Eiseman, Peter R.

Interactive Topology Generation and Grid Viewer for GridPro / az3000

Cheng Z., Eiseman P.R.

Published in 1996

A powerful interactive graphical manager has been established for GridPro® / az3000, the multiblock grid generation system which is fully automatic once a grid topology is given. While the generation of grid topology represents a substantially smaller task, the next question is how fast and easily can that task be accomplished. The answer given here comes with the use of interactive graphics. The subject of this paper is even more inclusive since the integrated use of graphics is employed to manage the entire grid generation process.

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Grid Generation for Aerospace Applications

Hauser J., Spel M., Muylaert J., Eiseman P.R.

Published in 1995

These lecture notes are the nonmathematical version of a general report on grid generation. These notes are the continuation of courses in grid generation that have been given at the Von Karman Institute in March, 1992, the International Space Course at the Technical University of Munich, October, 1993, and at the Istituto per Applicacione del Calculo (IAC), Rome, September, 1994.

With the advent of parallel computers, much more complex problems can be solved, provided computational grids of sufficient quality can be generated. The recent Aerothermodynamics Workshop at the European Space Agency (November 1994) has proved that grid generation is one of the pacing items in CFD.

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Examples of Grid Generation with Implicitly Specified Surfaces Using GridPro TM / az3000: Filleted Multi Tube Configurations

Cheng Z., Eiseman P.R.

Published in 1994

With examples, we illustrate how implicitly specified surfaces can be used for grid generation with GridPro/az3000. The particular examples address two questions: (1) How do you model intersecting tubes with fillets? and (2) How do you generate grids inside the intersected tubes? The implication is much more general. With the results in a forthcoming paper which develops an easy-to-follow procedure for implicit surface modeling, we provide a powerful means for rapid prototyping in grid generation.

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Applications of Multiblock Grid Generations with Automatic Zoning

Eiseman P.R., Cheng Z., Hauser J.

Published in 1994

This paper is centered on the applications of a new grid generation package entitled GridPro^TM / az3000. The effects of grid topology on various problems are shown. These have ranged from cases with only a few blocks to cases with thousands of blocks. More than five distinct CFD codes have been successfully run on the generated grids. The grids are smooth and nearly orthogonal.

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Applications of Multiblock Grid Generations with Automatic Zoning

Eiseman P.R., Cheng Z., Hauser J.

Published in 1994

This paper is centered on the applications of a new grid generation package entitled GridPro^TM / az3000. The effects of grid topology on various problems are shown. These have ranged from cases with only a few blocks to cases with thousands of blocks. More than five distinct CFD codes have been successfully run on the generated grids. The grids are smooth and nearly orthogonal.

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Multi-block Structured Grid Approach for Solving Flows over Complex Aerodynamics Configurations

Vatsa V.N., Sanetrik M.D., Parlette E.B., Eiseman P.R., Cheng Z.

Published in 1994

A finite-volume, central-difference code, developed for solving three-dimensional flows, is used to obtain viscous solutions for a high lift configuration of practical interest. A novel block-structured grid approach is employed to generate computational grids suitable for resolving high gradient regions without propagating the denser grids to the far-field. The number of mesh points required for such grids is considerably less than that with conventional block-structured grids, where high density grids propagate all the way to the far-field in order to maintain C₀ continuity (point-to-point match) across the block boundaries. Computational results are presented for a 3-element airfoil to demonstrate the feasibility of this approach for aerodynamic computations. The computed solutions compare well with experimental data and demonstrate the flexibility and flows over complex geometries.

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Algebraic Grid Generation with Control Points

Eiseman P.R., Choo Y.K., Smith R.E.

Published in 1992

This chapter discusses the use and application of the control point from of algebraic grid generation (CPF) and broadly indicates future benefits and corresponding developments. In the course this chapter, various enhancements to the theory will arise. In topological terms, the application will extend into a multiblock environment, and in operational terms, those applications will be executed with a number of automatic features.

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Control Point Grid Generation

Eiseman P.R.

Published in 1992

The control point form of algebraic grid generation is developed in a rigorous manner to illucidate the key attributes of the mathematical theory and is demonstrated graphically to visualize the type of action that is possible. Altogether, the algebraic coordinate transformation represents a flexible structure that is adaptable to various situations. This presents the capability to effectively free-form model the boundaries of objects in a field about which a numerical simulation is to be performed with the generated grid.

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Adaptive Grid Method for Unsteady Flow Problems

Bockelie M.J., Eiseman P.R.

Published in 1992

An adaptive grid solution method is described for computing the time accurate solution of an unsteady flow problems. The solution method consists of three parts: a grid point redistribution method; an unsteady Euler equation solver; and a temporal coupling routine that links the dynamic grid to the flow solver. The grid movement technique is a direct curve by curve method containing grid controls that generate a smooth grid that resolves the severe solution gradients and the sharp transitions in the solution gradients. By design, the temporal coupling procedure provides a grid that does not lag the solution in time. The adaptive solution method is tested by computing the unsteady inviscid solutions for a one-dimensional shock tube and a two-dimensional shock vortex interaction. Quantitative comparisons are made between the adaptive solutions, theoretical solutions and numerical solutions computed on stationary grids. Test results demonstrate the good temporal tracking of the solution by the adaptive grid, and the ability of the adaptive method to capture an unsteady solution of comparable accuracy to that computed on a stationary grid containing significantly more points than used in the adaptive grid.

Keywords: Solution adaptive, Time accurate, Unsteady flow

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Algebraic-Elliptic Grid Generation

Eiseman P.R., Lu N., Jiang M., Thompson J.F.

Published in 1991

The control point form (CPF) of algebraic grid generation has been combined with elliptic grid generation to obtain smooth grid with less computational time and storage space. By first elliptically generating a sparse net of control points, we generate dense grids algebraically from the control net with the help of CPF. This new strategy of algebraic-elliptical grid generation proves to be much faster in time and and smaller in storage space than directly generating the dense grid elliptically.

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Interactive Grid Generation

Eiseman P.R.

Published in 1991

The control point form of algebraic grid generation is developed in a rigorous manner to illustrate the key attributes of the mathematical theory and is demonstrated graphically to visualize the type of action that is possible. Altogether, the algebraic coordinate transformation represents a flexible structure that is adaptable to various situations. This presents the capability to effectively free-form model the boundaries of objects in a field about which a numerical simulation is to be performed with the generated grid.

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Control Point Forms for Interactive Grid Manipulation

Eiseman P.R.

Published in 1991

The mathematical structure of the control point forms for algebraic grid generation is presented. A fundamental prerequisite for the effective application of interactive grid manipulation is the simultaneous utilization of the control points with the constraints from the specified boundaries. The structure established is a commutative algebra generated from a collection of basic operators that is isomorphic to a collection of boundary elements that are ordered in terms of set theoretic intersections and unions.

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The Control and Application of Adaptive Grid Movement

Eiseman P.R., Bockelie M.J.

Published in 1989

This study describes aspects of the grid movement and the temporal coupling schemes for an adaptive grid solution method. An application of the adaptive method is presented by solving for the flow field of a shock vortex interaction.

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Applications of Mesh Generation to Complex 3D Configurations

Eiseman P.R., Smith R.E.

Published in 1989

Techniques and applications of algebraic grid generation are described. The techniques are univariate interpolations and transfinite assemblies of univariate interpolations. Because algebraic grid generation is computationally efficient, the use of interactive graphics in conjunction with the techniques is advocated. A flexible approach, wich works extremely well in an interactive environment, called the Control Point Form of Algebraic Grid Generation is described. The applications discussed are three-dimentional grids constructed about airplane and submarine configurations.

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Grid Generation for the Solution of Partial Differential Equations

Eiseman P.R., Erlebacher G.

Published in 1989

A general survey of grid generation is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for such meshed, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstrutured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic part, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes.

The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generations and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given.

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Interactive Grid Generation for Turbomachinery Flow Field Simulations

Choo Y.K., Eiseman P.R., Reno C.

Published in 1988

The control point form of agebraic grid generation presented here provides the means that are needed to generate well-structured grids for turbomachinery flow simulations. It uses a sparse collection of control points distributed over the flow domain. The shape and position of coordinate curves can be adjusted from these control points while the grid conforms precisely to all boundaries. An interactive program called TURBO, which uses the control point form, is being developed. Basic features of the code are discussed and sample grids are presented. A finite-volume LU implicit scheme is used to simulate flow in a turbine cascade on the grid generated by the program.

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A Control Point from of Algebraic GRID Generation

Eiseman P.R.

Published in 1988

Local control points are established within the context of algebaic grid generation. The method of generation is based upon a multidirectional assembly of multisurface transformations that incorporates the best features of tensor product and Boolean sum constructions. Upon assembly, the resultant capability is the capability to conform precisely to prescribed boundarieswhile being able to manipulate the grid through a sparsenet of control points.

Keywords: Grid, Net, Control points, Curve, Interpolation, Tensor, Transfinite

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Adaptive Triangular Mesh Generation

Erlebacher G., Eiseman P.R.

Published in 1987

A general adaptive algorithm is developed on triangular meshes. The adaptivity is provided by a combination of node addition, dynamic node connectivity and a simple node movement strategy. While the local restructuring process and the node addition mechanism take place in the physical plane, the nodes are displaced on a monitor surface, constructed from the salient features of the physical problem. An approximation to mean curvature detects changes in the direction of the monitor surface, and provides the pulling force on the nodes. Solutions to the axisymmetric Grand-Shafranov equation demonstrate the capturing, by triangles, of the plasma-vacuum interface in a free boundary equilibrium configuration.

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Adaptive Grid Generation

Eiseman P.R.

Published in 1987

A review of adaptive grid generation is presented with an emphasis on the basic concepts and the interrelationship between the various methods. The concepts are developed in a multifaceted progressive sense with enough detail so as to instill an operative spirit for the methods. The operational capabilities come from an explicit display of the necessary formulas for algorithmic construction. While virtually all adaptive procedures are aimed at problems with rapid solution variations, our main concern is the construction of methods that are not fundamentally restricted by the choice of problem or solution algorithm.

Moreover, to maintain a simple treatment for the computational data and to have access to many of the best solution algorithms, we consider coordinate transformations. As a consequence, particular attention is given to grid point motion that occurs in response to the influence from the solution data, regardless of how that data was obtained. After some introductory discussion on the utilization of solution data, the topic of grid point motion is addressed first in one dimension and then in higher dimensions.

The basic equidistribution process is first seen from a dozen different viewpoints in one dimension. This is further amplified with the practical notions of precise coefficient specification, the attraction to a given grid, and the action of evolutionary forces. With the definition of the metric, the direct extention into higher dimensions is developed with curve-by-curve methods. This is followed by finite volume methods and variational methods. With the various movement strategies established in a multidimensional context, the next consideration is the temporal coupling of the movement with the solution algorithm. This is undertaken in the discussion of temporal aspects.

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The Geometrical Construction of Pointwise Distributions on Curves

Eiseman P.R.

Published in 1987

A method is developed to generate desirable pointwise distributions along curves. This is accomplished with a simple geomenrical 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.

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The Local Redistribution of Points along Curves for Numerical Grid Generations

Eiseman P.R.

Published in 1986

A methodology is established to cluster points along curves in a manner which does not change the existing pointwise distribution outside of a specified region containing the cluster. In each instance, points are pulled from the perimeters of the region toward the cluster center. The result is a smooth expansion from each end followed by a compression into the center. Altogether, this represents a local redistribution of points which can be employed either interactively or automatically.

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A General Collapsing Technique for Three-Dimentional Algebraic Grid Generation

Marshall G., Eiseman P.R., Kuo J.T.

Published in 1986

We present a general collapsing technique for the generation of a regular lexicographically ordered grid within a three-dimensional object by using multilinear algebraic coordinate transformations. The method is applied to the grid generation of a topologically complex region consisting of various three-dimentional objects. The local lexicographically ordered object grid is transformed, with an appropriate assembling procedure, into a global lexicographically ordered object grid. The method presented possesses simplicity and at the same time a sufficient degree of generality, a considerable amount of grid control, and a desirable degree of global grid uniformity to make it competitive.

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Adaptive Grid Generation by Mean Value Relaxation

Eiseman P.R.

Published in 1985

A grid movement algorithm has been developed for the purpose of adaptively resolving numerical solutions to physical problems and, in addition, for grid clustering on arbitrary surfaces. Both the solutions and the arbitrary surfaces are represented by grid point data with a continuous definition provided by interpolation between points. Movement is applied relative to this representation. The algorithm comes from a local mean value construction to produce a finite difference molecule for movement. The mean value weights are of a general enough nature to provide for a generous number of clustering possibilities. The movement molecule is executed within an interactive cycle in the spirit of point Jacobi or Gauss-Seidel, and as a consequence, corresponds to the solution of some elliptic partial differential equation which satisfies a maximum (minimum) principle due to the mean value construction. From this principle, the movement will always preserve nonsingularity for the continuous transformation. For the discrete representation in the form of a grid, local geometric constraints are established to maintain this preservation.

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Alternating Direction Adaptive Grid Generation

Eiseman P.R.

Published in 1985

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 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.

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Grid Generation for Fluid Mechanics Computations

Eiseman P.R.

Published in 1985

The need for grid generation arose from the discrete requierements for the numerical simulation in fluid mechanics with the geomentric and topological complexity of physical regions and with the possibility of rapid solution variations at unpredictable locations. The general topological setting is examined to establish the overall formats for the desired grids without addressing the specific details required of the methods to meet the consequent variety of constraints.

The element of control over the grid is the most fundamental aspect needed to satisfy the constraints from topology, geometry, and solution variations. The most precise level of control is available from algebraic methods. This is established with the multisurface transformation in one direction and with the extension into multiple directions by means of Boolean operations. From a more relaxed level of precision, elliptic partial differential equation methods are developed from the Laplace system representing conformal conditions up to the Poisson system, where convexity controls are established and examined.

The general feedback cycle for adaptive movement is considered by using a monitor surface to consolidate the feedback data into a single object. With the natural objective of accurately representing this object, strategies for movement are considered. These cover methods from direct proportionality statements, Poisson systems, and variational formulations by taking the monitor surface either as a function over physical space or as a geometric entity. The basic control in the various methods is in the form of weight functions. With the placement in distinct spots in distinct formats, the consequent effects, although similar, are also distinct. How, when, and where control is applied give the separation between the methods and the important effects on the grid in the varying levels of response, directness, and precision.

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Coordinate Generation with Precise Controls over Mesh Properties

Eiseman P.R.

Published in 1982

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.

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High Level Continuity for Coordinate Generation with Precise Controls

Eiseman P.R.

Published in 1982

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.

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Orthogonal Grid Generation

Eiseman P.R.

Published in 1982

Subject to various constraints, orthogonal coordinate generation techniques have beed examined under the unifying context of the associated metric. The metric was seen to be the basic ingredient for both the generation process and the application of the results.

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Mesh Generation Using Algebraic Techniques

Eiseman P.R., Smith R.E.

Published in 1980

Coordinate transformations are powerful tools for the solution of the partial differential equations which describe physical phenomena. The use of transformations leads to well ordered discretizations of the physical domain thereby renders a simplification in a numerical solution process. The discretization is constrained by the underlying physics, the problem geometry and the topology of the region where the solution is to be obtained.

Algebraic methods provide precise controls for mesh generation. Methodologies for mesh construction can be based on a parameterized description of surfaces which consist of bounding surfaces and intermediate control surfaces. The surface locations determined by the respective surface parameterizations determine the nature of the transverse coordinatecurves which connect the bounding surfaces.

Relative to uniform conditions, precise control over the mesh placement in the physical domain can be accomplished by embedding distribution control functions in the surface parameterizations or in the transverse direction. Complex bounding topologies, especially in three dimensions, cause mesh construction difficulties. It is proposed that whenever feasible, the complex topology be simplified such as rendering the geometry quasi two-dimensional.

Precise controls are one of the major advantages of algebraic methods: they give the capability to prescribe specific desirable and helpful mesh formations.

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Conservation Laws of Fluid Dynamics - A Survey

Eiseman P.R., Stone A.P.

Published in 1980

The conservation law problem in fluid dynamics and a generalization to Riemannian manifolds is surveyed in this paper. In particular the fluid dynamic equations in coordinate free notation and also in curvilinear coordinates are reviewed, as well as methods of obtaining conservation law form for these equations. This form is of some importance for the numerical computation of certain flow fields.

The model of one dimensional time dependent, nonisentropic, adiabatic fluid flow is then used to obtain a notion of a conservation law on a Riemannian manifold. One defines a conservation law in this situation to be any exact differential 1-form whose image under a given linear operator h on 1-forms is also exact. In the study of conservation laws on manifolds a prominent role is played by the Nijenhuis tensor of h. Recent results concerning the existence of these conservation laws are reviewed.

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A Multi Surface Method of Coordinate Generation

Eiseman P.R.

Published in 1979

A fast, direct, multidimensional method of coordinate generation has been developed to match boundaries with coordinate surface and to control the manner in which other coordinate surfaces leave the boundaries. When applied to the numerical solution of boundary value problems, not only are complex domains mapped into sipmle ones, but also the mesh near the boundaries can be made to conform with a prescribed form. Particularly useful prescriptions are the specifications of mesh distributions and angles. Sample applications are the local modeling of boundary layer coordinates and the smooth attachment of one coordinate system to another. In the latter case, a number of coordinate systems can be used as building blocks to form one large mesh. Simple examples of these applications, and more, are presented. Since the method of coordinate generation is based on a direct use of the bounding surfaces and certain intermediate control surfaces, it is referred to as a multi-surface method.

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A Unification of Unidirectional Flow Approximations

Eiseman P.R.

Published in 1978

The physical properties, associated with a primary flow direction, depend upon the fundamental constitutive relationship of a fluid. For this reason, the approximation presented herein is a direct approximation of the stress tensor which is clearly a statement about the constitution of the fluid. Consequently, such an approximation will be independent of the choice of basic conservation laws governing transport phenomena.

Moreover, the approximation will be performed ina manner which preserves the tensor character of the stress. Consequently, the approximation will also be independent of coordinates. In this manner, the limitations of the previous approximations can be overcome. With the added generality, it should be noted that some terms of negligible size will be retained. However, benefits from the added generality will outweigh by far the expense of including these terms. Such benefits will be reflected in the wider range of applicability for a specific algorithm and in the quality of a approximation which can easily be increased because of the added flexibility.

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A Method For Computing Three Dimensional Flow in Ducts

Eiseman P.R., McDonald H.

Published in 1976

Of particular interest in the present study is the treatment of complex diffuser geometries. Here an approximate set of governing equations is derived for flow passages whose bounding walls lie in coordinate surfaces of a general system. A coordinate system analysis is then performed for the special case of a continuously between a circle and a mean rectangle.

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A Note on a Differential Concomitant

Eiseman P.R., Stone A.P.

Published in 1975

If h and k are vector 1-forms, the vanishing of the concomitant [h,k] is an integrability condition for cernain problems on manifolds. In the case that h = k the vanishing of the Nijenhuis tensor [h,h] implies d(tr h) is a conservation law for h, provided that tr h is not constant. When the trace of h is constant, a conservation law for h exists if one can find a vector 1-form k with nonconstant trace such that [h,k]=0.

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A Method for Computing Three-Dimentional Viscous Diffuser Flows

Eiseman P.R., McDonald H., Briley W.R., Olson R.E.

Published in 1975

A method for computing three-dimentional turbulent subsonic flow in curved diffusers is described. An approximate set of governing equations is derived for viscous flows which have a primary flow direction. The derivation is coordinate invariant, and the resulting equations are expressed in terms of tensors. General tube-like coordinates are developed for a general class of geometries applicable to subsonic diffusers. The coordinates are then particularized to diffusers having superelliptic cross sections whose shape can vary continuously between a circle and a near rectangle. The necessary metric information is derived for these superelliptic tube-like coordinates. Techniques for numerical solution of these equations by forward marching integration from upstream starting conditions are outlined.

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A Generalized Hodge Theory

Eiseman P.R., Stone A.P.

Published in 1974

For a given nonsingular vector one-form h with vanishing Nijenhuis tensor, there is an associated exterior derivative dh which satisfies a Poincare lemma and hence provides an h-dependent version of de Rham's theorem. The exterior derivative dh also has an adjoint δh with respect to the usual global inner product. This fact permits one to define a strongly elliptic self-adjoint second order differential operator Δh which is a generalization of the Laplace-Beltrami operator. Consequently one can then obtain a generalization of the classical Hodge decomposition theorem.

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A Generalization of the Green's Operator on a Compact Manifold

Eiseman P.R., Stone A.P.

Published in 1974

The Hodge theorem asserts the existence of a ρ-form λ such that Δλ = φ – H(φ), for an arbitrary C^∞ ρ-form φ. This equation has a unique solution orthogonal to all harmonic ρ-form, and the solution is denoted by G(φ). G is called the Green's operator. The purpose of this paper is to study generalized operators H_h and G_h where h is a vector l-form which is non singular and has vanishing Nijenhuis tensor [h,h].

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The Topology of Manifolds which Admit Covariant Constant 1 to 1 Tensor Fields

Eiseman P.R., Stone A.P.

Published in 1973

The harmonic theory due to H.V.D.Hodge can be extended to include the influence of 1-1 tensor fields h on compact, orientable, C^∞ Riemannian manifolds without boundary. In this paper it is assumed that h is covariant constant with respect to the Riemannian connexion of the manifold. The resulting expression that is obtained for Δ_h in terms of coordinates is of course much simpler than the expression obtained without the covariant constant assumption. However, the basic results are easily generalized provided one has enough patience.

From the local expression for Δ_h one obtains a quadratic form which can be used to determine the topology of the manifold. The quadratic form generalizes the corresponding form of Bochner and Lichnerowicz found in the paper of S. Golberg: Curvature and homology (1962). A by-product of this study is a statement about the existence of certain global conservation laws.

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The Numerical Solution of the Dynamical Equations in Curvilinear Coordinates

Eiseman P.R.

Published in 1973

Most of the practical fluid dynamic problems result in studies of flows which are constrained to curved boundaries. Such boundaries may even have some sharp corners. To circumvent certain computational difficulties, one must transform the governing equations of motion into a coordinate system where the boundaries are simple. Among the most useful transformations are those which are constructed from analytic functions and from simple homotopies.

In essence, the simplified boundaries are obtained at the expense of having a slightly more complicated system of equations. This slight complication offers very little resistance to a numerical solution. A numerical method is then developed to efficiently solve the transformed equations. The numerical scheme is a hybrid of the methods of Lax-Wendroff and MacCormack which is adapted to accommodate source terms. In particular, this time-dependent procedure has two steps which follow the noncentered pattern of MacCormack's method where the fluxes are adjusted in such a way that second derivatives are handled in the well-centered manner characteristic of the Lax-Wendroff method.

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A Higher Order DeRham Theorem

Eiseman P.R.

Published in 1971

In this paper we show that for a large class of algebras; there is, for each A in this class, a cochain complex (ΛΩ_q(A), d) where Ω_q(A) is the module of qth order differentials and d is the exterior derivative. Restricting ourselves to PL algebras on PL manifolds or to C^∞ algebras on C^∞ manifolds we obtain the cohomology of the underlying manifold from the cochain complex (ΛΩ_q(A), d). When q = 1 and A is an algebra of C^∞ functions, we just have the classical deRham theorem. As a non-trivial consequence of this general theorem, we find that the complex in question is isomorphic to the tensor product of a complex with q = 1 (the usual deRham complex when A is an algebra of C^∞ functions) and some acyclic complex.

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