a cellular automaton finite volume method for the simulation of dendritic and eutectic growth in binary alloys using an adaptive mesh refinement abstract a two dimensional model to simulate the dendritic and eutectic growth in binary alloys is developed a cellular automaton method is adopted to track the movement of the solid liquid interface the diffusion equation is solved in the solid and liquid phases by using an explicit finite volume method the computational domain is divided into square cells that can be hierarchically refined or coarsened using an adaptive mesh based on the quadtree algorithm such a mesh refines the regions of the domain near the solid liquid interface where the highest concentration gradients are observed in the regions where the lowest concentration gradients are observed the cells are coarsened the originality of the work is in the novel adaptive approach to the efficient and accurate solution of the posed multiscale problem the model is verified and assessed by comparison with the analytical results of the lipton glicksman kurz model for the steady growth of a dendrite tip and the jackson hunt model for regular eutectic growth several examples of typical microstructures are simulated and the features of the method as well as further developments are discussed

a return mapping algorithm for isotropic elastoplasticity a return mapping algorithm is presented for the numerical time integration of the constitutive equations for elastoplasticity with isotropic yield surfaces constructed from all three invariants of the stress tensor based on the first order backward euler difference formula bdf the governing equations for the stresses are solved in the space of the invariants and the discretized persistence parameter the stresses are recovered afterwards the solution concept is applied to a pressure independent yield function expressed in terms of the second and third invariant of the stress tensor the numerical performance of the method is demonstrated with two examples

quantitative phase field modeling of dendritic growth in two and three dimensions we report the results of quantitative phase field simulations of the dendritic crystallization of a pure melt in two and three dimensions these simulations exploit a recently developed thin interface limit of the phase field model a karma and w j rappel phys rev e 53 r3017 1996 which is given here a detailed exposition this limit makes it possible to perform efficient computations with a smaller ratio of capillary length to interface thickness and with an arbitrary interface kinetic coefficient simulations in one and two dimensions are first carried out to test the accuracy of phase field computations performed within this limit dendrite tip velocities and tip shapes are found to be in excellent quantitative agreement with exact numerical benchmarks of solvability theory obtained by a boundary integral method both with and without interface kinetics simulations in three dimensions exploit in addition to the asymptotics a methodology to calculate grid corrections due to the surface tension and kinetic anisotropies they are used to test basic aspects of dendritic growth theory that pertain to the selection of the operating state of the tip and to the three dimensional morphology of needle crystals without sidebranches for small crystalline anisotropy simulated values of ensuremath sigma are slightly larger than solvability theory predictions computed by the boundary integral method assuming an axisymmetric shape and agree relatively well with experiments for succinonitrile given the uncertainty in the measured anisotropy in contrast for large anisotropy simulated ensuremath sigma values are significantly larger than the predicted values this disagreement however does not signal a breakdown of solvability theory it is consistent with the finding that the amplitude of the mathrm cos 4 ensuremath varphi mode which measures the departure of the tip morphology from a shape of revolution increases with anisotropy this departure can therefore influence the tip selection in a way that is not accurately captured by the axisymmetric approximation for large anisotropy finally the tail shape at a distance behind the tip that is large compared to the diffusion length is described by a linear law r ensuremath sim z with a slope dr dz that is nearly equal to the ratio of the two dimensional and three dimensional steady state tip velocities furthermore the evolution of the cross section of a three dimensional needle crystal with increasing distance behind the tip is nearly identical to the evolution of a two dimensional growth shape in time in accord with the current theory of the three dimensional needle crystal shape

a classification of the face numbers of buchsbaum simplicial posets the family of buchsbaum simplicial posets generalizes the family of simplicial cell manifolds the h vector of a simplicial complex or simplicial poset encodes the combinatorial and topological data of its face numbers and the reduced betti numbers of its geometric realization novik and swartz showed that the h vector of a buchsbaum simplicial poset satisfies certain simple inequalities in this paper we show that these necessary conditions are in fact sufficient to characterize the h vectors of buchsbaum simplicial posets with prescribed betti numbers

variational algorithms and pattern formation in dendritic solidification we present a completely new variational algorithm for computing dendritic solidification this algorithm reproduces the gibbs thomson relation as a balance between bulk and surface energy and is able to operate in the infinite mobility limit with no unphysical time step restriction it may be used with arbitrary non smooth surface energy functions and may include finite kinetic mobility we perform computations with isotropic and anisotropic surface energy from a small irregular initial seed we generate radial tip splitting structures for isotropic energy and parabolic dendrites with side branching for anisotropic energy for anisotropic energy the final structure is determined by the material and environmental properties the initial shape is forgotten for the parabolic dendrite tips we obtain agreement with the ivantsov solutions within a few percent and proper dimensional scaling of lengths and velocities with surface energy

algorithms for phase field computation of the dendritic operating state at large supercoolings this paper deals with numerical solutions of the phase field model of solidification in two dimensions in the context of dendritic growth finite difference methods associated with vectorized algorithms are employed to solve the phase field equations the temperature equation is solved by an alternating direction implicit scheme and the equation for the phase field by an explicit euler scheme we identify a region of parameter space namely large supercoolings where we can compute dendritic morphologies that display steady state dendrite tip operating conditions that are independent of computational parameters

on the implementation of inelastic constitutive equations with special reference to large deformation problems abstract methods for finite element analysis of inelastic problems are reviewed problems with currently popular methods based on rate principles of virtual work are pointed out a formulation is presented based on the exact incremental virtual work equation the procedure for integration of the constitutive rate equations to obtain incremental constitutive equations is based on the assumption of a constant strain rate or straight strain path during the increment full newton raphson iteration is used to solve the resulting set of nonlinear equations the treatment of the constitutive equations in this newton raphson scheme leads to a linearized form which is substantially different from the classical rate equations the procedure is implemented in a finite element program and its effectiveness is shown in a number of examples

phase field computations of single needle crystals crystal growth and motion by mean curvature the phase field model for free boundaries consists of a system of parabolic differential equations in which the variables represent a phase or order parameter and temperature respectively the parameters in the equations are related directly to the physical observables including the interfacial width epsilon which we can regard as a free parameter in computation the phase field equations can be used to compute a wide range of sharp interface problems including the classical stefan model its modification to incorporate surface tension and or surface kinetic terms the cahn allen motion by mean curvature the hele shaw model etc also included is anisotropy in the equilibrium and dynamical forms generally considered by materials scientists by adjusting the parameters the computations can be varied continuously from single needle dendritic to faceted crystals the computational method consists of smoothing a sharp interface problem within the scaling of distinguished limits of the phase field e

non smooth multisurface plasticity and viscoplasticity loading unloading conditions and numerical algorithms rate independent plasticity and viscoplasticity in which the boundary of the elastic domain is defined by an arbitrary number of yield surfaces intersecting in a non smooth fashion are considered in detail it is shown that the standard kuhn tucker optimality conditions lead to the only computationally useful characterization of plastic loading on the computational side an unconditionally convergent return mapping algorithm is developed which places no restrictions aside from convexity on the functional forms of the yield condition flow rule and hardening law the proposed general purpose procedure is amenable to exact linearization leading to a closed form expression of the so called consistent algorithmic tangent moduli for viscoplasticity a closed form algorithm is developed based on the rate independent solution the methodology is applied to structural elements in which the elastic domain possesses a non smooth boundary numerical simulations are presented that illustrate the excellent performance of the algorithm

a level set simulation of dendritic solidification with combined features of front tracking and fixed domain methods a method combining features of front tracking methods and fixed domain methods is presented to model dendritic solidification of pure materials to explicitly track the interface growth and shape of the solidifying crystals a front tracking approach based on the level set method is implemented to easily model the heat and momentum transport a fixed domain method is implemented assuming a diffused freezing front where the liquid fraction is defined in terms of the level set function the fixed domain approach by avoiding the explicit application of essential boundary conditions on the freezing front leads to an energy conserving methodology that is not sensitive to the mesh size to compute the freezing front morphology an extended stefan condition is considered applications to several classical stefan problems and two and three dimensional crystal growth of pure materials in an undercooled melt including the effects of melt flow are considered the computed results agree very well with available analytical solutions as well as with results obtained using front tracking techniques and the phase field method

consistent tangent operators for rate independent elastoplasticity abstract it is shown that for problems involving rate constitutive equations such as rate independent elastoplasticity the notion of consistency between the tangent stiffness operator and the integration algorithm employed in the solution of the incremental problem plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon newton s method within the framework of closest point projection algorithms a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed to wit associative j 2 flow rules with general nonlinear kinematic and isotropic hardening rules as well as a class of non associative flow rules are considered the resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of newton s method whereas use of the socalled elastoplastic tangent in conjunction with a radial return integration algorithm a procedure often employed results in newton type of algorithms with suboptimal rate of convergence application is made to a set of numerical examples which include saturation hardening laws of exponential type

h vectors of gorenstein simplicial posets abstract as is well known h vectors of simplicial convex polytopes are characterized those h vectors satisfy dehn sommerville equations and some inequalities conjectured by p mcmullen and first proved by r stanley using toric geometry the boundary of a simplicial convex polytope determines a gorenstein simplicial poset but there are many gorenstein simplicial posets which do not arise this way however it is known that h vectors of gorenstein simplicial posets still satisfy dehn sommerville equations and that every component in the h vectors is non negative in this paper we prove that h vectors of gorenstein simplicial posets must satisfy one more subtle condition conjectured by r stanley and complete the characterization of h vectors of gorenstein simplicial posets our proof is purely algebraic but the idea of the proof stems from topology

second order phase field asymptotics for unequal conductivities we extend karma and rappel s improved asymptotic analysis of the phase field model to different diffusivities in solid and liquid we consider both second order classical asymptotics in which the interface thickness is taken much smaller than the capillary length and the new isothermal asymptotics in which the two lengths are considered comparable in the first case if the phase field model is required to be gradient flow for an entropy functional then for unequal diffusivities it is impossible to construct a phase equation with finite kinetics which converges with second order accuracy to a gibbs thomson equilibrium condition with infinitely fast kinetics in the second case some error terms are pushed to higher orders and it is easy to eliminate the remaining errors with finite phase kinetics

the phase field method in the sharp interface limit the phase field pf method for solidification phenomena is an open formulation based on a free energy functional two common choices for the pf potential here referred to briefly as the caginalp and kobayashi models are compared with respect to their numeric results within the classical sharp interface limit both qualitative and quantitative behavior are addressed and an assessment of the computational effort required to approximate a sharp interface problem is made it is shown that the specific form of the free energy potential does have a strong influence on the convergence of the pf results to their sharp interface limit compliance of the pf solutions with the linear kinetic model for the interface temperature is also investigated a simple one dimensional solidification problem in the presence of kinetic undercooling is solved by the pf model and also by a deforming grid method our results support the view that if care is exercised in formulating the phase temperature coupling there is a high degree of confidence in using the pf method for the numerical modeling of general solidification phenomena

an analysis of a phase field model of a free boundary a mathematical analysis of a new approach to solidification problems is presented a free boundary arising from a phase transition is assumed to have finite thickness the physics leads to a system of nonlinear parabolic differential equations existence and regularity of solutions are proved invariant regions of the solution space lead to physical interpretations of the interface a rigorous asymptotic analysis leads to the gibbs thompson condition which relates the temperature at the interface to the surface tension and curvature

numerical methods for microstructural evolutions in laser additive manufacturing abstract different methods are developed for simulation of microstructural evolution in metals and alloys subject to laser additive manufacturing the monte carlo mc method is combined with a new proposed two scales strategy for simulation of the solidification and the subsequent solid state phase transformation in ti 6al 4v the cellular automaton ca method shows its higher efficiency in comparison with mc method moore neighbor type with energy barrier is recommended for the ca simulation of grain growth of al 6061 the phase field pf model shows that different temperature gradients lead to different columnar grains in different layers of ti 32wt nb the computed structures are related to experimental observations

a three invariant cap viscoplastic rate dependent damage model for cementitious materials with return mapping integration in haigh westergaard coordinate space abstract we present a new strain rate dependent elasto viscoplastic damage constitutive model for cementitious materials by incorporating duvaut lions viscoplasticity generalized to multi surface plasticity c f simo et al 1988b simo and hughes 1998 followed by strain rate dependent damage initiation and evolution under multiaxial loading to our previous inviscid elastoplastic damage model paliwal et al 2017 we also present a unique numerical scheme in which the integration algorithm for modeling elastoplastic response utilizes fully implicit backward euler method in which the governing system of non linear equations are solved in the space of invariants based on haigh westergaard coordinate system hardening parameter and discretized plastic multiplier it uses the framework by matzenmiller and taylor 1994 which is modified in this work this is particularly useful as the yield criteria flow rule and the non linear hardening utilize functions that are dependent on the pressure effective shear stress and lode angle components of the stress tensor to the best of authors knowledge such integration algorithm in haigh westergaard coordinate space has never been developed for lode angle dependent yield criteria and the flow rule and non linear hardening plasticity furthermore we also develop an accurate form for the algorithmic elastoplastic damage tangent modulus consistent with the proposed return mapping technique numerical simulations of uniaxial tension and compression cases demonstrate that both the local integration point algorithm as well as the global algorithm using the algorithmic elastoplastic damage tangent modulus exhibit quadratic convergence finally the predictive capability of the proposed model is demonstrated by virtue of several numerical examples under various loading conditions covering different stress paths and different dynamic strain rates these predictions are also compared against experimental results and experimentally observed features from tests on various concrete specimens and demonstrate excellent agreement

on the numerical integration of three invariant elastoplastic constitutive models abstract we investigate the performance of a numerical algorithm for the integration of isotropically hardening three invariant elastoplastic constitutive models with convex yield surfaces the algorithm is based on a spectral representation of stresses and strains for infinitesimal and finite deformation plasticity and a return mapping in principal stress directions smooth three invariant representations of the mohr coulomb model such as the lade duncan and matsuoka nakai models are implemented within the framework of the proposed algorithm among the specific features incorporated into the formulation are the hardening softening responses and the tapering of the yield surfaces toward the hydrostatic axis with increasing confining pressure isoerror maps are generated to study the local accuracy of the numerical integration algorithm finally a boundary value problem involving loading of a strip foundation on a soil is analyzed with and without finite deformation effects to investigate the performance of the integration algorithm in a full scale non linear finite element simulation

adaptive mesh refinement computation of solidification microstructures using dynamic data structures we study the evolution of solidification microstructures using a phase field model computed on an adaptive finite element grid we discuss the details of our algorithm and show that it greatly reduces the computational cost of solving the phase field model at low undercooling in particular we show that the computational complexity of solving any phase boundary problem scales with the interface arclength when using an adapting mesh moreover the use of dynamic data structures allows us to simulate system sizes corresponding to experimental conditions which would otherwise require lattices greater than 217 217elements we examine the convergence properties of our algorithm we also present two dimensional time dependent calculations of dendritic evolution with and without surface tension anisotropy we benchmark our results for dendritic growth with microscopic solvability theory finding them to be in good agreement with theory for high undercoolings at low undercooling however we obtain higher values of velocity than solvability theory at low undercooling where transients dominate in accord with a heuristic criterion which we derive

computation of three dimensional dendrites with finite elements starting from an initial seed crystal inside an undercooled liquid the solid phase begins to grow rapidly and develops unstable growth patterns some growth directions are preferred because of anisotropic parameters in the physical model this results in the development of dendrites the physical model includes the heat equation for both the liquid and solid phases the gibbs thomson law couples velocity and curvature of the interface and the temperature we describe a numerical method that enables us to compute dendritic growth of crystals in two and three space dimensions the method consists of two coupled finite element algorithms the first one solves the heat equation the other operates on a discretization of the free boundary and computes the evolution of this moving interface the two methods work with totally independent grids by using timedependent locally refined and coarsened adaptive meshes in both methods we are able to reach a spatial resolution necessary to compute dendritic growth in two and three space dimensions

torus graphs and simplicial posets for several important classes of manifolds acted on by the torus the information about the action can be encoded combinatorially by a regular n valent graph with vector labels on its edges which we refer to as the torus graph by analogy with the gkm graphs we introduce the notion of equivariant cohomology of a torus graph and show that it is isomorphic to the face ring of the associated simplicial poset this extends a series of previous results on the equivariant cohomology of torus manifolds as a primary combinatorial application we show that a simplicial poset is cohen macaulay if its face ring is cohen macaulay this completes the algebraic characterisation of cohen macaulay posets initiated by stanley we also study blow ups of torus graphs and manifolds from both the algebraic and the topological points of view