grid generation techniques in computational fluid dynamics revue des differentes methodes on montre que les regions tridimensionnelles sont mieux traitees par fragmentation du champ en sous regions avec generation de grilles dans chaque sous region interet des grilles adaptatives couplees a des solutions physiques

alternating direction adaptive grid generation an alternating direction method has been developed to adaptively resolve numerical solutions to physical problems by moving the points of a coordinate grid with the solution or with the salient quantities derived from it a surface grid is defined over the grid on the physical region grid movement is computed on this abstractly defined solution surface with the use of pointwise weights and is then projected back to determine new locations in the physical region the weights cause clustering relative to the uniform conditions of surface arc length and are formed with magnitudes of quantities that are used for point attraction when the salient quantities have been used to form the surface the gradients are implicitly revolved and surface curvature becomes the primary remaining quantity in each coordinate direction the magnitude of normal curvature is used with arbitrary weights a movement cycle generally consists of a sweep through the coordinate curves in a given direction on a curve by curve basis and then in alternating directions to do the same thing until all directions have been covered

a 3 4713 approximation algorithm for the capacitated multicast tree routing problem given an underlying communication network represented as an edge weighted graph g v e a source node email protected v a set of destination nodes email protected v and a capacity k which is a positive integer the capacitated multicast tree routing problem asks for a minimum cost routing scheme for source s to send data to all destination nodes under the constraint that in each routing tree at most k destination nodes are allowed to receive the data copies the cost of the routing scheme is the sum of the costs of all individual routing trees therein improving on our previous approximation algorithm for the problem we present a new algorithm which achieves a worst case performance ratio of email protected where r denotes the best known approximation ratio for the steiner minimum tree problem since r is about 1 55 at the writing of the paper the ratio achieved by our new algorithm is less than 3 4713 in comparison the previously best ratio was email protected 3 5375

matrix traces on c algebra mn a abstract in this note for a c algebra a we define a matrix trace on the c algebra m n a to be a positive linear mapping m n a a such that u au a a m n a u u m n a and a 2 a 2 a 0 we give some basic properties of a matrix trace and prove that if a is a unital abelian c algebra then a map is an a module matrix trace on m n a if and only if there exists an element of a with 0 2 such that a tr a a a ij m n a where tr a i 1 n a ii

numerical grid generation in arbitrary surfaces through a second order differential geometric model abstract in this paper a set of second order partial differential equations for the generation of coordinates in a given surface has been developed and then solved numerically to demonstrate its potential for surface coordinate generation the proposed equations are not some arbitrarily chosen equations but are a consequence of the formulae of gauss and thus carry with them an explicit dependence on the geometric properties of the given surface furthermore these equations are easy to solve and require only the specification of the bounding curves to provide the dirichlet boundary conditions for their solution results of coordinate generation both in simply and doubly connected regions on some known surfaces with the option of coordinate redistribution have been presented extension of this technique to arbitrary surfaces seems to be straightforward

numerical generation and grid controls of boundary fitted conformal grids in multiply connected regions based on the study of grid generation methods in a simply connected region int j numer meth engng 1997 40 343 with the aid of a new transformation from a multiply connected region to a simply connected domain the grid generation in a multiply connected domain is carried out in this paper 2 d conformal and orthogonal grids are generated in a multiply connected region mostly involving a multiple blocks method developed in the present study to avoid grid distortion on the degenerated boundary caused by physical singularities controls of grids through specified points are implemented without losing the conformal properties further applications of this grids controlling technique for points inside the region the nested grids and the multiple blocks method are introduced and discussed comprehensively numerical examples are taken to demonstrate the validity and applicability of the theorems copyright 2006 john wiley sons ltd

fast generation of three dimensional computational boundary conforming periodic grids of c type a fast computer program grid3c has been developed to generate multilevel three dimensional c type periodic boundary conforming grids for the calculation of realistic turbomachinery and propeller flow fields the technique is based on two analytic functions that conformally map a cascade of semi infinite slits to a cascade of doubly infinite strips on different riemann sheets up to four consecutively refined three dimensional grids can be automatically generated and permanently stored on four different computer tapes grid nonorthogonality is introduced by a separate coordinate shearing and stretching performed in each of three coordinate directions the grids can be easily clustered closer to the blade surface the trailing and leading edges and the hub or shroud regions by changing appropriate input parameters hub and duct or outer free boundary can have different axisymmetric shapes a vortex sheet of arbitrary thickness emanating smoothly from the blade trailing edge is generated automatically by grid3c blade cross sectional shape chord length twist angle sweep angle and dihedral angle can vary in an arbitrary smooth fashion in the spanwise direction input coordinates must be cartesian while the output grid coordinates can be cartesian or cylindrical

size constrained tree partitioning approximating the multicast k tree routing problem in the multicast k tree routing problem a data copy is sent from the source node to at most k destination nodes in every transmission the goal is to minimize the total cost of sending data to all destination nodes which is measured as the sum of the costs of all routing trees this problem was formulated out of optical networking and has applications in general multicasting several approximation algorithms with increasing performance have been proposed in the last several years the most recent ones rely heavily on a tree partitioning technique in this paper we present a further improved approximation algorithm along the line the algorithm has a worst case performance ratio of 54 r 32 where r denotes the best approximation ratio for the steiner minimum tree problem the proofs of the technical routing lemmas also provide some insights into why such a performance ratio could be the best possible that one can get using this tree partitioning technique

a computational model for organism growth based on surface mesh generation the generation of curvilinear coordinate meshes has been a powerful tool in computational fluid dynamics calculation in the computational modelling of the fluid flows around complex bodies such as an airfoil or a complete aircraft this same technique may be applied to many other computational models in this work the approach is used as part of a computational model to generate simple geometries associated with biological forms or organisms the model adopted was first proposed by cummings and simulates morphogenesis in terms of the geometrical changes occurring during the growth and development of simple organisms this model depends on the generation of a curvilinear coordinate mesh on the surface of an organism previous work has concentrated on the model and its use in generating axisymmetric shapes that are simple models of elementary organisms in this work we describe how the model may be extended to geometrical symmetry breaking this paper describes the methodology of this extension and demonstrates it in the simulation of tentacle growth the resulting computational technique makes it possible to link models of cell bio chemistry and surface deformation

the determination of the poles of the mapping function and their use in numerical conformal mapping abstract let be the function which maps conformally a simply connected domain onto the unit disc this paper is concerned with the problem of determining the dominant poles of in compl and of using this information in order to obtain accurate numerical approximations to by means of the bergman kernel method

coordinate generation with precise controls over mesh properties abstract coordinate generation techniques with precise controls over mesh properties are mathematically developed the controls are precise because mesh properties can be explicitly specified within a local region independent of the mesh elsewhere the local regions can adjoin boundaries where a particular mesh form can greatly simplify a problem or can be used for a smooth juncture between distinct coordinate systems where in effect a branch cut with a prescribed geometry and mesh distribution can be obtained away from boundaries a local region can be given a particular mesh form to model internal objects or to simplify a problem

a compressed pca subspace method for anomaly detection in high dimensional data random projection is widely used as a method of dimension reduction in recent years its combination with standard techniques of regression and classification has been explored here we examine its use for anomaly detection in high dimensional settings in conjunction with principal component analysis pca and corresponding subspace detection methods we assume a so called spiked covariance model for the underlying data generation process and a gaussian random projection we adopt a hypothesis testing perspective of the anomaly detection problem with the test statistic defined to be the magnitude of the residuals of a pca analysis under the null hypothesis of no anomaly we characterize the relative accuracy with which the mean and variance of the test statistic from compressed data approximate those of the corresponding test statistic from uncompressed data furthermore under a suitable alternative hypothesis we provide expressions that allow for a comparison of statistical power for detection finally whereas these results correspond to the ideal setting in which the data covariance is known we show that it is possible to obtain the same order of accuracy when the covariance of the compressed measurements is estimated using a sample covariance as long as the number of measurements is of the same order of magnitude as the reduced dimensionality we illustrate the practical impact of our results in the context of predicting volume anomalies in internet traffic data

three dimensional grid generation using elliptic equations with direct grid distribution control boundary fitted computational meshes are generated by solving a system of elliptic partial differential equations the grid points at the boundary surfaces are arbitrarily distributed the interior grid distribution is controlled by a technique which insures a grid line orthogonality at the boundaries the grid is automatically adjusted while the grid line orthogonality at the boundaries is achieved in an iterative process this technique is demonstrated for three different configurations desirable cell size and shape are obtained for the boundary shapes which normally cause poor grid distribution

use of streamline coordinates in the numerical solution of compressible flow problems abstract iterative correction of streamline geometry is used as the basis of a numerical method for subsonic isentropic steady flow problems direct solution of coupled ordinary differential equations is applicable to the corresponding supersonic problem the method is formulated in three dimensions but for illustrative purposes is applied to a two dimensional jet involving a free surface condition the method may be particularly economical where intuition or experience provides a good starting guess for streamline geometry

boundary fitted coordinate systems for numerical solution of partial differential equations a review a comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given some general mathematical framework and error analysis common to such coordinate systems is also included the general categories of generating systems are those based on conformal mapping orthogonal systems nearly orthogonal systems systems produced as the solution of elliptic and hyperbolic partial differential equations and systems generated algebraically by interpolation among the boundaries also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation dynamically adaptive coordinate systems coupled with the physical solution and time dependent systems that follow moving boundaries are treated references reporting experience using such coordinate systems are reviewed as well as those covering the system development

general curvilinear coordinate systems the basic ideas of the construction and use of numerically generated boundary fitted coordinate systems for the numerical solution of partial differential equations are discussed with such coordinate systems all computation can be done on a fixed square grid in the rectangular transformed region regardless of the shape or movement of the physical boundaries a number of different types of configurations for the transformed region and the basic transformation relations from a cartesian system to a general curvilinear system are given the material of this paper is applicable to all types of coordinate system generation

a matrix trace inequality abstract in this note the matrix trace inequality for positive semidefinite matrices a and b tr ab m tr a 2 m tr b 2 m 1 2 is established where m is an integer the above inequality improves the result given by yang j math anal appl 250 2000 372 374

generation of boundary conforming grids around wing body configurations using transfinite interpolation a computational procedure for generating three dimensional nonorthogonal surface fitted mesh systems around wing fuselage configurations is presented the method is based on the concept of transfinite interpolation which has been extended to handle very general mapping function specifications at the boundaries thereby making it possible to generate single block mappings with geometry data specified only at the outer boundaries of the computational domain since it is a direct algebraic mapping technique the method is very inexpensive in terms of computer cost different types of possible mappings are compared with respect to resolution and economy of nodal points a procedure for a novel type of mapping designated type o o is described and several plots of generated grids demonstrate the capabilities of the method the singular lines inherent in every three dimensional mesh for this type of surface geometry are also discussed

a matrix trace inequality abstract in a short note y yang 2 solved a problem set by r bellman 1 it is not difficult to prove a slightly relaxed version of yang s result we shall use a different method

numerical generation of composite three dimensional grids by quasilinear elliptic systems a technique is presented for constructing a boundary conforming grid throughout a general three dimensional flow region as a composite of subregion grids each subregion grid is generated numerically by solving a quasilinear system of elliptic equations the boundary values represent nodal points in a quasi two dimensional grid that covers the curved surface bounding the subregion and are generated numerically by a modified elliptic system the boundary values are used to compute grid control parameters that are contained in the elliptic systems this provides flexible control over the distribution of grid points in the interior of the region in that the interior grid distribution is governed by the distribution of points on the boundary as well as by the boundary s geometric shape a primary feature of the technique is that the composite three dimensional grid remains both continuous and smooth across the surface of the juncture between any two adjoining subregions the present paper elucidates the details of the method and of its implementation and displays numerical results for both surface grids and space grids comprehensive results are displayed for a three dimensional grid about a simple wing body combination a numerical example is presented of a surface grid on a naca 0012 airfoil with high resolution of the wingtip region

a matrix trace inequality for products of hermitian matrices abstract in this note the following matrix trace inequality for products of hermitian matrices a and b tr ab 2 k tr a 2 k b 2 k is established where k is an integer the above inequality also partly answers a conjecture in bellman in proceedings of the 2nd international conference on general inequalities e f beckenbach ed pp 89 90 birkhauser basel 1980

adaptive gridding for finite difference solutions to heat and mass transfer problems abstract the authors purpose in writing this paper is to review some of their recent work in the calculation of optimal meshes for the solution of parabolic and elliptic partial differential equations pde they first explain our strategies for the adaptive placement of mesh points in addition they make some speculation as to promising avenues for future research in mesh adaptation finally they discuss examples of the application of adaptive gridding to problems of heat and mass transfer they draw these examples from our work in combustion modeling in obtaining numerical solutions of pdes the spatial derivatives are often approximated by discrete representations on a mesh network the accuracy of any numerical solution depends in an important way on the relationship of the location of the mesh points to changes in the dependent variables the authors objective is to investigate finite difference methods in which the mesh networks adapt themselves dynamically to obtain accurate solutions such methods represent an important advance in overcoming a major shortcoming of traditional fixed mesh methods which are often unable to resolve accurately steep fronts or sharp peaks

a trace class operator inequality abstract in this note we generalize the inequality about the trace of positive semidefinite matrix tr a b k tr a k tr b k to hilbert space and obtain a relevant inequality about positive trace class operator

high level continuity for coordinate generation with precise controls abstract the development of precisely controlled coordinate generation techniques is continued from a first study to include the higher order smoothness which is necessary for three dimensional applications in the first study the controls came from the use of local piecewise linear interpolants in the general multisurface transformation the consequent integration therein resulted in coordinates with continuity up to first derivatives and with the capability to prescribe uniformity either locally or globally for the family of transverse coordinate curves the admission of uniformity placed a constraint upon the general interpolants which was trivially satisfied in the piecewise linear case with smoother piecewise constructions the constraint is used herein along with a requirement for coordinate curves to have the most general possible curvature properties the result is a class of coordinate transformations with continuity extended up to higher derivatives that retains the precise local controls displayed in the first study and that can be used in three or more dimensions

a vectorized finite volume adaptive grid algorithm for navier stokes calculations an adaptive grid finite volume method has been used to solve the navier stokes equations for complete forebody and afterbody flowfields around blunt bodies the code which is applicable for axisymmetric or two dimensional flows allows the mesh to adjust during the computation to provide a closer spacing of mesh points in regions of high gradients thus minimizing the number of required computational points the solution technique is explicit utilizing a maximum time step advancement at each grid point to accelerate convergence to the steady state the code has been fully vectorized for efficient solution on the cyber 203 computer a very flexible rezoning routine is used to concentrate mesh points anywhere in the field either by a user defined weighting function or by allowing high gradient regions to adjust the grid the grid adjustment routine is implicit in nature and represents a very small portion of the total computational cost currently the code runs in approximately 1 6 x 10 5 seconds per grid point per iteration

routing algorithm for multicast under multi tree model in optical networks to establish a multicast connection in a wavelength routed optical network two steps are needed under the multi tree model one is to construct a set of light trees rooted at the source node such that in each of them at most a specified number of destination nodes are allowed to receive the data and every destination node must be designated in one of them to receive the data the other is to assign a wavelength to each of the produced light trees in such a way that two light trees must be assigned two distinct wavelengths if they use a common link in this paper we mainly study how to construct a multicast routing of minimal cost under the multitree model in optical networks where the routing cost is total costs of the produced light trees we propose a 4 approximation algorithm for this np hard problem

compactly supported fundamental functions for spline interpolation in this paper various ways of constructing locally supported fundamental splines leading to highly accurate local interpolation schemes are proposed and analyzed

a domain decomposition and overlapping method for the generation of three dimensional boundary fitted coordinate systems abstract a new method is presented for numerically generating boundary fitted coordinate systems for arbitrarily shaped three dimensional regions in the method the three dimensional region of interest is decomposed into several hexahedrons each of which has two grid surfaces overlapped with each of the neighboring hexahedrons based on this method a new computer program grid 3d has been developed which allows the generation of an unlimitted number of different types of coordinate systems application to a variety of geometries confirms that grid 3d is a convenient and efficient tool in the generation of boundary fitted coordinate systems even for a considerably complicated configuration consisting of many different components

transfinite mappings and their application to grid generation the two essential ingredients of any boundary value problem are the field equations which describe the physics of the problem and a set of relations which specify the geometry of the problem domain mesh generators or grid generators are preprocessors whicd decompose the problem domain into a large number of interconnected finite elements or curvilinear finite difference stencils a number of such techiques have been developed over the past decade to alleviate the frustation and reduce the time involved in the tedious manual subdividing of a complex shaped region or 3 d structure into finite elements our purpose here is to describe how the techniques of bivariate and trivariate blending function interpolation which were originally developed for and applied to geometric problems of computer aided design of sculptured surface and 3 d solids can be adapted and applied to the geometric problems of grid generation in contrast to other techniques which require the numerical solution of complex partial differential equations and hence a great deal of computing the transfinite methods proposed herein are computationally inexpensive

conformal grid generation this paper treats conformal mapping as it relates to the generation of grids to be used for flow simulation classical and contemporary mapping methods are discussed and compared in some detail numerous suggestions are included to help the reader avoid common pitfalls and the mapping methods believed most promising are identified

assessing the quality of curvilinear coordinate meshes by decomposing the jacobian matrix an algebraic decomposition of the jacobian matrix j x i email protected j which relates physical and computational variables is presented this invertible decomposition parameterizes the mesh by the physically intuitive qualities of cell orientation cell orthogonality cell volume and cell aspect ratio this decompostion can be used to analyze numerically generated curvilinear coordinate meshes and to assess the contribution of the mesh to the truncation error for any specific differential operator and algorithm this is worked out in detail for laplace s equation in nonconservative and conservative forms an analysis by g h k 1 of the mesh contribution to truncation error for the full potential code tair is given in abbreviated form the variables introduced here and their derivatives are also natural lagrange multipliers for adaptive mesh algorithms based on a variational principle

bf l _1 constrained normalized lms algorithms for adaptive beamforming we detail in this paper an l 1 norm linearly constrained normalized least mean square l 1 cnlms algorithm and its weighted version l 1 wcnlms applied to solve problems whose solutions have some degree of sparsity such as the beamforming problem in uniform linear arrays standard hexagonal arrays and non standard hexagonal antenna arrays in addition to the linear constraints present in the cnlms algorithm the l 1 wcnlms and the l 1 cnlms algorithms take into account an l 1 norm penalty on the filter coefficients which results in sparse solutions producing thinned arrays the effectiveness of both algorithms is demonstrated via computer simulations when employing these algorithms to antenna array problems the resulting effect due to the l 1 norm constraint is perceived as a large aperture array with few active elements although this work focuses the algorithm on antenna array synthesis its application is not limited to them i e the l 1 cnlms is suitable to solve other problems like sparse system identification and signal reconstruction where the weighted version the l 1 wcnlms algorithm presents superior performance compared to the l 1 cnlms algorithm

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

on matrix trace inequalities and related topics for products of hermitian matrices abstract alternative proofs of some simple matrix trace inequalities of bellman in general inequalities 2 proceedings 2nd international conference on general inequalities e f beckenbach ed pp 89 90 birkhauser basel 1980 neudecker j math anal appl 166 1992 302 303 and yang j math anal appl 133 1988 573 574 are considered and further properties of products of hermitian and positive semi definite matrices are investigated

three dimensional grid generation using poisson equations a method for constructing a boundary fitted grid system for arbitrary three dimensional 3 d configurations is presented the grid generation procedure is formulated through the poison equations a feature of the scheme is that grid lines in the grid system are smothly interconnected between the boundary co ordinates the control of the grid distributions in the interior of the system has implemented by the forcing functions appearing in the poisson equations and by the boundary coordinates reasonable cell sizes and shapes are obtained even for extreme boundary shapes which normally tend to cause poorly controlled grid distributions

algebraic grid generation three methods are described for transforming grids in bounded two and three dimensional physical domains into a uniform grid in a rectangular computational domain the methods are based on mathematical interpolation functions and do not require the solution of differential equations or the use of complex variables they are simply referred to as algebraic methods and are called transfinite interpolation the multisurface method and the two boundary technique the primary advantage of the methods is that they provide explicit control of physical grid shape and physical grid spacing secondly they require relatively few computations consequently the application of inter active computer graphics in conjunction with the methods is advocated for rapid generation of grids the basic mathematical structure of each method is described and particular attention is given to surface representation parameter variable generation and control function generation physical boundary topology and grid derivative requirements are presented

an efficient full potential implicit method based on characteristics for supersonic flows la methode utilise la theorie de la propagation de signal caracteristique pour simuler avec precision la structure de l ecoulement qui inclut des ondes de choc et un ecoulement transversal mixte elliptique hyperbolique

solution of viscous internal flows on curvilinear grids generated by the schwarz christoffel transformation abstract a method is presented for combining an accurate orthogonal curvilinear coordinate generation procedure with a fast accurate and stable forward marching viscous flow solution technique to solve for the flow field in arbitrary axisymmetric ducts in this method the coordinates are generated from the plane potential flow streamlines and potential lines using the schwarz christoffel transformation with a composite finite difference formula which is valid everywhere and which treats the poles exactly by analytic integration since the coordinate streamlines approximate the actual streamlines the equations of motion for viscous compressible flow can be parabolized so as to solve for both the boundary layers and core flow in a single streamwise pass the versatility of the method is demonstrated by two examples of viscous compressible swirling flow through complex radial gas turbine passages author

application of variational methods in the fixed and adaptive grid generation abstract the present paper is devoted to the development of a system of partial differential equations for the purpose of generating fixed and adaptive numerical grids the resulting equations are applicable to both two and three dimensional grid generation problems equations have also been presented for both the fixed and adaptive grids in curved surfaces

elliptic grid generation conclusions some modi cations for the restriction and prolongation process of multigrid methods have been presented that are necessary for the simulation of complex supersonic ow elds damping of the restricted residual error and blending between a second order central and a rst order upwind prolongation enable convergence in supersonic ows and resulted in strong reductions in cpu time the advantageof this kind of blending is the reduction to rst order prolongationat shock waves only

numerical generation of three dimensional coordinates between bodies of arbitrary shapes abstract this paper is devoted to the numerical solution of a set of second order elliptic partial differential equations for the generation of three dimensional curvilinear coordinates between two arbitrary shaped bodies the central idea of the method is to generate a series of surfaces between the given inner and the outer boundary surfaces and then to connect these surfaces in such a manner so as to have a sufficiently differentiable three dimensional coordinate net in the enclosed region it is important to state here that the proposed equations for the numerical solution form a consistent set of second order elliptic equations which are a consequence of the equations of gauss for a surface additional constraints are then imposed which besides yielding the simplest form of equations for numerical purposes also preserve the essential geometric properties of the generated surfaces author

the use of solution adaptive grids in solving partial differential equations the grid point distribution used in solving a partial differential equation using a numerical method has a substantial influence on the quality of the solution an adaptive grid which adjusts as the solution changes provides the best results when the number of grid points available for use during the calculation is fixed basic concepts used in generating and applying adaptive grids are reviewed in this paper and examples illustrating applications of these concepts are presented

an improved approximation algorithm for capacitated multicast routings in networks let g v e be a connected graph such that each edge email protected e is weighted by nonnegative real w e let s be a vertex designated as a source k be a positive integer and email protected v be a set of terminals the capacitated multicast tree routing problem cmtr asks to find a partition z 1 z 2 z of s and a set t 1 t 2 t of trees of g such that z i consists of at most k terminals and each t i spans z email protected s the objective is to minimize i 1 w t i where w t i denotes the sum of weights of all edges in t i in this paper we propose a 3 2 4 3 r approximation algorithm to the cmtr where r is the best achievable approximation ratio for the steiner tree problem

volume conserving smoothing for piecewise linear curves surfaces and triple lines abstract we present smoothing algorithms for piecewise linear curves surfaces and triple lines of intersection of surfaces that are based on the the idea of sequentially relaxing either individual nodes or edges in the mesh each relaxation is designed both to smooth the mesh and to conserve down to round off error the area or volume enclosed by the curve or surface for the case of smoothing surfaces and lines of intersection of surfaces each relaxation consists of a pure smoothing component and a volume conserving correction which is chosen to be of minimum norm since surfaces and triple intersection lines can be conservatively smoothed the algorithms are suitable for improving multimaterial grids used by physics simulations where exactly conserving the volume of each individual material may be a requirement or at least highly desirable the algorithms are also suitable for smoothing piecewise linear functions of one or two variables while simultaneously preserving their integrals we show examples of the application of the more powerful edge based algorithms to curve surface and multimaterial volume grids and to a thin film simulation

the direct variational grid generation method extended to curves the direct variational grid generation method is extended to weighted grids on smooth curves in r 3 by introducing a new weighted chord functional the new functional is positive definite at a global minimum but not everywhere convex lack of convexity appears to a significant obstacle in extending the direct variational approach to surface grid generation

conformal mappings for internal viscous flow problems abstract isolated occlusions in a straight channel or pipe can be described analytically in a fashion that allows efficient accurate computation of nonlinear flow fields through conformal transformations

upwind schemes and boundary conditions with applications to euler equations in general geometries abstract a unified treatment of several upwind shock capturing algorithms is presented each algorithm has a riemann initial value problem as its basis the treatment of boundaries involves solving the associated riemann initial boundary value problem the first author s algorithm applied to multidimensional euler equations in general geometries is then presented its worth is verified by various calculations which include mach 8 supersonic flow past a circular cylinder

generalized vortex lattice method for planar supersonic flow the generalized vortex lattice method for oscillating lifting surfaces in subsonic flow is extended to the supersonic planar regime

a hybrid differential algebraic method for three dimensional grid generation an effective method is presented for generating three dimensional boundary fitted grids the technique is based on a two step procedure involving the solution of differential equations and algebraic interpolations a coarse grid with an arbitrarily given point distribution at all the boundaries is first generated by solving the poisson equations with the forcing functions being automatically adjusted during the grid generation to achieve orthogonality at the boundaries a fine grid with a desired concentration of grid lines is then obtained by fitting tricubic spline functions on the coarse grid a primary feature of the method is that the mesh spacing can be quickly and explicitly changed while retaining the smoothness over the entire field and the boundary orthogonality applications to a variety of geometries demonstrate the performance of the method

elliptic grid generation various types of generating systems for boundary conforming coordinate systems based on the numerical solution of systems of elliptic partial differential equations are discussed particular emphasis is given to the determination of functions in these equations which control the distribution of the curvilinear coordinate lines in the field

vorticity jump in surface coordinates across a shock in nonsteady flow we obtain hayes formulas in the coordinates attached to an arbitrarily moving and deforming shock surface that occurs in nonsteady flows the formulas given here can be incorporated in a flow solver that uses a coordinate system attached to a moving and deforming shock surface

grid evolution in time asymptotic problems the selection of the proper coordinate system in solving any fluid flow or heat transfer problem is a very important consideration a new technique of moving mesh points in physical space is introduced so as to reduce the error in a computed asymptotic solution relative to that obtained using a fixed mesh the technique has been used to solve the simple viscous burgers equation in one and two dimensions substantial error reductions are demonstrated the treatment of boundary points and the effect of using different error criteria in generating grids are discussed

surface grid generation in the tangent plane abstract recent work has shown that parametrically formulated surface grid generation algorithms are severely limited by solution bifurcations of the discretized equations such bifurcations become apparent if the given surface has points of high curvature the present study demonstrates that discrete equations derived from an alternative physical variables formulation do not result in similar bifurcations however the natural formulation using physical variables is not entirely successful on surface of high curvature due to excessive truncation error in the solution grid it is shown that both bifurcations and truncation error are overcome by formulating the grid generation equations in terms of projections of the physical variables onto the surface tangent plane

an algebraic elliptic algorithm for boundary orthogonal grid generation to produce grids conforming to the boundary of a physical domain with boundary orthogonality features algebraic methods like transfinite interpolating schemes can be profitably used moreover the coupling of hermite type also interpolating prescribed boundary direction schemes with elliptic methods turns out to be effective to overcome the drawback of both algebraic and elliptic strategies thus in this paper we present an algorithm for the generation of boundary orthogonal grids which couples a mixed hermite algebraic method with a boundary orthogonal elliptic scheme numerical tests on domains with classical geometries show satisfactory performances of the algorithm and coupling effectiveness in achieving grid boundary orthogonality and smoothness

a synopsis of elliptic pde models for grid generation this paper is devoted to an analytical comparison of the various elliptic partial differential equation pde models which are in current use for grid generation these comparisons particularly between the equations from the laplace poisson system and the equations from a gaussian approach have yielded useful expressions connecting the 3d laplacians and the surface beltramians this effort specifically been successful when the transverse coordinate leaving the surface is orthogonal to the surface equations which are derivable from cartesian type poisson and those obtained by using the variational priciple in surface coordinates have also been considered

the numerical differentiation of discrete functions using polynomial interpolation methods a fortran subroutine package called dermod has been written for calculating numerical approximations for the spatial derivatives of a function defined only on a discrete set of data points the routines are designed to complement many existing codes for solving ordinary and partial differential equations and for the interpolation of tabular data we describe some new numerical differentiation algorithms discuss a mapping procedure for nonuniform grids and explain the program methodology used in designing the software

automatic algebraic coordinate generation a computer software system has been developed to automatically generate two dimensional coordinates from algebraic transformations for topologically complex regions a smooth assembly of the transformations can be used to automatically produce a composite mesh where a general gridded format is retained the algebraic mesh generation system consists of a collection of operator subroutines which are applied to an established data structure and which automatically perform the necessary parts of mesh construction from a sequence of multisurface transformations the system operators are discussed taking into account the data base the order of application direct surface generators geometric surface operators surface generators from existing surfaces transverse operators mesh operators assembly operators and data visualization operators attention is given to applications related to airfoils

the geometrical construction of pointwise distributions on curves a method is developed to generate desirable pointwise distributions along curves this is accomplished with a simple geometrical construction which provides a global parameter for curvature clustering together with other parameters for arbitrary local clustering specifications the level of available precision is considerable in that exact numbers of points can be assigned to both local clusters and to curvature simultaneously with specified spacing from the endpoints the basic construction simply involves the generation of an auxiliary curve along outward normal directions from the given one the distribution results when uniform arc length increments are taken along the auxiliary curve and are projected back along the normals to our given curve this construction can be applied either directly or in the form of equivalent weight functions moreover it is valid regardless of whether the curve lies in euclidian space or in surfaces and regardless of the dimensionality of the space in which the curve lies

computational conformal mapping for surface grid generation the paper describes the development and application of a new approach for formulating an elliptic generation system on parametrically defined surfaces the present derivation of the surface equations proceeds in two steps first conformal mapping of smooth surfaces onto rectangular regions is utilized to derive a first order system of partial differential equations analogous to beltrami s system for quasi conformal mapping of planar regions second a general elliptic generation system for three dimensional surfaces including forcing functions is formulated based on beltrami s system and quasi conformal mapping the resulting elliptic system is solved using an iterative method on arbitrary surfaces represented analytically by rational b splines the overall effect of this approach is a reliable and versatile elliptic method for generating and improving surface grids examples will be presented to demonstrate the application of the method in constructing practical grids

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

on one dimensional stretching functions for finite difference calculations abstract the class of one dimensional stretching functions used in finite difference calculations is studied for solutions containing a highly localized region of rapid variation simple criteria for a stretching function are derived using a truncation error analysis these criteria are used to investigate two types of stretching functions one is an interior stretching function for which the location and slope of an interior clustering region are specified the simplest such function satisfying the criteria is found to be one based on the inverse hyperbolic sine it was first employed by thomas et al aiaa j 10 1972 887 the other type of function is a two sided stretching function for which the arbitrary slopes at the two ends of the one dimensional interval are specified the simplest such general function is found to be one based on the inverse tangent the special case where the slopes were both equal and greater than one was first employed by roberts the general two sided function has many applications in the construction of finite difference grids examples of such applications are found in the listed references

theoretical foundation of the equations for the generation of surface coordinates references 1 alien h j and perkins e w characteristics of flow over inclined bodies of revolution naca rm a50l07 march 1951 fiechter m wirbelsysteme schlanker rotationskorper und die aerodynamischen kra fte dglr jahrbuch 1969 pp 77 85 hunt b l asymmetric vortex forces and wakes on slender bodies aiaa paper 82 1336 aug 1982 kompenhans j and hartmann k strftmungssichtbarmachung an einem hochangestellten ogivkreiszylinderrumpf mit hilfe der laser lichtschnittmethode dfvlr fb 86 45 1986 prandtl l and tietjens o applied hydroand aeromechanics mcgraw hill new york and london 1934 foeppl l wirbelbewegung hinter einem kreiszylinder sitz ber kdnigl bay akademie d wissenschaften math phys klasse jg 1913 munchen jan 1913 sun y c shen s f and zhu z a numerical simulation of the flow development behind a circular cylinder started from rest and a criterion for investigating unsteady separating flows ada mechanica vol 71 jan 1988 pp 1 20 dyer d e fiddes s p and smith j h b asymmetric vortex formation from cones at incidence a simple inviscid model aeronautical quarterly vol 33 nov 1982 pp 293 312 ng t on leading edge vortex and its control aiaa paper 89 3346 aug 1989 yi8 gao 0 gpo a sai3 a surface christoffel symbols of the second kind

an integral equation method for the numerical conformal mapping of interior exterior and doubly connected domains a numerical method based on the integral equation formulation of symm is described for computing approximations to the mapping functions which accomplish the following conformal maps a the mapping of a domain interior to a closed jordan curve onto the interior of the unit disc b the mapping of a domain exterior to a closed jordan curve onto the exterior of the unit disc c the mapping of a doubly connected domain bounded by two closed jordan curves onto a circular annulus the numerical method is based on approximating the unknown source density by cubic splines and singular functions and is particularly suited for the mapping of difficult domains having sharp corners

determination of surfaces using beltramians at the separation of two fluid media this paper describes a model formed of a set of elliptic partial differential equations involving beltramians of coordinates for generating the curved surfaces at the separation of two different media the main point is that the differential equations are formed in such a way that the sum of the local principal curvatures appears explicitly which in turn depends on the physical and chemical properties of the two media this dependence is quite obvious as far as the problem of surface tension is concerned the developed partial differential equations have been solved numerically for the simple case of a fluid in contact with a vertical wall and for generation of the surface of revolution due to the capillary effect in a tube of small radius these examples demonstrate the application of the method when the surface tension is important the method may be of value in tackling other problems such as the generation of synclastic and anticlastic surfaces provided that the local principal curvatures can be related to the coordinates by using some physical principle

surface grid generation based on elliptic pde models abstract elliptic grid generation methods for the construction of surface grids are discussed an improved elliptic surface grid generation method for parametric surfaces will be presented two methods of imposing grid orthogonality have been implemented several control function options are available for maintaining a desired distribution of grid points on the surface the overall effect of these improvements is a reliable and versatile elliptic method for generating and improving surface grids examples will be presented to demonstrate the application of the method in generating practical grids

steady viscous flows by compact differences in boundary fitted coordinates fourth order compact differencing is applied to the steady solution of two dimensional viscous incompressible flows at moderate reynolds number the physical region where the fluid flow occurs is mapped onto a rectangle by means of the boundary fitted coordinates transformation method the design of the general numerical algorithm rests upon discretization of velocity components and pressure field at the same grid points the validity of this procedure is assessed by the investigation of a stokes test problem fourth order numerical results are compared to the analytical solution second order results and chebyshev tau approximation it is shown that fourth order differences provide good precision particularly when the ability of generating irregular meshes is fully exploited standard problems as poiseuille and couette flows and the square cavity problem are solved the fourth order results on coarse mesh compare favourably with other techniques such as finite element method and second order differences a global convergence analysis was performed on a two dimensional problem with smooth boundary conditions one observes a rate of convergence of order four for the velocity components and of order three for the pressure for the square cavity problem with the two corner singularities the rates of convergence are decreased by almost an order of magnitude the solution of a plane constricted channel flow enhances the overall improvement in accuracy gained by the treatment of the geometrically complex region through the mapping technique

variational curve and surface grid generation abstract variational algorithms that control the lengths of grid lines cell areas and the orthogonality of grid lines can be used for generating boundary conforming grids on surfaces additional geometric control is provided by using a reference grid while solution adaptivity is achieved by using weights in a typical application the reference grid can be used to produce an exponential compression of the grid at a boundary while the solution adaptive weights are used to make the grid spacing inversely proportional to the gradient when the gradient is large of some solution being computed on the grid the grid is adapted on both the interior and boundary of the surface the algorithm performs these tasks with exceptional precision as demonstrated in the examples presented here

a general mapping procedure for variable area duct acoustics a general mapping procedure is described and applied to the study of noise propagation in variable area ducts the mapping provides a boundary fitted co ordinate system which is ideal for the finite difference solution of acoustic fields with irregular boundaries without the burden of large matrices required by finite element methods the procedure is first described in general and then applied to a particular two dimensional geometry under current experimental investigation this method should be ideally suited to the study of high frequency noise propagation in variable area ducts and in cases where the far field is included in the calculation procedure moreover the current approach can be directly extended to three dimensions resulting in numerical calculation over a rectangular parallelepiped in the transformed plane