semiautomatic construction of lattice boltzmann models a crucial step in constructing a lattice boltzmann model is the definition of a suitable set of lattice velocities and the correct assignment of the associated weights for high order models the solution of this problem requires a nontrivial effort this paper outlines the functioning of a publicly available python script which has been written to assist researchers in that task the speed of sound c s is considered as a parameter which can within limits be chosen at will under this premise the maxwell boltzmann constraint equations are a system of linear equations to determine the weights and hence amenable to numerical solution by standard linear algebra library routines by suitable contractions the tensor equations are mapped to a set of equivalent scalar equations which simplifies the treatment significantly for a user supplied set of velocity shells the software first checks if a solution for the weights exists and returns it if it also happens to be unique in such a case the software also calculates the range of c s values that yield positive weights standard models like d3q19 with a well defined special c s value then result as limiting cases where one of the weights vanishes in the case of an infinite set of solutions the user may find one particular solution by supplying a c s value and then minimizing one or several weights within the framework of standard linear programming some examples illustrate the feasibility and usefulness of the approach a number of models that have been discussed in the literature are nicely reproduced while the software has also been able to find some new models of even higher order

a level set approach to the computation of twinning and phase transition dynamics a computational method is proposed for the dynamics of solids capable of twinning and phase transitions in a two dimensional sharp interface model of twinning the stored energy function is a nonconvex potential with multiple wells the evolution of twin interfaces is governed by field equations and jump conditions of momentum balance and by a kinetic relation expressing the interface velocity as a function of the local driving traction and interfacial orientation a regularized version of the model is constructed based on the level set method a level set function which changes signs across the interface is introduced the evolution of this function is described by a hamilton jacobi equation whose velocity coefficient is determined by the kinetic relation jump conditions are thereby eliminated allowing finite difference discretization numerical simulations exhibit complex evolution of the interface including cusp formation needle growth spontaneous tip splitting and topological changes that result in microstructure refinement the results capture experimentally observed phenomena in martensitic crystals

hermite birkhoff interpolation on scattered data on the sphere and other manifolds abstract the hermite birkhoff interpolation problem of a function given on arbitrarily distributed points on the sphere and other manifolds is considered each proposed interpolant is expressed as a linear combination of basis functions the combination coefficients being incomplete taylor expansions of the interpolated function at the interpolation points the basis functions depend on the geodesic distance are orthonormal with respect to the point evaluation functionals and have all derivatives equal zero up to a certain order at the interpolation points a remarkable feature of such interpolants which belong to the class of partition of unity methods is that their construction does not require solving linear systems numerical tests are given to show the interpolation performance

the equilibrium of material forces in a 1d phase transition problem this paper aims at the use of material forces in estimating the interface location in an elastic one dimensional two phase problem a variational formulation based on variations of both the dependent and the independent variables results in the weak form of the momentum and the canonical momentum equations thus providing an appropriate framework for the use of the finite element method using this one can compute apart from the finite element solution to the equilibrium equation for the physical forces the corresponding one to the equilibrium equation for the material forces which provides the location of the interface in this way the total energy is minimized with respect to the deformations as well as to the interface location our theoretical expectations are confirmed by a numerical example concerning a one dimensional deformation induced phase transition problem with known analytical solution

attenuation factors in multivariate fourier analysis w gautschi s theory of attenuation factors for families of periodic functions in one variable is extended to families of functions in several variables again the linearity and the translation invariance of the operator which maps the data space onto the family are crucial special results are obtained for tensor product families and for the interpolation by translates of one generating function interesting examples are provided by box splines which include certain finite elements

domain evolution in ferroelectric materials a continuum phase field model and finite element implementation a numerical tool for simulating the polarization distribution in ferroelectric materials is presented using the concept of a phase field a continuum physics model is established which is descretized with finite elements the main feature of the numerical implementation is an implicit time integration of the non linear evolution equation representative examples and comparison to experimental measurements from the literature show the main features of the model

interpolation on uniform meshes by the translates of one function and related attenuation factors the exact fourier coefficients c pm are proportional to the discrete fourier coefficients dj f if pn is a translation invariant operator which depends only on the values of on an equidistant mesh of width 2it n the proportionality factors which depend only on p but not on are called attenuation factors and have been calculated for several operators p of spline type here we analyze first the interpolation problem which is produced by the functions j lirj n j 0 n 1 where a is a suitable 2 r periodic generating function it is essential that the associated interpolation matrix is of discrete convolution type thus we can derive conditions guaranteeing the unique solvability of the interpolation problem and representations of the interpolating function then the attenua tion factors may be expressed in terms of the fourier coefficients of a we point especially to the case where o is a reproducing kernel in a suitable hilbert space here we get attenuation factors of a new type which are generated by interpolation with analytic functions

strictly positive definite functions on the unit circle we study strictly positive definite functions on the unit circle in the euclidean space of dimension two we develop several conditions pertaining to the determination of such functions the major result is obtained by considering the set of real numbers as a vector space over the field of rational numbers and then applying the kronecker approximation theorem and weyl s criterion on equidistributions

multiscale interpolation on the sphere in this paper we study the convergence rate and inverse theorem for spherical multiscale interpolation in lp and sobolev norms the multiscale interpolation is constructed using a sequence of scaled compactly supported radial basis functions restricted to the unit sphere s n for the interpolation scheme the problem called native space barrier is considered in addition a bernstein type inequality is established to derive an inverse theorem for the multiscale interpolation and some numerical experiments to illustrate the theoretical results are given

shape topology optimization of stokes flow via variational level set method in this paper the conventional level set method is extended as an efficient approach for shape optimization the motion of the dynamic interface is governed by stokes equations although this method is not specifically designed for topology optimization it can handle topology changes easily furthermore a relatively smooth evolution can be maintained without re initialization the cost of this method is moderate the accuracy and efficiency of the proposed algorithm are illustrated by several numerical examples

general solution of lattices for cartesian lattice bhatanagar gross krook models we give the general solutions of lattices i e velocity sets and weights for the lattice bhatanagar gross krook lbgk models on two and three dimensional cartesian grids the solutions define the necessary and sufficient conditions so that the resulting lbgk model can accurately capture the dynamics of the moments retained in the distribution function in the parameter space of the weights the general solutions form low dimensional linear spaces from which minimal velocity sets are identified for the degrees of precision that are most relevant to the construction of high order lbgk models all well known lbgk lattices are found to be special cases of the given general solutions

a level set approach to domain wall kinetics and domain patterning in elastic ferroelectrics the level set method is a powerful tool for tracking a moving interface in ferroelectric materials the boundaries between domains of uniform polarization referred to as domain walls are viewed as interfaces in this work a variational level set method is used to formulate a model for the behavior of elastic ferroelectrics and more particularly to describe the domain formation and the domain wall kinetics a regularized version of the sharp interface theory is introduced by substitution of the singular surfaces with thin transition layers inserting so inhomogeneities into the energy density function it is proved that the forces driving domain wall motion are in fact inhomogeneity forces with this description the level set method becomes an appropriate framework to insert and exploit material forces the proposed model may interpret typical microstructures in 2 d elastic ferroelectrics with tetragonal crystal symmetry comparisons with phase field models and experiments show good agreement

low and high order accurate boundary conditions the present contribution focuses on the accuracy of reflection type boundary conditions in the stokesbrinkmandarcy modeling of porous flows solved with the lattice boltzmann method lbm which we operate with the two relaxation time trt collision and the brinkman force based scheme bf called bf trt scheme in parallel we compare it with the stokesbrinkmandarcy linear finite element method fem where the dirichlet boundary conditions are enforced on grid vertices in bulk both bf trt and fem share the same defect in their discretization a correction to the modeled brinkman equation appears given by the discrete laplacian of the velocity proportional resistance force this correction modifies the effective brinkman viscosity playing a crucial role in the triggering of spurious oscillations in the bulk solution while the exact form of this defect is available in lattice aligned straight or diagonal flows in arbitrary flow lattice orientations its approximation is constructed at boundaries we verify that such a brinkman viscosity correction has an even more harmful impact already at the first order it shifts the location of the no slip wall condition supported by traditional lbm boundary schemes such as the bounce back rule for that reason this work develops a new class of boundary schemes to prescribe the dirichlet velocity condition at an arbitrary wall boundary node distance and that supports a higher order accuracy in the accommodation of the trt brinkman solutions for their modeling we consider the standard bf scheme and its improved version called ibf this latter is generalized in this work to suppress or to reduce the viscosity correction in arbitrarily oriented flows our framework extends the one and two point families of linear and parabolic link wise boundary schemes respectively called b li and b mli which avoid the interference of the brinkman viscosity correction in their closure relations the performance of lbm and fem is thoroughly evaluated in three benchmark tests which are run throughout three distinctive permeability regimes the first configuration is a horizontal porous channel studied with a symbolic approach where we construct the exact solutions of fem and bf ibf with different boundary schemes the second problem refers to an inclined porous channel flow which brings in as new challenge the formation of spurious boundary layers in lbm that is numerical artefacts that arise due to a deficient accommodation of the bulk solution by the low accurate boundary scheme the third problem considers a porous flow past a periodic square array of solid cylinders which intensifies the previous two tests with the simulation of a more complex flow pattern the ensemble of numerical tests provides guidelines on the effect of grid resolution and the trt free collision parameter over the accuracy and the quality of the velocity field spanning from stokes to darcy permeability regimes it is shown that with the use of the high order accurate boundary schemes the simple uniform mesh based trt lbm formulation can even surpass the accuracy of fem employing hardworking body fitted meshes

high order lattice boltzmann models for wall bounded flows at finite knudsen numbers we analyze a large number of high order discrete velocity models for solving the boltzmann bhatnagar gross krook equation for finite knudsen number flows using the chapman enskog formalism we prove for isothermal flows a relation identifying the resolved flow regimes for low mach numbers although high order lattice boltzmann models recover flow regimes beyond the navier stokes level we observe for several models significant deviations from reference results we found this to be caused by their inability to recover the maxwell boundary condition exactly by using supplementary conditions for the gas surface interaction it is shown how to systematically generate discrete velocity models of any order with the inherent ability to fulfill the diffuse maxwell boundary condition accurately both high order quadratures and an exact representation of the boundary condition turn out to be crucial for achieving reliable results for poiseuille flow we can reproduce the mass flow and slip velocity up to the knudsen number of 1 moreover for small knudsen numbers the knudsen layer behavior is recovered

iterative methods based on polynomial interpolation filters to detect discontinuities and recover point values from fourier data in previous papers we proposed new filters based on polynomial interpolation to approximate the point values of a piecewise smooth function f on 0 1 from its fourier coefficients and derived error estimates we proved that if all the discontinuity points of f are nodes we can reconstruct point values of f accurately even close to the discontinuities we use the new filters to develop iterative methods for detecting the discontinuity points and therefore accurately approximate the point values of the function from its fourier coefficients

shape topology optimization for navier stokes problem using variational level set method we consider the shape topology optimization of the navier stokes problem a new algorithm is proposed based on the variational level set method by this algorithm a relatively smooth evolution can be maintained without re initialization and drastic topology change can be handled easily finally the promising features of the proposed method are illustrated by two benchmark examples

interpolation by periodic radial basis functions abstract on the unit circle s1 let d be the natural geodesic metric we investigate the possibility of interpolating arbitrary data on a set of nodes yi s1 by means of a function of the form x i 1n cif d x yi here f is a function from 0 to r and is subject to our choice the interpolation matrix a having elements aij f d yj yi is crucial to this problem in the basic case f x x we give necessary and sufficient conditions on the nodes for the invertibility of a for equally spaced nodes we give nearly complete conditions on f for the invertibility of a

generalized hermite interpolation and positive definite kernels on a riemannian manifold abstract in this paper we introduce a class of positive definite kernels defined on a closed compact riemannian manifold these kernels provide a grid free method for solving uniquely a generalized version of the hermite interpolation problem a version in which one fits a smooth surface to multi dimensional scattered data including data generated by derivatives fluxes or any other quantity one can obtain by integrating a function against a compactly supported distribution the positive definite kernels introduced here include the c members of the class of spherical positive definite functions introduced by schoenberg duke math j 9 1942 96 108 and shown to be strictly positive definite by xu and cheney proc amer math sec 116 1992 977 981 thus as a consequence of our results the generalized hermite interpolation problem is well poised i e a solution exists and is unique for the class of interpolants constructed out of c spherical positive definite functions we also provide similar results when the underlying manifold is the m dimensional torus

on the three dimensional central moment lattice boltzmann method a three dimensional 3d lattice boltzmann method based on central moments is derived two main elements are the local attractors in the collision term and the source terms representing the effect of external and or self consistent internal forces for suitable choices of the orthogonal moment basis for the three dimensional twenty seven velocity d3q27 and its subset fifteen velocity d3q15 lattice models attractors are expressed in terms of factorization of lower order moments as suggested in an earlier work the corresponding source terms are specified to correctly influence lower order hydrodynamic fields while avoiding aliasing effects for higher order moments these are achieved by successively matching the corresponding continuous and discrete central moments at various orders with the final expressions written in terms of raw moments via a transformation based on the binomial theorem furthermore to alleviate the discrete effects with the source terms they are treated to be temporally semi implicit and second order with the implicitness subsequently removed by means of a transformation as a result the approach is frame invariant by construction and its emergent dynamics describing fully 3d fluid motion in the presence of force fields is galilean invariant numerical experiments for a set of benchmark problems demonstrate its accuracy

approximation by periodic multivariate quasi projection operators abstract approximation properties of periodic quasi projection operators with matrix dilations are studied such operators are generated by a sequence of functions j and a sequence of distributions functions j error estimates for sampling type quasi projection operators are obtained under the periodic strang fix conditions for j and the compatibility conditions for j and j these estimates are given in terms of the fourier coefficients of approximated functions and provide analogs of some known non periodic results under some additional assumptions error estimates are given in other terms in particular using the best approximation a number of examples are provided

strictly positive definite kernels on subsets of the complex plane let z w x zw be a positive definite kernel and b a subset of in this paper we seek conditions in order that the restriction z w b x b zw be strictly positive definite since this problem has been solved recently in the cases in which b is either or the unit circle in our purpose here is twofold to present some results we obtained when attempting to solve the problem for the above and other choices of b and to acquaint the audience with some other questions that remain for two different classes of subsets we completely characterize the strict positive definiteness of the kernel we include a complete discussion of the case in which b is the unit circle of making a comparison with the classical problem of strict positive definiteness on the real circle

perturbed kernel approximation on homogeneous manifolds current methods for interpolation and approximation within a native space rely heavily on the strict positive definiteness of the underlying kernels if the domains of approximation are the unit spheres in euclidean spaces then zonal kernels kernels that are invariant under the orthogonal group action are strongly favoured in the implementation of these methods to handle real world problems however some or all of the symmetries and positive definiteness may be lost in digitalization due to small random errors that occur unpredictably during various stages of the execution perturbation analysis is therefore needed to address the stability problem encountered in this paper we study two kinds of perturbations of positive definite kernels small random perturbations and perturbations by dunkl s intertwining operators c dunkl y xu orthogonal polynomials of several variables encyclopedia of mathematics and its applications vol 81 cambridge university press cambridge 2001 we show that with some reasonable assumptions a small random perturbation of a strictly positive definite kernel can still provide vehicles for interpolation and enjoy the same error estimates we examine the actions of the dunkl intertwining operators on zonal strictly positive definite kernels on spheres we show that the resulted kernels are strictly positive definite on spheres of lower dimensions

on the search of more stable second order lattice boltzmann schemes in confined flows the von neumann linear analysis restricted by a heuristic selection of wave number vectors was applied to the search of explicit lattice boltzmann schemes which exhibit more stability than existing methods the relative stability of the family members of quasi incompressible collision kernels for the navier stokes equations in confined flows was analyzed the linear stability analysis was simplified by assuming a uniform velocity level over the whole domain where only the wave numbers of the first harmonic normal to the flow direction were permitted a singular equilibrium function that maximizes the critical velocity level was identified which was afterwards tested in particular cases of confined flows of interest validating the resulting procedure

on the least squares fit by radial functions to multidimensional scattered data this paper investigates some aspects of discrete least squares approximation by translates of certain classes of radial functions its specific aims are i to provide conditions under which the associated least squares matrix is invertible and ii to give upper bounds for the euclidean norms of the inverses of these matrices when they exist

optimizing the accuracy of lattice monte carlo algorithms for simulating diffusion the behavior of a lattice monte carlo lmc algorithm if it is designed correctly must approach that of the continuum system that it is designed to simulate as the time step and the mesh step tend to zero however we show for an algorithm for unbiased particle diffusion that if one of these two parameters remains fixed the accuracy of the algorithm is optimal for a certain finite value of the other parameter in one dimension the optimal algorithm with moves to the two nearest neighbor sites reproduces the correct second and fourth moments and minimizes the error for the higher moments at large times of the particle distribution and preserves the first two moments of the first passage time distributions in two and three dimensions the same level of accuracy requires simultaneous moves along two axes diagonal moves such moves attempting to cross an impenetrable boundary should be projected along the boundary rather than simply rejected we also treat the case of absorbing boundaries we discuss the relation between optimally accurate lmc algorithms and a particular case of lattice boltzmann lb algorithms for simulating diffusion and compare the computational efficiency of optimal lmc and optimal lb algorithms

diffuse reflection boundary condition for high order lattice boltzmann models with streaming collision mechanism for lattice boltzmann lb models their particle feature is ensured by the streaming collision mechanism however this mechanism is often discarded for high order lb models due to multi speed lattices here we propose a new way of implementing the diffuse reflection boundary condition that can maintain the characteristic streaming collision mechanism for high order models it is then tested on four lattice models namely d2q16 d2q17 d2q37 and d3q121 for isothermal and thermal couette flows and lid driven cavity flows our implementation is shown to be able to achieve second order accuracy globally

l2 approximation by the translates of a function and related attenuation factors let figure not available see fulltext be thek dimensional subspace spanned by the translates 2 j k j 0 1 k 1 of a continuous piecewise smooth complexvalued 2 periodic function for a given functionf l 2 its least squares approximants kf from figure not available see fulltext can be expressed in terms of an orthonormal basis iff is continuous s kf can be computed via its discrete analogue by fast fourier transform the discrete least squares approximant is used to approximate fourier coefficients and this complements the works of gautschi on attenuation factors examples of figure not available see fulltext include the space of trigonometric polynomials where is the de la vallee poussin kernel algebraic polynomial splines where is the periodic b spline and trigonometric polynomial splines where is the trigonometric b spline

strictly positive definite functions on spheres in euclidean spaces in this paper we study strictly positive definite functions on the unit sphere of the m dimensional euclidean space such functions can be used for solving a scattered data interpolation problem on spheres since positive definite functions on the sphere were already characterized by schoenberg some fifty years ago the issue here is to determine what kind of positive definite functions are actually strictly positive definite the study of this problem was initiated recently by xu and cheney proc amer math soc 116 1992 977 981 where certain sufficient conditions were derived a new approach which is based on a critical connection between this problem and that of multivariate polynomial interpolation on spheres is presented here the relevant interpolation problem is subsequently analyzed by three different complementary methods the first is based on the de boor ron general least solution for the multivariate polynomial interpolation problem the second which is suitable only for m 2 is based on the connection between bivariate harmonic polynomials and univariate analytic polynomials and reduces the problem to the structure of the integer zeros of bounded univariate exponentials finally the last method invokes the realization of harmonic polynomials as the polynomial kernel of the laplacian thereby exploiting some basic relations between homogeneous ideals and their polynomial kernels

on the comparison of two approaches to compute material forces for inelastic materials application to single slip crystal plasticity abstract the main goal of this contribution is the application of the material force method to dissipative materials whereby special emphasis is placed on the application to crystal plasticity balance equations on the material manifold as e g the material motion balance of momentum generally necessitate the computation of the spatial gradient of additional internal degrees of freedom or rather internal variables we therefore focus in this work on the comparison of two different numerical approaches to compute these gradient fields within a finite element setting namely a node point based and an integration point based formulation several numerical examples underline that the latter framework which is the convenient numerically comparatively cheap and well established approach in computational inelasticity turns out to be sufficiently accurate for the problem at hand

goal oriented r adaptivity based on variational arguments in the physical and material spaces abstract the error in a local quantity of interest depends on the error in the corresponding dual solution or generalized green s function this fact is used in dual weighted based goal oriented error estimation techniques and h or p adaptive algorithms this contribution is concerned with a novel approach for goal oriented r adaptivity based on energy minimization principles for the primal and the dual problems the minimization of the energy of the primal problem as well as the minimization of the energy of the dual problem with respect to a design function lead to the primal and dual material residuals respectively these residuals are error indicators for the discretization error on the current mesh we propose a goal oriented mesh optimization algorithm which ends in an optimized mesh with respect to a chosen quantity of interest by means of selected computational examples we demonstrate the capability of the proposed framework

a variational level set approach to multiphase motion a coupled level set method for the motion of multiple junctions of e g solid liquid and grain boundaries which follows the gradient flow for an energy functional consisting of surface tension proportional to length and bulk energies proportional to area is developed the approach combines the level set method of s osher and j a sethian with a theoretical variational formulation of the motion by f reitich and h m soner the resulting method uses as many level set functions as there are regions and the energy functional is evaluated entirely in terms of level set functions the gradient projection method leads to a coupled system of perturbed by curvature terms hamilton jacobi equations the coupling is enforced using a single lagrange multiplier associated with a constraint which essentially prevents a regions from overlapping and b the development of a vacuum the numerical implementation is relatively simple and the results agree with and go beyond the theory as given in 12 other applications of this methodology including the decomposition of a domain into subregions with minimal interface length are discussed finally some new techniques and results in level set methodology are presented

fast algorithms for periodic spline wavelets on sparse grids we consider boolean sums of univariate interpolation operators which define multivariate jth order blending interpolation operators on sparse grids sample spaces are defined as range of the blending operators sample and wavelet spaces have significantly lower dimension and good approximation order for certain function spaces fast decomposition and reconstruction algorithms for bivariate spline wavelets based on algorithms for univariate functions are described operation counts for the algorithms are given and it is shown that the complexity depends linearly on the dimension of sample spaces

interpolation by periodic radial functions let d be the natural geodesic metric on the circle s1 we study the interpolation of arbitrary data on a set of equally spaced nodes yi s1 by means of functions having the form l psto a i 1n ci d l yi in which is a function at our disposal a cardinal function is found and convergence questions can be answered with the aid of it