www.oeaw.ac.at
Tangential-Displacement and
Normal-Normal-Stress
Continuous Mixed Finite
Elements for Elasticity
J. Schöberl, A. Sinwel
RICAM-Report 2007-10
www.ricam.oeaw.ac.at
TANGENTIAL-DISPLACEMENT AND
NORMAL-NORMAL-STRESS CONTINUOUS
MIXED FINITE ELEMENTS FOR ELASTICITY
JOACHIM SCHÖBERL AND ASTRID SINWEL
Abstract. In this paper we introduce new finite elements to approximate the Hellinger Reissner formulation of elastictiy. The elements are the vector valued tangential continuous Nédélec elements
for the displacements, and symmetric, tensor valued, normal-normal
continuous elements for the stresses. These elements do neither
suffer from volume locking as the Poisson ratio approaches 12 , nor
suffer from shear locking when anisotropic elements are used for
thin structures. We present the analysis of the new elements, discuss their implementation, and give numerical results.
1. Introduction
Finite element simulation of mechanical problems has a long history,
and the theory is well developed. But, as material parameters or some
geometric dimensions become bad, the primal method may lead to very
poor results. This effect is known as locking. Here, many non-standard
methods such as mixed variational principles have been introduced [14].
We consider the Hellinger Reissner formulation, where the displacement vector as well as the stress tensor are approximated as independent unknowns. There are essentially the two possibilities to apply the
derivatives either to the displacements, or to the stresses. The first one
is easy to discretize, but leads usually to the standard method suffering
from locking. The second one needs stress tensor elements with continuous normal vectors, which are more difficult to construct [9, 1, 7, 10].
They may lead to better approximation properties. An alternative are
elements with weak symmetry [6, 26, 27, 12, 8]. In the present paper,
we develop a formulation, which is in between these two concepts. The
obtained finite elements have good approximation properties and are
easy to implement.
1991 Mathematics Subject Classification. 65N30.
Key words and phrases. Mixed Finite Elements, Elasticity, Locking.
Both authors acknowledges support from the Austrian Science Foundation FWF
within project grant Start Y-192, “hp-FEM: Fast Solvers and Adaptivity”.
1
2
JOACHIM SCHÖBERL AND ASTRID SINWEL
σnn
σnn
uτ
uτ
Figure 1. Triangular element
Figure 2. Tetrahedral element
We propose elements which have continuous tangential components
for the displacement vector u, and continuous normal-normal component of the stress tensor σ. The lowest order stable elements for a
triangle is drawn in Figure 1. The displacement element has one degree of freedom for the tangential component along edges. This is
exactly the lowest order Nédélec element. The normal component of
the normal stress vector is approximated as linear function along the
edges. There are exactly three dofs for each edge which are necessary
to fix the rigid body motions in 2D: The displacement is prescribed in
tangential direction, and the linear stress can equilibrate the normal
force and the angular momentum. The corresponding 3D element is
drawn in Figure 2. The tangential displacement is prescribed by one
value for each edge, and the normal component of the normal stress
vector is given as linear function on each face. Again, these six degrees
of freedom fix the rigid body motions: Displacements fix the tangential
displacement and the torsion, while the stresses compensate the normal force, and the two angular moments. The complete basis functions
including the internal bubbles will be given in Section 3
In Figure 3, the degrees of freedom of a quadrilateral element are
drawn. The goal is to discretize thin domains with anisotropic elements. Standard primal low order methods lead to bad results as the
aspect ratio of the element becomes large. In Section 6 we will analyze
the robustness of the corresponding Reissner Mindlin plate element.
Now, we motivate the anisotropic element by distinguishing stretching and bending degrees of freedom: The mean value of the horizontal
displacement prescribed by the bottom and top edge, and the mean
values of the stresses at the vertical edges prescribe a one dimensional
stretching deformation. The arising method is a mixed method with
continuous stresses and non-continuous displacements. It corresponds
to the presented elements, since this is a normal-normal continuous
MIXED FINITE ELEMENTS FOR ELASTICITY
3
σ nn
uτ
Figure 3. Quadrilateral element
1 × 1 stress tensor, and a displacement vector with non-continuous
normal components. A bending deformation is prescribed by the vertical deformation along the short edges, the differences of the horizontal
deformations, i.e., the rotations, and the differences of the horizontal
stresses, i.e., the bending moments. This leads to a stable element for
the Timoshenko beam model. A similar dimension reduction works out
for thin prismatic and hexahedral elements.
2. Different versions of the Hellinger Reissner
formulation
The equations of linear elasticity consist of the constitutive relation
(1)
Aσ = ε(u),
where u is the unknown displacement vector, ε(u) := 12 [∇u + (∇u)T ]
is the strain, σ is the unknown symmetric stress tensor. A is the
compliance tensor, which read as
1 D
1
Aσ =
σ +
tr(σ)I
2µ
λ+µ
for an isotropic material with Lamé coefficients µ and λ. Here, tr(τ )
denotes the trace of the tensor τ , and τ D is the deviator
1
τ D = τ − tr(τ )I.
d
The stress tensor shall be in equilibrium with the prescribed force
vector f , i.e.,
(2)
div σ = −f,
where the div-operator is applied to each row of the tensor. We assume
displacement boundary conditions
u = uD
at the part ΓD of the boundary, and traction boundary conditions
σn = g
4
JOACHIM SCHÖBERL AND ASTRID SINWEL
at the part ΓN of the boundary. We write σn for the normal stress
vector σn where n is the outer normal vector.
M
The primal mixed formulation is: find σ ∈ LSY
:= {τ ∈ Ld×d
:τ =
2
2
T
1 d
τ } and u ∈ [H ] such that u = uD on ΓD and
(3) R
R
M
Aσ
:
τ
−
ε(u) : τ = R0
∀ τ ∈ LSY
,
2
R
R
1
− ε(v) : σ
=
f · v + ΓN g · v
∀ v ∈ [H0,D ]d .
Since the stresses are L2 variables, they can be eliminated pointwise
from the first equation, and one ends up with the primal form: find
u ∈ [H 1 ]d such that u = uD and
Z
Z
Z
−1
1
A ε(u) : ε(v) = f · v +
g·v
∀ v ∈ [H0,D
(4)
]d .
ΓN
Alternatively, one may integrate by parts the off-diagonal terms, and
obtains the dual mixed formulation: find σ ∈ H(div)SY M := {τ ∈
M
LSY
: div τ ∈ Ld2 } and u ∈ [L2 ]d such that σn = g on ΓN and
2
R
R
R
∀ τ ∈ H(div)SY M ,
Aσ : τ +
u div τ = ΓD uD τn
(5)
R
R
v · div σ
= − f ·v
∀ v ∈ [L2 ]d .
The spaces imply the necessary continuity of conforming finite element sub-spaces. The primal mixed formulation requires continuous
elements for the displacements, but allows discontinuous elements for
the stresses. The dual mixed formulation needs elements with continuous normal-vector for the stresses, but the displacements may be
discontinuous. Our goal is to end up with elements, where the tangential component of the displacement and the normal component of the
normal stress vector are continuous.
The equilibrium equation (2) can be understood in different ways.
Posing the equation in L2 leads to the bilinear-form
Z
b(σ, v) = div σ · v,
and requires v ∈ L2 as well as div σ ∈ L2 . Testing the equilibrium
equation with displacements in H 1 , i.e., posing the equation in [H 1 ]∗ =
H −1 leads to the form
Z
b(σ, v) := hdiv σ, viH −1 ×H 1 = σ : ε(v).
This needs v ∈ H 1 , and div σ ∈ H −1 . The later one is clearly satisfied
by σ ∈ L2 .
MIXED FINITE ELEMENTS FOR ELASTICITY
5
Our new method tests the equilibrium equation with displacements
in H(curl):
hdiv σ, viH(curl)∗ ×H(curl) = hf, viH(curl)∗ ×H(curl)
∀ v ∈ H(curl)
The space is
H(curl) = {v ∈ [L2 ]d : curl v ∈ L2 },
where the differential operator curl is defined by
curl v = ∇ × v
for d = 3
and
∂vx ∂vy
−
for d = 2.
∂y
∂x
This allows right-hand sides of the form
Z
Z
hf, vi = f1 · v + f2 · curl v,
curl v =
where f1 is a usual volume force density, and f2 is a rotational force
density.
Lemma 1. The dual space of H(curl) is
H −1 (div) = {q ∈ [H −1 ]d : div q ∈ H −1 }.
Proof. Any function v ∈ H(curl) can be decomposed as
v = ∇ϕ + z
with ϕ ∈ H 1 and z ∈ [H 1 ]d and the corresponding bounds in the norms
(see [19], Theorem 3.4). The dual norm is
kqkH(curl)∗ =
'
hq, vi
v∈H(curl) kvkH(curl)
sup
hq, vi
v∈H(curl) inf v=∇ϕ+z kϕkH 1 + kzkH 1
sup
hq, ∇ϕ + zi
1
ϕ,z kϕkH 1 + kzkH
hq, ∇ϕi
hq, zi
+ sup
' sup
1
ϕ∈H 1 kϕkH 1
z∈[H 1 ]d kzkH
= sup
=
sup
ϕ∈H 1
hdiv q, ϕi
hq, zi
+ sup
1
kϕkH 1
z∈[H 1 ]d kzkH
= k div qkH −1 + kqkH −1
6
JOACHIM SCHÖBERL AND ASTRID SINWEL
The proper function space Σ for the stresses is
M
Σ = {σ ∈ LSY
: div σ ∈ H −1 (div)}
2
The constraint div σ ∈ H −1 is no new requirement, the only additional
constraint is div(div σ) ∈ H −1 . Thus, the space can be written also as
M
Σ = {σ ∈ LSY
: div div σ ∈ H −1 }
2
Throughout the following, let Th = {T } be a shape-regular triangulation of Ω, consisting of triangular or tetrahedral elements T in two
or three space dimensions, respectively. Let F denote the set of all
element interfaces (facets), i. e. edges in two or faces in three space dimensions, respectively. In 3d, we also define E as the set of all element
edges.
The following theorem provides that normal-normal continuous finite
elements are proper for the space Σ:
Theorem 2. Assume that σ is a piece-wise smooth symmetric tensor
field such that σnτ ∈ H 1/2 (∂T ) and nT σn is continuous across interfaces. Then there holds div σ ∈ H(curl)∗ .
Proof. Let v be a smooth test function. The divergence operator div :
H(div div) → H(curl)∗ is then defined as
Z
Z
o
X nZ
hdiv σ, vi = − σ : ∇v =
div σ · v −
σn · v .
Ω
T
T ∈T
∂T
We further investigate the boundary terms. Using the continuity of σnn
and uτ across element interfaces, we may reorder the surface integrals
facet by facet. We can write
XZ
XZ
σn · v =
σnn vn + σnτ vτ
T ∈T
∂T
T ∈T
=
∂T
XZ
F ∈F
σnn [vn ] +
F
≤
X
T
∂T
σnτ vτ
o
+
σnτ vτ
∂T
T ∈T
Using this relation, we obtain
Z
XnZ
hdiv σ, vi =
div σ · v −
T
XZ
XZ
F
F
[σnn ] vn
| {z }
=0
k div σkL2 (T ) kvkL2 (T ) + kσnτ kH 1/2 (∂T ) kvτ kH −1/2 (∂T )
T
C(σ) kvkH(curl)
MIXED FINITE ELEMENTS FOR ELASTICITY
7
By density, the continuous functional can be extended to the whole
space H(curl):
Z
o
XnZ
hdiv σ, vi =
div σ · v −
σnτ vτ
T
T
∂T
Remark 3. The condition σnτ ∈ H 1/2 is a continuity constraint at
vertices in 2d and at edges in 3d. If we slightly weaken the regularity,
the constraint disappears, and much simpler elements can be used. The
analysis in discrete norms in the next section will rectify this violation
of conformity.
Let us now include boundary conditions into the problem formulation. We assume that u is given on a non-trivial part ΓD ⊆ Γ of the
boundary, and surface tractions σn are prescribed on ΓN = Γ\ΓD . For
simplicity, let all data be homogenous,
u = 0 on ΓD ,
σn = 0 on ΓN .
Boundary conditions for uτ , σnn turn out to be essential, conditions
on un , σnτ are natural and can be included into the variational equations. More precisely, we search for (σ, u) ∈ Σ0 × V0 where
Σ0 = {σ ∈ H(div div) : σnn = 0 on ΓN }
V0 = {v ∈ H(curl) : vτ = 0 on ΓD }
If the solution of the elasticity problem is piecewise smooth, the
duality pairing hdiv σ, vi can be rewritten by means of domain and
boundary integrals. Now, we formulate the problem by means of these
integrals. For smooth solutions, this is an equivalent formulation of the
elastictiy problem.
Theorem 4. Let the solutions be piece-wise smooth. Then the elasticity
problem (1),(2) is equivalent to the following mixed problem: Find σ ∈
Σ0 and u ∈ V0 such that
R
R
P R
Aσ : τ + T
div
τ
·
u
−
τ
u
= 0
∀ τ ∈ Σ0
nτ
τ
T
∂T
R
R
P R
div σ · v − ∂T σnτ vτ
= − f ·v
∀ v ∈ V0
T
T
Proof. Again, we reorder the surface integrals on element interfaces
∂T in a face-wise mode. Then the second line turns out to be the
equilibrium equation, plus tangential continuity of the normal stress
vector:
XZ
XZ
(div σ + f )v +
[σnτ ]vτ = 0
∀v
T
T
F
F
8
JOACHIM SCHÖBERL AND ASTRID SINWEL
Since the space requires continuity of σnn , the normal stress vector is
continuous.
Element-wise integration by parts in the first line gives
XZ
XZ
(Aσ − ε(u)) : τ +
τnn [un ] = 0
∀τ
T
T
F
F
This is the constitutive relation, plus normal-continuity of the displacement. Tangential continuity of the displacement is implied by the space
H(curl).
3. Finite element error estimates
The displacement space H(curl) and the stress space H(div div) imply the necessary continuity of finite element spaces: to be conforming,
the tangential component of the displacement as well as the normalnormal tensor components need to be continuous.
For k ≥ 1, we define the finite element spaces
M
Σh := {τh ∈ LSY
: τh |T ∈ P k , τnn ∈ P max(1,k−1) continuous, τnn = 0 on ΓN }
2
Vh := {vh ∈ (L2 )d : vh |T ∈ P k , vτ continuous, vτ = 0 on ΓD }.
The space Vh is the second family of Nédélec. Up to our knowledge,
the space Σh is a new finite element space.
In the continuous setting, both v = H(curl) and Σ = H(div div) are
equipped with their natural norms. For the finite element analysis, we
choose different norms. We define the discrete, mesh dependent norm
for the displacements
X
X
2
kvk2Vh :=
kε(v)k2L2 (T ) +
h−1
F k[vn ]kL2 (F ) .
F ∈F
T
Note that, using the piecewise strain tensor instead of gradients, we
will not require Korn’s inequality [18, 22]. This will be useful when
treating anisotropic geometries or elements, for which the constant in
Korn’s inequality deteriorates. For the stress space, we use the L2 norm
kσkΣh := kσkL2 .
There the local mesh size hF is the average height perpendicular to F
of the two adjacent elements ∆F = {TF+ , FF− }. Then there holds
|∆F |
.
|F |
We can pose the finite dimensional problem
hF '
MIXED FINITE ELEMENTS FOR ELASTICITY
9
Find σ ∈ Σh , u ∈ Vh such that
a(σ, τ ) + b(τ, u) = 0
∀τ ∈ Σh
b(σ, v)
= −(f, v) ∀v ∈ Vh .
The bilinear forms a : Σ×Σ → R is defined via (6). In the homogenous,
isotropic case, the compliance tensor A is determined by the Lamé
constants λ and µ only. In this case a is given by (7). Here tr(τ ) denotes
the trace of the tensor τ , and τ D is the deviator τ D = τ − d1 tr(τ )I.
Z
a(σ, τ ) =
(6)
Aσ : τ dx
Ω
Z
(7)
=
Ω
1 D D
1
σ :τ +
tr(σ) tr(τ ) dx.
2µ
λ+µ
The second bilinear form b : Σ × V → R is generated by the divergence operator. For piecewise smooth functions τ, v it can be evaluated
by means of integrals over elements T and element boundaries ∂T . Using integration by parts, we can find two different formulations defining
b:
b(τ, v) = hdiv τ, vi
Z
XZ
div τ · v dx −
=
(8)
(9)
T
(10)
=
X
T
T
τnτ · vτ ds
∂T
Z
−
Z
τ : ε(v) dx +
T
τnn vn ds.
∂T
Combining these two forms into one, we also define the compound
bilinear form B : (Σ × V ) × (Σ × V ) → R by
B(σ, u; τ, v) = a(σ, τ ) + b(σ, v) + b(τ, u).
3.1. Discrete stability. In the continuous setting, we have continuity
and inf-sup stability of the bilinear form B on Σ0 × V0 . In this subsection, we are concerned with an analogous condition for the discrete
problem. It is sufficient to show that [14]
(1) Continuity of a, b
a(σ, τ ) ≤ α̃2 kσkΣh kτ kΣh
∀σ, τ ∈ Σh
b(σ, v) ≤ β̃2 kσkΣh kvkVh
∀σ ∈ Σh , v ∈ Vh
(2) Coercivity on the kernel
a(σ, σ) ≥ α̃1 kσk2Σh
∀σ ∈ Ker Bh
10
JOACHIM SCHÖBERL AND ASTRID SINWEL
(3) inf-sup stability
inf sup
v∈Vh σ∈Σh
b(σ, v)
≥ β̃1
kσkΣh kvkVh
We will see that both conditions are satisfied, but the stability constant
α̃1 depends on material parameters.
For isotropic material, the compliance tensor A is determined by
the Lamé constants λ and µ. As λ approaches infinity, the material
becomes incompressible. Then a is not uniformly coercive on the whole
space Σ0 . According to the second condition above, we only need
coercivity on the kernel of B. In the continuous case, Ker B is the set
of exactly divergence free functions. One can show that, for divergence
free functions σ, there holds
a(σ, σ) ≥ α1 kσk2Σ
where the constant α1 > 0 does not depend on λ [14].
The bilinear form a is not uniformly coercive on the discrete kernel
Ker Bh = {σ ∈ Σh : b(σ, v) = 0 ∀v ∈ Vh }.
To stabilize the problem, we add a further term to the bilinear form a
element by element. We define the stabilized form
X
h2 (div σ, div τ )0,T .
as (σ, τ ) = (Aσ, τ )0,Ω +
T
The solution σ satisfies − div σ = f . Therefore, to preserve consistency,
the term −h2 (f, div τ )0,T has to be added element-wise to the right
hand side. We obtain the stabilized problem
Find σ ∈ Σh , u ∈ Vh such that
P
s
a (σ, τ ) + hdiv τ, ui = − T h2 (f, div τ )0,T ∀τ ∈ Σh
hdiv σ, vi
= −(f, v)
∀v ∈ Vh .
We will see that the stabilized form
B s (σ, u; τ, v) = as (σ, τ ) + b(σ, v) + b(τ, u)
is inf-sup stable by proving an inf-sup condition for b and coercivity of
as on the discrete kernel, where the stability constants do not depend
on λ.
In the following, we use the H(curl) and the H(div div) conforming
transformations from the reference element T̂ to the physical element T ,
which are described more closely in Subsection 4.3. They are given by
v(x) = FT−T v̂(x̂)
1
σ(x) =
(FT σ̂(x̂)FTT )
JT2
MIXED FINITE ELEMENTS FOR ELASTICITY
11
Here FT is the Jacobian of the transformation ΦT : T̂ → T , and
JT = det FT denotes the Jacobi determinant. Note that the tangential components of vector fields and the normal-normal components of
tensor-valued functions are not changed by these transformations.
When treating anisotropic geometries, it will be useful to use different norms in the discrete spaces. For the displacement space, instead
of taking the piecewise defined gradient, we use the strain tensor ε(u)
on each element. As we will see, all stability estimates are independent
of the shape of the physical element.
Lemma 5. Assume T to be a shape-regular triangularization of the
domain Ω. Then the norm for the displacement space k·kVh is equivalent
to the broken H 1 semi norm
X
X
1
2
kε(v)k2L2 (T ) +
h−1
kvk2Vh ' kvkṼh :=
F kP [vn ]kL2 (F ) .
F ∈F
T
Moreover, for the discrete stress space we have
X
X
hF kσnn k2L2 (F ) .
kσk2Σh ' kσk2Σ̃h :=
kσk2L2 (T ) +
F ∈F
T
Proof. We prove the equivalence of the norms k · kVh and k · kṼh first.
This equivalence also holds true in the space of H(curl)-conforming
functions, which are piecewise smooth:
VT := {v ∈ V0 : v|T ∈ H 1 (T ) ∀T ∈ T }.
Clearly, there holds kvkṼh ≤ kvkVh We will now prove the opposite
bound, namely kvkVh kvkṼh . Let ΠT denote the element-wise projection onto the space of piecewise rigid body motions RM (T ) = {a + bx :
a ∈ Rd , b ∈ Rskw
d×d }. This projection is defined via
Z
Z
curl(v − ΠT v) dx = 0.
v − ΠT v dx =
T
T
As RM (T ) is a subspace of P 1 (T ), there holds for any facet F ∈ F
k(I − P 1 )[v]n kL2 (F ) = k(I − P 1 [v − ΠT (v)]n kL2 (F ) ≤ k[v − ΠT (v)]n kL2 (F ) .
Following [13], there holds
k[v − ΠT (v)]n kL2 (F ) X
T ∈∆F
hF kε(v)k2L2 (T ) .
12
JOACHIM SCHÖBERL AND ASTRID SINWEL
Combining these estimates, we obtain
X
2
h−1
F k[vn ]kL2 (F ) =
X
X
F
−1
1
2
1
2
h−1
F kP [vn ]kL2 (F ) + hF k(I − P )[v]n kL2 (F )
F
X
1
2
h−1
F kP [vn ]kL2 (F ) +
F
kε(v)k2L2 (T ) = kvkṼh
T
This relation proves the first equivalence.
For the stress space Σh , we transform σ element-wise to the reference element T̂ . There we use the equivalence of norms on a finitedimensional space, and transform back. This gives
kσk2L2 (Ω)
=
XZ
T
σ : σ dx
T
XZ
1
|FT σ̂FTT |2 |T | dx̂
4
T̂ JT
T
Z
XZ
X
T
2
'
(n̂ σ̂n̂) |T | dŝ +
=
T
=
∂ T̂
XZ
T
T
(nT σnT )2
∂T
T̂
1
|FT σ̂FTT |2 |T | dx̂
4
JT
|T |
ds + kσk2L2 (Ω) = kσk2Σ̃h
|F |
Lemma 6. The bilinear forms as : Σh × Σh → R, b : Σh × Vh → R are
continuous.
Proof. To show continuity of as in the discrete norms, we need the
inverse inequality
k div σkL2 (T ) h−1 kσkL2 (T )
∀σ ∈ Σh .
Then we can estimate
Z
X
1 D D
1
s
a (σ, τ ) =
σ :τ +
tr(σ) tr(τ ) dx +
h2 (div σ, div τ )L2 (T )
λ+µ
Ω 2µ
T
X
1
1
+
kσkL2 kτ kL2 +
kσkL2 (T ) kτ kL2 (T )
2µ λ + µ
T
= α̃2 kσkΣh kτ kΣh .
MIXED FINITE ELEMENTS FOR ELASTICITY
13
For the second bilinear form, we use the norm equivalences from above
and see
Z
X Z
b(σ, v) =
− σ : ε(v) dx +
σnn vn ds
T
≤
X
T
∂T
kσkL2 (T ) kε(v)kL2 (T ) +
T
X
1/2
hF kσnn kL2 (F ) h−1/2 k[vn ]kL2 (F )
F
≤ kσkΣ̃h kvkVh ≤ β̃2 kσkΣh kvkVh .
Lemma 7. The bilinear form b : Σh × Vh → R is inf-sup stable, there
exists a positive constant β̃1 > 0, β̃1 independent of h such that
(11)
inf sup
v∈Vh σ∈Σh
b(σ, v)
≥ β̃1 .
kσkΣh kvkVh
Proof. The finite element space Σh can be decomposed into two parts
Σh = Σfh + Σbh .
The first space, Σfh , is associated to the set of faces F. For each face
F ∈ F, there exists a family of functions (σjF )0≤j≤k such that
σjF F̄ = 0
∀F̄ ∈ F, F̄ 6= F,
f {σnn
: σ f ∈ Σfh } = P k (F )
F
The second space, Σbh , consists of element bubble functions, which
satisfy
b σnn
= 0 ∀F ∈ F.
F
To construct these spaces, we need tensor-valued basis functions Sif
corresponding to the facets of the element. These basis functions span
P 0 on a single element T , and the normal-normal component vanishes
on all facets but one, where it is constant. We will now define the basis
on the reference element T̂ . Therefore, let (λi ) denote the barycentric
coordinates of the vertices. In 2d, we propose
⊥
Ŝif = sym[∇λ⊥
i+1 ⊗ ∇λi+2 ], i = 1 . . . 3.
Calculating them explicitely yields
−2 1
0 −1
0 1
F
F
F
Ŝ1 =
, Ŝ2 =
, Ŝ3 =
.
1 0
−1 2
1 0
14
JOACHIM SCHÖBERL AND ASTRID SINWEL
In 3d, we need four face-based functions Ŝif and two additional element bubbles Ŝit to span the six-dimensional space of constant symmetric 3 × 3 tensor fields. We set them via
Ŝif = sym[(∇λi+1 × ∇λi+2 ) ⊗ (∇λi+2 × ∇λi+3 )]
Ŝ1t = sym[(∇λ1 × ∇λ2 ) ⊗ (∇λ3 × ∇λ4 )]
Ŝ2t = sym[(∇λ1 × ∇λ3 ) ⊗ (∇λ2 × ∇λ4 )].
Explicitely, they read as



−2 1 0
0
Ŝ1f =  1 0 0  , Ŝ2f =  1
0 0 0
−1



0 0 1
0
Ŝ4f =  0 0 0  , Ŝ1t =  0
1 0 0
−1



1 −1
0 0
0
−2 1  , Ŝ3f =  0 0 −1 
1
0
0 −1 2



0 −1
0 −1 0
0 1  , Ŝ2t =  −1 0 1  .
1 0
0
1 0
They can be transformed to the physical element T using the transformation given in Subsection 4.3. Using these transformations we can
define a piecewise constant tensor field S F for any facet F such that
F
Snn
|F̄ = δF F̄
∀F̄ .
Let v ∈ Vh be given. In order to prove the Lemma, we want to find
some σ ∈ Σh satisfying
Z
XZ
div σ · v dx −
σnτ · vτ ds ≥ β̃1 kσk0 kvkVh ,
b(σ, v) =
T
T
∂T
kσkΣh ≤ ckvkVh .
Therefore we decompose the stress function
σ = ασ b + βσ f
where α, β ∈ R are to be specified later. Let σ f be such that
f
σnn
|F = h−1
F [vn ]F
∀F ∈ F.
This is possible as [vn ] is a polynomial of degree k on each face, and
therefore we can simply set
X
F
σ f :=
h−1
F [vn ]F S .
F
Note that, when using k · kṼh , we can choose σ f of order one. The
approximation properties of Σh × Vh are still of optimal order when
using facet bubbles of order k − 1.
MIXED FINITE ELEMENTS FOR ELASTICITY
15
Next, we determine the “bubble part” σ b on each element T . In 2d,
the space of P 1 bubble functions on the reference element is spanned
by
Σ̂b := {ŜiF λi , i = 1, 2, 3}.
In 3d, we have to add some more functions
Σ̂b {ŜiF λi , i = 1 . . . 4} ∪ {ŜiT P 1 (T̂ ), i = 1, 2}.
Note that the space of constant symmetric tensor fields on the reference
element is spanned in 2d by {ŜiF }, in 3d by {ŜiF , ŜjT } respectively.
For simplicity, we will do the following calculations for the 2d case.
To treat three space dimensions, we would need to add the additional
bubble functions. Choose the “bubble part” σ b such that, on each
triangle resp. tetrahedron T , σ̂ b = JT2 (FT−1 σ b FT−T ) is given by
X
σ̂ b = JT2
(ε̂(v̂) : FT−T ŜiF FT−1 ) FT−T ŜiF FT−1 λi .
{z
}|
{z
}
|
i
∈P 1
∈P k−1
To achieve a bound for the L2 -norm of σ, we estimate the two parts
σ b , σ f separately by kvkVh . We will see in both cases, that the constant
arising is independent of the shape or the physical element. This will
prove useful when treating anisotropic geometries.
First, we treat σ f . Similar to the proof in Lemma 5, we obtain
X
f
hF kσnn
kσ f k2L2 (Ω) '
k2L2 (F )
F
=
X
2
h−1
F k[vn ]kL2 (F ) .
F
f
Here we used that σ consists of edge bubble functions only, therefore
the facet-contributions of σnn only form a norm on the discrete space.
For the bubble part σ b , there holds
XZ 1
b 2
kσ kL2 (Ω) = α
|FTT σ̂ b FT |2 JT dx̂
4
J
T̂ T
T
XZ X
'
|(ε̂(v̂) : FT−T ŜiF FT−1 )ŜiF λi |2 JT dx̂
T
=
T
T̂
XZ
T
=
T̂
i
XZ X
X
T
|(FT−1 ε̂(v̂)FT−T : ŜiF )ŜiF λi |2 JT dx̂
i
|FT−1 ε̂(v̂)FT−T |2 JT dx̂
T̂
kε(v)k2L2 (T ) .
16
JOACHIM SCHÖBERL AND ASTRID SINWEL
Combining the two inequalities, we obtain
kσk2L2 (Ω) ≤ αc1
X
kε(v)k2L2 (T ) + βc2
T
X
2
h−1
F k[vn ]kL2 (F )
F
We will need the following bound for the bubble part σ b on each
element
Z
σ b : ε(v) dx ≥ c3 kε(v)k2L2 (T ) .
T
Again, we prove this by transformation to the reference element T̂ . We
obtain
Z
b
Z
1
(FT σ̂(x̂)FTT ) : FT−1 ε̂(v̂)(x̂)FT−T JT dx̂
2
J
ZT̂ T
1
σ̂ : ε̂(v̂)JT dx̂
=
2
T̂ JT
Z X
=
(ε̂(v̂) : FT−T ŜiF FT−1 )FT−T ŜiF FT−1 λi : ε̂(v̂)JT dx̂
σ : ε(v) dx =
T
T̂
=
i
Z X
T̂
Z
ZT̂
=
(FT−1 ε̂(v̂)FT−T : ŜiF )2 λi JT dx̂
i
|FT−1 ε̂(v̂)FT−T |2 JT dx̂
|ε(v)|2 dx = kε(v)k2L2 (T )
T
where we used that the functions (Si )i form a basis for P02×2,SY M .
MIXED FINITE ELEMENTS FOR ELASTICITY
17
For σ defined as above we apply Young’s inequality and obtain
XZ
XZ
b(σ, v) =
ε(v) : σ dx −
σnn [vn ] ds
T
T
=
XZ
F
F
X
b
ασ : ε(v) + βσ f : ε(v) dx − β
T
T
≥
X
≥
X
F
Z
f
σnn
[vn ]ds
F
αc3 kε(v)k2L2 (T ) − βkσ f kL2 (T ) kε(v)kL2 (T ) + β
T
X
2
h−1
F k[vn ]kL2 (F )
F
αc3 kε(v)k2L2 (T ) −
−1/2
βc2 hF k[vn ]kL2 (∂T ) kε(v)kL2 (T )
+
T
β
X
2
h−1
F k[vn ]kL2 (F )
F
≥
βc2 ε2 X
kε(v)k2L2 (T )
2
T
T
X
βc2 X −1
2
− 2
h−1
hF k[vn ]k2L2 (F ) + β
F k[vn ]kL2 (F )
2ε F
F
X
αc3 kε(v)k2L2 (T ) −
Choosing β = 1, α = (1 + c22 )/(2c3 ), ε2 = 2c2 we get
1X
1X
b(σ, v) ≥
kε(v)k2L2 (T ) +
k[vn ]k2L2 (F )
2 T
2 F
≥
1
kvkVh kσkΣh .
2c(α, β)
To show coercivity of as , we need the following lemma. We assume
that ΓN is a non-trivial part of the boundary Γ = ∂Ω, where Neumann
boundary conditions are prescribed. The Dirichlet problem, i. e. ΓN =
∅, is treated separately.
Lemma 8. For any z ∈ L2 there exists some p ∈ [H 1 ]d , pτ |F ∈ P 1 for
all faces F ∈ F, satisfying
div p = z
p = 0
in Ω
on Γ\ΓN .
Proof. From Stokes theory it is well known that there exists a constant
γ > 0 such that
(div p, z)L2
inf sup
≥ γ.
z∈L2 p∈[H 1 ]d kpkH 1 kzkL2
18
JOACHIM SCHÖBERL AND ASTRID SINWEL
This implies the existence of p1 ∈ [H 1 ]d such that
div p1 = z,
p = 0 on Γ\ΓN .
We are now looking for a polynomial approximation to p1 . Therefore,
we consider the H 1 conforming finite element space Wh proposed by
[19]. The degrees of freedom of a function w ∈ Wh are the vertex values
and the mean values of the normal components of w on the element
faces. Restricted to a triangle T in 2D, w can be written as
w|T =
3
X
wi λi +
i=1
3
X
w̄j λj+1 λj+2 nj
j=1
whereas on a tetrahedron T in 3D w takes the form
4
4
X
X
wi λi +
w̄j λj+1 λj+2 λj+3 nj .
w|T =
i=1
j=1
Note that the tangential component of w ∈ Wh still is in P 1 .
From the Stokes problem it is known that there exists a Fortin operator Πh : [H 1 ]d → Wh such that
Z
(w − Πh w) · n dx = 0
∀F ∈ F
F
kΠh wk1 ≤ ckwk1
Using this operator, we define
p1,h = Πh p1
which satisfies, due to its construction
Z
Z
Z
div p1,h dx =
div p1 dx =
tr(σ) dx.
T
T
T
Next, we choose p2 by solving the following problem locally on each
triangle resp. tetrahedron T .
Find p2 ∈ H 1 , p2 |∂T = 0 such that
div p2 = z − div p1,h .
These local solutions exist, as the right hand side z − div p1,h has zero
mean value. Further p2 is in [H 1 ]d globally, as it vanishes on element
interfaces. Setting p = p1,h + p2 , we obtain
div p = div p1,h − div p2 = tr(σ),
i. e. the lemma is proven.
Using the result above, we can prove stability for the bilinear form
a in the finite element spaces.
s
MIXED FINITE ELEMENTS FOR ELASTICITY
19
Lemma 9. The bilinear form as : Σh ×Σh → R is coercive, there exists
a constant α̃1 independent of the Lamé constant λ such that
as (σ, σ) ≥ α̃1 kσk2Σh
(12)
∀σ ∈ Ker Bh
where Ker Bh is the discrete kernel
Ker Bh = {σ ∈ Σh : b(σ, v) = 0 ∀v ∈ Vh }.
(13)
Proof. For this proof we assume that the boundary part, where σn is
given, is non-trivial, |ΓN | =
6 0. For any τ ∈ Σ there holds
1
kτ k2L2 = kτ D k2L2 + k tr(τ )k2L2
d
and further
kτ k2L2 ≤ 2µa(τ, τ ) + k tr(τ )k2L2 .
Therefore, it is sufficient to show that for σ ∈ Ker Bh
a(σ, σ) ≥ ck tr(σ)k2L2 .
According to Lemma 8 there exists some p ∈ [H 1 ]d such that
div p = tr(σ),
p = 0
on Γ\ΓN ,
kpk1 ≤ ck tr(σ)kL2 .
For this p we may write
k tr(σ)k2L2
Z
=
tr(σ) div(p) dx
Ω
Z
σ : (div(p)I) dx
=
Ω
Z
= d
ZΩ
= d
ZΩ
≤ d
ZΩ
≤ d
Ω
σ : ∇p − (∇p)D dx
σ : ∇p dx − σ D : ∇p dx
σ : ∇p dx + dkσ D kL2 kpkH 1
σ : ∇p dx + dkσ D kL2 k tr(σ)kL2 .
20
JOACHIM SCHÖBERL AND ASTRID SINWEL
For the first term there holds
Z
XZ
XZ
σ : ∇p dx =
div σ · p dx −
(σn) · p ds
Ω
T
T
=
XZ
F
div σ · p dx −
T
T
F
XZ
F
[σnτ ] · pτ ds
F
= hdiv σ, pi
= hdiv σ, p − Ih pi
where we used that hσ, vh i = 0 for vh ∈ Vh in the last line. Let IV
denote the Nédélec interpolation operator, for which holds
Z
(14)
q(p − IV p) · τ dx = 0
∀q ∈ P 1 , E ∈ E
E
kp − IV pkL2 ≤ hkpkH 1 .
(15)
On each face F ∈ F pτ is in P 1 . As the interpolation operator IV is
polynomial preserving, the face integrals in the estimate above vanish,
XZ
[σnτ ](pτ − IV pτ ) ds = 0.
F
F
Therefore we may estimate
Z
XZ
div σ · (p − IV p) dx
σ : ∇p dx =
Ω
T
T
≤
X
k div σkL2 (T ) kp − IV pkL2 (T )
T
≤ c
X
hk div σkL2 (T ) kpk1
T
≤ c
X
hk div σkL2 (T ) k tr(σ)kL2 .
T
Collecting the results from above, we get
X
h2 k div σk2L2 (T ) + dkσ D k2L2 .
k tr(σ)k2L2 ≤ c
T
s
For the bilinear form a , this implies
Z
X Z
1 D2
1
s
2
a (σ, σ) =
|σ | +
tr(σ) dx +
h2 | div σ|2 dx
λ+µ
Ω 2µ
T
T
Z
X
1 D 2
≥
kσ kL2 +
h2 | div σ|2L2 (T ) dx
2µ
T
T
≥ c(µ)k tr(σ)k2L2 .
MIXED FINITE ELEMENTS FOR ELASTICITY
21
Remark 10. In the case of Dirichlet boundary conditions, it is not
possible to find p ∈ H 1 (Ω)d such that
div p = tr(σ),
p = 0
on Γ
kpkH 1 ≤ ck tr(σ)kL2 ,
as tr(σ) does not necessarily satisfy the complementarity condition
Z
tr(σ) dx = 0.
Ω
In this case, we choose a different norm for the discrete space Σh
1
kσk2Σh = min kσ − ρIk2L2 + kρIk2L2 .
ρ∈R
λ
We can find p ∈ H 1 (Ω)d0 such that
div p = tr(σ) − tr(σ),
p = 0
on Γ
kpk1 ≤ ck tr(σ)kL2 ,
In this case we can bound
k tr(σ) − tr(σ)kL2 ≤ c
X
hk div σkL2 (T ) + dkσ D kL2
T
Setting ρ = d1 tr(σ) we may estimate
1
kσk2Σh ≤ kσ − ρIk2L2 + kρIk2L2
λ
1
1
2
2
≤ c kσ − tr(σ)IkL2 + 2 k tr(σ) − tr(σ)kL2
d
d
!
X
≤ c kσ D k2L2 +
h2 k div σk2L2 (T )
T
s
≤ c(µ)a (σ, σ).
This proves stability for the Dirichlet case.
By the stability of the discrete saddle point problem and standard
interpolation error estimates we obtain the convergence result
Theorem 11. The finite element solution (uh , σh ) satisfies the error
estimate
ku − uh kVh + kσ − σh kΣh ≤ chm−1 kukH m
22
JOACHIM SCHÖBERL AND ASTRID SINWEL
for u ∈ H m with 1 ≤ m ≤ k + 1. The constant c is bounded uniformly
for λ → ∞.
Remark 12 (Hybridization). Discretization of the mixed system leads
to an indefinite system matrix. To avoid this, the continuity of σnn can
be broken and enforced by inter-element Lagrange multipliers [16, 5, 14].
Then all degrees of freedom for the finite element space Σh are bound to
only one element, and can be eliminated by static condensation. The
remaining matrix is symmetric and positive definite. The Lagrange
multipliers can be interpreted as normal displacements on the edges of
the triangles or faces of the tetrahedra, respectively.
4. Finite element basis functions
To construct finite elements for the spaces Vh ⊂ H(curl) and Σh ⊂
H(div div), we use the 2d-sequence
H1
∩
H 2 (T )
H(curl)
∩
1
2
H (T )
∇
−→
σ
div
div
T
−→
H(div div) −→ H −1 (div) −→ H −1
and in 3d
H1
∩
H 2 (T )
H(curl)
∩
2
1
[H (T )]
∇
−→
sym curl Q−1
−→
ε
curl
T
−→
H(curl curl) −→ H(div curl)
div
H(div div) −→
H −1 (div)
div
−→
H −1
with the operator Q, its inverse Q−1 and the stress operator σ given
by
1
QA = AT − tr(A)I,
Q−1 A = AT −
tr(A)I,
σ = Q−1 ε
d−1
The additional spaces used in the sequence are
H(curl curl) := {ε ∈ [L2 (Ω)]SY M : sym curl Q−1 curl ε ∈ [H −1 ]SY M }
H(div curl) := {δ ∈ [L2 (Ω)]3×3 : div sym curl Q−1 δ ∈ [H −1 ]3 }
The image of two successive operators is exactly the kernel of the
next, in 2d there holds
range(σ(∇·)) = Ker(div)
range(div σ(·)) = Ker(div)
In 3d, we use the relation
range(sym curl Q−1 curl) = Ker(div)
MIXED FINITE ELEMENTS FOR ELASTICITY
23
For the construction of shape functions on the reference elements, we
use Legendre polynomials (`i )0≤i≤k , which are orthogonal polynomials
in L2 (−1, 1) and span P k [−1, 1]. The integrated Legendre polynomials
(Li )2≤i≤k are defined by
Z x
Li (x) =
`i−1 (ξ) dξ.
−1
They are orthogonal with respect to the H 1 semi norm and vanish on
the interval bounds {−1, 1}. We will also need scaled Legendre polynomials (`Si )1≤i≤k and scaled integrated Legendre polynomials (LSi )2≤i≤k ,
which are defined for 0 < t ≤ 1 and −t ≤ x ≤ t. They are given by
x
`Si (x, t) = ti `i ( )
t
x
S
i
Li (x, t) = t Li ( ).
t
They satisfy LSi (−t, t) = LSi (t, t) = 0.
4.1. Triangular elements. Let the reference triangle T be defined by
T := {(x, y) | 0 ≤ x, y ≤ 1, x + y < 1}
with vertices V1 = (1, 0), V2 = (0, 1), V2 = (0, 0). Let λi , i = 1, 2, 3
denote the barycentric coordinates
λ1 = x, λ2 = y, λ3 = 1 − x − y.
For i, j ≥ 0, let
ui = LSi+2 (λ2 − λ1 , λ2 + λ1 )
vj = λ3 `j (2λ3 − 1).
Then ui is a polynomial of degree i + 2, which vanishes on edges
E1 , E2 . The polynomial vj is of degree j + 1 and vanishes on edge E3 .
The following H 1 conforming shape functions form a basis for P k (T )
[24, 28]
• Vertex-based functions: m = 1, 2, 3
φVm = λi
• Edge-based functions: m = 1, 2, 3, Em = [α, β], 0 ≤ i ≤ k − 2
m
φE
= LSi+2 (λα − λβ , λα + λβ )
i
• Cell-based functions: i, j ≥ 0, i + j ≤ k − 3
φTi,j = ui vj = LSi−2 (λ2 − λ1 , λ2 + λ1 )λ3 `j (2λ3 − 1)
24
JOACHIM SCHÖBERL AND ASTRID SINWEL
We define the local space of order k
T
k
m
Wk (T ) = span(φVm ) ⊕ span(φE
i ) ⊕ span(φi,j ) = P (T ).
Using this, H(curl) conforming shape functions can be defined as
below [24, 28]
• Edge-based functions: m = 1, 2, 3
– Low-order functions: m = 1, 2, 3, Em = [α, β]
0
ϕN
m = ∇λα λβ − λα ∇λβ
– High-order functions: 0 ≤ i ≤ k − 1
m
m
ϕE
= ∇φE
i
i
• Cell-based functions:
– Type 1, gradient fields: 0 ≤ i + j ≤ k − 2
ϕT,1
i,j = ∇(ui vj ) = ∇ui vj + ui ∇vj
– Type 2, 0 ≤ i + j ≤ k − 2
ϕT,2
i,j = ∇ui vj − ui ∇vj
– Type 3, 0 ≤ j ≤ k − 2
ϕT,3
= (∇λ1 λ2 − λ1 ∇λ2 )vj
j
The local H(curl) conforming FE-space is
Em
T
k
0
Vk (T ) = span(ϕN
m ) ⊕ span(ϕi ) ⊕ span(ϕi,j ) = P (T ).
There holds
∇Wk+1 (T ) ⊂ Vk (T ).
As a last step, we define H(div div) conforming shape functions on
the triangle T
• Edge-based functions, divergence free:
m = 1, 2, 3, Em = [α, β], 0 ≤ i ≤ k
m
ψiEm = σ(∇φE
i )
• Cell-based functions:
– Type 1, divergence free:
i, j ≥ 0, 0 ≤ i + j ≤ k − 1
T,1
ψi,j
= σ(∇(ui vj )) = ∇2 (ui vj ) − ∆(ui vj )I
= (∇2 ui vj + ∇ui ∇vjT + ∇vj ∇uTi + ui ∇2 vj )
−(∆ui vj + 2∇uTi ∇vj + ui ∆vj )I
MIXED FINITE ELEMENTS FOR ELASTICITY
25
– Type 2, i, j ≥ 0, 0 ≤ i + j ≤ k − 1
T,2
ψi,j
= (∇2 ui vj − ∇ui ∇vjT − ∇vj ∇uTi + ui ∇2 vj )
−(∆ui vj − 2∇uTi ∇vj + ui ∆vj )I
– Type 3, i ≥ 0, j ≥ 1, 1 ≤ i + j ≤ k − 1
T,3
ψi,j
= (∇2 ui vj − ui ∇2 vj ) − (∆ui vj − ui ∆vj )I
– Type 4, 0 ≤ j ≤ k − 1
ψjT,4 = σ((∇λ1 λ2 − λ1 ∇λ2 )vj )
From these basis functions, we derive the local H(div div) conforming
FE-space
T
Σk (T ) = span(ψiEm ) ⊕ span(ψi,j
) = P k (T ).
The way the basis was constructed implies
σ(Vk+1 (T )) ⊂ Σk (T ).
4.2. Tetrahedral elements. Let the reference tetrahedron T be defined by
T := {(x, y, z) | 0 ≤ x, y, z ≤ 1, x + y + z < 1}
with vertices V1 = (1, 0, 0), V2 = (0, 1, 0), V2 = (0, 0, 1), V3 = (0, 0, 0).
Let λi , i = 1 . . . 4 denote the barycentric coordinates
λ1 = x, λ2 = y, λ3 = z, λ4 = 1 − x − y − z.
For i, j ≥ 0, let
ui = LSi+2 (λ2 − λ1 , λ2 + λ1 )
vj = λ3 `Sj (λ3 − λ2 − λ1 , λ3 + λ2 + λ1 )
wl = λ4 `l (2λ4 − 1)
The polynomials ui vanish on faces F1 , F2 , whereas vj = 0 on F3 and
wk = 0 on F4 .
The following functions define a local basis for H 1 (T )
• Vertex based functions, m = 1 . . . 4
φVm = λm
• Edge based functions,m = 1 . . . 6, Em = [α, β], 0 ≤ i ≤ k − 2
m
φE
= LSi+2 (λα − λβ , λα + λβ )
i
• Face based functions, m = 1 . . . 4, Fm = [α, β, γ], 0 ≤ i + j ≤ k − 3
uα,β
= LSi+2 (λα −λβ , λα +λβ ), vjγ = λγ `S (λγ −λα −λβ , λγ +λα +λβ )
i
φFi,jm = LSi+2 (λα − λβ , λα + λβ )λγ `Sj (2λγ − λ, λ)
26
JOACHIM SCHÖBERL AND ASTRID SINWEL
• Cell based functions, 0 ≤ i + j + l ≤ k − 4
φTi,j,l = ui vj wl
The local space of order k is given by
Fm
T
k
m
Wk (T ) = span(φVm ) ⊕ span(φE
i ) ⊕ span(φi,j ) ⊕ span(φi,j,l ) = P (T ).
Again we use these functions to construct H(curl) conforming shape
functions
• Edge-based functions, m = 1 . . . 6, Em = [α, β]
– Low-order functions:
0
ϕN
m = ∇λα λβ − λα ∇λβ
– High-order functions: 0 ≤ i ≤ k − 1
m
m
ϕE
= ∇φE
i
i
• Face-based functions: m = 1 . . . 4, Fm = [α, β, γ],
uα,β
= LSi+2 (λα −λβ , λα +λβ ), vjγ = λγ `S (λγ −λα −λβ , λγ +λα +λβ )
i
– Type 1, 0 ≤ i + j ≤ k − 2
γ
ϕFi,jm ,1 = ∇(φFi,jm ) = ∇(uα,β
i vj )
– Type 2, 0 ≤ i + j ≤ k − 2
γ
α,β
γ
ϕFi,jm ,2 = ∇uα,β
i vj − ui ∇vj
– Type 3, 0 ≤ j ≤ k − 2
ϕFj m ,3 = (∇λα λβ − λα ∇λβ )vjγ
• Cell-based functions:
– Type 1, 0 ≤ i + j + l ≤ k − 3
T
ϕT,1
i,j,l = ∇(φi,j,l ) = ∇(ui vj wl )
– Type 2, 0 ≤ i + j + l ≤ k − 3
ϕT,2a
= ∇ui vj wl − ui ∇vj wl + ui vj ∇wl
i,j,l
ϕT,2b
= ∇ui vj wl + ui ∇vj wl − ui vj ∇wl
i,j,l
– Type 3, 0 ≤ j + l ≤ k − 3
ϕT,3
j,l = (∇λ1 λ2 − λ1 ∇λ2 )vj wl
We define the local H(curl) conforming FE-space
Em
Fm
T
k
0
Vk (T ) = span(ϕN
m ) ⊕ span(ϕi ) ⊕ span(ϕi,j ) ⊕ span(ϕi,j,l ) = P (T ).
As in 2D, there holds
∇Wk+1 (T ) ⊂ Vk (T ).
H(div div) conforming shape functions on the tetrahedron T can be
constructed as follows
MIXED FINITE ELEMENTS FOR ELASTICITY
27
• Face-based functions, divergence free
m = 1 . . . 4, Fm = [α, β, γ],
uα,β
= LSi+2 (λα − λβ , λα + λβ ), vjγ = λγ `S (λγ − λα − λβ , λγ +
i
λα + λβ )
1≤i+j ≤k+2
Fm ,1
γT T
ψi,j
= sym curl(curl(∇uα,β
∇v
)
))
i
j
• Cell-based functions:
– Type 1, divergence free, 0 ≤ i + j + l ≤ k
T,1a
ψi,j,l
= sym curl(curl(ui ∇vj ∇wlT )T ))
T,1b
= sym curl(curl(vj ∇ui ∇wlT )T )
ψi,j,l
T,1c
ψi,j,l
= sym curl(curl(wl ∇vj ∇uTi )T )
– Type 2, 0 ≤ i + j + l ≤ k
T,2a
ψi,j,l
T,2b
ψi,j,l
T,2c
ψi,j,l
= sym ∇vj × (∇2 ui ) × ∇wl − ui curl(curl(∇vj ∇wl )T ) , j, l ≥ 1
= sym ∇ui × (∇2 vj ) × ∇wl − vj curl(curl(∇ui ∇wl )T ) , l ≥ 1
= sym ∇vj × (∇2 wl ) × ∇ui − wl curl(curl(∇vj ∇ui )T ) , j ≥ 1
– Type 3, 0 ≤ i + j ≤ k
T,3a
0
ψi,j
= sym ∇vi × (∇φN
)
×
∇w
j
E12
T,3b
N0
ψi,j
= sym φE12 × (∇2 vi ) × ∇wj , i ≥ 1
The local H(curl) conforming FE-space is defined by
Fm
T
Σk (T ) = span(ψi,j
) ⊕ span(ψi,j,l
) = P k (T ).
Similar to the 2D case, we have
σ(Vk+1 (T )) ⊂ Σk (T ).
4.3. Global Finite Element Spaces. The global finite element spaces
Wh ⊂ H 1 , Vh ⊂ H(curl) and Σh ⊂ H(div div) can be derived from the
local spaces on the reference elements by element-wise transformation.
Therefore, let ΦT : T̂ → T be the mapping from the reference element T̂ to the physical element
T . ByFT we denote the Jacobian of
∂Φ
this transformation, FT (x̂) = ∂ x̂T,i
(x̂) . The Jacobi determinant is
j
ij
given by JT (x̂) = det(FT (x̂)). Normal and tangential vectors along the
element boundary transform as
τ = FT τ̂ , n = JT FT−T n̂.
28
JOACHIM SCHÖBERL AND ASTRID SINWEL
An H 1 conforming transformation from T̂ to T is given by
H 1 (T̂ ) → H 1 (T ),
ŵ 7→ w := ŵ ◦ Φ−1
T .
For H(curl), we use the covariant transformation is [21]
H(curl, T̂ ) → H(curl, T ),
v̂ 7→ v := FT−T v̂ ◦ Φ−1
T .
The covariant transformation is conforming for H(curl), as the tangential component of a vector field along the boundary of the element is
preserved.
The following transformation is conforming for H(div div)
H(div div, T̂ ) → H(div div, T ),
1
σ̂ 7→ σ := 2 (FT σ̂FTT ) ◦ Φ−1
T .
JT
For this transformation, the normal-normal component of the stress
field is preserved,
nT σn =
1 T −1
n FT FT σFTT FT−T n = n̂T σ̂n̂.
JT2
Using these transformations, and the local finite element spaces Wk (T̂ ),
Vh (T̂ ) and Σk (T̂ ) defined above, we get finite element spaces Wh , Vh and
Σh for H 1 , H(curl) and H(div div), respectively.
5. Non-conforming elements
Until now we have considered finite dimensional spaces Σh ⊂ Σ =
H(div div) and Vh ⊂ V = H(curl). A piecewise smooth function v is
in H(curl), if and only if the tangential component vτ is continuous
across element interfaces. In 2D, there are k + 1 degrees of freedom on
each edge, which are associated to the tangential displacement along
this edge. In this section, we will not assume that the finite element
space Vh is a subset of H(curl), but that the highest order edge basis
function may jump across element edges. Thereby we obtain a simplified element. It consists of more internal degrees of freedom, which can
be eliminated locally.
Let (σ, u) ∈ Σ × V denote the solution to the continuous problem,
and (σh , uh ) ∈ Σh × Vh the finite-element solution, where Vh is the
non-conforming space described above. We will see that the error
k(σ − σh , u − uh )kΣh ×Vh
MIXED FINITE ELEMENTS FOR ELASTICITY
29
uτ
uτ
highest order edge dofs
uτ
Figure 4. Conforming element
lower order edge dofs
Figure 5. Non-conforming element
can be bounded by the same order of convergence as in the conforming
case. The triangle inequality ensures
k(σ − σh , u − uh )kΣh ×Vh ≤ k(σ − IΣ σ, u − IV u)kΣh ×Vh +
k(IΣ σ − σh , IV u − uh )kΣh ×Vh
for suitable interpolation operators IV , IΣ . For the first term we know,
provided the solution is sufficiently smooth
k(σ − IΣ σ, u − IV u)kΣh ×Vh ≤ chk k(σ, u)kH k ×H k+1 .
For the second term, we have that (IΣ σ − σh , IV u − uh ) is a function
in the finite element space Σh × Vh . Therefore we can use the discrete
inf-sup stability of the bilinear form B
B(IΣ σ − σh , IV u − uh ; τh , vh )
k(τh , vh )kΣh ×Vh
τh ,vh
B(IΣ σ − σ, IV u − u; τh , vh )
≤ β̄ sup
+
k(τh , vh )kΣh ×Vh
τh ,vh
B(σ − σh , u − uh ; τh , vh )
β̄ sup
k(τh , vh )kΣh ×Vh
τh ,vh
k(IΣ σ − σh , IV u − uh )kΣh ×Vh ≤ β̄ sup
Again, the first term can be bounded by the approximation error and
is of order hk . For the second term, we use Galerkin orthogonality with
respect to σ, as Σh ⊂ Σ.
sup
τh ,vh
(16)
B(σ − σh , u − uh ; τh , vh )
B(σ − σh , u − uh ; 0, vh )
= sup
k(τh , vh )kΣh ×Vh
kvh kVh
vh
b(σ − σh , vh )
= sup
kvh kVh
vh
The edge basis functions associated to the tangential displacement are
continuous for order 0 to k − 1, only the highest order function jumps.
30
JOACHIM SCHÖBERL AND ASTRID SINWEL
Therefore, vh ∈ Vh can be divided into a conforming part v c , which
consists of the lower-order edge basis functions and the internal basis
functions, and into the non-conforming part. From the construction of
the basis for Vh we know that the restriction of this non-conforming
part to a triangle T is the gradient of an H 1 edge bubble function of
order k + 1. We may write
vh = v c + ∇h φ
v c ∈ H(curl)
3
X
i rT,i ∇φE
∇h φ T =
k+1 T
∀T ∈ T .
i=1
We use this and the fact that σh is a solution to the discrete problem
to obtain
b(σ − σh , vh ) = b(σ, vh ) − (f, vh )
= b(σ, v c ) − (f, v c ) +b(σ, ∇h φ) − (f, ∇h φ)
{z
}
|
=0
Z
XZ
∂φh
=
(div σ − f ) ·∇φ dx −
ds
σnτ ·
{z }
∂τ
T |
∂T
T
=0
XZ
∂φh
=
σnτ ·
ds.
∂τ
∂T
T
Therefore, we can estimate the right hand side of equation (16) by
sup
vh
b(σ − σh , vh )
b(σ − σh , ∇h φh )
≤ sup
kvh kVh
k∇h φh kVh
φh
P R
∂φh
T ∂T σnτ · ∂τ ds
= sup 1/2
2
P
φh
2 φ k2
−1 ∂φh k∇
+
h
h L2 (T )
T
∂n
L2 (∂T )
For an edge bubble function φh ∈ P k+1 (T ) there holds
2
2 2
∇ φh 2 + h−1 ∂φh ' ch−2 kφh kL2 (T ) .
∂n 2
L (T )
L (∂T )
Moreover, φh restricted to an edge is an integrated Legendre polynomial
of degree k + 1. This implies that ∂φh /∂τ is orthogonal to polynomials
MIXED FINITE ELEMENTS FOR ELASTICITY
31
of degree k − 1, and
X
b(σ − σh , vh )
sup
≤ c sup
inf
kvh kVh
s∈P k−1
vh
φh
T,F ⊂∂T
X
≤ c sup
φh
T,F ⊂∂T
≤ c
T,F ⊂∂T
X
F
h−2 kφh kL2
s∈P k−1
X
≤ c sup
φh
inf
h
(σnτ − s) ∂φ
ds
∂τ
−2
h kφh kL2
h
kσnτ − skL2 (F ) ∂φ
∂τ L2 (F )
R
inf
s∈P k−1
kσnτ − skL2 (F ) h−3/2 kφh kL2 (T )
h−2 kφh kL2
hk+1/2 kσnτ kH k (F )
T,F ⊂∂T
≤ chk kσkH k (Ω) .
This implies an optimal order of convergence of σ in the L2 norm for
the 2D non-conforming elements.
6. Anisotropic Estimates
Many technical constructions involve thin structures such as beams,
plates or shells. Traditionally lower dimensional models are derived and
treated by proper finite element methods. Here, the small thickness
parameter becomes explicite, and special care must be taken to avoid
the so called shear locking.
The concept of hierarchical finite element modeling treats the structure directly by anisotropic three dimensional finite elements, and the
accuracy of the model corresponds to the polynomial order in thickness
direction [11, 3, 17, 20], a posteriori estimates are in [25, 2].
Anisotropic error estimates for scalar equations are developed in [4].
Due to the combination of derivatives of the strain operator, these
techniques cannot be applied to conforming finite elements for elasticity problems. The goal of this section is to derive robust anisotropic
estimates for the elasticity problem discretized with the proposed new
elements.
Now, let ω ⊂ R2 be a domain identified with the cross-section of
a flat three dimensional domain, let I = (0, t), and set Ω = ω × I.
Furthermore, let T xy and T z be triangulations of ω and I, respectively.
Then T xy ⊗ T z gives the tensor-product mesh of the three-dimensional
domain. We may use just one element in thickness direction, or we
may use also a refined mesh to resolve several layers of a laminated
plate.
32
JOACHIM SCHÖBERL AND ASTRID SINWEL
We define Lagrangian and Nédélec finite element spaces on the cross
section and on the interval as
Lxy
:= {v ∈ H 1 (ω) : vT ∈ P k },
k
Nkxy := {v ∈ H(curl, ω) : vT ∈ [P k ]2 },
Pkxy := {v ∈ L2 (ω) : vT ∈ P k },
Lzk := {v ∈ H 1 (I) : vT ∈ P k },
Pkz := Nkz := {v ∈ L2 (I) : vT ∈ P k }.
xy
z
z
The spaces satisfy ∇Lxy
k+1 ⊂ Nk and ∇Lk+1 ⊂ Nk . Note that
z
Vh := Nkxy ⊗ Lzk+1 × Lxy
k+1 ⊗ Nk
gives the Nédélec space on the tensor product domain. Let Σxy
k be
the finite element space of symmetric 2 × 2 tensors with continuous
normal-normal components, and set


σxx σxy σxz
n
σ
σ
xx
xy
z
Σh := σ =  σxy σyy σyz  :
∈ Σxy
k ⊗ Nk ,
σxy σyy
σxz σyz σzz
o
σxz
xy
xy
z
z
∈ Pk ⊗ Nk , σzz ∈ Pk ⊗ Lk+2 .
σyz
By [15] there exist quasi-interpolation operators Ikxy : L2 (ω) → Lxy
k
such that
ku − Ikxy ukl,T hm−l |u|m,Te
0 ≤ l ≤ 1, 0 ≤ m ≤ k + 1, l ≤ m,
where Te is the union of the triangle T and its neighbours. In [23] quasixy
interpolation operators Qxy
k : L2 (ω) → Nk for the Nédélec spaces have
been introduced. They commute in the sense of
xy
∇Ik+1
= Qxy
k ∇,
and satisfy a corresponding interpolation error estimate
m−l
ku − Qxy
|u|m,Te
k ukl,T h
0 ≤ l ≤ 1, 0 ≤ m ≤ k, l ≤ m.
Note that the estimates for k·k1,T are not contained in [23], but are also
easily proven. We define according commuting interpolation operators
in thickness direction. On the tensor product domain we define the
interpolation operator
xy
z
z
Qk := Qxy
k ⊗ Ik+1 × Ik+1 ⊗ Qk
MIXED FINITE ELEMENTS FOR ELASTICITY
33
The in-plane deformation uxy := (ux , uy ) is interpolated by the Nédélec
operator in plane, and continuously in the thickness-direction, the
transversal displacement uz is interpolated vice versa.
Lemma 13. The product space interpolation operator Qk satisfies the
anisotropic error estimate
(17)
m
m
m
kε(u − Qk u)kL2 (T ) ≤ hm
xy k∇xy ε(u)kL2 (Te) + hz k∇z ε(u)kL2 (Te)
Proof. We bound the different components of the strain tensor as follows:
xy
z
kεxy (uxy − Ik+1
Qxy
k uxy )kT ≤ kεxy (uxy − Qk uxy )kT +
z
kεxy (Qxy
k (uxy − Ik+1 uxy ))kT
m
z
≤ hm
xy k∇xy εxy (uxy )kTe + kεxy (uxy − Ik+1 uxy )kTe
m
m
m
≤ hm
xy k∇xy εxy (uxy )kTe + hz k∇z εxy (uxy )kTe
The second term εxy,z involves the coupling of in-plane and transversal deformations and essentially relies on the commuting diagram:
z
z xy
2kεxy,z (u − Qk u)k0,T = k∇z (uxy − Qxy
k Ik+1 uxy ) + ∇xy (uz − Qk Ik+1 uz )k
z
= k(I − Qxy
k Qk ) ∇z uxy + ∇xy uz k
m
m
m
≤ hm
xy k∇xy εxy,z (u)kTe + hz k∇z εxy,z (u)kTe
xy
The last term, εz (uz − Qzk Ik+1
uz ) is bounded similar to the first one.
Anisotropic estimates for the stress tensor are also simply obtained.
A careful investigation of the estimates in Section 3 provides that all
stability estimates are independent of the aspect ratio of the elements.
For this, the weight of the jump-terms must be chosen as the element
size norm to the facet. Since the norm equivalence Vh ' Veh of Lemma 5
does not hold anymore, σnn must be chosen at the same order as [un ].
With the same arguments as in the isotropic case, we obtain the
Theorem 14 (anisotropic error estimates). Let uh and σh be the finite
element solutions on an anisotropic finite element mesh. Then there
holds the anisotropic error estimate
m
m
m
(18) ku − uh kVh + kσ − σh )kL2 ≤ chm
xy k∇xy ε(u)kΩ + chz k∇z ε(u)kΩ .
The constant c is independent of the aspect ratio of the elements.
34
JOACHIM SCHÖBERL AND ASTRID SINWEL
1
order 1
order 1, nc
order 2
order 2, nc
0.1
error stress
0.01
0.001
1e-04
1e-05
10
100
1000
10000
dofs
100000
1e+06
1e+07
Figure 6. Unit square, error kσ − σh kL2 (Ω) , ν = 0.3
7. Numerical Results
The first example we consider is the unit square. We suppose the
left side is fixed, all other boundaries are free. There is a constant
body force acting in vertical direction. We assume that the material
is homogenous, isotropic, linear elastic. We calculate stresses σh and
displacements uh using the method and elements described above. We
start from a coarse mesh, which is refined adaptively. As an error
estimator for the refinement we use a Zienkiewicz-Zhou type estimator.
For Figure 7, the material parameters are given by Young’s modulus
E = 1 and the Poisson ratio ν = 0.3. We plot the decrease of the
L2 -error of the stresses, kσ − σh kL2 (Ω) , as the number of unknowns
grows. We use both first and second order elements, and observe an
optimal order of convergence. The same behavior can be seen for the
non-conforming elements described in Section 5.
The elements do not suffer from volume locking, as the material
gets nearly incompressible. In Figure 7, we plot the L2 -error of the
stresses for a Poisson ratio ν = 0.4999. Again, we see the optimal
order convergence.
As a second example we consider a U-domain, where the left top
is fixed, and a surface load is acting on the right top. The domain
is discetized by 26 elements. We use curved elements of order 5 to
approximate the circular part of the U-domain. Also the finite element
spaces Σh , Vh are 5th order. In Figure 7 we plot the xx-component of
the stress tensor.
The last example is a cylindrical shell, where the radius of the cylinder is R = 0.5. The thickness of the shell is given as t = 0.005. In
Figure 7, we used P 1 elements for the stresses σ and P 2 elements for
MIXED FINITE ELEMENTS FOR ELASTICITY
35
1
order 1, nc
order 2, nc
0.1
error stress
0.01
0.001
1e-04
1e-05
10
100
1000
10000
dofs
100000
1e+06
1e+07
Figure 7. Unit square, nearly incompressible (ν = 0.4999)
Figure 8. U-domain, σxx
Figure 9. Cylindrical shell, σxy , σ ∈ P 1 , u ∈ P 2
u. We plot the shear stress σxy . In Figure 7, we increase the order by
one, i.e. P 2 for σh and P 3 for uh .
36
JOACHIM SCHÖBERL AND ASTRID SINWEL
Figure 10. Cylindrical shell, σxy , σ ∈ P 2 , u ∈ P 3
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Joachim Schöberl, Center for Computational Engineering Science,
RWTH Aachen, Germany
URL: http://www.cces.rwth-aachen.de
Astrid Sinwel, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz, Austria
URL: http://www.hpfem.jku.at