Adaptive Mesh

Three-dimensional welding analysis using an adaptive meshscheme q

H. Runnemalm a, S. Hyun b

a Volvo Aero Corporation, 461 81 Trollh�attan, Swedenb Department of Mechanical Engineering, Lule�a University of Technology, 971 87 Lule�a, Sweden

Received 10 May 1999

Abstract

One major problem arising in ®nite element analysis of welding is the long computer times required for a complete three-dimen-

sional analysis. An adaptive strategy for coupled thermo-mechanical analysis of welding is applied in order to reduce the computer

time. The paper describes a generic posteriori error formulation that evaluates both the thermal and the mechanical error distribution.

It is combined with a hierarchic remeshing strategy using a so-called graded element. The error indicator together with the known

movement of the local heat source is used to predict areas of re®nement. An increased accuracy is obtained with a reduced compu-

tational e�ort. Ó 2000 Elsevier Science S.A. All rights reserved.

1. Introduction

Experimental procedures are used for the development of weld processes and schedules, material se-lection and weld joint design when starting the manufacturing of a new product. The problems associatedwith these experimental procedures are costly and time consuming.

Therefore their use in an optimisation process is limited. Combining the experimental procedure withnumerical analysis will enhance the process of forming a welding procedure speci®cation (WPS). It willmake it possible to also consider residual stresses and deformations. It may even be possible to performsimulation as part of an optimisation process. However, then it is necessary that the required computa-tional time for a simulation is small.

Researchers have developed the ®nite element method (FEM) since early 1970 in order to predict thematerial response due to welding. The computer power has increased in several orders of magnitude. Still,the computational cost of a three-dimensional welding analysis is high if a ®ne mesh is used near the weld.

A number of improvements of the ®nite element method have been made during the last decade. Thedevelopment of grading elements is one of the improvements that signi®cantly have enhanced modelling ofwelding. It facilitates the creation of a graded mesh. This is needed due to the moving, concentrated heatsource and the local e�ects in the vicinity of the heat a�ected zone (HAZ). Formulation of grading ®niteelements has been proposed by McDill et al. [1] and the use of these elements has been demonstrated in anumber of welding applications [2±4]. Elements with variable number of nodes and using constraintequation to make the element conformable is described by Bathe [5]. This type of element was used byBrown and Song [6] in the analysis of welding.

www.elsevier.com/locate/cmaComput. Methods Appl. Mech. Engrg. 189 (2000) 515±523

q Correspondence to: L.E. Lindgren, Lulea University of Technology, 971 87 Lulea, Sweden. Tel.: +46 920 91306; fax: +46 920

99692; e-mail: [email protected].

0045-7825/00/$ - see front matter Ó 2000 Elsevier Science S.A. All rights reserved.

PII: S 0 0 4 5 - 7 8 2 5 ( 9 9 ) 0 0 3 0 4 - 7

In recent years three-dimensional models in analysis of welding have become more frequent, but stillgeometrical complex structures are rarely, if ever, seen. In 1988 the ®rst full scale three-dimensional modelof a weld using shell elements [7] was presented and in 1990 an analysis using solid element [8] of welds waspresented. The same application, a circumferential weld on a pipe was studied in these models. More re-cently researchers have presented results from multi-pass welding analysis of three-dimensional pipes [9,10].

2. Basic notation

The ®nite element formulation used in this work is implemented in an in-house code, SIMPLE. Thereason for using this code is mainly because it includes a large number of tools that are specially designed todeal with problems arising when setting up welding analysis. Some of them are;· Staggered thermo-mechanical solution process.· Automatic mesh re®nement and coarsening.· Error indication to control the adaptive meshing (this work).· Automatic assignment of properties to the element in the weldpath, either by the quiet or inactive ele-

ment technique [11].· The ability to assign cut-o� temperatures for material properties.· Geometry-based user input for model de®nition.· Analysis of the microstructure evolution for hypoeutectoid steels [12].

2.1. Welding analysis

Prediction of stresses induced in materials during welding using FEM has been a research task since thebeginning of 1970 [13,14]. A solution to this problem requires the consideration of several processes. Firstof all the thermal history of the weld must be predicted. Several authors have used FE analysis to predictaccurate thermal ®elds, some of those are [9,15,16]. Secondly the mechanical behaviour must be predictedaccurately. In this paper we follow the strategy outlined by Lindgren et al. [11]. The main steps of thestaggered coupled thermo-mechanical analysis performed in this study are shown in Box 1.

Box 1. Main steps in thermo-elasto±plastic analysis using a staggered approach.

Step 1. Initialize analysis and start increment stepping

Step 2. Assemble the conductivity and heat capacity matrix for thermal analysis

Use geometry of last converged mechanical solutionUse effective heat capacity matrixCompute thermal load

Step 3. Solve for the temperature ®eldIterate until convergence is reached (Step 2).

Step 4. Assemble the sti�ness matrix for mechanical analysis

Use temperature solution to calculate thermal strain �eth�Compute mechanical load

Step 5. Solve for deformations

Perform stress updatingCompute mid point strainCompute rotation matricesRotate all relevant quantities back to unrorate referenceUse an e�ective stress function (ESF) algorithm to compute stress increment.Rotate relevant quantities forward to con®guration at the end of the time step.

Evaluate residual forces of the solutionIterate until convergence is reached (Step 4).

Step 6. Go to next time step (Step 2)

516 H. Runnemalm, S. Hyun / Comput. Methods Appl. Mech. Engrg. 189 (2000) 515±523

Volume changes or transformation plasticity was assumed to be negligible as the material in the weldedparts was an austenitic stainless steel (AISI 304L). The viscous material behaviour at high temperature wasneglected. This means that the total strain rate was decomposed into three components i.e. elastic, plasticand thermal strain rates.

_e � _eel � _epl � _eth: �1�The material was considered to be isotropic and the yield criterion according to von Mises was used. TheGreen±Naghdi objective stress rate was used. We also assumed an isotropic hardening behaviour. Thematerial data at elevated temperature are uncertain and numerical problems become more severe at ele-vated temperature [17], therefore a mechanical cut o� temperature at 1000�C was used. The accumulatedplastic strain and thermal expansion was set to zero above this temperature. All mechanical materialproperties are evaluated at the cut-o� temperature if the computed temperature is above 1000�C. It hasbeen shown by Oddy et al. [18], that spurious stresses might be introduced into the model if the mechanicaland thermal ®elds are of di�erent orders. To prevent this we used a constant temperature evaluated at thecentre of each subdomain within the graded element.

3. The adaptive remeshing scheme

The concentrated heat input that appears in most welding applications requires a re®ned discretisation inthe neighbourhood of the weld. The obvious solution to this is to have a mesh with a ®ne region that ismoving with the heat source. This solution requires some kind of remeshing capability. Tetrahedral elementis frequently used in automatic meshing schemes, however it has been shown that linear hexahedral ele-ments are superior to linear tetrahedral elements in many applications [19,20]. The work in this paper buildsupon the graded hexahedral element described by McDill et al. [1]. The initial work of an automatedremeshing strategy for welding applications was performed by Lindgren et al. [2]. Their work includedremeshing of a moving region but did not use any error measure of the solution to control the remeshingscheme. They prescribed the re®nement/coarsening in the input ®le so that a smaller distance to the sourcegave smaller elements. The size of the element behind the heat source was also predetermined.

3.1. The error measures

Error measures based on gradient ®elds are employed in order to devise a procedure for automaticallyadjusting the element sizes in the FE analysis. The generic posteriori error estimator described by McDillet al. [21], is deployed in the present work. These errors are computed so that they are not in¯uenced bychanges of physical dimensions or reference values. Error measurements of gradient ®elds, stress and heat¯ux, are obtained by

eex;r � rex ÿ rfe; �2�

eex;q � qex ÿ qfe; �3�where rex; qfe is the exact stress and heat ¯ux and rfe; qfe is the stress and heat ¯ux by FE analysis.

The exact solutions are not available during the FE analysis. Therefore estimated stress errors ees areobtained from a smoothed continuous gradient ®eld r� and q�. This gradient ®eld is obtained by averagingthe nodal gradients using least square projections [22,23] from the gauss points.

Then estimated errors for element i are obtained as,

e�i�es;r � r�i�� ÿ r�i�fe ; �4�

e�i�es;q � q�i�� ÿ q�i�fe : �5�The L2 norms of stress and heat ¯ux errors for element i are expressed as:

ke�i�es;rkL2�

ZX�i��r�i��

�ÿ r�i�fe �T �r�i�� ÿ r�i�fe � dX

�1=2

; �6�

H. Runnemalm, S. Hyun / Comput. Methods Appl. Mech. Engrg. 189 (2000) 515±523 517

ke�i�es;qkL2�

ZX�i��q�i��

�ÿ q�i�fe �T �q�i�� ÿ q�i�fe � dX

�1=2

: �7�

Then the global error estimate for both stress and heat ¯ux are expressed as

keesk2 �XM

i�1

ke�i�es k2; �8�

where M is total number of elements in the old mesh.The local and global error estimates are normalized by the smoothed gradient norm. The smoothed

gradient norm for element i is expressed as:

kg�i�r kL2�

ZX�i��r�i�� T �r�i�� dX

� �1=2

; �9�

kg�i�q kL2�

ZX�i��q�i�� T �q�i�� dX

� �1=2

: �10�

And their corresponding global values are,

kgk2 �XM

i�1

kg�i�k2: �11�

Smoothed gradient norms (9) and (10) are used to normalize the element error estimates given by (6)and (7).

The normalized local error estimate for element i is expressed as

E�i� � ke�i�es kkgk : �12�

Then the normalized global error estimate is given as

E � keeskkgk �13�

The average error for the mesh is

kemk2 � keesk2

M�14�

and its normalized form is

Em � kemkkgk : �15�

The local normalized error value E�i� is used to determine where to remesh in the gradient ®eld.

3.2. The re®nement strategy

Using the error indicator described above makes it possible to create an optimal new mesh. Two de®-nitions of an optimal mesh, with respect to the error measure, can be found in the literature. The ®rst one isproposed by Zienkiewiczs and Zhu [24] and is based on equal distribution of the global error between allelements in the mesh. The second is proposed by Bugeda et al. [25] and is based on uniform distribution ofthe error density in the mesh. This paper will follow the ideas of Zienkiewicz and Zhu.

It is possible to relate the global accuracy (Eq. (16)) and the element accuracy (Eq. (17)) to the char-acteristic length �h� as


kek6Cehpÿm�1; �16�

ke�i�k6Ce�i�hpÿm�d ; �17�

where the characteristic element length �h� of an element is the maximum length of the diagonals in anelement, i refers to a separate element and d is 1, 2 or 3 depending on the dimension of the problem, p is theorder of the shape function used in the element and m is the highest order of differentiation in the strain±displacement relation.

Following the ideas of Li et al. [26] we can formulate a relation between the old and the new charac-teristic length of an element that is used for analysis in dimension d. We assume that we have a measure ofthe normalized element error and the normalized global error in the new mesh described as

E�2 �

XM�i�1

E��i�2; �18�

where the �� denotes values in the new mesh.If we assume that we have found a new mesh that is optimal, then the element error should be identical

for all elements in order to satisfy the assumption of equal distribution of the global error in each element.This yields according to Eq. (14)

E��i� � 1��M�p E�: �19�

Using Eq. (17) to compute the element error in the old and the new mesh gives:

E�i� � Chpÿm�d ; �20�

E��i� � C�h�pÿm�d

: �21�We assume that the constant C and C� are equal and rearrange Eqs. (19)±(21) to get an expression betweenthe new and the old element sizes.

h� � E�i�

E��M�p� �ÿ 1=pÿm�d� �

h; �22�

where h and h� are the old and the new characteristic length of an element, E� is the permissible error in anelement given by the user. M� is found by assuming equal distribution of the global error in each element.

M� �XM

i�1

E�i�es

E�

� � dpÿm�d

" #2�pÿm�d�=�2pÿ2m�d�

: �23�

The re®nement strategy is based on the local element error (17) which is believed to give advantageousresult over a strategy based on the global error (16), especially since the error in the mesh comes from arelatively small region when performing a welding analysis.

A combination of the errors from the thermal and the mechanical ®elds is used in welding analysis. Thisis necessary as the thermal and mechanical ®elds may have di�erent requirements on the element sizes. Themaximum of the thermal and mechanical errors in each element is used in the remeshing.

E�i�es � max E�i�r ;E�i�q

� �: �24�

The user speci®es a lower and an upper element size. Otherwise the automatic re®nement/coarseningprocedure may lead to extremely small elements or unnecessary large elements. Furthermore, it is necessaryto safeguard against too many elements. The users supply a maximum element error that we would prefer(Epref ) in the model. Thus, if the number of elements in the model increases too much, then we loosen the


error criteria up to an upper maximum element error (Emax). The variable allowable maximum elementerror is de®ned as

E� � Epref � �Emax ÿ Epref�e N ln 2=M� �; �25�

where N is the number of elements speci®ed by the user. The characteristic behaviour of Eq. (25) is shown inFig. 1. The curves correspond to N of 1000, 4000 and 8000 elements. Using this form of maximum elementerror is an ad hoc way of making the modelling part more ¯exible. It does not guarantee a maximumnumber of elements in the model, but adds on extra ¯exibility in completing an analysis. Still the analysiswill stop if the error goes to higher than Emax at the maximum possible size of the model.

4. Welding simulation using adaptive meshing

Two di�erent geometries have been analysed to highlight the characteristics of the proposed remeshingstrategy in welding applications. The ®rst analysis corresponds to circumferential butt welding of a tubewith thickness 2 mm, an inner diameter of 100 mm and a total length of 400 mm. Completion of thecircumferential weld was done in 7.26 s. Symmetry was assumed in the weld plane and the tube wasconsidered to be free to expand in all directions. The second analysis corresponds to a butt weld of a platewith dimensions 230� 86� 2 mm. Completion of a 230 mm long weld was done in 5.11 s. The plate was®xed in all except the welding direction along one of the long sides of the plate. The material was in bothcases AISI 304L, an austenitic stainless steel. Used material data in the analysis are all temperature de-pendent and can be found in [27]. The welding set-up was a bead on weld using a CO2 laser with 2700 W as

Fig. 1. The allowable global error in the model vs. number of elements.

Table 1

Re®nement parameters

Parameter Tube welding Plate welding

Minimum element size 1.5 mm 1.5 mm

Maximum element size 4.0 mm 4.0 mm

Preferable global error 20% 20%

Max global error 30% 30%

N (elements) 5000 5000


output power and a welding speed of 2.7 m/min. The laser equipment was assumed to deliver an overalle�ciency of 65%. This gave a net heat input of 1755 W.

The tube and the plate welding analysis was performed using the parameters given in Table 1. It is shownin Figs. 2a and 3a that the automatic mesh re®nement using heat ¯ux as error indication produced anelliptical formed zone in the vicinity of the weld spot. A distinct di�erent behaviour is found when usingpurely stress re®nement as parameter. It gives more re®ned elements behind the arc. This region can bedi�cult to predict by a manual procedure as in [2].

The number of elements in the model during the weld analysis is plotted in Fig. 4. It can be seen that thestress-driven mesh generation produced signi®cantly more elements than the heat ¯ux-driven. It can also beseen that the stress-driven analysis requires more elements even after the heat input is ®nished. This is dueto the residual stress ®eld that will remain in the welded component. The ¯ux-driven remeshing is creatingmuch lower number of elements in the model. This is of course due to a smoother gradient ®eld in thethermal analysis and also because body cools down to a uniform temperature.

A clear di�erence between the two cases when using stress in the error indicator can be seen in Fig. 4. Thetube analysis increases the number of elements during cooling where the plate analysis decreases thenumber of elements. This re¯ects a more complicated deformation behaviour in the pipe during cooling.

Fig. 2. Remeshing of a tube using (a) only heat ¯ux and (b) only e�ective stress.

Fig. 3. Remeshing of a plate using (a) only heat ¯ux and (b) only e�ective stress.


5. Conclusions

It has been shown that the suggested posteriori error estimate can be used to control the re®nementstrategy in welding applications. We have not been able to complete an analysis with equal accuracy with a®xed mesh due to lack of physical memory. The tube analysis (using the combined error estimate) required25 h on an IBM SP2 thin node/120 MHz using 512 Mb RAM.

It is obvious that the di�erent error measures produce totally di�erent meshes. Thus the combinedprocedure should be used in welding applications. It can also be seen that the mesh produced by theproposed error indicator di�ers to a signi®cant level compared to the mesh used in [2]. Their choice ofremeshing is very similar to the mesh produced by pure heat ¯ux based error indication which is more easyto foresee by a user. The behaviour of the adaptive mesh generation depends on geometry and constraint asshown. Thus the error-driven mesh generation is necessary when modelling complex three-dimensionalwelding problems as even an experienced user may ®nd it di�cult to mesh the model e�ciently.

Acknowledgements

The ®nancial support was provided by the Swedish Research Council (TFR) and the Polhem laboratoryat Lule�a University of Technology.

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