-
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: [email protected] (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