o. Pialsing,

Received 13 November 2013Received in revised form 30 May 2014Accepted 27 July 2014

Keywords:Thermo-mechanical modellingFinite element analysis

complex thermo-mechanical nite element analysismodels. The proposed diagramconsists of three differ-

cutive

ical part of the structural response through a combination of exper-

the treatment (welding or line heating) itself. An extensive reviewhas been conducted in [14].

In a fully uncoupled thermo-mechanical nite element model,the analysis is usually carried out in a staggered approach: the ther-mal problem is solved rst, followed by the solution of the mechan-ical problem. The latter mechanical analysis runs on the basis of thethermal results in order to account for the thermal stress and phase

ature range, a small constant time step and a nemesh are requiredl and themechan-ng of the trerature ran

Thus, for the entire analysis there should be an exact corrdence between the mesh density and the time step magbetween the two models. This requirement renders the whocedure of model development very complex and time-consuming.

1.2. The three major problems: mesh density, time step andconvergence of results

The rst problem arising during the thermo-mechanical model-ling is that the thermal and the mechanical models are completely

Corresponding author. Tel.: +30 210 45 81 656, +30 697 37 97 661.E-mail address: KaralisDimitris@TEEmail.gr (D.G. Karalis).

Computational Materials Science 95 (2014) 288301

Contents lists availab

M

lseimental and numerical simulations. The numerical part of theinvestigation still attracts high interest due to its extreme intricacyand the uncertainty in predicting the structural response prior to

in the areas of transformation for both the thermaical analysis. This allows for the accuratemonitoristress developed during the transformation temphttp://dx.doi.org/10.1016/j.commatsci.2014.07.0450927-0256/ 2014 Elsevier B.V. All rights reserved.ansientge [5].espon-nitudele pro-1.1. Introduction

The thermo-mechanical response of steel or aluminium platesduring welding or plate forming by line heating has been investi-gated by several researchers during the last decades. Most of theresearch is focused on either or both the thermal and the mechan-

formed by importing to the mechanical model the nodal tempera-tures at each time increment and calculating the thermal strain.From the aforementioned staggered approach it is deduced thatboth thermal and mechanical models must normally run with thesame analysis parameters, namely time step magnitude and meshdensity. If, for example, the material undergoes phase transforma-tion accompanied by volume change during a specic short temper-Convergence ow diagramRepair welding

1. The problem of modelling conseent phases and provides a step-by-step guide for the development of the nal thermo-mechanical model,taking into account convergence issues, mesh density and estimation of time step magnitude. In phase I, apreliminary thermo-mechanical analysis is carried out in order to get an idea of the model behaviour, therequired resources and the feasibility of the overall analysis. In phase II thenal thermalmodel is developedin full, taking into account the mechanical results obtained at the end of phase I, whereas in phase III thenal mechanical model is generated on the basis of a continuously modied thermal model. The proposedprocedure presented herein in the form of a ow diagram provides the capability for gradual output of thenumerical results (preliminary results, thermal results, mechanical results), while paying attention to thetime-consumingproblemof results convergence required for a numerically accurate analysis. The former isan important issue for large-scale complex simulation projects, whereas the latter provides evidence thatthe development of the numerical model has been realized on the basis of the modelling laws. For betterpresentation and understanding, the proposed procedure is applied to the study of a nite element analysisthermo-mechanical model, where increased intricacy generally exists.

2014 Elsevier B.V. All rights reserved.

phenomena changeeffects on the structural responseof the structure. This is per-Article history: In this paper the authors propose a practical ow diagram for the systematic development and solution ofA practical ow diagram for the solutionthermo-mechanical numerical models

D.G. Karalis a,, N.G. Tsouvalis b, V.J. Papazoglou b, D.IaHellenic Navy, Hellenic Naval Academy, Mechanics & Materials Division, Marine Materb Shipbuilding Technology Laboratory, School of Naval Architecture and Marine EngineerAthens 157 73, Greece

a r t i c l e i n f o a b s t r a c t

Computational

journal homepage: www.ef complex non-linear

antelis b

Laboratory, Hazjikyriakou Avenue, Piraeus 185 39, GreeceNational Technical University of Athens, 9 Heroon Polytechniou Avenue, Zografou,

le at ScienceDirect

aterials Science

vier .com/locate /commatsci

2. A typical example to explain the ow diagram

metal and aims at treating the existing weld close to the meltingtemperature. Such treatments are applied to repair in-situ crackedor defected welds (repair welding). The welded bracket is xed atits smaller side (see red1 triangles in Fig. 1 that refer to the xa-tions). After the material has cooled to ambient temperature, uni-form pressure is applied on the other side of the bracket (see redarrows in Fig. 1) tending to buckle the triangular reinforcing web.The latter pressure simulates the operational load present on thebracket after the completion of the treatment. The treated lengthlAB is equal to 128 mm, whereas the ange and web thicknessesare equal to 25 mm and 12.5 mm respectively. The power of thewelding torch was set equal to Q = 3770 W whereas the speed wasset equal to v = 6 mm/s. This simulation is quite complex involvingthe existence of extreme non-linearities as temperatures are raisedto the steel melting point. It represents a difcult-to-solve numericalanalysis, as thermal, mechanical and thermally-induced mechanical

Materials Science 95 (2014) 288301 289different in nature, as they model different physical phenomena.Therefore the mesh density selected for the solution of the thermalproblem is, in most cases, inappropriate for the solution of themechanical problem.

Secondly, the time step required for the accurate solution of themechanical analysis may be too large compared to the time steprequired for the accurate solution of the heat ow problem, where,for example, extreme temperature gradients are encountered. Thelatter is also valid in the opposite case as, at high temperatures, thestructure may exhibit extreme material non-linearities.

A third problem pertains to the results convergence criteria.The development of a numerical model by means of the nite ele-ment method is generally terminated when the analysis hasreached (a level of) results convergence. For example, classicalconvergence criteria are based on the stabilization of nodal results,such as temperatures or displacements with regard to mesh den-sity and time step. It is actually not worthy remeshing the modelor reducing the time step if the nodal results do not change valuesversus simulated time.

It should be emphasized at this point that in general there arefour types of convergence in nite element analyses:

i. convergence of equilibrium iterations due to non-linearities(e.g. material, contact or geometrical non-linearities),

ii. convergence in the solutions of the linearized systems ofalgebraic equations in case of iterative solvers,

iii. convergence of the results due to mesh renement andiv. convergence of the results due to time step reduction.

In most commercial nite element software platforms, specicoptimum values and tolerances are already pre-set in order tocontrol best the convergence of the equilibrium iterations due tonon-linearities and convergence of the equations in case of itera-tive solvers. In the present study, emphasis is given only to the lasttwo convergence types, namely time step and mesh renement, asthey are the main user-dependent parameters that strongly inu-ence the entire simulation and results convergence. The procedurefollowed towards the convergence of results governs directly theoverall simulated time, the numerical analysis cost and affectsthe accuracy of the results. For example, some of the complex sim-ulations presented in [6,7] have lasted a few days, time that couldhave been strongly increased if a few more additional analyseshave been required due to convergence issues. At this point, itshould be mentioned that in most publications dealing with com-plex thermo-mechanical simulations the convergence criteria havenot been described at all, as the authors provide only the modelssetup and the numerical results. Hence, the end reader of the afore-mentioned papers comes to understand that the authors havesomehow performed a convergence analysis prior to publishingthe results obtained by means of the nite element method. Thisconvergence analysis is of great interest as it is complicated,time-consuming and strongly user-dependent.

In sum, a common time step and mesh density are normallyrequired for both the thermal and the mechanical analysis. Thesetwo common parameters must allow both physical problems tobe modelled satisfactorily, but they must also provide an accept-able level of results convergence for both models.

1.3. The aim of this paper

The authors aim at proposing a practical ow diagram for thesystematic development and solution of complex FEA thermo-mechanical models. In this ow diagram a progressive develop-

D.G. Karalis et al. / Computationalment of several thermal and mechanical models will be presentedon the basis of different mesh densities and time steps, aiming atreaching the convergence of the thermal and mechanical results.2.1. The physical model

In order to discuss the proposed ow diagram, a thermo-mechanical simulation will be employed. The latter concerns theweld treatment of a welded bracket under load, by means of tung-sten inert gas (TIG) welding. The whole conguration of the simu-lation is presented in Fig. 1.

The bracket shown in Fig. 1 is made of typical carbon structuralsteel (containing 0.45% w/w carbon) and consists of a bent angeand a triangular reinforcing web welded on the ange. The weldsAC and AB exist along both sides of the web. Treatment is per-formed along the AB weld on the side towards the +z semi-axis(the one that is visible in Fig. 1) using a TIG torch without llerIt ought to be mentioned here that a ow diagram for the solutionof such staggered thermo-mechanical models is missing from theinternational literature and that the whole process is a real laby-rinth for both experienced and inexperienced users dealing withthermo-mechanical modelling. Please note that the aim of theauthors is to discuss the proposed ow diagram and present thesteps followed for creating the nal thermo-mechanical modelwith regard to mesh density and selection of time step and notto provide the mathematically-based analysis for its development.The latter has already been discussed in the literature [835]. Theimplementation of the proposed ow diagram requires a commer-cial thermal and mechanical or multi-physics FEA software pack-age for which code verication has been already performed.

Fig. 1. The welded bracket used as an example to present the ow diagram.1 For interpretation of color in Fig. 1, the reader is referred to the web version ofthis article.

mal and mechanical. Normally, during the rst thermal part, the

arise: how is the mesh renement and time step modication per-

Matformed, in order to achieve satisfactory convergence of the results?What is the philosophy behind this temporal and spatial rene-transient temperature distribution for the whole bracket is calcu-lated, whereas in the second mechanical part, the total transientdisplacements and stresses are calculated, including any residualstresses. Note that, during the second part of the analysis, apartfrom the thermal stresses derived from the thermal treatment,additional stresses are generated due to the externally appliedpressure. It is concluded from the above that the mesh densityand the time step of both models should be able to model all tran-sient phenomena related to the weld treatment (temperature dis-tribution, thermal stresses, residual stresses and distortion), aswell as the general mechanical response like local stresses raisedat the geometrical discontinuities of the structure and inducedby the externally applied forces and the weld treatment itself [1].

Typical questions during model development pertain to (a) theow diagram proposed in the following to reach the commonmeshdensity and time step that offer adequate convergence of theresults for both models, and (b) the overall time and computingresources required to complete the analysis. These questionsbecome more critical and difcult to be answered as the modelledphysical structure becomes more complicated and bigger in size[6,36,37]. A large-scale structure implies that each trial run ofthe thermo-mechanical analysis aimed at reaching an acceptablelevel of convergence will last at least for a considerable amountof time.

For the thermomechanical simulation of the aforementionedtreatment, a three dimensional nite element model was set upusing ALGOR nite element code [22]. A staggered approachwas employed by solving at rst for temperatures and then fordisplacements and stresses (uncoupled formulation). First-ordereight-nodded solid heat transfer elements were used for thethermal part and rst-order eight-nodded solid thermoplastic ele-ments (instead of second order elements with midside nodes [2,3])were used for the mechanical part in order to account for the worstscenario with respect to available element types. Mesh compatibil-ity was retained between the two analyses. The heat source wasmodelled by employing a moving Gaussian distribution. The kine-matics, the constitutive formulations, the modelling, as well as theboundary conditions were applied as per [6,7]. The temperatureeld was considered unaffected by the structural response. Thesteel was modelled as isotropic, having yield stress equal to380 MPa [6,7,38,39] and temperature dependent properties includ-ing plasticity and strain hardening. Cooling was implemented bymeans of conduction, convection and radiation. At the beginningof the simulation, the bracket is considered free from weldingresidual stresses. The stress free reference temperature of thematerial was set at ambient temperature (25 C).

3. The philosophy of the spatial and temporal renementaiming at results convergence

Prior to presenting the ow diagram some logical questionsphenomena coexist and strongly affect the solution and convergenceprocedure.

2.2. The numerical model

From the description of the aforementioned weld treatment, itis deduced that the numerical analysis consists of two parts: ther-

290 D.G. Karalis et al. / Computationalment? When convergence can be considered as satisfactory?The progressive time step reduction and the gradual mesh

renement play an important role affecting the accuracy of theentire simulation. Taking into account the variety of different ele-ments and analysis types that exist nowadays in most commercialnite element platforms, the development of an efcient set ofequations between the reduction of the time step and the gradualmesh renement that leads to results convergence is a verydifcult and triggering task. In the current proposed diagram, thisspatial and temporal renement is based on the repetitive execu-tion of the thermal and the post-mechanical model. This executionprovides feedback pertaining to the appropriateness of the spatialand temporal renement that was applied. The latter methodologyhas the advantage of applicability in most thermomechanicalsimulations except of casting simulation where it is not directlyapplicable due to material ow.

The gradual reduction of the initial time step that is applied bythe analyst is strongly affected by all the temperature and timedependent phenomena that take place during the entire simula-tion. It is well known, that in a typical transient non-linear thermo-mechanical analysis, temperature and time dependent magnitudesexist. Temperature dependent magnitudes can refer, for example,to the material properties, coefcient of heat convection and con-vection heat, radiation; whereas time dependent magnitudes canrefer to the moving heat source, heat convection, operational loads,pressures, existence of gaps, etc.

As a basis for the discussion of the next paragraphs, Fig. 2depicts typical examples of the temperature dependent heatcapacity, the thermal conductivity, the convection lm coefcient,the yield stress and the thermal dilatation of a typical mild steelthat undergoes several microstructural transformations dependingon the peak austenitization temperature (Tpeak) [6,7]. In the samegure, the time dependent moving heat source of a welding arcis also presented [6]. Furthermore, in Fig. 2f, the temperaturedepended axial stress response of an axially xed steel specimenthat undergoes phase change transformation is shown [5,6].

In nite element simulations, the temperatures in the thermalanalysis and the displacements in the mechanical analysis are cal-culated for every node of the model and are exported at every timestep. Therefore the applied gradual reduction of the initial timestep value should nally:

i. Provide small temperature differences at every node of themodel between all successive analysis steps, so that the tem-perature dependent phenomena are accurately modelled.For example, a very small time step can result in very smallnodal temperature differences between all successive analy-sis steps. It is up to the researcher to decide, whether thelatter temperature difference can accurately model thenon-linear material properties at the areas of solid statetransformations (see Fig. 2a, c and d) or whether it is enoughto accurately account for the convection heat losses (seeFig. 2b). On the other hand, a relatively bigger time stepcan provide larger nodal temperature differences betweensuccessive analysis steps and thus hiding or articiallyminimizing the effects of phase change on the transientmechanical response of the structure. Taking into accountthat the steel phase transformation temperature range isapproximately DTtr = 300 C, a practical guideline is to selectthe maximum allowable time step that provides tempera-ture differences of maximum 30 C, or 10% ofDTtr. This smalltemperature difference will later provide the basis for anaccurate mechanical analysis where small thermallyinduced stress differences are also required. Therefore theselected maximum allowable time step should additionallykeep the thermally-induced stress differences between suc-

erials Science 95 (2014) 288301cessive analysis steps in a stress analysis smaller than asmall percentage of the material yield stress. In currentpaper the value of 5% of the material yield stress is

MatD.G. Karalis et al. / Computationalsuggested. Similar guidelines can also be applied for thestrain. In conclusion, the maximum allowable temperaturedifference required for the realistic modelling of the temper-ature dependent phenomena denes the maximum allow-able time step value to be used. The latter will be obtainedafter the repetitive execution and post-processing of thethermal and mechanical models in all three phases of theow diagram that will be proposed later.

ii. Allow the time dependent magnitudes to be adequatelytaken into account during the entire analysis. For example,if the actual velocity of the moving source is high (seeFig. 2e), a relative small time step is required in order to

Fig. 2. An example of temperature and time dependent magnitudes in a typical thermconductivity, (b) surface convection lm coefcient, (c) material yield stress, (d) materesponse of an axially xed steel specimen that undergoes phase change transformationerials Science 95 (2014) 288301 291accurately capture the steep shape of the source alonglength a1. Furthermore, if instant cooling of the weld metalis applied during welding (e.g. underwater welding), only avery small time step value will be able to capture the instantchange of the heat transfer coefcient, and thus correctlycalculate the transient heat transfer phenomena. Taking intoaccount that in most conventional welding simulations (a)the size of the moving heat source is equal to several milli-metres along the three axes, (b) the torch speed is equal toseveral millimetres per second and (c) no forced convectionexists, a simple guideline is to set the initial value of timestep not larger than 1 s. Normally, the latter value is later

omechanical analysis, reprinted from [6,7]. (a) Material heat capacity and thermalrial thermal dilatation, (e) three dimensional moving heat source, (f) axial stress.

Matexpected to be strongly reduced to a small percentage of itsinitial value in order to satisfy the convergence criteria.Alternatively, a second practical guideline for the selectionof the initial value of time step pertains to divide the totalduration of the steepest part of the curve of the time depen-dent magnitude into minimum 3 different equal time steps.In conclusion, the steepest part of the curve of the timedependent magnitudes denes the maximum allowabletime step value to be applied. The latter will be deducedafter the iterative execution and post-processing of the ther-mal and mechanical results.

From the discussion above it is concluded that the nal timestep value that provides satisfactory convergence of the resultsshould be the minimum of the two maximum time steps derivedfrom items (i) and (ii) above.

As far as the gradual renement of the initial mesh density isconcerned, similar rules and observations that were described pre-viously are valid. More specically, the applied gradual renementof the initial mesh density should aim at providing enough ele-ments at the areas of interest in order to accurately model thematerial that is affected by:

i. The abrupt change of the temperature dependent magni-tudes. For example, the narrow heat affected zone that isgenerated between the weld pool and the base metal isstrongly affected by the temperature (and phase) dependentmaterial properties. A few elements along this zone wouldnot sufce to accurately obtain the highly transient phenom-ena that occur in this area. A practical guideline pertains toemploy an initial mesh density of minimum 3 elementsalong the heat affected zone. In conclusion, the mesh wherean abrupt change of the temperature dependent magnitudestakes place should be highly rened.

ii. The abrupt change of the time dependent magnitudes. Forexample, in order to accurately apply the power of the heatsource along length a1 (see Fig. 2e), many elements arerequired to be present along this length. It is obvious thatonly one or two elements along this area would not sufce.A practical guideline pertains to employ an initial mesh den-sity of 3 elements per the shortest length among a1, b or c(see Fig. 2e) for the whole area where the arc distributionis applied. In conclusion, the mesh where an abrupt changeof the applied time dependent magnitudes takes placeshould be highly rened.

iii. The stress concentrations generated by both residual orapplied operational loads. The former is of great importancein case of welding residual stress analysis and requires arened mesh along the three axes especially at the vicinityof the weld metal and the heat affected zone where residualstresses present strong variability. The latter stress concen-trations can be derived from a static stress analysis.

Similarly to the time step comments, the above discussionshows that the nal mesh density should at least satisfy items(i), (ii) and (iii) above, in order to provide satisfactory and accuratemodelling.

The third issue pertains to the criteria of the results conver-gence acceptance. This dilemma that is set at every sub-step ofthe analysis (more specically at every rhombus of the ow dia-gram that will be proposed later) strictly depends on theresearcher and the way he deals with the scope of the analysis,the areas of interest and the magnitudes being monitored. As

292 D.G. Karalis et al. / Computationalstated previously, it is actually not worthy remeshing the modelor reducing the time step if the nodal displacements (or temper-atures) do not change values versus simulated time in a dis-placement (or temperature) analysis. The same observation isalso valid for the stresses or heat uxes or other magnitudesof interest. It must be mentioned here, that the convergencestudy must be realized on the basis of the predicted results: aliterature research [33] has shown that authors have often usedreaction forces or displacements to perform their convergencetests and subsequently made predictions for other results suchstresses or strains. Furthermore the tolerance of the results con-vergence criteria may strongly differ among researchers; a 15%difference between the temperatures of two successive thermalanalyses results may seem enough for the termination of therepetitive execution of the models in a general heat treatmentsimulation; whereas it may be dealt as a considerable differencein the case of a material phase-change response analysis. Fur-thermore, the criteria of the results convergence depend alsoon the area of interest. In a residual stress analysis for example,the researcher is mainly focused on the weld metal and the heataffected zone; thus the residual stress results on these zonesmust converge satisfactorily when the researcher decides to ter-minate his investigation (or the repetitive loop of the ow dia-gram that will follow). On the other hand, in a residualdisplacement analysis, the area of interest is usually the farend of the plates that present the maximum distortion. Fromthe discussion above it is derived that (a) the aim of the thermo-mechanical simulation, (b) the response of the area of interest ofthe model and (c) the magnitudes being monitored, should bethe governing parameters that will allow the termination orthe continuation of the convergence study. Here the authorswould like to emphasize on the fact that an accurate thermal-stress analysis requires a very accurate thermal analysis. It isthus very important for the analyst to have the thermal analysisaccurately performed and the thermal results converged to anacceptable level. Finally, in addition to what was discussed pre-viously, a good practice in a thermal analysis where extremethermal gradients exist is to monitor the minimum temperaturecalculated by the software at every time step. For example, in atypical welding simulation without preheating, the minimumcalculated transient temperature of the model should alwaysbe at least the initial material temperature. The execution ofthe analysis with an inappropriate time step in combinationwith a rough mesh can provide smaller or negative transientminimum temperatures compared to the initial temperature ofthe material.

To summarize the discussion that deals with the convergencecriteria, in current paper the analysis will be terminated whenthe relative difference of the magnitudes being monitored (tem-peratures, stresses and displacements) between two successiveanalyses is less than 10%. The latter value provides an acceptablelevel of convergence in engineering terms taking into account thatwelding simulations are very complex non-linear thermomechan-ical problems. Of course a smaller value can also be applied, whichrepresents a more strict criterion but has a negative impact oncomputer resources and cost.

4. The proposed ow diagram

The proposed ow diagram consists of three different phases(phases IIII), each one containing several sub-steps and executionloops. For the readers convenience, each phase and sub-step arediscussed separately and are supported by gures of the relevantnite element model. In the proposed ow diagram a progressivedevelopment of several thermal and mechanical models will bepresented on the basis of different mesh densities and time steps,

erials Science 95 (2014) 288301aiming at reaching the convergence of the thermal and mechanicalresults. In current FEA simulation the main magnitudes being mon-itored are:

Mati. The transient temperature distribution at the mid-length ofthe treated zone, which provides evidence that the desiredmaximum temperature has been reached.

ii. The temperature distribution during cooling, in order to con-rm that the residual stresses refer to the totally cooled con-dition of the material.

iii. The transient von Mises stress at the mid-length of the trea-ted zone, which provides an indication of the plasticallydeformed zones.

iv. The residual von Mises stress after the material has cooled toambient temperature and prior to the application of theexternal load, in order to assess the resistance of the struc-ture against brittle fracture and susceptibility to environ-mental (season) cracking.

v. The thermally-induced residual displacement after thematerial has cooled to ambient temperature and prior tothe application of the external load.

vi. The nal von Mises stress, which provides the operationalstress.

vii. The nal displacement of the structure, which provides itsshape during operation.

4.1. Phase I: preliminary thermal and mechanical analysis

4.1.1. The linear loopPhase I pertains to the preliminary investigation carried out

prior to the main analyses (phases II and III) aiming at obtaininga general idea of the model behaviour, checking the feasibility ofthe analysis and getting feedback regarding the time required tocomplete the analysis and computer resources needed. The preli-minary analysis is considered to be very important, as it is worthknowing whether the analysis lasts a day or a week or whetherthe non-linearities encountered will allow the completion orcancellation of the simulation. Phase I contains two sub-steps,the preliminary linear and non-linear thermomechanical analyses.The ow diagram of phase I is shown in Fig. 3.

The preliminary linear transient thermal analysis, followed by alinear transient (or static) mechanical analysis including externalforces and thermal stresses at several time steps, can provide someanswers to all of the questions raised, namely the location of areasof greatest interest, mesh density and time step magnitude. At thislinear sub-step it is important to mention that both hand or analyt-ical solutions and user experience can help determine the areas ofgreatest importance or interest and can thus contribute to theselection and estimation of a proper initial mesh density for bothmodels. As a general guide, a ner mesh is required in thermo-mechanical modelling at areas close to any heat input, heat sinks,geometrical discontinuities and in areas of material, boundary andgeometrical non-linearities. Thus, the initial design of the meshdensity must take into account at least the aforementioned factors.Initial time step selection is also discussed in [25,26]; a relativesmall value has been suggested in previous sections in order tocapture the linear transient heat transfer phenomena to a satisfy-ing level. In the case of welding, for example, the speed of thewelding torch and the size of the arc distribution selected for mod-elling the heat input [40] can help determine an initial value of thetime step. Notice that, in most thermo-mechanical simulations(excluding repair welding of dynamically excited structures),modal effects are not taken into account, which facilitates initialtime step estimation [13,22,26]. It has to be emphasized here thatat this step it is not important to accurately and precisely estimatethe aforementioned initial values. On the basis of the ow diagramshown in Fig. 3, the repetitive execution of the linear models will

D.G. Karalis et al. / Computationalnally lead to the time step value and mesh density that providesatisfying convergence of the preliminary results. Normally a grad-ual decrease of the initial time step and a mesh renement at thecritical areas should be applied. Notice that, as stated previously,this sub-step aims at obtaining a general idea of the model behav-iour; thus a few repetitive executions should sufce.

In our example, the linear sub step has started by employing aninitial time step of Dt = 1 s and a coarse uniform mesh containingNE = 4208 elements. From the executions of the linear thermalanalyses it was derived that the maximum thermal gradientsappear close to the weld line (line AB, see Fig. 1), whereas themaximum stress gradients derived from the linear static stressanalysis, including both external forces and thermal stresses atspecic time steps, appear along the weld and at the triangularreinforcing web. After the termination of the linear sub-step, thenal locally rened mesh contained NE = 6079 elements and thenal time step had been gradually reduced to Dt = 0.1 s. The loopof the linear step was terminated when the relative difference ofthe maximum thermally-induced residual displacement (prior tothe application of the external pressure) between two successiveanalyses was less than 20%. The same criterion was also appliedfor the von Mises stress and the maximum temperature developedat mid-length of the treated zone. The latter value of 20% is largercompared to the desirable value of 10% that was described in 3but is considered as sufcient for the preliminary stage of the anal-ysis. At this point, the maximum temperature difference of everynode between all successive time steps was less than 85 C whichrefers to about 30% of DTtr (which is larger compared to the desir-able value of 10%). The nal linear analysis results are depicted inFig. 4.

4.1.2. The non-linear loopFollowing the ow diagram depicted in Fig. 3, the initial execu-

tion of the non-linear analyses should be carried out on the basis ofthe time step value and mesh density derived from the previouslinear sub-step. Thus, the thermal model is executed followed bythe execution of the non-linear mechanical one. For this reason,the calculated temperatures obtained from the thermal analysisare imported into the mechanical model; the external forces arethen applied, after the weld treated bracket has cooled to roomtemperature. This staggered execution is performed a few times(see also Fig. 3), each time with a modied time step and meshdensity. Normally a gradual decrease of the initial time step andan increase of mesh density at the critical areas should be applied.Notice that, at this point, the model modications are performedon the basis of both the thermal and the mechanical results, asthe common mesh density and time step must take into accountboth heat ux and stress gradients. At the end of this loop themodel is expected to be able to model the non-linear transientphenomena, to some extent at least, as it contains a crude meshthat is locally rened to a small degree close to the areas of highstress and heat ux gradients. It should be emphasized at this pointonce again that high accuracy of temperatures and displacementsis not required at this phase (accuracy issues of the calculatedresults will be discussed in the following phases) and, therefore,a relative small number of loops is suggested. At the end of phaseI, apart from the preliminary level of the results convergenceobtained, the user can reach some basic conclusions regardingthe demands of the overall numerical simulation, computerresources, time requirements and cost. It is also important to men-tion that, at this stage of the ow diagram, it is up to the analyst todecide whether to undertake the simulated project or not.

In our example, the nal thermal and mechanical model at theend of phase I contained NE = 7587 elements, whereas the timestep of the analysis was equal to Dt = 0.08 s. The loop of the non-linear step was terminated when the relative difference of the

erials Science 95 (2014) 288301 293maximum thermally-induced residual displacement between twosuccessive analyses was less than 15%. The same criterion was alsoapplied for the von Mises stress and the maximum temperature

Mat294 D.G. Karalis et al. / Computationaldeveloped at mid-length of the treated zone. Notice that at thispoint, the relative difference approximates the target value of10% that is required to terminate the convergence study but is still

Fig. 3. Flow diagraerials Science 95 (2014) 288301not acceptable in engineering terms. At the end of the non-linearsub-step the maximum temperature difference of every nodebetween all successive time steps was equal to 75 C which refers

m of phase I.

MatD.G. Karalis et al. / Computationalto about 25% of DTtr. The maximum von Mises stress difference ofevery node along the treated zone between all successive timesteps was equal to 27 MPa which refers to 7% of the yield stress.The results of the non-linear analysis are illustrated in Fig. 5. Fromthe results presented previously it is deduced that further execu-tions are required until the termination of the project as the con-vergence of the results being monitored is larger than 10% andthe temperature and the stress differences between all successivetime steps are larger than 10% of DTtr and 5% of yield stressrespectively.

The basic differences between the model shown in Fig. 4 andthe updated one shown in Fig. 5 are concentrated around the areasof high gradients, like those at both sides of the weld AB, at theheat-affected zone and around points B and C (see also Fig. 1).More specically, the model shown in Fig. 5 contains a ner meshalong both sides of weld AB at its heat affected zone, along thefree edge of the triangular web and close to points B and C, wherehigher thermal and stress gradients develop.

Fig. 4. The results at the end of the linear loop (NE = 6079 elements, Dt = 0.1 s). (a) Trans(b) temperature distribution during cooling (Tmax = 33 C, Tmin = 25 C), (c) von Mises stavon Mises static stress due to the applied operational pressure only (rmax = 205 MPa,pressure only (dmin = 0 mm, dmax = 0.515 mm, scale factor 10).erials Science 95 (2014) 288301 2954.2. Phase II: nalizing the thermal model and obtaining the thermalresults

Phase II pertains to the main analysis of the thermal problem. Itaims at calculating the transient and residual temperature distri-bution of the structure under investigation. The ow diagram ofphase II is shown in Fig. 6.

At the beginning of this phase, the thermal model is executedusing the mesh density and the time step derived from phase I.Depending on the convergence criteria, the models mesh is pro-gressively rened until the desired level of accuracy in the areasof interest is attained [33]. For example, the gradual increase ofmesh density as we approach the weld line AB will allow betterestimation of the size of the heat affected zones where phasetransformations occur during the overall treatment. Note thatthe mesh of the thermal model must also be rened but toa lesser degree at areas of high stress gradients of the mechan-ical model, as observed at the end of phase I. Notwithstanding

ient temperature at the mid-length of the treated zone (Tmax = 864 C, Tmin = 25 C),tic stress at the mid-length of the treated zone (rmax = 1063 MPa, rmin = 0 MPa), (d)rmin = 0.5 MPa), (e) nal displacement magnitudes due to the applied operational

Mat296 D.G. Karalis et al. / Computationalthat the latter local mesh renement does not necessarily con-tribute to the accuracy of the thermal analysis results, it will con-tribute to the faster solution and convergence of the mechanicalmodel carried out in phase III. As long as thermal results conver-gence has been attained for specic mesh density, the thermalmodel is re-executed by reducing the time step in order to con-rm that changes of the time step do not signicantly affect thetemperature results. This procedure may require a few moreloops to complete in order to provide a better estimation ofthe maximum temperatures reached in the heat affected zonesand to calculate the cooling rates in the transformation areas,necessary for the post thermal-stress analysis (phase III). At thisstage of phase II, further remeshing or time step reduction arenot expected to strongly affect the results, thus the nal thermalresults, such as maximum temperatures and cooling rates, canbe obtained. Notice that mesh renement and step time reduc-tion can be performed simultaneously for the case of analysts

Fig. 5. The results at the end of phase I (NE = 7587 elements, Dt = 0.08 s). (a) Transienttemperature distribution during cooling (Tmax = 34 C, Tmin = 25 C), (c) transient von Mivon Mises residual stress (rmax = 393 MPa, rmin = 0.5 MPa), (e) nal von Mises stressdmax = 0.522 mm, scale factor 10).erials Science 95 (2014) 288301that are very experienced with thermomechanical modelling. Hereit has to be emphasized that the thermal results include the tem-perature ranges and time steps of any microstructural transforma-tions realized in specic areas of the model. These microstructuralchanges can play an important role in the mechanical response ofthe structure analysed in phase III; thus this stage requires max-imum attention [15,39]. It should be mentioned at this pointthat normally a distinct or some deviation (if any) is expectedbetween numerical and experimental thermal results due to theunknowns involved in the analysis (e.g. heat input, arc efciency,material properties, heat loss); thus adaptation or calibration ofthe thermal model may be necessary, see Refs. [4,41]. The adapta-tion procedure may require a few more re-executions of the ther-mal model with modied input data. These adapted thermalnumerical results, later used for post-mechanical analysis (phaseIII), are expected to more accurately address the problem of thetransient and residual response of the structure.

temperature at the mid-length of the treated zone (Tmax = 954 oC, Tmin = 25 C), (b)ses stress at the mid-length of the treated zone (rmax = 360 MPa, rmin = 3 MPa), (d)(rmax = 382 MPa, rmin = 0.6 MPa), (f) nal displacement magnitudes (dmin = 0 mm,

Materials Science 95 (2014) 288301 297D.G. Karalis et al. / ComputationalOn the basis of the aforementioned discussion, the mesh of ourexample required severe renement along both sides of weld ABand its heat-affected zone in order to capture the high temperaturegradients and cooling rates (AB line, see Fig. 1). Local remeshinghas also been performed along the webs free edge and areas Band C, where high stress gradients appeared during the end ofphase I. The latter remeshing is carried out in order to preparethe mesh density for the mechanical analysis that will follow inphase III. The mesh density, consisting of NE = 20,080 elements atthe end of phase II and the nal non-linear thermal analysis resultsobtained using Dt = 0.037 s are presented in Fig. 7. The nal loop ofphase II was terminated when the relative difference of the maxi-mum temperature at mid-length of the treated zone between twosuccessive analyses was less than 10%. The latter value satises thecriterion that was described in 3 for the termination of thethermal part. At the end of phase II, the maximum temperaturedifference of every model node between all successive time steps

Fig. 6. Flow diagrawas equal to 30 C (or 10% of DTtr) which is also within the desir-able range. From the results depicted in Fig. 7 it is also deducedthat the maximum temperature calculated by the software(Tmax = 1390 C) was not above the melting point of the materialwhich is equal to 1537 C [42]. Thus no melted zone was created.If the actual treatment had resulted in the formation of a meltedzone then the thermal model developed during this phase shouldhave been further adapted with respect to the experimental oractual results (if any), as indicated in Fig. 6 in order to compensatefor the unknown parameters that are involved in the numericalanalysis. The duration of the analysis of the nal convergedthermal model (single run) was 3.5 CPU-hours2 whereas its sizewas equal to 3 GB.

m of phase II.

2 CPU Intel Core i3 M350 @ 2.27 GHz (dual core, dual thread), RAM 4 GB, HDD,Windows 64 bit OS.

4.3. Phase III: nalizing the mechanical model by modifying thethermal model

Phase III aims at nalizing the mechanical model for themechanical analysis. Contrary to the procedure described in phaseII for the nalization of the thermal part, the modications per-

formed on the mechanical model during this phase require thethermal model to be modied as well. Thus the user should modifyboth the thermal and the mechanical models which is a time-con-suming and computer-demanding procedure. The latter stronglyaffects the overall analysis cost. The ow diagram of phase III isdepicted in Fig. 8.

Fig. 7. The nal mesh density of the thermal model as obtained at the end of phase II (NE = 20,080 elements, Dt = 0.037 s). (a) Transient temperature at the mid-length of thetreated zone (Tmax = 1390 C, Tmin = 25 C), (b) temperature distribution during cooling (Tmax = 39 C, Tmin = 25 C).

298 D.G. Karalis et al. / Computational Materials Science 95 (2014) 288301Fig. 8. Flow diagram of phase III.

Analysis starts by executing the mechanical model on the basisof the mesh density and time step derived at the end of phase II forthe thermal model. Note that, at this point, the mesh density as cal-culated from the thermal model can capture to some degree thestress gradients of the mechanical analysis. As long as the mechan-ical model does not meet the convergence criteria, it is furthermodied and executed on the basis of a more detailed mesh inthe areas of interest. Again, as suggested in phase II, a relative widerange of different mesh densities must be tested [33]. For this pur-pose, the thermal model is necessarily further rened and executedas well, focusing also on the areas of high stress gradients derivedfrom the mechanical analysis. Again, attention of mesh renementis paid to the areas of the isolated boundary conditions, the nodeswhere the heat input is delivered, the geometrical discontinuitiesof the model, the areas of microstructural transformations andthe areas where non-linearities are observed. Please note that themodication and re-execution of the thermal model should notprovide better accuracy of thermal results in engineering terms,as the nal temperature results have already been obtained atthe end of phase II; thus, the thermal model is further modied

only in order to provide the appropriate basis for the executionof the mechanical analysis. It ought to be mentioned here thatthe current loop offers a good chance for the researcher to conrmthat the thermal model has met the convergence criteria requiredfor the termination of phase II.

As long as the mechanical results have converged to the desiredlevel (which implies for example that maximum displacementsstabilization has been observed with respect to mesh renementin a displacement analysis), both thermal and mechanical modelscan be further executed with their time step modied in order tocheck the sensitivity of the mechanical results on time step. Thelatter loop process may require a few successive executions untilthe nal time step values are stabilized. Simultaneous mesh rene-ment and time step reduction can also be applied for the case ofexpert analysts. At the end of this phase, mechanical results liketransient displacements and stresses and residual stresses can beobtained. Once again, note that at the end of this phase the thermalmodel has been further modied but the nal thermal results havealready been obtained at the end of phase II. The thermal modelobtained at the end of phase II should provide almost the same

se III

D.G. Karalis et al. / Computational Materials Science 95 (2014) 288301 299Fig. 9. The nal mesh density of the mechanical model as obtained at the end of pha

the treated zone (Tmax = 1516 C, Tmin = 25 C), (b) temperature distribution during coolintreated zone (rmax = 385 MPa, rmin = 0 MPa), (d) von Mises residual stress (rmax = 423 Mnal displacement magnitudes (dmin = 0 mm, dmax = 0.55 mm, scale factor 10).(NE = 42,773 elements, Dt = 0.025 s). (a) Transient temperature at the mid-length of

g (Tmax = 39 C, Tmin = 25 C), (c) transient von Mises stress at the mid-length of thePa, rmin = 0.2 MPa), (e) nal von Mises stress (rmax = 407 MPa, rmin = 0.4 MPa), (f)

able to capture the residual stresses developed along the treatedweld (see Fig. 9d). These stresses are also visible in the opera-

Mattional condition after the external pressure has been applied(see Fig. 9e). Furthermore, as it is derived from the thermalresults, the difference of the calculated maximum temperaturesat mid-length of the treated zone between the thermal analysesof phases II and III (1390 C and 1516 C respectively) was lessthan 10%, which on the basis of the criteria described in 3 con-rms the termination of phase II. Of course, in case of morestrict criteria (for example 5% of temperature differences) phaseII should have been further executed. The duration of the analy-sis of the nal converged mechanical model (single run) was 333CPU-hours (Footnote 2), whereas it size was equal to 98 GB.

5. Discussion and conclusions

In conclusion, on the basis of the proposed diagram the analysthas managed to determine the exact model density and time step(that were unknown at the end of phase I, see Fig. 4) in order toaccurately capture the thermo-mechanical response of this specictreatment by performing the minimum number of computer runs(end of phases II and III, see Figs. 7 and 9 respectively) while beingable to provide a preliminary answer with respect to the analysisfeasibility at the end of phase I. Furthermore, in the example pre-sented in this study, focus was given only on the seven itemsdescribed at the beginning of 4. Focusing on different magnitudesand at different time steps may have required more or less numberof numerical executions compared to these applied here (phases IIII). In any case, the criteria of convergence acceptance strictlydepend on the researcher. Summarizing, in current paper theauthors have adopted the following criteria for the terminationof the proposed ow diagram:

i. Relative difference of the maximum thermally-inducedresidual displacement between two successive analyses lessthan 10%. The same criterion was also applied for the vonMises stress and the maximum temperature developed atmid-length of the treated zone.accuracy in engineering terms as the one derived at the end ofphase III, but due to the lesser degree of freedom it contains, it isquicker to execute and can thus be used for further investigation.

In our example, strong stress gradients that appeared duringthe mechanical analysis along the free edge of the web, on bothsides of welds AB and AC and at the geometrical discontinu-ities (areas B, C) have led to further local mesh renement ofthe model. The model at the end of phase III containingNE = 42,773 elements and the nal non-linear analysis resultsobtained using Dt = 0.025 s are shown in Fig. 9. The nal loopof phase III was terminated when the relative difference of themaximum thermally-induced residual displacement betweentwo successive analyses was less than 8%. The same criterionwas also applied for the von Mises stress and the maximumtemperature developed at mid-length of the treated zone. Atthe end of phase III, the maximum temperature difference ofevery model node of the thermal analysis between all successivetime steps was reduced to 13 C (or 4.3% of DTtr) whereas themaximum von Mises stress difference of every node along thetreated zone between all successive time steps of the treatmentwas equal to 12 MPa (or 3% of the yield stress). From the aboveit is derived that all criteria described in 3 were satised; thustermination of the overall analysis was well applied. Here it hasto be mentioned, that the converged model presented in Fig. 9 is

300 D.G. Karalis et al. / Computationalii. Maximum temperature difference of every model nodealong the treated zone between all successive time stepsequal to or smaller than 10% of DTtr.iii. Maximum von Mises stress difference of every node alongthe treated zone between all successive time steps equal toor smaller than 5% of the yield stress of the material.

It should be mentioned at this point that the overall simulationdescribed in Section 4 could have been performed directly in a sin-gle phase by employing a very dense mesh throughout the wholemodel and a very small time step for the entire simulated time.This cursory methodology, although being direct and simple,does not provide evidence of results convergence, is not feasiblefor large scale models and requires the largest computer resourceswith respect to memory capacity and running CPU-hours. On theother hand, it should be also stated that the proposed ow diagramis certainly not unique and converged results may have beenobtained by carrying out a sensitivity analysis probably in a differ-ent manner instead of using the aforementioned diagram as asystematic solution guide. It is again emphasized that there is verylimited literature available on how to perform a systematic modeldevelopment for thermomechanical simulations.

Concluding, in this paper a practical ow diagram for the sys-tematic model development and solution of complex non-linearthermo-mechanical nite element analysis models is presented.The proposed diagram consists of three phases. In phase I, a preli-minary thermo-mechanical analysis is carried out in order to getan idea of the model behaviour, cost and feasibility of the overallanalysis. During this phase, both thermal and mechanical modelsare progressively modied until a preliminary level of results con-vergence is met. In phase II the nal thermal model is developed infull, taking also into account the mechanical results encountered atthe end of phase I, whereas in phase III the nal mechanical modelis generated on the basis of a continuously modied thermalmodel. The proposed procedure, which has been presented in theform of a ow diagram, allows for the gradual output of the numer-ical results (preliminary results, thermal results, mechanicalresults), assuring at the same time that these results are the out-come of converged analyses. The gradual output of the numericalresults is an important issue for large-scale simulation projects;whereas converged analyses is evidence that the development ofthe numerical models has been run on the basis of modelling laws.

Acknowledgements

The authors acknowledge that the necessity of a ow diagramfor the systematic development and solution of complex non-lin-ear thermomechanical models arose during the work within theEU funded project Shipbuilding Low Cost, Versatile and SafeWelding by YAG Laser Applications SHIPYAG (contract numberG3RD-CT-2000-00251). The authors gratefully acknowledge thatpart of the work presented in this paper was funded by the afore-mentioned European program.

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D.G. Karalis et al. / Computational Materials Science 95 (2014) 288301 301

A practical flow diagram for the solution of complex non-linear thermo-mechanical numerical models1 The problem of modelling consecutive phenomena1.1 Introduction1.2 The three major problems: mesh density, time step and convergence of results1.3 The aim of this paper

2 A typical example to explain the flow diagram2.1 The physical model2.2 The numerical model

3 The philosophy of the spatial and temporal refinement aiming at results convergence4 The proposed flow diagram4.1 Phase I: preliminary thermal and mechanical analysis4.1.1 The linear loop4.1.2 The non-linear loop

4.2 Phase II: finalizing the thermal model and obtaining the thermal results4.3 Phase III: finalizing the mechanical model by modifying the thermal model

5 Discussion and conclusionsAcknowledgementsReferences