Stress-based finite element methods in linear
and nonlinear solid mechanics
Benjamin Müller‡ and Gerhard Starke
‡
‡
Fakultät für Mathematik, Universität Duisburg-Essen, 45127 Essen, Germany
Abstract A comparison of stress-based finite element methods is
given for the prototype problem of linear elasticity and then extended to finite-strain hyperelasticity. Of particular interest is the
accuracy of traction forces in reasonable Sobolev norms with an
emphasis on uniform approximation behavior in the incompressible
limit. The mixed formulation of Hellinger-Reissner type leading to
a saddle-point problem as well as a first-order system least squares
approach are investigated and the strong connections between these
two methods are studied. In addition, we also discuss stress reconstruction techniques based on displacement approximations by
nonconforming finite elements.
1
Introduction
The accurate resolution of stresses associated with numerical simulations in
solid mechanics is of paramount importance in many applications. Large
stress components may cause plastic behavior or even damage and need
therefore to be approximated well. In particular, if surface traction forces
are of interest, finite element approximations in spaces which allow the safe
evaluation of boundary traces need to be used. For standard displacementbased or (in the incompressible case) displacement-pressure approaches, the
associated stresses are only contained in L2 which means that the (normal
component of the) boundary traces are not defined. Variational principles
which involve stresses in H(div)-like saddle point formulations of HellingerReissner-type or first-order system least squares approaches overcome this
problem directly. Another option is to reconstruct stresses in H(div) from
sufficiently accurate L2 approximations in analogy to the flux reconstruction procedures described, e.g., in Luce and Wohlmuth (2004); Nicaise et al.
(2008); Braess et al. (2009); Cai and Zhang (2012); Ern and Vohralı́k (2015).
For an equilibration approach to stress reconstruction in two-dimensional
linear elasticity see also Parés et al. (2006). Particularly attractive in the
1
context of incompressible elasticity are the quadratic nonconforming elements introduced by Fortin and Soulie (1983) and, in three space dimensions, Fortin (1985). Flux and stress reconstruction procedures working in
an element-wise way were studied for these elements by Kim (2012).
The history of mixed finite element methods of saddle-point type for the
approximation of stresses in H(div) in linear elasticity models goes back for
at least 30 years with early contributions by Arnold et al. (1984a), Arnold
et al. (1984b) and Stenberg (1988) among others, see (Boffi et al., 2013,
Chp. 9) for more details. Later, this approach also received much attention
in the engineering community, see e.g. Klaas et al. (1995). For the class
of first-order system least squares methods the state-of-the-art is presented
in Bochev and Gunzburger (2009) with a focus of fluid rather than solid
mechanics. The H(div)-based stress formulation which will be our starting
point in this contribution was studied in Cai and Starke (2004) for the linear
elasticity case and extended to hyperelastic material models in Müller et al.
(2014).
The investigation of hyperelastic models in Section 5 will be presented
in detail for the specific example of a neo-Hookean material law. For background on the analytical and numerical treatment of hyperelasticity, we refer
to Ciarlet (1988) and LeTallec (1994). Concerning a priori finite element
error estimates associated with such models, see Carstensen and Dolzmann
(2004). Our focus in Section 5 of this contribution will again be on approaches which remain robust in the incompressible limit. Similar to the
linear elasticity case this may be achieved either by adding an auxiliary pressure variable (cf. Auricchio et al. (2013)) or by inverting the stress-strain
relation, cf. (Wriggers, 2008, Sect. 10.3).
The elasticity problems under our consideration are based on an open,
bounded and connected domain Ω ⊂ IRd (d = 2, 3) with Lipschitz-continuous
boundary which constitutes the reference configuration of the undeformed
state. The boundary is divided into two disjoint subsets ΓD and ΓN , for
simplicity, both assumed to be non-empty. On ΓD , homogeneous displacement boundary conditions u = 0 are imposed, while surface traction forces
σ · n = g are prescribed on ΓN . The linear elasticity model may then be
written as the first-order system
div σ + f = 0
σ − Cε(u) = 0
(1)
in Ω subject to the above boundary conditions with ε(u) = (∇u+(∇u)T )/2
and
Cε = 2µε + λ(tr ε)I .
(2)
2
The system (1) may be derived from minimizing the energy associated with
the deformed system given by
Z
Z
Z
ψ(ε(v)) dx −
f · v dx −
g · v ds ,
(3)
Ω
Ω
ΓN
where the stored energy function is given by
ψ(ε) = µ|ε|2 +
λ
(tr ε)2 .
2
(4)
The necessary conditions for a stationary point of (3) are then equivalent
to (1).
While we can always scale the units such that µ is on the order of 1,
an important issue is the behavior of the formulations in the incompressible limit λ → ∞. It is already apparent from (2) that a naive numerical
approach to the above minimization problem will cause problems for incompressible or nearly incompressible materials. One possible remedy consists
in replacing λ(trε) by a new variable p which has the physical interpretation
of a pressure. Another option is to use the inverse C −1 instead of C in the
variational formulation. A straightforward calculation shows that
λ
1
1
1
λ→∞ 1
σ−
tr(σ)I
→
σ − tr(σ)I =
dev σ,
C −1 σ =
2µ
2µ + dλ
2µ
d
2µ
i.e. the operator C −1 remains well-defined in the incompressible limit, where
it constitutes the orthogonal projection onto the trace-free matrices dev.
Since C −1 itself is not invertible any more in the incompressible limit, we
also write A instead of C −1 in order to avoid missunderstandings. The
first-order system (1) turns into
div σ + f = 0
Aσ − ε(u) = 0 .
(5)
Of course, any variational approach based on (5) needs to use the stress
σ as an independent variable in the formulation. Such approaches will be
presented in the next sections.
We will make use of norms and inner products associated with different
spaces throughout this paper. Since L2 (Ω) (and its vector and matrix variants L2 (Ω)d and L2 (Ω)d×d , respectively) occurs most often, we abbreviate
the associated norm simply by k · k and the corresponding inner product by
( · , · ). Since we assume ΓD 6= ∅ (more precisely, a subset of ∂Ω of positive
measure), Korn’s inequality is valid in the form
k∇vk ≤ CK kε(v)k for all v ∈ HΓ1D (Ω)d .
3
(6)
Our general regularity assumption is that Ω ⊂ IRd , ΓN ⊂ ∂Ω and ΓD ⊂ ∂Ω
are such that, for any f ∈ L2 (Ω)d the solution of (5) satisfies (σ, u) ∈
H α (Ω)d×d × H 1+α (Ω)d such that
kσkH α (Ω) + kukH 1+α (Ω) ≤ CR kf k
(7)
holds for some constant CR > 0 and some α > 0.
2 Stress-based mixed formulation based on the
Hellinger-Reissner principle
This section is focussed on the approximation of stresses in the Sobolev
space H(div, Ω)d . The subspaces
HΓN (div, Ω)d = {τ ∈ H(div, Ω)d : τ · n = 0 on ΓN } ,
HΓ0N (div, Ω)d = {τ ∈ HΓN (div, Ω)d : div τ = 0}
will also be used. From the stress-strain relation ε(u) = C −1 σ, integration
by parts leads to
(C −1 σ, τ ) + (u, div τ ) + (γ, as τ ) = 0 ,
(8)
for all τ ∈ HΓN (div, Ω)d , where as τ = (τ − τ T )/2 denotes the asymmetric
part and γ is a new variable introduced for as ∇u. Together with the two
equations
(div σ + f , v) = 0 for all v ∈ L2 (Ω)d ,
(as σ, θ) = 0 for all θ ∈ L2 (Ω)d×d,as ,
(9)
where L2 (Ω)d×d,as denotes the subspace of L2 (Ω)d×d with vanishing symmetric part, the mixed variational formulation of Hellinger-Reissner consists
in finding (σ, u, γ) ∈ (σ N +HΓN (div, Ω)d )×L2 (Ω)d ×L2 (Ω)d×d,as such that
(8) and (9) hold. An alternative way of deriving this mixed variational formulation consists in viewing it as the KKT conditions for the minimization
of the energy (C −1 σ, σ)/2 subject to the constraints (9). In this context, u
and γ are Lagrange parameters for the momentum balance and symmetry
conditions, respectively, in (9). For the well-posedness of the system (8),
(9), the following result is of crucial importance.
Theorem 2.1. Assume that ΓN ⊆ ∂Ω consists of a finite number of connected components each of which has positive (d − 1)-dimensional measure.
Then,
kτ k . kdev τ k
(10)
holds for all τ ∈ HΓ0N (div, Ω)d .
4
Theorem 2.1 follows from the more general result of Theorem 3.1 in
Section 3. A direct and simple proof for the two-dimensional case is given
in the following.
Proof. (for d = 2). Without loss of generality assume that ΓN ( ∂Ω and
that ΓN is connected with |ΓN | > 0 (just replace ΓN by one of its connected
components, cut something off if ΓN = ∂Ω).
Using (Girault and Raviart, 1986, Thm. I.3.1), we can write τ = curl φ
with φ ∈ H 1 (Ω)2 . The boundary conditions (curl φ) · n = 0 imply φ to
be constant on ΓN , which we may choose to be zero, i.e., φ ∈ HΓ1N (Ω)2 .
Therefore, φ satisfies Korn’s inequality (6) and we obtain
kτ k = kcurl φk = k∇φk . kε(φ)k
1
(∂2 φ1 + ∂1 φ2 )
∂1 φ1
2
k
=k 1
(∂2 φ1 + ∂1 φ2 )
∂2 φ2
12
(∂ φ + ∂1 φ2 )
−∂1 φ1
k
=k 2 2 1
∂2 φ2
− 12 (∂2 φ1 + ∂1 φ2 )
∂ φ −∂1 φ1
= kdev 2 1
k = kdev curl φk = kdev τ k .
∂2 φ2 −∂1 φ2
Theorem 2.1 implies
(C −1 τ , τ ) ≥
1
kdev τ k2 & kτ k2 for all τ ∈ HΓ0N (div, Ω)d .
2µ
(11)
Since HΓ0N (div, Ω)d contains the null space of the constraints (9), the required coercivity condition is satisfied. As a second ingredient to the wellposedness, the inf-sup condition has to be established for (9), see (Boffi
et al., 2013, Prop. 9.3.2).
For the discretization of (8), (9), finite element spaces Πh ⊂ HΓN (div, Ω)d ,
Zh ⊂ L2 (Ω)d and Θh ⊂ L2 (Ω)d×d,as are inserted into (8), (9) leading to
HR
HR
a mixed finite element approximation (σ HR
h , zh , γ h ). To this end, various finite element combinations which satisfy the discrete inf-sup condition
have been proposed, starting with the famous PEERS element Arnold et al.
(1984a). For a systematic treatment of this topic see (Boffi et al., 2013,
Chp. 9). It is interesting to note that for k ≥ 1, the triple of finite element
spaces
(Πh , Zh , Θh ) = RTk (Th )d × DPk (Th )d × Pk (Th )d×d,as
5
is inf-sup stable (see Boffi et al. (2009) and (Boffi et al., 2013, Expl. 9.4.1)).
An important property of this approach is that the momentum balance is
best possible, i.e.,
kdiv σ HR
+ f k = kf − π h f k = inf kf − zh k .
h
zh ∈Zh
From the ellipticity and the inf-sup conditions, optimal order accuracy also
follows for the stress approximation with respect to the L2 (Ω)-norm, i.e,
kσ − σ HR
h k.
3
inf kσ − τ h k . hα kσkH α (Ω) .
τ h ∈Πh
(12)
Stress-displacement first-order system least squares
In this section, we consider the first-order system least squares approach
based on
div σ + f
R(σ, u) :=
,
(13)
Aσ − ε(u)
i.e., the minimization of
F(τ , v) := kR(τ , v)k2 = kdiv τ + f k2 + kAτ − ε(v)k2
(14)
among all τ ∈ σ N + HΓN (div, Ω)3 and v ∈ HΓ1D (Ω)3 . The minimizer (σ, u)
of (14) satisfies
(div σ, div τ ) + (Aσ − ε(u), Aτ ) = −(f , div τ ) ,
−(Aσ − ε(u), ε(v)) = 0
(15)
for all (τ , v) ∈ HΓN (div, Ω)3 × HΓ1D (Ω)3 which constitutes a linear variational problem. The well-posedness of (15) follows from the coercivity and
continuity of the bilinear form
B((σ, u); (τ , v)) := (div σ, div τ ) + (Aσ − ε(u), Aτ − ε(v))
(16)
with respect to HΓN (div, Ω)3 × HΓ1D (Ω)3 uniformly in the incompressible
limit. This property was shown under our assumptions on Ω, ΓN and ΓD in
Cai and Starke (2004). A consequence of its validity in the incompressible
limit is the following result.
Theorem 3.1. Assume that ΓN ⊆ ∂Ω consists of a finite number of connected components each of which has positive (d − 1)-dimensional measure.
Then,
kτ k . kdev τ k + kdiv τ k
(17)
holds for all τ ∈ HΓN (Ω)d .
6
The result of Theorem 3.1 was proved in Arnold et al. (1984b) for the case
ΓN = ∅ under the additional constraint (tr τ , 1) = 0 (see also (Boffi et al.,
2013, Prop. 9.1.1)) and in the general two-dimensional case in Carstensen
and Dolzmann (1998).
The discrete first-order system least squares approximation is obtained
by minimizing (14) among all τ h = σ N + Πh and vh ∈ Vh , where Πh ⊂
HΓN (div, Ω)3 and Vh ⊂ HΓ1D (Ω)3 are suitable finite element spaces. The
N
LS
approximate solution σ LS
h ∈ σ + Πh , uh ∈ Vh is determined by
LS
LS
(div σ LS
h , div τ h ) + (Aσ h − ε(uh ), Aτ h ) = −(f , div τ h ) ,
LS
−(Aσ LS
h − ε(uh ), ε(vh )) = 0
(18)
for all (τ h , vh ) ∈ Πh × Vh . Due to the coercivity and continuity of the
underlying bilinear form we obtain a quasi-optimal approximation, i.e.,
kσ − σ LS
h kdiv,Ω .
ku − uLS
h k1,Ω
inf kσ − τ h kdiv,Ω ,
τ h ∈Πh
. inf ku − vh k1,Ω .
(19)
vh ∈Vh
In particular, using, for some l ≥ 1, Raviart-Thomas spaces of degree l − 1
for Πh combined with standard conforming finite elements of degree l for
Vh , one gets
l
kσ − σ LS
h kdiv,Ω . h (|σ|l,Ω + |div σ|l,Ω ) ,
l
ku − uLS
h k1,Ω . h |u|l+1,Ω ,
if σ ∈ H l (Ω)3 with div σ(= −f ) ∈ H l (Ω) and u ∈ H l+1 (Ω)3 is satisfied. It
may also be worth noting that within the first-order least squares approach,
piecewise linear conforming displacement approximations are of optimal order uniformly in the incompressible limit. Of course, this requires the simultaneous computation of stress approximations in the lowest-order RaviartThomas spaces which may be considered too costly if these quantities are
not of particular interest. If the solution is less regular, then the optimal
approximation order may be retained with adaptively refined triangulations
based on using the local evaluation of the functional as an a posteriori error
estimator, cf. Cai et al. (2005). For domains with curved boundaries in association with the higher-order case l > 1, parametric finite element spaces
would be needed in order to retain the optimal approximation order. This
would involve the parametric Raviart-Thomas spaces studied in Bertrand
et al. (2014) for Πh in combination with standard isoparametric elements,
cf. (Brenner and Scott, 2008, Sect. 10.4).
7
It is important to keep in mind that the two terms in the functional
defined by (14) need to be scaled appropriately in order to get reasonable
approximations. This is due to the fact that the constants involved in
the above estimates must not become exceedingly large. The two main
ingredients which influence these constants are the Lamé parameter µ in the
material law (2) and CK in Korn’s inequality (6). If both are on the order
of one, then the scaling in (14) is adequate. This can be achieved by the
choice of suitable units for measuring forces and lengths. Our computational
experience suggests that it is generally less harmful to weight the momentum
balance term too strong than too weak with respect to the above rules.
In contrast to the mixed approximation σ HR
h , our least-squares approximation σ LS
does
not
satisfy
the
momentum
balance exactly if f ∈ div Πh .
h
We will now show that, in fact, the momentum balance term in the functional (14) converges faster than the overall functional. The proof is inspired
by the techniques used in Brandts et al. (2006) for the investigation of the
relations between saddle point and least-squares formulations for the firstorder system formulation of the Poisson equation.
Theorem 3.2. Under our regularity assumptions, the momentum balance
accuracy associated with the first-order system least squares approximation
satisfies
α
LS
kdivσ LS
kσ − σ LS
h +f k . h
h k + kε(u) − ε(uh )k + inf kf −zh k. (20)
zh ∈Zh
Proof. With fh = π h f ∈ Zh , the triangle inequality leads to
LS
LS
kdivσ LS
h +f k ≤ kdivσ h +fh k+kf −fh k = kdivσ h +fh k+kf −π h f k. (21)
The first term on the right hand side in (21) can be written as
kdiv σ LS
h + fh k = sup
zh ∈Zh
= sup
zh ∈Zh
(div
(div σ LS
h + fh , zh )
kzh k
σ LS
h
+ f , zh )
(div (σ LS
h − σ), zh )
= sup
.
kzh k
kzh k
zh ∈Zh
(22)
For any zh ∈ Zh , the following auxiliary boundary value problem may be
defined: Find Ξ ∈ HΓN (div, Ω)3 and η ∈ HΓ1D (Ω)3 such that
div Ξ = zh ,
AΞ − ε(η) = 0
8
(23)
holds. Let ΞHR
∈ Πh be the mixed finite element approximation of Hellingerh
Reissner type to (23) and let η h ∈ Vh be any approximation to η, then
(div (σ − σ LS
h ), zh )
= (div (σ − σ LS
h ), div Ξ)
LS
LS
= (div (σ − σ LS
h ), div Ξ) + (A(σ − σ h ) − ε(u − uh ), AΞ − ε(η))
HR
= (div (σ − σ LS
h ), div (Ξ − Ξh ))
HR
LS
+ (A(σ − σ LS
h ) − ε(u − uh ), A(Ξ − Ξh ) − ε(η − η h ))
HR
LS
= (A(σ − σ LS
h ) − ε(u − uh ), A(Ξ − Ξh ) − ε(η − η h ))
holds due to (15), (18) and the fact that div ΞHR
= div Ξ = zh is satisfied.
h
Combining this with (22) leads to
LS
LS
kdiv σ LS
h + fh k ≤ kA(σ − σ h ) − ε(u − uh )k
kA(Ξ − ΞHR
h ) − ε(η − η h )k
sup
kdiv Ξk
Ξ
LS
LS
. kσ − σ h k + kε(u − uh )k
(24)
ΞHR
h k
. hα
kΞ −
+ kε(η − η h )k
sup
kdiv Ξk
Ξ
LS
kσ − σ h k + kε(u − uLS
h )k
due to (12) and our general regularity assumption from Section 1.
Theorem 3.2 states that the error associated with momentum balance
converges of higher order. In particular, if f ∈ H α (Ω)d is assumed for the
right-hand side, then
α
LS
kdiv (σ − σ LS
kσ − σ LS
h )k . h
h k + kε(u) − ε(uh )k + kf k
(25)
holds. One implication of (25) is concerned with the approximation of
boundary traces. For the approximation of the resultant traction forces,
LS
LS
h(σ − σ LS
h ) · n, eiL2 (∂Ω) = (div (σ − σ h ), e) . kdiv (σ − σ h )k kek (26)
holds for any constant displacement field e ∈ IRd . A further implication,
which is seen best directly in (24), is that the second term in the least
LS
squares functional (14) dominates if (σ LS
h , uh ) is inserted. This property
9
can be of use, in particular, in the study of the functional as an a posteriori
error estimator.
4 Stress reconstruction for displacement-pressure
approaches
The most commonly used approach to compute finite element approximations for the linear elasticity model is based on minimizing the energy in (3)
among all v ∈ HΓ1D (Ω)d . The solution u ∈ HΓ1D (Ω)d satisfies
Z
Z
Ω
Z
f · v dx +
(2µ ε(u) : ε(v) + λ (div u) (div v)) dx =
Ω
g · v ds
ΓN
or, in short notation,
2µ (ε(u), ε(v)) + λ (div u, div v) = (f , v) + hg, viL2 (ΓN )
(27)
for all v ∈ HΓ1D (Ω)d . Obviously, this formulation becomes problematic as
the Lamé parameter λ tends to ∞ which is the case for incompressible
materials. One possible remedy is to introduce a new pressure-like variable
p = λ div u which leads to the saddle-point problem
2µ (ε(u), ε(v)) + (p, div v) = (f , v) + hg, viL2 (ΓN )
1
(div u, q) − (p, q) = 0
λ
(28)
for all v ∈ HΓ1D (Ω)d and q ∈ L2 (Ω). This saddle-point problem is a regular perturbation of the Stokes problem modelling incompressible fluid flow
(which coincides with the limiting case λ = ∞) and as such can be treated
with any inf-sup stable finite element pair (Vh , Qh ) for the Stokes equations, cf. (Boffi et al., 2013, sect. 4.3). The resulting finite-dimensional
saddle-point problem is then to find uh ∈ Vh and ph ∈ Qh such that
2µ (ε(uh ), ε(vh )) + (ph , div vh ) = (f , vh ) + hg, vh iL2 (ΓN )
1
(div uh , qh ) − (ph , qh ) = 0
λ
(29)
holds for all vh ∈ Vh and qh ∈ Qh . Since we are interested in the approximation quality of the stresses σ(u, p) = 2µε(u) + p I computed from
approximations to u and p, combinations seem favourable, where the error
kε(u − uh )k converges at the same order as kp − ph k. Such a combination
is given, for example, by the Taylor-Hood elements (continuously quadratic
10
for Vh with continuously linear for Qh ) and their higher-order generalizations, cf. (Boffi et al., 2013, sect. 8.8). Another possibility is the use
of the quadratic nonconforming elements introduced in Fortin and Soulie
(1983) for Vh combined with discontinuous piecewise linears. Since these
elements have some favourable properties with respect to the associated
derived stresses σ(uh , ph ), we will investigate them more closely.
The quadratic nonconforming elements by Fortin-Soulie
With respect to a triangulation Th of Ω with the corresponding set of
sides (edges for d = 2, faces for d = 3) denoted by Sh , the quadratic
nonconforming finite element space is defined by
VhF S = {vh ∈ L2 (Ω)d : vh |T ∈ P2 (T )d for all T ∈ Th ,
hJvh KS , siL2 (S) = 0 for all s ∈ P1 (S) , S ∈ Sh ∩ Ω ,
(30)
d
hvh , siL2 (S) = 0 for all s ∈ P1 (S) , S ∈ Sh ∩ ΓD } ,
where J · KS denotes the jump across the side S. It is necessary to go to
quadratic nonconforming elements since the linear nonconforming elements
by Crouzeix-Raviart do not satisfy the discrete Korn’s inequality, in general,
if ΓN 6= ∅. That such an inequality, which reads
X
X
k∇vh k2L2 (T ) .
kε(vh )k2L2 (T ) for all vh ∈ Vh ,
(31)
T ∈Th
T ∈Th
holds for Vh = VhF S (under our assumption that ΓD 6= ∅) is a consequence
of (Brenner, 2003, Thm. 3.1). The validity of (31) is required for the
well-posedness of the variational formulation which now consists in finding
S
2
uh ∈ VhF S and ph ∈ QF
h := {qh ∈ L (Ω) : qh |T ∈ P1 (T )} such that
2µ
X
(ε(uh ), ε(vh ))L2 (T )
T ∈Th
+
X
(ph , div vh )L2 (T ) = (f , vh ) + hg, vh i0,ΓN
T ∈Th
X
T ∈Th
(div uh , qh )L2 (T ) −
(32)
1
(ph , qh ) = 0
λ
S
is valid for all vh ∈ VhF S , qh ∈ QF
h . As was already described in the
original papers by Fortin and Soulie (1983) and Fortin (1985), the quadratic
C
TH
nonconforming space can be written as VhF S = VhT H + BN
is
h , where Vh
(component-wise) the standard space of conforming quadratic elements and
11
C
BN
denotes (again component-wise) a suitable space of non-conforming
h
bubble functions. In the two-dimensional case, this non-conforming bubble
space is given by
C,2
BN
= {bh ∈ L2 (Ω)2 : bh |T ∈ P2 (T )2 for all T ∈ Th ,
h
hvh , siL2 (S) = 0 for all s ∈ P1 (S)2 , S ∈ Sh } ,
C,2
i.e., there is exactly one non-conforming bubble function in BN
per trih
C,2
angle. We denote the corresponding one-dimensional space by BN
(T ).
h
In the three-dimensional case, the non-conforming bubble space is given by
C,3
BN
= {bh ∈ L2 (Ω)3 : bh |T ∈ P2 (T )3 for all T ∈ Th ,
h
hvh , siL2 (S) = 0 for all s ∈ P1 (S)3 , S ∈ Sh }
C,2
+ {bh ∈ L2 (Ω)3 : bh |T ∈ P2 (T )3 for all T ∈ Th , vh |S ∈ BN
(S)
h
and hJvh KS , siL2 (S) = 0 for all s ∈ P1 (S)3 , S ∈ Sh } .
C,3
The first part of BN
consists of exactly one non-conforming bubble funch
C,3
tion per tetrahedra, again denoted by BN
(T ). The second part is made
h
up of two-dimensional non-conforming bubble functions BhN C,2 (S) for each
face S ∈ Sh extended suitably into the two neighboring tetrahedra. It should
C
be kept in mind that the representation VhF S = VhT H + BN
is not a direct
h
sum. Globally constant functions can be expressed in two different ways in
these subspaces, in general. Moreover, in the three-dimensional case, the
representation of conforming piecewise linear functions is not unique.
The following result was also already contained in the original papers by
Fortin and Soulie (1983) and Fortin (1985) including the proof given below.
Proposition 4.1. Assume that f ∈ L2 (Ω) is piecewise constant with respect
to the triangulation Th of Ω ⊂ IRd , d = 2 or 3 and that g ∈ L2 (ΓN ) is
piecewise linear with respect to the subset of sides Sh,N = Sh ∩ΓN associated
with the Neumann boundary. If we denote by Sh,i = Sh ∩Ω the subset of sides
e h (uh , ph ) =
interior to the domain, then the (piecewise linear) stresses σ
S
2µε(uh ) + ph I computed from the solution (uh , ph ) ∈ VhF S × QF
of (29)
h
satisfy
e h (uh , ph ) = 0 piecewise for all T ∈ Th ,
f + div σ
(33)
e h (uh , ph ) · n, ei iL2 (S) = 0 for all S ∈ Sh,N ,
hg − σ
hJe
σ h (uh , ph ) · nKS , ei iL2 (S) = 0 for all S ∈ Sh,i ,
where ei ∈ IRd denotes the i-th unit vector.
12
(34)
C
Proof. Inserting a nonconforming bubble function bT ∈ BN
with support
h
restricted to T as test function into (32) leads to
0 = hg, bT i0,ΓN ∩∂T + (f , bT )L2 (T ) − (e
σ h (uh , ph ), ε(bT ))L2 (T )
e h (uh , ph ), bT )L2 (T )
= hg, bT i0,ΓN ∩∂T − he
σ h (uh , ph ) · n, bT i0,∂T + (f + div σ
e h (uh , ph ), bT )L2 (T ) ,
= (f + div σ
where the fact was used that hs, bT iL2 (S) = 0 for all s ∈ P1 (S)d , S ⊂ ∂T .
e h (uh , ph ), constant on T , must therefore vanish.
The term f + div σ
For all test functions vh ∈ VhF S , we therefore get from (32) that
X
0 = hg, vh iL2 (ΓN ) + (f , vh ) −
(e
σ h (uh , ph ), ε(vh ))L2 (T )
T ∈Th
=
X
e h (uh , ph ) · n, vh iL2 (S) −
hg − σ
S∈Sh,N
X
hJe
σ h (uh , ph ) · nKS , vh iL2 (S)
S∈Sh,i
holds. We pick one of the sides S ∈ Sh and choose the test function vh ∈
VhF S in such a way that in the sum above only the term associated with this
particular side does not vanish. In two dimensions, this is achieved using a
conforming piecewise quadratic function that vanishes on all edges besides
S. In three dimensions, the non-conforming bubble function corresponding
to the face S has the desired properties (note that Je
σ h (uh , ph ) · nKS is of
degree 1 on all faces). The symmetry properties of the chosen test functions
with respect to S finally implies (34).
e h (uh , ph ) proven in Proposition 4.1 can be used to
The properties of σ
d
get an efficient stress reconstruction σ R
h ∈ H(div, Ω) by Raviart-Thomas
1
elements of next-to-lowest order Πh . We will now explain how such a construction can be done in an element-wise fashion. In the two-dimensional
case this is equivalent to the technique described in Kim (2012). The stress
1
reconstruction σ R
h ∈ Πh is determined on each element T ∈ Th by the
following conditions:
σR
σ h (uh , ph )}}S · n for all S ⊂ ∂T ,
h T · n = {{e
(35)
R
div σ h = π 1h f ,
T
T
where {{ · }}S stands for the average value on S between the two adjacent
elements (set {{e
σ h (uh , ph )}}S · n = g on all sides S ⊂ ΓN ) and π 1h denotes
2
the L (Ω) projection onto the piecewise linear (possibly discontinuous) functions on Th . The first line in (35) coincides with the standard interpolation
conditions on the sides S ⊂ ∂T for next-to-lowest order Raviart-Thomas elements, cf. (Boffi et al., 2013, Example 2.5.3). It remains to be shown that,
13
in the situation encountered here, the remaining d interpolation conditions
in (Boffi et al., 2013, Example 2.5.3) are equivalent to the second line in
(35). To this end, note that
R
(div σ R
σ h (uh , ph ) · n, ei iL2 (∂T )
h , ei )L2 (T ) = hσ h · n, ei iL2 (∂T ) = he
e h (uh , ph ), ei )L2 (T ) = (π 0h f , ei )L2 (T )
= (div σ
holds, where (34) is used
first
line and (33) in the second line. This
in the
= π 0 f and the second condition of (35) consists
means that π 0h (div σ R
)
h T
h T
of only d linear equations at most which may be used to satisfy the remaining
interpolation conditions.
The construction is rather simple and consists of the following
two steps:
R,0 (i) Compute, on each element T , an affine function σ h ∈ P1 (T )d which
T
satisfies the first set of conditions in (35). These are d(d + 1) conditions
for d(d + 1) coefficients and amounts to the assignment of the appropriate
degrees of freedom depending on the finite element basis used. This results
in an approximation σ R,0
∈ H(div, Ω)d with σ R,0
· n = g on ΓN and
h
h
R,0
piecewise constant divσ h (which may also be interpreted as approximation
in the BDM1 space, cf. Kim (2012)).
(ii) Update for better divergence approximation (if f is not constant on T )
by adjusting the coefficients associated with the interior degrees of freedom.
The above reconstruction results in a stress approximation with similar
properties as for the Hellinger-Reissner formulation using the Boffi-BrezziFortin elements studied at the end of Section 2. In particular, the momentum balance error is minimized and optimal order approximation of the
stress is achieved with respect to the L2 (Ω) norm.
Computational comparison for incompressible linear elasticity
We close the part of this contribution associated with linear elasticity
by some two-dimensional computational results in order to provide some
insight on the actual behavior of the methods introduced above.
Example 1. The underlying domain is a quadrilateral with vertices
at (0, 0), (0.48, 0.44), (0.48, 0.6) and (0, 0.44), commonly known as Cook’s
membrane. It is fixed (u = 0) at the left edge of the boundary (x1 = 0)
while a uniform traction force pointing upwards (σ · n = (0, 1)) is applied
at the right edge (x1 = 0.48). At the remaining part of the boundary it
is kept in equilibrium (σ · n = 0). All the computations are done for the
incompressible limit λ = ∞ while µ is set to 1. Figure 1 shows the initial
triangulation with 44 elements. The results on a sequence of uniform refinements starting from this initial triangulation are compared for different
methods.
14
Figure 1. Initial triangulation for Cook’s membrane
Table 1 shows the resultant traction force
Z
n · (σ · n) ds
ΓD
in normal direction acting on the fixed left boundary part calculated from
different finite element approximations of displacement-pressure type. Due
to the divergence theorem the exact value is 0. Obviously, the evaluation of
the Taylor-Hood and P2/P0 (piecewise constant pressure) approximations
on the boundary does not reproduce this resultant traction force exactly
while this is the case for the Fortin-Soulie approximations in accordance
with Proposition 4.1. Another interpretation of these results is that the
piecewise linear stress approximations in L2 (Ω)d×d are not suitable for their
evaluation on the boundary, in general. For the non-conforming Fortin-
15
l
0
1
2
3
4
5
6
|Th |
44
176
704
2816
11264
45056
180224
Taylor-Hood
3.8467 · 10−2
2.8816 · 10−2
2.0867 · 10−2
1.4829 · 10−2
1.0383 · 10−2
7.1988 · 10−3
4.9631 · 10−3
P2/P0
3.3047 · 10−2
2.4458 · 10−2
1.7671 · 10−2
1.2547 · 10−2
8.7743 · 10−3
6.0760 · 10−3
4.1841 · 10−3
Fortin-Soulie
−4.0246 · 10−16
2.3592 · 10−16
2.3384 · 10−15
2.5180 · 10−14
−1.6708 · 10−14
−2.0714 · 10−13
−3.4016 · 10−13
Table 1. Resultant normal traction for displacement-pressure methods
Soulie approximations, the trace on the Dirichlet boundary coincides with
those associated with the recovered stress in H(div, Ω)d and does therefore
produce a good approximation of the boundary tractions.
Figure 2. Approximation of normal traction near singularity
Figure 2 shows the quality of the normal traction approximation for different displacement-pressure elements (after six uniform refinements) in the
neighborhood of the singularity at the left upper vertex of Cook’s membrane. The shaded graph is associated with the Taylor-Hood element pair,
the dotted lines are for P2/P0 and the solid straight lines for the FortinSoulie elements. The black curve represents the correct traction force distribution and was computed on an adaptively refined triangulation by the
16
least squares approach of Section 3. Away from the singularity all approximations are quite accurate while severe differences are visible in the quality
how well the singular behavior is resolved. The Fortin-Soulie does perform
much better and therefore justifies its larger number of degrees of freedom.
l
0
1
2
3
4
5
|Th |
44
176
704
2816
11264
45056
HR (BBF)
−1.6676 · 10−13
−1.0358 · 10−12
1.5558 · 10−11
3.2884 · 10−10
1.8848 · 10−10
8.9723 · 10−9
LS (RT1/P2)
1.9608 · 10−4
9.2856 · 10−5
4.3819 · 10−5
2.0679 · 10−5
9.7529 · 10−6
4.6026 · 10−6
recov. from FS
7.6328 · 10−16
−9.7145 · 10−17
−4.8260 · 10−15
−1.5449 · 10−14
−6.0172 · 10−14
−2.2891 · 10−14
Table 2. Resultant normal traction for stress-based methods
Table 2 lists the same quantities as Table 1 but this time compares
the different methods from Sections 2, 3 and 4. Due to the exact momentum conservation, the approximations with the Boffi-Brezzi-Fortin elements based on the Hellinger-Reissner principle produce the resultant traction forces perfectly (up to roundoff errors). The first-order system least
squares approach does not compute the resultant traction force exactly but
to quite acceptable accuracy while the numbers associated with the recovered stresses from the Fortin-Soulie elements are again correct up to working
precision.
Figure 3 shows the distribution of the normal traction at the left boundary near the singularity for the three different approaches of Table 2 after
five uniform refinements. The shaded graph belongs to the first-order system
least squares approach which performs slightly worse than the two alternatives. The dotted lines are associated with the Hellinger-Reissner principle
using the finite element combination of Boffi-Brezzi-Fortin and seem to resolve the singularity slightly better than the stresses recovered from the
Fortin-Soulie elements (solid straight lines).
Considering linear elasticity computations, the stress reconstruction approach is quite attractive since the global system that needs to be solved
involves fewer unknowns and the reconstructed stresses are of a similar accuracy as those obtained with a mixed method of saddle-point or least-squares
type. The situation may, however, be different for more complicated models where the stress is involved more directly. This is the case, for instance,
in the context of inelastic behavior caused by stress components exceeding a certain limit where the direct treatment of stresses in the variational
formulation is advantageous (cf. Reddy (1992) for a mixed approach of
17
Figure 3. Approximation of normal traction near singularity
saddle-point type, Starke (2007); Schwarz et al. (2009) for a least-squares
type approach). A comparison of the different approaches for the nonlinear
problems arising in association with hyperelastic material models will be
given in Section 5.
5
Extension to finite-strain Hyperelasticity
In the previous sections, the linear elasticity model was considered which is
derived under the assumption of small strains. Now we switch to the more
general case of finite strains with hyperelastic material models. More details
on the validity of these models and their mathematical aspects can be found,
e.g. in (Ciarlet, 1988, Chp. 4). Based on the deformation gradient given
by F = F (u) := I + ∇u, the left and right Cauchy-Green strain tensors are
defined as B = B(u) := F (u)F (u)T and C = C(u) := F (u)T F (u), respectively. This nonlinear dependence of strains to displacements constitutes the
geometrically nonlinear nature of this model. In addition, there is also a
nonlinearity in the material law describing the relation between stresses and
strains. This originates from a stored energy function ψ : IR3×3
sym → IR which
generalizes (4) and is no longer quadratic. Again, we restrict ourselves to
a homogeneous material which means that ψ does not explicitly depend on
the location x ∈ Ω.
18
Minimizing the total energy
Z
Z
Z
I(v) :=
ψ(C(v)) dx −
f · v dx −
Ω
Ω
g · v ds
(36)
ΓN
among all admissible displacements v ∈ V for some suitable space V is
again equivalent to finding a solution u ∈ V of the variational problem
(P (u), ∇v) = (f , v) + hg, vi0,ΓN for all v ∈ V ,
(37)
where P (u) := ∂F ψ(C)(u) denotes the first Piola-Kirchhoff stress tensor.
We assume that our problem is sufficiently regular so that we can choose
V = WΓ1,p
(Ω)3 for p > 2 as our solution space for (37). In that case, we
D
may also write (37) as a first-order system as
− div P = f
in Ω
P = ∂F ψ(C)
P · n = g on ΓN ,
in Ω
(38)
u = 0 on ΓD .
The first equation in (38) is an immediate consequence of the physically
necessary conservation of linear momentum for a static problem. Conservation of angular momentum for a static problem leads additionally to the
symmetry of P (u)F (u)T which is implicitly contained in the formulations
(37) and (38).
For homogeneous isotropic materials it is possible to express the stored
energy function ψ by a function ψ̃ : R3 → R, depending on three terms
I1 , I2 , I3 : R3×3 → R, i.e.,
ψ(C) = ψ̃(I1 (C), I2 (C), I3 (C)),
C = FTF,
(39)
with the principal invariants I1 (C) := tr(C), I2 (C) := tr(Cof C) and
I3 (C) := det C (cf. (Simo, Thm. 31.1 and Ex. 31.2)). Introducing the
so-called Kirchhoff stress tensor τ := P F T , a simple calculation then leads
to
!
∂ ψ̃
∂ ψ̃
∂ ψ̃ 2
∂ ψ̃
I3 (B)I + 2
+
I1 (B) B − 2
B =: G(B)
(40)
τ =2
∂I3
∂I1
∂I2
∂I2
or, equivalently, for the second Piola-Kirchhoff stress tensor Σ := F −1 P :
!
∂ ψ̃
∂ ψ̃
∂ ψ̃
∂ ψ̃
+
I1 (C) I − 2
C +2
I3 (C)C −1 := G̃(C), (41)
Σ=2
∂I1
∂I2
∂I2
∂I3
19
where G and G̃ are mappings from strains into stresses, similar as the fourthorder elasticity tensor C in linear elasticity.
In the following we assume that G(I) = 0 = G̃(I), i.e. the reference
configuration is stress-free, and that G, G̃ 0 are continuously differentiable in
the identity matrix with G 0 (I)[E] = 21 CE = G̃ 0 (I)[E], i.e. consistency of the
nonlinear model with the model of linear elasticity (cf. (Ciarlet, 1988, Sect.
3.8)). Since the elasticity tensor C itself is an isomorphism, the mappings
G 0 (I) = 12 C and G̃ 0 (I) = 21 C are also isomorphisms. Thus the local inversion
theorem (cf. (Ciarlet, 1988, Thm. 1.2-4)) is applicable and guarantees that
the inverse mappings G −1 (τ ) and G̃ −1 (Σ) are well-defined in a neighborhood
of τ = 0 and Σ = 0, respectively. Using these considerations we can modify
the strong formulation (38) into
div P + f = 0
G
−1
T
(P F (u) ) − B(u) = 0
in Ω,
in Ω,
(42)
P · n = g on ΓN , u = 0 on ΓD
using the representation in B or into
G̃
−1
−1
(F (u)
div P + f = 0
in Ω,
P ) − C(u) = 0
in Ω,
(43)
P · n = g on ΓN , u = 0 on ΓD
using the representation in C. Both systems are at least well-defined for
small stresses.
5.1
A least squares finite element method for isotropic hyperelastic materials
Since G 0 (I) = G̃ 0 (I) = 21 C, the implicit function theorem tells us that
(G −1 )0 (0) = (G̃ −1 )0 (0) = 2C −1 . This means that we encounter the same
problem as in the linear case, namely, that G −1 and G̃ −1 are not invertible
anymore in the incompressible limit. Due to this observation we use the
notation A and à instead of G −1 and G̃ −1 in (42) and (43). With this in
mind we introduce for P = P N + P̂ ∈ W q (div; Ω)3 + WΓqN (div; Ω)3 (with
P N · n = g on ΓN ), u ∈ WΓ1,p
(Ω)3 and f ∈ Lq (Ω)3 , p, q ≥ 4 sufficiently
D
large, the nonlinear operators
div P + f
,
R(P , u) :=
A(P F (u)T ) − B(u)
(44)
div P + f
R̃(P , u) :=
Ã(F (u)−1 P ) − C(u)
20
for (42) and (43), respectively. Based on these operators we define nonlinear
least squares functionals
F(P , u) := kR(P , u)k2 = kdiv P + f k2 + kA(P F (u)T ) − B(u)k2
(45)
for the formulation in B and
F̃(P , u) := kR̃(P , u)k2 = kdiv P + f k2 + kÃ(F (u)−1 P ) − C(u)k2 (46)
for the formulation in C.
5.2
Gauss-Newton iterative method
In the following we restrict ourselves to the minimization problem (45)
corresponding to the B-formulation. All further steps below can be handled similarly for the C-formulation. The minimization of (45) is carried out iteratively solving a sequence of linearized least squares problems.
Since the operator R(P , u) is continuously differentiable with respect to
(P , u), we can linearize it around a given approximation (P (k) , u(k) ) ∈
P N +WΓqN (div; Ω)3 ×WΓ1,p
(Ω)3 . The resulting linearized least squares funcD
(k)
tional depending on (P , u(k) ) is given by
F lin (Q, v) = F lin (Q, v; R(P (k) , u(k) ))
:= kR(P (k) , u(k) ) + R0 (P (k) , u(k) )[Q, v]k2 .
(47)
The minimizer (Q(k) , u(k) ) of (47) is then sought in a suitable normed function space ΠΓN × VΓD , provided that the values q and p are chosen sufficiently large such that R(P (k) , u(k) ) and also R0 (P (k) , u(k) )[Q, v] are contained in L2 (Ω)3 × L2 (Ω)3×3 . The linearized minimization problem (47) is
equivalent to the variational problem of finding (Q(k) , v(k) ) ∈ ΠΓN × VΓD
such that
B((Q(k) , v(k) ), (Q̂, v̂)) = − R(P (k) , u(k) ), R0 (P (k) , u(k) )[Q̂, v̂]
(48)
holds for all (Q̂, v̂) ∈ ΠΓN × VΓD . The bilinear form in (48) is defined on
ΠΓN × VΓD and given by
B((Q, v), (Q̂, v̂)) := R0 (P (k) , u(k) )[Q, v], R0 (P (k) , u(k) )[Q̂, v̂] . (49)
For the numerical implementation, a finite dimensional space Πh × Vh ⊂
(0)
(0)
ΠΓN × VΓD is chosen. Starting with an initial guess (P h , uh ) ∈ P N +
WΓqN (div; Ω)3 × WΓ1,p
(Ω)3 and setting k = 0, the discrete analogue of (48)
D
21
(k)
(k)
is then solved in Πh × Vh to obtain the correction term (Qh , vh ). Afterwards the new iterate is set to
(k+1)
(P h
(k+1)
, uh
(k)
(k)
(k)
(k)
) = (P h , uh ) + α(k) (Qh , vh )
(k)
(k)
where α(k) > 0 describes the step length in the direction (Qh , vh ). The
described approach is the well-known Gauss-Newton method combined with
a line search strategy, cf. (Nocedal and Wright, 2006, Chp. 3 and Sect.
10.3). For instance, for the determination of a suitable step length one can
use a backtracking line search approach, cf. (Nocedal and Wright, 2006, Alg.
(k+1)
(k+1)
(k+1)
3.1). The new iterate (P h
, uh
) automatically satisfies P h
·n = g
(k+1)
on ΓN and uh
= 0 on ΓD . An alternative approach for minimizing
the linearized problems of the form (47) in finite dimensional spaces is the
usage of the Levenberg-Marquardt method which replaces the line search
with a trust-region method, cf. (Nocedal and Wright, 2006, Chp. 4 and
Sect. 10.3).
Considering (48) and (49) one has to evaluate the nonlinear operator R
locally for given (P h , uh ) at each quadrature point for a numerical implementation. Due to the representations in (44) the problematical part is the
evaluation of A. A remedy, which works independently of the used stored energy function, is to solve the problem G(B) = τ h for given τ h := P h F (uh )T
with the help of Newton’s method. Assuming a finite λ and a sufficiently
small τ h , the sequence of Newton iterations is given by
B(j+1) = B(j) + ∆(j) ,
where ∆(j) ∈ R3×3 is the solution of
G 0 (B(j) )[∆(j) ] = τ h − G(B(j) ).
(50)
The initial guess B(0) = I ∈ R3×3 is at least for small (P h , uh ) reasonable,
since for (P h , uh ) = (0, 0) the solution is given by B = I. The equation
(50) can be solved with the help of a linear system of equations with nine
unknowns, where the matrix on the left-hand side depends on the old approximation B(j) and the right-hand side depends on (P h , uh ) and B(j) .
Applying Newton’s method on each quadrature point and on each element
of the triangulation is numerically expensive. For a special neo-Hookean
material which we consider in the next section it is possible to evaluate the
operator A locally without using Newton’s method. Moreover, one can take
the limit λ → ∞ in the operator A and can even set λ = ∞ in this model.
22
5.3
Least squares formulation for neo-Hookean model
The method described in Section 5.1 works generally for an arbitrary
isotropic stored energy function ψ, provided that the stresses are not too
large such that invertibility of the operators G and G̃ in (40) and (41) is
ensured. In this section we consider an isotropic material of neo-Hookean
type described by
ψ̃N H (I1 , I3 ) = αI1 + βI3 −
γ
ln(I3 ),
2
α, β, γ > 0,
with stored energy function
ψN H (C) = α tr(C) + β det(C) −
γ
ln(det C)
2
via (39) (cf. (Ciarlet, 1988, sect. 4.10)). With the derivatives
∂ ψ̃N H
∂I2
= 0,
∂ ψ̃N H
∂I3
=β−
γ
2I3
(51)
∂ ψ̃N H
∂I1
= α,
and equations (40) and (41) we achieve
GN H (B) = 2αB + (2β det B − γ)I,
G̃N H (C) = 2αI + (2β det C − γ)C −1 .
(52)
−1
−1
With AN H = GN
H and ÃN H = G̃N H denoting the corresponding inverses,
we end up with the nonlinear operators
div P + f
RN H (P , u) :=
,
AN H (P F (u)T ) − B(u)
div P + f
R̃N H (P , u) :=
ÃN H (F (u)−1 P ) − C(u)
(53)
in the Neo-Hooke case. The derivatives of (52) are given by
0
GN
H (B)[E] = 2αE + 2β(Cof B : E)I,
−1
0
G̃N
− (2β det C − γ)C −1 EC −1 .
H (C)[E] = 2β(Cof C : E)C
(54)
1
0
The conditions GN H (I) = 0 and GN
H (I)[E] = 2 CE (or G̃N H (I) = 0 and
1
0
G̃N H (I)[E] = 2 CE, respectively) lead to a linear system of equations for
the determination of α, β, γ which is uniquely solvable through
α=
µ
,
2
β=
λ
,
4
23
γ =µ+
λ
.
2
(55)
The derivatives in (54) can be directly inverted. After inserting the coefficients (55) in (54), the inverses are given by
λ
1
0
−1
Σ−
(Cof B : Σ)I ,
GN H (B) [Σ] =
µ
2µ + λ tr(Cof B)
1
0
−1
G̃N
[Σ] =
(56)
H (C)
λ
µ + 2 (1 − det C)
λ(det C)2
C Σ−
(Cof C −1 : Σ)C −1 C.
2µ + λ(1 + 2 det C)
The inverses are very helpful for the direct calculation of
R0N H (P , u)[Q, v]
div Q
=
A0N H (P F (u)T )[QF (u)T + P (∇v)T ] − (∇v)F (u)T − F (u)(∇v)T
and R̃0N H (P , u)[Q, v], respectively. Inserting B = I = C in (56) for finite λ
−1
−1
0
0
[Σ], as expected. For λ → ∞
[Σ] = 2C −1 Σ = G̃N
leads to GN
H (I)
H (I)
the first equation in (56) becomes
1
1
0
−1
Σ−
(Cof B : Σ)I
GN H (B) [Σ] =
µ
tr(Cof B)
and coincides for B = I with 2 AΣ from the linear elasticity case. The well0
−1
posedness of G̃N
for given strain C := ÃN H (Σ) and the identity
H (C)
0
−1
G̃N H (C) = 2A for λ → ∞ will be discussed later.
Local evaluation of AN H and ÃN H
We have seen at the end of Section 5.1 that we must evaluate AN H (P F (u)T )
in the B-formulation in each quadrature point. Analogously we have to
evaluate ÃN H (F (u)−1 P ) locally using the formulation in C. For both formulations one can evaluate AN H (P F (u)T ) and ÃN H (F (u)−1 P ) directly
without using Newton’s method as described in the sequel:
For the B-formulation on the one hand, given any stress tensor τ ∈ R3×3 ,
the corresponding strain B := AN H (τ ) ∈ R3×3 can be determined via
1
dev τ
1
B = dev B + tr(B)I =
+ tr(B)I
3
µ
3
(57)
with tr(B) solution of
(tr(B))3 + S tr(B) + T = 0,
24
(58)
depending on the coefficients
18µ
9
tr(Cof (dev τ )) +
,
2
µ
λ
2µ
1
2
det(dev τ ) − 1 −
T = 27
−
tr(τ ) .
µ3
λ
3λ
S=
(59)
A detailed derivation of (57) and (58) can be found in Müller et al. (2014).
3
2
If the discriminant D := S3 + T2
is positive, the cubic equation (58)
has only one real solution and we obtain exactly one reasonable strain which
corresponds to the given stress.
For the C-formulation on the other hand , given any stress tensor Σ ∈ R3×3 ,
the corresponding strain C := ÃN H (Σ) ∈ R3×3 can be determined via
C = ρ(Σ − µI)−1 ,
(60)
provided that Σ − µI is invertible. The parameter ρ in (60) is solution of
ρ3 + S ρ + T = 0
(61)
with coefficients
2
S := − det(Σ − µI),
λ
2µ
T := − 1 +
det(Σ − µI) ,
λ
(62)
cf. (Wriggers, 2008, Sect. 10.3). Provided that the discriminant of (61) is
positive, we get again one real solution for ρ and a unique real strain tensor
C corresponding to the given stress tensor Σ can be easily computed. One
remarkable fact in the cubic equations (58) and (61) is that we can take
also the limit λ → ∞ here. In fact, we can also set λ = ∞. In this case all
fractions with λ in the denominator in the coefficients (59) and (62) vanish.
With this in mind, we can come back to the discussion of the well-posedness
0
−1
of G̃N
for given strain C := ÃN H (Σ) in the incompressible limit
H (C)
λ → ∞. Inserting (62) into (61) we obtain
2
2µ
ρ3 = det(Σ − µI) ρ + 1 +
det(Σ − µI)
λ
λ
and with the help of (60) we can conclude that
2
2µ
−1
det C = det ÃN H (Σ) = ρ3 det ((Σ − µI) ) = ρ + 1 +
λ
λ
holds. Due to
p
λ
λ
2
2µ
λ→∞
µ + (1 − det C) = µ +
− ρ−
= −ρ → − 3 det(Σ − µI) ,
2
2
λ
λ
25
the second equation of (56) remains well-posed for λ → ∞ with
0
−1
0
−1
G̃N
[Q] = G̃N
[Q]
H (ÃN H (Σ))
H (C)
(det C)2
1
−1
−1
C
Q
−
= p
(Cof
C
:
Q)C
C.
1 + 2 det C
− 3 det(Σ − µI)
Inpthe case Σ = 0 (corresponding to C = ÃN H (0) = I) this leads to
0
−1
= 2A from the linear elasticity
− 3 det(0 − µI) = µ and hence G̃N
H (I)
case for λ → ∞. Combining the neo-Hookean model (51) with the firstorder systems (42) and (43) we have thus established a formulation which
allows us to consider fully incompressible materials.
Analysis of the formulation in B
Based on the convex sets
Π∞ := {Q ∈ L∞ (Ω)3 : kQkL∞ (Ω) ≤ δ} ∩ (P N + WΓ4N (div; Ω)3 ),
V∞ := {u ∈ W 1,∞ (Ω)3 : k∇ukL∞ (Ω) ≤ δ} ∩ WΓ1,4
(Ω)3
D
(63)
again with P N ∈ W ∞ (div; Ω)3 satisfying P N · n = g on ΓN , one can prove
for the first-order system operator RN H (cf. (44) with A = AN H locally
defined by (57), (58) and (59)) the estimates
kRN H (Q̂, v̂) − RN H (Q, v)k2 . kQ̂ − Qk2H(div; Ω) + kv̂ − vk2H 1 (Ω)
kRN H (Q̂, v̂) − RN H (Q, v)k2 & kQ̂ − Qk2H(div; Ω) + kv̂ − vk2H 1 (Ω) ,
(64)
provided that (Q̂, v̂), (Q, v) ∈ Π∞ × V∞ with sufficient small δ, cf. (Müller
et al., 2014, Thm. 4.4). In particular, (64) holds uniformly for λ → ∞.
Inserting the exact solution (P , u) ∈ Π∞ × V∞ with RN H (P , u) = 0 as
(Q, v) and an approximation (P h , uh ) ∈ Π∞ ×V∞ as (Q̂, v̂) in (64) directly
leads to
k(P − P h , u − uh )k2V . FN H (P h , uh ) . k(P − P h , u − uh )k2V
(65)
with V := HΓN (div; Ω)3 × HΓ1D (Ω)3 . The coercivity and continuity of the
nonlinear least squares functional FN H (P h , uh ) = kRN H (P h , uh )k2 in (65)
justifies its use as an a posteriori error estimator. The left inequality in (65)
implies the reliability while the right inequality stands for the efficiency.
Since the constants in (65) are independent of λ, the approach (45) combined
with the neo-Hookean stored energy function (51) is (Poisson) locking-free.
Equation (65) also leads to a priori error estimates. For instance, if we
3
combine, for some l ≥ 1, Raviart-Thomas elements Πlh := (RT l−1 (Th )) ⊂
26
Π∞ ⊂ H(div; Ω)3 for the approximation of P h with continuous elements
3
Vhl := (Pl (Th )) ⊂ V∞ ⊂ H 1 (Ω)3 for the approximation of uh and let
(P h , uh ) be the minimizer of FN H (Qh , vh ) among all (Qh , vh ) ∈ Πlh ×Vhl ⊂
H(div; Ω)3 × H 1 (Ω)3 , then we obtain
1
k(P − P h , u − uh )kV . (FN H (P h , uh )) 2
n
o
1
= inf (FN H (Qh , vh )) 2 : (Qh , vh ) ∈ Πlh × Vhl
. k(P − ρh P , u − rh u)kV
12
. hl kP k2H l (Ω) + kdiv P k2H l (Ω) + kuk2H l+1 (Ω)
(66)
under the assumption that P ∈ Π∞ ∩ H l (Ω)3×3 with div P ∈ H l (Ω)3 ,
u ∈ V∞ ∩H l+1 (Ω)3 and (P h , uh ) ∈ Π∞ ×V∞ holds. Here ρh and rh denote
the usual interpolation operators for the Raviart-Thomas and the standard
conforming finite elements, respectively, cf. (Boffi et al., 2013, Sects. 2.2 and
2.5). Under these assumptions the square of the error k(P − P h , u − uh )k2V
− 2l
and the least squares functional FN H both are proportional to h2l ∼ nt d
as h → 0, where nt denotes the number of elements in the triangulation Th .
Accuracy of balance of momentum in the nonlinear case
We investigate the generalization of Theorem 3.2 to the hyperelastic
situation. To this end, we assume that the linearized problem
div σ + f = 0 ,
R02 (P , u)[σ, υ]
(67)
= 0(= R2 (P , u))
has the following regularity properties, similar to those stated at the end of
Section 1 for the linear elasticity problem: For any f ∈ L2 (Ω)3 , the solution
(σ, υ) ∈ HΓN (div, Ω)d × HΓ1D (Ω)d of (67) satisfies (σ, υ) ∈ H α (Ω)d×d ×
H 1+α (Ω)d with
kσkH α (Ω) + kυkH 1+α (Ω) ≤ CR kf k
(68)
for some constant CR > 0 and α > 0.
Theorem 5.1. Under our regularity assumptions, the momentum balance
accuracy associated with the first-order system least squares approximation
for the Neo-Hooke model satisfies
α
LS
kdivP LS
kP − P LS
h +f k . h
h k + k∇u − ∇uh k + inf kf −zh k. (69)
zh ∈Zh
27
Proof. Starting as in the proof of Theorem 3.2, we arrive at
kdiv P LS
h + f k ≤ sup
zh ∈Zh
(div (P LS
h − P ), zh )
+ kf − π h f k .
kzh k
(70)
LS
2
Recalling that (P LS
h , uh ) ∈ Πh × Vh minimizes kR(P h , uh )k (and that
R(P , u) = 0), we have
LS
LS
0
LS
(R(P , u) − R(P LS
h , uh ), R (P h , uh )[Qh , vh ]) = 0
(71)
for all (Qh , vh ) ∈ Πh × Vh , where
R0 (P , u)[Q, v]
div Q
=
A0 (P F (u)T )[QF (u)T + P (∇v)T ] − F (u)(∇v)T − ∇vF (u)T
div Q
=:
.
R02 (P , u)[Q, v]
We replace the auxiliary boundary value problem (23) by the following:
Find Ξ ∈ HΓN (div, Ω)3 and η ∈ HΓ1D (Ω)3 such that
div Ξ = zh ,
R02 (P , u)[Ξ, η]
=0
(72)
holds. For arbitrary Ξh ∈ Πh with div Ξh = zh and η h ∈ Vh , we obtain
from (71) that
(div (P − P LS
h ), zh )
= (div (P − P LS
h ), div Ξ)
= (div (P − P LS
h ), div Ξ)
LS
0
+ (R2 (P , u) − R2 (P LS
h , uh ), R2 (P , u)[Ξ, η])
= (div (P − P LS
h ), div Ξ − div Ξh )
LS
LS
0
0
LS
+ (R2 (P , u) − R2 (P LS
h , uh ), R2 (P , u)[Ξ, η] − R2 (P h , uh )[Ξh , η h ])
LS
LS
LS
0
0
= (R2 (P , u) − R2 (P LS
h , uh ), R2 (P , u)[Ξ, η] − R2 (P h , uh )[Ξh , η h ])
holds. Combining this with (70) leads to
LS
LS
kdiv P LS
h + fh k ≤ kR2 (P , u) − R2 (P h , uh )k
sup
Ξ
LS
kR02 (P , u)[Ξ, η] − R02 (P LS
h , uh )[Ξh , η h ]k
.
kdiv Ξk
28
(73)
The second term in (73) can be bounded further as
LS
kR02 (P , u)[Ξ, η] − R02 (P LS
h , uh )[Ξh , η h ]k
≤ kR02 (P , u)[Ξ − Ξh , η − η h ]k
+
k(R02 (P , u)
−
LS
R02 (P LS
h , uh ))[Ξh , η h ]k
(74)
.
The first term in (74) may be bounded using (Müller et al., 2014, Lemma
4.3) to get
kR02 (P , u)[Ξ − Ξh , η − η h ]k . kΞ − Ξh k + kη − η h k . hα kdiv ηk
using our regularity assumption. For the second term in (74),
LS
k(R02 (P , u) − R02 (P LS
h , uh ))[Ξh , η h ]k
LS
∞ (Ω) + k∇(u − u
∞ (Ω)
. kP − P LS
k
)k
(kΞh k + k∇η h k)
L
L
h
h
. hα kdiv ηk
can be shown in a similar way using the expression for R0N H below (56).
This finishes the proof of (69).
Stress reconstruction in the nonlinear case In analogy to (28) one
may consider a displacement-pressure finite element approach to the nonlinear variational problem (37) associated with hyperelasticity. In the case
of a neo-Hookean model with
λ
2
P (u) = µ F (u) − F (u)−T +
(det F (u)) − 1 F (u)−T ,
(75)
2
a pressure-like variable
p=
λ
2
(det F (u)) − 1
2
may be introduced leading to the system
µ F (u) − F (u)−T , ∇v
+ p F (u)−T , ∇v = (f , v) + hg, vi0,ΓN
1
1
(det F (u))2 − 1 , q − (p, q) = 0
2
λ
(76)
to hold for all v and q in suitable test spaces. In order to allow for general
u ∈ HΓ1D (Ω)3 and to keep the pressure space as big as possible, one may
29
rewrite (75) slightly as
Cof F (u)
P (u) = µ F (u) −
det F (u)
λ
1
+
det F (u) −
Cof F (u)
2
det F (u)
(77)
and define the pressure variable as
λ
1
p=
det F (u) −
.
2
det F (u)
This is motivated by the fact that, for u ∈ H 1 (Ω)3 and under the additional
assumption that Cof F (u) ∈ L2 (Ω)3×3 , we have det F (u) ∈ L1 (Ω), see
(Ciarlet, 1988, Thm. 7.6-1). The assumption on the co-factor is not as
restrictive as it seems since it is usually fulfilled for the solutions associated
with edge singularities at re-entrant corners. For the same reason, one may
also assume that (det F (u))−1 ∈ L∞ (Ω) which suggests the well-posedness
of the following system: Find u ∈ HΓ1D (Ω)3 and p ∈ L1 (Ω) such that
Cof F (u)
, ∇v
µ F (u) −
det F (u)
+(p Cof F (u), ∇v) = (f , v) + hg, vi0,ΓN
1
1
1
det F (u) −
, q − (p, q) = 0
2
det F (u)
λ
(78)
for all v ∈ HΓ1D (Ω)3 and q ∈ L∞ (Ω). In the small-strain limit, (79) (as well
as (76)) turns into the displacement-pressure formulation of linear elasticity
(28).
The discrete problem consists in finding uh ∈ Vh and ph ∈ Qh such that
Cof F (uh )
µ F (uh ) −
, ∇vh
det F (uh )
+(ph Cof F (uh ), ∇vh ) = (f , vh ) + hg, vh i0,ΓN (79)
1
1
1
det F (uh ) −
, qh − (ph , qh ) = 0
2
det F (uh )
λ
for all vh ∈ Vh and qh ∈ Qh . It is natural to use the same combination
of spaces Vh and Qh as in the linear case although the compatibility is
not guaranteed. However, the investigations in Auricchio et al. (2010) and
Auricchio et al. (2013) indicate that at least some of these finite element
30
spaces (like Taylor-Hood) are also safe to use in the hyperelastic case for
incompressible materials. The stability and approximation properties of the
Fortin-Soulie elements (in combination with piecewise linear pressure) for
hyperelastic models with incompressible materials needs still to be investigated.
But even if the approximation quality is similar to the linear elasticity
case, the stress reconstruction procedure is more involved. The stress
e h (uh , ph ) = µ F (uh ) − F (uh )−T + ph F (uh )−T
P
associated with a piecewise quadratic uh and piecewise linear ph is certainly
not piecewise linear and therefore does not satisfy the properties of Proposition 4.1. The element-wise stress reconstruction procedure by Kim (2012)
is therefore not immediately applicable in the nonlinear case. The stress
reconstruction can be expected to be more involved for hyperelastic material models speaking in favour of our first-order system approach which
produces accurate stress approximations simultaneously.
5.4
Computational results
For the confirmation of our theoretical results, we consider some two- and
three-dimensional examples. Although the three-dimensional situation is,
of course, more realistic from an application point of view, two-dimensional
computations have the advantage that the asymptotic behavior is more
accessible.
Two-dimensional numerical tests
We start with the two-dimensional case and use a plane strain configuration, meaning that the displacement components u1 and u2 of u depend
only on x1 and x2 and the component u3 is constant. In this situation the
deformation gradient becomes


1 + ∂1 u1
∂2 u1
0
1 + ∂2 u2 0 .
F = F (u) = I + ∇u =  ∂1 u2
0
0
1
Due to the definition of the Cauchy-Green strain tensors B = F F T and
C = F T F , they have the same structure as F . Moreover, also the stress
tensor P has this structure (see (40) and (41)). Our numerical simulations
in the plane strain situation may therefore be based on a two-dimensional
domain with planar Raviart-Thomas elements for the first two rows of P
and piecewise polynomial functions without any continuity requirement for
31
the remaining nonzero entry P33 of P . For the displacement components u1
and u2 we can use continuous piecewise polynomial functions. In fact, we
use next-to-lowest-order Raviart-Thomas elements RT1 (Th ) for the plane
stress field, piecewise linear discontinuous elements DP1 (Th ) for P33 and
conforming, piecewise quadratic, finite elements P2 (Th ) for each displacement component. Thus our finite dimensional
space is altogether given by
Πh × Vh := RT1 (Th )2 × DP 1 (Th ) × P2 (Th )2 .
Example 2. In this example we consider again the polygonal Cook’s
membrane domain as in Section 4. With respect to the vertices (0, 0),
(48, 44), (48, 60) and (0, 44), the left part of the boundary is again used as
ΓD := {(0, x2 ) : 0 < x2 < 44} and the remaining boundary as ΓN . The
volume force is set to f = 0 and for the surface force, g = 0 is prescribed
on the upper and lower part of ΓN and g = (0, γ load )T with γ load ∈ R on
the right part ΓR := {(48, x2 ) : 44 < x2 < 60} of ΓN . We choose µ = 1 and
λ = ∞ as Lamé constants, i.e. we simulate a fully incompressible material
which is the numerical most challenging case. We use the formulation in B
and as load value γ load = 0.05. Before we present some numerical results,
note that the exact solution (P , u) of this problem is not in Π∞ ×V∞ , since
at the point (0, 44) the boundary condition changes from hard clamped
(u = 0) to stress-free (P · n = 0) and the interior angle is larger than
the critical one, cf. Rössle (2000). Although the regularity assumptions
in (63) are not satisfied for the exact solution, we see in Table 3 that the
nonlinear least squares functional FN H works reasonable as a posteriori
error estimator.
nt
186
275
390
559
821
1211
1796
2622
dim Πh
2378
3525
5010
7189
10583
15633
23208
33918
dim Vh
784
1150
1620
2314
3374
4954
7324
10656
FN H (P h , uh )
2.9972 · 10−2
1.4042 · 10−2
6.7178 · 10−3
3.2427 · 10−3
1.5525 · 10−3
7.3322 · 10−4
3.3695 · 10−4
1.4855 · 10−4
(order)
(1.939)
(2.110)
(2.023)
(1.916)
(1.930)
(1.973)
(2.165)
Table 3. Convergence rates of FN H with adaptive refinement (2d)
32
Figure 4. Adaptively refined triangulation for Cook’s membrane
In the last column of Table 3 the approximation order
(l+1)
(l+1)
(l)
(l)
log FN H P h , uh
− log FN H P h , uh
(l)
(l+1)
log nt − log nt
(l)
(l)
of FN H is listed. Here (P h , uh ) := (P h , uh ) denotes the approximated
(l)
solution and nt := nt the number of elements on level l ∈ N ∪ {0}. One
observes that the optimal convergence rate of 2 is achieved using adaptive
33
nt
186
744
2976
11904
47616
dim Πh
2378
9592
38528
154432
618368
FN H (P h , uh )
2.9972 · 10−2
1.3800 · 10−2
6.4895 · 10−3
3.0743 · 10−3
1.4538 · 10−3
dim Vh
784
3056
12064
47936
191104
(order)
(0.559)
(0.544)
(0.539)
(0.540)
Table 4. Convergence rates of FN H with uniform refinement (2d)
refinement. Figure 4 shows the mesh after four adaptive refinement steps
resulting in a triangulation with 821 triangles.
In Table 4 the approximation order using uniform refinement is illustrated. Obviously the optimal convergence rate is not reached and adaptive
refinement is superior. This is as expected due to the singularity at (0, 44).
nt
186
275
390
559
821
1211
1796
2622
adaptive refinement
kdiv(P − P h )k2 (order)
8.3534 · 10−9
1.9602 · 10−9
(3.707)
4.5544 · 10−10
(4.177)
1.0488 · 10−10
(4.079)
2.3596 · 10−11
(3.881)
4.9448 · 10−12
(4.021)
9.4024 · 10−13
(4.212)
1.4687 · 10−13
(4.907)
nt
186
744
2976
11904
47616
uniform refinement
kdiv(P − P h )k2 (order)
8.3534 · 10−9
1.9315 · 10−9
(1.056)
4.4487 · 10−10
(1.059)
1.0116 · 10−10
(1.068)
2.2323 · 10−11
(1.090)
Table 5. Improved convergence rates for balance of momentum (2d)
In Table 5 we can confirm numerically that the convergence rate of the
term kdiv(P − P h )k2 = kdiv P h + f k2 is approximately doubled, regardless
of using uniform or adaptive refinement. Furthermore the values itself are
close to zero which means that the approximations satisfy the conservation
of linear momentum quite well.
Besides the convergence rates we are interested in the quality of the
surface traction forces resulting from our stress approximations. The distribution of the normal component of the traction force acting at the left
boundary is shown in Figure 5. For a closer
R investigation of the accuracy of
these quantities we focus on the integral ΓD P · n ds which constitutes the
resultant force acting on the left hand boundary segment. Due to f = 0,
34
Figure 5. Normal traction at left boundary for Cook’s membrane
γ load = 0.0005
7.9750 · 10−3
7.9881 · 10−3
7.9944 · 10−3
7.9973 · 10−3
7.9987 · 10−3
R
Table 6. Comparison of ΓD P21 ds
nt
186
744
2976
11904
47616
the divergence theorem implies
Z
Z
Z
P · n ds = −
P · n ds = −
ΓD
ΓN
ΓR
γ load = 0.05
7.9764 · 10−1
7.9886 · 10−1
7.9945 · 10−1
7.9974 · 10−1
7.9988 · 10−1
for different load values
g ds =
0
−γ load |ΓR |
.
(80)
With the outward normal n = (−1, 0)T of ΓD , the load values γ load ∈
{0.0005, 0.05} and |ΓR | = 16 it follows immediately that
(
Z
8 · 10−3 , γ load = 0.0005
load
P21 ds = 16γ
=
8 · 10−1 , γ load = 0.05
ΓD
holds for the second entry in (80). One observes in Table 6 that the least
squares approach does in both cases produce quite satisfactory approximations to the resultant force.
35
Three-dimensional numerical tests
For fully three-dimensional examples we use the finite-dimensional spaces
3
3
Πh × Vh := (RT1 (Th )) × (P2 (Th )) on a tetrahedral decomposition of the
given domain.
Example 3. We consider a three-dimensional Cook membrane problem.
For this purpose we expand the two-dimensional domain of Example 2 in
x3 -direction with thickness 5. Thus the three-dimensional polyhedral domain is defined through the vertices (0, 0, 0), (48, 44, 0), (48, 60, 0), (0, 44, 0),
(0, 0, 5), (48, 44, 5), (48, 60, 5) and (0, 44, 5). We split the boundary Γ = ∂Ω
into the left lateral face ΓD := {(0, x2 , x3 ) : 0 < x2 < 44, 0 < x3 < 5} and
ΓN consisting of the remaining five lateral faces. We clamp the body on ΓD
and apply a surface force g = (0, γ load , 0)T with load value γ load ∈ R on the
right part of the boundary ΓR := {(48, x2 , x3 ) : 44 < x2 < 60, 0 < x3 < 5}.
On the other parts of ΓN no surface forces act (g = 0). As body force density we use f = 0, choose γ load = 0.05 and Lamé constants µ = 1, λ = ∞,
i.e. we consider again a fully incompressible material.
Figure 6 shows the mesh after three adaptive refinement steps resulting in
a triangulation with 2892 tetrahedra. The concentration of the refinement
in the vicinity of the singularity at the edge at x1 = 0 and x2 = 44 is
clearly visible. In Tables 7 and 8 the numerically obtained convergence
rates corresponding to FN H (P h , uh ) and kdiv(P − P h )k2 using adaptive
and uniform refinement, respectively, can be compared. One observes in
Table 7 that we obtain good convergence rates, close to the optimal value
4
3 , for the nonlinear functional using adaptive refinement. Moreover we see,
similar as in the two-dimensional example, that the convergence for the
balance of momentum is significantly faster than for the overall functional.
Moreover, the value kdiv(P − P h )k2 on each considered level is again close
to zero, i.e. linear momentum is conserved quite well.
Similar as in theRtwo-dimensional example we consider again the boundary integral values ΓD P · n ds. Due to |ΓR | = 16 · 5 = 80 the exact values
are

 

  
Z
0
Val1
0
−1
Val2  := −
P ·  0  ds = 80γ load  = 4
ΓD
0
0
Val3
0

following the same calculations as in the two-dimensional derivation. We
can observe in Table 9 that our least squares approach yields already on a
coarse mesh good approximations to the resultant forces and converges to
the correct values.
36
Figure 6. Adaptively refined triangulation for the 3D Cook’s membrane
nt
880
1410
1928
2892
dim Πh
22968
37161
50859
76734
dim Vh
4104
6321
8607
12576
FN H (P h , uh )
3.8682 · 10−1
2.0062 · 10−1
1.3179 · 10−1
8.1998 · 10−2
(order)
(1.393)
(1.343)
(1.170)
kdiv(P − P h )k2
2.3313 · 10−7
4.8949 · 10−8
1.8969 · 10−8
5.7679 · 10−9
(order)
(3.311)
(3.030)
(2.936)
Table 7. Convergence rates of FN H and kdiv(P − P h )k2 with adaptive
refinement (3d)
nt
880
7040
dim Πh
22968
186912
dim Vh
4104
30384
FN H (P h , uh )
3.8682 · 10−1
1.3719 · 10−1
(order)
(0.498)
kdiv(P − P h )k2
2.3313 · 10−7
3.3031 · 10−8
(order)
(0.940)
2
Table 8. Convergence rates of FN H and kdiv(P − P h )k with uniform
refinement (3d)
Acknowledgement.
The work reported here was supported by the German Research Foundation (DFG) under grant STA 402/11-1. The authors would also like to
thank Jörg Schröder and Alexander Schwarz for many discussions on the
37
nt
880
1410
1928
2892
nt
880
7040
adaptive refinement
Val1
Val2
1.7462 · 10−2 3.9723 · 100
6.6751 · 10−3 3.9872 · 100
3.0716 · 10−3 3.9921 · 100
1.8159 · 10−3 3.9959 · 100
uniform refinement
Val1
Val2
1.7462 · 10−2 3.9723 · 100
6.7975 · 10−3 3.9895 · 100
Val3
−1.1473 · 10−4
8.6541 · 10−6
5.3635 · 10−6
−6.7773 · 10−5
Val3
−1.1473 · 10−4
−3.8141 · 10−6
Table 9. Values of boundary integrals on ΓD (3d)
subject in the past years, especially related to the topic of Section 5.
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