) x. 3

5 Stable mixe elements 2D composite elts: Johnson Mercier 78 cf. Fraeijs e Veubeke 65; Watwoo Hartz 68 Arnol Douglas Gupta D polynomial elements, Arnol Winther 2002: nonconf. 3D very complicate... 4

6 Weak symmetry To avoi the problems arising from the symmetry of the stress tensor, it can be ( impose weakly (σ, u, p) = argcrit 1 ) 2 Aτ : τ + v iv τ + τ : p + f v H(iv,Ω;M) L 2 (Ω;V) L 2 (Ω;K) Fraejis e Veubeke 75, Arnol-Brezzi-Douglas 84 (PEERS), Stenberg, Morley, Major result of these talks: new elements in 2D an 3D These were obtaine via a new homological viewpoint... 5

7 The elasticity complexes A key to eveloping stable elements for elasticity (with strongly impose symmetry) is the elasticity complex: T C (Ω, V) isplacement ɛ C (Ω, S) strain J C (Ω, S) stress iv C (Ω, V) 0 loa J = curl c curl r, secon orer T is the space of infinitesmal rigi motions For weakly impose symmetry the relevant sequence is T C (V K) (gra, I) C (M) J C (M) 0 iv A skw C (V K) 0 where J is efine to be zero on skew matrices. 6

8 Outline 1. Mixe finite elements for elasticity 2. Exterior calculus 3. Discrete exterior calculus 4. Construction of the elasticity complex 7

9 De Rham complex e Rham complex: 0 R Λ 0 (Ω) Λ 1 (Ω) Λ n (Ω) 0 ω Λ k (Ω) ω x is k-linear alternating form on T x Ω x Ω L 2 e Rham complex: 0 R HΛ 0 (Ω) HΛ 1 (Ω) HΛ n (Ω) 0 HΛ k (Ω) = { ω L 2 Λ k (Ω) ω L 2 Λ k+1 (Ω) } Polynomial e Rham complexes 0 R H r Λ 0 Hr 1 Λ 1 Hr n Λ n 0 0 R P r Λ 0 Pr 1 Λ 1 Pr n Λ n 0 8

10 Koszul complex Koszul ifferential κ : Λ k+1 Λ k : (κω) x (v 1,..., v k ) = ω x (x, v 1,..., v k ) κ : P r Λ k P r+1 Λ k 1 (c.f. : P r+1 Λ k 1 P r Λ k ) 0 R P r Λ 0 κ Pr 1 Λ 1 κ κ Pr n Λ n 0 Koszul complex (κ + κ)ω = (r + k)ω ω H r Λ k H r Λ k = H r+1 Λ k 1 κh r 1 Λ k+1 κ is a contracting chain homotopy 9

11 P r Λ k an P + r Λ k Using the Koszul ifferential, we efine a special space of polynomial ifferential k-forms between P r Λ k an P r+1 Λ k : P + r Λ k := P r Λ k + κh r Λ k+1 Note that P + r Λ 0 = P r+1 Λ 0 an P + r Λ n = P r Λ n Go mae P r Λ k an P + r Λ k, all the rest is the work of man. 10

12 The case Ω R 3 0 R Λ 0 (Ω) 0 R C (Ω) Λ 1 (Ω) gra C (Ω, R 3 ) Λ 2 (Ω) curl C (Ω, R 3 ) Λ 3 (Ω) 0 iv C (Ω) 0 smooth e Rham complex 0 R H 1 (Ω) gra H(curl) curl H(iv) iv L 2 (Ω) 0 L 2 e Rham complex 0 R P r (Ω) gra P r 1 (Ω, R 3 ) curl P r 2 (Ω, R 3 ) iv P r 3 (Ω) 0 polynomial e Rham complex 0 R P r (Ω) x P r 1 (Ω, R 3 ) x P r 2 (Ω, R 3 ) x P r 3 (Ω) 0 Koszul complex 11

13 Outline 1. Mixe finite elements for elasticity 2. Exterior calculus 3. Discrete exterior calculus 4. Construction of the elasticity complex 12

14 Piecewise polynomial ifferential forms T a triangulation of Ω V by n-simplices P r Λ k (T ) := { ω HΛ k (Ω) ω T P r Λ k (T ) T T } P r + Λ k (T ) := { ω HΛ k (Ω) ω T P r + Λ k (T ) T T } P r + Λ 0 (T ) = P r+1 Λ 0 (T ) H 1 Lagrange elts P r + Λ n (T ) = P r Λ n (T ) L 2 n = 3: P r + Λ 1 (T ) H(curl) n = 3: P r Λ 1 (T ) H(curl) n = 3: P r + Λ 2 (T ) H(iv) n = 3: P r Λ 2 (T ) H(iv) iscontinuous elts Neelec 1st kin elts Neelec 2n kin elts Raviart Thomas elts Brezzi Douglas Marini elts 13

15 Degrees of freeom T an n-simplex, (T ) = set of faces of imension, 0 n DOF for P r Λ k (T ): u u v, v P + r 1+k Λ k (f), f (T ), k n f DOF for P r + Λ k (T ): u u v, v P r +k Λ k (f), f (T n ), k n f 14

16 Discrete exact sequences For every r 0, the P + r Λ k spaces give an exact piecewise polynomial subcomplex: 0 R P + r Λ 0 (T ) P + r Λ 1 (T ) For n = 3, r = 0 these are the Whitney elements: P + r Λ n (T ) 0 R gra curl iv 0 For all r, the natural projections Π k r+ : Λ k (Ω) P + r Λ k (T ) relate this to the e Rham sequence commutatively: 0 R Λ 0 (Ω) Π 0 r+ 0 R P + r Λ 0 (T ) Λ 1 (Ω) Λ n (Ω) 0 Π 1 r+ Π n r+ P + r Λ 1 (T ) P + r Λ n (T ) 0 15

17 Other iscrete exact sequences Another exact sequence ening at P r Λ n (T ) uses the P s Λ k spaces of increasing egree: (Demkowicz-Varepetyan 99): R P r+n Λ 0 (T ) P r+n 1 Λ 1 (T ) P r Λ n (T ) 0 In fact, these are just the extreme cases. There are 2 n 1 pw polynomial e Rham sequences in n imensions. All relate to the e Rham complex through a commuting iagram. For n = 3 the other two are: R P r+2 Λ 0 (T ) P r+1 Λ 1 (T ) P + r Λ 2 (T ) P r Λ 3 (T ) 0 R P r+2 Λ 0 (T ) P + r+1 Λ1 (T ) P r+1 Λ 2 (T ) P r Λ 3 (T ) 0 16

18 The four sequences ening with P 0 Λ 3 (T ) in 3D R gra curl iv 0 R gra curl iv 0 R gra curl iv 0 R gra curl iv 0 17

19 Outline 1. Mixe finite elements for elasticity 2. Exterior calculus 3. Discrete exterior calculus 4. Construction of the elasticity complex 18

20 Bernstein Gelfan Gelfan construction, I 1. Start with the e Rham sequence with values in W := K V: W Λ 0 (Ω; W) ( 0 0 ) Λ 1 (Ω; W) ( 0 0 ) Λ 2 (Ω; W) ( 0 0 ) Λ 3 (Ω; W) 0 2. For any x R 3 efine K x : V K by K x v = 2 skw(xv T ) an K : Λ k (Ω; V) Λ k (Ω; K) by (Kω) x (v 1,..., v k ) = K x [ω x (v 1,..., v k )]. 3. Define automorphisms Φ : Λ k (W) Λ k (W) by ( ) ( ) I K Φ =, Φ 1 I K = 0 I 0 I 4. Define A = Φ ( ) 0 0 Φ 1 to get a moifie e Rham sequence: Φ(W) Λ 0 (W) A Λ 1 (W) A Λ 2 (W) A Λ 3 (W) 0 19

21 Bernstein Gelfan Gelfan construction, II 5. Note that A = is given by ( S 0 ), where S = K K : Λ k (Ω; V) Λ k+1 (Ω; K) (Sω) x (v 1,..., v k+1 ) = µ sign(µ)k vµk+1 ω x (v µ1,..., v µk ). Properties: S is algebraic; for k = 1, S is an isomorphism; S = S (K K) = K = (K K) 6. Define subspaces Γ k Λ k (Ω; W) satisfying A(Γ k ) Γ k+1 an projections π k : Λ k (Ω; W) Γ k satisfying π k+1 A = Aπ k : Γ 0 = Λ 0 (Ω; W), π 0 = i, Γ 3 = Λ 3 (Ω; W), π 3 = i, Γ 1 = { (ω, µ) Λ 1 (Ω; W) : ω = Sµ }, Γ 2 = { (ω, µ) Λ 2 (Ω; W) : ω = 0 } ( ) ( ) π 1 I 0 = S 1 : Λ 1 (Ω; W) Γ 1, π = 0 S 1 : Λ 2 (Ω; W) Γ 2. I 20

22 Bernstein Gelfan Gelfan construction, III 6. The following iagram commutes (use S = S): Φ(W) Λ 0 (W) π 0 A Λ 1 (W) A Λ 2 (W) π 1 π 2 A Λ 3 (W) 0 π 3 Φ(W) Γ 0 A Γ 1 A Γ 2 A Γ 3 0 Therefore, the subcomplex on the bottom row is exact. 7. This subcomplex may be ientifie with the elasticity complex. 21

23 Bernstein Gelfan Gelfan construction, conclue Γ 0 A Γ 1 A Γ 2 A Γ 3 = = = = Λ 0 ( 0, S 0 ) (K V) Λ 1 1 S1 1 (Ω; K) 1 Λ 2 ( S 2, 2 ) (Ω; V) T Λ 3 (K V) With the ientifications Λ 0 (K V) C (V K) Λ 1 (K) C (M) Λ 2 (K) C (M) Λ 3 (K V) C (V K) this becomes the elasticity sequence T C (V K) (gra, I) C (M) J C (M) (iv,skw) T C (V K) 0 22