adaptive grid generation from harmonic maps on reimannian manifolds abstract in this paper we describe a new method for generation of solution adaptive grids based on harmonic maps on riemannian manifolds the reliability of the method is assured by an existence and uniqueness theorem for one to one maps between multidimensional multiconnected domains we formulate an adaptive riemannian metric consistent with this theorem several examples demonstrating application of the developed procedure are provided

variational barrier method of adaptive grid generation in hyperbolic problems of gas dynamics application of the harmonic mapping using a variational approach to generate moving adaptive grids in the hyperbolic problems of gas dynamics is considered using the example of a three point model of adaptation the possibility to generate an unfolded mesh with strong grid lines condensing in the vicinity of discontinuities of the control monitor function is demonstrated the algorithm of redistributing the boundary nodes is suggested and consists of using constrained minimization of the discrete harmonic functional when constraints define the boundary of the domain in real computations due to mesh adaptation it is possible to reduce the errors caused by shock waves smearing over the cells by many factors of ten modeling of the two dimensional 2 d supersonic gas flow in the channel has shown that the same accuracy on the adaptive grid with the same structure as the quasi uniform mesh can be acheived while requiring less cpu memory by a factor of 25 and less running time by a factor of 50 to 60 computational tests of the steady transonic and supersonic flow over an airfoil demonstrate the ability of the method to control mesh sizes across shocks

fractional iterates for n dimensional maps this paper constitutes an extension of results of the paper by narayaninsamy int j bifurcations chaos 6 1 1996 55 67 we study here the solutions of the problem of the fractional iteration for n dimensional maps

equidistribution schemes poisson generators and adaptive grids equidistribution of a weight function over a mesh is the main concept employed in the recent development of adaptive grid schemes this idea is reviewed in this paper and several examples of grids produced using equidistribution are presented the shortcomings of this approach are identified and its usefulness is evaluated next the traditional poisson grid generators are written as nonlinear equidistribution schemes in this form the relationship between the numerical solution of a physical problem the source terms in the grid generation equations and the weight functions in equidistribution schemes are identified examples are given illustrating the case of providing a useful adaptive grid with this approach

high order essentially nonsocillatory schemes for hamilton jacobi equations hamilton jacobi h j equations are frequently encountered in applications e g in control theory and differential games h j equations are closely related to hyperbolic conservation laws in one space dimension the former is simply the integrated version of the latter similarity also exists for the multidimensional case and this is helpful in the design of difference approximations in this paper high order essentially nonoscillatory eno schemes for h j equations are investigated which yield uniform high order accuracy in smooth regions and sharply resolve discontinuities in the derivatives the eno scheme construction procedure is adapted from that for hyperbolic conservation laws the schemes are numerically tested on a variety of one dimensional and two dimensional problems including a problem related to control optimization and high order accuracy in smooth regions good resolution of discontinuities in the derivatives and convergence to viscosity solutions are observed

level set based deformation methods for adaptive grids a new method for generating adaptive moving grids is formulated based on physical quantities level set functions are used to construct the adaptive grids which are solutions of the standard level set evolution equation with the cartesian coordinates as initial values the intersection points of the level sets of the evolving functions form a new grid at each time the velocity vector in the evolution equation is chosen according to a monitor function and is equal to the node velocity a uniform grid is then deformed to a moving grid with desired cell volume distribution at each time the method achieves precise control over the jacobian determinant of the grid mapping as the traditional deformation method does the new method is consistent with the level set approach to dynamic moving interface problems

a variational form of the winslow grid generator a structured grid generator based on a new approximation of dirichlet s functional is developed an unconstrained minimization process guarantees that all quadrilateral grid cells are convex at each iteration numerical results are presented in comparison with those for the original form of the winslow method a generalization to the case of adaptive grids based on harmonic maps between surfaces is considered

moving mesh methods based on moving mesh partial differential equations several versions of a moving mesh method are developed based on a mesh spatial smoothing technique and on the moving mesh pdes derived in a previous paper these versions are quite simple and easy to program they are applied to three bench mark one dimensional problems which show different solution behaviour the numerical results clearly demonstrate that the present methods are capable of accurately tracking rapid spatial and temporal transitions

a three critical points theorem revisited abstract in this paper the following result is proved let x be a reflexive real banach space i r an interval x r a sequentially weakly lower semicontinuous c 1 functional bounded on each bounded subset of x whose derivative admits a continuous inverse on x j x r a c 1 functional with compact derivative assume that lim x x j x for all i and that there exists r such that sup i inf x x x j x inf x x sup i x j x then there exist a non empty open set a i and a positive real number r with the following property for every a and every c 1 functional x r with compact derivative there exists 0 such that for each 0 the equation x j x x 0 has at least three solutions in x whose norms are less than r

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

adaptive zoning for singular problems in two dimensions abstract winslow s method for the automatic generation of computation meshes is extended to adaptively vary the zone sizes and orthogonality of grid lines in the resulting mesh through simple analysis and numerical examples the adaptive mesh is shown to give significant increases in accuracy in the computation of singular problems

multiple positive solutions of singular and nonsingular discrete problems via variational methods we employ the critical point theory to establish the existence of multiple solutions of some regular as well as singular discrete boundary value problems 2004 elsevier ltd all rights reserved msc 34b16 39a10

adaptive mesh methods for one and two dimensional hyperbolic conservation laws we develop efficient moving mesh algorithms for one and two dimensional hyperbolic systems of conservation laws the algorithms are formed by two independent parts pde evolution and mesh redistribution the first part can be any appropriate high resolution scheme and the second part is based on an iterative procedure in each iteration meshes are first redistributed by an equidistribution principle and then on the resulting new grids the underlying numerical solutions are updated by a conservative interpolation formula proposed in this work the iteration for the mesh redistribution at a given time step is complete when the meshes governed by a nonlinear equation reach the equilibrium state the main idea of the proposed method is to keep the mass conservation of the underlying numerical solution at each redistribution step in one dimension we can show that the underlying numerical approximation obtained in the mesh redistribution part satisfies the desired tvd property which guarantees that the numerical solution at any time level is tvd provided that the pde solver in the first part satisfies such a property several test problems in one and two dimensions are computed using the proposed moving mesh algorithm the computations demonstrate that our methods are efficient for solving problems with shock discontinuities obtaining the same resolution with a much smaller number of grid points than the uniform mesh approach

computing iterative roots of polygonal functions based on the iterative root theory for monotone functions an algorithm for computing polygonal iterative roots of increasing polygonal functions was given by j kobza in this paper we not only give an algorithm for roots of decreasing polygonal functions but also generalize kobza s results to the general n furthermore we extend our algorithms for polygonal pm functions a class of non monotonic functions

conditions of nondegeneracy of three dimensional cells a formula of a volume of cells numerical solutions of partial differential equations in three dimensions often use hexahedral cells composing a computational grid after the grid is constructed it is first of all verified whether the grid is unfolded or not for such a test in this paper conditions of nondegeneracy are found for hexahedral cells which are given by eight corner points and generated by the trilinear map from a unit cube to a region defined by these points the conditions include necessary conditions and two sets of sufficient conditions they are found as conditions of positivity of the jacobian of the trilinear map thus the conditions which guarantee the invertibility of the trilinear map from a unit cube to a hexahedron are given how general the nondegeneracy conditions are is shown by a numerical experiment formulas of the jacobian of the trilinear map are obtained and following from them as a separate result a formula of a volume of a cell is obtained

iteration and the solution of functional equations for functions analytic in the unit disk this paper considers the classical functional equations of schroeder f o p xf and abel f o p f 1 and related problems of fractional iteration where 9p is an analytic mapping of the open unit disk into itself the main theorem states that under very general conditions there is a linear fractional transformation 4 and a function a analytic in the disk such that a oa p and that with suitable normalization 4 and a are unique in particular the hypotheses are satisfied if 4p is a probability generating function that does not have a double zero at 0 this intertwining relates solutions of functional equations for t to solutions of the corresponding equations for d for example it follows that if 4p has no fixed points in the open disk then the solution space of f o p xf is infinite dimensional for every nonzero x although the discrete semigroup of iterates of t usually cannot be embedded in a continuous semigroup of analytic functions mapping the disk into itself we find that for each z in the disk all sufficiently large fractional iterates of p can be defined at z this enables us to find a function meromorphic in the disk that deserves to be called the infinitesimal generator of the semigroup of iterates of p if the iterates of p can be embedded in a continuous semigroup we show that the semigroup must come from the corresponding semigroup for 4 and thus be real analytic in t the proof of the main theorem is not based on the well known limit technique introduced by koenigs 1884 but rather on the construction of a riemann surface on which an extension of 9p is a bijection much work is devoted to relating characteristics of p to the particular linear fractional transformation constructed in the theorem

variational methods for non differentiable functionals and their applications to partial differential equations the interest in extending the theory of nonlinear partial differential equations pde to pde with discontinuous nonlinearities dnde has been increasing daily the reasons are twofold 1 many free boundary problems may be reduced to boundary value problems of dnde cf chang 16 71 chang and jiang 9 and fleishman and mahar 121 2 sometimes in dealing with a pde problem it is beneficial to put the original pde into a large category for instance dnde cf chang 7 simply speaking a dnde is as follows

multigrid navier stokes calculations for three dimensional cascades a vertex based finite volume method for solving the three dimensional compressible reynolds averaged navier stokes equations is presented for calculating turbomachinery cascade flows a discretization scheme for the viscous terms is proposed the baldwin lomax algebraic turbulence model is used the scheme is verified against laminar and turbulent flows over a flat plate two and three dimensional computations were carried out for a low pressure turbine cascade at design and off design conditions results are compared with inviscid solutions and experimental data

on fractional iterates of a brouwer homeomorphism embeddable in a flow we present a method for finding continuous and consequently homeomorphic orientation preserving iterative roots of a brouwer homeomorphism which is embeddable in a flow to obtain the roots we use a countable family of maximal parallelizable regions of the flow which is a cover of the plane the maximal parallelizable regions are unions of equivalence classes of an appropriate equivalence relation we show that if an equivalence class is invariant under the nth iterate of a brouwer homeomorphism g then it is invariant under g we use this fact to prove that each maximal parallelizable region of the flow must be invariant under all homeomorphic orientation preserving iterative roots of the given brouwer homeomorphism

three dimensional elliptic grid generation with fully automatic boundary constraints a new procedure for generating smooth uniformly clustered three dimensional structured elliptic grids is presented here which formulates three dimensional boundary constraints by extending the two dimensional counterpart presented by the author earlier this fully automatic procedure obviates the need for manual specification of decay parameters over the six bounding surfaces of a given volume grid the procedure has been demonstrated here for the mars science laboratory msl geometries such as aeroshell and canopy as well as the inflatable aerodynamic decelerator iad geometry and a 3d analytically defined geometry the new procedure also enables generation of single block grids for such geometries because the automatic boundary constraints permit the decay parameters to evolve as part of the solution to the elliptic grid system of equations these decay parameters are no longer just constants as specified in the conventional approach but functions of generalized coordinate variables over a given bounding surface since these decay functions vary over a given boundary orthogonal grids around any arbitrary simply connected boundary can be clustered automatically without having to break up the boundaries and the corresponding interior or exterior domains into various blocks for grid generation the new boundary constraints are not limited to the simply connected regions only but can also be formulated around multiply connected and isolated regions in the interior the proposed method is superior to other methods of grid generation such as algebraic and hyperbolic techniques in that the grids obtained here are c 2 continuous whereas simple elliptic smoothing of algebraic or hyperbolic grids to enforce c 2 continuity destroys the grid clustering near the boundaries

non monotonic iterative roots extended from characteristic intervals it has been treated as a difficult problem to find iterative roots of non monotonic functions for some pm functions which do not increase the number of forts under iteration a method was given to obtain a non monotonic iterative root by extending a monotone iterative root from the characteristic interval in this paper we prove that every continuous iterative root is an extension from the characteristic interval and give various modes of extension for those iterative roots of pm functions

review of some adaptive node movement techniques in finite element and finite difference solutions of partial differential equations abstract this article summarizes the results of a literature search for adaptive numerical methods of solving partial differential equations the methods discussed involve the adaptive movement of nodes so as to obtain a low level of solution truncation error while minimizing the number of nodes used in the calculation such methods are applicable to the solution of nonstationary flow problems that contain moving regions of rapid change in the flow variables surrounded by regions of relatively smooth variation flows with shock waves contact surfaces slip streams phase change interfaces and boundary layers can be modelled with great precision by these methods it will be shown that significant economies of execution can be attained if nodes are moved so that they remain concentrated in regions of rapid variation of the flow variables

an adaptive grid method and its application to steady euler flow calculations an adaptive remeshing procedure based on a cell volume deformation method is presented starting with an initial grid this method offers direct cell volume control through the specification of the transformation jacobian grid points are moved with appropriate grid velocities so that the specified cell volume distribution can be achieved at the end of the grid movement without adding or removing grid points the grid velocities are determined by solving a scalar poisson equation this method is applied to solving the compressible euler equations computational test cases of transonic flow over an airfoil are presented and demonstrate the desired control of grid sizes across shock waves

a method of constructing adaptive hexahedral moving grids a variational method of constructing an adaptive moving grid composed of hexahedral cells is considered the method is based on minimizing a functional defined on the riemannian manifold in space of variables including the physical space coordinates and components of a vector valued monitor function the mesh is generated in the manifold and then it is projected down to the physical region to provide the adaptive grid the integrand of the functional is the non dimensional ratio of the invariants of the metric the functional depends on metric elements of two metrics the curvilinear mesh generated in the manifold induces the monitor metric another control metric defined in a special way is responsible for additional cell shape control the issues of non degeneracy conditions for a hexahedral mesh are considered examples of adaptive grids are presented

nonlinear algebraic systems with discontinuous terms abstract using a multiple critical points theorem for locally lipschitz continuous functionals we establish the existence of at least three distinct solutions for a parametric discrete differential inclusion problem involving a real symmetric and positive definite matrix applications to tridiagonal fourth order and partial difference inclusions are presented

nonlinear elliptic eigenvalue problems with discontinuities abstract in this paper we study the existence of solution for two different eigenvalue problems the first is nonlinear and the second is semilinear our approach is based on results from the nonsmooth critical point theory in the first theorem we prove the existence of at least two nontrivial solutions when is in a half axis in the second theorem based on a nonsmooth variant of the generalized mountain pass theorem we prove the existence of at least one nontrivial solution for every r

a strongly coupled time marching method for solving the navier stokes andk ω turbulence model equations with multigrid many researchers use a time lagged or loosely coupled approach in solving the navier stokes equations and two equation turbulence model equations in a time marching method the navier stokes equations and the turbulence model equations are solved separately and often with different methods in this paper a strongly coupled method is presented for such calculations the navier stokes equations and two equation turbulence model equations in particular thek equations are considered as one single set of strongly coupled equations and solved with the same explicit time marching algorithm without time lagging a multigrid method together with other acceleration techniques such as local time steps and implicit residual smoothing is applied to both the navier stokes and the turbulence model equations time step limits due to the source terms in thek equations are relieved by treating the appropriate source terms implicitly the equations are also strongly coupled in space through the use of staggered control volumes the method is applied to the calculation of flows through cascades as well as over isolated airfoils convergence rate is greatly improved by the use of the multigrid method with the strongly coupled time marching scheme

second order godunov type scheme for reactive flow calculations on moving meshes the method of calculating the system of gas dynamics equations coupled with the chemical reaction equation is considered the flow parameters are updated in whole without splitting the system into a hydrodynamical part and an ode part the numerical algorithm is based on the godunov s scheme on deforming meshes with some modification to increase the scheme order in time and space the variational approach is applied to generate the moving adaptive mesh at every time step the functional of smoothness written on the graph of the control function is minimized the grid lines are condensed in the vicinity of the main solution singularities e g precursor shock fire zones intensive transverse shocks and slip lines which allows resolving a fine structure of the reaction domain the numerical examples relating to the chapman jouguet detonation and unstable overdriven detonation are considered in both one and two space dimensions

multiplicity result for a discrete eigenvalue problem with discontinuous nonlinearities using a three critical points theorem for nondifferentiable functionals we investigate a class of second order difference equation with discontinuous nonlinearities a new multiplicity result is obtained

a note on iterative roots of pm functions abstract in this paper we study iterative roots of pm functions a special class of non monotone functions problem 2 in w zhang pm functions their characteristic intervals and iterative roots ann polon math lxv 1997 119 128 is solved partly and theorem 4 in that paper is generalized

a variational hexahedral grid generator with control metric a variational method of constructing a spatial structured grid composed of hexahedral cells is presented the method executes a minimization of the variational functional the integrand of the functional is a ratio of the orthogonal invariants the functional depends on two metrics one metric is induced by a curvilinear mesh generated in the physical domain and the other control metric given in the canonical domain is responsible for an additional cell shape control for instance for condensing the coordinate surfaces and orthogonalization of the grid lines towards the domain boundary generally defining the control metric allows to generate an arbitrary given non folded mesh in the physical domain for every cell the functional is discretized on ten tetrahedra forming two dodecahedrons with the same vertices which span the hexahedral cell the discrete functional possesses an infinite barrier on the boundary of the set of non folded dodecahedral cells that ensures the construction of the non folded grid composed of such cells in the most practical cases the hexahedral grid with the same nodes is non folded as well the method of boundary nodes redistribution is considered examples of the grid construction are reported

curvature flow and entropy conditions applied to grid generation we describe a numerical technique to generate logically rectangular body fitted interior and exterior grids the technique is based on solving a hamilton jacobi type equation for a propagating level set function using techniques borrowed from hyperbolic conservation laws coordinate grid lines are kept smooth through curvature terms which regularize the equation of motion and upwind difference schemes which satisfy the correct entropy conditions of front propagation the resulting algorithm can be used to generate two and three dimensional interior and exterior grids around reasonably complex bodies which may contain sharp corners and significant variations in curvature the technique may also be easily extended to problems of boundary fitted moving grids

three critical points for perturbed nonsmooth functionals and applications abstract a recently established three critical points theorem of ricceri relating to continuously g teaux differentiable functionals based on a minimax inequality and on a truncation argument is extended to motreanu panagiotopoulos functionals as an application multiplicity results also yielding a uniform bound on the norms of solutions are obtained for variational hemivariational inequalities depending on two parameters

on the node fractional iterates let m be a bounded convex set of r n let f m r n be a continuous non bijective map we study solutions g of the functional equation g m x g g m 1 x f x m 2 this functional equation represents the problem of the fractional iteration for f

grid adaption based on modified anisotropic diffusion equations formulated in the parametric domain a new grid adaption algorithm for problems in computational fluid dynamics is presented the basic equations are derived from a variational problem formulated in the parametric domain of the mapping that defines the existing grid modification of the basic equations provides desirable properties in boundary layers the resulting modified anisotropic diffusion equations are solved for the computational coordinates as functions of the parametric coordinates and these functions are numerically inverted numerical examples show that the algorithm is robust that shocks and boundary layers are well resolved on the adapted grid and that the flow solution becomes a globally smooth function of the computational coordinates

three solutions to a perturbed nonlinear discrete dirichlet problem using critical point theory we study the existence of at least three solutions for a perturbed nonlinear dirichlet boundary value problem for difference equations depending on two positive parameters

an eigenvalue dirichlet problem involving the p laplacian with discontinuous nonlinearities a multiplicity result for an eigenvalue dirichlet problem involving the p laplacian with discontinuous nonlinearities is obtained the proof is based on a three critical points theorem for nondifferentiable functionals

existence and multiple solutions for a nonlinear system with a parameter abstract in this paper the critical point theory and the morse theory are employed to discuss the system of nonlinear equations of the form a u f u where the matrix a is not necessarily positive definite some existence and multiple solutions are obtained