in a recent paper , presented a coupled continuum field formulation for the interaction of electric effects , mechanical response and self - diffusion .their approach drew upon earlier work by larch and cahn  , nix and co - workers  , gnin  , bower and freund  , to name but a few . in reference , a review of these works and others was presented , computational techniques were developed based on the finite element method , and several initial and boundary value problems involving diffusional creep ( nabarro - herring and coble creep ) were solved .the formulation introduced in  was extended to interdiffusion , with dopants in silicon as motivation , by garikipati and bassman  . in the current paper , we present a further extension of that same formulation to account for the interaction of grain - boundary migration with stress - driven self - diffusion and electromigration in polycrystalline solids .different strategies used to model grain - boundary motion in a computational setting are described in the literature .sun and suo  developed a two - dimensional finite element formulation , based on the idea that the energy dissipated during the motion of the boundary must equal the reduction in the free energy of the system .this formulation was used to study several problems including grain growth in a thin film and the competition between surface grooving and grain - boundary migration .a similar variational formulation was developed by cocks and gill  and used to model the evolution of a large network of grains in two dimensions .another model was developed by , based on the variational formulation of reitich and soner  and using the level set method of osher and sethian  .the level set method is also used in the present paper as an interface - capturing technique . however , the goal of the current work is not merely to simulate grain - boundary motion , but to capture fully the interaction between this motion and other microscale phenomena that take place in pure polycrystalline materials , namely stress - mediated self - diffusion and electromigration .this distinguishes the current work from the existing literature .  in the current work, we recast the level - set equation in a simpler form by assuming that the level - set function remains a signed distance to the migrating grain boundary ( as with the original level - set equation , the use of an extensional velocity field helps maintain this signed - distance function ) . tested the resulting level - set formulation extensively and concluded that it is both accurate and robust despite its remarkable simplicity ( for some interface - evolution problems, this approach reduces the original level - set equation , a nonlinear hyperbolic pde , to an ode that is almost trivial to solve ) .they conducted several numerical experiments to assess the ability of the simplified level - set scheme to capture the correct solution , particularly in the presence of discontinuities in the extensional velocity and/or in the gradient of the level - set function .they also examined the convergence properties of the method and its performance in a variety of problems , including curvature flow and problems where the simplified level - set equation takes the form of a hamilton - jacobi equation with convex or non - convex hamiltonian .discretizations based on structured and unstructured finite - element meshes of bilinear quadrilateral and linear triangular elements were shown to perform equally well .they also found that sufficient accuracy is available through a standard galerkin formulation without resorting to any stabilization or discontinuity - capturing techniques .in addition , a variant of the simplified level - set formulation mentioned above was employed by for representing the evolution of phase boundaries over unstructured finite - element meshes . here, we treat a complex coupled problem of which grain - boundary migration is only one facet , in addition to mechanics , self - diffusion and electromigration .we apply this simplified level - set scheme to the problem of grain - boundary migration within this context , and we demonstrate its implementation as an integral part of the wider computational framework used to solve the coupled problem under consideration .  the remainder of this paper is organized as follows . in section [ sec : coup ] , we summarize the formulation for coupled mass transport and mechanics .then we examine the thermodynamics and kinetics of grain - boundary migration , and show how this phenomenon interacts with mass transport and mechanics in polycrystals . in section[ sec : num ] , we formulate the grain - boundary migration problem using the level set method and we describe the computational methods used in the implementation of this formulation .numerical examples are presented in section [ sec : resu ] . a summary is provided and conclusions are drawn in section [ sec : conc ] .the thermodynamics is posed in a continuum setting , with motivation provided by atomic processes .a schematic of the lattice is shown in fig . [fig : lattice ] .it shows atoms , vacancies , and a free surface ; the latter is a source and sink for vacancies .a grain boundary could serve as a source or sink also .free surfaces and grain boundaries are treated as regions of finite width , and respectively .we consider crystalline materials in which the dangling bonds around a vacancy cause an inward relaxation of the surrounding lattice ( see fig .[ fig : lattice ] ) . the resulting vacancy relaxation strain can be expressed as where is the atomic volume , ( with ) is the volume of a vacancy , is the vacancy concentration , is the vacancy concentration at thermodynamic equilibrium under vanishing external stress and is the second - order isotropic tensor .the creep strain resulting from the accumulation or depletion of atoms at a free surface or grain boundary with unit normal , , can be expressed as a non - phenomenological evolution equation for was derived and discussed in detail by .the thermal strain is given by where is the temperature , is a reference temperature and is the linear coefficient of thermal expansion .the elastic strain is obtained by subtracting these inelastic strain contributions from the total strain , ; i.e. the stress is obtained from the elastic strain and the ( generally anisotropic ) fourth - order elasticity tensor , , as given a state with local stress , , the incremental elastic strain energy density is then given as , where is the increment in elastic strain . with this background , the incremental internal energy density , corresponding to a state , and increments and is where is the entropy density and is the vacancy formation energy .the specific form of , the entropic dependence of , is not important to the development that follows .the density of work done by external agents can be expressed as here , the first term of the right - hand side is the classical stress - power term . the second term accounts for the apparent work performed by the electrostatic potential , , during electromigration ; in this phenomenological treatment , is the apparent charge ascribed to each vacancy .the last term accounts for the work done against the stress when vacancies are created at a free surface or grain boundary .the numerical factor appearing in this term arises from geometrical considerations .additionally , the requirement that this term be active only at sources and sinks is enforced using the indicator function , , defined as    the total entropy density is given by ,\ ] ] since the formulation is isothermal , the specific form of the vibrational term , , is unimportant .the second term is the entropy density due to mixing ( see kittel and kroemer  for details ) , is the boltzmann constant , is the concentration of atoms , and is the lattice site concentration . assuming that remains fixed in any material volume , i.e.  , the incremental entropy density corresponding to an increment in vacancy concentration , , at given stress and temperature can be expressed as    at the given state of stress and temperature , the incremental gibbs free energy density corresponding to increments and is defined as with this , the constitutive relations can be obtained in a systematic fashion as outlined in the following section .  * remark * since free surfaces and grain boundaries are considered , there are accompanying surface and grain boundary energies , and .these energies are taken to be independent of the strain and vacancy concentration for this formulation , and therefore do not appear in the incremental gibbs free energy density .anisotropic elasticity is assumed and the relation between the stress , , and the elastic strain , , is given by ( [ eq : str ] ) .the relation between the current density , , and the electric potential , , is given by ohm s law : where is the electric resistivity .the vacancy chemical potential is defined in the usual fashion  : where is a constant reference potential .applying ( [ eq : muv ] ) to ( [ eq : ine][eq : gib ] ) gives the coupling with mechanics is evident through the strain- and stress - dependent terms .the vacancy flux is obtained from ( [ eq : muv2 ] ) via the relation where is the vacancy diffusivity .this relation has been derived from an atomic basis by bardeen  .the term is the mobility of a vacancy and , the force acting on it , is the driving force for mass transport .the constitutive relations established above are incorporated in balance laws for mechanics , electric flow and mass transport , leading to a coupled system of governing differential equations .   neglecting dynamic effects and body forces ,the mechanics problem is governed by the quasistatic equilibrium equation and appropriate boundary conditions :    where is the domain of interest and the boundary subsets and have essential and natural boundary conditions specified , respectively .these subsets satisfy and .the electric flow problem is governed by the charge conservation equation with dirichlet and neumann boundary conditions :    where and .the mass - transport problem is governed by the continuity equation for vacancies , and appropriate initial and boundary conditions :    where and are the boundary subsets on which concentration and flux boundary conditions are specified , respectively . here , the boundary subsets satisfy and .the effectiveness of vacancy sources and sinks is characterized by the relaxation time , .the equilibrium vacancy concentration , , is defined by in ( [ eq : muv2 ] ) .the current location of a migrating grain boundary determines the value of the indicator function , ( see eq .( [ eq : chi ] ) ) ; i.e. it determines whether vacancy sources / sinks are active and whether creep strain can accumulate at a given point , .furthermore , information about the location of the boundary is needed to calculate the value of the vacancy formation energy , , and the activation energy for diffusion , , everywhere in the domain of interest .these properties are assumed to vary linearly over the width of a boundary region as shown in fig .[ fig : interfaces ] .the foregoing illustrates the influence of grain - boundary migration on mass transport . due to the tight coupling between mass transport and mechanics, the migration of the grain boundary also affects the stress . in turn, mass transfer across the grain boundary causes one grain to grow at the expense of its neighbor and thus leads to grain - boundary migration .[ br] , [ ] grain a [ ] grain b [ ] [ ] [ ] [ ]  and across boundary regions.,title="fig:",width=288 ]   generally , interface migration in polycrystals is driven by the accompanying decrease in the free energy of the system .the thermodynamic driving force for such a process , acting on a unit area of the interface , is thus defined as where is the increase in the total gibbs free energy of the system brought about by a motion of the interface , during which the interface sweeps through the volume . neglecting triple junctions ,the gibbs free energy of a polycrystal can be expressed as the sum of two contributions : where is the gibbs free energy density as defined in ( [ eq : gib ] ) and is the energy per unit area of an interface , .the term _ interface _ is used here to refer to free surfaces as well as grain boundaries , and the summation in ( [ eq : m_gfe ] ) is over all such interfaces in the polycrystal .  in situations where the gibbs free energy density , , suffers a decrease across a grain boundary, the total free energy of the system can be reduced if the grain with the smaller value of ( evaluated at the grain boundary ) were to grow at the expense of its neighbor .thus , a driving force acts on the grain boundary . for example, during recrystallization , annealed grains grow at the expense of cold - worked grains in which large dislocation densities lead to high values of .the misorientation between two adjacent grains of an elastically anisotropic material subjected to a directional load causes one grain to store a smaller amount of strain energy per unit volume than its neighbor .this leads to strain - induced grain - boundary migration .electromigration also causes atoms to jump across grain boundaries and thus leads to the migration of these boundaries  .  under the current formulation, the driving force discussed above can be characterized as follows : when a single atom hops across the grain boundary , it exchanges positions with a vacancy .therefore , the volume change associated with this hopping event is .the decrease in the gibbs free energy of the system is equal to the difference in the chemical potential of atoms across the boundary ; i.e. , or equivalently .hence , from eq .( [ eq : m_dforc ] ) , and assuming that the chemical potential gradient across the boundary is essentially linear  , the driving force , , can be expressed as where is the unit normal to the grain boundary and is its width .[ br] , due to surface tension , .,title="fig:",width=288 ]  from eq .( [ eq : m_gfe ] ) , it is clear that the total gibbs free energy of a polycrystal can also be lowered by decreasing the total surface area of the grain boundaries in the system .when a curved interface moves away from its center of curvature , sweeping through an increment of volume , the free energy of the system increases by where is the ( constant ) specific surface energy of the interface and and are its principal radii of curvature .this expression can be traced back to herring  .it follows that , in the current two - dimensional formulation , the driving force acting on a unit area of a ( cylindrical ) grain boundary , due to this effect , can be expressed as where is the radius of curvature and the negative sign indicates that drives the boundary to migrate _ toward _ its center of curvature ( see fig .[ fig : m_pgamma ] ) . in a polycrystal comprising only annealed grains , the action of leads to normal grain growth ; i.e. the growth of large grains at the expense of smaller ones .by contrast , in the early stages of recrystallization , small annealed grains grow at the expense of the surrounding matrix of cold - worked material .in this case , the boundaries surrounding the annealed grains move _ away _ from their respective centers of curvature under the effect of .   assuming that grain - boundary migration takes place as a result of individual atoms hopping across the boundary , the migration velocity can be expressed in the following form : where is in the direction of the local unit normal , , which is assumed to point away from the boundary s center of curvature and is the grain - boundary mobility .this classical result , derived by turnbull  using absolute reaction rate theory , holds provided that , a condition which is always met in grain growth and recrystallization  .the results of molecular dynamics simulations of curvature - driven  and strain - induced  grain - boundary migration in bicrystals agree with eq .( [ eq : m_kine ] ) , which can be modified as follows , to account for both types of driving forces discussed in section [ sec : migr_theo_ther ] :    in section [ sec : migr_theo_ther ] , it was established that diffusion of atoms across a grain boundary due to the local gradient in the atomic chemical potential causes the boundary to migrate .the driving force for boundary migration in this case , denoted , is given by ( [ eq : m_pg ] ) .an expression for the corresponding mobility , , can be obtained by stipulating that , in this case , the migration velocity in the direction of the local unit normal , , should be given by ( see porter and easterling  ) where and are , respectively , the local atomic and vacancy fluxes .also recall that is the atomic volume and is the volume of a vacancy . since atoms and vacancies move by exchanging positions ; i.e.  , we have finally , by combining ( [ eq : m_mg2 ] ) , ( [ eq : m_pg ] ) and ( [ eq : flux0 ] ) , we obtain since grain - boundary migration under the sole influence of involves _ transport _ across the boundary , and since , in the present treatment , mass transport is assumed to take place through the exchange of positions between atoms and vacancies , no lattice sites are transferred across the boundary in this process .by contrast , migration under the influence of consists of the transfer of lattice sites across the grain boundary  , via the process of atoms detaching from one grain and attaching to the one on the opposite side of the boundary .it is hence unreasonable to expect the mobilities , and , of these two distinctly different processes to be the same .as mentioned earlier , normal grain growth can be attributed to the action of the driving force .the fact that grain growth is a thermally activated process suggests an arrhenius - type relationship between the mobility , , and the temperature : where the pre - exponential factor , , and the activation energy for grain - boundary migration , , are dependent on the misorientation angle and the axis of rotation  .  finally , by combining ( [ eq : m_force2d ] ) , ( [ eq : m_kine2 ] ) , ( [ eq : m_mg2 ] ) and ( [ eq : m_tempdep1 ] ) , we obtain the following expression for the migration velocity : this section , we focus on the level - set formulation of the grain - boundary migration problem , and we present a detailed description of the computational techniques used in its implementation . to solve the coupled problem , the level - set formulation is integrated into the computational framework that was first introduced in reference  .this computational framework is based on an operator - split solution scheme and relies on the finite element method to solve the mechanics , mass - transport and electric - flow problems individually .while reference  addresses mainly the physics of stress - driven mass transport in polycrystals , it also contains details of the computational framework used therein . here , these details are omitted ; however , the operator - split solution scheme is outlined briefly in section [ sec : stagg ] to show how the level - set formulation is incorporated into the computational framework used .evolving interfaces can be tracked using the level set method , originally introduced by osher and sethian  .a comprehensive review of the method and the computational algorithms used in its implementation can be found in  .[ ] [ ]   consider an evolving grain boundary , , which divides the domain of interest , , into two disjoint open subsets , and .this situation is depicted in fig .[ fig : lsm ] .the boundary can be parameterized with the aid of the scalar function , defined on , provided that the following conditions are satisfied for all  :  [ eq : lscondi ]   the term _ level set _ refers to a set of points with a fixed value of , i.e. an iso - contour of ; the _ zero level set _ represents the grain boundary .accordingly , the unit normal to a given level set can be defined locally as where denotes the euclidean norm .this expression can be evaluated at any point on the zero level set to obtain the local unit normal to the boundary , .this definition implies that always points into the region .the evolution of the level - set field is governed by where is the ( scalar ) local propagation velocity of the level set passing through point . to track the motion of the grain boundary, we require that where is obtained at any point on the grain boundary from ( [ eq : m_kine3 ] ) .although this requirement does not place any restrictions on the choice of away from , the solution procedure is simplified greatly if is an _velocity field ; i.e. if this first - order partial differential equation can be solved for , at all points , using ( [ eq : fnvn ] ) as a boundary condition . a simpler alternative strategy for constructing the extensional fieldis discussed in section [ sec : nume_leve_velo ] below .the level - set function , , can be initialized as the signed distance from as follows : .\ ] ] it is noted that this initial condition satisfies ( [ eq : lscondi ] ) .it also implies that initially , importantly , this desirable mathematical property of the level - set field is preserved when the velocity field is extensional .this can be shown ( see  ) by noting that combining ( [ eq : proof1 ] ) and ( [ eq : lscont3 ] ) and assuming that and are smooth , we obtain thus , if is extensional ( i.e.  ) and is initially a signed - distance function ( i.e.  and ) , we have which implies that ( [ eq : lsncond ] ) holds for all . hence , eq . ( [ eq : lscont3 ] ) reduces to which governs the evolution of from the initial condition ( [ eq : lsinit ] ) .recall that the mechanics , mass - transport and electric - flow problems are solved using the finite element method , and that part of the coupling between these three sub - problems , on one hand , and the grain - boundary migration sub - problem , on the other , is through the indicator function , . in this setting , the level - set update is computed at each node via a generalized trapezoidal rule : where is the position vector of node and , with .it is noted that when explicit time integration is used ( ) , the update operation is trivial since , and hence , are determined from the known solution at and do not depend on .it is also noted that no stabilization is required due to the use of the simplified level - set equation ( [ eq : lscont4 ] ) in lieu of ( [ eq : lscont3 ] ) .the advantages as well as the performance and stability characteristics of the resulting level - set formulation are studied in detail by .  additionally , is defined more precisely as follows : where is the heaviside function .[ br]   the approach adopted here for constructing the extensional level - set propagation velocity field , , is based on the notion that for any , if is such that it is clear that the resulting velocity field satisfies ( [ eq : lsext ] ) when as depicted in fig .[ fig : n_lsm3 ] .this is not always the case however ; for instance , if ( see fig . [ fig : n_lsm3 ] ) , then is such that it follows that , for all , , i.e.  , which clearly satisfies ( [ eq : lsext ] ) also .the above arguments apply in as well .  due to the accumulation of numerical error, the level - set field may develop perturbations ; i.e.  may deviate from unity in some regions within .the field must be re - initialized to neutralize these perturbations and retain accuracy by maintaining ( see eqs .( [ eq : proof1][eq : nochange ] ) ) .box [ box : lsalgorithm ] shows the algorithm used to implement the velocity projection scheme outlined above and the re - initialization scheme based on eq .( [ eq : lsinit ] ) .it is noted that , to preserve the integrity of the solution , the re - initialization procedure should not change the current location of . in other words, the re - initialized field should have the same zero level set as the original ( perturbed ) field .the re - initialization scheme described here does not satisfy this condition strictly , i.e. it introduces error into the solution .re - initialization should therefore be used judiciously .this issue is examined in detail by . an alternative re - initialization procedure designed to minimize this type of erroris described in detail in  .it must be noted however that , in some cases including curvature - driven migration , quadratic convergence in is achieved with the level - set update formula ( [ eq : lscont4 ] ) , and importantly , this optimal convergence rate is preserved by the present re - initialization scheme ( see  for details ) .the algorithm in box [ box : lsalgorithm ] was first introduced by and was previously used by garikipati and rao  .it is adopted here for its simplicity , despite being relatively expensive the complexity of the present algorithm is at best , where is the number of elements intersected by the zero level set and is the total number of nodal points in the mesh ( see  ) .a more efficient algorithm , such as the fast marching method  with complexity , could be employed for velocity projection and level - set field re - initialization on cartesian grids .the local migration velocity , , on the grain boundary is dependent on the local curvature , , which is defined as follows : from ( [ eq : lsnorm ] ) and ( [ eq : lsncond ] ) , it is clear that , and the curvature can hence be expressed as where is the number of spatial dimensions . since the value of is updated at the finite element nodes , it is convenient to use the shape functions to evaluate the spatial derivatives in the above expression . however , since _shape functions are used for simplicity and robustness , eq .( [ eq : n_ls_kappa1 ] ) can not be used to evaluate directly . to overcome this difficulty, we introduce a ` smoothed ' normal vector field , weakly related to by where is an arbitrary weighting function .equivalently , the nodal values of each component , , of the smoothed normal vector can be obtained by minimizing the following discretized functional : ^{2}\,{\mathrm{dv}},\ ] ] where is the number of elements in the model , denotes an element domain , is the number of nodes per element and is the shape function associated with node .this leads to a matrix equation of the form , where is the global vector containing the nodal values of the component .the global mass matrix , , and right - hand side vector , , are obtained from the corresponding element arrays via the usual assembly process .the element arrays in this case are given by  [ eq : n_lssmoo_emat ]   finally , the curvature is evaluated as follows :   * remark * using this least - squares technique to smooth the stress field leads to a mixed problem in stress and displacement form ( see  ) . by analogy, using this approach to compute the curvature is formally equivalent to a two - field mixed formulation for and .* remark * similar techniques have been used previously to evaluate the curvature of an evolving interface ; e.g. see chessa and belytschko  .the coupled problem is solved using an operator - split algorithm .the sequence of operations carried out in one time step is shown in box [ box : staggered ] to illustrate how the level - set formulation is incorporated into this solution scheme ..material properties of al ( thin film ) used in the analysis [ cols="<,^,^",options="header " , ]   numerical results , obtained by solving the coupled initial and boundary value problem , are presented here with the aim of highlighting some of the advantages of the current approach .a wide , long segment of an aluminum interconnect line is modeled .the values of the material parameters used for al are given in table [ tab : v_mattab ] .the segment consists of two pure al crystals separated by a tilt grain boundary ( misorientation about ) .the line is assumed to operate at , with a reference temperature , .rigid passivation material surrounding the line prevents vacancies from crossing the upper and lower boundaries of the domain ( see fig . [fig : m_vacdist]a ) ; i.e. the condition holds at these boundaries .periodic boundary conditions are imposed on the vacancy concentration at the left and right boundaries of the segment and an electrostatic potential difference , , is applied between these two extremities .vacancies drift along the electric field , , pointing to the right .  in the first example , the migrating grain boundary consists , initially , of two straight ( planar ) sections , which are joined by a circular ( cylindrical ) section .the straight sections form angles with the upper and lower boundaries of the domain as shown in fig . [fig : m_vacdist]a .since the vacancy formation energy , , is low inside grain - boundary regions where vacancy sources are also present , vacancies accumulate in such regions resulting in high values of the local vacancy concentration , .the location of the grain boundary is revealed locally by the maximum - valued contour of .[ ] vacancy concentration [ ] .( b )  .( c ) .,title="fig:",width=432 ] 0.4 in .( b )  .( c ) .,title="fig:",width=432 ] 0.4 in .( b )  .( c ) .,title="fig:",width=432 ] ( -12.4,14.12)(a ) ( -12.4,8.27)(b ) ( -12.4,2.40)(c )  figure [ fig : m_vacdist ] shows the evolution of the vacancy concentration contours in the line .it is clear that , during the first seconds , the central cylindrical section of the boundary migrates to the right ; i.e. toward its center of curvature , while the planar sections are relatively less mobile ( fig .[ fig : m_vacdist]b ) .it must be emphasized that the interaction between mechanics , mass transport , electric effects and grain - boundary motion is accounted for , and the driving force for boundary migration , due to stress - driven diffusion and electromigration , is included in the calculations .however , its effect is overshadowed by the dominant driving force due to the curvature of the boundary .[ ] flux magnitude [ ] , in the line ( first example).,title="fig:",width=432 ]  , in the neighborhood of the grain boundary at ( first example ) .also see fig .[ fig : m_fluxdist ] for magnitude of .,width=307 ]  [ ] vacancy concentration [ ] .( b )  .( c ) .,title="fig:",width=432 ] 0.4 in .( b )  .( c ) .,title="fig:",width=432 ] 0.4 in .( b )  .( c ) .,title="fig:",width=432 ] ( -12.4,14.12)(a ) ( -12.4,8.27)(b ) ( -12.4,2.40)(c )  after seconds , the curvature is approximately the same everywhere on the grain boundary ( fig .[ fig : m_vacdist]c ) . at this stage , a steady state prevails and the boundary continues to travel toward the right without undergoing any further changes in shape .it is noted that the level - set calculations remain stable .the boundary remains smooth and does not develop any spurious cusps or ripples .the equilibrium angles between the grain boundary and the sidewalls are maintained as the solution progresses .  a contour plot of the magnitude of the vacancy flux at is shown in fig .[ fig : m_fluxdist ] .it is clear that a strong vacancy flux exists in the vicinity of the grain boundary . a vector plot of the vacancy - flux field in the neighborhood of the grain boundary is shown in fig .[ fig : m_vecflux ] and although the field is complicated in this neighborhood , it can be seen that vacancies tend to drift in the same direction as the migrating boundary , thus preventing the appearance of a ` trail ' of vacancies or other oscillations in the numerical solution in the boundary s wake .it is important to note that the flux field evolves continually as the boundary migrates .it is also notable that the formulation captured this aspect of the coupling between mass transport and grain - boundary migration without additional terms being added to the expression of the vacancy flux to account , specifically , for the effect of moving boundaries .the second example is concerned with the case where a curved grain boundary evolves into a planar configuration to reduce the free energy of the system . here , the equilibrium angles at the grain boundary - sidewall intersections are set to and the initial geometry of the boundary is different and less regular than in the first example .it is clear from fig .[ fig : m_vacdist2 ] , which shows the evolution of the vacancy concentration contours in this case , that the boundary flattens as the solution progresses .it is noted that the equilibrium angles are also preserved in this case .it is also noted that in this case , perturbations in the solution lead to the formation of spurious ` shadow ' regions ( , ; see fig .[ fig : n_lsm3 ] ) and re - initialization becomes necessary in these regions to maintain accuracy and to preserve the contact angles .although the numerical stability characteristics of the level - set formulation and those of the operator - split algorithm are not examined in a formal setting , no spatial or temporal oscillations are observed in the numerical solution of the example problems presented , as long as the time - step size is within the cfl limit ; i.e.  , where is the mesh parameter . a detailed study of the performance and stability characteristics of the present level - set formulation is presented by .details regarding the advantages , applications and numerical stability characteristics of operator - split schemes can be found in  and references therein .in this paper , we present the following contributions .  * _ _ .the coupled continuum formulation presented here was developed to model stress - driven self - diffusion and electromigration in polycrystals while accounting fully for the interaction between these mass transfer processes and the motion of grain boundaries .the formulation accounts for two distinct thermodynamic driving forces acting on a grain boundary ; one due to the boundary s own curvature and another engendered by mass transfer across the boundary via stress - driven self - diffusion and electromigration . +the level set method has _previously _ been used to pose grain - boundary migration as a time - dependent field problem governed by a pure advection equation .standard numerical schemes resort to spatial stabilization techniques ( e.g. upwinding schemes , galerkin / least - squares ) to attenuate the spurious oscillations known to appear in the numerical solution of equations of this type . here , on the other hand , the level - set equation is reduced , using the mathematical properties of signed distance functions and extensional velocity fields , to a simpler form which obviates the need for these stabilization techniques .this leads to a remarkably simple explicit scheme for advancing the solution in time .the algorithm used to construct the extensional velocity field and to simultaneously re - initialize the level - set field is presented .we also provide the description of an -projection technique used to compute the curvature of the grain boundary .the numerical examples presented indicate that the strong coupling in the problem is captured adequately and that the numerical implementation allows the solution of the coupled initial and boundary value problem to be advanced in time in a stable fashion to obtain physically meaningful results .we would like to acknowledge useful discussions with of the university of michigan .support of this work by the national science foundation ( grant cms0075989 ) is gratefully acknowledged .k. garikipati , l. c. bassman , and m. d. deal , `` a lattice - based micromechanical continuum formulation for stress - driven mass transport in polycrystalline solids , ''_ j. mech .phys.olids_ , vol .49 , no . 6 , pp .12091237 , 2001 .k. garikipati and l. c. bassman , `` atomically - based field formulations for coupled problems of composition and mechanics , '' in _ mrs symposium proceedings , multiscale modeling of materials2000 _ ( l. p. kubin , j. l. bassani , k. cho , h. gao , and r. l. b. selinger , eds . ) , vol .653 , ( warrendale ) , pp . z9.6.1z9.6.6 , materials research society , 2001 .a. f. bower and l. b. freund , `` finite element analysis of electromigration and stress induced diffusion in deformable solids , '' in _ mrs symposium proceedings , materials reliability in microelectronics v _( a. s. oates , k. gadepally , r. rosenberg , w. f. filter , and a. l. greer , eds . ) , vol .391 , ( pittsburgh ) , pp . 177188 , materials research society , 1995 .l. xia , a. f. bower , z. suo , and c. f. shih , `` a finite element analysis of the motion and evolution of voids due to strain and electromigration induced surface diffusion , '' _j. mech .phys.olids_ , vol .45 , no . 9 , pp . 14731493 , 1997 .m. upmanyu , d. j. srolovitz , l. s. shvindlerman , and g. gottstein , `` misorientation dependence of intrinsic grain boundary mobility : simulation and experiment , '' _ acta mater ._ , vol .47 , no .14 , pp . 39013914 , 1999 .m. sussman and e. fatemi , `` an efficient , interface - preserving level set redistancing algorithm and its application to interfacial incompressible flow , '' _ siam j. sci ._ , vol . 20 , no . 4 , pp .11651191 , 1999 .j. chessa and t. belytschko , `` an enriched finite element method and level sets for axisymmetric two - phase flow with surface tension , '' _ int .j. numer .methods engrg ._ , vol .58 , no . 13 , pp . 20412064 , 2003 .f. armero and j. c. simo , `` a priori stability estimates and unconditionally stable product formula algorithms for nonlinear coupled thermoplasticity , '' _ int .j. plast ._ , vol . 9 , no . 6 , pp .749782 , 1993 .

Abstract: this work is based upon a coupled , lattice - based continuum formulation that was previously applied to problems involving strong coupling between mechanics and mass transport ; e.g. diffusional creep and electromigration  . here we discuss an enhancement of this formulation to account for migrating grain boundaries . the level set method is used to model grain - boundary migration in an eulerian framework where a grain boundary is represented as the zero level set of an evolving higher - dimensional function . this approach can easily be generalized to model other problems involving migrating interfaces ; e.g. void evolution and free - surface morphology evolution . the level - set equation is recast in a remarkably simple form which obviates the need for spatial stabilization techniques . this simplified level - set formulation makes use of velocity extension and field re - initialization techniques . in addition , a least - squares smoothing technique is used to compute the local curvature of a grain boundary directly from the level - set field without resorting to higher - order interpolation . a notable feature is that the coupling between mass transport , mechanics and grain - boundary migration is fully accounted for . the complexities associated with this coupling are highlighted and the operator - split algorithm used to solve the coupled equations is described .   _ key words _ : grain - boundary migration , electromigration , stress - mediated self - diffusion , level set method , finite element method  captionsize overcaptionskip8@ 8@ plus 4@ minus 2@  [2 ]  tempboxa tempboxa > captionsize # 1 : # 2  to   fs fs  makefntext[1 ] 1em .0exmakefnmark#1


the natural flux of people s changing social ties and interests generates a dynamic social network .this invisible network can be observed by capturing daily or weekly snapshots of user activities on social media platforms and massively multiplayer online games ( mmogs ) .it is informative to study changes in the network at the _ community level _ , as well as the individual level .communities are emergent groups that are created as people form highly connected subnetworks with their families , co - workers , and friends . often communities are formed by participants with the same goals , interests , or a geographic location .for instance , in mmogs , network communities may emerge from guilds of players with common economic interests or alliances who share strategic goals . as the network changes , user groups can grow , shrink , or disappear , causing drastic changes in the total number of network communities .community detection can help us understand the hidden social structure of the user populations , but the dynamic aspect of networks can pose problems for standard algorithms. our method uses stochastic optimization to find the best community structure , assuming that the nodes are modeled as rational players who seek to maximize their personal utility while playing a _ community membership game _ with neighboring nodes . in this game , the active agent decides to join or leave different communities ; agents receive benefits from being part of the same community as their network neighbors but are penalized for joining too many communities .the nash equilibrium of the current game corresponds to the community structure of the current snapshot . as the network evolves, agents usually find it advantageous to modify their community membership strategy .this article examines the performance of varying the amount of information propagated from prior snapshots .even in dynamic networks , there are many nodes that retain the same community membership or rejoin their former communities . thus propagating information from previous snapshotscan provide more favorable initialization conditions for the stochastic optimization procedure .much of the power of the d - gt framework ( originally introduced in ) lies in its potential for customization ; however , without guidance end users can be overwhelmed by myriad choices .our aim in this article is to present a comprehensive analysis of the algorithm s performance in common scenarios ; while retaining the same basic game - theoretic model , we evaluate different variations of our procedure .first , we examine the performance of different utility functions , a similarity - based utility function  vs. the use of a personalized modularity function  .then we compare different initialization approaches in which the following information is propagated : 1 ) no information ( * d - gts * ) ; 2 ) the union of the community membership information over all snapshots ( * d - gt * ) ; 3 ) community membership information from the previous snapshot ( * d - gtp * ) ; 4 ) ground truth information for a small seed set ( * d - gtg * ) . our game - theoretic model is robust to minor changes in the procedure and most variants outperform the benchmarks .our experiments were conducted on networks created from different dynamic processes : internet routers ( as - oregon graph , the as - internet routers graph ) , shifting organization structure ( enron email dataset ) , citation graphs from arxiv ( hep - ph ) , and player interactions in massively multiplayer online games ( travian messages , travian trades ) .results were compared against five other methods : labelrankt , ilcd , oslom , infomap and louvain in terms of normalized mutual information ( nmi ) , modularity , and the number of detected communities .the next section presents an over - view of related work on community detection in dynamic networks .section [ sec : method ] describes our problem formulation and our proposed method ( * * d**ynamic * * g**ame * * t**heoretic community detection ) . experimental results are provided in section [ sec : results ] , before we conclude the article ( section [ sec : conclusion ] ) .the problem of community detection has been widely studied in the literature compared to the other research areas in the social networks  . in particular , in static networks, community detection has appeared in multiple disciplines including sociology and computer science .this has yielded a diverse set of approaches ranging from traditional network structure based algorithms  , optimization techniques  , label propagation  , propinquity  and information diffusion  . detecting community structure in dynamic networks , on the other hand ,has attracted less research attention due to the complexity of the problem and dearth of good datasets .there are some community detection algorithms originally design - ed for static networks that continue to perform well in dynamic datasets .for instance , lancichinetti et al.s oslom ( order statistics local optimization method ) works on single snapshots but also benefits from information from previous network partitions . like d - gt , oslom s optimization procedure can be initialized with the partition from the previous snapshot ; it aims to optimize cluster significance with respect to a global null model  .we use this method as one of our benchmarks , along with two other static community detection algorithms , louvain and infomap  .these algorithms perform well on many static community detection problems and have the benefit of being fast to compute on a single network snapshot .other network properties have also been used to perform dynamic community detection ; for instance hui et al . proposed a distributed method for community detection in which modularity was used as a measure instead of the objective function .qca ( quick community adaptation ) is a modularity - based approach that focuses explicitly on the changes in the network structure , rather than recomputing community structure from scratch at each time step  . in this article ,we evaluate the use of personal modularity as an alternative gain function to neighborhood similarity .some studies have focused on studying the evolution of communities over time .for instance , identified subsets of nodes , natural communities " , that were stable to small perturbations of the input data .communities detected in later snapshots were matched to earlier snapshots using the natural community tree structure .palla et al . proposed an innovative method for detecting communities in dynamic networks based on the k - clique percolation technique ; in their approach , communities are defined as adjacent k - cliques , that share nodes .  machine learning has also been employed to model changes in community structure ; for instance , predict transitions in community structure by learning supervised machine learning classifiers .this requires data on past transitions to train the classifiers , which limits its applicability to certain datasets . a data mining approach to detect clusters on time - evolving graphs ; community discovery and change detection are performed using the minimum description length ( mdl ) paradigm .  rather than independently detecting communities at each snapshot and matching them , another option is to make a local decision to add nodes to existing communities when new edges appear in the network .one of our benchmarks , ilcd ( intrinsic longitudinal community detection )  , updates the community structure of the network based on time - stamped sets of edges .nodes are added to communities if its mean number of second neighbors and robust second neighbors exceeds the current average for the community . however ,this model is limited to certain types of network changes , and can not handle interconnected pairs of nodes being simultaneously added or the removal of edges .optimization can be used to identify minimum cost community assignments in dynamic graphs .facetnet is a framework for analyzing communities in dynamic networks based on an optimization of snapshot costs .it is guaranteed to converge to a local optimal solution ; however , its convergence speed is slow , and it needs to be initialized with the number of communities which is usually unknown in practice . modeled dynamic community detection as a multi - objective optimization problem .their approach is parameter free and uses evolutionary clustering to optimize a dual objective function .the first objective selects for highly modular structures at the current time step , and the second minimizes the differences between community structures in the current and previous time steps .d - gt also uses a stochastic optimization procedure , but all of the agents individually optimize their utilities based on local network information .the markov cluster algorithm ( mcl )  identifies graph clusters by computing the probabilities of random walks through the graph ; flow simulations are performed by alternating expansion ( matrix squaring ) with inflation operations ( hadamard powers ) . due to its mathematical simplicity ,it is popular for community detection in many domains but is slow and often overestimates the number of communities in the dataset .regularized - mcl  is a variant of mcl that prevents overfitting by taking into account neighbor flows .it can also be used within a multi - level framework ( multi - level regularized mcl ) to speed up computation by executing a sequence of coarsening operations on the graph before executing r - mcl .labelrankt  , shares some similarities with the mcl techniques described above while improving upon them in several ways ; it is a label propagation approach in which the inflation operation is applied to the label distribution matrix rather than to the adjacency matrix .each node requires only local information during propagation making it more scalable than mcl and amenable to parallelization .due to labelrankt s strong performance and good implementation , it was selected as the best label propagation benchmark for our work .  our proposed method ( d - gt ) attempts to simulate the decision - making process of the individuals creating the communities , rather than focusing on statistical correlations between labels of neighboring nodes .we believe that exploiting game theory for dynamic community detection yields more realistic , fine - grained communities since intrinsically game theory is a good representation for expressing the behavior of individuals and strategic interactions among them  .  in previous work, we have demonstrated the success of game - theoretic approaches in static community detection across several domains , including detecting guilds in massively multiplayer online games  and predicting trust between users on e - commerce sites .many of these domains featured overlapping communities in static networks  ; however in this article the datasets are dynamic , but not overlapping .d - gt makes a hard assignment at the termination of the stochastic optimization procedure by selecting the community assignment with the highest utility function . in this article, we investigate the performance of different gain functions and initialization procedures , on a variety of evaluation metrics ( modularity , nmi , number of detected communities ) .the strength of the d - gt framework is its versatility ; our results show that substantial performance improvements can be achieved by customizing it for the problem at hand .first , we provide the formal definition of the problem and the notations used throughout the paper . given snapshots of a dynamic network and their corresponding underlying graphs , with vertices and edges , where t=1, ... ,m , we aim to detect community structure of the network .table [ tb : symb ] shows the symbols and definitions used throughout the paper .we leverage a dynamic agent - based model to detect communities by optimizing each user s utility through a stochastic search process  . in this article, we evaluate four different variants of the procedure : d - gt ( * * d**ynamic * * g**ame * * t**heoretic community detection ) , d - gtp ( * d - gt * with passing one * * p**revious snapshot ) d - gts ( * d - gt * with * * s**eparate runs ) and d - gtg ( * d - gt * with passing * * g**round truth ) .[ tb : symb ]  .definition of symbols . [ cols="<,<",options="header " , ]   table [ tb : runtime ] shows the running time of all algorithms on our largest datasets , the as - internet dataset ( with the highest number of snapshots ) and hep - ph ( with the highest number of nodes ) .overall , the running times provided by all algorithms were quite reasonable , as there are 733 snapshots in the internet dataset and over 10,000 and 30,000 nodes in oregon and hep - ph respectively .infomap is the fastest algorithm on two datasets ( internet and hep - ph ) and performs almost as well as oslom on third one ( oregon ) .all of the algorithms are sufficiently fast to run on large datasets .this article analyzes the performance of our game theoretic community detection algorithm , d - gt , on dynamic social networks .these social networks are very common in massively multiplayer online games , such as travian , where players self - organize into rapidly changing guilds and alliances .we show that d - gt s initialization procedure in combination with the similarity gain function is very effective at recovering the true community structure of the network , both in terms of predicting the number of communities and the players community membership vectors .it outperforms other dynamic community detection methods including labelrankt , ilcd , and oslom . in cases where the communities of a small number of players ( e.g. the guild leadership )is known d - gt can leverage the information to improve the nmi performance .when simply optimizing modularity , considering earlier community membership is less important and static community detection algorithms ( infomap and louvain ) perform well at this task , but all variants of d - gt offer competitive performance .one nice aspect of d - gt is that it is an easily extensible framework . in future workwe plan to experiment with other utility functions , loss functions , and update rules ; for instance , q - learning could be used as the update rule for the agents instead of the community membership game .we also believe that it is feasible to reduce the runtime of d - gt by creating approximate versions of the gain function that can be calculated based on local edge updates .we would like to thank drs .nitin agarwal and rolf t. wigand at university of arkansas for providing the travian dataset .

Abstract: most real - world social networks are inherently dynamic , composed of communities that are constantly changing in membership . to track these evolving communities , we need dynamic community detection techniques . this article evaluates the performance of a set of game theoretic approaches for identifying communities in dynamic networks . our method , d - gt ( * * d**ynamic * * g**ame * * t**heoretic community detection ) , models each network node as a rational agent who periodically plays a community membership game with its neighbors . during game play , nodes seek to maximize their local utility by joining or leaving the communities of network neighbors . the community structure emerges after the game reaches a nash equilibrium . compared to the benchmark community detection methods , d - gt more accurately predicts the number of communities and finds community assignments with a higher normalized mutual information , while retaining a good modularity .   example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore