Computational Modeling and Sub-Grid Scale Stabilization of...

Arch Computat Methods EngDOI 10.1007/s11831-014-9094-z

Computational Modeling and Sub-Grid Scale Stabilizationof Incompressibility and Convection in the Numerical Simulationof Friction Stir Welding Processes

C. Agelet de Saracibar · M. Chiumenti · M. Cervera ·N. Dialami · A. Seret

Received: 27 May 2013 / Accepted: 27 May 2013© CIMNE, Barcelona, Spain 2014

Abstract This paper deals with the computational model-ing and sub-grid scale stabilization of incompressibility andconvection in the numerical simulation of the material flowaround the probe tool in a friction stir welding (FSW) process.Within the paradigmatic framework of the multiscale sta-bilization methods, suitable pressure and convective deriv-ative of the temperature sub-grid scale stabilized coupledthermomechanical formulations have been developed usingan Eulerian description. Norton-Hoff and Sheppard-Wrightthermo-rigid-viscoplastic constitutive material models havebeen considered. Constitutive equations for the sub-gridscale models have been proposed and an approximation ofthe sub-grid scale variables has been given. In particular,algebraic sub-grid scale (ASGS) and orthogonal sub-gridscale (OSGS) methods for mixed velocity, pressure and tem-perature P1/P1/P1 linear elements have been considered.Furthermore, it has been shown that well known classical

C. Agelet de Saracibar (B) · M. Chiumenti · M. CerveraETS Ingenieros de Caminos, Canales y Puertos, UniversidadPolitécnica de Cataluña (UPC), Edificio C1, Campus Norte,UPC, Jordi Girona 1-3, 08034 Barcelona, Spaine-mail: [email protected]

C. Agelet de Saracibar · M. Chiumenti · M. Cervera ·N. Dialami · A. SeretInternational Center for Numerical Methods in Engineering(CIMNE), Gran Capitán s/n, 08034 Barcelona, Spaine-mail: [email protected]

M. Cerverae-mail: [email protected]

N. Dialamie-mail: [email protected]

A. SeretEcole Nationale Supérieure des Mines de Saint-Étienne,158 cours Fauriel, 42023 St-Étienne Cedex 2, Francee-mail: [email protected]

stabilized formulations, such as the Galerkin least-squares(GLS) for incompressible (or quasi-incompressible) prob-lems or the Streamline Upwind/Petrov-Galerkin (SUPG)method for convection dominant problems, can be recoveredas particular cases of the multiscale stabilization frameworkconsidered. Using a product formula algorithm for the solu-tion of the coupled thermomechanical problem, the resultingalgebraic system of equations has been solved using a stag-gered procedure in which a mechanical problem, defined bythe linear momentum balance equation, under quasi-staticconditions, and the incompressibility equation, is solved firstat constant temperature. Then a thermal problem, defined bythe energy balance equation, is solved keeping constant themechanical variables, i.e. velocity and pressure. The com-putational model has been implemented in an enhanced ver-sion of the finite element software COMET, developed bythe authors at the International Center for Numerical Meth-ods in Engineering (CIMNE). Two numerical examples havebeen considered. The first one deals with the numerical sim-ulation of a coupled thermomechanical flow in a 2D rec-tangular domain. Steady-state and transient conditions havebeen considered. The goal of this numerical example hasbeen the comparison between different sub-grid scale sta-bilization methods for the velocity and temperature equa-tions. In particular, using a GLS stabilization method for thepressure equation, a comparison between SUPG and OSGSconvective stabilization methods has been performed. Addi-tionally, using a SUPG stabilization method for the tem-perature equation, a comparison between GLS and OSGSpressure stabilization methods has been done. The secondexample deals with the 3D numerical simulation of a rep-resentative FSW process. Numerical results obtained havebeen compared with experimental results available in the lit-erature. A good agreement on the temperature distributionhas been obtained and predicted peak temperatures compare

123

C. Agelet de Saracibar et al.

well, both in value and position, with the experimental resultsavailable.

Keywords Variational multiscale methods · Stabilizedfinite element methods · Algebraic sub-grid scales ·Orthogonal sub-grid scales · Stabilizationof incompressibility and convection · Thermally coupledrigid-plastic solids · Friction stir welding

1 Introduction, Motivation and Goals

Thermally coupled incompressible rigid-plastic materialmodels are of particular practical interest in the numericalsimulation of different metal forming processes, in whichthe plastic strains are much higher than the elastic ones,and therefore the elastic strains can be neglected. Further-more, for some of those applications the plastic strains andplastic strain rates are extremely high and Eulerian or Arbi-trary Lagrangian-Eulerian (ALE) formalisms, instead of aLagrangian one, are far more convenient from the numer-ical point of view. Those situations arise, for instance, inthe numerical simulation of friction stir welding (FSW)processes [7,33].

The formulation of thermally coupled incompressiblerigid-plastic material models using Eulerian or ALE for-malisms is of particular interest from the numerical point ofview for different reasons. The need to stabilize the standardGalerkin finite element approximation comes from two mainsources. Due to the incompressibility constraints, pressureinstabilities appear if we wish to use equal velocity-pressureinterpolations. On the other hand, Eulerian or ALE for-malisms introduce convection terms in the governing equa-tions and convection instabilities may arise for convectiondominant problems. As it is now well known, both sourcesof instabilities can be overcome by using sub-grid scale finiteelement stabilized formulations [57–60].

Sub-grid scale finite element stabilized methods have beenwidely used in computational fluid dynamics (CFD) and theywere initially developed by Hughes et al. for the Stokes prob-lem [57].

Sub-grid scale models were first introduced by Hughes[58]. The Variational Multiscale (VMS) method—a par-adigm for computational mechanics—was introduced byHughes et al. [59]. See also Hughes et al. [60] for a pre-sentation of multiscale and stabilized methods. The basicidea was to split the exact continuous unknowns into twocomponents or scales: the finite-element component and thesub-grid scale or subscale component. The subscales repre-sent the component of the exact continuous solution whichcannot be captured by the finite element partition. The par-ticular approximation used for these sub-grid scales definesthe numerical model.

Within the CFD framework, VMS methods have beenused by Codina [37] for the diffusion-convection reactionproblem, Codina [39] to stabilize incompressibility and con-vection through orthogonal subscales, first introducing theorthogonal sub-grid scale (OSGS) method, Codina [41] forgeneralized stationary incompressible flows, Codina [42] fortransient incompressible flows using dynamic orthogonalsubscales, Hughes et al. [60] for the incompressible Navier-Stokes equations, Codina et al. [43] for transient incompress-ible flows using dynamic subscales, Codina and Principe [44]for thermally coupled incompressible flows using nonlineardynamic subscales, Principe [68], Principe and Codina [69],and Avila et al. [14] for the low Mach number flow equa-tion, and Codina et al. [45] for turbulent thermally coupledincompressible flows.

Codina et al. [43] showed that it was worth to track thesubscales in time in a variational multiscale approach to thetransient incompressible Navier–Stokes equations and to takeinto account all their contributions in the convective term.They pointed out two main reasons. The first and very sim-ple reason was that it leaded to global momentum conserva-tion, a rare property. The second one was the door opening tomodel turbulence. Tracking the subscales from the point ofview of the time integration scheme showed several advan-tages. First, the resulting formulation leaded in a natural wayto the correct behavior of the stabilization parameters withthe time step, while steady-state solutions were not depend-ing on it. Moreover, the conflict about the design of the sta-bilization terms for time dependent problems (either at thesemi-discrete or the fully discrete level) disappeared, sincespace discretization (scale splitting) and time discretizationcan be commuted. Numerical results showed that the methodwas stable and the improvement in accuracy with respect toquasi-static subscales was notorious.

Codina and Principe [44] showed that the use of dynamicsubscales had also several benefits in the numerical sim-ulation of thermally coupled incompressible flows, suchas improved time stability and accuracy, particularly whenδt → 0, correct behavior of the stabilization parameters withthe time step size, commutation of space discretization (sub-scale splitting) and time discretization, and the coupling ofvelocity and temperature subscales was dealt with in a naturalway. Numerical results confirmed the conclusions drawn forisothermal flows, yielding a more accurate formulation, inwhich oscillations originated by initial transients were elim-inated and numerical dissipation was minimized.

Within the framework of Computational Solid Mechan-ics (CSM) sub-grid scale stabilization methods have beenused by Chiumenti et al. [29] for incompressible linearelasticity, Cervera et al. [22] for incompressible plastic-ity at small strains, Christ et al. [35] and Chiumenti et al.[30] for incompressible linear elasticity and small strainsJ2 plasticity, Cervera et al. [23] for small strains J2 con-

123

Computational Modeling and Sub-Grid Scale Stabilization

tinuum damage models, Agelet de Saracibar et al. [4] forinfinitesimal and finite strain J2 plasticity, and Agelet deSaracibar et al. [5] for finite deformation J2 plasticity. Theyused sub-grid scale finite element stabilized mixed displace-ment/pressure formulations, in which the sub-grid scale pres-sure was neglected. Algebraic sub-grid scale (ASGS) andorthogonal sub-grid scale (OSGS) methods were used.

Cervera et al. [25,26] used sub-grid scale finite elementstabilized mixed stress/displacement and strain/displacementformulations for nonlinear (damage) solid mechanics mod-els. Chiumenti et al. [32] used sub-grid scale finite ele-ment stabilized mixed strain/displacement and displace-ment/pressure formulations for Mode-I and Mode-II strainlocalization, using Rankine and J2 elasto-plasticity mod-els, respectively. Sub-grid scale finite element stabilizationmethods, using continuum damage models and a mixedstrain/displacement formulation, have been used by Cerveraet al. [26,27] to address the problem of strain localizationfor tensile and mixed-mode cracking within the so-calledsmeared crack approach. Numerical examples showed thatthe resulting procedure was remarkably robust, it was notrequired the use of auxiliary tracking techniques and theresults obtained were not suffering from spurious mesh biasdependence.

Chiumenti et al. [31] used a sub-grid scale finite elementstabilized mixed displacement/pressure/temperature for thenumerical simulation of Shaped Metal Deposition (SMD)processes.

Agelet de Saracibar et al. [6–8], Chiumenti et al. [33],and Dialami et al. [52] used a sub-grid scale finite elementstabilized mixed velocity/pressure/ temperature formula-tion for coupled thermo-rigid-plastic models, using Eulerianand Arbitrary Lagrangian Eulerian (ALE) formalisms, forthe numerical simulation of friction stir welding (FSW)processes. They used ASGS and OSGS methods and quasi-static sub-grid scales, neglecting the sub-grid scale pressureand using the finite element component of the velocity in theconvective term of the energy balance equation.

Chiumenti et al. [34] used a novel stress-accurate FE tech-nology for highly non-linear analysis with incompressibilityconstraints typically found in the numerical simulation ofFSW processes. They used a mixed linear piece-wise inter-polation for displacement, pressure and stress fields, respec-tively, resulting in an enhanced stress field approximationwhich enables for stress accurate results in nonlinear com-putational mechanics.

Friction stir welding (FSW) is a new method of weldingin solid state, created and patented by The Welding Insti-tute (TWI) in 1991 [77]. In FSW a cylindrical, shoulderedtool with a profiled probe is rotated and slowly plunged intothe joint line between two pieces of sheet or plate mater-ial, which are butted together. The parts have to be clampedonto a backing bar in a manner that prevents the abutting

joint faces from being forced apart. Once the probe has beencompletely inserted, it is moved with a small tilt angle inthe welding direction. The shoulder applies a pressure on thematerial to constrain the plasticised material around the probetool. Due to the advancing and rotating effect of the probeand shoulder of the tool along the seam, an advancing sideand a retreating side are formed and the softened and heatedmaterial flows around the probe to its backside where thematerial is consolidated to create a high-quality solid-stateweld. The maximum temperature reached is of the order of80 % of the melting temperature. Despite the simplicity ofthe procedure, the mechanisms behind the process and thematerial flow around the probe tool are very complex. Thematerial is extruded around the rotating tool and a vortex flowfield near the probe due to the downward flow is induced bythe probe thread. The process can be regarded as a solid phasekeyhole welding technique since a hole to accommodate theprobe is generated, then filled during the welding sequence.The material flow depends on welding process parameters,such as welding and rotation speed, pressure, etc., and on thecharacteristics of the tools, such as materials, design, etc.

The first applications of FSW have been in aluminium fab-rications. Aluminium alloys that are difficult to weld usingconventional welding techniques, are successfully weldedusing FSW. The weld quality is excellent, with none of theporosity that can arise in fusion welding, and the mechanicalproperties are at least as good as the best achievable by fusionwelding. Being a solid-state welding process, the structurein the weld nugget is free of solidifying segregation, beingsuitable for welding of composite materials. The process isenvironmentally friendly, because no fumes or spatter aregenerated, and there is no arc glare or reflected laser beamswith which to contend. Another major advantage is that, byavoiding the creation of a molten pool which shrinks signif-icantly on re-solidification, the distortion after welding andthe residual stresses are low. With regard to joint fit up, theprocess can accommodate a gap of up to 10 % of the materialthickness without impairing the quality of the resulting weld.As far as the rate of processing is concerned, for materialsof 2 mm thickness, welding speeds of up to 2 m/min can beachieved, and for 5 mm thickness up to 0.75 m/min. Recenttool developments are confidently expected to improve onthese figures.

Friction stir welding has been used to weld all wroughtaluminium alloys, across the AA-2xxx, AA-5xxx, AA-6xxxand AA-7xxx series of alloys, some of which are borderingon being classed as virtually unweldable by fusion weldingtechniques. The process can also weld dissimilar aluminiumalloys, whereas fusion welding may result in the alloyingelements from the different alloys interacting to form dele-terious intermetallics through precipitation during solidifi-cation from the molten weld pool. Friction stir welding canalso make hybrid components by joining dissimilar materials

123


such as aluminium and magnesium alloys. The thicknesses ofAA-6082-T6 that have so far been welded have ranged from1.2 to 50 mm in a single pass, to more than 75 mm whenwelding from both sides. Welds have also been made in pres-sure die cast aluminium material without any problems frompockets of entrapped high pressure gas, which would vio-lently disrupt a molten weld pool encountering them.

The original application for friction stir welding was thewelding of long lengths of material in the aerospace, ship-building and railway industries. Examples include large fueltanks and other containers for space launch vehicles, cargodecks for high-speed ferries, and roofs for railway carriages.FSW is used already in routine, as well as in critical appli-cations, for the joining of structural components made ofaluminium and its alloys. Indeed, it has been convincinglydemonstrated that the process results in strong and ductilejoints, sometimes in systems which have proved difficultusing conventional welding techniques. The process is mostsuitable for components which are flat and long (plates andsheets) but can be adapted for pipes, hollow sections andpositional welding.

The computational modeling of FSW processes is a com-plex task and it has been a research topic of increasing interestin computational mechanics during the last decades.

Thermal models for the numerical simulation of FSWprocesses were used by McClure et al. [66], Colegrove etal. [46], and Khandkar and Khan [62,63].

Bendzsak et al. [15,16] used the Eulerian code Stir3D tomodel the flow around a FSW tool, including the tool threadand tilt angle in the tool geometry and obtaining complexflow patterns. The temperature effects on the viscosity wereneglected.

Dong et al. [53] developed a simplified model for thenumerical simulation of FSW processes, taking into accountboth the friction heating and plastic work in the modellingof the heat flow phenomena, predicting the development of aplastic strain around the weld zone in the initial stage of weld-ing. However, they did not consider the longitudinal move-ment of the tool.

Xu et al. [79] and Xu and Deng [80,81] developed a3D finite element procedure to simulate the FSW processusing the commercial Finite Element Method (FEM) soft-ware ABAQUS, focusing on the velocity field, the materialflow characteristics and the equivalent plastic strain distrib-ution. The authors used an Arbitrary Lagrangian-Eulerian(ALE) formulation with adaptive meshing and considerlarge elasto-plastic deformations and temperature-dependentmaterial properties. However, the authors did not perform afully coupled thermo-mechanical simulation, superimposingthe temperature map obtained from the experiments as a pre-scribed temperature field to perform the mechanical analysis.The numerical results were compared to experimental dataavailable, showing a reasonable good correlation between the

equivalent plastic strain distributions and the distribution ofthe microstructure zones in the weld.

Ulysse [78] presented a fully coupled 3D FEM visco-plastic model for FSW of thick aluminium plates using thecommercial FEM code FIDAP. The author investigated theeffect of tool speeds on the process parameters. It was foundthat a higher translational speed leads to a higher weldingforce, while increasing the rotation speed has the oppositeeffect. Reasonable agreement between the predicted and themeasured temperature was obtained and the discrepancieswere explained by an inadequate representation of the con-stitutive behavior of the material for the wide ranges of strain-rate, temperatures and strains typically found during FSW.

Askari et al. [13] used the CTH hydrocode coupled toan advection-diffusion solver for the energy balance equa-tion. The CTH code, developed by Sandia National Labora-tories, uses the finite volume method to discretize the domain.The elastic response was taken into account in this case. Theresults proved encouraging with respect to gaining an under-standing of the material flow around the tool. However, sim-plified friction conditions were used.

Chen and Kovacevic [28] developed a 3D FEM model tostudy the thermal history and thermo-mechanical phenomenain the butt-welding of aluminium alloy AA-6061-T6 usingthe commercial FEM code ANSYS. Their model incorpo-rated the mechanical reaction between the tool and the weldmaterial. Experiments were conducted and an X-ray diffrac-tion technique was used to measure the residual stress in thewelded plate. The welding tool (i.e. the shoulder and pin) inthe FEM model was modeled as a heat source, with the nodesmoved forward at each computational time step. This simplemodel severely limited the accuracy of the stress and forcepredictions.

Colegrove et al. [46,47] used the commercial Computa-tional Fluid Dynamics (CFD) software FLUENT for a 2D and3D numerical investigation on the influence of pin geometryduring FSW, comparing different pin shapes in terms of mate-rial flow and welding forces on the basis of both a stick anda slip boundary condition at the tool/work-piece interface. Inspite of the good obtained results, the accuracy of the analy-sis was limited by the assumption of isothermal conditions.Seidel and Reynolds [73] also used the CFD commercialsoftware FLUENT to model the 2D steady-state flow arounda cylindrical tool.

Schmidt and Hattel [72] presented the development of a3D fully coupled thermo mechanical finite element model inABAQUS/Explicit using the ALE formulation. The flexibil-ity of the FSW machine was taken into account by connect-ing the rigid tool to a spring. The work-piece was modeledas a cylindrical volume with inlet and outlet boundary condi-tions. A rigid back-plate was used. The contact forces weremodeled using a Coulomb friction law, and the surface wasallowed to separate. Heat generated by friction and plastic

123


deformation was considered. The simulation modeled thedwell and weld phases of the process.

An ALE formulation for the numerical simulation of FSWprocesses was also used by Zhao [82].

Nikiforakis [67] used a finite difference method to modelthe FSW process. Despite the fact that he was only present-ing 2D results, the model proposed had the advantage ofminimizing the calibration of model parameters, taking intoaccount a maximum of physical effects. A transient and fullycoupled thermo-fluid analysis was performed. The rotationof the tool was handled through the use of the overlappinggrid method. A rigid-viscoplastic material law was used andsticking contact at the tool work piece interface was assumed.Hence, heating was due to plastic deformation only.

Heurtier et al. [56] used a 3D semi-analytical coupled ther-momechanical FE model to simulate FSW processes. Themodel uses an analytical velocity field and considers heatinput from the tool shoulder and plastic strain of the bulkmaterial. Trajectories, temperature, strain, strain rate fieldsand micro-hardness in various weld zones were computedand compared to experimental results obtained on an AA2024-T351 alloy FSW joint.

Buffa et al. [18] using the commercial finite element soft-ware DEFORM-3D, proposed a 3D Lagrangian, implicit,coupled thermo-mechanical numerical model for the sim-ulation of FSW processes, using a rigid-viscoplastic mate-rial description and a continuum assumption for the weldseam. The proposed model is able to predict the effect ofprocess parameters on process variables, such as the tem-perature, strain and strain rate fields, as well as materialflow and forces. A reasonable good agreement between thenumerically predicted results, on forces and temperature dis-tribution, and experimental data was obtained. The authorsfound that the temperature distribution about the weld line isnearly symmetric because the heat generation during FSWis dominated by rotating speed of the tool, which is muchhigher than the advancing speed. On the other hand, thematerial flow in the weld zone is non-symmetrically distrib-uted about the weld line because the material flow duringFSW is mainly controlled by both advancing and rotatingspeeds.

De Vuyst et al. [48–51] used the coupled thermo-mechanical finite element code MORFEO to simulate theflow around simplified tool geometries for FSW process.The rotation and advancing speed of the tool were modeledusing prescribed velocity fields. An attempt to consider fea-tures associated to the geometrical details of the probe andshoulder, which had not been discretized in the finite ele-ment model in order to avoid very large meshes, was takeninto account using additional special velocity boundary con-ditions. In spite of that, a mesh of roughly 250,000 nodesand almost 1.5 million of linear tetrahedral elements wasused. A Norton-Hoff rigid-viscoplastic constitutive equa-

tion was considered, with averaged values of the consistencyand strain rate sensitivity constitutive parameters determinedfrom hot torsion tests performed over a range of temperaturesand strain rates. The computed streamlines were comparedwith the flow visualization experimental results obtainedusing copper marker material sheets inserted transversallyor longitudinally to the weld line. The simulation resultscorrelated well when compared to markers inserted trans-versely to the welding direction. However, when comparedto a marker inserted along the weld centerline only qualitativeresults could be obtained. The correlation may be improvedby modeling the effective weld thickness of the experiment,using a more realistic material model, for example, by incor-porating a yield stress or temperature dependent properties,refine velocity boundary conditions or further refining themesh in specific zones, such as for instance, under the probe.The authors concluded that it is essential to take into accountthe effects of the probe thread and shoulder thread in orderto get realistic flow fields.

Shercliff et al. [74] developed microstructural models forfriction stir welding of 2000 series aluminium alloys.

López et al. [65] and Agelet de Saracibar et al. [9] devel-oped numerical algorithms to optimize material model andFSW process parameters using neural networks. They pro-posed a new model for the dissolution of precipitates in fullyhardened aluminium alloys and they optimized the mastercurve and the effective activation energy. Furthermore, theydeveloped an algorithm to optimize the advancing and rota-tion speed, taking as weld quality criteria the minimization ofthe maximum hardness drop at the transversal section underthe pin.

Santiago et al. [71] developed a simplified computationalmodel taking into account the real geometry of the tool, i.e.the probe thread, and using an ALE formulation. They con-sidered also a simplified friction model to take into accountdifferent slip/stick conditions at the pin shoulder/work-pieceinterface.

Chiumenti et al. [33], Dialami et al. [52], and Chiumentiet al. [34] developed an apropos kinematic framework for thenumerical simulation of FSW processes. They considered acombination of ALE, Eulerian and Lagrangian descriptionsat different zones of the computational domain and they pro-posed an efficient coupling strategy. Within this approach, aLagrangian formulation was used for the pin, an ALE for-mulation was used at the stir zone of the work-piece, and anEulerian formulation was used in the remaining part of thework-piece. The stir zone was defined as a circular domainclose to the pin. The finite element mesh in the stir zone wasrotating attached to the pin. The resulting apropos kinematicsetting efficiently permitted to treat arbitrary pin geometriesand facilitates the application of boundary conditions. Theformulation was implemented in an enhanced version of thefinite element code COMET [21] developed by the authors

123


at the International Center for Numerical Methods in Engi-neering (CIMNE).

Bussetta et al. [19,20] compared a fluid approach using anapropos kinematic framework, developed by Chiumenti et al.[33], Dialami et al. [52], and Chiumenti et al. [34], and theirsolid approach using an ALE formalism, in the numericalsimulation of FSW processes with a non-cylindrical pin.

This paper deals with the computational modeling andsub-grid scale stabilization of incompressibility and convec-tion for the numerical simulation of the material flow aroundthe probe tool in a FSW process.

Within the paradigmatic framework of the multiscale sta-bilization methods, suitable sub-grid scale finite element sta-bilization methods to stabilize the incompressibility con-straint and the convection term arising in coupled thermo-mechanical formulations when a Eulerian description andmixed P1/P1/P1 linear velocity/linear pressure/linear tem-perature finite elements are used. Norton-Hoff and Sheppard-Wright thermo-rigid-viscoplastic constitutive material mod-els are considered. Quasi-static sub-grid scales are con-sidered and approximations of the sub-grid scale variablesand stabilization parameters are given. In particular, alge-braic sub-grid scale (ASGS) and orthogonal sub-grid scale(OSGS) methods are presented. Furthermore, it is shownthat well known classical stabilized formulations, such as theGalerkin Least-Squares (GLS) for incompressible (or quasi-incompressible) problems or the Streamline Upwind/Petrov-Galerkin (SUPG) for convection dominant problems, can berecovered as particular cases of the multiscale stabilizationframework considered.

Within the framework of fractional step methods, a prod-uct formula algorithm, arising from an isothermal split, isintroduced to solve the coupled thermomechanical problem[1–3,10–12]. The resulting algebraic system of equations issolved using a staggered procedure, in which a mechani-cal problem, defined by the quasi-static version of the linearmomentum balance equation and the incompressibility equa-tion, is solved first at constant temperature. Then a thermalproblem, defined by the energy balance equation, is solvednext, keeping constant the mechanical variables, i.e. velocityand pressure.

One of the goals of the paper is to compare the two sub-grid scale finite element stabilization methods implemented,ASGS and OSGS. Another goal of the paper is to assessthe performance of the ASGS and OSGS methods in a 3Dnumerical simulation of a FSW process.

The outline of the paper is as follows. Section 2 dealswith the sub-grid scale stabilized mixed formulation for thenumerical simulation of the flow of the material around a toolprobe in a FSW processes. Some key features of the FSWprocess are addressed and their implications in the compu-tational model are first identified. The strong form of thegoverning equations and constitutive equations are provided,

and the mixed variational form of the governing equationsis given. Next, the multiscale stabilization method is intro-duced and an approximation of the sub-grid scale variablesand the stabilization parameters is given. ASGS and OSGSfinite element stabilization methods are considered and it isshown how classical GLS and SUPG methods can be recov-ered as a particular case of the ASGS method.

An isothermal operator split and a product formula algo-rithm (PFA), defining mechanical and thermal problems, areintroduced in Sect. 3. The ASGS and OSGS finite elementstabilized formulation for the mechanical and thermal prob-lems are presented.

Section 4 introduces a finite element partition of themechanical and thermal problems and the resulting algebraicsystems of equations are written in matrix form. Finally aconvenient and efficient staggered solution algorithm for theOSGS method is presented.

Two representative numerical examples are shown inSect. 5. The first one is a numerical simulation of a coupledthermomechanical flow in a 2D rectangular domain. Steady-state and transient conditions are considered. An assessmentof the ASGS and OSGS stabilization methods implementedis performed. The second example shows a 3D numericalsimulation of a FSW process. Numerical results are com-pared with the experimental results provided by Zhu andChao [83]. Calculations are performed using an enhancedversion of the finite element program COMET [21] devel-oped by the authors at the International Center for Numer-ical Methods in Engineering (CIMNE) in Barcelona. Pre-and post-processing is done with GiD [54], also developedat CIMNE. Finally some concluding remarks are drawn.

2 Sub-Grid Scale Stabilized Mixed Formulation

2.1 Computational Modelling of FSW Processes

The flow of the material around a FSW tool is characterizedby a Reynolds number which is much smaller than 1, typ-ically of the order 10−4, due to the small length scale, thelow velocities and the very high viscosity of the material.For these values of the Reynolds number, the inertial forcesof the linear momentum balance equation can be neglectedand a quasi-static analysis can be performed.

The deformation of the material taking place around aFSW tool is extremely high. The computational modeling ofthe material flow around a FSW tool using a Lagrangian for-mulation requires continuous remeshing to avoid extremelydistorted mesh elements. Therefore, the use of alternative for-mulations, such as ALE or Eulerian formulations, is a betterchoice. In this work we will use an Eulerian formulation.

Transient thermal conditions will be considered. On theother hand, the Peclet number for a FSW process typically

123


ranges from 101 to 103. For this range of values of the Pecletnumber, the convective term of the spatial energy balanceequation cannot be neglected.

Coupled thermo-mechanical rigid-visco-plastic consti-tutive material models, such as the Norton-Hoff or theSheppard-Wright, will be considered [6–8,19,20,33,34,52].

Heat transfer by conduction, convection and radiation areconsidered. Heat transfer by conduction is considered at thecontact interface between the work-pieces and the tool, andthe work-pieces and the back-plate. Heat transfer by convec-tion, governed by the Newton law, and radiation, governedby the Stefan-Boltzmann law, are considered at the surfacesof the work-pieces and the tool which are in contact with theenvironment [33,34].

The resulting coupled thermo-mechanical problem willbe solved using a product formula algorithm, arising from anisothermal split, leading to a staggered solution algorithm.A mechanical problem, involving mechanical variables asunknowns, is defined at constant temperature and a thermalproblem, involving the temperature as unknown, is definedat constant configuration. A pressure stabilized mixed lin-ear velocity/linear pressure finite element interpolation for-mulation will be used to solve the mechanical problem anda convection stabilized linear temperature interpolation for-mulation will be used to solve the thermal problem [33,34].

2.2 Local Formulation

Let � ⊂ Rndim , with ndim = 2, 3, be the computational

domain of interest in which the deformation takes placeduring the time interval [0, T ], and let ∂� be its bound-ary. Let us consider the following partitions of the bound-ary ∂� = ∂v� ∪ ∂σ �, ∂� = ∂ϑ� ∪ ∂q�, such that∅ = ∂v� ∩ ∂σ �, ∅ = ∂ϑ� ∩ ∂q�. Using a mixedvelocity/pressure/temperature spatial formulation, consider-ing quasi-static conditions and incompressibility, the Initialand Boundary Value Problem (IBVP) to be considered isdefined by the local spatial form of the linear momentum bal-ance equation, mass continuity equation and energy balanceequation, together with appropriate Dirichlet and Neumannboundary conditions, and initial conditions. The IBVP con-sists in finding a velocity vector field v, a pressure scalar fieldp, and a temperature scalar field ϑ such that

−∇ p − ∇ · s (v, ϑ) = b in �, t ∈ [0, T ]∇ · v = 0 in �, t ∈ [0, T ]c∂tϑ + cv · ∇ϑ + ∇ · q (ϑ)

= r + D in �, t ∈ [0, T ]v = v on ∂v�, t ∈ [0, T ](p1 + s (v, ϑ)) n = t on ∂σ �, t ∈ [0, T ]ϑ = ϑ on ∂ϑ�, t ∈ [0, T ]q (ϑ) · n = q on ∂q�, t ∈ [0, T ]ϑ = ϑ0 in �, t = 0

(1)

In these equations, s(v, ϑ) is the deviatoric part of theCauchy stress tensor, q(ϑ) is the spatial heat flux per unitof surface, b is the body force per unit of spatial volume,r is the internal heat source rate per unit of spatial volume,D = D(v, ϑ) is the dissipation rate per unit of spatial volume,v is the prescribed velocity vector field, t is the prescribedtraction vector field, ϑ is the prescribed temperature, q is theprescribed outward normal heat flux per unit of surface, ϑ0

is the initial temperature, c = c(ϑ) is the heat capacity, ∇ isthe spatial nabla vector differential operator, and 1 is the unitsecond-order tensor.

Heat transfer by thermal contact, convection and radiationis considered according to the following expressions

qc = hcgϑ

qconv = hconv (ϑ − ϑenv)

qrad = σε(ϑ4 − ϑ4

env

)(2)

where qc is the heat transfer by thermal contact, qconv isthe heat transfer by convection, qrad is the heat transferby radiation, hc is the contact heat transfer coefficient, gϑ

is the thermal contact gap, hconv is the convection heattransfer coefficient, ϑenv is the environmental temperature,σ = 5.6704E-08 Wm-2K-4 is the Stefan-Boltzmann con-stant, and ε is the relative emissivity.

Material mechanical response is modeled using an incom-pressible thermo-rigid-plastic material model. Taking intoaccount incompressibility, the constitutive equation for thedeviatoric part of the Cauchy stress tensor and the dissipationrate per unit of volume can be written as

s (v, ϑ) = 2μ (v, ϑ) dev(∇sv

) = 2μ (v, ϑ)∇sv

D (v, ϑ) = s (v, ϑ) : dev(∇sv

)

= 2μ (v, ϑ) dev(∇sv

) : dev(∇sv

)

= 2μ (v, ϑ) ∇sv : ∇sv (3)

where μ = μ(v, ϑ) is the equivalent viscosity of the materialmodel, ∇s(·) is the spatial symmetric gradient operator, anddev(·) is the deviatoric operator.

Here two target thermo-rigid-plastic material models,Norton-Hoff and Sheppard-Wright, will be considered andtheir constitutive equations are given in the next sub-section.

Material thermal response is modeled using Fourier lawgiven by

q (ϑ) = −κ (ϑ)∇ϑ (4)

where κ = κ(ϑ) is the thermal conductivity.Substituting the mechanical and thermal constitutive

equations into the governing equations, the IBVP can be writ-ten as

123


−∇ p − ∇ · (2μ∇sv) = b in �, t ∈ [0, T ]∇ · v = 0 in �, t ∈ [0, T ]c∂tϑ + cv · ∇ϑ − ∇

· (κ∇ϑ) = r + D in �, t ∈ [0, T ]v = v on ∂v�, t ∈ [0, T ](p1 + 2μ∇sv) n = t on ∂σ �, t ∈ [0, T ]ϑ = ϑ on ∂ϑ�, t ∈ [0, T ]−κ∇ϑ · n = q on ∂q�, t ∈ [0, T ]ϑ = ϑ0 in �, t = 0

(5)

2.3 Constitutive Equations

Two mechanical constitutive incompressible thermo-rigid-plastic material models, suitable for the numerical simula-tion of friction stir welding (FSW) processes, are considered:Norton-Hoff and Sheppard-Wright [33,34].

(i) Norton-Hoff material model: The constitutive equation forthe rigid-plastic Norton-Hoff material model is given by

s (v, ϑ) = 2μ (v, ϑ) dev(∇sv

) = 2μ (v, ϑ)∇sv

:= K (ϑ)(√

3 ˙ε (v))m(ϑ)−1 ∇sv (6)

where K (ϑ) is a temperature dependent consistency parame-ter, 0 ≤ m(ϑ) ≤ 1 is a temperature dependent rate sensitivityparameter and ˙ε(v) is the equivalent strain rate defined as

˙ε (v)=(

2

3dev

(∇sv) : dev

(∇sv))1/2

=(

2

3∇sv : ∇sv

)1/2

(7)

Then, the equivalent viscosity for the Norton-Hoff mate-rial model takes the form

μ (v, ϑ) := 1

2K (ϑ)

(√3 ˙ε (v)

)m(ϑ)−1(8)

(ii) Sheppard-Wright material model: The constitutive equa-tion for the rigid-plastic Sheppard-Wright material model isgiven by

s (v, ϑ) = 2μ (v, ϑ) dev(∇sv

) = 2μ (v, ϑ)∇sv

:= 2σe( ˙ε (v) , ϑ

)

3 ˙ε (v)∇sv (9)

where σe( ˙ε(v), ϑ) is the strain rate and temperature depen-dent yield stress defined as

σe( ˙ε (v) , ϑ

) = 1

αlog

⎡⎣

(Z

( ˙ε (v) , ϑ)

A

)1/n

+√√√√1 +

(Z

( ˙ε (v) , ϑ)

A

)2/n⎤⎥⎦ (10)

where α, A and n are material parameters, and Z( ˙ε(v), ϑ) isthe Zener-Hollomon parameter, representing the temperaturecompensated equivalent strain rate, given by

Z( ˙ε (v) , ϑ

) = ˙ε (v) exp

(Q

Rϑ

)(11)

where Q is the activation energy and R = 8.314 Jmol−1K−1

is the universal constant for an ideal gas.Then, the equivalent viscosity for the Sheppard-Wright

material model takes the form

μ (v, ϑ) := σe( ˙ε (v) , ϑ

)

3 ˙ε (v)(12)

2.4 Variational Formulation

To define the variational setting let us introduce some stan-dard notation. Let L p(�) be the space of functions whosep power (1 ≤ p < ∞) are integrable in a domain �, beingL∞ (�) the space of bounded functions in �. Let Hm (�)

be the space of functions whose derivatives of order up tom ≥ 0 (integer) belong to L2 (�). The space H1

0 (�) con-sists of functions in H1 (�) vanishing on the boundary. Abold character is used to denote the vector counterpart of allthese spaces. The L2 (�) inner product is denoted as (·, ·)and the L2 (∂�) inner product is denoted as (·, ·)∂�.

Let us introduce the infinite dimensional functional spacesW := V × P × T and W0 := V0 × Q × T0 where

V ={

v ∈ H2 (�) |v = v on ∂u�}

P ={

p ∈ H1 (�)}

T ={ϑ ∈ H2 (�) |ϑ = ϑ on ∂ϑ�

}

V0 ={δv ∈ L2 (�) |δv = 0 on ∂u�

}

Q ={δp ∈ L2 (�)

}

T0 ={δϑ ∈ L2 (�) |δϑ = 0 on ∂ϑ�

}(13)

The variational form of the IBVP consists in finding avelocity vector field v ∈ V , pressure scalar field p ∈ P andtemperature scalar field ϑ ∈ T such that

− (δv,∇ p) − (δv,∇ · (

2μ∇sv)) = (δv, b) ∀δv

(δp,∇ · v) = 0 ∀δp

(δϑ, c∂tϑ) + (δϑ, cv · ∇ϑ) − (δϑ,∇ · (κ∇ϑ))

= (δϑ, r) + (δϑ,D) ∀δϑ (14)

where δv ∈ V0, δp ∈ Q, δϑ ∈ T0.Integrating by parts some of the terms above, the varia-

tional form of the IBVP consists in finding a velocity vector

123


field v ∈ V , pressure scalar field p ∈ P and temperaturescalar field ϑ ∈ T such that

(∇ · δv, p) + (∇sδv, 2μ∇sv) = Fv (δv) ∀δv

(δp,∇ · v) = 0 ∀δp

(δϑ, c∂tϑ) + (δϑ, cv · ∇ϑ) + (∇δϑ, κ∇ϑ) = Fϑ (δϑ) ∀δϑ

(15)

where

Fv (δv) := (δv, b) + (δv, t

)∂σ �

Fϑ (δϑ) := (δϑ, r) + (δϑ,D) − (δϑ, q) ∂q� (16)

and δv ∈ V0, δp ∈ P, δϑ ∈ T0 where, now the spaces offunctions W := V × P × T and W0 := V0 × P × T0 aredefined as

V ={

v ∈ H1 (�) |v = v on ∂v�}

P ={

p ∈ L2 (�)}

T ={ϑ ∈ H1 (�) |ϑ = ϑ on ∂ϑ�

}

V0 ={δv ∈ H1

0 (�) |δv ∈ H1 (�) and δv = 0 on ∂v�}

T0 ={δϑ ∈ H1

0 (�) |δϑ ∈ H1 (�) and δϑ = 0 on ∂ϑ�}

(17)

2.5 Galerkin Finite Element Projection

The standard Galerkin projection of the variational form ofthe IBVP is now straightforward. Let us consider a finiteelement partition {�e} , e = 1, . . . , ne of the computationaldomain �, where ne is the number of elements in the parti-tion. We can now construct conforming finite element spacesfor the velocity, pressure and temperature in the usual man-ner. We will assume that they are all built from continuouspiecewise polynomials of the same degree k.

Let us introduce the finite element spaces Wh := Vh ×Ph × Th ⊂ W and W0,h := V0,h × Ph × T0,h ⊂ W0 whereVh ⊂ V,Ph ⊂ P, Th ⊂ T ,V0,h ⊂ V0 and T0,h ⊂ T0 aredefined as

Vh ={

vh ∈ H1 (�) |vh = vh on ∂v�}

Ph ={

ph ∈ L2 (�)}

Th ={ϑh ∈ H1 (�) |ϑh = ϑh on ∂ϑ�

}

V0,h ={δvh ∈ H1 (�) |δvh = 0 on ∂v�

}

T0,h ={δϑh ∈ H1 (�) |δϑh = 0 on ∂ϑ�

}(18)

The spatial discrete variational form of the IBVP consistsin finding a velocity vector field vh ∈ Vh , pressure scalar

field ph ∈ Ph and temperature scalar field ϑh ∈ Th such that

(∇ · δvh, ph) + (∇sδvh, 2μh∇svh) = Fv (δvh) ∀δvh

(δph,∇ · vh) = 0 ∀δph

(δϑh, ch∂tϑh) + (δϑh, chvh · ∇ϑh) + (∇δϑh, κh∇ϑh)

= Fϑ (δϑh) ∀δϑh (19)

where δvh ∈ V0,h, δph ∈ Ph, δϑh ∈ T0,h .It is well know that the spatial discrete variational prob-

lem defined above may suffer from two types of numericalinstabilities: the compatibility required for the velocity andpressure finite element spaces posed by the inf-sup conditionand the dominance of the (nonlinear) convective term overthe viscous one when the equivalent viscosity is small [17].

In this work we adopt a stabilized finite element formula-tion based on the sub-grid scales method first introduced byHughes [58] and as a Variational Multiscale (VMS) stabiliza-tion method by Hughes et al. [59]. The sub-grid scale finiteelement stabilization method allows the use of equal veloc-ity/pressure order interpolation, thus avoiding or bypassingthe need to satisfy the inf-sup or Babuska-Brezzi condition[17], and avoids the oscillations which arise in convectivedominant problems. The basic idea is to split the exact con-tinuous unknowns into two components, corresponding totwo different scales, the resolvable coarse scale and the unre-solved fine scale. The first one is captured by the Galerkinfinite element projection and is denoted as the finite elementscale. The second one cannot be captured by the finite ele-ment solution and is denoted as the sub-grid scale. The goalis to find an approximate solution for the sub-grid scales andto include them into the discrete finite element solution.

Within the sub-grid scales framework, different stabiliza-tion formulations can be considered. Here we will restrictour attention to two approaches denoted as algebraic sub-grid scale (ASGS) and orthogonal sub-grid scale (OSGS).In the ASGS method, the velocity, pressure and tempera-ture sub-grid scales are taken proportional to the correspond-ing residuals of the velocity, pressure and temperature equa-tions, given in terms of the linear momentum balance, incom-pressibility and energy balance equations, respectively. In theOSGS method, only the component of these residuals whichis L2 orthogonal to the corresponding finite element space isconsidered. This idea was first introduced by Codina in [39]as an extension of a stabilization method originally intro-duced for the Stokes problem in [36] and fully analyzed forthe stationary Navier-Stokes equations in [40].

The main features of the sub-grid scale finite element sta-bilization formulation proposed in this work are the follow-ing: (i) quasi-static sub-grid scales, instead of dynamic sub-grid scales, are considered, neglecting the time derivativeof the sub-grid scale temperature; (ii) only sub-grid scalevelocity and temperature are introduced, assuming that thesub-grid scale pressure is zero; (iii) the stabilized velocity,

123


instead of its finite element component, is used in the con-vective term of the sub-grid scale temperature; (iv) trackingof the sub-grid scales is not done in the temperature depen-dent material properties and the dissipation rate, and they areapproximated by their corresponding finite element compo-nents; (v) ASGS and OSGS sub-grid scale finite elementstabilization methods are considered.

2.6 Sub-Grid Scales Stabilization

Within the paradigmatic sub-grid scale stabilization frame-work introduced by Hughes et alet al. [57], it is consideredthat the continuous unknown fields can be split in two com-ponents, corresponding to different scales or levels of reso-lution: a coarse one, which is captured by the finite elementpartition, and a fine one, which cannot be captured by thefinite element partition.

In order to get a stable finite element solution for the dis-crete finite element formulation, it is necessary to include,somehow, the effect of both scales in the approximation. Thecoarse or finite element scale can be appropriately solved bya standard Galerkin finite element approximation, while thefine or sub-grid scale can be included, at least locally, toenhance the stability of the Galerkin finite element approxi-mation.

Let us consider the following splits of the infinite dimen-sional spaces as W = Wh ⊕ W and W0 = W0,h ⊕ W0,where W = V × P × T is any suitable space to com-plete Wh in W and W0 = V0 × P0 × T0 is any suit-able space to complete W0,h in W0, where V, P, T aresub-grid scale spaces for the velocity, pressure and tem-perature, respectively, and V0, P0, T0 are the sub-grid scalespaces for the variations of the velocity, pressure and tem-perature, respectively. Obviously the sub-grid scale spacesare infinite-dimensional spaces, but once the final method isformulated they will be approximated by finite-dimensionalspaces, although we will keep the same symbols in order tosimplify the notation.

Let us consider the following sub-grid scale split of theexact continuous velocity, pressure and temperature fields

v = vh + v, p = ph + p, ϑ = ϑh + ϑ (20)

where the components with subscripts h belong to the cor-responding finite element spaces, and the components withthe tilde belong to the corresponding sub-grid spaces. Thenvh ∈ Vh, ph ∈ Ph, ϑh ∈ Th are the components of thevelocity, pressure and temperature on the (coarser) finite ele-ment scale, and v ∈ V, p ∈ P, ϑ ∈ T are the componentsof the velocity, pressure and temperature on the (finer) sub-grid scale. These additional components are what we willcall sub-grid scale or subscale velocity, pressure and tem-perature. We will assume that the sub-grid scales vanish at

the inter-element boundaries ∂�e and thus W ≈ W0. Thishappens, for instance, if they are approximated using bubblefunctions [58], or if one assumes that their Fourier modescorrespond to high wave numbers [39].

Substituting the sub-grid scale split of the velocity andtemperature into the equivalent viscosity, thermal conductiv-ity and heat capacity, the following expressions are obtainedand the following approximations are introduced

μ = μ(

vh + v, ϑh + ϑ)

≈ μ (vh, ϑh) := μh

κ = κ(ϑh + ϑ

)≈ κ (ϑh) := κh

c = c(ϑh + ϑ

)≈ c (ϑh) := ch (21)

Substituting the sub-grid scale split of the velocity andtemperature into the mechanical and thermal constitutiveequations, and taking into account the above approximations,yields

s (v, ϑ) = 2μ (v, ϑ) dev∇sv = 2μ (v, ϑ) ∇sv

≈ 2μ (vh, ϑh)∇svh + 2μ (vh, ϑh)∇s v

:= 2μh∇svh + 2μh∇s v

q (ϑ) = −κ (ϑ)∇ϑ

≈ −κ (ϑh)∇ϑh − κ (ϑh)∇ϑ

:= −κh∇ϑh − κh∇ϑ (22)

Similarly, the following approximation is introduced forthe dissipation rate per unit of volume

D (v, ϑ) = s (v, ϑ) : dev(∇sv

) = 2μ (v, ϑ)

dev(∇sv

) : dev(∇sv

)

= 2μ (v, ϑ)∇sv : ∇sv

≈ 2μ (vh, ϑh)(∇svh + ∇s v

) : (∇svh + ∇s v)

≈ 2μ (vh, ϑh)∇svh : ∇svh

:= 2μh∇svh : ∇svh := D (vh, ϑh) := Dh (23)

Neglecting the sub-grid scale pressure, p = 0, consider-ing quasi-static sub-grid scales, neglecting the dynamic sub-grid scale temperature, setting ∂t ϑ = 0, and substituting thesub-grid scale split of the velocity and temperature into thevariational formulation of the IBVP, two systems of vari-ational equations are obtained. The first one represents theprojection of the governing equations onto the correspondingdiscrete finite element spaces, and is given by

(∇ · δvh, ph) + (∇sδvh, 2μh∇svh) + (∇sδvh, 2μh∇s v

)

= Fv (δvh) ∀δvh

(δph,∇ · vh) + (δph,∇ · v) = 0 ∀δph

(δϑh, ch∂tϑh) + (δϑh, ch (vh + v) · ∇ϑh)

+(δϑh, ch (vh + v) · ∇ϑ

)+ (∇δϑh, κh∇ϑh)

+(∇δϑh, κh∇ϑ

)= Fϑ (δϑh) ∀δϑh (24)

123


The second system of equations represents the projectionof the governing equations onto the corresponding infinite-dimensional sub-grid scale spaces, and is given by

(∇ · δv, ph) + (∇sδv, 2μh∇svh) + (∇sδv, 2μh∇s v

)

= Fv (δv) ∀δv(δϑ, ch∂tϑh

)+

(δϑ, ch (vh + v) · ∇ϑh

)

+(δϑ, ch (vh + v) · ∇ϑ

)+

(∇δϑ, κh∇ϑh

)

+(∇δϑ, κh∇ϑ

)= Fϑ

(δϑ

)∀δϑ (25)

Note that modeling the gradients of the sub-grid scales ismore involved than modeling the sub-grid scales themselves.Therefore integrating some terms by parts at the element leveland taking into account that the sub-grid scales vanish at theinter-element boundaries, the first system of equations canbe conveniently written as

(∇ · δvh, ph) + (∇sδvh, 2μh∇svh)

− (∇ · (2μh∇sδvh

), v

) = Fv (δvh) ∀δvh

(δph,∇ · vh) − (∇δph, v) = 0 ∀δph

(δϑh, ch∂tϑh) + (δϑh, ch (vh + v) · ∇ϑh)

−(∇ · (ch (vh + v) δϑh) , ϑ


−(∇ · (κh∇δϑh) , ϑ

)= Fϑ (δϑh) ∀δϑh (26)

On the other hand, for the second system of equations itis convenient to write them in terms of the correspondingfinite element residuals of the governing equations. There-fore integrating some terms by parts at the element level andtaking into account that the sub-grid scales vanish at theinter-element boundaries, the second system of equationscan be conveniently written as

− (δv,∇ · (

2μh∇s v)) = (

δv, rv,h) ∀δv

−(δϑ, ch (vh + v) · ∇ϑ − ∇ ·

(κh∇ϑ

))

=(δϑ, rϑ,h

)∀δϑ (27)

where the finite element residuals of the velocity and temper-ature equations, arising from the local governing equations,are defined as

rv,h := ∇ ph + ∇ · (2μh∇svh

) + bh

rϑ,h := ch∂tϑh + ch (vh + v) · ∇ϑh − ∇ · (κh∇ϑh)

− rh − Dh (28)

Note that the sub-grid scale velocity appears in the convectiveterm of the temperature residual.

The idea now is to find an approximate solution for the sec-ond system of variational equations, finding an approximatediscrete solution for the sub-grid scales within each elementof the finite element partition. Once the approximation for the

sub-grid scales has been found, they are substituted into thefirst system of variational equations, resulting in an enhancedstable mixed variational formulation of the IBVP.

If Pv and Pϑ denote the projections onto the sub-gridscale velocity and temperature spaces, respectively, the sec-ond system of equations can be written as

−Pv

(∇ · (2μh∇s v

)) = Pv

(rv,h

)

−Pϑ

(ch (vh + v) · ∇ϑ − ∇ ·

(κh∇ϑ

))= Pϑ

(rϑ,h

)

(29)

These equations need to be solved within each element, atthe element level, using homogeneous velocity and temper-ature Dirichlet boundary conditions.

Using the same arguments as in [39], now extended tothermally coupled problems, the sub-grid scales are approx-imated within each element of the finite element partitionas

1

τv

v = Pv

(rv,h

) = Pv

(∇ ph + ∇ · (2μh∇svh

) + bh)

1

τϑ

ϑ = −Pϑ

(rϑ,h

) = −Pϑ (ch∂tϑh

+ch (vh + v) · ∇ϑh − ∇ · (κh∇ϑh) − rh − Dh)

(30)

where the stabilization parameters introduced above areapproximated as

τv =(

c12μh

h2

)−1

= 1

c1

h2

2μh

τϑ =(

c1κh

h2 + c2ch |vh + v|

h

)−1

(31)

where h is the element size, and c1 and c2 are algorithmicconstants.

Note that, taking into account that we have assumed thatthe material parameters and the dissipation rate were notdependent on the sub-grid scales, the resulting system ofequations for the sub-grid scales is linear, the stabilizationparameter for the velocity do not depends on the sub-gridscales, while the stabilization parameter for the temperaturedepends on the sub-grid scale velocity.

Introducing the stabilized velocity, pressure and tempera-ture given by

vstabh := vh +v=vh +τvPv

(∇ ph +∇ · (2μh∇svh

)+bh)

pstabh := ph

ϑ stabh := ϑh + ϑ = ϑh − τϑ Pϑ (ch∂tϑh

+ chvstabh · ∇ϑh − ∇ · (κh∇ϑh) − rh − Dh)

(32)

123


the quasi-static sub-grid scale stabilized mixed variationalformulation of the IBVP can be written as(∇ · δvh, pstab

h

)+ (∇sδvh, 2μh∇svh

)

−(∇ · (

2μh∇sδvh), vstab

h − vh

)= Fv (δvh) ∀δvh

(δph,∇ · vh) −(∇δph, vstab

h − vh

)= 0 ∀δph

(δϑh, ch∂tϑh) +(δϑh, chvstab

h · ∇ϑh


−(∇ ·

(chvstab

h δϑh

), ϑ stab

h − ϑh

)

−(∇ · (κh∇δϑh) , ϑ stab

h − ϑh

)= Fϑ (δϑh) ∀δϑh (33)

Taking into account that vstabh := vh + v is divergence

free, the following expression holds

∇ ·(

chvstabh δϑh

)= ∇ch · vstab

h δϑh + chδϑh∇ · vstabh

+ chvstabh · ∇δϑh

= ∇ch · vstabh δϑh + chvstab

h · ∇δϑh (34)

and the quasi-static sub-grid scale stabilized mixed varia-tional formulation of the IBVP can be written as(∇ · δvh, pstab

h


)

−(∇ · (

2μh∇sδvh), vstab

h − vh



h − vh

)= 0 ∀δph


h · ∇ϑh


−(

chvstabh · ∇δϑh, ϑ stab

h − ϑh

)

−(∇ch · vstab

h δϑh, ϑ stabh − ϑh

)

−(∇ · (κh∇δϑh) , ϑ stab

h − ϑh

)= Fϑ (δϑh) ∀δϑh

(35)

2.6.1 P1/P1/P1 Linear Velocity/Linear Pressure/LinearTemperature Elements

For P1/P1/P1 linear velocity/linear pressure/linear tempera-ture elements, the terms ∇svh and ∇ϑh , as well as ∇sδvh

and ∇δϑh , are constants within an element and the followingexpressions hold

∇ · (∇svh) = 0, ∇ · (∇sδvh

) = 0,

∇ · (∇ϑh) = 0, ∇ · (∇δϑh) = 0 (36)

On the other hand, note that the equivalent viscosity is ahighly non-linear function of the equivalent strain rate, whichis a function of ∇svh , and, in general, of the temperature.Note that for P1/P1/P1 linear velocity/linear pressure/lineartemperature elements, the equivalent strain rate is constantwithin an element. Then, for P1/P1/P1 linear velocity/linear

pressure/linear temperature elements, the equivalent viscos-ity will be constant within an element only if it is not a func-tion of the temperature or if being a function of the temper-ature, the temperature is constant within the element.

Then, if the equivalent viscosity and the thermal conduc-tivity are temperature dependent, the following expressionshold for P1/P1/P1 linear velocity/linear pressure/linear tem-perature elements

∇ · (2μh∇svh

) = 2μh∇ · ∇svh + 2∇svh∇μh = 2∇svh∇μh,

∇ · (2μh∇sδvh

) = 2μh∇ · ∇sδvh +2∇sδvh∇μh =2∇sδvh∇μh,

∇ · (κh∇ϑh) = κh∇ · ∇ϑh + ∇κh · ∇ϑh = ∇κh · ∇ϑh,

∇ · (κh∇δϑh) = κh∇ · ∇δϑh + ∇κh · ∇δϑh = ∇κh · ∇δϑh

(37)

The quasi-static sub-grid scales for P1/P1/P1 linear veloc-ity/linear pressure/linear temperature elements are given by,

1

τv

v = Pv

(rv,h

) = Pv

(∇ ph + 2∇svh∇μh + bh)

1

τϑ

ϑ = −Pϑ

(rϑ,h

) = −Pϑ (ch∂tϑh + ch (vh + v) · ∇ϑh

−∇κh · ∇ϑh − rh − Dh) (38)

The stabilized velocity, pressure and temperature forP1/P1/P1 linear velocity/linear pressure/linear temperature ele-ments, take the form

vstabh := vh + v = vh + τvPv


pstabh := ph

ϑ stabh := ϑh + ϑ = ϑh

−τϑ Pϑ

(ch∂tϑh +chvstab

h · ∇ϑh −∇κh · ∇ϑh −rh −Dh

)

(39)

The quasi-static sub-grid scale stabilized mixed varia-tional formulation for P1/P1/P1 linear velocity/linear pres-sure/linear temperature elements, reads(∇ · δvh, pstab

h


)

−(

2∇sδvh∇μh, vstabh − vh



h − vh

)= 0 ∀δph


h · ∇ϑh


−(


h − ϑh

)

−(∇ch · vstab

h δϑh, ϑ stabh − ϑh

)

−(∇κh · ∇δϑh, ϑ stab

h − ϑh

)= Fϑ (δϑh) ∀δϑh (40)

Remark 1 Temperature independent equivalent viscosity,constant thermal conductivity and constant heat capacity.

123


Let us consider now P1/P1/P1 linear velocity/linear pres-sure/linear temperature elements with temperature indepen-dent equivalent viscosity, constant thermal conductivity, andconstant heat capacity. Note that the equivalent viscosity is afunction of the equivalent strain rate, which is a function ofthe symmetric spatial gradient of the velocity, and the tem-perature. Using a linear interpolation for the velocity and thetemperature, the spatial gradient of the velocity is constantwithin an element, and its spatial gradient will be equal tozero if the viscosity is temperature independent. Then, if theequivalent viscosity is temperature independent, the thermalconductivity is constant, and the heat capacity is constant,the following expressions hold at the element level

∇μh = 0, ∇κh = 0, ∇ch = 0 (41)

The quasi-static sub-grid scales for P1/P1/P1 linear veloc-ity/linear pressure/linear temperature elements, temperatureindependent equivalent viscosity, and constant thermal con-ductivity, take the form

1

τv

v = Pv

(rv,h

) = Pv (∇ ph + bh)

1

τϑ

ϑ = −Pϑ

(rϑ,h

)

= −Pϑ (ch∂tϑh + ch (vh + v) · ∇ϑh − rh − Dh)

(42)

and the stabilized velocity, pressure and temperature, takethe form

vstabh := vh + v = vh + τvPv (∇ ph + bh)

pstabh := ph


−τϑ Pϑ

(ch∂tϑh + chvstab

h · ∇ϑh − rh − Dh

)(43)

The quasi-static sub-grid scale stabilized mixed varia-tional formulation for P1/P1/P1 linear velocity/linear pres-sure/linear temperature elements, temperature independentequivalent viscosity, constant thermal conductivity, and con-stant heat capacity, reads(∇ · δvh, pstab

h


) = Fv (δvh) ∀δvh


h − vh

)= 0 ∀δph


h · ∇ϑh


−(


h − ϑh


(44)

Substituting the quasi-static sub-grid scales into the aboveexpression, the sub-grid scale stabilized mixed variationalformulation reads


(δph,∇ · vh) − τv

(∇δph, Pv (∇ ph)

)

−τv

(∇δph, Pv (bh)

)= 0 ∀δph


h · ∇ϑh


+τϑ

(chvstab

h · ∇δϑh, Pϑ (ch∂tϑh))

+τϑ

(chvstab

h · ∇δϑh, Pϑ

(chvstab

h · ∇ϑh

))

−τϑ

(chvstab

h · ∇δϑh, Pϑ (rh))

−τϑ

(chvstab

h · ∇δϑh, Pϑ (Dh))

= Fϑ (δϑh) ∀δϑh

(45)

2.6.2 Quasi-Static Algebraic Sub-grid Scale (ASGS)

For the algebraic sub-grid scale (ASGS) method, the projec-tions onto the sub-grid scale spaces velocity and temperaturespaces are defined as Pv = I and Pϑ = I , respectively,yielding

1

τv

v = Pv

(rv,h

) = rv,h = ∇ ph + 2∇svh∇μh + bh

1

τϑ

ϑ = −Pϑ

(rϑ,h

) = −rϑ,h

=−(

ch∂tϑh +chvstabh ·∇ϑh −∇κh ·∇ϑh −rh −Dh

)

(46)

The ASGS stabilized velocity, pressure and temperaturefor P1/P1/P1 linear velocity/linear pressure/linear tempera-ture elements take the form

vstabh := vh + v = vh + τv


pstabh := ph

ϑ stabh := ϑh + ϑ

= ϑh −τϑ

(ch∂tϑh +chvstab

h ·∇ϑh −∇κh ·∇ϑh −rh −Dh

)

(47)

Remark 2 Temperature independent equivalent viscosity,constant thermal conductivity and constant heat capacity.The quasi-static ASGS sub-grid scales for P1/P1/P1 linearvelocity/linear pressure/linear temperature elements, withtemperature independent equivalent viscosity, and constantthermal conductivity, take the form

1

τv

v = Pv

(rv,h

) = rv,h = ∇ ph + bh

1

τϑ

ϑ = −Pϑ

(rϑ,h

) = −rϑ,h

= −(

ch∂tϑh + chvstabh · ∇ϑh − rh − Dh

)(48)

123


and the ASGS stabilized velocity, pressure and temperaturetake the form

vstabh := vh + v = vh + τv (∇ ph + bh)

pstabh := ph

ϑ stabh :=ϑh +ϑ =ϑh −τϑ

(ch∂tϑh +chvstab

h · ∇ϑh −rh −Dh

)

(49)

The quasi-static ASGS sub-grid scale stabilized mixedvariational formulation reads


(δph,∇ · vh) − τv (∇δph,∇ ph)

−τv (∇δph, bh) = 0 ∀δph


h · ∇ϑh


+τϑ

(chvstab

h · ∇δϑh, ch∂tϑh

)

+τϑ

(chvstab

h · ∇δϑh, chvstabh · ∇ϑh

)

−τϑ

(chvstab

h · ∇δϑh, rh

)

−τϑ

(chvstab

h · ∇δϑh,Dh

)= Fϑ (δϑh) ∀δϑh (50)

Remark 3 Use of the finite element velocity, instead of thestabilized one, in the convective term of the temperatureequation.

Setting τv = 0 in the stabilized variational form of the energybalance equation, i.e. setting vstab

h = vh in the convectiveterm of the sub-grid scale temperature, the stabilized velocity,pressure and temperature read

vstabh := vh + v = vh + τv (∇ ph + bh)

pstabh := ph

ϑ stabh := ϑh + ϑ

= ϑhτϑ

(ch∂tϑh + chv·

h∇ϑh − rh − Dh)

(51)

The quasi-static ASGS sub-grid scale stabilized mixed vari-ational formulation reads


(δph,∇ · vh)

−τv (∇δph,∇ ph)

−τv (∇δph, bh) = 0 ∀δph


+τϑ (chvh · ∇δϑh, ch∂tϑh)

+τϑ (chvh · ∇δϑh, chvh · ∇ϑh)

−τϑ (chvh · ∇δϑh, rh)

−τϑ (chvh · ∇δϑh,Dh) = Fϑ (δϑh) ∀δϑh (52)

Remark 4 Classical GLS/SUPG stabilization method. Clas-sical GLS pressure stabilization and SUPG convection sta-bilization methods can be recovered as a particular case

of the ASGS stabilization method neglecting the followingterms:

τv (∇δph, bh) = 0

τϑ (chvh · ∇δϑh, ch∂tϑh) = 0

τϑ (chvh · ∇δϑh, rh) = 0

τϑ (chvh · ∇δϑh,Dh) = 0 (53)

The quasi-static ASGS sub-grid scale stabilized mixedvariational formulation corresponds to the classical GLS/SUPG stabilized mixed variational formulation and reads


(δph,∇ · vh) − τv (∇δph,∇ ph) = 0 ∀δph


+τϑ (chvh · ∇δϑh, chvh · ∇ϑh) = Fϑ (δϑh) ∀δϑh

(54)

2.7 Quasi-Static Orthogonal Sub-Grid Scale (OSGS)

For the orthogonal sub-grid scale (OSGS) method, the pro-jections onto the sub-grid scale velocity and temperaturespaces are defined as Pv = P⊥

h and Pϑ = P⊥h , where

P⊥h = I − Ph represents the L2 orthogonal projection, i.e.

the L2 projection onto the space which is orthogonal to theappropriate finite element one, and Ph represents the L2 pro-jection onto the appropriate, velocity or temperature, finiteelement space. The sub-grid scales turn out to be orthogonalto this finite element space.

Note that, assuming constant heat capacity, the orthogonalprojection of the transient thermal term is zero, yielding

P⊥h (ch∂tϑh) = 0 (55)

The quasi-static OSGS sub-grid scales for P1/P1/P1 linearvelocity/linear pressure/linear temperature elements take theform

1

τv

v = Pv

(rv,h

) = P⊥h

(rv,h

)

= P⊥h


1

τϑ

ϑ = −Pϑ

(rϑ,h

) = −P⊥h

(rϑ,h

)

= −P⊥h (ch (vh + v) · ∇ϑh − ∇κh · ∇ϑh − rh − Dh)

(56)

The OSGS stabilized velocity, pressure and temperaturefor P1/P1/P1 linear velocity/linear pressure/linear tempera-ture elements, take the form

vstabh := vh + v = vh + τvP⊥

h


pstabh := ph

123



−τϑP⊥h (ch (vh + v) · ∇ϑh − ∇κh · ∇ϑh − rh − Dh)

(57)

Remark 5 Temperature independent equivalent viscosity,constant thermal conductivity and constant heat capacity.

The quasi-static OSGS sub-grid scales for P1/P1/P1 linearvelocity/linear pressure/linear temperature elements, withtemperature independent equivalent viscosity, and constantthermal conductivity, take the form

1

τv

v = Pv

(rv,h

) = P⊥h

(rv,h

) = P⊥h (∇ ph + bh)

1

τϑ

ϑ = −Pϑ

(rϑ,h

) = −P⊥h

(rϑ,h

)

= −P⊥h (ch (vh + v) · ∇ϑh − rh − Dh) (58)

and the OSGS stabilized velocity, pressure and temperaturetake the form


h (∇ ph + bh)

pstabh := ph

ϑ stabh := ϑh + ϑ = ϑh − τϑP⊥

h

(chvstab

h · ∇ϑh − rh − Dh

)

(59)

The quasi-static OSGS sub-grid scale stabilized mixedvariational formulation reads



(∇δph,P⊥

h (∇ ph))

−τv

(∇δph,P⊥

h (bh))

= 0 ∀δph


h · ∇ϑh


+τϑ

(chvstab

h · ∇δϑh,P⊥h

(chvstab

h · ∇ϑh

))

−τϑ

(chvstab

h · ∇δϑh,P⊥h (rh)

)

−τϑ

(chvstab

h · ∇δϑh,P⊥h (Dh)


(60)

Remark 6 Use of the finite element velocity, instead of thestabilized one, in the convective term of the temperatureequation.

Setting τv = 0 in the stabilized variational form ofthe energy balance equation, i.e. setting vstab

h = vh inthe convective term of the sub-grid scale temperature, thesub-grid scale stabilized velocity, pressure and temperatureread


h (∇ ph + bh)

pstabh := ph

ϑ stabh := ϑh + ϑ = ϑh − τϑP⊥

h

(chv·

h∇ϑh − rh − Dh)

(61)

The quasi-static OSGS sub-grid scale stabilized mixedvariational formulation reads


Box 1 Quasi-static sub-gridscale stabilization. P1/P1/P1linear velocity/linearpressure/linear temperatureelement. Temperature dependentviscosity, thermal conductivityand heat capacity

123


Box 2 Quasi-static sub-gridscale stabilization. P1/P1/P1linear velocity/linearpressure/linear temperatureelement. Constant heat capacity,constant thermal conductivityand temperature independentviscosity

Box 3 Quasi-static sub-grid scale stabilization. P1/P1/P1 linear veloc-ity/linear pressure/linear temperature element. Constant heat capac-ity, constant thermal conductivity and temperature independent viscos-

ity, finite element velocity in the convective terms, and projections ofbody forces, temperature transient term, heat source and dissipation areneglected


(∇δph,P⊥

h (∇ ph))

−τv

(∇δph,P⊥

h (bh))

= 0 ∀δph


+τϑ

(chvh · ∇δϑh,P⊥

h

(chv·

h∇ϑh))

−τϑ


h (rh))

−τϑ


h (Dh))

= Fϑ (δϑh) ∀δϑh (62)

Remark 7 OSGS counterpart of the classical GLS/SUPGstabilization method.

The OSGS counterpart of the GLS/SUPG stabilizationmethod can be obtained neglecting the following terms inthe stabilized variational equations given above:

τv

(∇δph,P⊥

h (bh))

= 0

τϑ


h (rh))

= 0

τϑ


h (Dh))

= 0 (63)

The quasi-static OSGS sub-grid scale stabilized mixedvariational formulation corresponds to the OSGS counterpart

123


of the GLS/SUPG stabilized mixed variational formulationand reads



(∇δph,P⊥

h (∇ ph))

= 0 ∀δph


+τϑ


h

(chv·

h∇ϑh)) = Fϑ (δϑh) ∀δϑh

(64)

2.7.1 Summary

A summary of the main results obtained so far on the space-discrete setting is shown in Boxes 1, 2, 3. Box 1 collectsthe stabilized variational equations and stabilized unknownsfor the ASGS and OSGS method using P1/P1/P1 linearvelocity/linear pressure/linear temperature elements, assum-ing temperature dependent viscosity, thermal conductivity,and heat capacity. Box 2 collects the corresponding expres-sions for the particular simplified case of constant thermalconductivity, constant heat capacity and temperature inde-pendent viscosity. Finally, as a particular case of the expres-sions shown in Box 2, in Box 3 the stabilized velocity appear-ing in the convective terms of the temperature equation issubstituted by its finite element component, and the projec-tions of body forces, temperature transient term, heat sourceand dissipation are neglected. In this case, the classical GLSand SUPG stabilization methods are recovered as a particularcase of the ASGS method.

3 Time Discrete Setting and Product FormulaAlgorithm

The sub-grid scale stabilized mixed variational formulationof the IBVP can be solved using a staggered algorithm arisingfrom an isothermal operator split of the governing equationsand a Product Formula Algorithm (PFA). Within this frame-work, a mechanical problem at constant temperature, with thevelocity and pressure as mechanical variables, and a thermalproblem at constant velocity and pressure, with the temper-ature as thermal variable, may be defined [1–3,10–12].

Within a time discrete setting, a staggered algorithm isdefined such that for an arbitrary time step, the isothermalmechanical problem is solved first, keeping constant the tem-perature from the previous time step. Once the mechanicalvariables have been updated, the thermal problem is solvednext, keeping constant the updated mechanical variables[1–3,10–12].

P1/P1/P1 linear velocity/linear pressure/linear tempera-ture elements are considered. A first-order Backward-Euler

(BE) time integration scheme is used to integrate the localtime variation of the temperature.

3.1 Mechanical Problem

The fully discrete mechanical problem which arises from theapplication of a PFA to the isothermal operator split of thegoverning equations is defined by the discrete stabilized vari-ational forms of the linear momentum balance and incom-pressibility equations given by(∇ · δvh, pstab

h,n+1

)+ (∇sδvh, 2μh,n+1∇svh,n+1

)

−(

2∇sδvh,n+1∇μh,n+1, vstabh,n+1 − vh,n+1

)

= Fv,n+1 (δvh) ∀δvh(δph,∇ · vh,n+1

) −(∇δph, vstab

h,n+1 − vh,n+1

)= 0 ∀δph

(65)

The fully discrete residual of the velocity equation at timen + 1 is given by

rv,h,n+1 := ∇ ph,n+1 + 2∇svh,n+1∇μh,n+1 + bh,n+1 (66)

and the discrete velocity stabilization parameter at time n+1is given by

τv,n+1 =(

c12μh,n+1

h2

)−1

= 1

c1

h2

2μh,n+1(67)

Note that for the mechanical problem, the temperatureis kept constant and equal to the temperature at the end ofthe previous time step, denoted as ϑh,n . Therefore, for themechanical problem, the equivalent viscosity μh,n+1 at timen + 1 is evaluated as μh,n+1 := μ

(vh,n+1, ϑh,n

).

3.1.1 Algebraic Sub-Grid Scales (ASGS)

The fully discrete sub-grid scale velocity at time n + 1 forthe ASGS is defined as

vn+1 = τv,n+1rv,h,n+1

= τv,n+1(∇ ph,n+1+2∇svh,n+1∇μh,n+1+bh,n+1

)

(68)

The fully discrete sub-grid scale stabilized velocity andpressure at time n + 1 for the ASGS are defined as

vstabh,n+1 := vh,n+1 + τv,n+1rv,h,n+1

:= vh,n+1

+τv,n+1(∇ ph,n+1 + 2∇svh,n+1∇μh,n+1 + bh,n+1

)

pstabh,n+1 := ph,n+1 (69)

Substituting the fully discrete stabilized velocity and pres-sure at time n+1 into the corresponding stabilized variational

123


formulation of the linear momentum balance and incom-pressibility equations yields(∇ · δvh, ph,n+1

) + (∇sδvh, 2μh,n+1∇svh,n+1)

−τv,n+1(2∇sδvh∇μh,n+1,∇ ph,n+1

)

−τv,n+1(2∇sδvh∇μh,n+1, 2∇svh,n+1∇μh,n+1

)

−τv,n+1(2∇sδvh∇μh,n+1, bh,n+1

)


)

−τv,n+1(∇δph,∇ ph,n+1

)

−τv,n+1(∇δph, 2∇svh,n+1∇μh,n+1

)

−τv,n+1(∇δph, bh,n+1

) = 0 ∀δph (70)

For implementation purposes, a compact and convenientalternative expression can be obtained in terms of the resid-uals, yielding(∇ · δvh, ph,n+1

) + (∇sδvh, 2μh,n+1∇svh,n+1)

−τv,n+1(∇sδvh, 2∇μh,n+1 ⊗ rv,h,n+1

)


)

−τv,n+1(∇δph, rv,h,n+1

) = 0 ∀δph (71)

3.1.2 Orthogonal Sub-Grid Scales (OSGS)

The fully discrete sub-grid scale velocity at time n + 1 forthe OSGS is defined as

vn+1 = τv,n+1P⊥h

(rv,h,n+1

)

= τv,n+1P⊥h

(∇ ph,n+1

+2∇svh,n+1∇μh,n+1 + bh,n+1)

(72)

The fully discrete sub-grid scale stabilized velocity andpressure at time n + 1 for the OSGS are defined as

vstabh,n+1 := vh,n+1 + τv,n+1P⊥

h

(rv,h,n+1

)

:= vh,n+1 + τv,n+1P⊥h

(∇ ph,n+1

+ 2∇svh,n+1∇μh,n+1+bh,n+1)

pstabh,n+1 := ph,n+1 (73)

Substituting the fully discrete stabilized velocity and pres-sure at time n+1 into the corresponding stabilized variationalformulation of the linear momentum balance and incom-pressibility equations yields(∇ · δvh, ph,n+1

) + (∇sδvh, 2μh,n+1∇svh,n+1)

−τv,n+1

(2∇sδvh∇μh,n+1,P⊥

h

(∇ ph,n+1))

−τv,n+1


h

(2∇svh,n+1∇μh,n+1

))

−τv,n+1


h

(bh,n+1

))

= Fv,n+1 (δvh) ∀δvh

(δph,∇ · vh,n+1

)

−τv,n+1

(∇δph,Ph⊥ (∇ ph,n+1

))

−τv,n+1

(∇δph,P⊥

h

(2∇svh,n+1∇μh,n+1

))

−τv,n+1

(∇δph,P⊥

h

(bh,n+1

)) = 0 ∀δph (74)

For implementation purposes, a compact and convenientalternative expression can be obtained in terms of the resid-uals, yielding(∇ · δvh, ph,n+1

) + (∇sδvh, 2μh,n+1∇svh,n+1)

−τv,n+1

(∇sδvh, 2∇μh,n+1 ⊗ P⊥

h

(rv,h,n+1

))


)

−τv,n+1

(∇δph,Ph⊥ (

rv,h,n+1)) = 0 ∀δph (75)

Orthogonal projection of the residual of the velocity equa-tion can be written as

P⊥h

(rv,h,n+1

) := rv,h,n+1

−Ph(rv,h,n+1

) := rv,h,n+1 − πv,h,n+1 (76)

where we have introduced πv,h,n+1 ∈ L2 (�) as the L2 pro-jection of the discrete residual of the velocity equation at timen + 1 onto the corresponding finite element space.

Adding the corresponding variational equation for the L2

projection of the residual of the velocity equation onto thefinite element space, the extended stabilized variational for-mulation of the mechanical problem can be written as(∇ · δvh, ph,n+1

) + (∇sδvh, 2μh,n+1∇svh,n+1)

−τv,n+1(∇sδvh, 2∇μh,n+1 ⊗ (

rv,h,n+1 − πv,h,n+1))


)

−τv,n+1(∇δph, rv,h,n+1 − πv,h,n+1

) = 0 ∀δph(δπv,h, rv,h,n+1

) − (δπv,h,πv,h,n+1

) = 0 ∀δπv,h (77)

where δπv,h ∈ L2 (�) is the test function for the L2 projec-tion of the residual of the velocity equation onto the corre-sponding finite element space.

3.2 Thermal Problem

The fully discrete thermal problem which arises from theapplication of a PFA to the isothermal operator split of thegoverning equations is defined by the stabilized variationalform of the energy balance equation given by

1

�t

(δϑh, ch,n+1

(ϑh,n+1 − ϑh,n

))

+(δϑh, ch,n+1vstab

h,n+1 · ∇ϑh,n+1

)

123


+ (∇δϑh, κh,n+1∇ϑh,n+1)

−(

ch,n+1vstabh,n+1 · ∇δϑh, ϑ stab

h,n+1 − ϑh,n+1

)

−(δϑhvstab

h,n+1 · ∇ch,n+1, ϑstabh,n+1 − ϑh,n+1

)

−(∇κh,n+1 · ∇δϑh,, ϑ

stabh,n+1 − ϑh,n+1

)

= Fϑ,n+1 (δϑh) ∀δϑh (78)

The fully discrete residual of the temperature equation attime n + 1 is given by

rϑ,h,n+1 := 1

�tch,n+1

(ϑh,n+1 − ϑh,n

)

+ ch,n+1vstabh,n+1 · ∇ϑh,n+1

−∇κh,n+1 · ∇ϑh,n+1 − rh,n+1 − Dh,n+1 (79)

The discrete temperature stabilization parameter at timen + 1 is given by

τϑ,n+1 =⎛⎝c1

κh,n+1

h2 + c2

ch,n+1

∣∣∣vstabh,n+1

∣∣∣h

⎞⎠

−1

(80)

Note that the convective terms and the time-discrete stabiliza-tion parameter for the temperature at time n+1 are computedwith the discrete stabilized velocity at time n + 1. Thereforethe stabilized temperature at time n + 1 is a function of thestabilized velocity at time n + 1.

3.2.1 Algebraic Sub-Grid Scales (ASGS)

The fully discrete sub-grid scale temperature at time n + 1for the ASGS is given by

ϑn+1 = −τϑ,n+1rϑ,h,n+1 (81)

The fully discrete sub-grid scale stabilized temperature attime n + 1 for the ASGS is given by

ϑ stabh,n+1 := ϑh,n+1 − τϑ,n+1rϑ,h,n+1 (82)

Substituting the fully discrete sub-grid scale stabilizedtemperature at time n + 1 into the corresponding stabilizedvariational formulation of the energy balance equation yields

1

�t

(δϑh, ch,n+1

(ϑh,n+1 − ϑh,n

))


h,n+1 · ∇ϑh,n+1

)+ (∇δϑh, κh,n+1∇ϑh,n+1

)

+τϑ,n+11

�t

(ch,n+1vstab

h,n+1 · ∇δϑh, ch,n+1(ϑh,n+1 − ϑh,n

))

+τϑ,n+1

(ch,n+1vstab

h,n+1 · ∇δϑh, ch,n+1vstabh,n+1 · ∇ϑh,n+1

)

−τϑ,n+1

(ch,n+1vstab

h,n+1 · ∇δϑh,∇κh,n+1 · ∇ϑh,n+1

)

−τϑ,n+1

(ch,n+1vstab

h,n+1 · ∇δϑh, rh,n+1 + Dh,n+1

)

+τϑ,n+11

�t

(δϑhvstab

h,n+1 · ∇ch,n+1, ch,n+1(ϑh,n+1 − ϑh,n

))

+τϑ,n+1

(δϑhvstab

h,n+1 · ∇ch,n+1, ch,n+1vstabh,n+1 · ∇ϑh,n+1

)

−τϑ,n+1

(δϑhvstab

h,n+1 · ∇ch,n+1,∇κh,n+1 · ∇ϑh,n+1

)

−τϑ,n+1

(δϑhvstab

h,n+1 · ∇ch,n+1, rh,n+1 + Dh,n+1

)

+τϑ,n+11

�t

(∇κh,n+1 · ∇δϑh, ch,n+1(ϑh,n+1 − ϑh,n

))

+τϑ,n+1

(∇κh,n+1 · ∇δϑh, ch,n+1vstab

h,n+1 · ∇ϑh,n+1

)

−τϑ,n+1(∇κh,n+1 · ∇δϑh,∇κh,n+1 · ∇ϑh,n+1

)

−τϑ,n+1(∇κh,n+1 · ∇δϑh, rh,n+1 + Dh,n+1

)

= Fϑ,n+1 (δϑh) ∀δϑh (83)

For implementation purposes, a compact and convenientalternative expression can be obtained in terms of the residualof the temperature equation, yielding

1

�t

(δϑh, ch,n+1

(ϑh,n+1 − ϑh,n

))


h,n+1 · ∇ϑh,n+1

)

+ (∇δϑh, κh,n+1∇ϑh,n+1)

+τϑ,n+1

(∇δϑh, ch,n+1vstab

h,n+1rϑ,h,n+1

)

+τϑ,n+1

(δϑh, vstab

h,n+1 · ∇ch,n+1rϑ,h,n+1

)

+τϑ,n+1(∇δϑh,∇κh,n+1rϑ,h,n+1

)

= Fϑ,n+1 (δϑh) ∀δϑh (84)

Note that we need to store and update just the finite elementcomponent of the temperature at time n.

Remark 8 Constant heat capacity and constant thermal con-ductivity.

If the heat capacity and the thermal conductivity are con-stants, the following expressions hold

∇ch,n+1 = 0

∇κh,n+1 = 0 (85)

Using the expressions given above, the discrete sub-grid scalestabilized formulation takes the form

1

�t

(δϑh, ch,n+1

(ϑh,n+1 − ϑh,n

))


h,n+1 · ∇ϑh,n+1

)+ (∇δϑh, κh,n+1∇ϑh,n+1

)

+τϑ,n+11

�t

(ch,n+1vstab

h,n+1 · ∇δϑh, ch,n+1(ϑh,n+1 − ϑh,n

))

+τϑ,n+1

(ch,n+1vstab

h,n+1 · ∇δϑh, ch,n+1vstabh,n+1 · ∇ϑh,n+1

)

−τϑ,n+1

(ch,n+1vstab

h,n+1 · ∇δϑh, rh,n+1 + Dh,n+1

)

= Fϑ,n+1 (δϑh) ∀δϑh (86)

123


3.2.2 Orthogonal Sub-Grid Scales (OSGS)

The fully discrete sub-grid scale temperature at time n + 1for the OSGS is given by

ϑn+1 = −τϑ,n+1P⊥h

(rϑ,h,n+1

)(87)

The fully discrete sub-grid scale stabilized temperature attime n + 1 for the OSGS is given by

ϑ stabh,n+1 := ϑh,n+1 − τϑ,n+1P⊥

h

(rϑ,h,n+1

)(88)

Substituting the fully discrete stabilized temperature at timen + 1 into the corresponding stabilized variational formula-tion of the energy balance equation yields

1

�t

(δϑh, ch,n+1

(ϑh,n+1 − ϑh,n

))


h,n+1 · ∇ϑh,n+1

)+ (∇δϑh, κh,n+1∇ϑh,n+1

)

+τϑ,n+11

�t

(ch,n+1vstab

h,n+1 · ∇δϑh, P⊥h

(ch,n+1

(ϑh,n+1 − ϑh,n

)))

+τϑ,n+1

(ch,n+1vstab


(ch,n+1vstab

h,n+1 · ∇ϑh,n+1

))

−τϑ,n+1

(ch,n+1vstab


(∇κh,n+1 · ∇ϑh,n+1))

−τϑ,n+1

(ch,n+1vstab


(rh,n+1 + Dh,n+1

))

+τϑ,n+11

�t

(δϑhvstab

h,n+1 · ∇ch,n+1, P⊥h

(ch,n+1

(ϑh,n+1 − ϑh,n

)))

+τϑ,n+1

(δϑhvstab

h,n+1 · ∇ch,n+1, P⊥h

(ch,n+1vstab

h,n+1 · ∇ϑh,n+1

))

−τϑ,n+1

(δϑhvstab

h,n+1 · ∇ch,n+1, P⊥h

(∇κh,n+1 · ∇ϑh,n+1))

−τϑ,n+1

(δϑhvstab

h,n+1 · ∇ch,n+1, P⊥h

(rh,n+1 + Dh,n+1

))

+τϑ,n+11

�t

(∇κh,n+1 · ∇δϑh, P⊥

h

(ch,n+1

(ϑh,n+1 − ϑh,n

)))

+τϑ,n+1

(∇κh,n+1 · ∇δϑh, P⊥

h

(ch,n+1vstab

h,n+1 · ∇ϑh,n+1

))

−τϑ,n+1

(∇κh,n+1 · ∇δϑh, P⊥

h

(∇κh,n+1 · ∇ϑh,n+1))

−τϑ,n+1

(∇κh,n+1 · ∇δϑh, P⊥

h

(rh,n+1 + Dh,n+1

))

= Fϑ,n+1 (δϑh) ∀δϑh (89)

For implementation purposes, a compact and convenientalternative expression can be obtained in terms of the residualof the temperature equation, yielding

1

�t

(δϑh, ch,n+1

(ϑh,n+1 − ϑh,n

))


h,n+1 · ∇ϑh,n+1

)

+ (∇δϑh, κh,n+1∇ϑh,n+1)

+τϑ,n+1


h,n+1P⊥h

(rϑ,h,n+1

))

Box 4 Mechanical problem:ASGS and OSGS stabilizedvariational formulation forP1/P1/P1 linear velocity/linearpressure/linear temperatureelements

123


+τϑ,n+1

(δϑh, vstab

h,n+1 · ∇ch,n+1P⊥h

(rϑ,h,n+1

))

+τϑ,n+1

(∇δϑh,∇κh,n+1P⊥

h

(rϑ,h,n+1

))

= Fϑ,n+1 (δϑh) ∀δϑh (90)

The orthogonal projection of the residual of the temperatureequation can be written as

P⊥h

(rϑ,h,n+1

) := rϑ,h,n+1 − Ph(rϑ,h,n+1

)

:= rϑ,h,n+1 − πϑ,h,n+1 (91)

where we have introduced πϑ,h,n+1 ∈ L2 (�) as the L2 pro-jection of the discrete residual of the temperature equation attime n + 1 onto its corresponding finite element space.

Adding the corresponding variational equation for the L2

projection of the residual of the temperature equation onto itscorresponding finite element space, the extended stabilizedvariational formulation of the thermal problem can be writtenas

1

�t

(δϑh, ch,n+1

(ϑh,n+1 − ϑh,n

))


h,n+1 · ∇ϑh,n+1

)

+ (∇δϑh, κh,n+1∇ϑh,n+1)

+τϑ,n+1


h,n+1

(rϑ,h,n+1 − πϑ,h,n+1

))

+τϑ,n+1

(δϑh, vstab

h,n+1 · ∇ch,n+1(rϑ,h,n+1 − πϑ,h,n+1

))

+τϑ,n+1(∇δϑh,∇κh,n+1

(rϑ,h,n+1 − πϑ,h,n+1

))

= Fϑ,n+1 (δϑh) ∀δϑh(δπϑ,h, rϑ,h,n+1

) − (δπϑ,h, πϑ,h,n+1

) = 0 ∀δπϑ,h (92)

where δπϑ,h ∈ L2 (�) is the test function for the L2 pro-jection of the residual of the temperature equation onto itscorresponding finite element space.

Remark 9 Constant heat capacity and constant thermal con-ductivity.

Box 5 Thermal problem: ASGS and OSGS stabilized variational formulation for P1/P1/P1 linear velocity/linear pressure/linear temperatureelements

123


If the heat capacity and the thermal conductivity are constant,the following expressions hold

P⊥h

(ch,n+1

(ϑh,n+1 − ϑh,n

)) = 0

∇ch,n+1 = 0

∇κh,n+1 = 0 (93)

Using the expressions given above, the discrete sub-grid scalestabilized formulation takes the form

1

�t

(δϑh, ch,n+1

(ϑh,n+1 − ϑh,n

))


h,n+1 · ∇ϑh,n+1

)

+ (∇δϑh, κh,n+1∇ϑh,n+1)

+τϑ,n+1

(ch,n+1vstab

h,n+1 · ∇δϑh

P⊥h

(ch,n+1vstab

h,n+1 · ∇ϑh,n+1

))

−τϑ,n+1

(ch,n+1vstab

h,n+1 · ∇δϑh,P⊥h

(rh,n+1 + Dh,n+1

))

= Fϑ,n+1 (δϑh) ∀δϑh (94)

3.3 Summary

Boxes 4 and 5 collect the main expressions for the dis-crete ASGS and OSGS stabilized variational formulation forthe mechanical and thermal problems, respectively, usingP1/P1/P1 linear velocity/linear pressure/linear temperatureelements.

4 Finite Element Implementation

4.1 Mechanical Problem

4.1.1 Algebraic Sub-Grid Scale (ASGS)

Once the finite element discretization has been performed,the matrix form of the nonlinear algebraic system of equa-tions emanating from the discrete ASGS sub-grid scale finiteelement stabilized variational form of the governing equa-tions of the mechanical problem can be written as

Rv,n+1 := Rv (Vn+1, Pn+1) = 0

Rp,n+1 := Rp (Vn+1, Pn+1) = 0 (95)

where Rv,n+1 and Rp,n+1 are the global nodal residual vec-tors arising from the velocity and pressure equations at timen + 1, respectively, and Vn+1 and Pn+1 are the vectors ofnodal velocity and nodal pressure unknowns at time n + 1,respectively.

The vectors of nodal mechanical unknowns Vn+1 andPn+1 can be obtained through an incremental-iterative solu-tion method. Using a Newton-Raphson algorithm, the non-

linear algebraic systems of equations has to be linearized atthe iteration i + 1 of the time step n + 1, yielding

Rv

(V(i)

n+1, P(i)n+1

)+ DRv

(V(i)

n+1, P(i)n+1

)· �V(i)

n+1

+ DRv

(V(i)

n+1, P(i)n+1

)· �P(i)

n+1 = 0

Rp

(V(i)

n+1, P(i)n+1

)+ DRp

(V(i)

n+1, P(i)n+1

)· �V(i)

n+1

+ DRp

(V(i)

n+1, P(i)n+1

)· �P(i)

n+1 = 0 (96)

where the vectors of nodal increment of velocities �V(i)n+1

and nodal increment of pressures �P(i)n+1 are defined as

�V(i)n+1 = V(i+1)

n+1 − V(i)n+1

�P(i)n+1 = P(i+1)

n+1 − P(i)n+1 (97)

Typical element entries for a node A of the global nodalresidual vectors Rv,n+1 and Rp,n+1 are given by

RAv

(V(i)

n+1, P(i)n+1

)=

∫

�e

∇Nv,A p(i)h,n+1d�

+∫

�e

BTA2μ

(i)h,n+1∇sv(i)

h,n+1d�

−∫

�e

τ(i)v,n+1BT

A

(2∇μ

(i)h,n+1 ⊗ r(i)

v,h,n+1

)d�

−∫

�e

Nv,Abh,n+1d�

−∫

∂σ �e

Nv,A th,n+1d∂σ �

RAp

(V(i)

n+1, P(i)n+1

)=

∫

�e

Np,Ar (i)p,h,n+1d�

−∫

�e

τ(i)v,n+1∇Np,A · r(i)

v,h,n+1d� (98)

where Nv,A and Np,A are the velocity and pressure interpo-lation shape functions for node A, respectively, and BA is thevelocity symmetric gradient interpolation matrix for node A.

4.1.2 Orthogonal Sub-Grid Scale (OSGS)

Once the finite element discretization has been performed,the matrix form of the nonlinear algebraic system of equa-tions emanating from the discrete OSGS sub-grid scale finiteelement stabilized variational form of the governing equa-tions of the mechanical problem can be written as

Rv,n+1 := Rv

(Vn+1, Pn+1, v,n+1

) = 0

Rp,n+1 := Rp(Vn+1, Pn+1, v,n+1

) = 0

Rπv,n+1 := Rπv

(Vn+1, Pn+1, v,n+1

) = 0 (99)

123


where Rv,n+1, Rp,n+1, Rπv,n+1 are the global nodal resid-ual vectors arising from the velocity, pressure, and velocityresidual projection equations, respectively, at time n + 1,and Vn+1, Pn+1, v,n+1 are the vectors of nodal unknowns,velocity, pressure, and velocity residual projection, respec-tively, at time n + 1.

The solution of the above system of equations can beobtained through a robust and efficient staggered algorithmby solving instead the slightly modified system of equationsgiven by

Rv,n+1 := Rv

(Vn+1, Pn+1, v,n

) = 0

Rp,n+1 := Rp(Vn+1, Pn+1, v,n

) = 0

Rπv,n+1 := Rπv

(Vn+1, Pn+1, v,n+1

) = 0 (100)

The idea is to obtain first Vn+1 and Pn+1 by solving thefirst two systems of equations, keeping constants v,n at thetime step n. Once Vn+1 and Pn+1 have been obtained, theyare substituted into the last two equations to obtain v,n+1

at the time step n + 1.The unknowns Vn+1 and Pn+1 can be obtained using an

incremental iterative Newton-Raphson algorithm, through anexact linearization of the first two systems of equations givenby

Rv

(V(i)

n+1, P(i)n+1, v,n

)

+DRv

(V(i)

n+1, P(i)n+1, v,n

)· �V(i)

n+1

+DRv

(V(i)

n+1, P(i)n+1, v,n

)· �P(i)

n+1 = 0

Rp

(V(i)

n+1, P(i)n+1, v,n

)

+DRp

(V(i)

n+1, P(i)n+1, v,n

)· �V(i)

n+1

+DRp

(V(i)

n+1, P(i)n+1, v,n

)· �P(i)

n+1 = 0 (101)

where the vectors of nodal increment of velocities �V(i)n+1

and nodal increment of pressures �P(i)n+1 are defined as

�V(i)n+1 = V(i+1)

n+1 − V(i)n+1

�P(i)n+1 = P(i+1)

n+1 − P(i)n+1 (102)

Typical element entries for a node A of the global nodalresidual vectors Rv,n+1 and Rp,n+1 are given by

RAv

(V(i)

n+1, P(i)n+1, v,n

)=

∫

�e

∇Nv,A p(i)h,n+1d�

+∫

�e

BTA2μ

(i)h,n+1∇sv(i)

h,n+1d�

−∫

�e

τ(i)v,n+1BT

A

(2∇μ

(i)h,n+1 ⊗

(r(i)v,h,n+1 − πv,h,n

))d�

−∫

�e

Nv,Abh,n+1d�

−∫

∂σ �e

Nv,A th,n+1d∂σ �

RAp

(V(i)

n+1, P(i)n+1, v,n

)=

∫

�e

Np,Ar (i)p,h,n+1d�

−∫

�e

τ(i)v,n+1∇Np,A ·

(r(i)v,h,n+1 − πv,h,n

)d� (103)

where Nv,A and Np,A are the velocity and pressure interpo-lation shape functions for node A, respectively, and BA is thevelocity symmetric gradient interpolation matrix for node A.

Once Vn+1 and Pn+1 have been obtained they are substi-tuted into the last two systems of equations to get v,n+1 bysolving the uncoupled algebraic linear system

Rπv,n+1 := Rπv

(Vn+1, Pn+1, v,n+1

) = 0 (104)

The computation of the projection v,n+1 can be trans-formed in a straight-forward operation by considering appro-priate lumped mass matrices, leading to an efficient solu-tion algorithm, without loss of precision or robustness[4,5,22,23,29,30,35].

4.2 Thermal Problem

4.2.1 Algebraic Sub-Grid Scale (ASGS)

Once the finite element discretization has been performed,the matrix form of the nonlinear algebraic system of equa-tions emanating from the discrete ASGS sub-grid scale finiteelement stabilized variational form of the governing equationof the thermal problem can be written as

Rϑ,n+1 := Rϑ (Tn+1) = 0 (105)

where Rϑ,n+1 screte global nodal residual vector arising fromthe temperature equation at time n+1, and Tn+1 is the vectorof nodal temperature unknowns at time n + 1.

The vector of nodal temperature unknowns can be obtainedthrough an incremental-iterative solution method. Using aNewton-Raphson algorithm, the nonlinear algebraic systemof equations has to be linearized at the iteration i + 1 of thetime step n + 1. The exact linearization of the above systemof equations can be written as

Rϑ

(T(i)

n+1

)+ DRϑ

(T(i)

n+1

)· �T(i)

n+1 = 0 (106)

where the vector of nodal increment of temperature un-knowns is defined as

�T(i)n+1 = T(i+1)

n+1 − T(i)n+1 (107)

123


For the Quasi-static Sub-grid Scales (QSGS), a typicalelement entry for a node A of the global nodal residual vectorRϑ,n+1 is given by

RAϑ

(T(i)

n+1

)= 1

�t

∫

�e

Nϑ,Ac(i)h,n+1

(ϑ

(i)h,n+1 − ϑh,n

)d�

+∫

�e

Nϑ,Ac(i)h,n+1vstab

h,n+1 · ∇ϑ(i)h,n+1d�

+∫

�e

∇Nϑ,Aκ(i)h,n+1∇ϑ

(i)h,n+1d�

+∫

�e

τ(i)ϑ,n+1∇Nϑ,A · c(i)

h,n+1vstabh,n+1r (i)

ϑ,h,n+1d�

+∫

�e

τ(i)ϑ,n+1 Nϑ,Avstab

h,n+1 · ∇c(i)h,n+1r (i)

ϑ,h,n+1d�

+∫

�e

τ(i)ϑ,n+1∇Nϑ,A · ∇κ

(i)h,n+1r (i)

ϑ,h,n+1d�

−∫

�e

Nϑ,Arh,n+1d�

−∫

�e

Nϑ,AD(i)h.n+1d�

+∫

∂q�e

Nϑ,Aqh,n+1d∂q� (108)

where Nϑ,A is the temperature interpolation shape functionfor node A.

4.2.2 Orthogonal Sub-Grid Scale (OSGS)

Once the finite element discretization has been performed,the matrix form of the nonlinear algebraic system of equa-tions emanating from the discrete OSGS sub-grid scale finiteelement stabilized variational form of the governing equationof the thermal problem can be written as

Rϑ,n+1 := Rϑ

(Tn+1, ϑ,n+1

) = 0

Rπϑ ,n+1 := Rπϑ

(Tn+1, ϑ,n+1

) = 0 (109)

where Rϑ,n+1 and Rπϑ ,n+1 are the global nodal residual vec-tors arising from the temperature and temperature residualprojection equations at time n + 1, respectively, and Tn+1

and ϑ,n+1 are the vectors of nodal unknowns, temperatureand temperature modified residual projection at time n + 1,respectively.

The solution of the above system of equations can beobtained through a robust and efficient staggered algorithmby solving the slightly modified systems of equations givenby

Rϑ,n+1 := Rϑ

(Tn+1, ϑ,n

) = 0

Rπϑ ,n+1 := Rπϑ

(Tn+1, ϑ,n+1

) = 0 (110)

The idea is to obtain first Tn+1 by solving the first systemof equations, keeping constant ϑ,n at the time step n. OnceTn+1 has been obtained, it is substituted into the second equa-tion to obtain ϑ,n+1 at the time step n + 1.

The vector of nodal temperature unknowns Tn+1 canbe obtained using an incremental iterative Newton-Raphsonalgorithm, through an exact linearization of the first systemof equations given by

Rϑ

(T(i)

n+1, ϑ,n

)+ DRϑ

(T(i)

n+1, ϑ,n

)· �T(i)

n+1 = 0

(111)

where the vector of nodal increment of temperatures �T(i)n+1

is defined as

�T(i)n+1 = T(i+1)

n+1 − T(i)n+1 (112)

Once Tn+1 has been obtained it is substituted into the sec-ond system of equations to get ϑ,n+1 by solving the linearsystem

Rπϑ ,n+1 := Rπϑ

(Tn+1, ϑ,n+1

) = 0 (113)

Similarly as for the mechanical problem, the computationof the projection ϑ,n+1 can be transformed in a straight-forward operation by considering an appropriate lumpedmass matrix, leading to an efficient solution algorithm, with-out loss of precision or robustness [4,5,22,23,29,30,35].

A typical element entry for a node A of the global nodalresidual vector Rϑ,n+1 is given by

RAϑ

(T(i)

n+1, ϑ,n

)= 1

�t

∫

�e

Nϑ,Ac(i)h,n+1

(ϑ

(i)h,n+1 − ϑh,n

)d�

+∫

�e

Nϑ,Ac(i)h,n+1vstab

h,n+1 · ∇ϑ(i)h,n+1d�

+∫

�e

∇Nϑ,Aκ(i)h,n+1∇ϑ

(i)h,n+1d�

+∫

�e

τ(i)ϑ,n+1∇Nϑ,A · c(i)

h,n+1vstabh,n+1

(r (i)ϑ,h,n+1 − πϑ,h,n

)d�

+∫

�e

τ(i)ϑ,n+1 Nϑ,Avstab

h,n+1 · ∇c(i)h,n+1


)d�

+∫

�e

τ(i)ϑ,n+1∇Nϑ,A · ∇κ

(i)h,n+1


)d�

123


−∫

�e

Nϑ,Arh,n+1d�

−∫

�e

Nϑ,AD(i)h,n+1d�

+∫

∂q�e

Nϑ,Aqh,n+1d∂q� (114)

where Nϑ,A is the temperature interpolation shape functionfor node A.

5 Numerical Examples

In this section we present two numerical examples that illus-trate the performance of the formulation presented above.The first example deals with the numerical simulation of acoupled thermomechanical flow in a 2D rectangular domain.Both steady-state and transient conditions are considered.The main goal of this numerical example is to compare thebehavior of the different pressure and convection stabiliza-tion methods. First, using GLS method to stabilize the pres-sure, temperature distributions using SUPG and OSGS con-vection stabilization methods are compared. Secondly, usingSUPG method to stabilize the convective term of the temper-ature equation, pressure and velocity contours using GLS andOSGS pressure stabilization methods are compared. The sec-ond numerical example shows the 3D numerical simulationof a FSW process. ASGS (GLS for the pressure equation andSUPG for the temperature equation) and OSGS (both for thepressure and the temperature equations) stabilization meth-ods are used. The Newton-Raphson method, combined witha line search procedure, is used to solve the nonlinear equa-tions emanating from the fully discrete stabilized variationalequations. Calculations are performed with an enhanced ver-sion of the finite element code COMET [21] developed bythe authors at the International Center for Numerical Meth-ods in Engineering (CIMNE) in Barcelona. Pre- and post-processing is done with GiD [54], also developed at CIMNE.

5.1 Coupled Thermomechanical Flow in a 2DRectangular Domain

The first numerical example deals with the coupled thermo-mechanical analysis of the fluid flow and thermal convectionalong a 2D rectangular domain.

The geometry of the domain, length 4 m and height 1 m,is shown on Fig. 1. Mechanical and thermal boundary con-ditions are schematically shown on Fig. 2. Velocity is pre-scribed to zero along the lateral sides of the tube. The inflowvelocity is prescribed in x-direction using a parabolic veloc-ity given by V = 1.252E−02 (y − 0.5)2 + 3.13E−03 m/s.

Fig. 1 Geometry of the 2D rectangular domain

Fig. 2 Prescribed velocity and temperature boundary conditions on the2D rectangular domain

Table 1 Material thermal properties

Mass density(Kg m−3)

Heat capacity(J Kg−1◦C−1)

Thermal conductivity(W m−1◦C−1)

7.80E+03 0.5E+03 50

Table 2 Norton-Hoff material model parameters

Material Model Consistency parameter(MPa s)

Rate sensitivityparameter

Norton-Hoff 100 1

Fig. 3 Finite element mesh consisting of 20 × 20 linear quadrilateralelements

The temperature is prescribed to 50 and 30◦C at the inflowand outflow, respectively.

A Norton-Hoff rigid visco-plastic constitutive model isconsidered. The thermal material properties and Norton-Hoff material model parameters are given in Tables 1 and 2,respectively.

The geometry is discretized with a finite element meshconsisting of 20×20 linear quadrilateral elements and 441nodal points, as shown in Fig. 3.

Two simulations, steady-state and dynamic for 70 timesteps of 10 s are considered. First, the GLS stabilizationmethod is used for the mechanical part and the thermal partis stabilized using SUPG and OSGS stabilization methods.The temperatures at time steps 10, 40, 70 and the steady-statetemperature along a line at the center of the tube are shownin Fig. 4.

123


Fig. 4 Comparison betweenthe temperature distribution atthe center line at different times(10, 40 and 70) and at the steadystate, obtained using GLSpressure stabilization methodand OSGS and SUPGconvection stabilization methods

Fig. 5 Pressure contour linesusing SUPG convectionstabilization method and GLSpressure stabilization method

Fig. 6 Pressure contour linesusing SUPG convectionstabilization method and OSGSpressure stabilization method

Temperatures provided by the SUPG and OSGS methodslook very similar and only some slight differences can beseen, showing that OSGS provides slightly better results forthe steady-state case.

Next, the SUPG stabilization method is used for the ther-mal part and the mechanical part is stabilized once with the

GLS and once with the OSGS stabilization methods. Thepressure contour lines for both cases are shown in Figs. 5and 6.

A difference on the pressure contours obtained using GLSand OSGS stabilization methods is clearly shown at theinflow and outflow.

123


Fig. 7 Velocity profiles at theinflow, outflow and center usingSUPG convection stabilizationmethod and GLS pressurestabilization method

Fig. 8 Velocity profiles at theinflow, outflow and center usingSUPG convection stabilizationmethod and OSGS pressurestabilization method

Fig. 9 Velocity contour linesusing SUPG convectionstabilization method and GLSpressure stabilization method

The velocity profiles at the inflow, outflow and center ofthe tube, using GLS and OSGS stabilization methods, areshown on Figs. 7 and 8, respectively. The velocity contourlines for both cases are illustrated in Figs. 9 and 10.

The figures clearly show that there is also an influence ofthe stabilization method on the velocity fields.

5.2 3D Transient Coupled Thermo-Mechanical NumericalSimulation of a FSW Process

This example shows the 3D transient coupled thermo-mechanical numerical simulation of a FSW process. A finiteelement discretization of the work-pieces, cylindrical tool

123


Fig. 10 Velocity contour linesusing SUPG convectionstabilization method and OSGSpressure stabilization method

Table 3 Material thermal properties for the work-pieces, tool and back-plate

Body Mass density(Kg m−3)

Heat capacity(J Kg−1◦C−1)

Thermal conductivity(W m−1◦C−1)

Convection heattransfer coefficient(W m−2◦C−1)

Relativeemissivity

Work-pieces 8.00E+03 0.51E+03 21.4 10 0.17

Tool 7.85E+03 0.46E+03 43.0 10 0.80

Back-plate 7.85E+03 0.46E+03 43.0

Table 4 Material parameters forthe Sheppard–Wright materialmodel for the work-pieces

Material Model A α(Pa−1) n Q(J mol−1)

AISI 304 L Sheppard-Wright 8.3E+15 1.2E - 08 4.32 4.01E+05

and back-plate has been considered in the simulation. A sim-plified geometrical model for the tool has been used, avoidingthe finite element discretization of the shoulder scroll and theprobe thread.

Geometrical data, process parameters, material proper-ties and experimental results are taken from Zhu and Chao[83]. The diameters of the tool shoulder and tool probe are19.05 mm (3/4 inches) and 6.35 mm (1/4 inches), respec-tively. The height of the tool shoulder and the depth of thetool probe are 50 mm and 3 mm, respectively. The lengthof the work-pieces, along the welding direction, is 300 mm.The total width of the two work-pieces is 200 mm and thethickness is 3.18 mm (1/8 inches). The rotational velocity andadvancing velocity of the tool are 500 rpm and 101 mm/min(4 inches/min), respectively. Material thermal properties areshown in Table 3.

The material of the work-pieces is an AISI 304 L and ithas been modeled using a Sheppard-Wright material model.Material parameters for the Sheppard-Wright material modelhave been taken from Jorge Jr. and Balancin [61] and they

are shown in Table 4. The tool and the back-plate have beenmodeled as thermo-rigid bodies.

Table 5 shows the boundary conditions used in the simu-lation for the mechanical and thermal problems.

The prescribed tool advancing velocity has been imposedas a prescribed advancing velocity in the opposite directionto the work-pieces and back-plate. The prescribed tool rota-tional velocity has been applied to the tool.

Full stick friction conditions and thermal contact bound-ary conditions on the tool shoulder/work-pieces and toolprobe/work-pieces contact interfaces have been considered,assuming an infinity value for the heat transfer coefficient.Thermal contact boundary conditions, with a contact heattransfer coefficient of 5.0E +3 W m−2 ◦C−1, have been alsoconsidered on the contact surface between the work-pieces(bottom surface) and the back-plate (top surface). Convec-tion/radiation boundary conditions have been considered onthe top and lateral surfaces of the tool and on the top sur-face area of the work-pieces which is not in contact withthe tool shoulder. Adiabatic boundary conditions have been

123


Table 5 Boundary conditionsfor the mechanical and thermalproblems

Body Boundary Mechanical problem Thermal problem

Work-pieces Inlet Prescribed minus advancingtool velocity

Prescribed temperature = 25◦C

Outlet Zero prescribed traction vector Adiabatic

Sides Prescribed minus advancingtool velocity

Adiabatic

Bottom Prescribed minus advancingtool velocity

Thermal contact, HTC

Top Zero normal velocity and zerotangential components of thetraction vector

Convection + radiation

Shoulder Frictional stick Thermal contact, HTC =∞Probe Frictional stick Thermal contact, HTC =∞

Tool Side Rigid with prescribed rotationaltool velocity at all the nodesof the tool

Convection + radiation

Top Convection + radiation

Shoulder Thermal contact, HTC = ∞Probe Thermal contact, HTC = ∞

Back-plate Inlet Rigid with prescribed minusadvancing tool velocity at allthe nodes of the back-plate

Prescribed temperature = 25◦C

Outlet Adiabatic

Sides Adiabatic

Bottom Adiabatic

Top Thermal contact, HTC

considered on the outlet and the two external lateral surfacesof the work-pieces, as well as on the outlet, bottom and lat-eral surfaces of the back-plate. On the inlet surfaces of thework-pieces and the back-plate, the temperature has beenprescribed to the environmental one. The initial and environ-mental temperature is 25◦C.

A finite element mesh consisting of 116,414 linear tetrahe-dra and 21,494 nodal points has been considered in the sim-ulation. Prescribed advancing and rotational velocities havebeen imposed in an incremental way, using 1,000 time steps.A uniform time step of 0.1 sec has been considered. Thefull FSW numerical simulation has been done using 2,000time steps. Computing time for the whole simulation, usinga personal computer with 2 Gb RAM, was around 74 h.

Figure 11 shows a view of the finite element mesh of thetool, work-pieces and back-plate. Figure 12 shows a detailof the finite element mesh used for the tool (left) and work-pieces (right) in the stir area.

Figure 13 shows the temperature map distribution at theend of the numerical simulation. It can be clearly seen theconvection effect due to the advancing tool speed. A detailof the temperature map on the tool and work-pieces at thewelding line section at the end of the simulation is shown inFig. 14. Extremely high temperature gradients through thethickness of the work-pieces arise below the tool shoulderarea.

Fig. 11 Finite element mesh of the tool, work-pieces and back-plate

Figures 15 and 16 show a comparison between the exper-imental results reported by Zhu and Chao [83] and thenumerical results, using ASGS (GLS) and OSGS stabiliza-tion methods, obtained for the temperature along five dif-ferent lines parallel to the welding line on the top surface(z=3.18 mm) and four different lines parallel to the weldingline on the bottom surface (z=0.0 mm) of the work-pieces,respectively.

The five lines considered on the top of the surface(z=3.18 mm) of the work-pieces are located at y=12 mm,y=15.5 mm, y=18 mm, y=21 mm and y=27.5 mm of thewelding line. As it is shown in Fig. 5, numerical results

123


Fig. 12 Details of the finite element mesh of the tool (left) and work-pieces (right)

Fig. 13 Temperature mapdistribution at the end of thesimulation

Fig. 14 Zoom of thetemperature map distribution ata cut of the welding line sectionat the end of the simulation

obtained compare qualitatively well with the experimen-tal ones. The peak temperature for the lines located aty=12 mm, the one which is closest to the welding line, isremarkably well caught by the numerical simulation, both

in position and value. Peak temperatures for lines located aty=15.5 mm, y=18 mm and y=21 mm are slightly underes-timated by the numerical simulation, but the values at x=0 fitwell with the experimental results. On the other hand, ASGS

123


Fig. 15 Temperatures at theend of the simulation, along fourdifferent lines parallel to thewelding line on the top surfaceof the work-pieces. Comparisonbetween the experimentalresults (symbols) reported byZhu and Chao [83] and thenumerical results (lines), usingASGS (GLS) and OSGSstabilization methods

Fig. 16 Temperatures at theend of the simulation, along fourdifferent lines parallel to thewelding line on the bottomsurface of the work-pieces.Comparison between theexperimental results (symbols)reported by Zhu and Chao [83]and the numerical results (lines),using ASGS (GLS) and OSGSstabilization methods

(GLS) and OSGS stabilization methods yield basically thesame results.

The four lines considered on the bottom surface(z=0.0 mm) of the work-pieces are located at y=14 mm,y=17 mm, y=21 mm and y=27 mm of the welding line.As it is shown in Fig. 6, numerical results obtained com-pare qualitatively well with the experimental ones. Onceagain, the peak temperature for the line located at y=14mm, the one which is closest to the welding line, iscaught remarkably well by the numerical simulation, both inposition and value. Similarly, peak temperatures for lineslocated at y=17 mm and y=27 mm are also quite remark-

ably well caught, both in position and value, while the peaktemperature for the line located at y=21 mm is slightlyunderestimated by the numerical simulation. Temperaturesat x=0 fit pretty well with the experimental results. ASGS(GLS) and OSGS stabilization methods yield virtually iden-tical results.

6 Conclusions

In this paper, different aspects related to the computationalmodeling and the sub-grid scale finite element stabilization of

123


incompressibility and convection in the numerical simulationof friction stir welding processes have been addressed. Twosuitable rigid-thermoplastic constitutive models, Norton-Hoff and Sheppard-Wright, have been introduced. Within theparadigmatic framework of the multiscale methods, suitableASGS and OSGS stabilization methods have been introducedfor a fully incompressible, quasi-static, transient coupledthermomechanical formulation, using an Eulerian descrip-tion. Classical GLS and SUPG stabilization methods can berecovered as a particular case of the sub-grid scale stabi-lization framework developed. An assessment of the sub-grid scale finite element stabilization methods has been per-formed. Two numerical examples have been considered. Thefirst one deals with numerical simulation of a coupled ther-momechanical flow in a 2D rectangular domain. Steady-state and transient conditions have been considered. Usingthe GLS method to stabilize the pressure, temperature dis-tributions using SUPG and OSGS convection stabilizationmethods have been compared. Numerical results show thatOSGS stabilization method yields slightly better temperatureresults. On the other hand, using the SUPG method to stabi-lize the convective term of the temperature equation, pressureand velocity contours using GLS and OSGS pressure stabi-lization methods have been also compared. Numerical resultsshow that OSGS stabilization method yield better results,particularly at the inflow and outflow sections. Finally, thesecond example shows a 3D numerical simulation of a FSWprocess. ASGS (GLS and SUPG) and OSGS stabilizationmethods have been used. Numerical results have been com-pared with experimental ones available. A good agreementon the temperature distribution along different lines parallelto the welding line on the bottom and top surfaces of thework-pieces has been obtained and predicted peak temper-atures compare well, both in value and position, with theexperimental results available.

Acknowledgments This work has been supported by the EuropeanCommission under the STREP project of the VI Framework Programme“Detailed Multi-Physics Modeling of Frictional Stir Welding” (DEEP-WELD), the European Research Council under the Advanced Grant:ERC-2009-AdG “Real Time Computational Mechanics Techniques forMulti-Fluid Problems”, the Spanish Ministerio de Educación y Cienciaunder the PROFIT project CIT-020400-2007-82: “Nuevas Herramientaspara Optimizar el Proceso de Soldadura por Fricción” (FSWNET) andthe project of the Plan Nacional de I + D + I (2004–2007) “SimulaciónNumérica del Proceso de Soldadura Mediante Batido por Fricción”(FSW)

References

1. Agelet de Saracibar C (1998) Numerical analysis of coupled ther-momechanical contact problems. Computational model and appli-cations. Arch Comput Methods Mech 5:243–301. doi:10.1007/BF02897875

2. Agelet de Saracibar C, Cervera M, Chiumenti M (1999) On theformulation of coupled thermoplastic problems with phase-change.Int J Plast 15:1–34. doi:10.1016/S0749-6419(98)00055-2

3. Agelet de Saracibar C, Cervera M, Chiumenti M (2001)On the constitutive modeling of coupled thermomechanicalphase-change problems. Int J Plast 17:1565–1622. doi:10.1016/S0749-6419(00)00094-2

4. Agelet de Saracibar C, Chiumenti M, Valverde Q, Cervera M (2004)On the orthogonal sub-grid scale pressure stabilization of smalland finite deformation J2 plasticity, In: Agelet de Saracibar C (ed)Monograph series on computational methods in forming processes,monograph no. 2, CIMNE, Barcelona, Spain

5. Agelet de Saracibar C, Chiumenti M, Valverde Q, Cervera M(2006) On the orthogonal sub-grid scale pressure stabilization offinite deformation J2 plasticity. Comput Methods Appl Mech Eng195:1224–1251. doi:10.1016/j.cma.2005.04.007

6. Agelet de Saracibar C, Chiumenti M, Santiago D, Dialami N,Lombera G (2010) On the numerical modeling of FSW processes.In: Proceedings of the international symposium on plasticity and itscurrent applications, plasticity 2010, St. Kitts, St. Kitts and Nevis,January 3–8

7. Agelet de Saracibar C, Chiumenti M, Santiago D, Cervera M,Dialami N, Lombera G (2010) A computational model for thenumerical simulation of FSW processes. In: Barlat F, Moon YH,Lee MG (eds) NUMIFORM 2010: proceedings of the 10th inter-national conference on numerical methods in industrial formingprocesses, Dedicated to Professor O. C. Zienkiewicz (1921–2009),Pohang, South Korea, 13–17 June 2010, AIP Conference Proceed-ings, vol 1252, 2010, pp. 81–88. doi:10.1063/1.3457640

8. Agelet de Saracibar C, Chiumenti M, Cervera M, Dialami N, San-tiago D, Lombera G (2011) Advances in the numerical simulationof 3D FSW processes. In: Proceedings of the international sym-posium on plasticity and its current applications, plasticity 2011,Puerto Vallarta, Mexico, January 3–8

9. Agelet de Saracibar C, López R, Ducoeur B, Chiumenti M,de Meester B (2013) Un modelo numérico para la simulaciónde disolución de precipitados en aleaciones de aluminio conendurecimiento utilizando redes neuronales. Revista Internacionalde Métodos Numéricos para Cálculo y Diseño en la Ingeniería29(1):29–37. doi:10.1016/j.rimni.2012.02.003

10. Armero F, Simo JC (1991) A new unconditionally stable frac-tional step method for non-linear coupled thermomechanical prob-lems. Int J Numer Methods Eng 35:737–766. doi:10.1002/nme.1620350408

11. Armero F, Simo JC (1992) Product formula algorithms for nonlin-ear coupled thermo-plasticity: formulation and non-linear stabilityanalysis, SUDAM Report #92-4. Division of Applied Mechanics,Stanford University, Palo Alto, CA, USA, Department of Mechan-ical Engineering

12. Armero F, Simo JC (2003) A priori stability estimates and uncondi-tionally stable product formula algorithms for non-linear coupledthermoplasticity. Int J Plast 9(6):749–782

13. Askari A, Silling S, London B, Mahoney M (2003) Modeling andanalysis of friction stir welding processes. In: Proceedings of the4th International Symposium on Friction Stir Welding (4ISFSW),GKSS Workshop, Park City, Utah, USA, May 14–16

14. Avila M, Principe J, Codina R (2010) Finite element dynamicalsub-grid scale approximation of low Mach number flow equations.In: Dvorkin E, Goldschmit M, Storti M (eds) Proceedings of theAsociación Argentina de Mecánica Computacional, Buenos Aires,Argentina, 15–18 Noviembre 2010, Mecánica Computacional, volXXIX, pp 7967–7983

15. Bendzsak G, North T, Smith C (2000) An experimentally vali-dated 3D model for friction stir welding. In: Proceedings of the2nd International Symposium on Friction Stir Welding (2ISFSW),Gothenburg, Sweden, June 27–29

123

http://dx.doi.org/10.1007/BF02897875

http://dx.doi.org/10.1007/BF02897875

http://dx.doi.org/10.1016/S0749-6419(98)00055-2

http://dx.doi.org/10.1016/S0749-6419(00)00094-2

http://dx.doi.org/10.1016/S0749-6419(00)00094-2

http://dx.doi.org/10.1016/j.cma.2005.04.007

http://dx.doi.org/10.1063/1.3457640

http://dx.doi.org/10.1016/j.rimni.2012.02.003

http://dx.doi.org/10.1002/nme.1620350408



16. Bendzsak G, North T, Smith C (2000) Material properties relevantto 3-D FSW modeling. In: Proceedings of the 2nd InternationalSymposium on Friction Stir Welding (2ISFSW), Gothenburg, Swe-den, June 27–29

17. Brezzi F, Fortin M (1991) Mixed and hybrid finite element methods.Springer Series in Computational Mathematics, vol 15. Springer,New York

18. Buffa G, Hua J, Shivpuri R, Fratini L (2006) A continuum-basedfem model for friction stir welding—model development. MaterSci Eng A 419:389–396. doi:10.1016/j.msea.2005.09.040

19. Bussetta P, Dialami N, Boman R, Chiumenti M, Agelet de SaracibarC, Cervera M, Ponthot JP (2013) Comparison of a fluid and a solidapproach for the numerical simulation of friction stir welding with anon-cylindrical pin. In: Proceedings of the international conferenceon coupled problems in engineering 2013, Santa Eulalia, Ibiza,Spain, June 17–19

20. Bussetta P, Dialami N, Boman R, Chiumenti M, Agelet de SaracibarC, Cervera M, Ponthot JP (2013) Numerical simulation of frictionstir welding process with different pin geometries based on a com-bined ALE/remeshing formulation. Steel Research International.doi:10.1002/srin.201300182

21. Cervera M, Agelet de Saracibar C, Chiumenti M (2002) COMET—a coupled mechanical and thermal analysis code. Data Input Man-ual. Version 5.0, Technical Report IT-308, CIMNE, Barcelona,Spain, http://www.cimne.com/comet

22. Cervera M, Chiumenti M, Valverde Q, Agelet de Saracibar C (2003)Mixed linear/linear simplicial elements for incompressible elastic-ity and plasticity. Comput Methods Appl Mech Eng 192:5249–5263. doi:10.1016/j.cma.2003.07.007

23. Cervera M, Chiumenti M, Agelet de Saracibar C (2004) Shear bandlocalization via local J2 continuum damage mechanics. ComputMethods Appl Mech Eng 193:849–880. doi:10.1016/j.cma.2003.11.009

24. Cervera M, Chiumenti M, Agelet de Saracibar C (2004) Softening,localization and stabilization: capture of discontinuous solutionsin J2 plasticity. Int J Numer Anal Methods Geomech 28:373–393.doi:10.1002/nag.341

25. Cervera M, Chiumenti M, Codina R (2010) Mixed stabilized finiteelement methods in nonlinear solid mechanics. Part I: formulation.Comput Methods Appl Mech Eng 199:2559–2570. doi:10.1016/j.cma.2010.04.006

26. Cervera M, Chiumenti M, Codina R (2010) Mixed stabilized finiteelement methods in nonlinear solid mechanics. Part II: strain local-ization. Comput Methods Appl Mech Eng 199:2571–2589. doi:10.1016/j.cma.2010.04.005

27. Cervera M, Chiumenti M, Codina R (2011) Mesh objective mod-eling of cracks using continuous linear strain and displacementinterpolations. Int J Numer Methods Eng 87:962–987. doi:10.1002/nme.3148

28. Chen CM, Kovacevic R (2003) Finite element modeling offriction stir welding - thermal and thermomechanical analy-sis. Int J Mach Tools Manuf 43:1319–1326. doi:10.1016/S0890-6955(03)00158-5

29. Chiumenti M, Valverde Q, Agelet de Saracibar C, Cervera M (2002)A stabilized formulation for incompressible elasticity using lineardisplacement and pressure interpolations. Comput Methods ApplMech Eng 191:5253–5264. doi:10.1016/S0045-7825(02)00443-7

30. Chiumenti M, Valverde Q, Agelet de Saracibar C, Cervera M (2004)A stabilized formulation for incompressible plasticity using lineartriangles and tetrahedra. Int J Plast 20:1487–1504. doi:10.1016/j.ijplas.2003.11.009

31. Chiumenti M, Cervera M, Salmi A, Agelet de Saracibar C, DialamiN, Matsui K (2010) Finite element modeling of multi-pass weld-ing and shaped metal deposition processes. Comput Methods ApplMech Eng 199:2343–2359. doi:10.1016/j.cma.2010.02.018

32. Chiumenti M, Cervera M, Agelet de Saracibar C (2010) A mixedstabilized finite element formulation for strain localization analysis.In: Proceedings of the 11th Pan-American Congress of AppliedMechanics—PACAM XI, January 4–8, Foz do Iguaçu, PR, Brazil

33. Chiumenti M, Cervera M, Agelet de Saracibar C, Dialami N (2013)Numerical modelling of friction stir welding processes. ComputMethods Appl Mech Eng 254:353–369. doi:10.1016/j.cma.2012.09.013

34. Chiumenti M, Cervera M, Agelet de Saracibar C, Dialami N (2013)A novel stress-accurate FE technology for highly non-linear analy-sis with incompressibility constraint. Application to the numericalsimulation of the FSW process, AIP conference proceedings. Pro-ceedings of the international conference on numerical methods informing processes, NUMIFORM 2013, Shenyang, China, 2013.NUMIFORM 2013, AIP Conference Proceedings, vol 1532, 2013,pp. 45–56

35. Christ D, Cervera M, Chiumenti M, Agelet de Saracibar C (2003)A mixed finite element formulation for incompressibility usinglinear displacement and pressure interpolations, monograph no.77, CIMNE, Barcelona, Spain

36. Codina R, Blasco J (1997) A finite element formulation for thestokes problem allowing equal velocity-pressure interpolation.Comput Methods Appl Mech Eng 143:373–391. doi:10.1016/S0045-7825(96)01154-1

37. Codina R (1998) Comparison of some finite element meth-ods for solving the diffusion-convection-reaction equations.Comput Methods Appl Mech Eng 156:185–210. doi:10.1016/S0045-7825(97)00206-5

38. Codina R, Blasco J (2000) Stabilized finite element method fortransient Navier-Stokes equations based on pressure gradient pro-jection. Comput Methods Appl Mech Eng 182:287–300. doi:10.1016/S0045-7825(99)00194-2

39. Codina R (2000) Stabilization of incompressibility and convec-tion through orthogonal sub-scales in finite element methods.Comput Methods Appl Mech Eng 190:1579–1599. doi:10.1016/S0045-7825(00)00254-1

40. Codina R, Blasco J (2000) Analysis of a pressure-stabilized finiteelement approximation of the stationary Navier-Stokes equations.Numer Math 87:59–81. doi:10.1007/s002110000174

41. Codina R (2001) A stabilized finite element method for generalizedstationary incompressible flows. Comput Methods Appl Mech Eng190:2681–2706. doi:10.1016/S0045-7825(00)00260-7

42. Codina R (2002) Stabilized finite element approximation of tran-sient incompressible flows using orthogonal subscales. Com-put Methods Appl Mech Eng 191:4295–4321. doi:10.1016/S0045-7825(02)00337-7

43. Codina R, Principe J, Guasch O, Badia S (2007) Time depen-dent subscales in the stabilized finite element approximation ofincompressible flow problems. Comput Methods Appl Mech Eng196:2413–2430. doi:10.1016/j.cma.2007.01.002

44. Codina R, Principe J (2007) Dynamic subscales in the finite elementapproximation of thermally coupled incompressible flows. Int JNumer Methods Fluids 54:707–730. doi:10.1002/fld.1481

45. Codina R, Principe J, Avila M (2010) Finite element approximationof turbulent thermally coupled incompressible flows with numeri-cal sub-grid scale modelling. Int J Numer Methods Heat Fluid Flow20(5):492–516. doi:10.1108/09615531011048213

46. Colegrove P, Painter M, Graham D, Miller T (2000) Three dimen-sional flow and thermal modelling of the friction stir weldingprocess. In: Proceedings of the 2nd International Symposium onFriction Stir Welding (2ISFSW), Gothenburg, Sweden, June 27–29

47. Colegrove P, Shercliff H, Threadgill P (2004) Modelling the fric-tion stir welding of aerospace alloys. Proceedings of the 5th Inter-national Symposium on Friction Stir Welding (5ISFSW), Metz,France, September 14–16

123

http://dx.doi.org/10.1016/j.msea.2005.09.040

http://dx.doi.org/10.1002/srin.201300182

http://www.cimne.com/comet




http://dx.doi.org/10.1002/nag.341







http://dx.doi.org/10.1016/S0890-6955(03)00158-5

http://dx.doi.org/10.1016/S0890-6955(03)00158-5

http://dx.doi.org/10.1016/S0045-7825(02)00443-7

http://dx.doi.org/10.1016/j.ijplas.2003.11.009

http://dx.doi.org/10.1016/j.ijplas.2003.11.009




http://dx.doi.org/10.1016/S0045-7825(96)01154-1

http://dx.doi.org/10.1016/S0045-7825(96)01154-1

http://dx.doi.org/10.1016/S0045-7825(97)00206-5

http://dx.doi.org/10.1016/S0045-7825(97)00206-5

http://dx.doi.org/10.1016/S0045-7825(99)00194-2

http://dx.doi.org/10.1016/S0045-7825(99)00194-2

http://dx.doi.org/10.1016/S0045-7825(00)00254-1

http://dx.doi.org/10.1016/S0045-7825(00)00254-1

http://dx.doi.org/10.1007/s002110000174

http://dx.doi.org/10.1016/S0045-7825(00)00260-7

http://dx.doi.org/10.1016/S0045-7825(02)00337-7

http://dx.doi.org/10.1016/S0045-7825(02)00337-7


http://dx.doi.org/10.1002/fld.1481

http://dx.doi.org/10.1108/09615531011048213


48. De Vuyst T, D’Alvise L, Simar A, de Meester B, Pierret S (2005),Finite element modelling of friction stir welding aluminium alloysplates—inverse analysis using a genetic algorithm. Weld World49(3/4):44–55

49. De Vuyst T, D’Alvise L, Simar A, de Meester B, Pierret S (2004)Inverse analysis using a genetic algorithm for the finite elementmodelling of friction stir welding. In: Proceedings of the 5th Inter-national Symposium on Friction Stir Welding (5ISFSW), Metz,France, September 14–16

50. De Vuyst T, D’Alvise L, Robineau A, Goussain JC (2006) Materialflow around a friction stir welding tool—experiment and simula-tion. In: Proceedings of the 8th international seminar on numericalanalysis of weldability, Graz, Austria, September 25–27

51. De Vuyst T, D’Alvise L, Robineau A, Goussain JC (2006) Sim-ulation of the material flow around a friction stir welding tool.In: Proceedings of the 6th International Symposium on FrictionStir Welding (6ISFSW), Saint-Sauveur, Quebec, Canada, October10–13

52. Dialami N, Chiumenti M, Cervera M, Agelet de Saracibar C (2013)An apropos kinematic framework for the numerical modellingof friction stir welding. Comput Struct 117:48–57. doi:10.1016/j.compstruc.2012.12.006

53. Dong P, Lu F, Hong JK, Cao Z (2001) Coupled thermome-chanical analysis of friction stir welding process using simpli-fied models. Sci Technol Weld Join 6(5):281–287. doi:10.1179/136217101101538884

54. GiD: The Personal Pre and Post processor, CIMNE, 2011. http://www.gidhome.com

55. Heurtier P, Desrayaud C, Montheillet F (2002) A thermomechani-cal analysis of the friction stir process. Mater Sci Forum 396:1537–1542. doi:10.4028/www.scientific.net/MSF.396-402.1537

56. Heurtier P, Jones MJ, Desrayaud C, Driver JH, Montheillet F, Alle-haux D (2006) Mechanical and thermal modeling of friction stirwelding. J Mater Process Technol 171:348–357. doi:10.1016/j.jmatprotec.2005.07.014

57. Hughes TJR, Franca L, Balestra M (1986) A new finite elementformulation for computational fluid dynamics: V. Circumventingthe Babuska-Brezzi condition: A stable Petrov-Galerkin formula-tion of the Stokes problem accommodating equal order interpola-tions. Comput Methods Appl Mech Eng 59:85–99. doi:10.1016/0045-7825(86)90025-3

58. Hughes TJR (1995) Multiscale phenomena: green’s function, theDirichlet-to-Neumann formulation, subgrid scale models, bubblesand the origins of stabilized formulations. Comput Methods ApplMech Eng 127:387–401. doi:10.1016/0045-7825(95)00844-9

59. Hughes TJR, Feijóo GR, Mazzei L, Quincy JB (1998) The varia-tional multiscale method—a paradigm for computational mechan-ics. Comput Methods Appl Mech Eng 166:3–24. doi:10.1016/S0045-7825(98)00079-6

60. Hughes TJR, Scovazzi G, Franca L (2004) Multiscale and stabilizedmethods. In: Stein E, de Borst R, Hughes TJR (eds), Encyclope-dia of computational mechanics. Wiley, Chichester, 2004. doi:10.1002/0470091355

61. Jorge AM Jr, Balancin O (2005) Prediction of steel flow stressesunder hot working conditions. Mater Res 8(3):309–315. doi:10.1590/S1516-14392005000300015

62. Khandkar M, Khan J (2001) Thermal modeling of overlap frictionstir welding for Al-alloys. J Mater Process Manuf Sci 10:91–105.doi:10.1177/1062065602010002613

63. Khandkar M, Khan J, Reynolds A (2003) Prediction of tempera-ture distribution and thermal history during friction stir welding:input torque based model. Sci Technol Weld Joining 8(3):165–174.doi:10.1179/136217103225010943

64. Langerman M, Kvalvik E (2003) Modeling plasticized aluminumflow and temperature fields during friction stir welding. In: Pro-ceedings of the 6th ASME-JSME Thermal Engineering Joint Con-

ference, Hapuna Beach Prince Hotel, Kohala Coast, Hawaii Island,Hawaii, USA, March 16–20

65. López R, Ducoeur B, Chiumenti M, de Meester B, Agelet deSaracibar C (2008) Modeling precipitate dissolution in hard-ened aluminium alloys using neural networks. Int J Mater Form1(1):1291–1294. doi:10.1007/s12289-008-0139-4

66. McClure JC, Tang W, Murr LE, Guo X, Feng Z, Gould JE (1998)A thermal model of friction stir welding. In: Proceedings of the5th international conference on trends in welding research, PineMountain, Georgia, USA, June 1–5, 1998, pp 590–595

67. Nikiforakis N (2005) Towards a whole system simulation of FSW.In: Proceedings of the 2nd FSW modelling and flow visualisationseminar, GKSS Forschungszentrum, Geesthacht, Germany, Janu-ary 31—February 1

68. Principe J (2008) Subgrid scale stabilized finite elements for lowspeed flows, Ph.D. Thesis. Technical University of Catalonia,Barcelona, Spain

69. Principe J, Codina R (2008) A stabilized finite elementapproximation of low speed thermally coupled flows. Int JNumer Methods Heat Fluid Flow 18(7/8):835–867. doi:10.1108/09615530810898980

70. Principe J, Codina R (2009) Mathematical models for ther-mally coupled low speed flows. Adv Theor Appl Mech 2:93–112 http://www.m-hikari.com/atam/atam2009/atam1-4-2009/principeATAM1-4-2009.pdf

71. Santiago D, Lombera G, Urquiza S, Agelet de Saracibar C, Chiu-menti M (2010) Modelado termo-mecánico del proceso de FrictionStir Welding utilizando la geometría real de la herramienta. RevistaInternacional de Métodos Numéricos para Cálculo y Diseño enIngeniería 26:293–303. doi:10.1016/j.rimni.2012.02.003

72. Schmidt H, Hattel J (2004) Modelling thermo mechanical condi-tions at the tool/matrix interface in friction stir welding. In: Pro-ceedings of the 5th International Symposium on Friction Stir Weld-ing (5ISFSW), Metz, France, September 14–16

73. Seidel TU, Reynolds AP (2003) Two-dimensional friction stirwelding process model based on fluid mechanics. Sci Technol WeldJoining 8(3):175–183. doi:10.1179/136217103225010952

74. Shercliff HR, Russell MJ, Taylor A, Dickerson TL (2000)Microstructural modeling in friction stir welding of 2000 seriesaluminium alloys. Mécanique & Industries 6(2005):25–35. doi:10.1051/meca:2005004

75. Shi Q, Dickerson T, Shercliff H (2003) Thermo-mechanical FEmodeling of friction stir welding of AL-2024 including tool loads.In: Proceedings of the 4th International Symposium on FrictionStir Welding (4ISFSW), Park City, Utah, USA, May 14–16

76. Song M, Kovacevic R (2003) Numerical and experimental studyof the heat transfer process in friction stir welding. J Eng Manuf217(Part B):73–85

77. Thomas WM, Nicholas ED, Needham JC, Murch MG, Temple-Smith P, Dawes CJ (1991) Friction stir butt welding. GB PatentNo. 9125978.8, International Patent No. PCT/GB92/02203

78. Ulysse P (2002) Three-dimensional modeling of the friction stir-welding process. Int J Mach Tools Manuf 42:1549–1557. doi:10.1016/S0890-6955(02)00114-1

79. Xu S, Deng X, Reynolds AP, Seidel TU (2001) Finite elementsimulation of material flow in friction stir welding. Sci. Technol.Weld. Joining 6(3):191–193. doi:10.1179/136217101101538640

80. Xu S, Deng X (2003) Two and three-dimensional finite elementmodels for the friction stir welding process. In: Proceedings of the4th International Symposium on Friction Stir Welding (4ISFSW),Park City, Utah, USA, May 14–16

81. Xu S, Deng X (2004) Two and three-dimensional finite elementmodels for the friction stir welding process, University of SouthCarolina, Department of Mechanical Engineering, Columbia,South Carolina 29208, USA

123

http://dx.doi.org/10.1016/j.compstruc.2012.12.006

http://dx.doi.org/10.1016/j.compstruc.2012.12.006

http://dx.doi.org/10.1179/136217101101538884

http://dx.doi.org/10.1179/136217101101538884

http://www.gidhome.com

http://www.gidhome.com

http://dx.doi.org/10.4028/www.scientific.net/MSF.396-402.1537

http://dx.doi.org/10.1016/j.jmatprotec.2005.07.014


http://dx.doi.org/10.1016/0045-7825(86)90025-3

http://dx.doi.org/10.1016/0045-7825(86)90025-3

http://dx.doi.org/10.1016/0045-7825(95)00844-9

http://dx.doi.org/10.1016/S0045-7825(98)00079-6

http://dx.doi.org/10.1016/S0045-7825(98)00079-6

http://dx.doi.org/10.1002/0470091355

http://dx.doi.org/10.1002/0470091355

http://dx.doi.org/10.1590/S1516-14392005000300015

http://dx.doi.org/10.1590/S1516-14392005000300015

http://dx.doi.org/10.1177/1062065602010002613

http://dx.doi.org/10.1179/136217103225010943

http://dx.doi.org/10.1007/s12289-008-0139-4

http://dx.doi.org/10.1108/09615530810898980

http://dx.doi.org/10.1108/09615530810898980

http://www.m-hikari.com/atam/atam2009/atam1-4-2009/principeATAM1-4-2009.pdf

http://www.m-hikari.com/atam/atam2009/atam1-4-2009/principeATAM1-4-2009.pdf

http://dx.doi.org/10.1016/j.rimni.2012.02.003

http://dx.doi.org/10.1179/136217103225010952

http://dx.doi.org/10.1051/meca:2005004

http://dx.doi.org/10.1051/meca:2005004

http://dx.doi.org/10.1016/S0890-6955(02)00114-1

http://dx.doi.org/10.1016/S0890-6955(02)00114-1

http://dx.doi.org/10.1179/136217101101538640


82. Zhao H (2005) Friction stir welding (FSW) simulation usingan Arbitrary Lagrangian-Eulerian (ALE) moving mesh approach,Ph.D. Dissertation, West Virginia University, Morgantown, WestVirginia, USA, 2005. http://hdl.handle.net/10450/4367

83. Zhu XK, Chao YJ (2004) Numerical simulation of transient tem-perature and residual stresses in friction stir welding of 304L stain-less steel. J Mater Process Technol. 146(2):263–272. doi:10.1016/j.jmatprotec.2003.10.025

123

http://hdl.handle.net/10450/4367



Computational Modeling and Sub-Grid Scale Stabilization of...

Documents

Transcript of Computational Modeling and Sub-Grid Scale Stabilization of...