Computational modeling of growthA critical review, a classification of concepts and two new consistent approaches

E. Kuhl, A. Menzel, P. Steinmann

Abstract The present contribution is dedicated to thecomputational modeling of growth phenomena typicallyencountered in modern biomechanical applications. We setthe basis by critically reviewing the relevant literature andclassifying the existing models. Next, we introduce a geo-metrically exact continuum model of growth which is not apriori restricted to applications in hard tissue biomecha-nics. The initial boundary value problem of biomechanics isprimarily governed by the density and the deformationproblem which render a nonlinear coupled system ofequations in terms of the balance of mass and momentum.To ensure unconditional stability of the required timeintegration procedure, we apply the classical implicit Eulerbackward method. For the spatial discretization, we suggesttwo alternative strategies, a node-based and an integrationpoint–based approach. While for the former, the discretebalance of mass and momentum are solved simultaneouslyon the global level, the latter is typically related to a stag-gered solution with the density treated as internal variable.The resulting algorithms of the alternative solution tech-niques are compared in terms of stability, uniqueness,efficiency and robustness. To illustrate their basic features,we elaborate two academic model problems and a typicalbenchmark example from the field of biomechanics.

Keywords Growth, Bone remodeling, Finite elementtechnologies, Stability, Convergence

1IntroductionSince the formulation of the first continuum model ofgrowth which was presented more than a quarter of acentury ago by Cowin and Hegedus [8], the modeling andsimulation of biomechanical processes has experienced anenormously growing interest. In contrast to traditionalengineering materials, biomaterials, in particular hard andsoft tissues, show the ability to adapt not only theirexternal shape but also their internal microstructure toenvironmental changes. The functional adaption of hardtissues to changes in the mechanical loading situation hasbeen known for more than a century and is often referredto as Wolff’s law of bone remodeling [53]. Comprehensive

overviews on the experimental findings of growth phe-nomena can be found e.g. in the monographs of Pauwels[45], Fung [12, 13], Taber [48], Carter and Beaupre [2] andHumphrey [27].

The first continuum theory of growth for hard tissueshas been presented under the name of theory of adaptiveelasticity by Cowin and Hegedus [8]. Within their theory,the biological structure is considered as an open systemwhich is allowed to constantly exchange mass, momentum,energy and entropy with its environment. While thisexchange is accounted for exclusively in terms of volumesource terms in the original model, the more enhancedmodel by Epstein and Maugin [10] additionally allows foran exchange in terms of surface fluxes, see also Kuhl andSteinmann [31]. Thereby, the flux of mass is typicallyattributed to the migration of cells while mass sourcesstem from cell growth and shrinkage, cell death, celldivision or cell enlargement. A completely different ap-proach to hard and soft tissue mechanics falls within theframework of the theory of porous media, see e.g. Ehlersand Markert [9], Kuhn and Hauger [36], Humphrey andRajagopal [27] or Steeb and Diebels [47]. Thereby, theexchange of mass, momentum, energy and entropy takesplace between the individual constituents of the mixture,while the mass, momentum, energy and entropy of theoverall mixture remain constant. Nevertheless, since themechanical behavior of hard tissues is primarily governedby the response of the calcified bone matrix, we shallconfine attention to the solid phase alone and make use ofthe open system framework in the sequel.

Driven by the development of modern computer tech-nologies, the newly derived theories of growth were soonsupplemented by finite element based numerical simula-tions. Although the initial class of models suffered fromnumerical instabilities, the first results in the area of boneremodeling were quite encouraging, see e.g. Huiskes et al.[24, 26], Carter et al. [4], Beaupre et al. [1], Weinans et al.[49, 50] or Harrigan and Hamilton [17, 18] or the recentoverview by Hart [21]. However, these first models werebasically restricted to the mechanics of hard tissues suchas bones. In contrast to hard tissues, which are typicallynot subjected to large strains, the modeling of soft tissuesrequires a geometrically nonlinear kinematic description,see e.g. the first publication by Rodriguez et al. [46] or therecent works of Cowin [6], Holzapfel et al. [22], Chen andHoger [5], Gasser and Holzapfel [14] and Lubarda andHoger [40].

The first attempts of our own group to simulate bio-logical growth processes within the geometrically exact

Computational Mechanics 32 (2003) 71–88 � Springer-Verlag 2003

DOI 10.1007/s00466-003-0463-y

71

Received: 18 October 2002 / Accepted: 27 May 2003

E. Kuhl, A. Menzel, P. Steinmann (&)Chair of Applied Mechanics, University of Kaiserslautern,P.O. Box 3049, D-67653 Kaiserslautern, Germanye-mail: ps@rhrk.uni-kl.de

framework are documented in Kuhl and Steinmann [31–35]. In the present work, we focus on the comparison ofdifferent finite element techniques for biomechanicalgrowth processes. In particular, two alternative classes ofmodels will be derived and elaborated: a node-based andan integration point–based approach. While the former isbased on the monolithic solution of the balance of massand momentum for open systems and offers the potentialto incorporate a mass flux, the latter is based on astaggered solution strategy by introducing the density asinternal variable on the quadrature point level. In view ofmodern finite element technologies, both approaches willbe combined with a fully implicit Euler backward timeintegration scheme and are embedded in an incrementaliterative Newton-Raphson solution technique supple-mented by a consistent linearization of the governingequations. Rather than analyzing the features of the sug-gested approaches in fully three-dimensional applicationsas done by Kuhl and Steinmann [34], we shall focus on thesystematic study of one-dimensional model problemsand on the analysis of the classical two-dimensionalbenchmark problem of the proxima femur in the presentcontribution.

This contribution is organized as follows. We set thestage by reviewing the existing literature on finite elementbased numerical modeling of growth in Chapter 2. Chapter3 then introduces the basic equations of growth within theframework of finite strain kinematics. The correspondingfinite element formulation is derived in Chapters 4 and 5,whereby the former focuses on a node-based approachwhile the latter is characterized through an integrationpoint–based treatment of the density. Both finite elementchapters incorporate the spatial discretization, the con-sistent linearization, a characteristic flow-chart of thealgorithm and the discussion of two academic modelproblems. The applicability of the two different classes ofsolution strategies to a classical example from biome-chanics is studied in Chapter 6. Chapter 7 presents a finaldiscussion of the derived results.

2Different remodeling algorithms revisedWe begin our study with a review of the existing literatureon the computational modeling of growth which datesback to the mid eighties. Surprisingly, an enormous bodyof literature is related to the analysis of stability anduniqueness of the suggested models. Often, these proper-ties are attributed to the final finite element solution al-though they typically originate from the ill-posedness ofthe underlying continuous problem. To clarify the notionsof stability and uniqueness, we shall make a cleardistinction between the continuous problem, the time-discrete problem and the fully discrete finite elementproblem.

2.1Stability and uniqueness of continuous modelThe modeling of growth is basically characterized throughtwo fundamental equations: the balance of mass and thebalance of momentum. While the computational treatmentof the balance of momentum, i.e. the deformation

problem, is rather standard in modern finite elementtechnologies, the numerical solution of the balance ofmass, i.e. the density problem, can be carried out in var-ious different ways. Conceptually speaking, the evolutionof the material density has to be calculated according tosome driving force, the so-called biological stimulus. Afteranalyzing different mechanically induced stimuli, e.g.based on the current strain [8], the actual stress state [4] orthe dissipated energy in the form of damage [39], it wasagreed upon, that the free energy density of the calcifiedtissue represented the most reasonable candidate. [29, 30,49, 50]. However, it soon turned out, that the use of thefree energy density alone resulted in instabilities due to thepositive feedback effect it had on the remodeling process.As documented by Carter et al. [4], Harrigan and Hamil-ton [16–19], Harrigan et al. [20] and Weinans et al. [49] itproduced unstable 0–1-type solutions of either no bone orfully dense calcified bone which were physically mean-ingless. Several different remedies were suggested, e.g. theintroduction of a so-called dead-zone [1, 2, 50], the non-local averaging based on spatial influence functions [41,42], the a posteriori smoothing of either the mechanicalstimulus or the solution itself [11, 28, 43, 44] or the use ofmore sophisticated micromechanically motivated models[23, 25]. Nevertheless, the simplest and maybe also mostefficient modification was motivated by pure mathematicalreasoning. In a rigorous analysis on stability and unique-ness, Harrigan and Hamilton [17, 18] derived a modifiedremodeling rate equation which a priori guaranteed stable,unique and path-independent solutions for a particularchoice of parameters. We shall thus apply a biomechanicalstimulus in the sense of Harrigan and Hamilton in thesequel.

2.2Stability and uniqueness of semi-discrete modelThe resulting set of equations defines the continuousinitial boundary value problem which has to be discretizedin time and space. Traditionally, finite difference schemesare applied for the temporal discretization. Motivated bythe computational efficiency of explicit time integrationschemes, the first models were based on the classical ex-plicit Euler forward method, see e.g. [1, 4, 26, 49, 50]. Incontrast to the unconditionally stable implicit Eulerbackward scheme, the Euler forward method is only con-ditionally stable and poses restrictions on the time stepsize. These were discussed in detail by Levenston [37, 38]and Cowin et al. [7]. However, we do not believe, that theuse of an explicit time marching scheme is the source,at least not the only one, for the particular instabilitiesreported in the literature. Nevertheless, to avoid thispotential drawback of explicit schemes, Harrigan andHamilton [17] suggested the use of the Euler backwardmethod which was also applied by Nackenhorst [43, 44]and will be the method of choice for our furtherdeviations.

2.3Stability and uniqueness of fully discrete modelThe temporal discretization renders a highly nonlinearsemi-discrete coupled system of equations. Its spatial

72

discretization is usually embedded in the finite elementframework. Upon all the existing finite element solutionstrategies, we can basically distinguish three different ap-proaches to evaluate the balance of mass and momentum:a partitioned sequential solution, the partitioned staggeredsolution and the monolithic simultaneous solution. Mostof the pioneering work falls within the first category, seee.g. Huiskes et al. [24, 26], Carter et al. [4], Beaupre et al.[1] or Weinans et al. [49, 50]. Driven by the need forcomputational efficiency that was a relevant issue in themid eighties, the solution was traditionally carried out in atwo-step-strategy. First, the balance of momentum wassolved in the classical way. Only then, the density wasdetermined in a mere post-processing step. The constitu-tive properties were adjusted according to the currentdensity and a new set of material parameters was assignedto each element before the next iteration step was carriedout. The structure of the applied commercial finite elementcodes basically prescribed the algorithmic setup. Since thedensity was assumed to be elementwise constant withinthis algorithm, this first strategy has been termed element-based approach in the literature.

Unfortunately, most of the simulations of the element-based approach produced unphysical checkerboard-typesolutions which were attributed to the fact that the densitywas interpolated in a C�1-continuous fashion. This defi-ciency of the original models led to the reconsideration ofthe finite element realization in the mid nineties. In asystematic case study, Jacobs et al. [28] compared the earlyelement-based approach with an integration point-basedapproach and a node-based approach. For the integrationpoint-based approach, the balance of mass and momen-tum are still elaborated in a partitioned way, however,their evaluation is carried out in a staggered sense, see alsoWeng [52]. In the integration point-based approach, whichoriginates back to the computational modeling of inelas-ticity, the density is introduced as an internal variable onthe integration point level. Nevertheless, since the inte-gration point-based approach is still based on a discretepointwise representation of the density field, it did notyield any remarkable improvements.

If the density field is assumed to be at least C0-con-tinuous within the entire domain of consideration, thedensity has to be introduced as nodal degree of freedomon the global level. According to Jacobs et al. [28] andFischer et al. [11], only this node-based approach couldguarantee a physically meaningful solution. In completeanalogy to the temperature field in thermo-elastic prob-lems, the density is treated as an independent field whichcan be determined simultaneously with the deformationfield. The monolithic solution of the balance of mass andmomentum is computationally more expensive; thus it isnot surprising that this solution strategy only becameprominent after sufficient computational facilities hadbecome available.

Within the present work, we shall derive a class of node-based and integration point-based finite element formu-lations, which are based on a monolithic simultaneous andon a partitioned staggered solution strategy, respectively.However, it will turn out, that the approach that had beentermed element-based in the literature can be understood

as a special case of either of the two categories. We willillustrate, that, provided that the spatial discretization iscarried out in a consistent way, neither of the three ap-proaches leads to unstable solutions if the underlyingcontinuous problem is stable and unique in the sense ofHarrigan and Hamilton [16–18].

3Theory of growthWithin the present chapter, we introduce the basic equa-tions of the continuum theory of growth: the fundamentalkinematics, the relevant balance equations and the con-stitutive assumptions. To lay the basis for the finite ele-ment formulations derived in the following chapters, wecast the governing equations into their weak format whichis then discretized in time.

3.1KinematicsTo set the stage, we briefly summarize the underlyingkinematics of geometrically nonlinear continuummechanics. Let B0 and Bt denote the reference and cur-rent configuration occupied by the body of interest at timet0 and t 2 R, respectively. The kinematic description isbasically characterized through the deformation map umapping the material placement x of a physical particle inthe material configuration B0 to its spatial placement X inthe spatial configuration Bt.

x ¼ uðX; tÞ : B0 � R! Bt ð1ÞThe corresponding deformation gradient F defines thelinear tangent map from the material tangent space TB0 tothe spatial tangent space TBt.

F ¼ ruðX; tÞ : TB0 ! TBt ð2ÞIts determinant defines the related Jacobian J asJ ¼ det F > 0. Moreover, we introduce the left Cauchy–Green tensor b ¼ F � Ft as a characteristic spatial strainmeasure. In what follows, Dt ¼ otf�gjX will denote thematerial time derivative of a quantity f�g at fixed materialplacement X. Accordingly, the spatial velocityv ¼ DtuðX; tÞ can be understood as the material timederivative of the deformation map u. Recall, that itsmaterial gradient rv is identical to the material timederivative of the deformation gradient F as DtF ¼ rv. Inthe sequel, we shall apply a formulation which is entirelyrelated to the material frame of reference. Thus, rf�g andDivf�g denote the gradient and the divergence of any fieldf�g with respect to the material placement X.

3.2Balance equationsHaving introduced the basic kinematic quantities, we nowturn to the discussion of the balance equations for themechanics of growth. While in classical continuummechanics the amount of matter contained in a fixed ref-erence volume B0 typically does not change no matter howthe body is moved, deformed or accelerated, the density q0

of the reference body no longer represents a conservationproperty within the theory of growth. Rather, its rate of

73

change is determined by a possible in- or outflux of massR and an amount of locally created material R0.

Dtq0 ¼ Div RþR0 ð3ÞIn a similar way, the balance of momentum balances theweighted rate of change of the mass-specific momentumdensity p which is nothing but the spatial velocityp ¼ DtuðX; tÞ ¼ v with the reduced momentum flux �PPt

and the reduced momentum source �bb0.

q0Dtp ¼ Div �PPt þ �bb0 ð4ÞNote, that the above equation represents the so-called‘mass-specific’ version of the balance of momentum whichis particularly useful in the context of growth since itcontains no explicit dependence on changes in mass. Thenature of growth thus manifests itself exclusively in termsof the Neumann boundary conditions for the reducedmomentum flux �PPt � N ¼ tclosed þ�ttopen and the definitionof the reduced momentum source �bb0 ¼ bclosed

0 þ �bbopen0 in

which the additional growth-dependent contributions �ttopen

and �bbopen0 are taken into account. A detailed derivation of

the balance equations for the mechanics of growth withinthe framework of open system thermodynamics can befound in Kuhl and Steinmann [31, 35].

Remark 3.1 Within the classical literature of biome-chanics, the balance of mass (3) is typically referred to as‘biological’ or ‘homeostatic equilibrium’ while the balanceof momentum (4) represents the equation of ‘mechanicalequilibrium’.

3.3Constitutive equationsFinally, the set of governing equations has to be closed byintroducing appropriate constitutive assumptions for themass flux R, the mass source R0, the reduced momentumflux �PPt and the reduced momentum source �bb0. Parallelingthe definition of the flux of concentrations according toFick’s law, the mass flux R

R ¼ R0rq0 ð5Þis related to the gradient of the density rq0 weighted by amass conduction coefficient R0 whereby the latter has theunit of a length squared divided by the time. FollowingHarrigan and Hamilton [17, 18], we propose a constitutiveequation for the mass source R0 which is governed by thefree energy W0 ¼ q0W.

R0 ¼ cq0

q�0

� ��m

W0 �W�0

� �ð6Þ

Herein, q�0 and W�0 denote the reference value of the den-sity and of the free energy, respectively, while m is anadditional material parameter. Moreover, the additionalparameter c, which is of the unit time divided by the lengthsquared, basically governs the speed of the adaption pro-cess. In the context of hard tissue mechanics, the freeenergy W0 is typically characterized through the elasticfree energy, e.g. of Neo-Hooke typeWneo

0 ¼ k ln2 J þ l½b : 1� 3� 2 ln J�� �

=2, weighted by therelative density ½q0=q

�0�

n such that W0 ¼ q0=q�0

� �nWneo

0 , seee.g. Carter and Hayes [3] or Gibson and Ashby [15].

Herein, k and l are the classical Lame constants. More-over, the exponent n typically varies between 1 � n � 3:5according to the actual porosity of the open-pored groundsubstance. This particular choice of the free energy definesthe reduces momentum flux as �PPt ¼ q0DFW.

�PPt ¼ q0

q�0

� �n

�PPneot ð7Þ

with �PPneot ¼ lFþ ½k ln J � l�F�t½ �. The reduced momen-tum flux �PPt can thus be interpreted as the classical Neo-Hookean first Piola Kirchhoff stress tensor �PPneot weightedby the actual relative density ½q0=q

�0�

n. For the sake ofsimplicity, the reduced momentum source �bb0 is assumedto vanish identically.

�bb0 ¼ 0 ð8Þ

Remark 3.2 The introduction of a mass flux hasbeen studied in detail by Kuhl and Steinmann [33].Note, however, that the incorporation of a massflux which essentially smoothes sharp solutions andenables the simulation of size effects requires at least aC0-continuous interpolation of the density correspond-ing to its discretization as global unknown on the nodallevel.

Remark 3.3 In the context of biomechanics, the drivingforce of the evolution of mass is classically referred toas ‘biological stimulus’. Herein, we have suggested afree-energy based stimulus of the form ½q0=q

�0��mW0.

Correspondingly, the reference free energy W�0 can beunderstood as ‘attractor stimulus’. However, alternativestress- or strain based stimuli as discussed e.g. by Weinansand Prendergast [51]. Even energy-dissipation basedstimuli can be found in the recent literature, see e.g.Levenston and Carter [39].

Remark 3.4 The choice of the exponent m determines thestability of the resulting algorithm. With m ¼ 1 we obtainthe classical model of Beaupre et al. [1]. Nevertheless,Harrigan and Hamilton [17, 18] choose m > n to guar-antee uniqueness and stability of the solution.

3.4Strong formThe reference density q0 and the deformation u furnishthe primary unknowns of the mechanics of growth. Theyare governed by the scalar-valued balance of mass (3)and by the vector-valued mass-specific balance ofmomentum (4) which can be cast into the residualstatements

Rqðq0;uÞ ¼ 0 in B0

Ruðq0;uÞ ¼ 0 in B0ð9Þ

with the residuals Rq and Ru defined in the following form.

Rq ¼ Dtq0 � Div R �R0

Ru ¼ q0Dtp� Div �PPt � �bb0

ð10Þ

Herein, the boundary oB0 of the material domain can bedecomposed into disjoint parts oBq

0 and oBr0 for the

74

density problem and equivalently into oBu0 and oBt

0 forthe deformation problem. While Dirichlet boundaryconditions are prescribed on oBq

0 and oBu0 ,

q0� qpresc0 ¼ 0 on oBq

0

u� upresc ¼ 0 on oBu0

ð11Þ

Neumann boundary conditions can be given for the massflux and the tractions on oBr

0 and oBt0,

R � N� ½rclosed þ �rropen� ¼ 0 on oBr0

�PPt � N� ½tclosed þ�ttopen� ¼ 0 on oBt0

ð12Þ

with N denoting the outward normal to oB0.

3.5Weak formAs a prerequisite for the finite element discretization, thecoupled set of equations has to be reformulated in weakform. To this end, the residual statements of the balance ofmass and momentum (9) and the corresponding Neumannboundary conditions (12) are tested by the scalar- andvector-valued test function dq and du, respectively.

Gqðdq; q0;uÞ ¼ 0 8dq in H01ðB0Þ

Guðdu; q0;uÞ ¼ 0 8du in H01ðB0Þ

ð13Þ

The weak forms Gq and Gu expand into the followingexpressions.

Gq ¼ZB0

dqDtq0 dV þZB0

rdq � R dV

�Z

oBr0

dq½rclosed þ �rropen�dA�ZB0

dqR dV

Gu ¼ZB0

du � q0Dtp dV þZB0

rdu : �PPt dV

�Z

oBt0

du � ½tclosed þ�ttopen�dA�ZB0

du � �bb0 dV

ð14Þ

3.6Temporal discretizationFor the temporal discretization of the governing equations(13), we partition the time interval of interest T into nstep

subintervals ½tn; tnþ1� as

T ¼[nstep�1

n¼0

½tn; tnþ1� ð15Þ

and focus on a typical time slab ½tn; tnþ1� for whichDt :¼ tnþ1 � tn > 0 denotes the actual time increment.Assume, that the primary unknowns q0n and un and allderivable quantities are known at the beginning of the ac-tual subinterval tn. In the spirit of implicit time marchingschemes, we now reformulate the set of governing equa-tions in terms of the unknowns q0nþ1 and unþ1.

Gqnþ1ðdq; q0nþ1;unþ1Þ ¼ 0 8 dq in H0

1ðB0ÞGu

nþ1ðdu; q0nþ1;unþ1Þ ¼ 0 8 du in H01ðB0Þ

ð16Þ

Without loss of generality, we shall apply the classicalEuler backward time integration scheme in the sequel. Incombination with the following approximations of the firstorder material time derivatives Dtq0 and Dtp as

Dtq0 ¼1

Dt½q0nþ1 � q0n�

Dtp ¼1

Dt½pnþ1 � pn�

ð17Þ

we obtain the following semi-discrete weak forms of thebalance of mass and momentum.

Gqnþ1 ¼

ZB0

dqq0nþ1 � q0n

Dtþrdq � Rnþ1 dV

�Z

oBr0

dq½rclosednþ1 þ �rr

opennþ1 �dA�

ZB0

dqR0nþ1dV

Gunþ1 ¼

ZB0

du � q0

pnþ1 � pn

Dtþrdu : �PPt

nþ1 dV

�Z

oBt0

du � ½tclosednþ1 þ�tt

opennþ1 �dA�

ZB0

du � �bb0nþ1dV

ð18ÞWe now turn to the spatial discretization of the above setof equations. To this end, we suggest two alternative finiteelement techniques which differ by the treatment of thebalance of mass. While the balance of mass is evaluatedglobally in the node-based approach, it is solved locally onthe integration point-level in the second approach.

4Node-based approachThe node-based approach derived in the present chapter isessentially characterized through a C0-continuousinterpolation of the density field q0 in combination withthe standard C0-continuous interpolation of the defor-mation field u. Just like the displacements in classicalfinite element approaches, the density is introduced as anodal degree of freedom which is solved for on the globallevel. This can either be done in a partitioned way bysolving the balance of mass and momentum sequentiallyas proposed in the related literature or in a monolithic wayby evaluating both balance equations simultaneously. Wesuggest the latter approach which is believed to be themost consistent strategy within a modern finite elementcontext. Note, that C0-continuity of the density field ismandatory, if density gradients are incorporated in theformulation, e.g. through the constitutive equation of themass flux.

4.1Spatial discretizationIn the spirit of the finite element method, the domain ofinterest B0 is discretized into nel elements Be

0.

75

B0 ¼[nel

e¼1

Be0 ð19Þ

The geometry Xh of each subset is interpolated element-wise in terms of the local basis functions NX and the dis-crete node point positions Xj of all j ¼ 1; . . . ; neX elementnodes as XhjBe

0¼PneX

j¼1 NjXXj. According to the isopara-

metric concept, the trial function uh is interpolated on theelement level with the same basis function Nu as the ele-ment geometry. Following the Bubnov–Galerkin approach,similar basis functions are applied to interpolate the testfunctions dqh and duh.

dqhjBe0¼Xneq

i¼1

Niq dqi 2 H0

1ðB0Þ

duhjBe0¼Xneu

j¼1

Nju duj 2 H0

1ðB0Þ

q0hjBe

0¼Xneq

k¼1

Nkq qk 2 H1ðB0Þ

uhjBe0¼Xneu

l¼1

Nlu ul 2 H1ðB0Þ

ð20Þ

While the element set of deformation nodes j ¼ 1; . . . ; neu

typically corresponds to the set of node point positionsj ¼ 1; . . . ; neX , the element set of density nodesi ¼ 1; . . . ; neq can generally be chosen independently. Thediscretization of the gradients of the test functions rdqh

and rduh and the gradients of the trial functions rqh andruh is straightforward and results in the followingexpressions.

rdqhjBe0¼Xneq

i¼1

dqirNiq

rduhjBe0¼Xneu

j¼1

duj �rNju

rqh0jBe

0¼Xneq

k¼1

qkrNkq

ruhjBe0¼Xneu

l¼1

ul �rNlu

ð21Þ

With the above discretizations, the discrete algorithmicbalance of mass and momentum can be rewritten as

RqI ðq0h

nþ1;uh

nþ1Þ ¼ 0 8 I ¼ 1; . . . ; nnq

RuJ ðq0

hnþ1;u

hnþ1Þ ¼ 0 8 J ¼ 1; . . . ; nnu

ð22Þ

whereby the discrete residua RqI and Ru

J expand into thefollowing forms.

RqI ¼ A

nel

e¼1

ZBe

0

Niqq0nþ1 � q0n

DtþrNi

q � Rnþ1 dV

�Z

oBer0

Niq½rclosed

nþ1 þ �rropennþ1 �dA�

ZBe

0

NiqR0nþ1dV

RuJ ¼ A

nel

e¼1

ZBe

0

Njuq0

pnþ1 � pn

DtþrNj

u � �PPnþ1 dV

�Z

oBte0

Nju½tclosed

nþ1 þ�ttopennþ1 � dA�

ZBe

0

Nju�bb0nþ1 dV

ð23ÞTherein, the operator A symbolizes the assembly ofall element contributions at the element density nodesi ¼ 1; . . . ; neq and the element deformation nodesj ¼ 1; . . . ; neu to the overall residuals at the globaldensity and deformation node points I ¼ 1; . . . ; nnq andJ ¼ 1; . . . ; nnu.

4.2LinearizationThe discrete residual statements characterizing themechanics of growth (22) represent a highly nonlinearcoupled system of equations which can be solved effi-ciently within the framework of a monolithic incrementaliterative Newton-Raphson solution strategy. To this end,we perform a consistent linearization of the governingequations at time tnþ1

RqI

kþ1nþ1 ¼ Rq

Iknþ1 þ dRq

I¼:

0 8 I ¼ 1; . . . ; nnq

RuJ

kþ1nþ1 ¼ Ru

Jknþ1 þ dRu

J ¼:

0 8 J ¼ 1; . . . ; nnu

ð24Þ

whereby the iterative residua dRqI and dRu

J take thefollowing format.

dRqI ¼

Xnnq

K¼1

KqqIK dqK

þXnnu

L¼1

KquIL � duL

dRuJ ¼

Xnnq

K¼1

KuqJK dqK

þXnnu

L¼1

KuuJL � duL

ð25Þ

In the above definitions, we have introduced the iterationmatrices

KqqIK ¼

oRqI

oqK

KquIL ¼

oRqI

ouL

KuqJK ¼

oRuJ

oqK

KuuJL ¼

oRuJ

ouL

ð26Þ

which take the following abstract representations.

KqqIK ¼ A

nel

e¼1

ZBe

0

Niq

1

DtNk

q dV �ZBe

0

Niqoq0

R0Nkq dV

þZBe

0

rNiq� R0rNk

q dV

KquIL ¼ A

nel

e¼1�ZBe

0

NiqoFR0�rNl

u dV

76

KuqJK ¼ A

nel

e¼1

ZBe

0

rNju�oq0

�PPtNkq dV

KuuJL ¼ A

nel

e¼1

ZBe

0

Njuq0

1

Dt2INl

u dV

þZBe

0

rNju�oF

�PPt�rNludV

ð27Þ

Note, that the specification of the partial derivatives of themass source R0 and the reduced momentum flux �PPt withrespect to the primary unknowns q0 and u depends on theparticular choice of the constitutive equations. For theconstitutive equations suggested in Sect. 3.3, we obtain thefollowing expressions,

oq0R0 ¼ c½n�m� q0

q�0

� ��m 1

q0

W0

oFR0 ¼ cq0

q�0

� ��m

�PPt

oq0�PPt ¼ n

1

q0

�PPt

oF�PPt ¼ q0

q�0

� �n

½lI ���Iþ kF�t � F�t � ½k ln J � l�F�t�F�1�

ð28Þwhereby the component representations of the non-stan-dard dyadic products read f��gijkl ¼ f�gik � fgjl andf��gijkl ¼ f�gil � fgjk. Finally, the solution of the sys-tem of equations (24) renders the iterative update for theincrements of the global unknowns qI and uJ .

DqI ¼ DqI þ dqI 8 I ¼ 1; . . . ; nnq

DuJ ¼ DuJ þ duJ 8 J ¼ 1; . . . ; nnuð29Þ

Remark 4.1 Within the node–based approach, we typicallysolve the balance of mass and momentum simultaneouslyon the global level assuming a C0–continuous interpola-tion of the density q0 and the deformation u. The onlynode-based finite element formulation that relaxes thiscontinuity requirement is the Q1P0 element, which isbased on an element-wise constant and thus C�1-con-tinuous density interpolation. Recall, that for this element,which is classically applied for constrained problemsarising in computational fluid dynamics or in in-compressible elastodynamics, the constantly interpolateddegree of freedom, in our case the density, can be elimi-nated locally on the element level. This procedure, whichbares strong resemblance to the classical static condensa-tion, results in the modified residual ~RRu

J and the modifiedstiffness matrix ~KKuu

JL .

~RRuJ ¼ Ru

J � KuqJK Kqq

IK½ ��1Rq

I

~KKuuJL ¼ Kuu

JL � KuqJK Kqq

IK½ ��1Kqu

IL

In the biomechanical literature, the algorithm resultingfrom an element-wise constant density distribution hasbeen termed ‘‘element-based approach’’.

Remark 4.2 Within the biomechanical context, the evo-lution of the density q0 and the deformation u inducestime scales which typically differ by orders of magnitude.To avoid the related numerical difficulties caused by theresulting ill–conditioned system matrices, the balance ofmomentum is usually evaluated in a quasi–static sense, i.e.the mechanical forces are interpreted as an average dailyloading on the biological structure. Consequently, the firstterms in the discrete momentum residualN

juq0½p0nþ1 � p0n�Dt and in the tangent operator

Njuq0=Dt2INl

u in Eqs. (23)2 and (27)4 vanish identically.

Remark 4.3 When consistent dynamic matrices are ap-plied to interpolate the time-dependent contributions, i.e.the Ni

q½q0nþ1 � q0n�Dt and the Niq1=DtNk

q terms in Eqs.(23)1 and (27)1, the numerical solution might tend to de-velop spurious oscillations near sharp fronts. The use oflumped dynamic matrices typically reduces these numer-ical artifacts.

4.3Algorithmic flowchartA typical finite element-based solution algorithm resultingfrom the node-based approach is sketched in the flowchartin Table 1. It illustrates that the balance of mass and thebalance of momentum are solved simultaneously on theglobal level.

Remark 4.4 It is worth noting that the numerical com-putation of the Q1P0 element with an elementwise con-stant density distribution requires some modifications inthe algorithmic realization. In this particular case, thebalance of mass and momentum are evaluated sequen-tially. First the midpoint density is advanced in time byperforming a local Newton iteration on the element level.Only then, the balance of momentum is evaluated at eachintegration point with the element density given. Since thedeformation unþ1 is now the only unknown on the globallevel, the related residual and the tangential stiffness haveto be modified according to Remark 4.1. Note, that withinthe Q1P0 approach the evolution of the element density isdriven by the average element mass source.

q0nþ1 � q0n

Dt¼ 1

V

ZBe

0

R0ðq0nþ1;unþ1ÞdV

Table 1. Algorithm of node-based approach

77

Since the driving force R0 is a nonlinear function of thedeformation unþ1, it has to be evaluated numerically by anappropriate numerical integration before the elementdensity can be determined through the suggested localNewton iteration.

4.4Illustration in terms of model problemTo illustrate the features of the node-based approach, weshall now elaborate its behavior with respect to two aca-demic model problems upon which the first relates to adiscontinuous solution whereas the solution of the secondis supposed to be continuous. These two examples some-how represent the extreme cases we assume to encounterin realistic biomechanical applications. In both cases, weanalyze a similar tension specimen of unit length with alength to width ratio of 1%. The elastic material parame-ters are chosen to E ¼ 1 and m ¼ 0:2, which correspond tok ¼ 0:2778 and l ¼ 0:4167. The growth–related parame-ters take values of q�0 ¼ 1, W�0 ¼ 2, c ¼ 1, n ¼ 2 and m ¼ 3,whereby stability is guaranteed as m > n. A unit tensileload is applied to both ends of the specimen. In combi-nation with the chosen set of parameters, this load gen-erates stretches of about 250%. To guarantee convergencewithin the global Newton iteration, the load has to beapplied incrementally in ten steps of 0.1 each. After theloading phase, the load is held constant for another 50time steps of Dt ¼ 0:1 while the solution convergestowards the biological equilibrium state.

Figure 1 depicts the four applied finite element formu-lations as typical representatives of the node–based ap-proach, whereby the white circles characterize the densitynodes whereas the dark circles mark the deformationnodes. While the classical Q1Q1 and Q2Q2 element arebased on an equal order interpolation of the density andthe deformation field, the Q1P0 element and the S2Q1element interpolate the density one order lower than thedeformation. All elements except for the Q2Q2 elementwhich is evaluated numerically at 3 � 3 integration pointsmake use of a Gauss–Legendre quadrature based on 2 � 2integration points.

The resulting number of degrees of freedom and thenumber of integration points as characteristic measures ofcomputational efficiency are given in Table 2. Thereby, theindividual columns are related to a discretization with 10,

40 and 100 elements, thus representing a typicalh-refinement. The rows which correspond to the differentelement formulations can be interpreted as a sort ofp-refinement.

4.4.1Discontinuous model problemFirst, we trigger a discontinuous solution by varying thevalue of the attractor stimulus W�0 from W�0 ¼ 2:0 at bothends in discrete steps of DW�0 ¼ �0:25 towards W�0 ¼ 1:0in the middle of the specimen. Obviously, the localreduction of the attractor stimulus has a direct influenceon the creation of new material. The decrease of its value iscompensated by a considerable local increase in density, asillustrated in Fig. 2. The quantitative results of the dis-continuous model problem are summarized in Tables 3and 4, which show the relative change in density½q0 � q�0�=q�0 in the center of the specimen and the end-point displacement u, respectively. The depicted densitydistributions illustrate that the Q1Q1 element producesspurious oscillations close to sharp fronts. It converges tothe discontinuous solution upon mesh refinement, how-ever, the artificial overshoots in the solution remain. Theelement of highest order, the Q2Q2 element, performsworst upon all elements, compare also Tables 3 and 4. Thecorresponding density distribution is too smooth by farand the results are only valuable upon a considerable meshrefinement. Since the Q2Q2 element is the most expensiveelement from a computational point of view, it can beclassified as non-reasonable for practical use. For thisparticular discontinuous type of solution, the Q1P0element with a C�1-continuous density interpolationperforms best upon all elements tested. With anelementwise constant density distribution, it is able tocapture sharp fronts by construction. The S2Q1 elementshows a less oscillatory behavior than the Q1Q1 element.Unlike the Q2Q2 element, it is able to capture disconti-nuities in a reasonable way. It converges fast uponrefinement and is also not too expensive from acomputational point of view.

4.4.2Continuous model problemThe second example is based upon a continuous solutionwhich is generated by smoothly varying the width of thespecimen from 1% of the specimen length at both ends to-wards 0.5% in the middle of the bar. Thereby, the materialparameters are kept constant throughout the specimen. Thecharacteristics of the different elements for a continuoussolution can be concluded from Fig. 3 and Tables 5 and 6.Obviously, the local reduction of the cross section is directlycompensated by the creation of material in the center of thespecimen. Remarkably, all node-based formulations of aC0-continuous density interpolation perform equally well forthis sufficiently smooth problem. Already for the discreti-zation with 40 elements, the results of the Q1Q1 element, theQ2Q2 element and the S2Q1 element have converged to areasonable extend, since their continuous density interpo-lations are typically designed to capture continuous solu-tions. While the Q1P0 element yields useless results for thecoarsest mesh, it converges rapidly upon mesh refinement

Fig. 1. Different finite elements – node-based approach

Table 2. Degrees of freedom and integration points

nel = 10 nel = 40 nel = 100

Q1Q1 66ð40Þ 246ð160Þ 606ð400Þ

Q2Q2 189ð90Þ 729ð360Þ 1809ð900Þ

Q1P0 44ð90Þ 164ð360Þ 404ð900Þ

S2Q1 128ð40Þ 488ð160Þ 1208ð400Þ

78

even though it applies only a C�1-continuous densityinterpolation, see also Tables 5 and 6.

5Integration point–based approachIn contrast to the previous approach, we shall now relax theassumption of C0 continuity for the density field q0 andallow for a discrete pointwise density representation whilethe deformation field u is, of course, still required to be C0-continuous. Consequently, the balance of mass can beevaluated locally on the integration point level, whereas theglobal system of equations is expressed in terms ofthe deformation only. By introducing the density as internalvariable on the integration point level, we solve the balanceof mass and momentum in a staggered sense. Recall, that by

Fig. 2. Discontinuous model problem – density distribution of node–based approach – Q1Q1, Q2Q2, Q1P0 and S2Q1 elements

Table 3. Discontinuous model problem – relative change indensity ½q0 � q�0�=q�0 at midpoint

nel = 10 nel = 40 nel = 100

Q1Q1 0.321502 0.297437 0.305828Q2Q2 0.256541 0.302062 0.306116Q1P0 0.300028 0.306185 0.306171S2Q1 0.318560 0.305875 0.306171

Table 4. Discontinuous model problem – endpoint displacement u

nel = 10 nel = 40 nel = 100

Q1Q1 2.32251 2.37135 2.36629Q2Q2 2.37124 2.37151 2.37156Q1P0 2.32239 2.37122 2.36624S2Q1 2.37221 2.37181 2.37166

Fig. 3. Continuous model problem – density distribution of node-based approach – Q1Q1, Q2Q2, Q1P0 and S2Q1 elements

79

relaxing the continuity requirement for the density, we apriori exclude the possibility of incorporating a mass flux.

5.1Spatial discretizationIn complete analogy to the previous section, the spatialdiscretization is based on the partition of the domain ofinterest B0 into nel elements Be

0

B0 ¼[nel

e¼1

Be0 ð30Þ

on which the element geometry Xh is interpolated asXhjBe

0¼PneX

i¼1 NiXXi. According to the isoparametric con-

cept in combination with the Bubnov–Galerkin technique,similar basis functions N are applied for the interpolationof the test and trial functions duh and uh.

duhjBe0¼Xneu

j¼1

Njuduj 2 H0

1ðB0Þ

uhjBe0¼Xneu

l¼1

Nluul 2 H1ðB0Þ

ð31Þ

Consequently, the spatial gradients of the test and trialfunctions rduh and ruh can be expressed in the follow-ing form.

rduhjBe0¼Xneu

j¼1

duj �rNju

ruhjBe0¼Xneu

l¼1

ul �rNlu

ð32Þ

The algorithmic balance of momentum can thus be statedas

RuJ ðuh

nþ1Þ ¼ 0 8 J ¼ 1; . . . ; nnu ð33Þ

whereby the discrete residual RuJ is defined in the following

form.

RuJ ¼ A

nel

e¼1

ZBe

0

Njuq0

pnþ1 � pn

DtþrNj

u � �PPnþ1 dV

�Z

oBte0

Nju½tclosed

nþ1 þ�ttopennþ1 �dA�

ZBe

0

Nju�bb0nþ1dV

ð34Þ

5.2LinearizationSimilar to the previous chapter, the nonlinear equilibriumequation (33) is solved within the incremental iterativeNewton–Raphson iteration scheme requiring a consistentlinearization at time tnþ1.

RuJ

kþ1

nþ1¼ Ru

Jk

nþ1þ dRu

J ¼:

0 8 J ¼ 1; . . . ; nnu ð35ÞThe iterative residual dRu

J

dRuJ ¼

Xnnu

L¼1

KuuJL �duL ð36Þ

can be expressed in terms of the iteration matrix

KuuJL ¼

oRuJ

ouL

ð37Þ

which takes the interpretation of the global tangentialstiffness matrix.

KuuJL ¼ A

nel

e¼1

ZB0

Njq0

1

DtINl dV þ

ZB0

rNj � dF�PPt � rNl dV

ð38ÞTherein, dF

�PPt denotes the consistent tangent operator

dF�PPt ¼ oF

�PPt � oq0�PPt oq0

R0

� ��1oFR0 ð39Þ

whereby oF�PPt, oq0

�PPt and oFR0 were already given in Eq.(28), while oq0

R0 can be expressed in the following form.

oq0R0 ¼ c½n�m� 1

q0

q0

q�0

� ��m

W0Dt � 1 ð40Þ

The iterative update for the incrementals of the globalunknowns uJ

DuJ ¼ DuJ þ duJ 8 J ¼ 1; . . . ; nnu ð41Þcan finally be determined in terms of the solution duJ ofthe linearized system of equations (35).

Remark 5.1 Note, that the structure of the consistenttangent operator introduced in Eq. (39) resembles thestructure of the modified element stiffness matrix afterstatic condensation defined in Remark 4.1.

dF�PPt ¼ oF

�PPt � oq0�PP t oq0

R0

h i�1oFR0

~KKuuJL ¼ Kuu

JL � KuqJK ½Kqq

IK ��1

KquIL

The difference between the terms oq0R0 and oq0

R0

somehow reflects the influence of the algorithmic

Table 5. Continuous model problem – relative change in density½q0 � q�0�=q�0 at midpoint

nel = 10 nel = 40 nel = 100

Q1Q1 0.509938 0.525144 0.527496Q2Q2 0.527441 0.527699 0.534161Q1P0 0.470753 0.513734 0.522864S2Q1 0.526425 0.528890 0.529023

Table 6. Continuous model problem – endpoint displacement u

nel = 10 nel = 40 nel = 100

Q1Q1 2.92605 2.92755 2.92780Q2Q2 2.92732 2.92785 2.92793Q1P0 2.92554 2.92752 2.92780S2Q1 2.92792 2.92794 2.92794

80

treatment. By replacing the continuous term oq0R0 with its

algorithmic counterpart oq0R0 we ensure the consistent

linearization of the discrete constitutive equations.

5.3Algorithmic flowchartTable 7 illustrates a typical flowchart resulting from theintegration point-based approach. The balance equationsare solved in a staggered way, characterized through alocal Newton iteration for the actual integration pointdensity embedded in the global Newton iteration for thedeformation field.

Remark 5.2 Similar to the node-based approach, the reali-zation of an elementwise constant density distributionbased on a selective reduced integration of the scalar fieldrequires some algorithmic modifications. In this case, thebalance of mass and momentum are solved sequentially.Just like for the Q1P0 element, the actual midpoint densityhas to be determined in a local Newton iteration before thebalance of momentum is evaluated on the integration pointlevel. In contrast to the Q1P0 element, however, the densityevolution is now driven by the midpoint mass source.

q0nþ1 � q0n

Dt¼ R0ðq0nþ1;unþ1Þ

��n¼0

As the mass source R0 is a nonlinear function of thedeformation unþ1, its midpoint value generally differsfrom its element average. Consequently, the Q1P0element and the Q1sri element generally render differentresults. The significance of this difference stronglydepends on the inhomogeneity of the discrete deforma-tion field unþ1 and vanishes identically for a homoge-neous deformation.

5.4Illustration in terms of model problemWe now turn to the illustration of the integration point–based approach in terms of the discontinuous and the

continuous model problem defined in chapter 4. Theanalyzed element formulations are depicted in Fig. 4.

While the Q1 and the Q2 element are classically inte-grated with 2 � 2 and 3 � 3 quadrature points, respec-tively, the Q1sri element is based on a selective reducedone–point–integration of the density part in combinationwith a 2 � 2 integration of the deformation part. The S2fri

element, however, applies a fully reduced 2 � 2 integra-tion of both contributions.

The number of degrees of freedom and integrationpoints, which is considerably lower than in the previousnode-based approach, is depicted in Table 8.

5.4.1Discontinuous model problemThe resulting density distribution, the relative change indensity ½q0 � q�0�=q�0 in the middle of the specimen and theendpoint displacement u are given in Fig. 5 and Tables 9and 10, respectively. Note, that in order to illustrate thetypical features of the integration point–based approach,the integration point values are plotted in a non-smootheddiscontinuous fashion. Unlike in classical finite elementpost–processing, where the integration point values aretypically extrapolated to the nodes and then averaged, weplot the actual constant integration point value in theentire area that is assigned to the particular quadraturepoint. As expected, all elements with the density asinternal variable perform excellent for this particulardiscontinuous problem. By construction, the integrationpoint–based approach is ideally suited for discontinuoussolutions with sharp fronts.

5.4.2Continuous model problemEven for the continuous problem, the integration point–based elements perform remarkably well. Not only theclassical Q1 element and the Q2 element but also the Q1sri

element and the S2fri element converge fast upon meshrefinement, compare also Fig. 6 and Tables 11 and 12.Thereby, the computationally cheapest approaches,namely the Q1 element and the selective reduced inte-grated Q1sri element show only minor differences. Both ofthem can thus be classified as extremely fast and efficient

Table 7. Algorithm of integration point-based approach

Fig. 4. Different elements – integration point-based approach

Table 8. Degrees of freedom and integration points

nel = 10 nel = 40 nel = 100

Q1 44ð40Þ 164ð160Þ 404ð400Þ

Q2 126ð90Þ 486ð360Þ 1206ð900Þ

Q1sri 44ð10=40Þ 164ð40=160Þ 404ð100=400Þ

S2fri 106ð40Þ 406ð160Þ 1006ð400Þ

81

for practical use. Nevertheless, due to the standard com-putational structure of the related numerical algorithm, weprefer the classical Q1 element over the Q1sri element forpractical applications.

Remark 5.3 This particular model problem still shows arather homogeneous deformation in each element. Thus,the difference between the Q1P0 element and the Q1sri

element is only of minor nature. However, by comparingthe resulting maximum density and the endpointdisplacements, slight deviations of both approaches canbe detected.

6Representative exampleFinally, the derived classes of algorithms shall be com-pared in terms of a representative example from biome-

Fig. 5. Discontinuous model problem – density distribution of integration point-based approach – Q1, Q2, Q1sri and S2fri elements

Table 9. Discontinuous model problem – relative change indensity ½q0 � q�0�=q�0 at midpoint

nel = 10 nel = 40 nel = 100

Q1 0.300028 0.306185 0.306418Q2 0.306443 0.306186 0.306188Q1sri 0.300028 0.306168 0.306171S2fri 0.306504 0.306282 0.306171

Table 10. Discontinuous model problem – endpoint displace-ment u

nel = 10 nel = 40 nel = 100

Q1 2.32239 2.37122 2.36624Q2 2.37095 2.37146 2.37154Q1sri 2.32239 2.37135 2.36630S2fri 2.37156 2.37155 2.37155

Fig. 6. Continuous model problem – density distribution of integration point-based approach – Q1, Q2, Q1sri and S2fri elements

82

chanics. To this end, we study the classical benchmarkproblem of bone remodeling of the proxima femur sub-jected to an average daily loading situation. A detailedspecification of the problem is given by Carter and Beau-pre [2]. For the model suggested herein, we chose theelastic parameters to E ¼ 500 and m ¼ 0:2 correspondingto k ¼ 138; 8889 and l ¼ 208; 3333. while the additionalgrowth-related parameters are chosen to q�0 ¼ 1:2,W�0 ¼ 0:01, c ¼ 1, n ¼ 2 and m ¼ 3. The geometry andthe loading conditions are depicted in Fig. 7. Typically,three different loading situations can be identified. Loadcase 1 corresponds to the load condition for the midstancephase of gait, while load cases 2 and 3 represent theextreme cases of abduction and adduction defined inTable 13.

The different elements and the corresponding numberof degrees of freedom and integration points as atypical measure for the computational efficiency is given inTable 14. Figures 8–11 illustrate the distribution of therelative changes in density ½q0 � q�0�=q�0 as a result of thedifferent spatial discretization techniques. The first set of

Fig. 8 illustrates the behavior of the Q1Q1 element withrespect to h-refinement. The results do not differ consid-erably upon refinement of the mesh. Remarkably, alreadythe coarsest mesh with 658 elements seems to be sufficientto capture the typical biomechanical characteristics of thesolution: the development of a dense system of compres-sive trabeculae carrying the stress from the superiorcontact surface to the calcar region of the medial cortex, asecondary arc system of trabeculae through the infero-medial joint surface into the lateral metaphyseal region,the formation of Ward’s triangle and the development of adense cortical shaft around the medullary core. Conse-quently, we shall restrict our further analyses to thecoarsest 658-element mesh.

Figure 9 contrasts the results of the different node-basedfinite element formulations. Remarkably, the Q1Q1 element,the Q2Q2 element, the Q1P0 element and the S2Q1 elementrender nearly identical results. Unlike reported in the re-lated literature, the Q1P0 formulation with an elementwiseconstant density interpolation does not show any spatialinstabilities. Due to the choice of a well–posed continuumformulation in combination with an implicit time integra-tion scheme, all finite element formulations, even the onebased on an elementwise constant density interpolation,render stable and unique results.

Figures 10 and 11 show the integration point-basedcounterpart of the solution which result from the classicalQ1 element, the Q2 element, the selectively reduced inte-grated Q1sri element and the fully reduced integrated S2fri

element. While Fig. 10 depicts the discrete values of theinternal variables for each area assigned to thecorresponding quadrature point, Fig. 11 illustrates theclassical post-processing result that is obtained by anextrapolation of the integration point values to the nodesin combination with an averaging of these nodal values.The Q1 element and the S2fri element apply the samenumber of integration points per element. However, the

Fig. 7. Proxima femur – loading conditions

Table 13. Proxima femur – loading conditions

Load case Value (N) Direction () Value (N) Direction ()

1 2317 24 703 282 1158 )15 351 )83 1548 56 468 35

Table 14. Number of elements, degrees of freedom and integra-tion points

nel ndof nip

Q1Q1 658 2175 2632Q1Q1 2632 8295 10528Q1Q1 5922 18363 23688Q2Q2 658 8295 5922Q1P0 658 1450 2632S2Q1 658 4939 2632Q1 658 1450 2632Q2 658 5530 5922Q1sri 658 1450 658/2632S2fri 658 4214 2632

Table 11. Continuous model problem – relative change in den-sity ½q0 � q�0�=q�0 at midpoint

nel = 10 nel = 40 nel = 100

Q1 0.470754 0.513734 0.522864Q2 0.514584 0.526201 0.529164Q1sri 0.470753 0.513734 0.522864S2fri 0.503670 0.522526 0.526427

Table 12. Continuous model problem – endpoint displacement u

nel = 10 nel = 40 nel = 100

Q1 2.92548 2.92751 2.92780Q2 2.92732 2.92785 2.92793Q1sri 2.92599 2.92765 2.92786S2fri 2.92789 2.92794 2.92794

83

solution of the classical Q1 element seems to be slightlyoscillatory while the S2fri solution is rather smooth due toits quadratic interpolation of the deformation field. Sincethe solution is rather homogeneous within each element,

the integration point-based Q1sri element renders nearlyidentical results as the node-based Q1P0 element. Thedifferences of the individual integration point-basedelements vanish upon smoothing as illustrated in Fig. 11.

Fig. 9. Proxima femur –density distribution of node-based approach – 658 Q1Q1,Q2Q2, Q1P0 and S2Q1 ele-ments

Fig. 8. Proxima femur – density distri-bution of node-based approach – 658,2632 and 5922 Q1Q1 elements

84

By making use of the traditional post-processing step astypically applied for the graphic representation of internalvariables, all results of the integration point-based

approach are nearly alike. Moreover, they all roughlycorrespond to the results of the node-based approachdepicted in Fig. 9.

Fig. 10. Proxima femur –density distribution of inte-gration point-based approach– 658 Q1, Q2, Q1sri and S2fri

elements

Fig. 11. Proxima femur –density distributionintegration point-basedapproach – 658 Q1, Q2, Q1sri

and S2fri elements-smoothed

85

Remark 6.1 Within the present approach, neither thenode-based Q1P0 element nor the integration point-basedQ1sri element tend to develop the spatial instabilities re-ported in the literature. To document the fact, that thesespatial instabilities are already introduced on the contin-uum level and then, of course, carry over to the temporallyand spatially discrete problem, we analyzed the similarproblem described above, only now with the parametersn ¼ 2 and m ¼ 0. According to Harrigan and Hamilton[17, 19], this set of parameters is supposed to produce anon-unique and unstable solution. The numerical resultdepicted in Fig. 12 shows the typical discontinuous 0–1type of solution with the classical tendency towardschecker-boarding. The given density distribution corre-sponds to the last converged state within the simulation;the calculation stopped after half of the prescribedtime period because of the loss of convergence withinthe global Newton iteration caused by ill-conditionedsystem matrices.

7DiscussionThe basic concern of the present work was the comparisonof different computational strategies to model growthprocesses typically encountered in modern biomechanicalapplications. Motivated by the huge body of literature onthis highly active branch of research, we began by com-paring the existing formulations in terms of appropriatedclassifications. It turned out, that the notions of stabilityand uniqueness were often attributed to the temporal or tothe spatial discretization although the cause of instabilityactually originated back to the ill-posedness of theunderlying continuous problem. We thus presented a well-posed continuum model embedded in the framework offinite deformations. Unlike the existing small strain for-mulations, the derived geometrically exact model is not apriori restricted to hard tissue mechanics but is potentiallyable to simulate the behavior of soft tissues. In contrast tomost existing formulations in the literature, we suggested

the use of an implicit time stepping scheme which isunconditionally stable and thus poses no additionalrestrictions on the choice of the time step size.

The analysis of different spatial discretization tech-niques constituted the main part of this contribution.While the deformation field is treated in the classical way,we suggested two alternative discretization strategies forthe density field, a node-based and an integration point-based approach. For the former, the density is introducedas a global unknown on the nodal level, whereas it istreated as an internal variable on the integration pointlevel in the latter approach. The traditional element-basedapproach can then be classified as a special case of eitherthe node-based or the integration point-based approach.For the geometrically exact formulation analyzed herein,the corresponding finite element formulations, namely theQ1P0 element and the Q1sri element, showed a slightlydifferent behavior, while for the linear elastic model ap-plied in the literature, both formulations should be abso-lutely identical. In the context of modern finite elementtechnologies, the discretized equations were linearizedconsistently and their solution was embedded in anincremental iterative Newton-Raphson procedure.

Both alternative discretization strategies were finallycompared numerically in terms of a discontinuous and acontinuous model problem and the classical application ofbone remodeling in the proxima femur. As expected, thenode-based elements performed better for continuoussmooth solutions while the integration point-based ele-ments proved advantageous for discontinuous problemswith sharp fronts. Despite of this difference, both strate-gies yielded remarkably similar results upon meshrefinement. In most of the cases, the analyzed lower orderelements, i.e. the Q1P0 element, the classical Q1 elementand the Q1sri element, were superior over the higher orderelements. They proved computationally cheap, i.e. cheapin memory, storage and computer time, and thus turnedout to be extremely fast. Especially the computationallymost expensive Q2Q2 element performed poor close tosharp fronts and is thus not recommended for furtherpractical use.

Another remarkable difference between the two alter-native discretization techniques is the size of the overallproblem, which is directly related to the computer timerequired for the solution. For small scale problems, bothstrategies needed approximately the same calculation time.While the node-based approach basically suffers fromlarger system matrices, the integration point-based ap-proach requires a local Newton iteration at the integrationpoint level. Nevertheless, the influence of the formerbecomes more and more pronounced upon mesh refine-ment. Provided that both strategies give similar results,this lack of computational efficiency of the node-basedapproach might be an important drawback for realisticlarge scale simulations. However, one has to keep in mind,that the node-based approach becomes unavoidable, ifhigher order gradients of the density are to be incorpo-rated, e.g. by the incorporation of a mass flux or by theneed to reproduce the classical size effect.

Finally, we would like to conclude by pointing out,that the results of the present study are not necessarily

Fig. 12. Proxima femur – density distribution of node-basedapproach – Q1P0 element – unstable continuum model withn ¼ 2 and m ¼ 0

86

restricted to the modeling of growth but rather hold in amore general sense. Any multi-field problem in continuummechanics, e.g. convection–diffusion in chemo-mechani-cal applications, the classical thermo-elasticity or prob-lems arising in inelasticity such as damage or plasticity,can be treated in an analogous way. As long as no gradi-ents of the additional field, e.g. the concentration, thetemperature, the damage variable or the plastic multiplier,are incorporated in the formulation, one is free to chooseeither a C0-continuous node-based approach or a discretepointwise representation within the integration point-based approach. Upon mesh refinement, both strategiesshould converge to the same solution, provided that thedifferent interpolation orders of the individual fields arechosen appropriately. As soon as higher order spatialgradients enter the formulation, e.g. through diffusion,heat conduction, gradient damage or gradient plasticity,the node-based C0-continuous approach becomes man-datory. In this sense, the present work is believed not onlyto yield a contribution to the computational modeling ofgrowth in particular but also to the numerical simulationof multi-field problems in general.

References1. Beaupre GS, Orr TE, Carter DR (1990) An approach for time–

dependent bone modelling and remodelling. J. Orthop. Res. 8:651–670

2. Carter DR, Beaupre GS (2001) Skeletal Function and Form –Mechanobiology of Skeletal Development, Aging andRegeneration. Cambridge University Press

3. Carter DR, Hayes WC (1977) The behavior of bone as a two-phase porous structure. J. Bone Jt. Surgery 59-A: 785–794

4. Carter DR, Orr TE, Fhyrie DP (1989) Relationships betweenloading history and femoral can–cellous bone architecture.J. Biomech. 22: 231–244

5. Chen Y-C, Hoger A (2000) Constitutive functions of elasticmaterials in finite growth and deformation. J. Elasticity 59:175–193

6. Cowin SC (1996) Strain or deformation rate dependent fintegrowth in soft tissues. J. Biomech. 29: 647–649

7. Cowin SC, Arramon YP, Luo GM, Sadegh AM (1998) Remarkson the technical note entitled Temporal stability of node-based internal bone adaption. J. Biomech. 31: 103–104

8. Cowin SC, Hegedus DH (1976) Bone remodelling I: Theory ofadaptive elasticity. J. Elasticity 6: 313–326

9. Ehlers W, Markert B (1998) Modellierung fluidgesattigter,viskoelastischer poroser Festkorper in der Biomechanik.In: Die Methode der Finiten Elemente in der Biomechanik,Biomedizin und angrenzenden Gebieten. Workshop 1998,Universitat Ulm

10. Epstein M, Maugin GA (2000) Thermomechanics ofvolumetric growth in uniform bodies. Int. J. Plasticity 16:951–978

11. Fischer KJ, Jacobs CR, Levenston ME, Carter DR (1997)Observations of convergence and uniqueness of node–basedbone remodeling simulations. Annals Biomed. Eng. 25: 261–268

12. Fung YC (1990) Biomechanics – Motion, Flow, Stress, andGrowth. Springer Verlag, Berlin – Heidelberg – New York

13. Fung YC (1993) Biomechanics – Mechanical Properties ofLiving Tissues. Springer Verlag, Berlin – Heidelberg – NewYork, second edition

14. Gasser TC, Holzapfel GA (2002) A rate-independent elasto-plastic constitutive model for biological fiber-reinforcedcomposites at finite strains: Continuum basis, algorithmic

formulation and finite element implementation. Comput.Mech. 29: 340–360

15. Gibson LJ, Ashby MF (1982) The mechanics ofthree–dimensional cellular materials. Proc Royal Soc London,A 382: 43–59

16. Harrigan TP, Hamilton JJ (1992) An analytical and numericalstudy of the stability of bone remodelling theories: Depen-dence on microstructural stimulus. J. Biomech. 25: 477–488.Corrigendum 26: 365–366 (1993)

17. Harrigan TP, Hamilton JJ (1992) Optimality condition forfinite element simulation of adaptive bone remodeling. Int.J. Solids Struct. 29: 2897–2906

18. Harrigan TP, Hamilton JJ (1993) Finite element simulation ofadaptive bone remodelling: A stability criterion and a timestepping method. Int. J. Numer. Meth. Eng. 36: 837–854

19. Harrigan TP, Hamilton JJ (1994) Necessary and sufficientconditions for global stability and uniqueness in finite ele-ment siumlations of adaptive bone remodeling. Int. J. SolidsStruct. 31: 97–107

20. Harrigan TP, Hamilton JJ, Reuben JD, Tone A, Viceconti M(1996) Bone remodelling adjcent to intromedullary stems: Anoptimal structures approach. Biomaterials 17: 223–232

21. Hart RT (2001) Bone modeling and remodeling: Theories andcomputation. In: Bone Biomechanics Handbook, Cowin SC,(ed.) chapter 31, pp. 1–42. CRC-Press, Boca Raton

22. Holzapfel GA, Gasser TC Ogden RW (2000) A new constitu-tive framework for arterial wall mechanics and a comparativestudy of material models. J. Elasticity 61: 1–48

23. Huiskes R (2000) Challenges in Bone Biomechanics. Intree-rede, Technische Universiteit Eindhoven.

24. Huiskes R, Chao EYS (1983) A survey of finite elementanalysis in orthopedic biomechanics: The first decade.J. Biomech. 16: 385–409

25. Huiskes R, Ruimerman R, van Lenthe GH Janssen JD (1990)Effects of mechanical forces on maintenance and adaption ofform in trabecular bone. Nature 405: 704–706

26. Huiskes R, Weinans H, Grootenboer HJ, Dalstra M, Fudala B,Slooff TJ (1987) Adaptive bone–remodeling theory applied toprosthetic–design analysis. J. Biomech. 20: 1135–1150

27. Humphrey JD (2002) Cardiovasular Solid Mechanics. SpringerVerlag, Berlin – Heidelberg – New York

28. Jacobs CR, Levenston ME, Beaupre GS, Simo JC, Carter DR(1995) Numerical instabilities in bone remodeling simula-tions: The advantages of a node–based finite elementapproach. J. Biomech. 28: 449–459

29. Krstin N, Nackenhorst U, Lammering R (1999). Zur konsti-tutiven Beschreibung des anisotropen beanspruchungsadap-tiven Knochenumbaus. In: Die Methode der Finiten Elementein der Biomechanik, Biomedizin und angrenzenden Gebieten.Workshop 1999, Universitat Ulm

30. Krstin N, Nackenhorst U, Lammering R (2000) Zurkonstitutiven Beschreibung des anisotropen beanspruchung-sadaptiven Knochenumbaus. Technische Mechanik 20: 31–40

31. Kuhl E, Steinmann P (2002) Mass–and volume specific viewson thermodynamics for open systems. Proce. Royal Soc.London (Accepted for publication)

32. Kuhl E, Steinmann P (2002) Material forces in open sys-tem mechanics. Comput. Meth. Appl. Mech. Eng.(for publication)

33. Kuhl E, Steinmann P (2002) Theory and numerics of geo-metrically nonlinear open system mechanics. Int. J. Numer.Meth. Eng. (Accepted for publication)

34. Kuhl E, Steinmann P (2003) Computational modeling ofhealing: An application of the material force method. Bio-mechanics and Modeling in Mechanobiology, (Accepted forpublication)

35. Kuhl E, Steinmann P (2003) On spatial and material settingsof thermo-hyperelastodynamics for open systems. ActaMechanica 160: 179–217

87

36. Kuhn W, Hauger W (2000) A theory of adaptive growth ofbiological materials. Arch. Appl. Mech. 70: 183–192

37. Levenston ME (1997) Temporal stability of node-basedinternal bone adaption simulations. J. Biomech. 30: 403–407

38. Levenston ME (1998) Author’s response to comments byCowin et al. regarding Levenston (1997). J. Biomech. 31: 105–106

39. Levenston ME, Carter DR (1998) An energy dissipation-basedmodel for damage stimulated bone adaption. J. Biomech. 31:579–586

40. Lubarda VA, Hoger A (2002) On the mechanics of solids witha growing mass. Int. J. Solids Struct. 39: 4627–4664

41. Mullender MG, Huiskes R (1995) A proposal for theregulatory mechanisms of Wolff’s law. J. Orthop. Res. 13:503–512

42. Mullender MG, Huiskes R, Weinans A (1994) A physiologicalapproach to the simulation of bone remodeling as a self-organizational control process. J. Biomech. 27: 1389–1394

43. Nackenhorst U (1997) Ein effizientes Finite ElementVerfahren zur Simulation des beanspruchungsadaptivenKnochenwachstums. In: Die Methode der Finiten Elemente inder Biomechanik, Biomedizin und angrenzenden Gebieten.Workshop 1997, Universitat Ulm

44. Nackenhorst U (1997) Numerical simulations of stress stim-ulated bone remodeling. Technische Mechanik 17: 31–40

45. Pauwels (1965) Gesammelte Abhandlungen zur funktionellenAnatomie des Bewegungsapparates. Springer Verlag, NewYork – Berlin – Heidelberg

46. Rodriguez EK, Hoger A, McCulloch AD (1994) Stress-dependent finite growth in soft elastic tissues. J. Biome-chanics 27: 455–467

47. Steeb H, Diebels S (2002) A thermodynamic-consistent modeldescribing growth and remodeling phenomena. Comput. Mat.Sci.

48. Taber LA (1995) Biomechanics of growth, remodeling, andmorphogenesis. ASME Appl. Mech. Rev. 48: 487–545

49. Weinans H, Huiskes R, Grootenboer HJ (1992) The behaviorof adaptive bone-remodeling simulation models. J. Biomech.25: 1425–1441

50. Weinans H, Huiskes R, Grootenboer HJ (1994) Effects of fitand bonding characteristics of femoral stems on adaptivebone remodeling. J. Biomech. Eng. 116: 393–400

51. Weinans H, Prendergast PJ (1996) Tissue adaption as a dy-namic process far from equilibrium. Bone 19: 143–149

52. Weng S (1997) Ein Evolutionsmodell zur mechanischenAnalyse biologischer Strukturen. Dissertation, Mitteilungenaus dem Institut fur Mechanik, Ruhr-Universitat Bochum,Bochum

53. Wolff J (1892) Das Gesetz der Knochentransformation.Hirschwald, Berlin

88