Adaptive Mesh Technique for Thermal–Metallurgical Numerical

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng (in press)Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2083

Adaptive mesh technique for thermalmetallurgical numericalsimulation of arc welding processes

M. Hamide, E. Massoni and M. Bellet,

Ecole des Mines de Paris, CEMEF, UMR CNRS 7635, BP 207, Sophia Antipolis Cedex 06904, France

SUMMARY

A major problem arising in finite element analysis of welding is the long computer times required for acomplete three-dimensional analysis. In this study, an adaptative strategy for coupled thermometallurgicalanalysis of welding is proposed and applied in order to provide accurate results in a minimum computertime. The anisotropic adaptation procedure is controlled by a directional error estimator based on localinterpolation error and recovery of the second derivatives of different fields involved in the finite elementcalculation. The methodology is applied to the simulation of a gastungsten-arc fusion line processed ona steel plate. The temperature field and the phase distributions during the welding process are analyzedby the FEM method showing the benefits of dynamic mesh adaptation. A significant increase in accuracyis obtained with a reduced computational effort. Copyright q 2007 John Wiley & Sons, Ltd.

Received 28 August 2006; Revised 22 March 2007; Accepted 29 March 2007

KEY WORDS: finite elements; welding; heat transfer; phase transformation; mesh adaptation; anisotropicmetric; error estimation

1. INTRODUCTION

The accuracy of a numerical solution obtained by the finite element method depends on the spatialdiscretization of the physical domain. In general, the desired element sizes in different directionsare influenced by the physical and geometrical features of the problem which can vary significantlyin time and space. In many physical problems, including welding, the solution exhibits anisotropicfeatures creating a demand for elements which are aligned with the solutions anisotropy. Inrealistic cases, the information required to compute the desired solution field to an acceptablelevel of accuracy is unknown a priori. An efficient approach to overcome this difficulty consistsin applying an adaptative procedure in which the errors arising from spatial discretization arecontrolled within a specified tolerance. An anisotropic adaptative procedure modifies the mesh insuch a way that the local mesh resolution becomes adequate in all directions.

Correspondence to: M. Bellet, Ecole des Mines de Paris, CEMEF, UMR CNRS 7635, BP 207, Sophia AntipolisCedex 06904, France.

E-mail: [email protected]

Copyright q 2007 John Wiley & Sons, Ltd.

M. HAMIDE, E. MASSONI AND M. BELLET

The concentrated heat input that appears in most welding applications requires a refined dis-cretization in the neighborhood of the molten region below the moving electrodes, where strongaxial and transverse thermal gradients prevail. In addition, some induced solid-state phase changeoften generate residual gradients of phase fractions. The capture of such thermal and metallurgicalgradients requires some kind of remeshing capability in order to continuously maintain or regeneratea finely discretized region moving with the heat source. The initial work on an automated remesh-ing strategy for welding applications was performed by Lindgren et al. [1]. Their work includedremeshing of a moving region but did not use any error estimation to guide the remeshing schemeand control the accuracy of the solution produced. They prescribed the refinement/coarsening inthe input file so that a smaller distance to the source gave smaller elements. The size of the elementbehind the heat source was also predetermined. Recently, Runnemalm and Hyun [2] proposed anadaptative strategy that evaluates both the thermal and the mechanical error distribution using aZienkiewiczZhu error estimator [3]. It is combined with a hierarchic remeshing strategy using aso-called graded element. In this approach, the directionality of the error estimation is ignored,resulting in isotropic adaptative remeshing.

In the present paper, following the approach initiated by Fortin [4] and Alauzet et al. [5], we placeparticular emphasis on the anisotropic mesh adaptation process generated by a directional errorestimator based on the recovery of the second derivatives of the different fields involved in the finiteelement solution. The goal of this approach is to achieve a mesh-adaptative strategy minimizingthe directional error estimation in the mesh. As shown in this paper, this approach allows us torefine the mesh, stretch and orient the elements in such a way that, along the adaptation process,accurate controlled solutions are obtained while keeping the number of unknowns affordably low.

The organization of the paper is as follows. Section 2 introduces the numerical model that is usedto solve and describe the welding process. In this paper, the analysis is limited to coupled thermalmetallurgical simulations of welding. Section 3 presents the overall anisotropic mesh adaptationprocedure: the anisotropic error estimator, together with the procedures to get the recovered Hessianmatrix are described. In this section we discuss the Hessian strategy and review the concept of amesh metric field. Finally, in Section 4, the application of different anisotropic adaptative strategiesto welding simulations is presented and the results obtained are discussed.

2. WELDING ANALYSIS

During welding, the interaction of the heat source and the material leads to rapid heating, melting,and the formation of the weld pool. When the heat source moves away, the weld pool cools andsolidifies. Depending on the welded alloys, as the temperature decreases, various solid-state phasetransformations take place resulting in the final microstructure of the weldment. The properties of aweldment, such as strength, ductility, toughness, and corrosion resistance are significantly affectedby its microstructure. Thus, it is important to understand the temperature and microstructureevolution during welding: this is the purpose of the present model. In the next two sections, a briefoverview of the thermalmetallurgical simulations of welding is given. The next two paragraphsgive a brief overview of the thermalmetallurgical model.

2.1. Heat transferAssuming thermal equilibrium at the microscopic scale, the transient heat transfer in a multiphasesolid continuum is governed by the following volume averaged equation (for more details see

Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (in press)DOI: 10.1002/nme

ADAPTIVE MESH TECHNIQUE IN THE 3D THERMALMETALLURGICAL SIMULATION

Appendix A):

Ht

(T )= k=2,N

i>1i =k

gik k>1k = j

gk j

Hk (1)

where g denotes the volume fraction and the index k denotes either the liquid (k = 1) and (k = 2, N )for different metallurgical phases that may exist in the solid state (see next section). The volumetricenthalpy function H is defined as the integral of the heat capacity with respect to temperature:

H(T )= T

0cp d + glLv (2)

where gl is the liquid volume fraction, the density, cp is the specific heat, and Lv denotes thelatent heat of fusion/solidification per unit of volume.

The term on the right-hand side of Equation (1) includes latent heat effects associated with solidphase transformations. The enthalpy change associated with the i j transformation is equal to:Hi j = gi (Hj Hi ).

This formulation permits then to take into account the energy changes associated with thesolid-state phase changes, while taking advantage of the stability and robustness of the enthalpyformulation for the liquidsolid phase change.

2.2. Solid-state phase transformation modelA series of phase transformations take place in both the fusion zone (FZ) and the heat-affectedzone (HAZ) during welding of low alloyed steels. A typical microstructural history of theFZ is (ferrite, pearlite) austenite liquid austenite (ferrite, pearlite, bainite,martensite),while a typical microstructure evolution in the HAZ corresponds to (ferrite, pearlite)austenite (ferrite, pearlite, bainite,martensite). In the HAZ, the transformation during heating(austenization) is of importance because it affects the kinetics of phase transformations duringcooling.

During heating, the calculation of austenite formation for an arbitrary thermal evolution is basedon the Leblond model for low carbon steel [6]. The rate of transformation is described accordingto the expression:

ga = geqa (T ) ga(T )

(T )(3)

(T )= (T Ae3)

(4)

geqa (T )= T Ae1Ae3 Ae1 (5)

The transformation is described by Equation (3), where ga is the volume fraction of austenite andga its time derivative, Ae3 the transformation equilibrium temperature, geqa the phase equilibriumdefined in Equation (5), is a function of temperature T , given by Equation (4) and and arematerial constants.



The model assumes that austenite starts to form when the temperature is above the austenitetransformation equilibrium temperature Ae1. In the present work, the influence of grain size onfurther transformations during cooling is neglected. Therefore, the evolution of grain size duringaustenization is not modeled.

During cooling, the simulation of diffusive metallurgical transformations is based on TTT (timetemperature transformation) diagrams with a Scheils additivity rule for incubation or nucleationand the JohnsonMehlAvrami (JMA) equation for growth. The use of the TTT diagram in thecase of a non-isothermal transformation is done considering its decomposition into successiveisothermal incremental transformations.

For incubation, the prediction of the beginning of the transformation of austenite into ferritepearlite or bainite is achieved using the so-called additivity principle, following Scheils rule, assuggested by Denis et al. [7] and Fernandes et al. [8]. The considered transformation is supposedto start when t

0

dt(T )

= 1 (6)

where (T ) is the incubation time to transform the fraction g isothermally at temperature T .For growth, a JMA law (Equation (7)) is used to compute the fraction of ferrite, pearlite or

bainite transformed:gk(t, T )= 1 eb(T )tn(T ) (7)

where gk describes the fraction transformed in an isothermal reaction as a function of time t andb, n are temperature-dependent coefficients, to be determined from TTT diagrams (cf. [7, 8] fordetails).

The calculation of the martensitic transformation is based on KostinenMarburger equation [9,Equation (8)] and is dependent on the maximum austenite fraction gmaxa and temperature. It isassumed that the transformation starts at the martensitic start temperature, Ms, being a materialconstant which may depend on: composition, cooling rate, stress state

gm(t, T )= gmaxa (1 e(MsT )) (8)

3. MESH ADAPTATION

Adaptative finite elements problems are generally based on isotropic, a posteriori error estimates.Basically, a posteriori error estimators can be classified into two types. The first of these wasintroduced by Babuska and Rheinboldt [10, 11] and is based on evaluating the residuals of theapproximate solution. Therefore, it strongly depends on the problem operator. The second approach,proposed by Zienkiewicz and Zhu [3], estimates the error in gradient-based norms, using somerecovery process (nowadays often called ZZ error estimators).

Recently, error estimators have been proposed for anisotropic meshes [12], the goal beingto reach a given level of accuracy with fewer vertices. The anisotropic interpolation estimatesintroduced in [13] were used, together with a ZienkiewiczZhu error estimator to approach theerror gradient.

In the present paper, we place particular emphasis on the anisotropic mesh adaptation processgenerated by a directional error estimator based on the recovery of the second derivatives of thefinite element solution (the so-called Hessian strategy).



3.1. The a posteriori error estimator

To obtain directional information of the error we use the Hessian strategy [5, 14], a method in whichthe fields second derivatives are used to extract information on the error distribution. The Hessiancan be computed from any scalar component of the solution fields, in our case, the temperature orthe different phase fractions. As shown further, this directional information can be converted intoa mesh metric field which prescribes the desired element size and orientation.

Let us present the method as briefly as possible, since the details can be found in References[5, 15, 16]. Denoting u the exact solution of the consistent scalar field and uh its discretization,we use an indirect approach to estimate the error u uh. It has been proved that for ellipticproblems, the finite element error can be bounded by the interpolation error (Ceas lemma [17]):

u uhcu hu (9)where hu is the linear interpolate of u on the mesh and c is a constant independent of the currentmesh.

Here, we assume that this relation still holds in the class of problems envisaged. Actually, similaranalysis based on the interpolation error shows (practically) that the link between the interpolationerror and the approximation error is even stronger than the bound given by Ceas lemma [17].Therefore, the interpolation error appears a reasonable way of defining an error estimate accordingto [4].

The function u being supposed sufficiently smooth can be expanded into a Taylor series. Thenthe interpolation error has a upper bound proportional to the second derivatives of u [14, 17]. Toexpress this upper bound, let us consider the following assumptions and notations:

The function u is regular enough and is associated with the solution of our welding problem. K =[a, b, c, d] denotes a tetrahedron of the finite element mesh. The P1 interpolation of u on element K is an affine function on K . hu coincides with u at the vertices of K .To bound the error u hu, we use its Taylor expansion with integral rest at a vertex of K

(for instance a) with respect to any interior point x in K :

(u hu)(a)= (u hu)(x) + xa (u hu)(x) + 1

0(1 t)(ax [Hu(x + txa)]ax) dt

where Hu denotes the second derivative of the variable u. Let us assume now that the maximalerror is achieved at point x (closer to a than to b, c, or d), so (u hu)(x)= 0, then we have(xa (u hu)(x))= 0. As hu coincides with u at the vertices of K ((u hu)(a)= 0), wehave

e(x)= 1

0(1 t)(ax [Hu(x + txa)]ax) dt (10)

Let a be the projection of a on the tetrahedron surface, according to the direction ax (a is locatedon the face opposite to a). There exists a positive real number such that ax = aa. As a is closerto x than any other vertex of K , then 3/4 [5]. Then, we obtain

|e(x)| 916

10

(1 t)(aa [Hu(x + txa)]aa) dt (11)



Finally

|e(x)| 932

maxyK |(aa [Hu(y)]aa)| (12)

Relationship (12) is not useful in practice as the bound depends on the extremum x which is notknown a priori. However, it can be reformulated as follows [5]:

eK c maxxK maxv (v |Hu(x)|v) (13)

where c = 9/32, v is any vector joining two interior points in K and |Hu | is the absolute value ofthe Hessian matrix of the solution (i.e. consisting of absolute eigenvalues).

The bound of the previous relation is difficult to compute. As we can replace all vectors includedin K with a combination of the edges of K , a new upper bound error can finally be obtained [5]:

eK c maxxK maxe (e |Hu(x)|e) (14)

where e denotes one of the six tetrahedron edges.The Hessian strategy involves the computation of the symmetric matrix of second derivatives

that can be decomposed as |Hu | = R||RT, where R is the eigenvectors matrix and = diag(k) isthe diagonal matrix of eigenvalues. The directions associated with the eigenvectors k are referredto as principal directions and the eigenvalues k are then equivalent to the second derivatives alongthe local principal directions.

3.2. Metric definitionPerforming anisotropic mesh adaptation requires a way of defining the desired element size distri-bution over the domain. Mesh metric tensors are used to represent an anisotropic mesh size fielddefining the desired mesh anisotropy at a point (see, for example, [5, 18]). The concept of a meshmetric field is used to represent the desired size field as a tensor over the domain.

The Hessian strategy is based on the idea that a high magnitude of an eigenvalue implies a higherror in the direction associated with the corresponding eigenvector, so a small element size wouldbe desired in this direction. Conversely, a low eigenvalue magnitude in a particular eigendirectionsuggests that the element size could be large along this direction.

To achieve a suitable mesh resolution in different directions, a uniform distribution of localerrors is applied in the principal directions which leads to ch2k k = , where is the user specifiedtolerance for the error and hk is the desired size in the kth principal direction. So, the edges e ofthe adapted mesh must check = c(e Me).

The stated goal of the mesh adaptation algorithm is to yield a mesh with regular elements inthe metric space where each edge e must satisfy the following relation (see, for example, [18]):

e Me = 1 (15)A mesh with all its edges satisfying the above relationship is commonly referred to as a unit mesh.A mesh metric tensor M is then obtained at each node by calculating a scaled eigenspace of therecovered Hessian matrix as M = RRT, where R is the eigenvector matrix and = (c/) is thediagonal matrix of scaled eigenvalues.

Truncation values hmin and hmax are specified to limit mesh sizes. One reason for truncating theelement size, in terms of edge lengths, is to avoid singular metrics. For example, it is necessary



Figure 1. Two-dimensional schematic of the intersection of mesh metric tensors represented by ellipses.

to apply hmax in case of a null or quasi-null eigenvalue in the direction where the solution doesnot vary. The modified eigenvalues of the Hessian matrix then become

k = min(

max

(c

|k |, 1h2max

),

1h2min

)(16)

where is the prescribed error.When several variable fields u are considered concurrently, the previous approach leads to a

metric for each variable and we chose to take the intersection of the different metrics. In practice,the intersection of metrics is achieved by the simultaneous reduction of two quadratic forms whichis a valid operation since the metric tensors are positive definite, for details see [5]. We illustratethe procedure from a geometrical point of view in Figure 1. It can be shown that this processallows one to compute a common basis for the two quadratic forms that can be used to determinethe ellipsoid with maximum volume contained in the geometrical intersection of the two candidateellipsoids. The ellipsoid representing the final intersected metric is the one with the maximumvolume contained in the common volume of all the candidates and therefore respects the sizerequirements of the different metrics.

3.3. Relative error

To combine various variables together to construct a metric, it is often reasonable to consider arelative bound on the error. In order to have dimensionless error, we define the following estimation:u hu|u|

,K c maxxK maxeEk(

e |Hu(x)||u| e)

(17)

where |u| = max(|u|, u,) with is a cut-off to avoid numerical difficulties (in case of anull or quasi-null value of u).

3.4. Hessian evaluation

To compute the Hessian matrix Hu of the P1 field u, we reconstruct in two steps the secondderivatives at each node P by using the computed solution from the patch S of all elements Ksurrounding node P . This is done as follows.



In the first step, we recover the gradient at node P by taking the volume weighted average ofgradients on elements in the patch S surrounding the node P . Note that u being a P1 field, itsgradient is constant on each element K :

h(uh)(P)= 1KS |K |

( KS

|K |(uh)|K)

(18)

It can also be noticed that this is equivalent to a lumped-mass approximation of a least-squaresreconstruction of the gradient for linear elements.

In the second step, the same procedure is now applied to the P1 field h(uh) to obtain therecovered Hessian matrix:

(Hu)i j (P)= 1KS |K |

( KS

|K |(

x j

(h

uhxi

)))(19)

3.5. Transfer of variablesState variables should be transferred from the old to the new mesh after remeshing. In the contextof thermalmetallurgical simulation of welding, these variables consist of enthalpy and phasefractions (liquid and solid). Since they are defined at nodes, a direct interpolation can be used.For each node of the new mesh, its location in the old mesh is determined (element and localisoparametric coordinates). The new values at this node are then obtained by interpolation in theelement of the old mesh. It should be noted that the temperature field can be determined, in asecond step, from the new value of the enthalpy and phase fraction.

4. APPLICATION TO WELDING SIMULATION

4.1. Welding conditions and material properties

We consider a thick plate of A508 steel, the dimensions of which are given in Figure 2(a). Thetemperature-dependent thermophysical properties are given in Figure 3.

The welding parameters chosen for this analysis are as follows. Welding process: gastungsten-arc welding (GTAW); welding current I = 150 A, welding voltage V = 10 V and traveling speed

Figure 2. (a) Geometry of the specimen (dimensions in mm) and comparison point A and (b) finite elementreference mesh of one half of the plate.



Figure 3. Thermophysical properties used in analysis (SI units).

of 1 mm s1. The weld efficiency is assumed to be = 0.65. The associated heat input, I V ,moving with the electrode, is simulated by a simple surface flux with uniform distribution withina disc of radius 5 mm.

4.2. Finite element model

For the simulation study, only one-half of the plate is analyzed due to symmetry. The boundaryconditions of the welding process incorporate heat transfer boundary conditions. The symme-try surface is defined as under adiabatic boundary conditions. On all other surfaces, bound-ary conditions of convection and radiation with the environment are applied with a convectivecoefficient h = 12 W m2 K1 and emissivity coefficient = 0.75 and an external temperatureText = 25C.

To evaluate the efficiency of our adaptative procedure, we first obtain results on a fine mesh(Figure 2(b)). The mesh size along the electrode path has been fixed after a preliminary studyinvolving different meshes of different mesh sizes in this region. The value 1 mm for the meshsize has been fixed after checking the stationarity of the temperature solution with mesh size. Theresult obtained is then used for purpose of comparison. Several simulations have been performedwith adaptative remeshing. The calculations differ by the spatial discretization, all other conditionsbeing identical. Two types of simulation with remeshing have been carried out:

Thermal-driven mesh adaptation: The adaptative technique is based on the thermal error distri-bution.

Thermometallurgical-driven mesh adaptation: The automatic mesh refinement is based on boththermal and metallurgical error distributions.

4.3. Thermal-based mesh adaptation

The reference FE model without remeshing consists of 14 329 nodes and 68 891 linear tetrahedralelements and is presented in Figure 2(b). As indicated above, the minimum mesh size is 1 mmalong the electrode path, the maximum mesh size being 10 mm. The initial FE model used incalculations with remeshing consists of 6842 tetrahedral elements (1683 nodes).



Figure 4. Thermal-based mesh adaptation (= 0.01): (a) anisotropic FEM mesh; (b) zoom on refinedzone with anisotropic remeshing; (c) temperature distribution at time 95 s (K); and (d) zoom on

refined zone with isotropic remeshing.

The remeshing is performed at each time step (dt = 1 s). See Figures 2(b) and 4(a) for the FEmesh. As expected, the adaptative remeshing generates more refined elements in the neighborhoodof the thermal source and coarser elements in its trail. It can also be seen in Figure 4 that anisotropicelements aligned with the heat flow are created around the FZ. It should be noted that the aspectratio of the elements reach values from 1 to 10 in this region (the allowed aspect ratio for thissimulation was hmax/hmin = 10). The minimum mesh size is 1 mm.

As shown in Table I, the calculation on the fine reference mesh (without remeshing) required6 h and 25 min of CPU time. Two calculations with anisotropic remeshing have been performed,one with a prescribed error = 0.01 and a second one with = 0.005. The CPU time was 1 h and1 min for = 0.01 and 1 h and 52 min for = 0.005. The final number of elements in the secondcalculation is much higher than in the first one with the same truncation element size values. Asexpected, the calculation with = 0.005 generates a larger refined zone in the neighborhood of thethermal source than the calculation with = 0.01.



Table I. Refinement parameters (Nbe denotes the number of elements, himpmax the prescribed maximumelement size, himpmin the prescribed minimum element size, hmax the maximum element size, hmin the

minimum element size).

Initial Final himpmin h

impmax hmin hmax

Nbe Nbe (mm) (mm) (mm) (mm) CPU timeFine reference mesh 68 891 68 891 1 10 1 10 6 h 25 minCoarse reference mesh 11 439 11 439 2 10 2 10 58 minAnisotropic adapted mesh, = 0.01 6842 5866 1 10 0.95 11.5 1 h 1 minIsotropic adapted mesh, = 0.01 6842 10 685 1 10 0.95 11.5 1 h 57 minAnisotropic adapted mesh, = 0.005 6842 11 012 1 10 0.9 10.6 1 h 52 minIsotropic adapted mesh, = 0.005 6842 46 906 1 10 0.9 10.6 4 h 19 minNote: Calculation run on a Pentium 4 PC, 2 GHz with 2 Gb RAM.

Figure 5. Thermal-based mesh adaptation (= 0.01): (a) temperature evolution at Point A and(b) temperature distribution at time 95 s (K).

In order to demonstrate the efficiency of the remeshing procedure due to its anisotropic (i.e.directional) character, we performed additional comparisons between anisotropic and isotropicremeshings, using the same objective accuracy and the same truncation values hmin and hmax.The results are reported in Table I. It can be seen that the number of elements required to producethe same level of interpolation error is significantly different between isotropic and anisotropicmeshes. With = 0.01, the anisotropic mesh uses only 5866 elements, about half of the 10 685elements that are required by the isotropic mesh. The computation time is reduced by a factor 2,when using remeshing. A similar trend is found when prescribing a more stringent accuracylevel (= 0.005). A zoom on both meshes near heat source (Figure 5(b) and (d)) reveals thatthe elements of the isotropic mesh in the region are small and almost equilateral, whereas theanisotropic elements are stretched orthogonally to the temperature gradient.

An example of temperature distribution is given in Figure 5(c). The temperature evolutionat Point A in the different analyses is shown in Figure 5(a). The first observation from theplots in Figure 5(a) is that the results are significantly smoother in the time domain than inthe spatial domain, as shown in Figure 5(b). This illustrates that the spatial noise associated with



Figure 6. Thermal-based mesh adaptation (= 0.01): (a) bainite distribution at time 95 s and (b) timeevolution of phases proportions at Point A.

Figure 7. Evolution of the temperature difference T = |T Tref|, at Point A, where Tref is the temperatureobtained on the reference fine mesh.

the Hessian recovery does not globally pollute the solution in time, suggesting that the primaryfields (temperature and phase fractions) remain unaffected (Figure 6). The adaptative techniquemakes the FEM mesh much denser, so that the temperature distribution is more accurate than withcoarse mesh (see Figures 5 and 7).

We observe in Figure 5(a) that the temperature evolution Point a A converges to the referencetemperature evolution when reducing the prescribed error . From Table I and Figure 5(a), it canbe seen that for a comparable accuracy of the results, the use of an adaptative meshing procedurereduces CPU costs by a factor 6. This shows the efficiency of the proposed approach.

4.4. Thermometallurgical-based mesh adaptation

Comparing Figures 4 and 8 a clear difference between the obtained two meshes when usingonly the thermal error distribution (Figure 4(a)) or both the thermal and the metallurgical error



Figure 8. A thermometallurgical-based mesh adaptation (= 0.01): (a) FEMmesh and (b) zoom on refined zone.

Table II. Refinement parameters and results for thermalmetallurgical adaptation.

Initial Nbe Final Nbe himpmin (mm) h

impmax (mm) CPU time

Fine reference mesh 68 891 68 891 1 10 6 h 25 minAdapted mesh, error 0.01 6842 15 816 1 50 2 h 22 min

Note: Calculation run on a Pentium 4 PC, 2 GHz with 2 Gb RAM.

Figure 9. A thermometallurgical-based mesh adaptation (= 0.01): (a) temperature evolution at Point Aconverges and (b) temperature distribution at time 95 s (K).

distributions is evidenced. It is shown in Figure 4(a) that the automatic mesh refinement usingtemperature as error indication produced an elliptical zone in the vicinity of the FZ. A distinctbehavior is found when guiding the mesh adaptation with both phase proportion (in the present



Figure 10. A thermometallurgical-based mesh adaptation (= 0.01): (a) time evolution of phasesproportions at Point A and (b) bainite distribution at time 95 s.

Figure 11. Profiles of bainite volume fraction in a cross-section located at X = 0.095 m at time 250 s.

case, the bainite volume fraction) and temperature. In this latter case, the mesh is much denser inthe wake of the heat source in order to provide a better representation of steep gradients of phasefraction. It can be seen that the thermometallurgical-driven mesh generation produces significantlymore elements than the thermal-driven one (see Table II, Figures 4 and 8). This is due to theresidual gradients of phases fractions that remain in the welded component after welding. Thethermal-driven remeshing creates a much lower number of elements in the model. This is of coursedue to a smoother gradient field in the thermal analysis and also because the plate cools down toa uniform temperature.

Comparing Figures 10(a) and 6(b), it can be seen that the phase time evolutions in a givenpoint are not significantly affected by mesh adaptation, in agreement with the same trend fortemperature (Figures 9(a) and 5(a)). Regarding the spatial distribution of the phases, the impact



is much more significant, as expected. Figure 11 shows the residual profile of the bainite volumefraction in a transverse section, using different meshes. It can be seen that the phase distributionis much more accurate than with an adaptation based only on the thermal error distribution. Thegradients of phase fraction are better described than with the reference fine mesh. The coupledthermalmetallurgical results in an optimal description of the distribution of the different phases.This result is extremely important in terms of the prediction and assessment of the quality ofweldments, for which an accurate determination of the phase fractions prevailing in the HAZ is akey factor. Beyond the simple thermalmetallurgical approach considered here, this result is alsovaluable in view of further thermalmetallurgicalmechanical calculations which are necessary topredict the risk of failure during welding and the residual stresses and deformations.

5. CONCLUSION

In this paper, adaptive remeshing procedures have been presented and applied in the context ofcoupled thermalmetallurgical simulation of welding. The method is based on anisotropic meshadaptivity dictated by directional error estimators. Those estimators, based on the Hessian recoveryof P1 field, are used to construct a mesh metric field that provides information on the local meshresolution desired in different directions. The method allows to easily combine metric tensors forvariables of different types and nature.

In practice, the calculation results show that the temperature field and the distributions of phasefractions with adaptive mesh converge to the results obtained with reference mesh. For the casetested here, the calculation time comparison shows that the adaptive mesh technique can reducethe calculation time by almost one-third. It also reduces the data-storage requirement substantially.For some applications, both points are key factors in determining whether a successful FE simulationcan be completed or not.

Larger savings may be expected for application with longer welds as the zone associated withlarge gradients will be smaller relative to the total length of the weld. In the framework of thermal-metallo-mechanical simulation, the current logic for deciding the size of the elements in relationto thermal and metallurgical fields should be combined with mechanical fields: this should bestudied in future works. Future studies will incorporate a moving mesh strategy based on errorestimation to reduce the number of remeshing steps and accelerate the efficiency of the adaptativemethod.

APPENDIX A: VOLUME AVERAGING FOR HEAT TRANSFER

In the welding context, metallurgical transformations highly depend on temperature history. Con-versely, the impact of phase transformations (liquidsolid and solidsolid) on heat transfer shouldbe considered.

The application of the spatial averaging method to the equation of energy conservation in aelementary representative volume (REV) of the multiphase material [19], yields for each phase kthat may exist in the REV:

(gkkhk)t

+ (gkkhkvk) (qk)= Qk (A1)



where gk denotes the volume fraction of phase k (k = 1, N ), k the density of phase k, hk itsenthalpy per unit mass, vk its intrinsic average velocity, qk the intrinsic average heat flux vectorin phase k, and Qk the heat exchange rate with other phases.

Summing these equations for the different phases k = 1, N , and assuming a uniform temperatureon the REV and the Fourier law for heat conduction, we get, using the convention of summationfor repeated indices:

(gkkhk)t

+ (gkkhkvk) (gkkTk)= 0 (A2)

where denotes the heat conductivity.Neglecting advection effects, and denoting H the enthalpy per unit volume, we get

(gk Hk)t

(gkkTk)= 0 (A3)

Noting now that Hk/t = (Hk/T )T /t = (cp)kT /t , and denoting = gkk the averageheat conductivity and cp= gk(cp) the average heat capacity, this leads to

gkt

+ C pTt (Tk)= 0 (A4)

Let us define now some notations regarding the different phase changes that may occur in theREV. For each phase k, the rate of change of the volume fraction can be expressed by

gkt

= i =k

gik j =k

gk j (A5)

in which gik denotes the rate of transformation of phase i into phase k, using the followingconvention: gik>0 when phase i is partially transformed into phase k, and gik = 0 otherwise.

In what follows, we will separate the fusion/solidification from the other phase changes occurringin the solid state. The liquid phase will then be identified by the index k = 1, and the differentsolid phases by k = 2, N . Equation (A4) then becomes(

i =1gi1

1 = jg1 j

)H1 +

k=2,N

(i =k

gik k = j

gk j

)Hk

+cpTt (Tk)= 0 (A6)(i =1

gi1 1 = j

g1 j

)H1 +

k=2,N

(1 =k

g1k k =1

gk1

)Hk

+ k=2,N

i>1i =k

gik k>1k = j

gk j

Hk + cpTt (Tk)= 0 (A7)



The two first terms deal with fusion or solidification, while the third one encompasses solid-statephase transformations only. Rearranging the two first terms, denoting now the liquid phase withthe index l 1, and putting the term of solid-state transformations on the right-hand side, we get

cpTt +

k=2,N

(i =1

gi1 i =k

gik

)(Hl Hk) (Tk)

= k=2,N

i>1i =k

gik k>1k = j

gk j

Hk (A8)

The first term can be reasonably approximated by Lvgl/t , with Lv the latent heat of fusion perunit volume. We then obtain

cpTt + Lvglt

(Tk)= k=2,N

i>1i =k

gik k>1k = j

gk j

Hk (A9)

For the sake of simplification, we approximate the two first terms by the time derivative of H , afunction of the temperature only, which is defined as follows and can be seen as a pseudo-averageenthalpy:

H(T )= T

0cp d + glLv (A10)

In this expression, T0 is a reference temperature and the averaging of the heat capacity cp isdefined a priori, that is with fixed predetermined volume fractions of the different phases for eachtemperature. This approximation finally yields:

Ht

(T )= k=2,N

i>1i =k

gik k>1k = j

gk j

Hk (A11)

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