Robust adaptive remeshing strategy for large deformation, transient impact simulations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2006; 65:2139–2166Published online 21 November 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1531

Robust adaptive remeshing strategy for large deformation,transient impact simulations

Tobias Erhart1,‡, Wolfgang A. Wall2,∗,† and Ekkehard Ramm1,§

1Institute of Structural Mechanics, University of Stuttgart, Pfaffenwaldring 7, 70569 Stuttgart, Germany2Chair for Computational Mechanics, Technical University of Munich,

Boltzmannstr. 15, 85748 Garching, Germany

SUMMARY

In this paper, an adaptive approach, with remeshing as essential ingredient, towards robust and efficientsimulation techniques for fast transient, highly non-linear processes including contact is discussed. Thenecessity for remeshing stems from two sources: the capability to deal with large deformations thatmight even require topological changes of the mesh and the desire for an error driven distributionof computational resources. The overall computational approach is sketched, the adaptive remeshingstrategy is presented and the crucial aspect, the choice of suitable error indicator(s), is discussed inmore detail. Several numerical examples demonstrate the performance of the approach. Copyright �2005 John Wiley & Sons, Ltd.

KEY WORDS: adaptive remeshing; error indicators; finite strain plasticity; impact; contact

1. INTRODUCTION

For the development of robust simulation techniques for general transient and non-linear solidmechanics problems including large deformations the question of remeshing is essential. Thenecessity for remeshing is thereby twofold. Solid mechanics problems are typically treated ina Lagrangean framework. Hence large deformations result at least in heavy mesh distortion oreven in full topology changes and remeshing is already needed simply to be able to computethe whole process. The second argument towards adaptive remeshing is the aim of a qualitycontrolled simulation or at least an ‘optimal’ distribution of the used computational resources.Therefore, the application of remeshing increases robustness and efficiency as well as the

∗Correspondence to: Wolfgang A. Wall, Chair for Computational Mechanics, Technical University of Munich,Boltzmannstr. 15, 85748 Garching, Germany.

†E-mail: [email protected], http://www.lnm.mw.tum.de/Members/wall‡E-mail: [email protected]§E-mail: [email protected]

Received 18 February 2005Revised 6 September 2005

Copyright � 2005 John Wiley & Sons, Ltd. Accepted 6 September 2005

2140 T. ERHART, W. A. WALL AND E. RAMM

quality of solutions. All of these features are essential when targeting sophisticated industrialapplications.

The specific class of problems focussed in this contribution excel themselves through alarge complexity. They are mainly fast transient plane strain or axisymmetric problems thatare treated in an explicit way. Adopted material models range from simple elastic up to largestrain thermo-elastic-viscoplastic models. The problems exhibit large deformations and mayeven include severe mesh topology changes. Frictional contact situations are an essential aspectof the physical process.

Adaptive methods have proven their capacity towards quality control of numerical simulationsas well as for increasing robustness and efficiency. Obviously one would hope for the chanceto base such an adaptive approach on a real estimation of the error, preferably with givenlower and upper bounds [1]. However, for such kind of complex highly non-linear problemsthat we are dealing with in this study, rigorous error estimators are still missing. Hence qualitycontrol has to be done through an error assessment via indicators [2]. Different approaches foradaptivity in complex settings have been proposed in the literature.

In 1987, Zienkiewicz and Zhu [3] presented an error estimator for linear problems based onthe comparison of non-smoothed and smoothed stresses. They use the fact that local recoveryprocedures lead to improved results compared to actual ones from finite element solution. Thelevel of improvement is then used to estimate the error. Since no bounds for the true error canbe proven for arbitrary problems, the ZZ-method gives only an error indication in such cases.Nevertheless, this indicator and similar ones are often used for adaptive remeshing proceduresand good results can be achieved even in non-linear calculations. For problems involvingplastic response or localization, Belytschko and Tabbara [4] used the original ZZ-indicator aswell as a modified version, where they compared strains instead of stresses. Fourment andChenot [5] developed another modification by using a correction term in order to satisfy thebalance equations with the smoothed solution as well. They applied their indicators to industrialproblems like forming processes. Peric et al. [6] expanded the ZZ-indicator to problems withfinite elasto-plastic strains and large deformations in transient analysis. For instance, errorindicators based on the plastic dissipation or the rate of plastic work are used to solve metalforming processes with an adaptive remeshing strategy. Common to all the above methods isthat gradients of a predetermined quantity are chosen as a basis for an indicator-driven adaptivestrategy.

Another approach is the use of indicators, which directly apply physical relevant quantitiesinstead of gradients to control the mesh density distribution. One of the first approaches stemsfrom Ortiz and Quigley [7], who used the variation of the velocity field per element as anindicator for localization, wave propagation or problems with large deformations. A similarindicator was developed by Batra and Ko [8]. The integral of the second invariant of thedeviatoric strain-rate tensor serves here as a suitable quantity to identify local phenomena likeshear bands in high strain-rate deformations. The same indicator is used by Camacho and Ortiz[9] for different industrial applications as impact, perforation and penetration of metallic bodies.Their approach is well suited for a number of non-linear, transient problems. Other quantities,namely the effective strain and the effective strain rate, are used by Petersen and Martins[10] to get adaptive meshes for the numerical simulation of metal sheet forming processes. Asa last example, the indicator of Marusich and Ortiz [11] should be mentioned. They also usea physically motivated indicator for mesh density control and found that the rate of plasticwork is very useful for adaptive simulation of high-speed machining.

Copyright � 2005 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 65:2139–2166

ROBUST ADAPTIVE REMESHING STRATEGY 2141

Another possibility for feeding an adaptive strategy is the use of geometric indicators. Anexample can be found in Reference [12] where some distortion measures are used in orderto decide if the mesh has to be refined locally. However the applicability of such simple,geometric indicators is limited since physical behaviour of the structure is not at all taken intoaccount.

One main focus of the present contribution is the question of suitable error indicators forthe discussed class of problems. In our approach, two categories of indicators, gradient-basedas well as local quantity-based, are used for adaptive remeshing. For the first category, wecompute local gradients of the solution at finite element nodes using the superconvergent patchrecovery technique [13]. Then, these gradients are directly transformed to desired element sizesfor a new spatial discretization. This results in a refinement or coarsening in regions of high orlow gradients (h-adaptivity). For the second category, i.e. local quantity indicators, no gradientsof the solution have to be determined. Instead, absolute values of relevant physical quantitiesare used to control the mesh density. The adaptive remeshing then aims at distributing elementsso that every element experiences roughly the same amount of ‘physical action’. Both types ofindicators as well as the question of the ‘relevant quantities’ are tightly connected to the specificphysical process or even to the specific example at hand. The different indicators will be usedseparately or in combination. At the boundary the resulting mesh density will be enhanced viaa geometric indicator. To this end the geometric parameter ‘curvature’ is used as a geometricrefinement indicator for strongly curved boundaries. It is only activated for refinement and notfor coarsening of the ‘mechanical mesh density’.

The paper is structured as follows. The general solution procedure with spatial and timediscretization, inelastic material modelling and contact treatment is given in Section 2. InSection 3 the overall remeshing strategy including mesh quality check, indicator-driven meshdensity calculation, mesh generation and transfer of state variables is outlined. The errorassessment by gradient-based, local quantity and geometric indicators is presented in detailin Section 4. Numerical examples and applications are discussed in Sections 5–7.

2. GENERAL SOLUTION PROCEDURE

Before presenting our strategy for adaptive remeshing and the detailed investigation oferror assessment via indicators, the overall solution procedure of our computational setup isintroduced.

2.1. Two-dimensional spatial discretization

The weak form of the initial boundary value problem with displacement field as primary variableu, acceleration field a, Cauchy stresses �, test function �u, volume force �b and traction treads ∫

�(�)

�u · �a dv = −∫

�(�)

gradx�u : � dv +∫

�(�)

�u · �b dv +∫

� (��)

�u · t da (1)

with current, deformed configuration �(�) and Neumann-boundary �(��). The spatial dis-cretization by FEM leads to the well-known semi-discretized form of equation of motion for



a system without damping

Md(t) + f int(d(t)) = fext(t) (2)

with constant lumped mass matrix M and time-dependent vectors of nodal displacements d,nodal accelerations d, internal forces f int and external forces fext. Damping is not considered,since its influence is very small in impact mechanics. For axisymmetrical and plane strain cases,two different bilinear quadrilateral elements are utilized in our approach: a reduced integratedelement with stabilization (‘RI4’) based on the ideas of Belytschko and Bindeman [14] anda selective reduced integrated element with modified deformation gradient (‘SI4’) originatingfrom de Souza Neto et al. [15]:

• Element ‘RI4’: The bilinear quadrilateral element is integrated with only one integrationpoint, making it very fast and efficient. To avoid unstable zero energy modes, a stabilizationtechnique is needed, which in our case is based on the assumed strain method [14]. Withoutdetailed investigation of the formulation, it can be stated that no artificial parameters areneeded since this stabilization is physically based. In addition, volumetric and shear lockingare diminished with this element formulation. Nonetheless, it should be mentioned, that itcannot be guaranteed hourglass modes will not appear for arbitrary meshes and non-linearcases. Therefore, quality control of the solution (mesh geometry, energy balance) shouldalways come along with the application of this element.

• Element ‘SI4’: It is slower but more robust than ‘RI4’. The intention of this elementformulation is to avoid volumetric locking by integrating deviatoric parts with 4 andvolumetric parts with only 1 integration point (‘selective reduced integration’ or ‘meandilatational method’). The starting point is the multiplicative decomposition of thedeformation gradient F = FdevFvol into deviatoric part Fdev = J−1/3F and volumetric partFvol = J 1/31 with Jacobian J := det F. Then, at all four integration points Q, the new,modified deformation gradient FQ is made up of the unchanged deviatoric part (FQ)devand an element-constant volumetric part (F0)vol evaluated at the centre of the element [15]

FQ = (FQ)dev(F0)vol =(

det[F0]det[FQ]

)1/3

FQ (3)

The implementation of this method is very simple, because the ‘normal’ deformationgradient FQ only has to be replaced by its modified counterpart FQ. The applicationof element ‘SI4’ is independent of the material model at hand and therefore no extraparameter is needed.

2.2. Explicit time integration

A one-step, first-order accurate explicit time integration procedure called ‘modified Eulermethod’ (MEM) [16] is used to integrate Equation (2) in time. Starting from the discreteequations of motion at time tn

Mdn + f int(dn) = fextn (4)

current accelerations dn are obtained by

dn = M−1[fextn − f int(dn)] (5)



The use of a diagonalized (lumped) mass matrix M simplifies the inversion and therefore,only vector analysis is needed for the computation of dn. Since nodal velocities dn and nodaldisplacements dn are known from the last time step, the corresponding values for the next timestep can be calculated by the MEM as follows:

dn+1 = dn + �t dn (6)

dn+1 = dn + �t dn+1 = dn + �t dn + �t2dn (7)

The analysis of the spectral stability of the MEM leads to the same stability criterion as forthe well-known and often used ‘central difference method’ (CDM) [17]

�t� 2

�(8)

i.e. the size of the eigenfrequency � of the observed system limits the time step �t . Sinceit would be very costly or even impossible to compute � for large, non-linear systems, it ismore reasonable and common to reduce the multi-dimensional problem (in our case: planestrain/axisymmetry) to the one-dimensional elastic case, where the critical time step �tcritdepends on minimal element length lmin and elastic wave speed ce [18]:

�tcrit = lmin

ce, ce =

[E(1 − �)

(1 + �)(1 − 2�)�

]1/2

(9)

with Young’s modulus E, Poisson ratio � and density �. Alternatively, more enhanced criticaltime step estimates can be found in Reference [19]. With this critical time step estimate—oftencalled CFL condition (Courant, Friedrichs and Lewy [20])—an adaptive time stepping can easilybe realized: In regular intervals, the current time step size is set to a value little smaller thangiven by this stability limit. Additionally, the energy balance always should be checked at theend of explicit calculations, especially in highly non-linear situations. Stability limit (8) usuallyresults in very small time steps for common problems. Nonetheless, explicit time integrationis well suited for transient impact problems, since the necessary small time steps are neededanyway in order to resolve the physically occurring high-frequency parts (wave propagation)of the solution.

The presented time integration scheme possesses no numerical dissipation for linear systemsas it is the case for CDM. How far this statement applies to generally non-linear behaviourcannot be shown analytically. However, practical experiences have shown slight numericaldissipative behaviour in highly non-linear situations.

2.3. Inelastic material behaviour: finite strain thermo-visco-plasticity for metals

Our implementation of finite strain plasticity is based on the multiplicative split of the defor-mation gradient into elastic and plastic parts F = FeFp and an isotropic Eulerian formulationin eigenvalues [21–23]. Using a spectral decomposition of the Finger-tensor b = FFT and theKirchhoff stress tensor � = J�, a general return mapping scheme with elastic predictor andclosest point projection algorithm formulated in principal logarithmic strains � = 1/2 ln(b) =[�1, �2, �3]T and in principal Kirchhoff stresses � = [�1, �2, �3]T is applied [24]. With thisapproach, which is valid for isotropic materials, the functional form of the return mapping



is identical to the algorithm of the infinitesimal theory [22]. Therefore, the whole scheme withelastic predictor, plastic corrector, etc. is not shown here, but only the most important parts ofour model are presented briefly.

The stress–strain relation associated with quadratic free energy function in principal log-arithmic strains �e takes the following form:

� = �

��e= tr(�e)1 + 2�dev(�e) = tr(�e)1 + 2�(�e − tr(�e)1) (10)

with bulk modulus , shear modulus � and the definitions tr(�e) = �e1 +�e2 +�e3 and 1 = [1, 1, 1]T.The elastic region of the classical J2-plasticity is bounded by the von Mises yield function f ,which is formulated in Kirchhoff stresses here

f (�, q) = ||s|| −√

23q(�p, ˙�p

, T )�0 (11)

This means, that plastic flow begins, when the norm of the Kirchhoff stress deviator ||s|| = √2J2

with s = �−1/3 tr(�)1 reaches the critical value of equivalent stress � = √2/3q(�p, ˙�p

, T ), whichis a function of equivalent plastic strain �p, equivalent plastic strain rate ˙�p

and temperature T .Equation (11) is now well suited for the so-called Johnson–Cook model [25], which is oftenused for metals subjected to transient impact loading [8, 9, 26, 27]. In this approach the yieldlimit q is multiplicatively composed of a static, a dynamic and a thermal part:

q(�p, ˙�p, T ) = (�y + Et(�

p)n)

(1 + C ln

(1 + ˙�p

�0

))(1 − (T ∗)m) (12)

In this empirical formula, �y indicates the static yield limit and material parameters Et andn describe the strain hardening behaviour. The increase of strength due to high strain rates,i.e. the viscosity, is considered through the second term of Equation (12). Factor C quantifiesthe strain rate dependency, whereas the rate of plastic strain ˙�p

is normalized by �0 = 1.0[1/s].With the last part in (12), the effect of decreasing strength for high temperatures is reproduced:The homologous temperature T ∗ is a measure for the mobility of crystal grid componentsand is modelled as a function of current temperature T , room temperature T0 and meltingtemperature Tm

T ∗ = T − T0

Tm − T0(13)

This non-dimensional variable ranges from 0 (current = room temp.) to 1 (current = melt. temp.).Thermal softening is controlled by the exponent m. Since transient impact and penetrationprocesses are of very short duration, adiabatic temperature changes are assumed in our study.In addition, for the local temperature evolution, the empirical assumption is used, that most(90–100%) of the plastic work W p is transformed into heat

T = �d

�cv

W p = �d

�cv

(J−1�T�p), 0.9 � �d � 1.0 (14)

with grade of dissipation �d and specific heat capacity cv .The return mapping algorithm finally ends up in a local Newton iteration, where plastic

strain and plastic strain rate are the implicitly considered unknowns, whereas the temperatureis accounted for in an explicit manner, i.e. it is computed from previous time step values.



2.4. Contact formulation

An essential part of transient impact simulations is the treatment of contact between severalmoving and deforming bodies. The boundary of a plane domain, discretized by finite elements,is characterized by boundary nodes and outer element edges (boundary segments). In ourapproach, node-to-segment contact is used, i.e. boundary nodes of a domain (slave) slide alongthe boundary segments of another domain (master) and vice versa, if master and slave areinterchanged. With appropriate spatial discretization formulations [28], this method leads to themodified semi-discretized form of equation of motion

Md(t) + f int(d(t)) + fcon(d(t)) = fext(t) (15)

Comparison with Equation (2) shows, that contact forces fcon additionally appear on the left-hand side, which can be regarded as correction forces to guarantee non-penetration. Form (15)is now separated in a contact independent part and a contact dependent part [29]. Therefore,accelerations are split into two terms dn = dint

n + dconn , where Equation (5) is responsible for

the first term. The contact dependent part is then defined as

dconn = M−1[fcon(dn)] (16)

Time integration (6) and (7) is now modified in a way, that the non-contact part is treatedexplicitly as before, whereas the contact part is implicitly accounted for

dn+1 = dn + �t dintn + �t dcon

n+1 = dpredn+1 + �t dcon

n+1 (17)

dn+1 = dn + �t dn + �t2dintn + �t2 dcon

n+1 = dpredn+1 + �t2dcon

n+1 (18)

In the algorithm, this additive split can be used in such a way that accelerations dintn , predictor

velocities dpredn and predictor displacements dpred

n are calculated first without contact consider-ations. After that, possible penetrations have to be ‘repaired’ and emerging contact forces andcorresponding nodal accelerations are determined. Finally, all kinematic variables are corrected.

The contact forces are calculated via the Lagrange multiplier method [28], where generally anon-linear system of equations has to be solved iteratively, e.g. by a Newton–Raphson method.On the other hand, simplifying assumptions can be applied to linearize the problem in anincrement, which leads to directly solvable linear systems of equation. Such a fast and robustapproach is presented in the following.

A direct determination of contact forces is possible, if concentrated nodal masses and linearrelations between contact forces and correction accelerations are used [30]. This approach isdemonstrated here with the most simple constellation of a ‘contact group’, consisting of oneslave node xS and one master segment between boundary nodes xM

1 and xM2 parametrized by

∈ [0, 1] (see Figure 1). In one time step, it is assumed that a penetration �gN < 0 developes,which has to be corrected by corresponding nodal forces. The non-penetration condition readsas follows in this case:

�gN + [�dS − (1 − )�dM1 − �dM

2 ] · nM != 0 (19)

with correction displacements �dS, �dM1 , and �dM

2 of the involved nodes. The position of theclosest point xM to slave node xS on the master segment is defined by , where nM is the



∆gN

xS

xM1

xM2

n M

t M

Slave

Master

x M

FM1

FM2

FS

1

Figure 1. Contact group with one slave node and one master segment.

normal vector pointing outwards from the master body. With Newton’s second law of motion,nodal displacements are linearly related to contact nodal forces through

�dS = 1

2

�t2

mS FS, �dMI = 1

2

�t2

mMI

FMI , I = 1, 2 (20)

with nodal masses mS and mMI . The master nodal forces FM

I are calculated as

FM1 = (1 − )FS, FM

2 = FS (21)

The use of Coulomb’s friction law results in the following expression for the slave nodal forceFS in case of ‘slip’:

FS = [nM − �k sign(vrel)tM]pN (22)

where pN = FS · nM, �k, tM and vrel are contact pressure, kinetic friction coefficient, tangentvector and relative velocity given by

vrel = [(1 − )dM1 + dM

2 − dS] · tM (23)

in which the predictor velocities are used. In case of ‘stick’, the term �k sign(vrel) in Equation(22) has to be replaced by �s, which is denoted as static friction coefficient. Insertion ofEquations (20)–(23) in condition (19) finally provides a linear equation for the determinationof pN , which in turn allows the calculation of FS, FM

I , �dS and �dMI .

Normally, contact groups consist of several master segments and slave nodes. If the presentedprocedure is applied analogously to such larger contact groups, one obtains a set of linearequations for the determination of contact forces fcon(d).

3. ADAPTIVE REMESHING STRATEGY

The essential ingredients of an adaptive remeshing strategy are (i) a mesh quality check fortriggering automatic remeshing, (ii) a reasonable assessment of discretization errors and deriva-tion of a corresponding mesh density distribution, (iii) an automatic mesh generation tool forgraded meshes and (iv) methods for the transfer of state variables from old to new discretiza-tion. Theoretical definitions and numerical treatment of these aspects are dealt with in this



chapter. Good descriptions of similar, alternative strategies can be found in References [9, 6]or [31].

3.1. Mesh quality check

In an adaptive strategy, there are basically two possibilities for reaching a remeshing decisionin the algorithm:

• given times are defined before the calculation, or• mesh quality is checked in regular intervals and automatic remeshing is triggered if some

finite elements are of ‘bad quality’.

In this context we ignore the possibility of remeshing decisions based on solution quality,since a real quantification of solution errors is not available for our class of problems. Differentgeometrical quality measures for finite elements are existing. Usually, various criteria like aspectratio, angle deviation, etc. are combined to one resultant quality measure, in order to graspdifferent distortion phenomena at the same time [32, 33].

In our approach, mesh quality is evaluated for each element by a combination of twogeometrical criteria, the so-called ‘corner angle criterion’ and the ratio of the radius of theinscribed circle to the radius of the circumscribed circle. For a quadrilateral element, thecorner angle criterion CA� is defined as the scaled maximum deviation of one of the fourinner angles �c to an optimal angle of 90◦ (see Figure 2)

CA� = 1 − maxc=1...4

{ |�c − 90◦|90◦

}, CA� ∈ [0, 1] (24)

This criterion is scaled in such a way that it takes 1 for an optimal and 0 for an irregularelement. To derive the inner/outer circle criterion IU�, the quadrilateral is decomposed intofour triangles and then the radii ri and rc of the inscribed and the circumscribed circle arecalculated for each triangle (Figure 2). Then the minimal ratio is multiplied with an appropriatecoefficient to achieve the same quality range [0, 1] as before

IU� = 1

(√

2 − 1)min

t=1...4

{(ri

rc

)t

}, IU� ∈ [0, 1] (25)

rc

ri

inner/outer circle criterion

IU

Corner angle criterion

CA

α3

α3

α1 α2

decomposition ofquad into four

triangles

Figure 2. Definitions for quality measures for quadrilateral elements.



If the combination of these two measures, e.g. the mean value

QUAL� = 12 (IU� + CA�), QUAL� ∈ [0, 1] (26)

falls below a prescribed threshold value for a prescribed number of elements, the remeshingprocedure starts automatically. With this mesh quality check, it is guaranteed, that the FEsolution is prevented from occurrence of negative Jacobian problems.

3.2. Error assessment and mesh density distribution

Since no real error estimators with guaranteed, sharp lower and upper limits are available forcomplex problems like in our transient, non-linear impact calculations, the only possibility forerror assessment is to use refinement indicators based on heuristic considerations. As will beexplained in detail in Section 4, gradient-based error indicators, local quantity-based refinementindicators, geometrical indicators as well as combinations of them are possible. It has to bepointed out, that these indicators cannot quantify the error but give a qualitative informationwhere the finite element mesh has to be refined or coarsened in order to get an optimaldistribution of computational resources. The magnitude of refinement depends on prescribedparameters such as maximal allowed size of indicator per element or minimal element length.In Section 4, it will be described, how a mesh density distribution is derived from the errorindicators. The final result of this procedure is a scalar element size value hnew

e at the nodesof the examined mesh, which is the basic information for generation of a new discretization.

3.3. Mesh generation

Generating a graded mesh needs two sources of information: the boundary of the domain andthe mesh density inside the domain mostly given on a background mesh. In our case of largedeformations and changing topologies the background mesh is preferably the old mesh of thepreceding time step, for which the error assessment is done and the mesh density distributionis calculated. Secondly, a polygonal description of the boundary has to be provided. Boundarynodes must be generated adaptively, i.e. subject to the calculated mesh density. Special carehas to be taken of boundaries in contact situations. The re-ordering of nodes on convexlycurved boundaries during remeshing can result in sudden gaps between contacting bodies,which results in the so-called ‘remeshing shocks’: oscillations induced by unphysically highcontact forces. This is remedied by a smooth contact formulation combined with a procedurefor precise surface recovery [34]. After that, a new ‘almost-all-quad’ mesh is generated insidethe domain by an ‘advancing front method’ starting from these previously generated boundarynodes and based on the given mesh density found at the nodes of the old mesh [35–37]. Inexceptional cases, e.g. for sharp corners, triangular elements (CST—constant strain triangles)are also allowed. This has the advantage, that no ‘bad quads’ have to be generated and thereforethe time span until next remeshing is increased.

3.4. Transfer of state variables

Since results are determined in an iterative or in an incremental process for non-linear ortransient problems, a transfer of state variables between old and new spatial discretizationis essential [6, 9]. Examples of variables, which have to be ‘mapped’ are strains, stresses,velocities, boundary conditions, etc., i.e. a mapping strategy for both nodal and integration



point data is needed. The location and number of nodes and integration points changes totallyin case of complete remeshing of a domain. Therefore, the location of new points in the oldmesh has to be determined by an intelligent search algorithm (e.g. directional search [36]) andthereafter, state variables have to be transferred by a suitable mapping method. Especially inthe context of explicit time integration the quality of the mapping scheme is crucial. Sincemapping cannot be supplemented by equilibrium iterations in these cases mapping errors maypropagate and pollute the whole simulation. Other aspects like minimization of diffusion shouldbe met as in implicit situations. The following different mapping methods have been evaluated:

• Inter/extrapolation method: Simple procedure, where variables z are interpolated/extra-polated using shape functions NI of the background elements. For nodal values, simpleinterpolation is sufficient:

z( ) =nN∑I=1

NI ( )zI (27)

with nN being the number of background element nodes. To transfer values betweenintegration points, a laborious, more error-prone way has to be taken, namely old dataare extrapolated from old integration points to old nodes first, then interpolated to newnodes and finally interpolated again to new integration points [6].

• Inverse distance weighted method: More than one element, namely an element patch inthe background mesh is used as a source of information. The values zQ at the correspond-ing background assembling points (e.g. quadrature points Q) are summed and inverselyweighted with their distance rQ to the considered point in the new mesh [12]

z =(

nQ∑Q=1

zQ

r2Q

)·(

nQ∑Q=1

1

r2Q

)−1

(28)

• (Weighted) Moving least square method: The well-known superconvergent patch recovery[13] is used for the transfer of state variables. This method aims at a locally (in a patcharound position (xQ, yQ)) enhanced, smoothed solution z∗(xQ, yQ), which is the resultof a least square minimization

z :nQ∑

Q=1w(xQ, yQ) · (zQ − z∗(xQ, yQ))2 → Minimum (29)

Weighting with a positive continuous function w(xQ, yQ), which takes the distances ofthe assembling points into account, is additionally possible for this method [38]. Here, abell-shaped curve with shape-parameter c is given as an example

w(xQ, yQ) = e−c2[(xQ/ max |xQ|)2+(yQ/ max |yQ|)2] (30)

In our approach, the interpolation method is used for the transfer of nodal values, i.e. thevelocities d, whereas integration point data like deformation gradient F, elastic Finger tensorbe and internal variables � are mapped by the weighted moving least square method (seeFigure 3). This strategy has proven to be suitable and efficient and was used by other authorsin a similar way [39, 40].



deformation gradient Felast. Finger tensor be

internal variables α

velocities d.

nodal values integration point values

interpolation method (Weighted) moving least square method

new mesh

old mesh

patch

Figure 3. Transfer of state variables from old to new spatial discretization.

4. ERROR INDICATION

As mentioned before, we classified the error indicators in three categories: gradient-based, localquantity and geometric indicators, respectively. In the following, all indicators that have beenstudied and implemented are presented in detail along with their associated refinement strategyand some remarks on the selection of specific indicators for a given problem.

4.1. Gradient-based indicators

Gradient-based indicators use the fact that there is a connection between the size of local gra-dients and the quality of solution. Here, ‘gradients’ not only mean stresses, but local variationsof arbitrary quantities like strains, plastic work, etc. The corresponding adaptive remeshingstrategy is characterized by an equi-distribution—with respect to the element length—of thosegradients in the domain of solution. This leads to a refinement in high gradient zones on theone hand, and to a coarsening in low gradient areas on the other. Now the question is, whichrelevant quantities for the gradient calculation should be used to indicate a meaningful error andsubsequently govern the mesh density? We have chosen two different elastic and one plasticvariable, whose gradients are tightly connected to some of the specific physical phenomena orprocesses occurring in complex problems like in our applications, depending on the specificexample at hand.

The first quantity is the von Mises stress

1 := �√

2(dev(�e) : dev(�e)) (31)

which especially tracks elastic shape changing behaviour, whereas the second one, the hydro-static stress

2 : = tr(�e) (32)



full integration:16 ass. points

red. integration:4 ass. points

boundary:2+2 ass. points

corner:1+3 ass. points

red. integration:3 ass. points

Figure 4. Various element patches for the calculation of a local gradient.

mainly grasps elastic volumetric changes. Plastic regions in a structure, e.g. localization zones,can be detected with the third quantity ‘plastic work’

3 :=∫ t

t0

J−1� : �p dt (33)

Gradients of these quantities need to be calculated at all nodes of the old mesh. Thesuperconvergent patch recovery technique in an element patch [13] is used to get informationabout the local gradient at the particular assembly node, i.e. the central node of the patch.According to this method, a polynomial expansion is used for the considered quantity k

k(xy) := P(x, y)a (34)

where the shape functions P(x, y) and the unknown coefficients a are given by

P(x, y) = [1, x, y, xy], a = [a1, a2, a3, a4]T (35)

for element patches with more than three assembling (i.e. integration) points. If there are onlythree points available, which might be the case in corner or boundary situations along with1-point quadrature (see Figure 4), slightly different vectors are used

P(x, y) = [1, x, y], a = [a1, a2, a3]T (36)

In the case of element patches with less than three assembly points (e.g. corner situationsalong with 1-point quadrature), adjacent elements have to be taken into account as shown inFigure 4. The next step is a least square fit between the polynomial approximation k andthe discrete values k,Q at the position of the n assembling points. The sum of the squareddistances

D(a) =n∑

Q=1(k(xQ, yQ) − k,Q)2 =

n∑Q=1

(P(xQ, yQ)a − k,Q)2 (37)

is minimized through

�D(a)

�a= 0 (38)

which leads to a system of linear equations for the unknown vector an∑

Q=1PT(xQ, yQ)P(xQ, yQ)a =

n∑Q=1

PT(xQ, yQ)k,Q (39)



After solution of (39), the norm of the gradient can be calculated at the position of node N

(central node of patch)

gk := ‖∇k(xN, yN)‖ = (∇k(xN, yN) · ∇k(xN, yN))1/2 (40)

The local element size of the new mesh has to be inversely proportional to the size of thelocal gradient due to the above-mentioned reasons. In addition, the magnitude of refinementneeds to be controlled in order to adapt the element sizes to the different indicator and/orexamples. This can be achieved by scaling the gradient with a prescribed value �k, whichdescribes the maximal allowed change of the considered quantity k per element. At the endthe new element size hnew

e,k results in

hnewe,k = �k

gk(41)

4.2. Local quantity indicators

On the basis of pure physical considerations, local refinement indicators can be developed thatcontrol the mesh density in the interior mesh. For this purpose, a physically relevant quantityis calculated for each element in the old mesh. The new mesh is then generated provided thatthis quantity gets equi-distributed so that every element experiences roughly the same ‘physicalaction’. This obviously results in refinement for high values and in coarsening for small valuesof this quantity per old element. For these indicators, no gradients need to be calculated. Inthe following we sketch some of these local quantity indicators that are given in the literatureand that we adopted for our study.

For problems of strain localization, Ortiz and Quigley [7] proposed an adaptive strategy,which is based on the equi-distribution of variation of the velocity field v over the elementsof the mesh. Such localization problems can be characterized by large differences between themechanical behaviour of the global structure and a local zone, e.g. shear bands, bringing alonga loss of ellipticity of the governing equations. For a two-dimensional element, this indicatorcan be expressed as the maximum deviation for the velocity vi of two nodes a and b

�1 := maxi

{maxa,b

|veia − ve

ib|}

, i = x, y, a, b = 1, . . . , nnodes (42)

This is a suitable indicator for the analysis of transient processes with wave propagation aswell as localization. It is however restricted to dynamic calculations. Singularities in quasi-staticzones of transient problems or poor approximations of quasi-static quantities are not treatedappropriately.

Another physical indicator was used by Batra and Ko [8] for shear bands in plane straincompression and by Camacho and Ortiz [9] for different impact and penetration calculations.The intention of this indicator is that the integral of the second invariant of the deviatoricstrain-rate tensor over an element

�2:=∫

�(�e)

√12 dev� : dev� dv (43)

is the same for all elements. This indicator is capable of resolving strain localizations indynamic calculations.



A third indicator, based on the equi-distribution of plastic power, was developed by Marusichand Ortiz [11] for simulation of high-speed machining

�3:=∫

�(�e)

Wp dv =∫

�(�e)

J−1� : �p dv (44)

This indicator is especially suited for non-linear, highly dynamic problems with large plasticdeformations, since it leads to refinement in high strain rate regions.

In order to be able to combine different indicators later, we adopt here a similar strategyas before. At first, the elementwise calculated quantities �l are related to the respective oldelement size hold

e

gl = �l

holde

(45)

Now we use the same refinement strategy as we did for the gradient-based error indicators,i.e. the new element size is inversely proportional to the ‘variation per element’ gl and it isscaled by a prescribed value ��l

hnewe,l = ��l

gl(46)

4.3. Geometric indicator

Strongly curved surfaces have to be discretized with sufficiently small elements to avoid sig-nificant ‘geometrical errors’. This is especially important with respect to mass conservationand in contact situations. The curvature of the boundary is used as a geometric indicator tocontrol this problem. However, this indicator is only activated if it suggests refinement of the‘mechanical’ indicator driven mesh density.

In differential geometry, the curvature of a plane graph at position P is defined by

= lims→0

ϑ

s(47)

where ϑ is the angle between the tangent in point P and the tangent in a close-by point P′.s is the arc-length between these two points (Figure 5 left). In case of a discrete boundarydescription, e.g. the polygonal boundary of a finite element mesh, the discrete curvature h atthe position of a finite element node is given in analogy to Equation (47) as

h := ϑ12 (s1 + s2)

(48)

Now, ϑ is the smaller angle between two boundary segments with lengths s1 and s2, respectively(Figure 5 right). If this measure is calculated for boundary nodes of the old mesh, the elementsize in the new mesh results in

hnewe,g = ϑtol

h(49)

where ϑtol is a prescribed value, which can be interpreted as a tolerance angle between twoadjacent boundary segments. A similar indicator was used by Camacho and Ortiz [9]. In general,



Figure 5. Curvature of a graph (left) and of a polygonal line (right).

through application of this indicator, there is no connection between internal element size andboundary element size. In such cases, a mesh smoothing technique is used to avoid largedifferences between element sizes in the interior and at the boundary.

4.4. Selection of indicators and final mesh density

The objective of the different indicators and possible application areas have been given above.For real and complex applications the question of the final selection of indicator remains.Which specific indicator or combination of different indicators should be used? The answer isboth trivial and difficult at the same time. It should be rather clear from the above discussionwhich indicators are not suited in which cases. The final selection however can only be basedon engineering intuition and on a deep insight into the physics of the problem or the class ofproblems. It is crucial to know what the governing phenomena and the dominating processesare. One should be aware that the answers not only depend on the problem itself but alsoon the respective quantities of interest of a specific simulation. Some clarifying examples aregiven in Sections 5–7.

To obtain the final mesh density, two further steps are needed. As already pointed out itis in some cases useful to combine different error indicators. For this purpose, the minimumof the different element sizes hnew

e,k (which belong to the five gradient-based indicators), hnewe,l

(which belong to the three local quantity indicators) and hnewe,g (the geometric indicator for the

boundary) has to be chosen as final element size

hnewe = min{hnew

e,k , hnewe,l , hnew

e,g } (50)

On the other hand, the element size has to be restricted by some prescribed threshold values,because undesirably small elements (time step size!) or large elements (approximation quality!)should be avoided

hmine �hnew

e �hmaxe (51)

5. BENCHMARK PROBLEM: TAYLOR BAR IMPACT

To demonstrate the effectiveness of our adaptive strategy and especially to examine the suit-ability of different error indicators for large deformation transient problems, the benchmark ofa metal bar impact, often called Taylor bar impact [41], is used. A cylindrical rod made ofsoft metal hits a rigid wall with high velocity. During the impact, elastic and slower plasticwaves are initiated, propagating in axial direction and reflecting at the bar tails. Therefore,



Figure 6. Taylor bar impact: model conditions and experimental deformations [43].

Figure 7. Taylor bar impact with indicator ‘gradient of von Mises stress’ 1: adaptivemeshes and von Mises stresses (0–700 N/mm2).

the initial kinetic energy is transformed into internal energy and large plastic deformations canbe observed.

For our analysis, a common parameter set [9, 42] is used: A copper rod with length 32.4mm,diameter 6.4mm and initial velocity of 227m/s impacts on a rigid wall (see Figure 6). Copper ismodelled as a linear elastic–plastic material with Young’s modulus E = 117 000N/mm2, Poissonratio � = 0.35, density � = 8930 kg/m3, yield limit �y = 400 N/mm2 and hardening modulusEt = 100 N/mm2 (n = 1 in Johnson–Cook model). Influence of strain rate and temperature arenot yet taken into account for this example. The process lasts 80 �s, i.e. after this period, thekinetic energy is totally transformed into plastic deformation.

Different kinds of calculations are compared. In four adaptive variants, different indicatorsfrom Section 4 are used: 1 (gradient of von Mises stress), 3 (gradient of plastic work),�1 (variation of velocity) and �3 (plastic power). The size of the bilinear, axisymmetricquadrilateral elements is bounded by hmin

e = 0.16 mm and hmaxe = 0.8 mm. In two non-adaptive

computations, constant mesh densities are used (fine: he = hmine , coarse: he = hmax

e ). In favourof better comparison, a constant time step size of �t = 0.004 �s is chosen for all calculationsand remeshing is done at time instances 6, 18, 38 and 68 �s.

The evolutions of deformation, of spatial discretization and of different state variables areshown in Figures 7–10 to illustrate the interrelation between adaptive meshes and the choice



Figure 8. Taylor bar impact with indicator ‘gradient of plastic work’ 3: adaptivemeshes and plastic energy (0–1500 N mm).

Figure 9. Taylor bar impact with indicator ‘variation of velocity’ �1: adaptivemeshes and velocity field (0–227 m/s).

of error indicators. Using the gradient of von Mises stress as parameter for error assessment inthis example is problematic insofar as new meshes are subject to fluctuations due to oscillatorybehaviour (Figure 7). Frequent changes of discretizations with very different element sizes canunpleasantly intensify the error which is made during transfer of state variables. In contrast, theindicator ‘gradient of plastic work’ causes a smooth and continuous growth of fine mesh regionsbeing consistent with the evolving plastic zone (Figure 8). The use of local velocity variation asrefinement indicator results in a decrease of mean mesh density with every remeshing (Figure9), which is inappropriate, if high quality of solution is needed at the end of the process. Finally,the indicator ‘rate of plastic work’ also depicts the connection between adaptive discretizationsand the corresponding physical/mechanical process: Refinement occurs, where rate of plastic



Figure 10. Taylor bar impact with indicator ‘plastic power’ �3: adaptive meshesand plastic strain rates (0–75 000 1/s).

Table I. Taylor bar impact: comparison of results.

Max. equiv.Length Mushroom plastic strain Calculation(mm) radius (mm) (dimensionless) time (s)

No adaptivity: coarse 21.45 7.03 1.98 92No adaptivity: fine 21.40 7.19 2.99 2442Adaptivity: ind. 1 21.42 7.18 2.67 683Adaptivity: ind. 3 21.42 7.20 2.98 697Adaptivity: ind. �1 21.41 7.20 2.09 723Adaptivity: ind. �3 21.41 7.20 2.08 842Kamoulakis (42) 21.47–21.66 7.02–7.12 2.47–3.24Zhu and Cescotto (44) 21.26–21.49 6.89–7.18 2.75–3.03Camacho and Ortiz (9) 21.42–21.44 7.21–7.24 2.97–3.25

strain is very high, i.e. where current material flow takes place. Therefore, the propagatingplastic wave is accompanied by a shift of fine mesh region (Figure 10).

In Table I, the final rod length, the final mushroom radius, the final maximum equivalentplastic strain and the required calculation time for our different adaptive and non-adaptivecases are listed. The results from Kamoulakis [42], Zhu and Cescotto [44] and Camachoand Ortiz [9] are also mentioned for comparison. As expected, the non-adaptive, coarse meshcomputation shows an identifiably stiffer behaviour compared to the non-adaptive, fine meshcase. On the other hand, the adaptive calculations produce results for final length and finalradius, which are in the range of the non-adaptive, fine mesh version, indicating the high qualityof solutions. Regarding the maximum equivalent plastic strain, the indicator ‘gradient of plasticwork’ leads to the best result, since the mushroom region is finely discretized throughout thewhole computation, which is not the case for the three other indicators (see Figure 11). Thisis an example for the fact, that the choice of indicator(s) should always take into account thequantity of interest.



Figure 11. Equivalent plastic strains (0–2.75) at t = 80 �s.

Figure 12. Taylor bar impact: evolution of number of elements (mean values in brackets).

The efficiency of adaptive procedures becomes visible by the comparison of calculationtimes (Table I). A significant time saving can be observed if compared with the non-adaptive,fine mesh case, which is associated with the number of used elements. This is clarified byFigure 12, in which the history of number of elements is illustrated for the different cases. Thenon-adaptive, fine mesh calculation with constant mesh density shows only slight decrease ofnumber of elements due to decrease of the cross-sectional area (axisymmetric volume remainsconstant!). The adaptive cases start with the same fine mesh (4000 elements). But then thenumber of elements drops drastically during the first remeshing and stays relatively low untilthe end of the computation.



6. HIGH-STRAIN-RATE COMPRESSION OF WHA BLOCK

In the second example, a prismatic metal block with rectangular cross-section undergoes highlydynamic compression with a strain rate of 5000 1/s [27]. Therefore, a first linearly increasingand then constant velocity of 50 m/s acts on the upper and lower boundary as shown in Figure13 on the left. The quarter system of the block is initially discretized with 325 quadrilateralplane strain elements and symmetric boundary conditions are used (Figure 13 right). Johnson–Cook material parameters for the tungsten heavy alloy (WHA) are taken from Batra and Peng[27] and are listed here in Table II.

The highly dynamic pressure loading causes large plastic strains, which again effects signif-icant temperature rise. The involved thermal softening finally results in adiabatic shear bands,i.e. strain localizations in consequence of heat. A detailed description of this phenomenologycan be found in Reference [45]. Here, this transient example is used to verify the coupling ofthe adaptive remeshing strategy and the presented thermoviscoplastic material model.

For error assessment, the local quantity indicator ‘variation of velocity’ from Reference [7]with corresponding scaling factor ��1 = 4.0 m/s is used as refinement indicator. This indica-tor is well suited for this example, since dynamically developing strain localizations can bedetected automatically. The value of the scaling factor describes the maximal allowed changeof the considered quantity per element (see Equation (41)). The element sizes are bounded byhmin

e = 0.04 mm and hmaxe = 0.4 mm for graded meshes. The minimal element length of 40 mm

therefore lies in the range of physically observable shear band thicknesses [46].Figure 14 shows the evolution of adiabatic shear bands in a time-frame between 30 and 42�s

represented by temperature distributions and appropriate adaptive meshes. The development oflocalizations results in expedient mesh refinements, which again provide high resolutions incorresponding regions.

Figure 13. Plane strain WHA block compression: geometry, loading and discretization.

Table II. Plane strain compression: Johnson–Cook material parameters for WHA.

� (kg/m3) E (N/mm2) � (–) �y (N/mm2) Et (N/mm2)18 600.0 412 900.0 0.29 1506.0 177.0

C (–) n (–) m (–) Tm (K) T0 (K) cv (J/kg K)0.016 0.12 0.5 1723.0 293.15 134.0



Figure 14. Dynamic WHA block compression: adaptive meshes and temperature distributions.

Figure 15. Dynamic WHA block compression: load–displacement curve and velocity field at t = 38�s.

The load–displacement curve and the velocity field at t = 38 �s is represented in Figure 15.The oscillations stem from the initial impact and the propagation and reflection of stress wavesin the metal block. At an upper edge displacement of about 3.5 mm, strain localization causes



global softening behaviour. The difference between a nonadaptive calculation with a coarsemesh (325 elements) and the adaptive remeshing computation (up to 4188 elements) in thepost-critical state points out the mesh dependency of the solution. Application of a regularizationmethod would be helpful to correct the ill-conditioning of the problem. But this is not thetopic of this study. Some regularization is achieved by consideration of viscous effects inEquation (12).

7. PENETRATION OF STEEL CYLINDER BY WHA LONG ROD

A cylindrical rod made of tungsten heavy alloy (WHA) hits an externally clamped cylindricalblock made of high-strength steel at an incident velocity of 1250 m/s. The required data ofthis experiment from Anderson et al. [26] and the initial two-dimensional discretization of thisaxisymmetric problem are shown in Figure 16. For both metals the temperature and strainrate dependent model of Section 2.3 is used, whereas the parameters in Table III come fromAnderson et al. [26] and Camacho and Ortiz [9]. This application is taken from the field ofcivil and military safety engineering.

Figure 16. Long rod penetration: geometry and initial discretization.

Table III. Long rod impact: Johnson–Cook material parameters.

� (kg/m3) E (N/mm2) � (–) �y (N/mm2) Et (N/mm2)Tungsten heavy alloy 17600.0 322400.0 0.30 1510.0 177.0High-strength steel 7850.0 199400.0 0.29 1500.0 569.0

C (–) n (–) m (–) Tm (K) T0 (K) cv (J/kg K)Tungsten heavy alloy 0.016 0.12 1.0 1723.0 293.15 134.0High-strength steel 0.003 0.22 1.17 1777.0 293.15 477.0



Figure 17. Long rod penetration: adaptive meshes and equivalent plastic strain distributions.

The WHA long rod is discretized with element edge lengths between hmine = 0.15 mm and

hmaxe = 1.0 mm at the beginning and during the adaptive computation. The element size limits

for the clamped steel cylinder are hmine = 0.15mm and hmax

e = 3.0mm. A constant static frictioncoefficient of �s = 0.15 mm and a constant kinetic friction coefficient of �k = 0.05 are assumedfor contact. For the adaptive calculation of this problem, the ‘gradient of plastic work’ is usedas error indicator for both components with scaling factor �3 = 30 N.

Figure 17 shows deformed meshes and appropriate equivalent plastic strain distributions attime instances t = 20, 40, 60 and 80 �s with a logarithmic spectrum as in Camacho and Ortiz[9]. The physical process gets obvious: the impactor penetrates the steel cylinder and producesa deep crater. At the same time, the melted tungsten is pressed outwards and the bar almostcompletely erodes. Plastic deformations are very high in both parts. These phenomena can beobserved in experiments [26] and in corresponding simulations [9] in the same way.



Figure 18. Long rod penetration: discretization and temperature distribution.

Figure 19. Trajectories and velocity histories of rod nose and tail.

In our calculation with repeated remeshings, the spatial discretization adaptively follows thetransient process. At the beginning, strong refinement in the process zone under the impactor isascertained. Beyond this area of high plastic deformation, the material behaves elastically andtherefore the mesh density is relatively low. The fine mesh region increases with proceedingpenetration depth. The error indicator ‘gradient of plastic work’ is apparently well suited tofollow the physics of this problem in an intelligent way.

The discretized contact zone is shown in more detail at t = 20 �s in Figure 18. The corre-sponding temperature distribution reveals that large plastic strains cause significant temperaturerise with high gradients. Especially the nose of the WHA rod nearly reaches its meltingtemperature and therefore behaves like a viscous fluid, which is laterally pressed outwards.

Anderson et al. [26] determined in their experiments the positions of front and rear endof the impactor rod at different time instances directly via X-raying (crosses in Figure 19left). The appropriate velocities were calculated indirectly by averaging and time differentiation(dashed lines in Figure 19 right). Comparison with our simulation results show very good



agreement. The displacements are nearly identical and also the velocity of the tail is matchedvery well. Slight deviations can be observed for the nose velocity, which is—among otherreasons—caused by the indirect determination of experimental results through averaging. Thehigh-frequent structural response in the process zone effects identifiable oscillations in thevelocity history, however these fluctuations are smaller than in Camacho and Ortiz [9].

8. CONCLUSION

A computational approach for the simulation of non-linear transient impact problems has beenpresented. A general solution procedure with appropriate spatial discretization for plane strainand axisymmetrical cases, explicit time integration scheme, a constitutive model for finite strainthermo-visco-plasticity and algorithmic treatment of frictional contact provided a sophisticatedbasis. To manage the large deformations of those fast transient, highly non-linear impact pro-cesses an adaptive remeshing strategy for the quality control of simulations was proposed. Themain parts of the procedure are a mesh quality check for automatic triggering of remeshing, anassessment of discretization errors via indicators, a mesh generation tool and mapping methodsfor the transfer of state variables from old to new finite element meshes. Different types of in-dicators are used to control an error driven distribution of computational resources. The choiceof an indicator or a combination of indicators mainly depends on two items. An engineeringintuition and an insight into the physics of the problem at hand are one essential base forthe selection of suitable indicator(s). On the other hand, the specific quantities of interest of asimulation also heavily influence this selection. Different numerical examples showed the ap-plicability and the effect of different indicators. The overall strategy has proven to be reliable,robust and efficient.

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Robust adaptive remeshing strategy for large deformation, transient impact simulations

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