Application of the Finite Element Exterior Calculus to the ... · Application of the Finite Element Exterior Calculus to the ... of linear elasticity I Stability of ... complex theory

Application of the Finite Element ExteriorCalculus to the Equations of Linear Elasticity

Richard S. Falk

Department of MathematicsRutgers 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

I Variational formulations of the equations of linear elasticity

I Stability of discretizations of saddle-point problems

I Connections to exact sequences – continuous and discrete

I Exact sequences for elasticity

I From de Rham to elasticity

I Stability of continuous formulation of elasticity with weaklyimposed symmetry

I Finite element methods for the equations of elasticity fromconnections to de Rham


Equations of linear elasticity

For N = 2, 3, equations of linear elasticity written as system:

Aσ = εu, div σ = f in Ω ⊂ RN .

• stressfield σ(x) ∈ S (symmetric matrices).• displacement field u(x) ∈ RN .• f = f (x) given body force.• A = A(x) : S 7→ S given, uniformly positive definite, compliancetensor (material dependent).

• εij(u) = [∂ui/∂xj + ∂uj/∂xi ]/2.• div of matrix field taken row-wise.• If body clamped on boundary ∂Ω of Ω, BC: u = 0 on ∂Ω.


Stress–displacement formulations

Strongly imposed symmetry:

Find (σ, u) ∈ H(div,Ω, S)× L2(Ω, RN) such that:

(Aσ, τ) + (div τ, u) = 0, τ ∈ H(div,Ω, S),

(div σ, v) = (f , v) v ∈ L2(Ω, RN).

Weakly imposed symmetry:

Find (σ, u, p) ∈ H(div,Ω, M)× L2(Ω, RN)× L2(Ω, K) such that:

(Aσ, τ) + (div τ, u) + (τ, p) = 0, τ ∈ H(div,Ω, M),

(div σ, v) = (f , v), v ∈ L2(Ω, RN),

(σ, q) = 0, q ∈ L2(Ω, K).

M = N × N matrices, K = skew symmetric matrices.


A simpler problem

Consider mixed formulation of Poisson’s equation ∆p = f .

(u, v) + (p,div v) = 0 ∀v ∈ H(div,Ω, RN),(div u, q) = (f , q) ∀q ∈ L2(Ω, R).

Approximation well understood and related to commuting diagramsfor de Rham sequence. In 2-D, if we have finite element spacesand bounded projection operators satisfying commuting diagram:

0 −−→ H1 curl−−→ H(div)div−−→ L2 −−→ 0yΠ1

h

yΠdh

yΠ0h

0 −−→ Shcurl−−→ Vh

div−−→ Qh −−→ 0

then mixed finite element stable, and get quasi-optimalapproximation.


Elasticity (Hilbert) complexes with strong symmetry

Find (σ, u) ∈ H(div,Ω, S)× L2(Ω, RN) such that

(Aσ, τ) + (u,div τ) = 0 ∀τ ∈ H(div,Ω, S),(div σ, v) = (f , v) ∀v ∈ L2(Ω, RN).

Corresponding complexes in this case:

0 −→ H1(R3)ε−→ H(J, S)

J−→ H(div, S)div−−→ L2(R3) → 0,

in 3-D, where Jσ = curl(curlσ)T .

0 −→ H2 J−→ H(div, S)div−−→ L2(R2) → 0

in 2-D, where

Jq =

(∂2q/∂y2 −∂2q/∂x∂y

−∂2q/∂x∂y ∂2q/∂x2

).


Approximation in 2D

Although elasticity problem only involves last two spaces incomplex

0 −→ H2 J−→ H(div, S)div−−→ L2(R2) → 0,

having full sequence gives clue how to choose discretization.

One looks for subcomplex of form

0 −→ QhJ−→ Σh

div−−→ Vh → 0.

For example, looking for simple finite element subspace of H2, oneis led to choosing Qh = Argyris space of C 1 quintics. SinceJQh ⊂ Σh, Σh must be a piecewise cubic space, and since Argyrisspace has 2nd derivative DOF at vertices, DOF for Σh will includevertex DOF (not usual for H(div) spaces).


Arnold-Winther elements

In 2002, Arnold-Winther constructed commuting diagrams of form:

0 −−→ C 2(R)J−−→ C 0(S)

div−−→ L2(R2) −−→ 0yI 2h

yI dh

yI 0h

0 −−→ QhJ−−→ Σh

div−−→ Vh −−→ 0

Simplest case of family of elements: Qh = Argyris space of C 1

quintics. Stress space Σh = p. cubic functions with p. lineardivergence (24 DOF). Displacements Vh = p. linear functions.

However, since I 2h involves point values of 2nd derivative and I 1

h

involves point values, these operators do not extend to boundedoperators in Hilbert spaces H2 and H(div, S).

In Bulletin, bounded cochain projections constructed. Givesstability for corresponding mixed finite element method by Hilbertcomplex theory (assumes Ω star-shaped).


Elasticity with weakly imposed symmetry

In 2-D, setting W = K× R2, relevant complex:

· · · −→ C∞(K)J−→ C∞(M)

(skwdiv)−−−→ C∞(W) → 0.

In 3-D, with W = K× R3, relevant complex:

· · · −→ C∞(M)J−→ C∞(M)

(skwdiv)−−−→ C∞(W) → 0.

Here J : C∞(M) 7→ C∞(M) denotes extension of previousoperator.

Jτ = curlS−1 curl τ, S algebraic


New approach to discretization of elasticity sequences:

I Use procedure on continuous level to derive elasticitysequence from multiple copies of de Rham sequence.

I Use this connection to establish stability for continuousformulation of elasticity

I To discretize, start from known good discretizations of deRham sequence.

I Determine conditions so that an analogue of stability proof forcontinuous problem will give stability of discrete problem.

To see structure more clearly, adopt notation of differential forms.

For simplicity, mostly consider 2D examples.


de Rham sequences with values in a vector space

Write 2-D de Rham sequence in form:

0 −→ Λ0 d0

−→ Λ1 d1

−→ Λ2 → 0.

Also consider sequences whose values lie in either V = Rn or K,space of skew-symmetric matrices. Both corresponding de Rhamsequences also exact, e.g.,

0 −→ Λ0(V)d0

−→ Λ1(V)d1

−→ Λ2(V) → 0.

Here Λk(V) consists of elements of form:

ω(x) =∑

I

fI (x)dxI

with coefficients fI ∈ C∞(Ω, V).


Weak symmetry elasticity sequence from de Rham (BGG)

Following ideas of Eastwood: Start from two de Rham sequences:

· · · −→ Λn−2(K)dn−2

−−−→ Λn−1(K)dn−1

−−−→ Λn(K) → 0,

· · · −→ Λn−2(V)dn−2

−−−→ Λn−1(V)dn−1

−−−→ Λn(V) → 0.

For both n = 2 and n = 3, spaces Λn−1(V) are spaces of stressesand can be identified with n × n matrices.

Let X = (x1, . . . , xn)T and define Kk : Λk(V) → Λk(K) by

Kkω = XωT − ωXT

Then define

Sk := dkKk − Kk+1dk : Λk(V) → Λk+1(K)


The operator Sk

Can show: Sk is an algebraic operator.

Two important operators: Sn−2 and Sn−1.Operator Sn−1 can be identified with skw, i.e., taking skew part ofmatrix (i.e., (W −W T )/2).

n = 2 : Sn−2

(ω1

ω2

)=

(0 ω2

−ω2 0

)dx1 +

(0 −ω1

ω1 0

)dx2

Easy to check S0 invertible. For n = 3, S1 more complicated, butstill algebraic and invertible.

Key property used to establish stability:

dn−1Sn−2 = −Sn−1dn−2

n = 2 : (div W )

(0 −11 0

)+ 2 skw curlW = 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:

· · · −→ Λn−2(K)dn−2

−−−→ Λn−1(K)dn−1

−−−→ Λn(K) → 0

Sn−2 Sn−1

· · · −→ Λn−2(V)dn−2

−−−→ Λn−1(V)dn−1

−−−→ Λn(V) → 0.

Since Sn−2 invertible, combine to one sequence: Let W = K× V.

· · · −→ Λn−2(K)dn−2S−1

n−2dn−2

−−−−−−−−−−→ Λn−1(V)(

Sn−1dn−1)−−−−→ Λn(W) → 0

After proper identifications, (n = 2), this is elasticity sequence

C∞(K)J−→ C∞(M)

(skwdiv)−−−→ C∞(W) → 0.


Approximation using Hilbert complex theory ?

Let W = K× V. Elasticity complex is:

· · · −→ Λn−2(K)dn−2S−1

n−2dn−2

−−−−−−−−−−→ Λn−1(V)(

Sn−1dn−1)−−−−→ Λn(W) → 0

Can find finite element subspaces of Λn−2(K), Λn−1(V), andΛn(W) with bounded projections.

However, operator S−1n−2 does not make sense on finite element

spaces, since spaces not of same dimension and Sn−1 does notmap to correct finite element space.

So cannot obtain a finite element subcomplex and hence cannotapply our theory directly from this formulation.


Go back one step

· · · −→ Λn−2(K)dn−2

−−−→ Λn−1(K)dn−1

−−−→ Λn(K) → 0

Sn−2 Sn−1

· · · −→ Λn−2(V)dn−2

−−−→ Λn−1(V)dn−1

−−−→ Λn(V) → 0.

· · · −→ Λn−2h (K)

dn−2

−−−→ Λn−1h (K)

dn−1

−−−→ Λnh(K) → 0

Sn−2,h Sn−1,h

· · · −→ Λn−2h (V)

dn−2

−−−→ Λn−1h (V)

dn−1

−−−→ Λnh(V) → 0.

Each discrete complex is subcomplex of corresponding continuouscomplex. Only operators Sn−1 and Sn−2 need to be approximated.


Stability of continuous problem

To establish stability for continuous problem, prove inf-sup:

Theorem: Given (ω, µ) ∈ L2Λn(Ω; K)× L2Λn(Ω; V), there existsσ ∈ HΛn−1(Ω; V) such that dn−1σ = µ, −Sn−1σ = ω. Moreover,we may choose σ so that

‖σ‖HΛ ≤ c(‖ω‖+ ‖µ‖),

for a fixed constant c , where

‖σ‖2HΛ = ‖σ‖2 + ‖dn−1σ‖2.


Outline of key part of proof

Look for σ of form: σ = dn−2% + η ∈ HΛn−1(Ω; V) .

By standard result,

Can find η ∈ H1Λn−1(Ω; V) satisfying dn−1η = µ,

and then τ ∈ H1Λn−1(Ω; K) satisfying dn−1τ = ω + Sn−1η.

Since Sn−2 isomorphism from H1Λn−2(Ω; V) onto H1Λn−1(Ω; K),have % ∈ H1Λn−2(Ω; V) with Sn−2% = τ .

Then dn−1σ = dn−1dn−2ρ + dn−1η = µ and

− Sn−1σ = −Sn−1dn−2%− Sn−1η = dn−1Sn−2%− Sn−1η

= dn−1τ − Sn−1η = ω.


Remarks about proof

(i) Although elasticity problem only involves 3 spacesHΛn−1(Ω; V), L2Λn(Ω; V), and L2Λn(Ω; K), proof brings in 2additional spaces: HΛn−2(Ω; V) and HΛn−1(Ω; K).

(ii) Although Sn−1 is only S operator arising in formulation, Sn−2

plays key role in proof. (dn−1Sn−2 = −Sn−1dn−2).

(iii) Do not fully use fact that Sn−2 is an isomorphism fromΛn−2(Ω; V) to Λn−1(Ω; K), only that it is a surjection.

(iv) Other slightly weaker conditions can be used in some places inthe proof: (needed for some choices of stable finite elementspaces).


Try to do analogous proof in discrete case

Let Λkh be discrete k–forms: ω(x) =

∑I fI (x)dxI , where fI are

piecewise polynomial functions with values in R.

Assume following discrete complex is exact

0 −→ Λ0h

d0

−→ Λ1h

d1

−→ Λ2h → 0

and that projection operators Πh onto Λkh are such that following

diagram commutes

0 −−→ Λ0 d−−→ Λ1 d−−→ Λ2 −−→ 0yΠh

yΠh

yΠh

0 −−→ Λ0h

d−−→ Λ1h

d−−→ Λ2h −−→ 0

Λkh : use standard FE spaces giving subcomplex of de Rham.


What is needed?

Define discrete version of operators Kk and Sk :

Kk,h = ΠhKk , Sk,h = ΠhSk .

Problem in using same stability proof on discrete level: can’texpect Sn−2,h to be an isomorphism from Λn−2

h (V) to Λn−1h (K).

However, looking at proof, only need:

(A) The operator Sn−2,h : Λn−2h (V) 7→ Λn−1

h (K) is onto.


Choosing finite element spaces to get stable discretization

Need to find combinations of discrete de Rham sequences for whichAssumption (A) (Sn−2,h : Λn−2

h (V) 7→ Λn−1h (K) onto) is satisfied.

Let n = 2. Given ω1 =

(0 −f1f1 0

)dx1 +

(0 −f2f2 0

)dx2 ∈ Λ1

h(K),

find ω0 = (g1, g2)T ∈ Λ0

h(V) such that

S0,hω0 = πhS0ω0 = πh

[(0 −g2

g2 0

)dx1 −

(0 −g1

g1 0

)dx2

]=

(0 −f1f1 0

)dx1 +

(0 −f2f2 0

)dx2.

Can be reduced to checking degrees of freedom of discrete spaces.


Simple stable choice

P1Λ0h(K)

d0

−→ P−1 Λ1h(K)

d1

−→ P0Λ2h(K) → 0

P2Λ0h(V)

d0

−→ P1Λ1h(V)

d1

−→ P0Λ2h(V) → 0.

A

AAAA• •

•

A

AAAA

HHY *

?A

AAAA

•d0

-d1

-

A

AAAA• •

•

•

• •

A

AAAA

HHYHHY

**

??A

AAAA

•d0

-d1

-

Top sequence: continuous P1, RT0, piecewise constants.Bottom sequence: continuous P2, BDM1, piecewise constants.


A simpler element

A

AAAA• •

•

A

AAAA

HHY *

?A

AAAA

•d0

-d1

-

A

AAAA• •

•

•

• •

A

AAAA

HHYHHY

**

??A

AAAA

•d0

-d1

-

Don’t use all the DOF in P2Λ0h(V) to map onto P−1 Λ1

h(K). Needonly P1Λ

0h(V) + 3 edge bubbles. Leads to reduced stress space and

same displacement space with same accuracy of approximation.

Analogous to stable Stokes element P1 + edge bubbles – P0.


Remarks on 3-D elements

For r ≥ 0, Use exact sequences:

Pr+1Λ0h(K)

d0

−→ P−r+1Λ1h(K)

d1

−→ P−r+1Λ2h(K)

d2

−→ PrΛ3h(K) → 0,

Pr+2Λ0h(V)

d0

−→ P−r+2Λ1h(V)

d1

−→ Pr+1Λ2h(V)

d2

−→ PrΛ3h(V) → 0.

First sequence is usual sequence for Nedelec elements of first kind.Note second sequence not usual sequence for the Pr spaces(second kind Nedelec elements).

Pr+3Λ0h(V)

d0

−→ Pr+2Λ1h(V)

d1

−→ Pr+1Λ2h(V)

d2

−→ PrΛ3h(V) → 0.

From FEEC, we know there are 2n−1 complexes in n dimensions.


Family of stable finite elements

Mixed elasticity with weakly imposed symmetry: Find(σ, u, p) ∈ Σh × Vh × Qh ⊂ H(div,Ω, M)× L2(Ω, V)× L2(Ω, K)such that

(Aσ, τ) + (div τ, u) + (τ, p) = 0, τ ∈ Σh,

(div σ, v) = (f , v), v ∈ Vh,

(σ, q) = 0, q ∈ Qh.

A family of elements: r ≥ 0, for n = 2 and n = 3:• Σh

∼= Pr+1Λn−1h (V)

• Vh∼= PrΛ

nh(V)

• Qh∼= PrΛ

nh(K)


Error estimates

Theorem: Suppose (σ, u, p) solves elasticity system and(σh, uh, ph) solves discrete elasticity system. Under reasonablehypotheses, and for appropriate projection operators Πh,

‖σ − σh‖+ ‖p − ph‖ ≤ C (‖σ −Πn−1h σ‖+ ‖p −Πn

hp‖),‖u − uh‖ ≤ C (‖σ −Πn−1

h σ‖+ ‖p −Πnhp‖+ ‖u −Πn

hu‖),‖dn−1(σ − σh)‖ = ‖dn−1σ −Πn

hdn−1σ‖.

For family just discussed, and 1 ≤ k ≤ r + 1,

‖σ − σh‖+ ‖p − ph‖+ ‖u − uh‖ ≤ Chk(‖σ‖k + ‖p‖k + ‖u‖k).


Connections to Earlier Work

Most early work on reduced symmetry elements (e.g., PEERS,Fortin, Morley, Stenberg) based on use of stable pair of Stokeselements.

Fits this framework with one basic modification, i.e., need to insertL2 projection (Πn

h ) into top discrete exact sequence.

Λ0h(K)

d0

−→ Λ1h(K)

Πnhd1

−−−→ Λ2h(K) → 0

Λ0h(V)

d0

−→ Λ1h(V)

d1

−→ Λ2h(V) → 0.

Spaces Λ1h(K) and Λ2

h(K) correspond to velocity and pressure spacefor Stokes.


PEERS (Arnold-Brezzi-Douglas)

For n = 2, letting B3 denote cubic bubble function, choose

Λ1h(V) = P−1 Λ1(Th; V) + dB3Λ

0(Th; V), Λ2h(V) = P0Λ

2(Th; V),

Λ2h(K) = P1Λ

2(Th; K) ∩ H1Λ2(K),

Choose two remaining spaces:

Λ0h(V) = (P1 + B3)Λ

0(Th; V), Λ1h(K) = S0Λ

0h(V).

A

AAAA• •

•

•(Stokes mini-element)

A

AAAA• •

•Πhd

1-

A

AAAA• •

•

•A

AAAA

HHY *

?

•-

d0

A

AAAA

•d1-

Known error estimate:

‖σ − σh‖0 + ‖p − ph‖0 + ‖u − uh‖0 ≤ Ch(‖σ‖1 + ‖p‖1 + ‖u‖1).


Farhoul-Fortin method – improved stress approximation

Choose

Λ1h(V) = P1Λ

1(Th; V), Λ2h(V) = P0Λ

2(Th; V),

Λ2h(K) = P1Λ

2(Th; K) ∩ H1Λ2(K),

and two remaining spaces as

Λ0h(V) = P2Λ

0(Th; V), Λ1h(K) = S0Λ

0h(V) ≡ P2Λ

1(Th; K)∩H1.

A

AAAA• •

•

•

• •(Taylor-Hood)

A

AAAA• •

•-

Πhd1

A

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•

•

• •

A

AAAA

HHYHHY

**

??

-d0

A

AAAA

•-

d1


Some recent papers on reduced symmetry elements

D. Boffi, F. Brezzi, M. Fortin, Reduced symmetry elements in linearelasticity, Commun. Pure Appl. Anal. 8 (2009), no. 1, 95-121.

B. Cockburn, J. Gopalakrishnan and J. Guzman, A new elasticityelement made for enforcing weak stress symmetry, Math. Comp.,79 (2010), 1331-1349.

J. Gopalakrishnan and J. Guzman, A second elasticity elementusing the matrix bubble, IMA J. Numer. Anal., 32 (2012), 352-272.

J. Guzman, A unified analysis of several mixed methods forelasticity with weak stress symmetry, J. Sci. Comp., 44 (2010),156-169.

R. Falk, Finite Elements for Linear Elasticity, in Mixed FiniteElements, Compatibility Conditions, and Applications, LectureNotes in Mathematics, Springer-Verlag, 1939 (2008), pp. 160-194.


Nonconforming elements with symmetric stresses

D. N. Arnold and R. Winther, Nonconforming mixed elements forelasticity, Dedicated to Jim Douglas, Jr. on the occasion of his75th birthday. Math. Models Methods Appl. Sci. 13 (2003), no.3, 295-307.

J. Gopalakrishnan and J. Guzman, Symmetric non-conformingmixed finite elements for linear elasticity, SIAM J. Numer. Anal.,49 (2011), no. 4, 1504-1520.


Methods using rectangular elements

D. N. Arnold and G. Awanou, Rectangular mixed finite elementsfor elasticity, Math. Models Methods Appl. Sci. 15 (2005), no. 9,1417-1429.

G. Awanou, A rotated nonconforming rectangular mixed elementfor elasticity, Calcolo 46 (2009), no. 1, 49-60.

G. Awanou, Two remarks on rectangular mixed finite elements forelasticity, to appear in J. of Scientific Computing.

G. Awanou, Rectangular Mixed Elements for Elasticity withWeakly Imposed Symmetry Condition, to appear in Advances inComputational Mathematics.


Summary I

• While not fully fitting FEEC theory for Hilbert complexes whenoriginally developed, use of full commuting diagrams for wholeelasticity sequence with strong symmetry enabled construction offirst stable pair of elements using polynomial shape functions(A-W).

0 −−→ C 2(R)J−−→ C 0(S)

div−−→ L2(R2) −−→ 0yI 2h

yI dh

yI 0h

0 −−→ QhJ−−→ Σh

div−−→ Vh −−→ 0


Summary II

• BGG construction of elasticity sequence (with weak symmetry)from de Rham sequence gave new understanding of structure andplayed key role in construction of new stable elements for elasticity(A-F-W).

· · · −→ Λn−1(K)dn−1

−−−→ Λn(K) → 0

Sn−2 Sn−1

· · · −→ Λn−2(V)dn−2

−−−→ Λn−1(V)dn−1

−−−→ Λn(V) → 0.

Key identity: dn−1Sn−2 = −Sn−1dn−2 all dimensions

• By slightly modifying approach, old methods fit new framework.


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Application of the Finite Element Exterior Calculus to the ... · Application of the Finite Element Exterior Calculus to the ... of linear elasticity I Stability of ... complex theory