Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
Finite Element Methods Based on a Three-Field
Formulation (Hu-Washizu) in Elasticity
Bishnu P. Lamichhane,
[email protected]
School of Mathematical and Physical Sciences, University of Newcastle, Australia
CARMA Retreat
18 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)
Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
Table of Contents
1
Introduction
The Boundary Value Problem of Elasticity
Finite Element Method
Hu-Washizu Formulation of Linear Elasticity
2
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Finite element discretization
3
Numerical Results
4
Numerical Results for Quadrilateral or Hexahedral Meshes
Bishnu P. Lamichhane,
[email protected]
Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu)
Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
The Boundary Value Problem of Elasticity
Finite Element Method
Hu-Washizu Formulation of Linear Elasticity
The Boundary Value Problem of Elasticity
Consider an elastic body in a bounded polyhedral domain Ω in
Rd , d ∈ {2, 3}. We want to compute the deformation and stress
on the elastic body under a body force f on Ω and a surface
force g N on a part ΓN of the boundary of Ω. The elastic body
is 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)
Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
The Boundary Value Problem of Elasticity
Finite Element Method
Hu-Washizu Formulation of Linear Elasticity
Standard Weak Formulation
∂u
Let H 1 (Ω) = {u ∈ L2 (Ω) : ∂x
∈ L2 (Ω), i = 1, · · · , d} be a Hilbert space with norm
i
qR
1
(u2 + k∇uk2 ) dx, and HD
(Ω) = {u ∈ H 1 (Ω) : u|ΓD = 0}. Let
kukH 1 (Ω) =
Ω
(u) := 12 (∇u + [∇u]t ). The standard weak formulation is to find u ∈ W so that
Z
Z
Z
(v) : C(u) dx =
f · v dx +
g N · v dx, v ∈ W .
Ω
Ω
ΓN
Defining a bilinear form B(·, ·) and a linear form `(·) as
Z
Z
Z
B(u, v) :=
(v) : C(u) dx, `(v) :=
f · v dx +
g N · v dx,
Ω
Ω
Ω
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 by
Lax-Milgram theorem.
Bishnu P. Lamichhane,
[email protected]
Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu)
Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
The Boundary Value Problem of Elasticity
Finite Element Method
Hu-Washizu Formulation of Linear Elasticity
Standard Weak Formulation
∂u
Let H 1 (Ω) = {u ∈ L2 (Ω) : ∂x
∈ L2 (Ω), i = 1, · · · , d} be a Hilbert space with norm
i
qR
1
(u2 + k∇uk2 ) dx, and HD
(Ω) = {u ∈ H 1 (Ω) : u|ΓD = 0}. Let
kukH 1 (Ω) =
Ω
(u) := 12 (∇u + [∇u]t ). The standard weak formulation is to find u ∈ W so that
Z
Z
Z
(v) : C(u) dx =
f · v dx +
g N · v dx, v ∈ W .
Ω
Ω
ΓN
Defining a bilinear form B(·, ·) and a linear form `(·) as
Z
Z
Z
B(u, v) :=
(v) : C(u) dx, `(v) :=
f · v dx +
g N · v dx,
Ω
Ω
Ω
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 by
Lax-Milgram theorem.
Bishnu P. Lamichhane,
[email protected]
Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu)
Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
The Boundary Value Problem of Elasticity
Finite Element Method
Hu-Washizu Formulation of Linear Elasticity
Existence, Uniqueness and Stability
Theorem
Let ` ∈ W ∗ , W ∗ is the dual space of W . Here the bilinear form B(·, ·) is
continuous, i.e.,
|B(u, v)| ≤ βkukH 1 (Ω) kvkH 1 (Ω) , β > 0
on W × W
and coercive on W , i.e.,
|B(u, u)| ≥ αkuk2H 1 (Ω) , α > 0
on W .
Thus the continuous problem has a unique solution, which depends continuously on
the right-hand side (well-posed).
Bishnu P. Lamichhane,
[email protected]
Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu)
Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
The Boundary Value Problem of Elasticity
Finite Element Method
Hu-Washizu Formulation of Linear Elasticity
Finite Element Method
The main idea: replace the continuous space W by a discrete one W h . Here the
subscript h refers to the fact that the discrete space W h is based on some finite
element mesh Th having mesh-size h. The discrete problem is: given ` ∈ W ∗ , find
uh ∈ W h such that
B(uh , v h ) = `(v h ), v h ∈ W h ,
which yields the algebraic system
A~u = f~,
A ∈ Rn×n ,
~u, f~ ∈ Rn , n := dim W h .
If W h ⊂ W and the continuous problem is well-posed, the discrete problem is also
well-posed due to Lax-Milgram theorem.
Bishnu P. Lamichhane,
[email protected]
Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu)
Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
The Boundary Value Problem of Elasticity
Finite Element Method
Hu-Washizu Formulation of Linear Elasticity
Finite Element Method
Ceá Lemma[1964]: Let W h ⊂ W . Let u and uh be the solutions of the continuous
and the discrete problem, respectively. Then, the following a priori estimate holds
β
ku − uh kH 1 (Ω) ≤ C inf ku − v h kH 1 (Ω) , C = ,
α
v h ∈W h
where β is the continuity constant, and α is the coercivity constant.
Bishnu P. Lamichhane,
[email protected]
Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu)
Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
The Boundary Value Problem of Elasticity
Finite Element Method
Hu-Washizu Formulation of Linear Elasticity
Volumetric Locking
Nearly incompressible material: λL is very large. Incompressible limit: λL → ∞. Note
that in Ceá Lemma, we have
β
ku − uh kH 1 (Ω) ≤ C inf ku − v h kH 1 (Ω) , C = ,
α
v h ∈W h
where |B(u, v)| ≤ βkukH 1 (Ω) kvkH 1 (Ω) , β > 0 and |B(u, u)| ≥ αkuk2H 1 (Ω) , α > 0.
Remember that
Z
Z
B(u, v) :=
(v) : C(u) dx =
(v) : (2µL (u) + λL div u1) dx,
Ω
and thus β = C1 λL .
finite element yields:
ku − uh kH 1 (Ω) ≤
Ω
Hence standard a priori result for linear, bilinear or trilinear
C1
C1
λL inf ku − v h kH 1 (Ω) =
λL hkukH 2 (Ω)
α
α
v h ∈W h
=⇒ poor accuracy, volumetric locking.
The key idea of solving this problem is to use a mixed formulation. We use here the
Hu-Washizu formulation, where stress, strain and displacement are unknown.
Bishnu P. Lamichhane,
[email protected]
Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu)
Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
The Boundary Value Problem of Elasticity
Finite Element Method
Hu-Washizu Formulation of Linear Elasticity
Volumetric Locking
Nearly incompressible material: λL is very large. Incompressible limit: λL → ∞. Note
that in Ceá Lemma, we have
β
ku − uh kH 1 (Ω) ≤ C inf ku − v h kH 1 (Ω) , C = ,
α
v h ∈W h
where |B(u, v)| ≤ βkukH 1 (Ω) kvkH 1 (Ω) , β > 0 and |B(u, u)| ≥ αkuk2H 1 (Ω) , α > 0.
Remember that
Z
Z
B(u, v) :=
(v) : C(u) dx =
(v) : (2µL (u) + λL div u1) dx,
Ω
and thus β = C1 λL .
finite element yields:
ku − uh kH 1 (Ω) ≤
Ω
Hence standard a priori result for linear, bilinear or trilinear
C1
C1
λL inf ku − v h kH 1 (Ω) =
λL hkukH 2 (Ω)
α
α
v h ∈W h
=⇒ poor accuracy, volumetric locking.
The key idea of solving this problem is to use a mixed formulation. We use here the
Hu-Washizu formulation, where stress, strain and displacement are unknown.
Bishnu P. Lamichhane,
[email protected]
Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu)
Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
The Boundary Value Problem of Elasticity
Finite Element Method
Hu-Washizu Formulation of Linear Elasticity
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. Q1 P0 ), 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 as
this can be treated as the mother of all these methods.
Bishnu P. Lamichhane,
[email protected]
Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu)
Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
The Boundary Value Problem of Elasticity
Finite Element Method
Hu-Washizu Formulation of Linear Elasticity
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:
strain-displacement:
equilibrium:
(Cd − σ, e)0
((u) − d, τ )0
(σ, (v))0
=
=
=
0,
0,
(f , v)0 ,
e∈D
τ ∈S
v∈V.
Note that d = (u) for the exact solution.
Sadddle point form: find (u, d, σ) ∈ V × S × S such that
ã((u, d), (v, e))
b((u, d), τ )
+ b((v, e), σ)
= `(v),
=
0,
(v, e) ∈ V × S,
τ ∈ S,
(1)
where
Z
ã((u, d), (v, e)) =
Z
d : Ce dx, and b((u, d), τ ) =
Ω
Bishnu P. Lamichhane,
[email protected]
((u) − d) : τ dx.
Ω
Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu)
Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
The Boundary Value Problem of Elasticity
Finite Element Method
Hu-Washizu Formulation of Linear Elasticity
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:
strain-displacement:
equilibrium:
(Cd − σ, e)0
((u) − d, τ )0
(σ, (v))0
=
=
=
0,
0,
(f , v)0 ,
e∈D
τ ∈S
v∈V.
Note that d = (u) for the exact solution.
Sadddle point form: find (u, d, σ) ∈ V × S × S such that
ã((u, d), (v, e)) + b((v, e), σ) = `(v),
b((u, d), τ )
=
0,
(v, e) ∈ V × S,
τ ∈ S,
(1)
where
Z
ã((u, d), (v, e)) =
Z
d : Ce dx, and b((u, d), τ ) =
Ω
Bishnu P. Lamichhane,
[email protected]
((u) − d) : τ dx.
Ω
Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu)
Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
Finite element discretization
Stabilized Hu-Washizu Formulation
We consider the standard Hu-Washizu formualtion: find (u, d, σ) ∈ V × S × S such
that
ã((u, d), (v, e))
b((u, d), τ )
+ b((v, e), σ)
= `(v),
=
0,
(v, e) ∈ V × S,
τ ∈ S.
(2)
The well-posedness of this saddle point problem is analyzed by using the standard
saddle point theory. The main difficulty in the discrete setting is to show that
(1) the bilinear form ã(·, ·) 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 matrix
corresponding to b(·, ·) has maximal rank);
Using some simple finite element spaces, it is not possible to satisfy these two
conditions simultaneously as the bilinear form ã(·, ·) is not elliptic on the whole space
V × S.
Bishnu P. Lamichhane,
[email protected]
Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu)
Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
Finite element discretization
Stabilized Hu-Washizu Formulation
We consider the standard Hu-Washizu formualtion: find (u, d, σ) ∈ V × S × S such
that
ã((u, d), (v, e))
b((u, d), τ )
+ b((v, e), σ)
= `(v),
=
0,
(v, e) ∈ V × S,
τ ∈ S.
(2)
The well-posedness of this saddle point problem is analyzed by using the standard
saddle point theory. The main difficulty in the discrete setting is to show that
(1) the bilinear form ã(·, ·) 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 matrix
corresponding to b(·, ·) has maximal rank);
Using some simple finite element spaces, it is not possible to satisfy these two
conditions simultaneously as the bilinear form ã(·, ·) is not elliptic on the whole space
V × S.
Bishnu P. Lamichhane,
[email protected]
Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu)
Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
Finite element discretization
Stabilized Hu-Washizu Formulation
We consider the standard Hu-Washizu formualtion: find (u, d, σ) ∈ V × S × S such
that
ã((u, d), (v, e))
b((u, d), τ )
+ b((v, e), σ)
= `(v),
=
0,
(v, e) ∈ V × S,
τ ∈ S.
(2)
The well-posedness of this saddle point problem is analyzed by using the standard
saddle point theory. The main difficulty in the discrete setting is to show that
(1) the bilinear form ã(·, ·) 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 matrix
corresponding to b(·, ·) has maximal rank);
Using some simple finite element spaces, it is not possible to satisfy these two
conditions simultaneously as the bilinear form ã(·, ·) is not elliptic on the whole space
V × S.
Bishnu P. Lamichhane,
[email protected]
Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu)
Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
Finite element discretization
Stabilized Hu-Washizu formulation
This gives us a motivation
to modify the bilinear form ã(·, ·) consistently by adding
R
the stabilization term Ω ((u) − d) : ((v) − e) dx so that we obtain the ellipticity on
the whole space V × S. Thus we define with some α > 0
Z
Z
a((u, d), (v, e)) =
d : Ce dx + α ((u) − d) : ((v) − e) dx.
Ω
Ω
Our modified saddle point problem is to find (u, d, σ) ∈ V × S × S such that
a((u, d), (v, e))
b((u, d), τ )
+ b((v, e), σ)
= `(v),
=
0,
(v, e) ∈ V × S,
τ ∈ S,
(3)
The first condition is met.
Bishnu P. Lamichhane,
[email protected]
Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu)
Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
Finite element discretization
Finite Element Discretization
We need three finite element spaces W h ⊂ W for the displacement and
Sh ⊂ L2 (Ω) for each component of the strain and Mh ⊂ L2 (Ω) for each
component of the stress.
The finite element space for the displacement is W h = [Vh ⊕ Bh ]d , where
Vh := {v ∈ H 1 (Ω) : v |T ∈ P1 (T ), T ∈ Th }
is the standard linear finite element on Th , and
Bh = {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
ϕi
xi
Bishnu P. Lamichhane,
[email protected]
Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu)
Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
Finite element discretization
Finite Element Discretization
1
Our goal here is to compute the constraint b((uh , dh ), τ h ) = 0 efficiently. We
want to solve the equation for dh :
Z
Z
Z
((uh ) − dh ) : τ h dx = 0 or
dh : τ h dx =
(uh ) : τ h dx.
Ω
2
Ω
Ω
PN
PN
Let dh = i=1 di ϕi , and τ h = i=1 τ i ψi , where {ϕi }N
i=1 is the set of standard
finite elements, and {ψi }N
i=1 forms a basis for M h . Then the above equation
leads to a linear system
Dd~ = ~e,
where the (i, j)th component of D is
Z
ϕi ψj dx.
Ω
3
N
If we choose two sets {ϕi }N
i=1 and {ψi }i=1 form a biorthogonal system, D will
be a diagonal matrix → highly efficient numerical method.
Bishnu P. Lamichhane,
[email protected]
Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu)
Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
Finite element discretization
Finite Element Discretization
1
Our goal here is to compute the constraint b((uh , dh ), τ h ) = 0 efficiently. We
want to solve the equation for dh :
Z
Z
Z
((uh ) − dh ) : τ h dx = 0 or
dh : τ h dx =
(uh ) : τ h dx.
Ω
2
Ω
Ω
PN
PN
Let dh = i=1 di ϕi , and τ h = i=1 τ i ψi , where {ϕi }N
i=1 is the set of standard
finite elements, and {ψi }N
i=1 forms a basis for M h . Then the above equation
leads to a linear system
Dd~ = ~e,
where the (i, j)th component of D is
Z
ϕi ψj dx.
Ω
3
N
If we choose two sets {ϕi }N
i=1 and {ψi }i=1 form a biorthogonal system, D will
be a diagonal matrix → highly efficient numerical method.
Bishnu P. Lamichhane,
[email protected]
Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu)
Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
Finite element discretization
Finite Element Discretization
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 , v h ) = `(v h ),
vh ∈ V B
h,
where
Z
Z
Πh (uh ) : CΠh (v h ) dx + α
Ah (uh , v h ) =
Ω
((uh ) − Πh (uh )) : ((v h ) − Πh (v h )) dx,
Ω
where dh = Πh (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 H 2 -regular. Then, we obtain an optimal a priori
estimate for the discretization error in the displacement
ku − uh k1,Ω ≤ Chkf k0,Ω .
(4)
where C < ∞ is independent of λ and h.
Bishnu P. Lamichhane,
[email protected]
Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu)
Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
Numerical Results for Cook’s Membrane
16
A
f
⌦
44
vertical displacement of point A
48
Hu-Washizu
standard
number of elements per side
Vertical tip displacement at T versus no. of elements, linear elasticity, E = 250 and
ν = 0.4999
Bishnu P. Lamichhane,
[email protected]
Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu)
Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
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
Bishnu P. Lamichhane,
[email protected]
Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu)
Introduction
Stabilized Hu-Washizu Formulation for Simplicial Meshes
Numerical Results
Numerical Results for Quadrilateral or Hexahedral Meshes
Numerical Results with Hexahedra
Nearly incompressible cylindrical (Mooney-Rivlin) shell under bending force
A nearly incompressible (neo-Hookean) torus under compression
Bishnu P. Lamichhane,
[email protected]
Finite Element Methods Based on a Three-Field Formulation (Hu-Washizu)