Application of the Finite Element Exterior Calculus to the Equations of Linear Elasticity Richard S. Falk Department of Mathematics Rutgers University June 12, 2012 Joint work with: Douglas Arnold, University of Minnesota Ragnar Winther, Centre of Mathematics for Applications, University of Oslo, Norway Richard S. Falk Finite Element Methods for Linear Elasticity
Outline Of Talk ◮ Variational formulations of the equations of linear elasticity ◮ Stability of discretizations of saddle-point problems ◮ Connections to exact sequences – continuous and discrete ◮ Exact sequences for elasticity ◮ From de Rham to elasticity ◮ Stability of continuous formulation of elasticity with weakly imposed symmetry ◮ Finite element methods for the equations of elasticity from connections to de Rham Richard S. Falk Finite Element Methods for Linear Elasticity
Equations of linear elasticity For N = 2 , 3, equations of linear elasticity written as system: in Ω ⊂ R N . A σ = ε u , div σ = f • stressfield σ ( x ) ∈ S (symmetric matrices). • displacement field u ( x ) ∈ R N . • f = f ( x ) given body force. • A = A ( x ) : S �→ S given, uniformly positive definite, compliance tensor (material dependent). • ǫ ij ( u ) = [ ∂ u i /∂ x j + ∂ u j /∂ x i ] / 2. • div of matrix field taken row-wise. • If body clamped on boundary ∂ Ω of Ω, BC: u = 0 on ∂ Ω. Richard S. Falk Finite Element Methods for Linear Elasticity
Stress–displacement formulations Strongly imposed symmetry: Find ( σ, u ) ∈ H ( div , Ω , S ) × L 2 (Ω , R N ) such that: ( A σ, τ ) + ( div τ, u ) = 0 , τ ∈ H ( div , Ω , S ) , v ∈ L 2 (Ω , R N ) . ( div σ, v ) = ( f , v ) Weakly imposed symmetry: Find ( σ, u , p ) ∈ H ( div , Ω , M ) × L 2 (Ω , R N ) × L 2 (Ω , K ) such that: ( A σ, τ ) + ( div τ, u ) + ( τ, p ) = 0 , τ ∈ H ( div , Ω , M ) , v ∈ L 2 (Ω , R N ) , ( div σ, v ) = ( f , v ) , ( σ, q ) = 0 , q ∈ L 2 (Ω , K ) . M = N × N matrices, K = skew symmetric matrices. Richard S. Falk Finite Element Methods for Linear Elasticity
A simpler problem Consider mixed formulation of Poisson’s equation ∆ p = f . ∀ v ∈ H ( div , Ω , R N ) , ( u , v ) + ( p , div v ) = 0 ∀ q ∈ L 2 (Ω , R ) . ( div u , q ) = ( f , q ) Approximation well understood and related to commuting diagrams for de Rham sequence. In 2-D, if we have finite element spaces and bounded projection operators satisfying commuting diagram: → L 2 − curl div → H 1 0 − − − − → H ( div ) − − − → 0 � Π 1 � Π d � Π 0 h h h curl div 0 − − → S h − − → − − → Q h − − → 0 V h then mixed finite element stable, and get quasi-optimal approximation. Richard S. Falk Finite Element Methods for Linear Elasticity
Elasticity (Hilbert) complexes with strong symmetry Find ( σ, u ) ∈ H ( div , Ω , S ) × L 2 (Ω , R N ) such that ( A σ, τ ) + ( u , div τ ) = 0 ∀ τ ∈ H ( div , Ω , S ) , ∀ v ∈ L 2 (Ω , R N ) . ( div σ, v ) = ( f , v ) Corresponding complexes in this case: → H 1 ( R 3 ) ε → H ( J , S ) J → H ( div , S ) div → L 2 ( R 3 ) → 0 , 0 − − − − − in 3-D, where J σ = curl ( curl σ ) T . → H 2 J → H ( div , S ) div → L 2 ( R 2 ) → 0 0 − − − − in 2-D, where � ∂ 2 q /∂ y 2 − ∂ 2 q /∂ x ∂ y � Jq = . ∂ 2 q /∂ x 2 − ∂ 2 q /∂ x ∂ y Richard S. Falk Finite Element Methods for Linear Elasticity
Approximation in 2D Although elasticity problem only involves last two spaces in complex → H 2 J → H ( div , S ) div → L 2 ( R 2 ) → 0 , 0 − − − − having full sequence gives clue how to choose discretization. One looks for subcomplex of form J div 0 − → Q h − → Σ h − − → V h → 0 . For example, looking for simple finite element subspace of H 2 , one is led to choosing Q h = Argyris space of C 1 quintics. Since JQ h ⊂ Σ h , Σ h must be a piecewise cubic space, and since Argyris space has 2nd derivative DOF at vertices, DOF for Σ h will include vertex DOF (not usual for H ( div ) spaces). Richard S. Falk Finite Element Methods for Linear Elasticity
Arnold-Winther elements In 2002, Arnold-Winther constructed commuting diagrams of form: J div → C 2 ( R ) → C 0 ( S ) → L 2 ( R 2 ) − 0 − − − − − − − → 0 � I 2 � I d � I 0 h h h J div 0 − − → Q h − − → Σ h − − → V h − − → 0 Simplest case of family of elements: Q h = Argyris space of C 1 quintics. Stress space Σ h = p. cubic functions with p. linear divergence (24 DOF). Displacements V h = p. linear functions. However, since I 2 h involves point values of 2nd derivative and I 1 h involves point values, these operators do not extend to bounded operators in Hilbert spaces H 2 and H ( div , S ). In Bulletin, bounded cochain projections constructed. Gives stability for corresponding mixed finite element method by Hilbert complex theory (assumes Ω star-shaped). Richard S. Falk Finite Element Methods for Linear Elasticity
Elasticity with weakly imposed symmetry In 2-D, setting W = K × R 2 , relevant complex: → C ∞ ( M ) ( skw div ) → C ∞ ( K ) J · · · − − − − − → C ∞ ( W ) → 0 . In 3-D, with W = K × R 3 , relevant complex: → C ∞ ( M ) ( skw div ) → C ∞ ( M ) J · · · − − − − − → C ∞ ( W ) → 0 . Here J : C ∞ ( M ) �→ C ∞ ( M ) denotes extension of previous operator. J τ = curl S − 1 curl τ, algebraic S Richard S. Falk Finite Element Methods for Linear Elasticity
New approach to discretization of elasticity sequences: ◮ Use procedure on continuous level to derive elasticity sequence from multiple copies of de Rham sequence. ◮ Use this connection to establish stability for continuous formulation of elasticity ◮ To discretize, start from known good discretizations of de Rham sequence. ◮ Determine conditions so that an analogue of stability proof for continuous problem will give stability of discrete problem. To see structure more clearly, adopt notation of differential forms. For simplicity, mostly consider 2D examples. Richard S. Falk Finite Element Methods for Linear Elasticity
de Rham sequences with values in a vector space Write 2-D de Rham sequence in form: → Λ 0 d 0 → Λ 1 d 1 → Λ 2 → 0 . 0 − − − Also consider sequences whose values lie in either V = R n or K , space of skew-symmetric matrices. Both corresponding de Rham sequences also exact, e.g., → Λ 0 ( V ) d 0 → Λ 1 ( V ) d 1 → Λ 2 ( V ) → 0 . 0 − − − Here Λ k ( V ) consists of elements of form: � ω ( x ) = f I ( x ) dx I I with coefficients f I ∈ C ∞ (Ω , V ). Richard S. Falk Finite Element Methods for Linear Elasticity
Weak symmetry elasticity sequence from de Rham (BGG) Following ideas of Eastwood: Start from two de Rham sequences: → Λ n − 2 ( K ) d n − 2 → Λ n − 1 ( K ) d n − 1 → Λ n ( K ) → 0 , · · · − − − − − − − → Λ n − 2 ( V ) d n − 2 → Λ n − 1 ( V ) d n − 1 → Λ n ( V ) → 0 . · · · − − − − − − − For both n = 2 and n = 3, spaces Λ n − 1 ( V ) are spaces of stresses and can be identified with n × n matrices. Let X = ( x 1 , . . . , x n ) T and define K k : Λ k ( V ) → Λ k ( K ) by K k ω = X ω T − ω X T Then define S k := d k K k − K k +1 d k : Λ k ( V ) → Λ k +1 ( K ) Richard S. Falk Finite Element Methods for Linear Elasticity
The operator S k Can show: S k is an algebraic operator. Two important operators: S n − 2 and S n − 1 . Operator S n − 1 can be identified with skw, i.e., taking skew part of matrix (i.e., ( W − W T ) / 2). � 0 � 0 � ω 1 � ω 2 � − ω 1 � n = 2 : = dx 1 + S n − 2 dx 2 ω 2 − ω 2 0 ω 1 0 Easy to check S 0 invertible. For n = 3, S 1 more complicated, but still algebraic and invertible. Key property used to establish stability: d n − 1 S n − 2 = − S n − 1 d n − 2 � 0 � − 1 n = 2 : ( div W ) + 2 skw curl W = 0 . 1 0 Much more complicated identity in 3-d. Richard S. Falk Finite Element Methods for Linear Elasticity
Elasticity sequence from de Rham sequence Picture is: → Λ n − 2 ( K ) d n − 2 → Λ n − 1 ( K ) d n − 1 → Λ n ( K ) → 0 · · · − − − − − − − ր S n − 2 ր S n − 1 → Λ n − 2 ( V ) d n − 2 → Λ n − 1 ( V ) d n − 1 → Λ n ( V ) → 0 . · · · − − − − − − − Since S n − 2 invertible, combine to one sequence: Let W = K × V . Sn − 1 ( dn − 1 ) d n − 2 ◦ S − 1 n − 2 ◦ d n − 2 → Λ n − 2 ( K ) → Λ n − 1 ( V ) → Λ n ( W ) → 0 · · · − − − − − − − − − − − − − − − After proper identifications, ( n = 2), this is elasticity sequence → C ∞ ( M ) ( skw div ) C ∞ ( K ) J − − − − → C ∞ ( W ) → 0 . Richard S. Falk Finite Element Methods for Linear Elasticity
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