stable computation of differentiation matrices and scattered node stencils based on gaussian radial basis functions stable computation of differentiation matrices and scattered node stencils based on gaussian radial basis functions

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 front tracking method for multiphase free boundary problems a front tracking method for the solution of multiphase free boundary problems is presented based on the method of lines with invariant imbedding this method is sufficiently general to allow the solution of problems with three or more phases and problems in which phases appear and disappear a discussion of the method for the three phase stefan problem and for the moving contact problem as well as a proof of convergence for the latter is included in addition the method is demonstrated for a free boundary problem with a combination of stefan and ablation boundary conditions and for the two dimensional contact problem numerical results indicate that this front tracking method is fast enough to solve a wide range of problems on a personal computer

multiscale approximation and reproducing kernel hilbert space methods we consider reproducing kernels k omega times omega to mathbb r in multiscale series expansion form i e kernels of the form k left boldsymbol x boldsymbol y right sum ell in mathbb n lambda ell sum j in i ell phi ell j left boldsymbol x right phi ell j left boldsymbol y right with weights lambda ell and structurally simple basis functions left phi ell i right here we deal with basis functions such as polynomials or frame systems where for ell in mathbb n the index set i ell is finite or countable we derive relations between approximation properties of spaces based on basis functions phi ell j 1 leq ell leq l j in i ell and spaces spanned by translates of the kernel span k boldsymbol x 1 cdot dots k boldsymbol x n cdot with x n boldsymbol x 1 dots boldsymbol x n subset omega if the truncation index l is appropriately coupled to the discrete set x n an analysis of a numerically feasible approx

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

stable computation of multiquadric interpolants for all values of the shape parameter abstract spectrally accurate interpolation and approximation of derivatives used to be practical only on highly regular grids in very simple geometries since radial basis function rbf approxima ions permit this even for multivariate scattered data there has been much recent interest in practical algorithms to compute these approximations effectively several types of rbfs feature a free parameter e g c in the multiquadric mq case f r r 2 c 2 the limit of c increasingly flat basis functions has not received much attention because it leads to a severely ill conditioned problem we present here an algorithm which avoids this difficulty and which allows numerically stable computations of mq rbf interpolants for all parameter values we then find that the accuracy of the resulting approximations in some cases becomes orders of magnitude higher than was the case within the previously available parameter range our new method provides the first tool for the numerical exploration of mq rbf interpolants in the limit of c the method is in no way specific to mq basis functions and can without any change be applied to many other cases as well

crystal growth and dendritic solidification we present a numerical method which computes the motion of complex solid liquid boundaries in crystal growth the model we solve includes physical effects such as crystalline anisotropy surface tension molecular kinetics and undercooling the method is based on two ideas first the equations of motion are recast as a single history dependent boundary integral equation on the solid liquid boundary a fast algorithm is used to solve the integral equation efficiently second the boundary is moved by solving a hamiltonjacobi type equation on a fixed domain formulated by osher and sethian for a function in which the boundary is a particular level set this equation is solved by finite difference schemes borrowed from the technology of hyperbolic conservation laws the two ideas are combined by constructing a smooth extension of the normal velocity off the moving boundary in a way suggested by the physics of the problem our numerical experiments show the evolution of complex crystalline shapes development of large spikes and corners dendrite formation and side branching and pieces of solid merging and breaking off freely

a fixed grid finite element technique for modelling phase change in steady state conduction advection problems a new fixed grid finite element technique for modelling phase change effects in steady state conduction advection problems is described the method is appropriate for materials that transform at a fixed critical temperature and is based on the assumption that the enthalpy experiences a jump discontinuity at this temperature yielding a singular specific heat this singularity is described using a dirac delta function and the appropriate finite element matrix equations are developed for isoparametric elements a simple first order upwinding scheme is used to avoid numerical oscillations in advection dominated problems the utility of the method is illustrated through the solution of a one dimensional test problem using eight node hexahedral isoparametric finite elements the technique may be applied to a wide range of finite element types and offers a number of advantages over other schemes for modelling phase change effects

comparison of finite element techniques for solidification problems the accuracies of the computed temperatures of a liquid in a corner region under freezing conditions are compared for various fixed grid finite element techniques using the analytical solution for this problem as a reference in the finite element formulation of the problem different time stepping schemes are compared the implicit euler backward algorithm combined with an iterative scheme and two three time level methods the lees algorithm and a dupont algorithm which are both applied as non iterative schemes furthermore different methods for handling the evolution of latent heat are examined an approximation method suggested by lemmon and one suggested by del giudice both using the enthalpy formulation as well as a fictitious heat flow method presented by rolph and bathe results of calculations performed with the consistent heat capacity matrix are compared with those performed with a lumped heat capacity matrix

a numerical method to compute the dissolution of second phases in ternary alloys abstract dissolution of stoichiometric multi component particles in ternary alloys is an important process occurring during the heat treatment of as cast aluminium alloys prior to hot extrusion a mathematical model is proposed to describe such a process in this model an equation is given to determine the position of the particle interface in time using two diffusion equations which are coupled by nonlinear boundary conditions at the interface some results concerning existence uniqueness and monotonicity are given furthermore for an unbounded domain an analytical approximation is derived the main part of this work is the development of a numerical solution method finite differences are used on a grid which changes in time the discretization of the boundary conditions is important to obtain an accurate solution the resulting nonlinear algebraic system is solved by the newton raphson method numerical experiments illustrate the accuracy of the numerical method the numerical solution is compared with the analytical approximation

monopole charges for arbitrary compact gauge groups and higgs fields in any representation the topological invariants of monopoles are described for an arbitrary compact gauge groupg and higgs field in any representation the results generalize those obtained recently for compact and simply connectedg and in the adjoint representation the cases when the residual symmetry group ish u 1 orh u 3 are worked out explicitly this latter is needed to accommodate fractional electric charge with monopoles having one dirac unit magnetic charge

on the stability of viscous free surface flow supported by a rotating cylinder using an adaptive finite element fe scheme developed recently by the authors we shed new light on the long standing fundamental problem of the unsteady free surface stokes flow exterior to a circular cylinder rotating about its horizontal axis in a vertical gravitational field for supportable loads we observe that the steadystate is more readily attained for near maximal fluid loads on the cylinder than for significantly sub maximal loads for the latter we investigate large time dynamics by means of a finite difference approximation to the thin film equations which is also used to validate the adaptive fe simulations applied to the full stokes equations for these significantly sub maximal loads conversely by comparing results of the two methods we assess the validity of the thin film approximation as either the load is increased or the rotation rate of the cylinder is decreased results are presented on the independent effects of gravity surface tension and initial film thickness on the decay to steady state finally new numerical simulations of load shedding are presented

topological charges in monopole theories let us consider a monopole theory with a compact simply connected gauge group and the higgs field in the adjoint representation using root theory we show that

highly efficient oscillation free solution of the transport equation over long times and large spaces transport problems with sharp transitions or high convection often stress even the most advanced computational methods beyond feasibility limits one promising line of attack involves transforming problems to a moving deforming coordinate system mdcs using mdcs as recently developed one may concentrate computational attention when and where it is most appropriate also possibly greatly reducing the effective magnitude of terms which cause trouble in principle mdcs may be used with finite difference finite element or other numerical methods according to preference in the particular finite element system advanced here mesh movement may be variable in time and space so that physically fixed boundaries may remain fixed in the transformed coordinate system a simple finite difference system is used in time when the method is applied to the standard transport equation test case great efficiency and accuracy are obtained in early time even very steep fronts may be represented well with no oscillations using space and time step sizes well beyond conventional peclet courant number constraints more important problems are solved involving transport over much longer times and larger spaces when diffusion has some chance to operate and hence a parabolic model is more warranted the same accuracy and stability are achieved as during early time using no more nodes and time steps overall than during early time runs this constitutes orders of magnitude savings in effort over what the application of conventional constraints would require relative efficiency from application of mdcs to other problems would depend on the particulars of each case

space time finite element techniques for computation of fluid structure interactions we describe the space time finite element techniques we developed for computation of fluid structure interaction fsi problems among these techniques are the deforming spatial domain stabilized space time dsd sst formulation and its special version and the mesh update methods including the solid extension mesh moving technique semmt also among these techniques are the block iterative quasi direct and direct coupling methods for the solution of the fully discretized coupled fluid and structural mechanics equations we present some test computations for the mesh moving techniques described we also present numerical examples where the fluid is governed by the navier stokes equations of incompressible flows and the structure is governed by the membrane and cable equations overall we demonstrate that the techniques we have developed have increased the scope and accuracy of the methods used in computation of fsi problems 2005 elsevier b v all rights reserved

mesh independent spectra in the moving finite element equations we derive the moving finite element mfe equations for the solution of a scalar evolutionary equation in d space dimensions d geqq 1 and introduce the elementwise approach to mfe this approa

boundary integral equation solution of moving boundary phase change problems boundary integral equation methods are presented for the solution of some two dimensional phase change problems convection may enter through boundary conditions but cannot be considered within phase boundaries a general formulation based on space time green s functions is developed using the complete heat equation followed by a simpler formulation using the laplace equation the latter is pursued and applied in detail an elementary noniterative system is constructed featuring linear interpolation over elements on a polygonal boundary nodal values of the temperature gradient normal to a phase change boundary are produced directly in the numerical solution the system performs well against basic analytical solutions using these values in the interphase jump condition with the simplest formulation of the surface normal at boundary vertices because the discretized surface changes automatically to fit the scale of the problem the method appears to offer many of the advantages of moving mesh finite element methods however it only requires the manipulation of a surface mesh and solution for surface variables in some applications coarse meshes and very large time steps may be used relative to those which would be required by fixed grid domain methods computations are also compared to original lab data describing two dimensional soil freezing with a time dependent boundary condition agreement between simulated and measured histories is good

temporal derivatives in the finite element method on continuously deforming grids when solving certain time dependent partial differential equations using a finite element technique on a deforming grid it is shown that there is a need to differentiate the trial solution with respect to the position of each of the moveable node points a result is presented which enables these nonstandard derivatives to be expressed in terms of the standard spatial derivatives of the trial functions provided that certain conditions are satisfied these conditions are investigated and shown to be satisfied for a large class of trial spaces even when the necessary conditions are not satisfied by all of the trial functions being used the result is shown to be still of great use in evaluating these awkward nodal derivatives 1 u x t y u x t a u

a best approximation property of the moving finite element method the moving finite element method mfe for the solution of time dependent partial differential equations pdes is a numerical solution scheme which allows the automatic adaption of the finite element approximation space with time an analysis of how this method models the steady solutions of a general class of parabolic linear source equations is presented it is shown that under certain conditions the steady solutions of the mfe problem can correspond to best free knot spline approximations to the true steady solution of the differential equation when using the natural norm associated with the problem hence a quantitative measure of the advantages of the mfe method over the usual fixed grid galerkin method is produced for these equations a number of numerical examples are included to illustrate these results

an interval for the shape parameter in radial basis function approximation the radial basis functions rbfs depend on an auxiliary parameter called the shape parameter great theoretical and numerical efforts have been made to find the relationship between the accuracy of the rbf approximations and the value of the shape parameter in many cases the numerical approaches are based on minimization of an estimation of the error function such as rippa s approach and its modifications these approaches determine a value for the shape parameter in this paper we propose a practical approach to determine an interval instead of a value without any minimization and estimation of error function the idea is based on adding a loop on the shape parameter but not to minimize an error norm suitable values of the shape parameter are determined by take into account the practical convergence behavior of the problem the proposed method is applied to some illustrative examples including one and two dimensional interpolation and partial differential equations the results of the method are compared with those of some other approaches to confirm the reliability of the method furthermore numerical stability of the method of line respect to the shape parameter is considered for time dependent pdes the results show that the proposed method can be used as an independent method to find a shape parameter and also as an alternative for investigating the validity of the values of the other methods

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

flair fluz line segment model for advection and interface reconstruction abstract a computational technique for solving fluid problems with free surfaces and interfaces is presented the conventional cell volume fraction approach is employed for tracking the interfaces however for surface advection and its reconstruction a new and more accurate flair flux line segment model for advection and interface reconstruction algorithm is developed the surface is approximated by a set of line segments fitted at the boundary of every two neighboring computational cells a criterion is developed for identifying the line segment orientation by inspecting the cell volume fractions the new cell volume fraction field is obtained by integrating the advected area underneath the interface line segment as an example this technique is applied to the capillary driven viscous flow of an initially elliptic two dimensional fluid zone the problem is posed mathematically as a solution of the navier stokes equations with moving free surface boundary conditions the damping motion of the fluid zone is observed through transport of the free surface which is related to the instantaneous internal velocity field under the influence of surface tension and viscous forces

simulation of free dendritic crystal growth in a gravity environment in this paper we simulate the evolution and free particle motion of an individual nucleus that grows into a dendritic crystal the melt flow and the convective heat transfer around the crystal are simulated as they settle due to gravity there is an intricate coupling between the settling and the evolution of the crystal the relative flow induced by the settling enhances the growth at the downward facing parts which in its turn affects the subsequent settling motion simulations have been done in two dimensions using a semi sharp phase field model the flow was constrained to a rigid body motion by using lagrange multipliers inside the solidified part the model was formulated using two different meshes one is a fixed background mesh which covers the whole domain the other is an adaptive mesh where the node points are also translated and rotated with the movement of the solid particle in the latter the dendritic growth is simulated by the semi sharp phase field method

sharp interface cartesian grid method iii solidification of pure materials and binary solutions a numerical technique is presented for computing dendritic growth of crystals from pure melts and binary solutions the governing equations are solved on a fixed cartesian mesh and the immersed phase boundary is treated as a sharp solid fluid interface the interface is tracked using a level set field a finite difference scheme is presented that incorporates the immersed phase boundary with only a small change to a standard cartesian grid poisson solver the scheme is simple to implement in three dimensions the results from our calculations show excellent agreement with two dimensional microscopic solvability theory for pure material solidification it is shown that the method predicts dendrite tip characteristics in excellent agreement with the theory the sharp interface treatment allows discontinuous material property variation at the solid liquid interface this facilitates sharp interface simulations of dendritic solidification of binary solutions

a stable gaussian radial basis function method for solving nonlinear unsteady convection diffusion reaction equations abstract we investigate a novel method for the numerical solution of two dimensional time dependent convection diffusion reaction equations with nonhomogeneous boundary conditions we first approximate the equation in space by a stable gaussian radial basis function rbf method and obtain a matrix system of odes the advantage of our method is that by avoiding kronecker products this system can be solved using one of the standard methods for odes for the linear case we show that the matrix system of odes becomes a sylvester type equation and for the nonlinear case we solve it using predictor corrector schemes such as adams bashforth and implicit explicit imex methods this work is based on the idea proposed in our previous paper 2016 in which we enhanced the expansion approach based on hermite polynomials for evaluating gaussian radial basis function interpolants in the present paper the eigenfunction expansions are rebuilt based on chebyshev polynomials which are more suitable in numerical computations the accuracy robustness and computational efficiency of the method are presented by numerically solving several problems

short note a volume of fluid approach for crystal growth simulation a new approach to simulating the dendritic growth of pure metals based on a recent volume of fluid vof method with plic piecewise linear interface calculation reconstruction of the interface is presented the energy equation is solved using a diffuse interface method which avoids the need to apply the thermal boundary conditions directly at the solid front the thermal gradients at both sides of the interface which are needed to obtain the front velocity are calculated with the aid of a distance function to the reconstructed interface the advection equation of a discretized solid fraction function is solved using the unsplit vof advection method proposed by lopez et al j comput phys 195 2004 718 742 extended to three dimensions by hernandez et al int j numer methods fluids 58 2008 897 921 and the interface curvature is computed using an improved height function technique which provides second order accuracy the proposed methodology is assessed by comparing the numerical results with analytical solutions and with results obtained by different authors for the formation of complex dendritic structures in two and three dimensions

a temperature based finite element solution for phase change problems a finite element procedure for solving multidimensional phase change problems is described the algorithm combines a temperature formulation with a finite element treatment of the differential equation and discontinuous integration within the two phase elements to avoid the necessity of regularization a new criterion for the computation of the iteration matrix is proposed it is based on a quasi newton correction of the jacobian matrix for conduction problems without change of phase a set of test problems with exact solution is analysed and demonstrates that the procedure can accurately evaluate the front position and temperature history with a reasonable computational effort

a new stable basis for radial basis function interpolation it is well known that radial basis function interpolants suffer from bad conditioning if the basis of translates is used in the recent work by pazouki and schaback 2011 5 the authors gave a quite general way to build stable and orthonormal bases for the native space n f w associated to a kernel f on a domain w r s the method is simply based on the factorization of the corresponding kernel matrix starting from that setting we describe a particular basis which turns out to be orthonormal in n f w and in 2 w x where x is a set of data sites of the domain w the basis arises from a weighted singular value decomposition of the kernel matrix this basis is also related to a discretization of the compact operator t f n f w n f w t f f x w f x y f y dy x w and provides a connection with the continuous basis that arises from an eigendecomposition of t f finally using the eigenvalues of this operator we provide convergence estimates and stability bounds for interpolation and discrete least squares approximation

equidistributing principles in moving finite element methods abstract recently miller and his co workers proposed a moving finite element method based on a least squares principle this was followed by a similar method by the present authors using a petrov galerkin approach in this paper the two methods are compared in particular it is shown that both methods move their nodes according to an approximate equidistributing principle this observation leads to a criterion for the placement of the nodes it is also shown that the penalty function designed by miller may also be used with the petrov galerkin method finally numerical examples are given illustrating the performance of the two methods

moving boundary moving mesh analysis of phase change using finite elements with transfinite mappings two dimensional heat conduction phase change problems are solved using a moving boundary moving mesh approach a transfinite mapping technique successfully controls interior mesh motion and numerical results compare well with analytical solutions calculations also agree well with two dimensional laboratory data for cases featuring time dependent boundary conditions

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

moving finite element methods for evolutionary problems i theory abstract in this paper the first of two on the subject we present a unified approach to moving and fixed finite element methods for evolutionary problems in terms of projections the central theoretical results are concerned with moving finite elements for one dimensional scalar problems particularly hyperbolic equations with shocks but the viewpoint extends to general systems in any number of dimensions

numerical approximation of a thermally driven interface using finite elements a two dimensional finite element model for dendritic solidification has been developed that is based on the direct solution of the energy equation over a fixed mesh the model tracks the position of the sharp solid liquid interface using a set of marker points placed on the interface the simulations require calculation of the temperature gradients on both sides of the interface in the direction normal to it at the interface the heat flux is discontinuous due to the release of latent heat during the solidification melting process two ways to calculate the temperature gradients at the interface evaluating their interpolants at gauss points were proposed using known one and two dimensional solutions to stable solidification problems the stefan problem it was shown that the method converges with second order accuracy when applied to the unstable solidification of a crystal into an undercooled liquid it was found that the numerical solution is extremely sensitive to the mesh size and the type of approximation used to calculate the temperature gradients at the interface i e different approximations and different meshes can yield different solutions the cause of these difficulties is examined the effect of different types of interpolation on the simulations is investigated and the necessary criteria to ensure converged solutions are established copyright 2003 john wiley sons ltd

computation of solid liquid phase fronts in the sharp interface limit on fixed grids a finite difference formulation is applied to track solid liquid boundaries on a fixed underlying grid the interface is not of finite thickness but is treated as a discontinuity and is explicitly tracked the imposition of boundary conditions exactly on a sharp interface that passes through the cartesian grid is performed using simple stencil readjustments in the vicinity of the interface attention is paid to formulating difference schemes that are globally second order accurate in x and t error analysis and grid refinement studies are performed for test problems involving the diffusion and convection diffusion equations and for stable solidification problems issues concerned with stability and change of phase of grid points in the evolution of solid liquid phase fronts are also addressed it is demonstrated that the field calculation is second order accurate while the position of the phase front is calculated to first order accuracy furthermore the accuracy estimates hold for the cases where there is a property jump across the interface unstable solidification phenomena are simulated and an attempt is made to compare results with previously published work the results indicate the need to begin an effort to benchmark computations of instability phenomena

numerical simulation of crystal growth in three dimensions using a sharp interface finite element method a sharp interface numerical model is presented to simulate thermally driven crystal growth in three dimensional space the model is formulated using the finite element method and works directly with primitive variables it solves the energy equation in a fixed volume mesh while explicitly tracking the motion of the solid liquid interface the three dimensional interface is represented by connected planar triangles that form a surface mesh to accurately capture the morphology of the growing dendrite the surface mesh is updated every few time steps so that the quality of the triangles is maintained and the size of the triangles is always kept in a range associated with the element size of the fixed volume mesh the interface curvature is calculated by a least squares paraboloid fitting to neighbouring nodes the model is validated through a comparison with an exact solution of a three dimensional stefan problem a mesh refinement study a mesh orientation test and a comparison with solvability theory it is shown that the interface position is tracked to second order accuracy simulations are performed under different combinations of the undercooling and surface energy the effects of these parameters on the growth and morphology of the dendrites are studied copyright 2006 john wiley sons ltd

parallel finite element computations in fluid mechanics we provide an overview of the role of parallel finite element computations in fluid mechanics we emphasize the class of flow problems involving moving boundaries and interfaces some of the computationally most challenging flow problems such as fluid object and fluid structure interactions as well as free surface and two fluid flows belong to this class in the development of the methods targeting this class of problems the computational challenges involved need to be addressed in such a way that 3d computation of complex applications can be carried out efficiently on parallel computers this requirement has to be one of the key factors in designing various components of the overall solution approach such as solution techniques for the discretized equations and mesh update methods for handling the changes in the spatial domain occupied by the fluid this overview includes a description of how the computational challenges are addressed and how the computational schemes can be enhanced with special preconditioning techniques 2005 elsevier b v all rights reserved

on strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method we consider a problem stemming from recent models of phase transitions in stimulus responsive hydrogels wherein a sharp interface separates swelled and collapsed phases extended finite element methods that approximate the local solution with an enriched basis such that the mesh need not explicitly fit the interface geometry are emphasized attention is focused on the weak enforcement of the normal configurational force balance and various options for evaluating the jump in the normal component of the solute flux at the interface we show that as the reciprocal interfacial mobility vanishes it plays the role of a penalty parameter enforcing a pure dirichlet constraint eventually triggering oscillations in the interfacial velocity we also examine alternative formulations employing a lagrange multiplier to enforce this constraint it is shown that the most convenient choice of basis for the lagrange multiplier results in oscillations in the multiplier field and a decrease in accuracy and rate of convergence in local error norms suggesting a lack of stability in the discrete formulation under such conditions neither the direct evaluation of the gradient of the approximation at the phase interface nor the interpretation of the lagrange multiplier field provide a robust means to obtain the jump in the normal component of solute flux fortunately the adaptation and use of local domain integral methodologies considerably improves the flux evaluations several example problems are presented to compare and contrast the various techniques and methods

a stable algorithm for flat radial basis functions on a sphere when radial basis functions rbfs are made increasingly flat the interpolation error typically decreases steadily until some point when runge type oscillations either halt or reverse this trend because the most obvious method to calculate an rbf interpolant becomes a numerically unstable algorithm for a stable problem in the case of near flat basis functions there will typically also be a separate point at which disastrous ill conditioning enters we introduce here a new method rbf qr which entirely eliminates such ill conditioning and we apply it in the special case when the data points are distributed over the surface of a sphere this algorithm works even for thousands of node points and it allows the rbf shape parameter to be optimized without the limitations imposed by stability concerns since interpolation in the flat rbf limit on a sphere is found to coincide with spherical harmonics interpolation new insights are gained as to why the rbf approach with nonflat basis functions often is the more accurate of the two methods

a characteristic finite element scheme for convective dispersive transport with non equilibrium reaction a finite element method coupled with the method of characteristics is applied to the problem of convective dispersive transport with non equilibrium reaction convective terms are calculated using moving particles together with influence element meshes for particle tracking whereas dispersion terms are calculated by means of finite element techniques the contribution from the term of non equilibrium reaction is incorporated into the algebraic equations by the use of analytical solutions to the equation of non equilibrium reaction for each time step and extended simpson s numerical integrations comparison of the numerical results from the proposed procedure with analytical solutions to the simultaneous partial differential equations shows good agreement for the wide spectrum ranging from dispersion to convection dominated transport with various reaction parameters

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

fractional hermite interpolation using rbfs in high dimensions over irregular domains with application abstract in the interpolation method in some cases one often has a number of data points and its derivatives which are obtained by sampling or experimentation in this case the problem of finding an approximating function passing through these points and coinciding with given values of its derivatives at these points is generally known as hermite interpolation the hermite interpolation is mostly a method of interpolating data points as a polynomial function that is faced with some challenges in high dimensions and on irregular domains radial basis functions take advantage of being flexible with respect to geometry easy to implement in higher dimensions and can also provide high order convergence so they can be applied as a suitable tool to high dimensional hermite interpolation problem on irregular domains in many applied systems commonly available derivatives information is presented using fractional order derivatives instead of integer ones for this purpose in this paper we assume that the values of an unknown function and its fractional derivatives at some distinct points are presented therefore we intend to apply a new approach which we call it as fractional hermite interpolation with radial basis functions in high dimensions optimal recovery conditions for the fractional hermite interpolant are investigated then the existence and uniqueness of the solution in this type of generalized interpolation are proved in order to increase the accuracy and stability of the method hilbert schmidt s theory has also been used the main advantages of the used method are its simplicity and efficiency in high dimensions and over irregular domains finally numerical results in one two and three dimensions and a real world problem are presented to support our theoretical analysis

numerical simulations of unsteady crystal growth an efficient algorithm is developed to numerically compute solutions of problems related to crystal growth the method is based on an integral equation formulation that involves an integration over the entire history of the growth a direct calculation of this memory integral becomes more costly as time increases but an indirect method is presented that has a fixed operation cost per timestep the one dimensional procedure is tested and applied to the problem of rapid directional solidification where the nonlinear development of a recently discovered oscillatory instability is followed

impossibility of fast stable approximation of analytic functions from equispaced samples it is shown that no stable procedure for approximating functions from equally spaced samples can converge exponentially for analytic functions to avoid instability one must settle for root exponential convergence the proof combines a bernstein inequality of 1912 with an estimate due to coppersmith and rivlin in 1992

a stable method for the evaluation of gaussian radial basis function solutions of interpolation and collocation problems radial basis functions rbfs are a powerful tool for approximating the solution of high dimensional problems they are often referred to as a meshfree method and can be spectrally accurate the best accuracy can often be achieved when the so called shape parameter of the basis functions is small which in turn tends to make the interpolation matrix increasingly ill conditioned to overcome such instability in the numerical method which arises for even the most stable problems one needs to stabilize the method in this paper we present a new stable method for evaluating gaussian radial basis function interpolants based on the eigenfunction expansion for gaussian rbfs this work enhances the ideas proposed in fasshauer and mccourt 2012 by exploiting the properties of the orthogonal eigenfunctions and their zeros we develop our approach in one and two dimensional spaces with the extension to higher dimensions proceeding analogously in the univariate setting the orthogonality of the eigenfunctions and our special collocation locations give rise to easily computable cardinal basis functions the accuracy robustness and computational efficiency of the method are tested by numerically solving several interpolation and boundary value problems in one and two dimensions high accuracy simple implementation and low complexity for high dimensional problems are the advantages of our approach on the down side our method is currently limited to the use of tensor products of unevenly spaced one dimensional data locations

surface integrals and monopole charges in non abelian gauge theories we derive a formula which gives all the magnetic charges topological invariants of a monopole in the adjoint representation of a non abelian gauge theory in terms of surface integrals at infinity

stable computations with gaussian radial basis functions radial basis function rbf approximation is an extremely powerful tool for representing smooth functions in nontrivial geometries since the method is mesh free and can be spectrally accurate a perceived practical obstacle is that the interpolation matrix becomes increasingly ill conditioned as the rbf shape parameter becomes small corresponding to flat rbfs two stable approaches that overcome this problem exist the contour pade method and the rbf qr method however the former is limited to small node sets and the latter has until now been formulated only for the surface of the sphere this paper focuses on an rbf qr formulation for node sets in one two and three dimensions the algorithm is stable for arbitrarily small shape parameters it can be used for thousands of node points in two dimensions and still more in three dimensions a sample matlab code for the two dimensional case is provided

the problem of global color in gauge theories the problem of global color which arose recently in monopole theory is generalized to arbitrary gauge theories a subgroup k of the unbroken gauge group g is implementable iff the gauge bundle reduces to the centralizer of k in g equivalent implementations correspond to equivalent reductions such an action is an internal symmetry for a given configuration iff the yang mills field reduces also the case of monopoles is worked out in detail

a constrained moving finite element solution of the one dimensional oxygen diffusion with absorption problem abstract we derive an adaptive finite element method for the numerical solution of a one dimensional implicit moving boundary problem the oxygen diffusion with absorption problem of crank and gupta the method which includes a moving grid and incorporates an iteration for the velocity of the moving boundary produces good results over a range of parameters we present numerical and graphical results in one dimension and indicate an extension of the method to two dimensions

a conserving discretization for the free boundary in a two dimensional stefan problem the dissolution of a disk likeal2cuparticle is considered a characteristic property is that initially the particle has a nonsmooth boundary the mathematical model of this dissolution process contains a description of the particle interface of which the position varies in time such a model is called a stefan problem it is impossible to obtain an analytical solution for a general two dimensional stefan problem so we use the finite element method to solve this problem numerically first we apply a classical moving mesh method computations show that after some time steps the predicted particle interface becomes very unrealistic therefore we derive a new method for the displacement of the free boundary based on the balance of atoms this method leads to good results also for nonsmooth boundaries some numerical experiments are given for the dissolution of anal2cuparticle in anal cualloy

bases for kernel based spaces since it is well known de marchi and schaback 2001 4 that standard bases of kernel translates are badly conditioned while the interpolation itself is not unstable in function space this paper surveys the choices of other bases all data dependent bases turn out to be defined via a factorization of the kernel matrix defined by these data and a discussion of various matrix factorizations e g cholesky qr svd provides a variety of different bases with different properties special attention is given to duality stability orthogonality adaptivity and computational efficiency the newton basis arising from a pivoted cholesky factorization turns out to be stable and computationally cheap while being orthonormal in the native hilbert space of the kernel efficient adaptive algorithms for calculating the newton basis along the lines of orthogonal matching pursuit conclude the paper

approximation of the heat equation in a variable domain with application to the stefan problem a discretization of the one dimensional heat equation in a time dependent domain is considered with particular emphasis on a generalization of the crank nicholson scheme for the discretization with respect to time application to a one dimensional stefan problem is given with an iterative algorithm to follow the moving boundary numerical results agree well with the theoretical error bounds

modeling the growth and interaction of multiple dendrites in solidification using a level set method abstract a level set method is presented to study the growth and interaction of multiple dendrites in solidification the method couples thermal and solute diffusion with propagation of multiple interfaces a single signed distance function is used to track the solid liquid interface with the aid of markers the value of which is the orientation angle for identification of different crystals the problem of evolving multiple crystal interfaces is reduced to two tasks 1 tracking one level set variable signed distance function and 2 determination of the marker for a newly solidified finite element nodal point tracking a single level set variable is implemented by solving the level set equation with interface velocity computed from an extended stefan equation using the marker information crystal orientation determination of the marker for a newly solidified finite element nodal point is implemented by using an algorithm modified from the fast marching technique both of these two steps are computationally efficient and the approach is suitable for incorporating effects of multiple crystals convergence and accuracy of this approach are demonstrated by using different grid spacings and comparing with results obtained from the multi phase level set method a parametric study is performed to investigate the effects of solidification speed and thermal gradient on the resulting solidification microstructure pattern numerical results of columnar to equiaxed transition cet qualitatively agree with an analytical estimation and are similar to previous numerical results obtained using a phase field method a convergence study is performed to determine the appropriate grid spacing for numerical simulation at lower surface tension cet occurs at a lower thermal gradient for a giving solidification speed secondary dendrite formation is more apparent with lower surface tension the differences and similarities between the three dimensional and two dimensional growth results are analyzed randomness in crystal orientation and required undercooling for nucleation are modeled and found to have a great effect on the microstructure pattern the efficiency of the present approach is finally demonstrated with an example that includes the growth of hundreds of crystals with consideration of randomness effects

front tracking finite element method for dendritic solidification 765 abstract a two dimensional finite element method capable of tracking sharp interfaces is developed the method is based on a fixed mesh of bilinear isoparametric elements and is extremely simple and easy to use the interfaces are tracked with a set of marker points that define their position at all times several different approaches to finding the interface position and velocity are discussed and their effect on the convergence rate is examined it is shown through numerical examples that the approximation to the interface position converges with second order accuracy in two dimensional problems as opposed to previously developed finite difference algorithms that are only first order accurate the approximation to the interface velocity is shown to be of first order a number of examples are examined including several that provide a careful comparison with previously published results

high order finite element methods for moving boundary problems with prescribed boundary evolution abstract we introduce a framework for the design of finite element methods for two dimensional moving boundary problems with prescribed boundary evolution that have arbitrarily high order of accuracy both in space and in time at the core of our approach is the use of a universal mesh a stationary background mesh containing the domain of interest for all times that adapts to the geometry of the immersed domain by adjusting a small number of mesh elements in the neighborhood of the moving boundary the resulting method maintains an exact representation of the prescribed moving boundary at the discrete level or an approximation of the appropriate order yet is immune to large distortions of the mesh under large deformations of the domain the framework is general making it possible to achieve any desired order of accuracy in space and time by selecting a preferred and suitable finite element space on the universal mesh for the problem at hand and a preferred and suitable time integrator for ordinary differential equations we illustrate our approach by constructing a particular class of methods and apply them to a prescribed boundary variant of the stefan problem we present numerical evidence for the order of accuracy of our schemes in one and two dimensions

finite element solution of maxwell s equations for hyperthermia treatment planning abstract a general galerkin formulation is given for the periodic form of maxwell s equations on a bounded inhomogeneous medium the formulation conserves energy exactly and its dispersion relation is studied solutions for test problems representative of hyperthermia applications show good agreement with analytic solutions a representative clinical example is shown

finite element analysis of diffusion with reaction at a moving boundary abstract we consider the model problem of mass transport by diffusion with chemical reaction at a moving boundary the objective is to model the chemical process by predicting the location of the boundary as the reaction proceeds in time a finite element analysis is formulated in one dimension using moving elements to follow the boundary motion and an iterative linearization method is developed to uncouple and solve the diffusion and reaction equations the semidiscrete system for the transport process is integrated using a predictor corrector scheme and its accuracy and stability are examined in numerical experiments numerical results are compared with an analytic solution for the quasi steady case in which diffusion is very rapid in comparison to reaction and to a perturbation solution for the unsteady case other supporting numerical studies are made to examine the sensitivity of the solution and boundary motion to changes in diffusion and reaction rate for both linear and nonlinear reactions

mesh deformation using the biharmonic operator the use of the biharmonic operator for deforming a mesh in an arbitrary lagrangian eulerian simulation is investigated the biharmonic operator has the advantage that two conditions can be specified on each boundary of the mesh this allows both the position and the normal mesh spacing along a boundary to be controlled which is important for two fluid interfaces and periodic boundaries at these boundaries we can simultaneously fix the position of the boundary and ensure that the normal mesh spacing is continuous across the boundary in addition results for deforming surfaces show that greater surface deformation can be tolerated when using biharmonic equations compared to approaches using second order partial differential equations a final advantage is that with the biharmonic operator the integrity of a grid in a moving boundary layer can be preserved as the boundary moves the main disadvantage of the approach is its increased computational expense

a direct method for solving two dimensional one phase stefan problems abstract a new algorithm is proposed for solving the stefan problem for the case when the initial region occupied by the medium in a given phase is a starshaped two dimensional domain the evolution of this region as time increases is determined by plotting the free boundary directly at each time step this is approximated by a polygon in the interior of which the heat equation is solved with piecewise linear finite elements numerical experiments indicate that the method is stable and that convergence is to be expected under suitable assumptions on the regularity of both the boundary and initial data of the problem

unified approach to simulation on deforming elements with application to phase change problems abstract several different approaches to simulation on deforming finite elements are shown to generate the same weighted residuals formulation for the evolution of the dependent variables control of node motion by means of mesh stretching in two dimensions is illustrated in the context of phase change problems simulation of problems with large phase boundary excursions shows good agreement with analytic solutions

solving poisson type equations with robin boundary conditions on piecewise smooth interfaces abstract we present two finite volume schemes to solve a class of poisson type equations subject to robin boundary conditions in irregular domains with piecewise smooth boundaries the first scheme results in a symmetric linear system and produces second order accurate numerical solutions with first order accurate gradients in the l norm for solutions with two bounded derivatives the second scheme is nonsymmetric but produces second order accurate numerical solutions as well as second order accurate gradients in the l norm for solutions with three bounded derivatives numerical examples are given in two and three spatial dimensions

fixed grid techniques for phase change problems a review summary the aim of this paper is to categorize the major fixed grid formulations and solution methods for conduction controlled phase change problems using a two phase model of a solid liquid phase change the basic enthalpy equation is derived starting from this equation a number of alternative formulations are obtained all the formulations are reduced to a standard form from this standard form finite element and finite volume discretizations are developed these discretizations are used as the basis for a number of fixed grid numerical solution techniques for solidification phase change systems in particular various apparent capacity and source based enthalpy methods are explored

non linear simulation of dendritic solidification of an undercooled melt two dimensional finite element simulations of solidification for quiescent undercooled pure metals are presented the full non linear transient heat equation is used with phase front tracking which is subject to local curvature and interfacial kinetics during early stages of the waveform instability the simulated solutions match the linear stability analysis with fidelity beyond the valid range of that analysis the numerical solution continues to demonstrate the physically observed exponential growth behaviour and characteristic spacing between fingers whereas the simulations show the sensitivity of dendritic growth to initial conditions as expected for an unstable process the overall pattern formation preserves the characteristic spacing the simulations are terminated after the onset of bifurcation thereafter the numerical model is inappropriate for physical comparison owing to the planar two dimensional limitation

the hermite spectral method for gaussian type functions although hermite functions were widely used for many practical problems numerical experiments with the standard normalized hermite functions psi n v worked poorly in the sense that too many hermite functions are required to solve differential equations in order to obtain accurate numerical solutions it is necessary to choose a scaling factor alpha and use psi n alpha v as the basis functions in this paper the scaling factors are given for functions that are of gaussian type which have finite supports m m the scaling factor used is max 0 leqq j leqq n gamma j m where gamma j j 0 n are the roots of psi n 1 v and n 1 is the number of the truncated terms used the numerical results show that after using this scaling factor only reasonable numbers of the hermite functions are required to solve differential equations

an implementation of a moving finite element method abstract the moving finite element method an adaptive gridding procedure for systems of partial differential equations whose solutions contain steep gradients has been implemented in a very straightforward way accurate results can be obtained using fewer nodes than that required by fixed mesh solution methods performance of the method using some minor changes to a standard ordinary differential equation algorithm is illustrated with solutions to burger s equation

an analysis of solidification problems by the boundary element method the problem of interest in this paper is the calculation of the motion of the solid liquid interface and the time dependent temperature field during solidification of a pure metal an iterative implicit algorithm has been developed for this purpose using the boundary element method bem with time dependent green s functions and convolution integrals the bem approach requires discretization of only the surface of the solidifying body thus the numerical method closely follows the physics of the problems and is intuitively very appealing the formulation and the numerical scheme presented here are general and can be applied to a broad range of moving boundary problems emphasis is given to two dimensional problems comparison with existing semi analytical solutions and other numerical solutions from the literature reveals that the method is fast accurate and without major time step limitations

multiphase dynamics in arbitrary geometries on fixed cartesian grids in this work a mixed eulerian lagrangian algorithm calledelafint eulerian lagrangian algorithm for interface tracking is developed further and applied to compute flows with solid fluid and fluid fluid interfaces the method is capable of handling fluid flows in the presence of both irregularly shaped solid boundaries and moving boundaries on a fixed cartesian grid the field equations are solved on the underlying fixed grid using a collocated variable pressure based formulation the moving boundary is tracked explicitly by the lagrangian translation of marker particles the moving boundary passes through the grid and the immersed boundary technique is used to handle its interaction with the underlying grid the internal solid boundaries are dealt with by using a cut cell technique particular attention is directed toward conservation and consistency in the vicinity of both phase boundaries the complex geometry feature has been tested for a variety of flow problems the performance of the immersed boundary representation is demonstrated in the simulation of newtonian liquid drops the combination of the two features is then employed in the simulation of motion of drops through constricted tubes the capabilities developed here can be useful for solving flow problems involving moving and stationary complex boundaries

a monolithic approach to fluid structure interaction using space time finite elements abstract the paper presents a simultaneous solution procedure for fluid structure interaction problems the structural motion is described by geometrically nonlinear elastodynamics the fluid is modeled by the incompressible navier stokes equations the space time finite element method is applied to both continua leading to an almost uniform discretization in which velocity variables are used for fluid and structure in order to enforce momentum conservation and geometric continuity at the interface a weighted residual formulation of coupling conditions is introduced the discretized model equations for fluid structure and coupling conditions are formulated in a single equation system the nonlinear system and the fluid mesh movement are solved in a single iteration loop two dimensional examples of membrane flutter and vortex excited plate vibrations demonstrate the efficiency of the methodology

stable evaluation of gaussian radial basis function interpolants we provide a new way to compute and evaluate gaussian radial basis function interpolants in a stable way with a special focus on small values of the shape parameter i e for flat kernels this work is motivated by the fundamental ideas proposed earlier by bengt fornberg and his coworkers however following mercer s theorem an l 2 mathbb r d rho orthonormal expansion of the gaussian kernel allows us to come up with an algorithm that is simpler than the one proposed by fornberg larsson and flyer and that is applicable in arbitrary space dimensions d in addition to obtaining an accurate approximation of the radial basis function interpolant using many terms in the series expansion of the kernel we also propose and investigate a highly accurate least squares approximation based on early truncation of the kernel expansion

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

the moving finite element method applications to general partial differential equations with multiple large gradients abstract the moving finite element mfe method has been reduced to practice in the automatic solution program dyla for general systems of transient partial differential equations pdes in 1 d several test examples are presented which illustrate the unique node movement and systematic control features which are intrinsic in the mfe method computational dilemmas of numerical diffusion gibbs overshooting and undershooting zone tangling and grid remap or re connection aliasing which occur frequently in conventional pde methods are essentially eliminated in the mfe method arbitrarily large gradients or shocks can be solved with extremely high resolution and accuracy for non coincident or even counterdirected propagating wavefronts boundary layers of arbitrarily small dimensions are solved with high accuracy simultaneously with the large scale structures in reactive and non reactive fluid calculations the mfe method requires a small fraction of the grid nodes which are used in conventional pde solution methods because the nodes migrate continuously and systematically to those positions where they are most needed in order to yield accurate pde solutions on entire problem domains courant friedrichs lewy time step limits are exceeded by wide margins by factors of two to several thousand finally the extension of the mfe method to 2 d is briefly discussed

an algorithm for the simulation of transient viscoelastic flows with free surfaces we propose a numerical procedure for solving a class of transient viscoelastic flows with free surfaces it is based on a galerkin finite element technique on deforming elements combined with a predictor corrector scheme the method is applied to the analysis of jet breakup caused by capillary forces non linear effects known to experimentalists are predicted and a detailed comparison with asymptotic results is carried out

generalized magnetic monopoles over contact manifolds a generalization of magnetic monopoles is given over an odd dimensional contact manifold and we discuss whether the yang mills higgs functional attains at generalized monopoles the absolute minimal value the topological invariant