Finite Element Methods Based on a Three-Field Formulation

IntroductionStabilized Hu-Washizu Formulation for Simplicial Meshes

Numerical ResultsNumerical Results for Quadrilateral or Hexahedral Meshes

Finite Element Methods Based on a Three-FieldFormulation (Hu-Washizu) in Elasticity

Bishnu P. Lamichhane, [email protected]

School of Mathematical and Physical Sciences, University of Newcastle, Australia

CARMA Retreat18 August, 2012

Joint Work with B.D. Reddy and A.T. McBride”The art of doing mathematics consists in finding that special case which

contains all the germs of generality” by D. Hilbert (1862-1943).

Bishnu P. Lamichhane, [email protected] Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu) in Elasticity



Table of Contents

1 IntroductionThe Boundary Value Problem of ElasticityFinite Element MethodHu-Washizu Formulation of Linear Elasticity

2 Stabilized Hu-Washizu Formulation for Simplicial MeshesFinite element discretization

3 Numerical Results

4 Numerical Results for Quadrilateral or Hexahedral Meshes




The Boundary Value Problem of ElasticityFinite Element MethodHu-Washizu Formulation of Linear Elasticity

The Boundary Value Problem of ElasticityConsider an elastic body in a bounded polyhedral domain Ω inRd, d ∈ 2, 3. We want to compute the deformation and stresson the elastic body under a body force f on Ω and a surfaceforce gN on a part ΓN of the boundary of Ω. The elastic bodyis supposed to be fixed on a part ΓD of its boundary, where∂Ω = ΓD ∪ ΓN . Useful in manufacture engineering.

Measured (black) and computed (red) impact on the wall is plotted.Bishnu P. Lamichhane, [email protected] Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu) in Elasticity




Standard Weak Formulation

Let H1(Ω) = u ∈ L2(Ω) : ∂u∂xi∈ L2(Ω), i = 1, · · · , d be a Hilbert space with norm

‖u‖H1(Ω) =√∫

Ω(u2 + ‖∇u‖2) dx, and H1

D(Ω) = u ∈ H1(Ω) : u|ΓD= 0. Let

ε(u) := 12 (∇u+ [∇u]t). The standard weak formulation is to find u ∈W so that∫

Ω

ε(v) : Cε(u) dx =

∫Ω

f · v dx+

∫ΓN

gN · v dx, v ∈W .

Defining a bilinear form B(·, ·) and a linear form `(·) as

B(u,v) :=

∫Ω

ε(v) : Cε(u) dx, `(v) :=

∫Ω

f · v dx+

∫Ω

gN · v dx,

our problem is to find u ∈W such that

B(u,v) = `(v), v ∈W .

The existence, uniqueness and stability of the solution of this problem is given byLax-Milgram theorem.





Standard Weak Formulation

Let H1(Ω) = u ∈ L2(Ω) : ∂u∂xi∈ L2(Ω), i = 1, · · · , d be a Hilbert space with norm

‖u‖H1(Ω) =√∫

Ω(u2 + ‖∇u‖2) dx, and H1

D(Ω) = u ∈ H1(Ω) : u|ΓD= 0. Let

ε(u) := 12 (∇u+ [∇u]t). The standard weak formulation is to find u ∈W so that∫

Ω

ε(v) : Cε(u) dx =

∫Ω

f · v dx+

∫ΓN

gN · v dx, v ∈W .

Defining a bilinear form B(·, ·) and a linear form `(·) as

B(u,v) :=

∫Ω

ε(v) : Cε(u) dx, `(v) :=

∫Ω

f · v dx+

∫Ω

gN · v dx,

our problem is to find u ∈W such that

B(u,v) = `(v), v ∈W .

The existence, uniqueness and stability of the solution of this problem is given byLax-Milgram theorem.





Existence, Uniqueness and Stability

Theorem

Let ` ∈W ∗, W ∗ is the dual space of W . Here the bilinear form B(·, ·) iscontinuous, i.e.,

|B(u,v)| ≤ β‖u‖H1(Ω)‖v‖H1(Ω), β > 0 on W ×W

and coercive on W , i.e.,

|B(u,u)| ≥ α‖u‖2H1(Ω), α > 0 on W .

Thus the continuous problem has a unique solution, which depends continuously onthe right-hand side (well-posed).





Finite Element Method

The main idea: replace the continuous space W by a discrete one W h. Here thesubscript h refers to the fact that the discrete space W h is based on some finiteelement mesh Th having mesh-size h. The discrete problem is: given ` ∈W ∗, finduh ∈W h such that

B(uh,vh) = `(vh), vh ∈W h,

which yields the algebraic system

A~u = ~f, A ∈ Rn×n, ~u, ~f ∈ Rn, n := dimW h.

If W h ⊂W and the continuous problem is well-posed, the discrete problem is alsowell-posed due to Lax-Milgram theorem.





Finite Element Method

Cea Lemma[1964]: Let W h ⊂W . Let u and uh be the solutions of the continuousand the discrete problem, respectively. Then, the following a priori estimate holds

‖u− uh‖H1(Ω) ≤ C infvh∈W h

‖u− vh‖H1(Ω), C =β

α,

where β is the continuity constant, and α is the coercivity constant.





Volumetric Locking

Nearly incompressible material: λL is very large. Incompressible limit: λL →∞. Notethat in Cea Lemma, we have


‖u− vh‖H1(Ω), C =β

α,

where |B(u,v)| ≤ β‖u‖H1(Ω)‖v‖H1(Ω), β > 0 and |B(u,u)| ≥ α‖u‖2H1(Ω), α > 0.Remember that

B(u,v) :=

∫Ω

ε(v) : Cε(u) dx =

∫Ω

ε(v) : (2µLε(u) + λL divu1) dx,

and thus β = C1λL. Hence standard a priori result for linear, bilinear or trilinearfinite element yields:

‖u− uh‖H1(Ω) ≤C1

αλL infvh∈W h

‖u− vh‖H1(Ω) =C1

αλL h‖u‖H2(Ω)

=⇒ poor accuracy, volumetric locking.The key idea of solving this problem is to use a mixed formulation. We use here theHu-Washizu formulation, where stress, strain and displacement are unknown.





Volumetric Locking

Nearly incompressible material: λL is very large. Incompressible limit: λL →∞. Notethat in Cea Lemma, we have


‖u− vh‖H1(Ω), C =β

α,

where |B(u,v)| ≤ β‖u‖H1(Ω)‖v‖H1(Ω), β > 0 and |B(u,u)| ≥ α‖u‖2H1(Ω), α > 0.Remember that

B(u,v) :=

∫Ω

ε(v) : Cε(u) dx =

∫Ω

ε(v) : (2µLε(u) + λL divu1) dx,

and thus β = C1λL. Hence standard a priori result for linear, bilinear or trilinearfinite element yields:

‖u− uh‖H1(Ω) ≤C1

αλL infvh∈W h

‖u− vh‖H1(Ω) =C1

αλL h‖u‖H2(Ω)

=⇒ poor accuracy, volumetric locking.The key idea of solving this problem is to use a mixed formulation. We use here theHu-Washizu formulation, where stress, strain and displacement are unknown.





Examples of Remedies

(1) Incompatible modes(Wilson-Taylor et al.)

(2) Assumed stress methods, Enhanced assumed strain (EAS) methods(Pian-Sumihara 84, Simo-Rifai 90, Braess et al. 04)

(3) Mixed Enhanced Strains (MES), Strain Gap Method (SGM)(Kasper-Taylor 00, Romano et al. 01)

(4) Displacement-pressure formulation (e.g. Q1P0), B-bar approach(Brezzi-Fortin, Hughes)

(4) Nodal Strain or Nodal Pressure(Bonet, Dohrmann, Puso, etc.)

A unified framework for analysis: Hu-Washizu formulation and its modifications asthis can be treated as the mother of all these methods.





Hu-Washizu Formulation of Linear Elasticity

Focus on formulations in which V is as before; new variables are in L2(Ω)Find (u, σ, d) ∈ V × S ×D := S such that (displacement, stress and strain)Standard Hu-Washizu formulation:[Hu 55,Washizu 82]

elastic law: (Cd− σ, e)0 = 0, e ∈Dstrain-displacement: (ε(u)− d, τ )0 = 0, τ ∈ Sequilibrium: (σ, ε(v))0 = (f ,v)0, v ∈ V .

Note that d = ε(u) for the exact solution.Sadddle point form: find (u,d,σ) ∈ V × S × S such that

a((u,d), (v, e)) + b((v, e),σ) = `(v), (v, e) ∈ V × S,b((u,d), τ ) = 0, τ ∈ S, (1)

where

a((u,d), (v, e)) =

∫Ω

d : Ce dx, and b((u,d), τ ) =

∫Ω

(ε(u)− d) : τ dx.





Hu-Washizu Formulation of Linear Elasticity

Focus on formulations in which V is as before; new variables are in L2(Ω)Find (u, σ, d) ∈ V × S ×D := S such that (displacement, stress and strain)Standard Hu-Washizu formulation:[Hu 55,Washizu 82]

elastic law: (Cd− σ, e)0 = 0, e ∈Dstrain-displacement: (ε(u)− d, τ )0 = 0, τ ∈ Sequilibrium: (σ, ε(v))0 = (f ,v)0, v ∈ V .

Note that d = ε(u) for the exact solution.Sadddle point form: find (u,d,σ) ∈ V × S × S such that


where

a((u,d), (v, e)) =

∫Ω

d : Ce dx, and b((u,d), τ ) =

∫Ω

(ε(u)− d) : τ dx.




Finite element discretization

Stabilized Hu-Washizu Formulation

We consider the standard Hu-Washizu formualtion: find (u,d,σ) ∈ V × S × S suchthat

a((u,d), (v, e)) + b((v, e),σ) = `(v), (v, e) ∈ V × S,b((u,d), τ ) = 0, τ ∈ S. (2)

The well-posedness of this saddle point problem is analyzed by using the standardsaddle point theory. The main difficulty in the discrete setting is to show that

(1) the bilinear form a(·, ·) is coercive on a suitable kernel space (the solution (u,d)is unique).

(2) the bilinear form b(·, ·) satisfies a uniform inf-sup condition ( the matrixcorresponding to b(·, ·) has maximal rank);

Using some simple finite element spaces, it is not possible to satisfy these twoconditions simultaneously as the bilinear form a(·, ·) is not elliptic on the whole spaceV × S.



























Stabilized Hu-Washizu formulation

This gives us a motivation to modify the bilinear form a(·, ·) consistently by addingthe stabilization term

∫Ω

(ε(u)− d) : (ε(v)− e) dx so that we obtain the ellipticity onthe whole space V × S. Thus we define with some α > 0

a((u,d), (v, e)) =

∫Ω

d : Ce dx+ α

∫Ω

(ε(u)− d) : (ε(v)− e) dx.

Our modified saddle point problem is to find (u,d,σ) ∈ V × S × S such that


The first condition is met.





Finite Element Discretization

We need three finite element spaces W h ⊂W for the displacement andSh ⊂ L2(Ω) for each component of the strain and Mh ⊂ L2(Ω) for eachcomponent of the stress.

The finite element space for the displacement is W h = [Vh ⊕Bh]d, where

Vh := v ∈ H1(Ω) : v|T ∈ P1(T ), T ∈ Th

is the standard linear finite element on Th, andBh = bT ∈ Pd+1(T ) : bT = 0 on ∂T. Here Th is the standard simplicial mesh.

A finite element mesh and a basis function in 2D A hanging node

xi

A finite element mesh and a basis function in 2D A hanging node

ϕi






1 Our goal here is to compute the constraint b((uh,dh), τh) = 0 efficiently. Wewant to solve the equation for dh:∫

Ω

(ε(uh)− dh) : τh dx = 0 or

∫Ω

dh : τh dx =

∫Ω

ε(uh) : τh dx.

2 Let dh =∑N

i=1 diϕi, and τh =∑N

i=1 τ iψi, where ϕiNi=1 is the set of standardfinite elements, and ψiNi=1 forms a basis for Mh. Then the above equationleads to a linear system

D~d = ~e,

where the (i, j)th component of D is∫Ω

ϕi ψj dx.

3 If we choose two sets ϕiNi=1 and ψiNi=1 form a biorthogonal system, D willbe a diagonal matrix → highly efficient numerical method.






1 Our goal here is to compute the constraint b((uh,dh), τh) = 0 efficiently. Wewant to solve the equation for dh:∫

Ω

(ε(uh)− dh) : τh dx = 0 or

∫Ω

dh : τh dx =

∫Ω

ε(uh) : τh dx.

2 Let dh =∑N

i=1 diϕi, and τh =∑N

i=1 τ iψi, where ϕiNi=1 is the set of standardfinite elements, and ψiNi=1 forms a basis for Mh. Then the above equationleads to a linear system

D~d = ~e,

where the (i, j)th component of D is∫Ω

ϕi ψj dx.

3 If we choose two sets ϕiNi=1 and ψiNi=1 form a biorthogonal system, D willbe a diagonal matrix → highly efficient numerical method.






Result: the finite element approximation converges uniformly to the exact solution.We also obtain the reduced problem of finding uh ∈ V B

h such that

Ah(uh,vh) = `(vh), vh ∈ V Bh ,

where

Ah(uh,vh) =

∫Ω

Πhε(uh) : CΠhε(vh) dx+α

∫Ω

(ε(uh)−Πhε(uh)) : (ε(vh)−Πhε(vh)) dx,

where dh = Πhε(uh). Now we formulate the main result:

Theorem

Assume that u and uh be the solutions of continuous and discrete problems,respectively, and he solution is H2-regular. Then, we obtain an optimal a prioriestimate for the discretization error in the displacement

‖u− uh‖1,Ω ≤ Ch‖f‖0,Ω. (4)

where C <∞ is independent of λ and h.




Numerical Results for Cook’s Membrane

48

16

44

number of elements per side

vert

ical

dis

plac

emen

t of

poi

nt A

A

Hu-Washizustandard

f

Vertical tip displacement at T versus no. of elements, linear elasticity, E = 250 andν = 0.4999




Numerical Results with Quadrilaterals: Cook’s Membrane

Vertical tip displacement at T versus no. of elements, linear (left), geononlinear(middle) and neo-Hookean (right), E = 250 and ν = 0.4999




Numerical Results with Hexahedra

Nearly incompressible cylindrical (Mooney-Rivlin) shell under bending force

A nearly incompressible (neo-Hookean) torus under compression


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Finite Element Methods Based on a Three-Field Formulation