AUGMENTED LAGRANGE METHODS FOR QUASI-INCOMPRESSIBLEMATERIALS - APPLICATIONS TO SOFT BIOLOGICAL TISSUE

[PREPRINT]

This is a preprint of an article published inInt. J. Numer. Methods Biomed. Eng. (IJNMBE), 29 (3) , pp. 332-350, 2013

http://dx.doi.org/10.1002/cnm.2504

S. Brinkhues†, A. Klawonn∗‡, O. Rheinbach∗‡, and J. Schroder††Institut fur Mechanik, Abt. Bauwissenschaften,

Fakultat fur Ingenieurwissenschaften,Universitat Duisburg-Essen, 45117 Essen, Germany.{sarah.brinkhues, j.schroeder}@uni-duisburg-essen.de

‡ Fakultat fur Mathematik,Universitat Duisburg-Essen, 45117 Essen, Germany.

{axel.klawonn, oliver.rheinbach}@uni-duisburg-essen.de

Abstract. Arterial walls in the healthy physiological regime are characterized by nearly incompressible, anisotropic,hyperelastic material behavior. Polyconvex material functions representing such materials typically incorporate a penaltyfunction to account for the incompressibility. Unfortunately, the penalty will affect the conditioning of the stiffness matrices.For high penalty parameters the performance of iterative solvers will degrade and when using direct solvers the quality of thesolutions will deteriorate. In this paper, an augmented Lagrange approach is used to cope with the almost incompressibilitycondition. Here, the penalty parameter can be chosen much smaller and as a consequence the arising linear systems ofequations have better properties. An improved convergence is then observed for the FETI-DP domain decompositionmethod which is used as an iterative solver.

1. Introduction. In this paper, we analyze the expansion of arterial walls using a FETI-DP domaindecomposition within a quasi-static process. Arterial walls are biological soft tissues and therefore almostincompressible. Their behavior is anisotropic as a consequence of embedded collagen fibers. Major bloodvessels are composed of distinct tissue layers, i.e., the intima (tunica intima), the media (tunica media),and the adventitia (tunica adventitia). We assume that the influence of the intima on the mechanicalproperties is negligible.

In Brands et al. [2] we have compared several material models describing the mechanical behaviorof arterial walls in order to study the mechanical response and the influence on the nonlinear iterationas well as on the FETI-DP domain decomposition method which we use to solve the linearized systemsof equations. It was found that for all material models considered here, the material parameters, whichinclude the parameters to adjust the almost incompressibility, have a substantial influence on the con-vergence of the iterative solution method. It is of course well known already in the context of linearelasticity that almost incompressibility can have a major impact on iterative solvers.

Here, we will take a different approach as opposed to [2]. We consider a single material model andinvestigate different ways to incorporate the almost incompressibility constraint. Partial results havealready been reported in a proceedings article [3]. In the classical approach, a penalty enforcing thegeometric constraint that the determinant of the deformation gradient F is close to 1, det(F ) ≈ 1, isadded to the free energy, using a weight ε1 which is referred to as the penalty parameter. Only forε1 approaching infinity, the constraint is fulfilled exactly. Unfortunately, using high penalty parametersresults in ill-conditioned linearized systems which affect the performance of iterative solvers such thatthe convergence rate degrades significantly. For direct solvers, the computed solutions may becomemeaningless.

Alternatively, if the augmented Lagrange approach, see Simo and Taylor [4] as well as Fortin and1

Fig. 1.1. Layered finite element model of an arterial segment discretized using 10-noded tetrahedra; Image from [1].The outer layers show the geometry of the reconstructed media, intima, and plaque (from left to right).

Fortin [5], is taken for the almost incompressibility constraint, in addition to the penalty term, a Lagrangemultiplier µT is introduced on each finite element T and µT (det(F ) − 1) is added element-wise to theenergy ψ.

The Lagrange multiplier will be computed iteratively by an Uzawa-like iteration

µT,k+1 = µT,k + ξk(det (F )− 1), (1.1)

where in our computations in Section 7 the sequence ξk will be chosen as a constant ξk = ξ.We formally collect all local multipliers µT,k in a vector µk = (µT,k)T . The iteration for µk may

be nested with the Newton iteration, see Figure 6.2 in Section 6, or may be carried out simultaneously,see Figure 6.3 in Section 6. In the nested approach (NAL), for each value of µk, a nonlinear system issolved using the Newton iteration until convergence. In the simultaneous approach (SAL), the value ofµk is changed according to (1.1) after each single Newton step. It is thus performed embedded in orsimultaneously with the Newton iteration. Note that due to the F -approach, see Section 4, the element-wise constraint det(F ) = 1 is enforced only in a mean sense to avoid locking effects, i.e., we will enforceθ = 1, for the element-wise scalar-valued variable θ, instead of det(F ) = 1, see Section 4.

There exist domain decomposition solvers that are robust in the almost incompressible linear elasticitysetting, see Klawonn et al. [6], Li and Widlund [7], Rheinbach [8], Dohrmann and Lehoucq [9], andothers. But for recent nonoverlapping algorithms such as FETI-DP and BDDC (Balancing DomainDecomposition by Constraints) methods the coarse spaces becomes quite large, especially in 3D. Newcoarse unknowns have to be introduced for every edge and face of each subdomain. There has been someprogress in reducing the size of the coarse space using a hybrid FETI-DP method [10] in 2D. In [10], asingle additional coarse unknown for each subdomain is sufficient.

Nevertheless, in this paper, we use a FETI-DP method with a coarse space for compressible elasticity.In Brands et al. [2], we have seen that this will result in an acceptable performance even in the nonlinearalmost incompressible setting. In nonlinear almost incompressible elasticity, before convergence, the theviolation of the incompressibility constraint will show local variations. Such a situation does not directlycorrespond to a linear elastic model with a single global Poisson ratio ν for the complete computationaldomain but rather to a varying incompressibility within the domain. Note that recent results for linearelasticity with varying Poisson ratio ν show that all FETI-DP and BDDC methods are robust withrespect to localized almost incompressibility as long as the distance of the incompressible inclusions tothe subdomain boundaries are bounded from below by a constant, see Gippert et al. [11, 12].

2. Continuum Mechanical Foundations. In the reference configuration, the computational do-main, i.e., the body of interest, is indicated by B ⊂ IR3, parametrized in X and the current configurationby S ⊂ IR3, parametrized in x. The non-linear function ϕt : B → S maps the points X ∈ B onto pointsx ∈ S at the time t ∈ IR. Here, the deformation gradient F and the right Cauchy-Green tensor C are

2

defined by

F = Gradϕ and C = F TF (2.1)

with the Jacobian J(X) := det F (X) > 0. We consider hyperelastic materials, where the existence of astrain energy function ψ is postulated and defined per unit reference volume. We focus on energy functionsin the form ψ = ψ(C,M (1),M (2)), where the last two tensorial arguments M (a) := A(a)⊗A(a)|a = 1, 2are the so-called structural tensors. These arguments characterize the anisotropy of the material, wherethe unit directions A(1) and A(2) are approximations of the collagen fiber bundle orientations in theartery. The formulation of anisotropic constitutive functions based on the concept of structural tensorswas first introduced in an attractive way by Boehler [13]. An overview of the theory of representationsof tensor functions can be found in the excellent review Zheng [14].

For the construction of specific constitutive equations we need the invariants of the deformationtensor and the additional structural tensors. The principle invariants of C are

I1 := tr (C), I2 := tr[CofC], I3 := det(C), (2.2)

where Cof(C) = det(C)C−T .For the modeling of the mechanical behavior of the arterial wall we assume that the response can

be approximated by two superimposed transversely isotropic models. This is based on the fact that onlyweak interactions between the two fiber directions are observed; for further arguments, see Holzapfelet al. [15]. Therefore, the additional (mixed) invariants of interest are

J(a)4 := tr[CM (a)], J

(a)5 := tr[C2M (a)] for a = 1, 2 , (2.3)

see, e.g., [16] and the references therein. The set of differential equations for the underlying boundaryvalue problem is governed by the local form of the balance of linear momentum for the quasi-static case

Div(P ) + f0 = 0 with P = FS , (2.4)

here, P characterizes the 1st Piola-Kirchhoff stress tensor. The 2nd Piola-Kirchhoff stress tensor iscomputed by S = 2∂Cψ and f0 is the body force vector.

3. Free Energy Functions for Soft Biological Tissue. Many biological tissues, such as arteries,are fiber enforced materials composed of an almost incompressible matrix substance with embeddedcollagen fibers. The arrangement of the fibers in arterial walls is characterized by two preferred directionshelically wound along the artery. The material behavior of the collagen fiber bundles is represented bythe superposition of two transversely isotropic models; see Holzapfel et al. [15]. Thus, the strain energiesare given by

ψ = ψiso(C) + ψti(C,M (1)) + ψti(C,M (2)) . (3.1)

There exist different possibilities to model the mechanical response of soft biological tissue; see, e.g.,[17, 15]. We are interested in polyconvex energy functions. For the construction of anisotropic, polyconvexfunctions to ensure the existence of minimizers, see, e.g., [18]. Here, we use the model due to [15], whichwas denoted model ψB in [2],

ψ = c1

(I1I

−1/33 − 3

)+

2∑a=1

k1

2k2

{exp

(k2

⟨J

(a)4 I

−1/33 − 1

⟩2)− 1

}(3.2)

+ ε1(Iε23 + I−ε23 − 2

)α.

3

0

0.5

1

1.5

2

0.98 0.985 0.99 0.995 1 1.005 1.01 1.015 1.02

ψP (I3)

I3

ε1 = 360;ε2 = 9

ε1 = 10;ε2 = 4

Fig. 3.1. Comparison of the penalty function ψP (I3) = ε1(Iε23 + I−ε2

3 − 2) for the two sets of parameters for themedia; see Table 3.1.

Model Layer c1 ε1 ε2 α k1 k2 βf

[kPa] [kPa] [-] [-] [kPa] [-] [◦]

adv. 7.17459 70.0 8.5 1 3.6865e-3 51.1512 54.7ψ Penaltymed. 9.22575 360.0 9.0 1 192.857 2626.84 43.9

adv. 7.17459 10.0 4.0 1 3.6865e-3 51.1512 54.7ψ Augmentedmed. 9.22575 10.0 4.0 1 192.857 2626.84 43.9

Table 3.1The parameter fit is performed under the assumption of incompressibility (det(F ) = 1). The penalty parameters ε1

and ε2 are not fitted but chosen a posteriori such that a good fit is still obtained; see Figure 3.2. The parameter βf is theangle between the fibers and the circumferential direction.

Here 〈b〉 denotes the Macauley brackets defined by 〈b〉 = (|b|+ b)/2, with b ∈ IR. The penalty term ψP =ε1

(Iε23 + I−ε23 − 2

)αwith the penalty parameters ε1 and ε2, models the incompressibility by penalizing

deviations from the constraint 1 = I3 = det C = (det(F ))2; see also Figure 3.1.The 4th-order elasticity tensor can be computed from ψ using

C = 4∂2ψ

∂C∂C= 4

∂

∂C

(∑

i

∂ψ

∂Li

Li∂C

)= 4

∑

i

(∂2ψ

∂2Li

∂Li∂C

⊗ ∂Li∂C

+∂ψ

∂Li

∂2Li∂C∂C

),

with the invariants {Li|1, 7} ={I1, I2, I3, J

(1)4 , J

(2)4 , J

(1)5 , J

(2)5

}. One of the contributions to the stiffness

matrix steming from the penalty term ψP is therefore

∂ψP∂I3

= ε1α(Iε23 + I−ε23 − 2

)α−1ε2

(Iε2−13 − I−ε2−1

3

)(3.3)

and

∂2ψP∂2I3

= ε1α(α− 1)(Iε23 + I−ε23 − 2

)α−2(ε2

(Iε2−13 − I−ε2−1

3

) )2

+ε1α(Iε23 + I−ε23 − 2

)α−1ε2

((ε2 − 1) Iε2−2

3 + (ε2 + 1)I−ε2−23

).

(3.4)

We can see that the influence of this term grows linearly with ε1 and exponentially with ε2. The conditionnumber of the stiffness matrix may deteriorate accordingly.

4

We consider an abdominal aorta and adjust our function to this biological material. We assume thatthe mechanical influence of the intima in atherosclerotic arteries is negligible. In the healthy part of theartery it is therefore neglected and in the diseased part it is considered as part of the plaque. In order tofind the parameters for our model, we adjust the stresses from the material models to the experimentalstresses in Holzapfel [19], where separated layers of an abdominal aorta were studied in uniaxial tensiontests. We have chosen to use a different adjustment strategy than in our earlier publication [2]. Here, thefit of the parameters is performed under the assumption that the material is incompressible. We thereforehave to enforce the incompressibility constraint separately, i.e., when using a penalty approach we haveto chose sufficiently large penalty parameters, see Table 3.1. In the augmented Lagrange approach, asufficiently accurate stopping criterion has to be chosen for the augmented Lagrange loop, i.e., we haveto chose a sufficiently small TOL, see Figure 6.2 and 6.3. The results of the parameter adjustment forthe material models are summarized in Table 3.1. The plaque is modelled, as in [2], as an isotropicMooney–Rivlin material,

ψMooney−Rivlin = β1I1 + η1I2 + δ1I3 − δ2 ln I3 , (3.5)

where β1 = 80.0 kPa, η1 = 250.0 kPa and δ1 = 2000.0 kPa. To obtain a stress-free state in the referenceconfiguration the parameter δ2 has to satisfy δ2 = β1 + 2η1 + δ1.

0

10

20

30

40

50

60

70

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4

axial(exp)axial(comp)circum(exp)

circum(comp)

ψ-Adv.

λ

σ

0

20

40

60

80

100

120

140

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35

axial(exp)axial(comp)circum(exp)

circum(comp)

ψ-Media

λ

σ

Fig. 3.2. Data fits while assuming incompressibilty. For the resulting parameters, see Table 3.1.

4. The Three-Field Method. In order to treat the quasi-incompressible material, we relax thepoint-wise incompressibility constraint and only penalize volumetric changes in a mean sense on every10-noded tetrahedral finite element. This is accomplished, as in [2], by applying a three-field formulation,known as the F -approach; see, e.g., Simo [20, Section 45].

Let J = det(F ), then we can introduce F as

F = J1/3F , F = J−1/3F .

From this definition, we clearly have det(F ) = 1. We then introduce a new, element-wise scalar variableθ, such that θ = J will be satisfied on every finite element in a mean sense, and define

F := θ1/3F , C := F T F

with F = F (ϕ, θ), C = C(ϕ, θ). We consider the following three-field Lagrangian using the multiplier π

ÃL(ϕ, θ, π) =∫

Bψ(C(ϕ, θ)) + π(J(ϕ)− θ)dx− Vext(ϕ),

where Vext(ϕ) is the potential energy of external forces; for more details, see Simo [20, Section 45]. Wethen choose a P2 − P0 − P0 mixed finite element discretization for ϕ, θ, and π, i.e., piecewise quadratic

5

elements for the deformation field ϕ and piecewise constant elements for the scalar fields θ and π. Staticcondensation of θ and π, which can be performed locally on each finite element, leads to a reducedproblem that we will then solve by a Newton iteration. In the case of the augmented Lagrange methodsour functional is

ÃL(ϕ, θ, π, µ) =∫

Bψ(C(ϕ, θ)) + π(J(ϕ)− θ)dx− Vext(ϕ) +

∫

Bµ(θ − 1)dx.

which is discretized by a P2 − P0 − P0 − P0-approach. Again θ and π are statically condensated. TheLagrange multiplier field µ is found using an Uzawa-like iteration; see Figures 6.2 and 6.3.

5. FETI-DP Method. Domain decomposition methods are divide-et-impera algorithms wherean approximate inverse is constructed from solving small problems on subdomains and a small globalproblem. Here, we use as a nonoverlapping domain decomposition method, see Figure 7.2, a member of thewell known family of FETI-DP methods; see [21, 22, 23, 24, 25, 26, 27] and the references therein. Domaindecomposition methods are naturally parallel algorithms, for a FETI-DP method parallel scalability hasbeen demonstrated for more than 65.000 processors cores [28].

Our approach is a classical Newton-Krylov-FETI-DP approach, i.e., we linearize first and then usethe FETI-DP domain decomposition method to solve the linearized systems. The augmented Lagrangeapproach may be then characterized as a augmented-Lagrange-Newton-Krylov-FETI-DP method.

FETI methods have been used to solve large structural mechanics problems on massively parallelmachines; see [29, 30, 26, 28]. An introduction to domain decomposition methods is given in the books [31,32, 33]. Finite Element Tearing and Interconnecting (FETI) domain decomposition methods were firstintroduced by Farhat and Roux [34]. The more recent FETI-DP methods were first introduced in Farhatet al. [21, 22] and further developed, e.g., in [23, 24, 25, 26].

In FETI-DP methods the computational domain B is partitioned into nonoverlapping subdomainsBi where one or several subdomains are assigned to each processor; see Figure 7.2. Each subdomainBi has a typical diameter H and its boundary is denoted by ∂Bi. Moreover, Bi, i = 1, . . . , N , is theunion of finite elements with matching finite element nodes on the boundaries of neighboring subdomainsacross the interface Γ :=

⋃i 6=j ∂Bi ∩ ∂Bj . We use a graph partitioner [35] to define the decomposition

of the body into subdomains. Note that in our present algorithm the definition of the subdomains is acompletely algebraic process and does not need access to geometric data. We make use of the weak formof the balance of linear momentum on each subdomain which is, neglecting body forces, given by

G =∫

Bi

δF : P (F ) dV −∫

∂Bi,σ

δu · t dA , (5.1)

with the traction vector t = PN acting on the Neumann boundary ∂Bi,σ, the outward unit normal Non ∂Bi,σ, the virtual displacement field δu, and the virtual deformation gradient δF .

For each subdomain Bi, we assemble the local stiffness matrices K(i) and load vectors f (i), i =1, . . . , N . We can write the linearized subdomain problems as

K(i)(∆D(i)) = f (i), (5.2)

where ∆D(i) denotes the local displacement increment on subdomain Bi in the current Newton step, seeSection 7.

These local problems will in general not have a unique solution as they may lack essential Dirichletboundary conditions. This is particularly the case for all interior subdomains, i.e., subdomains Bi where∂Bi ∩ ∂B = ∅.

Let us define the interface by Γ =N⋃

i=1

∂Bi \ ∂B. If we view the N local problems in (5.2) as N

independent problems then the solutions will be discontinuous across the interface Γ.6

Writing the subdomain matrices and right hand sides block form, we obtain

K =

K(1)

. . .K(N)

, ∆D =

∆D(1)

...∆D(N)

, f =

f (1)

...f (N)

.

The discrete problem can be formulated as minimization problem continuity constraint Bu = 0 onthe interface Γ. The matrices B = [B(1), . . . ,B(N)] have entries from 0, 1,−1. We introduce Lagrangemultipliers λ to enforce the continuity along the subdomain interface Γ and obtain the problem:

Find (∆D, λ), such that

{K (∆D) + BTλ = fB (∆D) = 0 .

(5.3)

This problem can be solved by eliminating the displacement variables ∆D and solving the resultingSchur complement system by a Krylov subspace method. In this paper we use GMRES as our Krylovsubspace method. Note that the iteration is performed only on the interface, which significantly reducesthe memory requirements for the GMRES method.

In FETI-DP methods some continuity constraints on primal displacement variables ∆DΠ are enforcedthroughout iterations. The local subdomain problems are invertible if a sufficient number of primalconstraints are chosen. These primal constraints also constitute a coarse problem for the method makingit numerically scalable.

After incorporating the primal constraints we obtain the saddle point system[

K BT

B 0

] [∆Dλ

]=

[f0

], (5.4)

where the matrix K and right hand side f are partially coupled in the primal variables,

K =

K(1)BB K

(1)T

ΠB

. . ....

K(N)BB K

(N)T

ΠB

K(1)

ΠB · · · K(N)

ΠB KΠΠ

, f =

f(1)B...

f(N)B

fΠ

.

Choosing a sufficient number of primal variables the matrix K becomes positive definite. We cannow reduce the system of equations to an equation in λ. It remains to solve

F fetiλ = d, (5.5)

where F feti = BK−1

BT .To precondition F feti , we use the classical Dirichlet preconditioner

M−1 := BDRTΓSRΓBT

D, (5.6)

where S is the Schur complement obtained by eliminating the interior variables in every subdomain, i.e.,

S =

S(1)

. . .S(N)

.

7

Here, RΓ is a restriction matrix, consisting of zeros and ones, that, when applied to a vector ∆D, removesthe interior variables from ∆D. The matrices BD are scaled variants of the jump operator B where,in the simplest case, the contribution from and to each interface node is scaled by the inverse of themultiplicity of the node. We define the multiplicity of a node as the number of subdomains it belongs to.For heterogeneous problems a more elaborate scaling, using an appropriate scaling factor, defined by thecoefficients ρi, is necessary; see, e.g., [23, p. 1532, Formula (4.3)] and [36, p. 1403, Formula (6)].

6. Algorithms. In our nonlinear scheme we solve a sequence of linear problems obtained fromNewton’s method, see, e.g., Figure 6.1. This is also referred to as (pseudo) time stepping or load stepping.To obtain a fair comparison between the different approaches, we have chosen a well-known automatictime stepping strategy. We increase ∆t when the number of Newton iterations is smaller than a certainthreshold and decrease ∆t if it is larger than a given threshold.

The simultaneous augmented Lagrange (SAL) approach, where the iteration for the Lagrange multi-plier is performed simultaneously with the Newton correction, can be viewed as an inexact Newton-likemethod. Thus, a quadratic convergence rate cannot be expected. Here, we have to chose the boundsfor the auto time stepping more generously. For all approaches the maximal time step size was boundedfrom above by a certain value. In Figures 6.1, 6.2, and 6.3, we describe the algorithms in detail.

7. Application and Numerical Results. In this section we report on simulations using the finiteelement model of an arterial segment with an embedded plaque, see Figure 7.1, obtained from IVUSultra sound data; see [37, 1]. The discretization uses 10-noded tetrahedral elements which results in1.3M degrees of freedom for the displacement. A significant portion of the unknowns is located insidethe plaque component. We have 300K tetrahedra generating 300K unknowns for the Lagrange multiplierµ in the augmented Lagrange method. A pressure of up to 33.33 kPa (≈ 250 mmHg) is applied to theinterior of the arterial segment.

For the computations we apply our simulation environment consisting of the Finite Element ProgramFEAP of R. L. Taylor, University of California, and our parallel implementation of the Finite ElementTearing and Interconnecting - Dual Primal method (FETI-DP) to simulate our 3D problems. The com-putations were performed on an AMD Opteron compute server with 16 cores at 2.5 Ghz. For our parallelsolver we use MPI, PETSc [38, 39, 40], UMFPACK [41] and the ACML.

The residual norm of the Newton step is computed as the Euclidean norm of the right hand sidevector. The stopping criterion for the Newton method is either the reduction of the absolute residualnorm to a value less than 10−5 or an absolute residual norm of less than 10−4 for three consecutiveNewton steps.

The size of the incremental load steps (pseudo time) is controlled by the automatic time steppingscheme; see Section 6. For the penalty approach we increase ∆t when the number of Newton iterationsis smaller than 6 and decrease ∆t when it is larger than 9. This choice produced the best results.In the nested augmented Lagrange approach (NAL), for the auto time stepping, we only consider thefirst Newton iteration within each augmented Lagrange iteration since this is the critical step. Thelater Newton steps typically converge faster compared to the first step. In the simultaneous augmentedLagrange (SAL) approach the bounds for the auto time stepping are chosen much larger, i.e., as 18 and36, see the discussion in Section 6. In all cases the maximal time step size was bounded from above by∆tmax = 0.4.

The parameter ξk = ξ is chosen by hand from trial and error using a small finite element model(SAL, TOL=10−2: ξ = 5.0 × 102; SAL, TOL=10−3: ξ = 1.0 × 103; NAL: ξ = 5.0 × 102). In this sense,our results are not sharp but rather give an upper bound on the performance of the augmented Lagrangeapproaches.

For the FETI-DP domain decomposition method, we use a decomposition into 224 subdomains; seeFigure 7.2. The continuity constraint in the FETI-DP method generates 400K unknows for the Lagrangemultiplier λ, see (5.4) and Figure 7.2. The FETI-DP Krylov iteration is stopped when the absoluteresidual is reduced to 5× 10−9.

8

The total cost of the computation can roughly be estimated by multiplying the number of globalNewton steps by the corresponding average number of (inner) FETI-DP iterations, see Tables 7.1, 7.2, 7.4,and 7.3. In Table 7.5 the data is collected in compact form for comparison.

Penalty

t Newton steps ∅ FETI-DP its

0.010 9 172.20.020 5 173.00.036 5 175.80.061 5 179.40.101 6 189.30.141 5 187.00.204 6 201.80.267 5 195.60.367 7 208.00.467 7 204.10.567 5 207.40.725 6 217.80.884 5 225.41.135 6 242.01.386 6 253.81.637 7 266.31.889 5 279.42.287 6 301.02.500 6 312.5

Σ 112 Total ∅ 220.8Table 7.1

Newton iteration for the penalty formulation (P). Pseudo-time t, number of Newton steps, average number of Kryloviterations per Newton step. TOL=10−2. Simulation until 33.33 kPa (250 mmHg = Pseudotime 2.5).

Our results show that the use of the augmented Lagrange method can significantly improve theproperties of the linearized systems occurring in the nonlinear solution scheme, i.e., the FETI-DP iterativemethod converges in a lower number of iterations. The convergence of the nonlinear scheme is alsoimproved, i.e., in our nonlinear scheme larger pseudo time steps ∆t can be chosen. Of course, an additionaliteration process for the Lagrange multiplier is introduced.

The comparison of Tables 7.1 and 7.2 show that in case of the simultaneous augmented Lagrangeapproach the additional cost for the augmented Lagrange iteration is more than amortized by the fasterconvergence of the nonlinear scheme and the linear iterative solver. The nested augmented Lagrangeapproach has a considerably higher cost; see Table 7.3.

Clearly, the linear systems resulting from the augmented Lagrange approaches are better conditioned.We should note here that ill-conditioning not only influences the convergence of iterative solvers but italso affects direct solvers, i.e., the quality of the solution will deteriorate.

Moreover, in the augmented Lagrange approaches the volumetric change is exactly controlled duringthe iteration process, i.e., we have satisfied element-wise the condition | det(F )− 1| ≤ 10−2 in Tables 7.2and 7.3. In the penalty approach the volumetric change produced by the chosen penalty parametersis only known a posteriori. In our example in Table 7.1 the solution using the penalty approach onlysatisfies |det(F )−1| ≤ 0.02084; see Table 7.5. In the augmented Lagrange approach, for a slightly highercost, we can also enforce | det(F )− 1| ≤ 10−3 by setting TOL=10−3; cf. Table 7.4 and 7.5.

9

Simultaneous AL, TOL=10−2

t Newton/AL steps ∅ FETI-DP its

0.010 9 99.30.026 4 100.50.051 5 101.40.091 5 102.40.154 6 104.20.254 7 105.30.412 9 109.10.664 13 118.81.062 14 138.21.462 17 158.81.862 18 179.72.261 18 202.52.500 16 215.9

Σ 141 Total ∅ 150.2Table 7.2

Simultaneous augmented Lagrange (SAL) iteration. Pseudo-time t, number of Newton/AL steps, average number ofKrylov iterations per Newton/AL step. TOL=10−2, ξ = 5.0× 102. Simulation until 33.33 kPa (250 mmHg = Pseudotime2.5).

Nested AL, TOL=10−2

t Newton steps ∅ FETI-DP its

0.010 10 99.20.020 5 99.80.036 6 101.20.061 6 102.20.101 10 103.10.164 18 104.00.227 18 104.10.327 23 106.00.427 25 109.00.585 53 115.40.744 40 122.20.902 42 130.71.061 40 138.91.312 45 151.61.563 49 163.71.815 49 182.52.066 57 191.42.317 65 206.22.500 57 216.6

Σ 618 Total ∅ 154.0Table 7.3

Nested augmented Lagrange (NAL) iteration. Pseudo-time t, number of Newton steps, average number of Kryloviterations per Newton/AL step. TOL=10−2, ξ = 5.0× 102. Simulation until 33.33 kPa (250 mmHg = Pseudotime 2.5).

10

Simultaneous AL, TOL=10−3

t Newton/AL steps ∅ FETI-DP its

0.010 9 99.30.026 6 100.30.051 8 101.30.091 10 102.20.154 15 103.00.254 13 104.20.412 15 108.90.664 18 119.81.062 37 143.51.462 38 164.61.862 37 188.12.261 43 212.32.500 35 233.1

Σ 284 Total ∅ 161.4Table 7.4

Simultaneous augmented Lagrange (SAL) iteration. Pseudo-time t, number of Newton/AL steps, average number ofKrylov iterations per Newton/AL step. TOL=10−3, ξ = 1.0× 103. Simulation until 33.33 kPa (250 mmHg = Pseudotime2.5).

Load steps Global Newton ∅ FETI-DP it min(det F ) max(det F )

Penalty 19 112 220.8 0.99265 1.02084(min = 170,max = 316)

TOL=10−2

Simult. AL (SAL) 13 141 150.2 0.99006 1.00997(min 98, 217)

Nested AL (NAL) 19 618 154.0 0.99039 1.00997(min = 98,max = 427)

TOL=10−3

Simult. AL (SAL) 13 248 161.4 0.99900 1.00099(min = 98,max = 368)

Table 7.5Comparison of the Methods. We can estimate the total cost roughly by multiplying the number of global Newton

iterations by the average number of FETI-DP iterations.

In the results in Tables 7.2, 7.3, and 7.4, we see that the number of Newton/AL-iterations increasesduring the simulation. This may be due to the fact that in the beginning of the simulation only a verysmall number of finite elements violate the element-wise condition | det(F )− 1| ≤ 10−2 and the numberof such elements increases during the simulation.

The results in Tables 7.1, 7.2, 7.3, and 7.4 also show an increase of the FETI-DP iterations duringthe simulation. We believe that this may in part be due to an increasing influence of the incompressibilityconstraint during the simulation but also results from the exponential stiffening behavior of the fibers. In[42], we have observed that the anisotropies introduced to the material wall models by the terms modeling

11

the fibers can have a visible impact on the convergence of the nonlinear iteration scheme as well as theconvergence of the iterative linear solver. Ideas described in [28] may improve the convergence of domaindecomposition solvers for such anisotropic problems.

For completeness we also report on the parallel scalability of the FETI-DP method obtained for asingle linearized system on a Cray XT6m. We use the arterial wall modell depicted in Figure 7.1 with1.3M degrees of freedom. Here, the model was decomposed into 112 subdomains which was more suitablefor the Cray supercomputer as opposed to the Opteron compute server used for the other computations.

Strong scalability benchmark from an arterial wall structureCray XT6m at CCSS, Universitat Duisburg-Essen

Cores It. Time7 99 400s14 99 231s28 99 79s56 99 47s112 99 34s

Table 7.6Strong scalability benchmark on the Cray XT6m at Universitat Duisburg-Essen: Parallel FETI-DP solver applied

to a single linear problem from the nonlinear simulation of an arterial wall structure with 1.3M displacement variables.Decomposition into 112 subdomains. In this benchmark, the iteration was stopped when the residual was reduced by afactor of 1e07.

Acknowledgements: The work of the authors was financially supported by the Deutsche Forschungs-gemeinschaft (DFG) under research grants KL 2094/2 and SCHR 570/7. We also acknowledge thescientific support by Prof. Dr. med. R. Erbel, Dr. med. D. Bose (both Universitatsklinikum Essen). Weacknowledge the use of the Cray XT6m supercomputer at Universitat Duisburg-Essen.

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Nonlinear Iteration (Penalty)

Set k = 0 and t0 = ∆t0;

Apply partial load tk · fload if full load not yet reached;

Use Newton iteration to solve the nonlinear problem.

Solve linearized problem by the FETI-DP method using GMRES;

Apply Newton correction;

Adapt load step size ∆tk+1, i.e.,

∆tk+1 = 101/5∆tk, ∆tk+1 = 10−1/5∆tk, or ∆tk+1 = ∆tk;

Set tk+1 = tk + ∆tk+1;

Fig. 6.1. Using the penalty approach for the incompressibility.

Nonlinear Iteration (Nested Augmented Lagrange)

Set k = 0 and t0 = ∆t0;

Apply partial load tk · fload if full load not yet reached;

Set Lagrange multiplier µ0 = 0;

Do

Use Newton iteration to solve the nonlinear problem for fixed

multiplier µk;

Solve linearized problem by the FETI-DP method using

GMRES;

Apply Newton correction;

Update Lagrange parameter for all elements T that violate the

condition |θT − 1| ≥ TOL, i.e., µT,k+1 = µT,k + ξk(θT − 1);

While elements T with |θT − 1| ≥ TOL exist;

Adapt load step size ∆tk+1, i.e., ∆tk+1 = 101/5∆tk,

∆tk+1 = 10−1/5∆tk, or ∆tk+1 = ∆tk;

Set tk+1 = tk + ∆tk+1;

Fig. 6.2. Nested augmented Lagrange (NAL) for the incompressibility [4, 5].

15

Nonlinear Iteration (Simultaneous Augmented Lagrange)

Set k = 0 and t0 = ∆t0;

Apply partial load tk · fload if full load not yet reached;

Set Lagrange multiplier µ0 = 0;

While Newton iteration has not converged and while elements

T with |θT − 1| ≥ TOL exist: Solve nonlinear problem with

simultaneous Newton iteration and iteration for µ;

Solve linearized problem by the FETI-DP method using GMRES;

Apply Newton correction and update Lagrange parameter for all

elements T that violate the condition |θT − 1| ≥ TOL, i.e.,

µT,k+1 = µT,k + ξk(θT − 1);

Adapt load step size ∆tk+1, i.e.,

∆tk+1 = 101/5∆tk, ∆tk+1 = 10−1/5∆tk, or ∆tk+1 = ∆tk .

Set tk+1 = tk + ∆tk+1;

Fig. 6.3. Simultaneous augmented Lagrange (SAL) for the incompressibility [4, 5].

Fig. 7.1. Cut through the atherosclerotic arterial segment with the embedded plaque; geometry obtained from IVUSdata, see Figure 1.1 and [1]. The finite element model has 1.3M displacement unknowns and 300K unknowns for theLagrange multiplier µ in the augmented Lagrange method.

Fig. 7.2. Exploded view of the nonoverlapping domain decomposition of the arterial segment into 224 nonoverlap-ping subdomains.

16