Publication list with abstracts

Publication List with Abstracts

Jonas T. Holdeman, Jr.

DISSERTATION AND THESIS

Electromagnetic Scattering of Charged Particles with Spin, J. T. Holdeman, Jr., Ph.D. Dissertation, Case Institute of Technology (Case-Western Reserve University), Cleveland Ohio (1966).

SUMMARY. The effect of the anomalous magnetic moment of charged particles (protons) in electromagnetic scattering was studied. The coulomb interaction was treated exactly, while the phase shift due to the interaction of the magnetic moment with the coulomb field was treated in the distorted-wave Born approximation. The phase shifts were found from the Schroedinger equation with relativistic corrections and from the Dirac equation with a Pauli moment interaction. The expressions for the two scattering amplitudes in terms of phase shifts were compared. Scattering amplitudes and electromagnetic polarization were calculated numerically and compared with other approximations.

A Direct Intensity Recording Microphotometer for Objective Prism Stellar Spectra, J. T. Holdeman, Jr., (unpublished) M.S. Thesis, Louisiana State Univ. (1960).

SUMMARY. Objective prism stellar spectrophotography provides a method for the rapid accumulation of large masses of stellar spectra.The spectral images are scanned with a microphotometer and the results displayed on a strip chart recorder. Since the relation between exposure and image density of photographic film is nonlinear, depending also on the development conditions and the plate batch, the plotted data must be corrected to obtain the original spectral intensities. This thesis describes the design and construction of an analog computer which computes the corrections to the measurements in real time using standard exposures on a calibrating photographic plate developed with the image plates to compute the inverse of the exposure-density transfer curve.

RECENT PAPERS AND PRESENTATIONS

“A velocity-stream function method for three-dimensional incompressible fluid flow,” J. T. Holdeman (March, 2011).

ABSTRACT. We describe a velocity-stream function method for computing incompressible fluid flow, extending earlier work in two- to three-dimensions. We present a strictly-divergence-free finite element in primitive variables in 3D as the curl of a (generalized) Hermite vector potential element with velocity degrees-of-freedom and show a method for its derivation. This linear velocity element is defined on a reference cube with 8 nodes, each node having 3 vector potential (stream function) and 3 velocity degrees-of-freedom. The method and element are applied to the lid-driven cavity problem and open duct flow in three dimensions. .

“Computation of incompressible thermal flows using Hermite finite elements,” J. T. Holdeman (June, 2010).

ABSTRACT. Using Hermit basis functions with the finite element method offers a remarkably simple way to compute non-isothermal buoyancy-driven incompressible flow. The Hermite bases we use simplify the governing equation and strongly enforce the continuity equation. For this problem, we use a fourth-order C¹ stream function defined on rectangles here, but other higher and lower-order Hermite elements on rectangles and triangles can easily be derived or modified from elements found in the plate-bending literature. Hermite elements are also used for the temperature and pressure. We conclude with results from application of the method to the square thermal cavity at moderate to high Rayleigh numbers.

“A Hermite finite element method for incompressible fluid flow,” J. T. Holdeman (July, 2009).

ABSTRACT. We describe some Hermite stream function and velocity finite elements and a divergence-free finite element method for the computation of incompressible flow. Divergence-free velocity bases defined on (but not limited to) rectangles are presented, which produce pointwise divergence-free flow fields (∇ . u_h≡ 0). The discrete velocity satisfies a flow equation which does not involve pressure. The pressure can be recovered as a function of the velocity if needed. The method is formulated in primitive variables and applied to the stationary lid-driven cavity and backward-facing step test problems.

“I. Some Lagrange Interpolation Functions for Solenoidal and Irrotational Vector Fields,” J. T. Holdeman (January, 2003).

ABSTRACT. Some remarkable new Lagrange interpolation functions on rectangular Cartesian meshes in two dimensions, and rectangular hexahedral meshes in three dimensions, are developed. The first examples are linearly-complete with a common (constant) divergence or a common (constant) curl on each element or subdomain of the mesh. These are extended to quadratic-complete functions, and then extended to orthogonal curvilinear coordinate systems, and affine meshes in those systems. After revisiting the equation of motion for incompressible flow, the functions, with suitable constraints, are used with the finite element method (FEM) to solve the incompressible Stokes equation.

“II. Some Hermite Interpolation Functions for Solenoidal and Irrotational Vector Fields,” J. T. Holdeman (January, 2003).

ABSTRACT. In a previous paper (I), some Lagrange interpolation functions for vector fields were introduced, with the property that they exhibited a constant divergence or constant curl in Cartesian coordinates. They were extended to curvilinear coordinates, where they exhibited a common or consistent (non-constant) divergence or curl in each subdomain of the mesh. It was shown how strongly-solenoidal fields would result using a simple constraint on each subdomain of the mesh. In application to unsteady and nonlinear partial differential equations, the constraint equations have to be satisfied at each time step or nonlinear iteration. It would be more efficient to solve the constraint equation once, producing solenoidal basis functions to interpolate the vector field. This approach leads from Lagrange to Hermite interpolation functions.

“Governing Equation for Incompressible Flow; Revisiting the Navier-Stokes Equation,” J. T. Holdeman, (October, 2003).

ABSTRACT. It has been recognized by mathematicians for almost seventy years that the incompressible Navier-Stokes equation can be orthogonally decomposed into an equivalent pressureless governing equation for the fluid motion and an equation for the pressure as a functional of the velocity field. It would be troubling indeed if this decomposition did not have some “physical” basis. The “physical” basis is to be found from a reexamination of the derivation of the governing equation from first principles. Correction of an oversight leads to a pair of complementary equations more fundamental than the composite Navier-Stokes equation. It is then shown how fluid mechanics and computational fluid dynamics can be reconciled with this (new?) context.

“Revisiting Incompressible Fluid Flow,” J. T. Holdeman, Southeastern Section of the American Physical Society, Auburn, AL, Oct. 31 - Nov. 2, 2002.

ABSTRACT. For over 150 years the incompressible Navier-Stokes equation (INSE) has been recognized as the governing equation of motion for viscous incompressible fluid flow. However, the derivation of this equation from first principles is incorrect. When applying Newton's second law to a small volume of fluid, it is traditional to assume that a pressure difference across the volume leads to a pressure gradient term. Since dynamic pressure differences propagate at infinite velocity in an incompressible medium, no dynamic pressure difference can be established, and no pressure gradient term can appear in the equation of motion.

The absence of a pressure gradient was noticed circa 1935 by J. Leray while developing existence and uniqueness proofs of solutions to the INSE. In the weak form of the INSE, the irrotational pressure gradient vanishes because it is orthogonal to the solenoidal velocity, but this observation has largely been dismissed as a “mathematical trick.” In fact, the INSE is not the dynamic equation it seems to be, but is the combination of a pressureless kinematic differential equation for the fluid motion, where incompressibility plays the role of a conservation law, and an algebraic equation (in time) for the “effective” pressure, which depends on conservative forces and the irrotational part of the fluid motion. That the pressureless momentum equation is a complete description of the incompressible flow will be demonstrated by showing results of the pressureless computation of a variety of fluid problems including “pressure-driven” flows.

“Recent Advances in the Finite Element Method for Incompressible Flow," J. T. Holdeman, Fourteenth U.S. National Congress of Theoretical and Applied Mechanics, Blacksburg, VA, June 23-28, 2002.

ABSTRACT. The recent availability of divergence-free finite elements permits the computation of incompressible fluid flow without the complication of the usual divergence constraint. Results of the computation of flow in two dimensions with a divergence-free velocity element that is the curl of a sufficiently-continuous Hermite stream function element are presented. The governing equation for this flow does not involve the pressure.

The usual Navier-Stokes momentum equation is not the dynamic equation involving pressure forces that it would seem to be. It is two separate orthogonal equations: a pressure-free one for the velocity, and one for the pressure as a functional of the velocity field and any conservative body forces. The pressure equation is of the pressure-Poisson form.

The weak form of this pressure-free velocity equation follows from the inner product of the Navier-Stokes momentum equation with a set of solenoidal test functions. From the Helmholtz decomposition theorem, the pressure gradient and conservative body force terms vanish by orthogonality. An integro-differential equation form results from application of the solenoidal projection operator to the momentum equation. These can be understood as forms of a kinematic governing equation for the velocity, where the incompressibility constraint plays the role of a conservation law. Absence of the pressure gradient can be understood at the elementary derivation level: a dynamic pressure gradient cannot be imposed on an infinitesimal fluid element because it would be instantly equilibrated due to the infinite speed of sound in an incompressible medium.

“Pressure-driven” flow would seem to present a dilemma for pressure-free computation, but net flow in these problems is governed by the stream function on the boundary. The Hermite elements, with stream function and velocity degrees of freedom, seem to be essential to application of this approach.

Computational results will be shown as examples of different boundary conditions. These include the lid-driven cavity, fully-developed and developing flow, flow over a backward-facing step and general Couette flow.

“An Hermite Finite Element Method for Incompressible Flow,” J. T. Holdeman, Finite Elements in Flow Problems 2000, International Association of Computational Mechanics & U.S. Association for Computational Mechanics, Austin, TX, April 30 - May 4, 2000.

REVISED ABSTRACT. Few would argue that incompressible computational fluid dynamics is a simple discipline, and there is no want of those who would make it easier. The difficulty is illustrated by the thousands of papers published on the subject over several decades, an effort that still continues. This presentation will describe a new finite element method which makes the computation of incompressible flow about as easy as it can get.

P. Gresho has remarked that attempts at divergence-free methods often conclude looking like stream-function methods. D. Griffiths introduced elements with a mixture of velocity and stream-function DOFs, and M. Fortin and C. Hall each found similar elements. We reconcile these efforts with a new fourth-degree Hermite stream-function element with stream-function and curl as DOFs, unisolvent over set including all cubic polynomials. Taking curl, Hermite vector elements allow velocity computation in primitive variables, decoupled from pressure, on rectangular partitions. An Hermite pressure element with potential and gradient DOFs, with (irrotational) gradient-vector test space, provides recovery of continuous pressures. Explicit orthogonal Helmholtz decomposition of solution space allows mapping to general convex quadrilaterals while preserving the pointwise-solenoidal property of vector fields. The Hermite elements were derived from Lagrange elements with constant divergence (curl), (viewed as imbeddings of the Hermite solenoidal (irrotational) spaces), and any of the methods of above-mentioned authors could be used.

“Divergence-Free Finite Elements and Related Spaces for the Incompressible Navier-Stokes Equation,” J. T. Holdeman, SIAM Annual Meeting, Atlanta GA, 1999.

ABSTRACT. New (cubic-complete) fourth-degree Hermite stream function and (quadratic-complete) third-degree Hermite velocity elements allow stable, accurate velocity computation, decoupled from pressure, on a rectangular partition. A new (cubic-complete) fourth-degree Hermite pressure element, with its (quadratic-complete) irrotational gradient-vector test space, provides recovery of continuous, accurate pressures. Explicit orthogonal Helmholtz decomposition of solution space allows mapping to general convex quadrilaterals while preserving the pointwise-solenoidal property of vector fields.

“A New Finite Element Method for Incompressible Fluid Flow,” J. T. Holdeman, Centennial Meeting of the American Physical Society, YB14, Atlanta, GA, March 1999.

ABSTRACT. We introduce a method producing stable, accurate and economical solutions of the incompressible Stokes and Navier-Stokes equations without modification of these equations. New Hermite finite elements defined on rectangles are introduced. These permit an orthogonal Helmholtz decomposition of the finite element test space into pointwise solenoidal and irrotational parts. The degrees of freedom are the field components, stream function and potential. This allows: (1) no incompressibility constraint to deal with, (2) velocity field independent of the pressure approximation, (3) pressure approximation can be chosen independent of velocity approximation, (4) div stability/inf- sup/LBB condition always satisfied in an optimal way, (5) velocity field is "data" for pressure determination, (6) cubic polynomial vector field elements which are quadratic complete and pointwise divergence-free, (7) explicit stream function for visualization, (8) smaller, stable algebraic system, (9) transformations to general convex quadrilaterals while preserving solenoidal and irrotational properties. Extensions to curvilinear coordinates and three dimensions have been found but will not be reported here.

“New Finite and Infinite Elements in Polar Coordinates Supporting Point-wise Divergence-free Vector Fields,” J. T. Holdeman, Thirteenth U.S. National Congress of Applied Mechanics, University of Florida, June 21-26, 1998.

ABSTRACT. Finite elements have been found which can produce strongly (pointwise) divergence-free fields in two and three dimensions using constraints which require only the essential condition that the net flow or flux from each element vanish. With affine geometric mappings from a master element on a square or cube to a structured mesh of parallelograms or parallelopipeds, these elements are linearly and quadratically complete in the sense that they exactly interpolate all linear or quadratic terms in the Taylor series expansion of vector fields which have constant or zero divergence. While transformations exist which map these elements to general quadrilaterals and hexahedra with straight or curved sides while preserving the pointwise divergence-free property, the mapped elements no longer interpolate linear and quadratic vector fields exactly. It is possible, however, to reformulate the elements in certain curvilinear coordinate systems so that all appropriate low order curvilinear terms in the series expansion of divergence-free fields are interpolated exactly on structured meshes.

As an example of this reformulation, linear and quadratic finite elements are presented in polar coordinates on the interval 0<r<infinity. The interval of definition is then extended to include the origin, with appropriate redefinition of the nodal field values at the origin. The interval is further extended to infinity with the introduction of linear and quadratic infinite elements satisfying inhomogeneous and homogeneous boundary conditions at infinity. Appropriate constraints to produce pointwise divergence-free vector fields on these infinite elements are presented. The effectiveness in interpolating far (and some not so far) fields for some simple problems with closed-form solutions in fluids and electromagnetics is shown.

Many of the results to be presented have also been extended to axisymmetric, cylindrical and spherical coordinate systems. The elements described may be characterized as associated with the divergence operator, but other families of elements have been found that are associated with the curl operator.

PUBLISHED PAPERS

“Torque on a Spinning Superconducting Sphere Inside a Superconducting Cylinder or Spherical Cap,” Louis B. Holdeman and Jonas T. Holdeman, Jr., J. Appl. Phys. 57 684-697 (1985).

ABSTRACT. A spinning superconducting sphere generates a magnetic field (the London field) while excluding external magnetic fields from its interior (the Meissner effect), so that a torque on the sphere results if there are magnetic or superconducting materials nearby. We calculate the torque on such a sphere inside a concentric, infinitely long superconducting cylinder. We also calculate the torque on such a sphere inside a concentric, spherical superconducting shell in which there is a single circular hole.

“Support-Electrode Torque on a Spherical Superconducting Gyroscope,” L. B. Holdeman and J. T. Holdeman, Jr., IEEE Trans. Magn., MAG-20 2042-2047 (1984).

SUMMARY. It has been observed that a precise measurement of the precession of a spherical gyroscope orbiting the earth could provide a test of general relativity.A spinning superconducting gyroscope produces a magnetic field, and the precession of the gyroscope can be measured by changes this field produces in superconducting loops encompassing the rotor. It is essential that no significant torques be coupled to the gyroscope through its magnetic field. Here we calculate the diamagnetic torque produced by superconducting support electrodes.

“Sensivity of a Soil-Plant-Atmosphere Model to Changes in Air Temperature, Dew Point Temperature, and Solar Radiation,” R. J. Luxmoore, Janice L. Stolzy and J. T. Holdeman, Agricultural Meteorology 23 115-129 (1981).

ABSTRACT. Air temperature, dew point temperature and solar radiation were independently varied in an hourly soil-plant-atmosphere model in a sensitivity analysis of these parameters. Results suggest that evapotranspiration in eastern Tennessee is limited more by meteorological conditions that determine the vapor-pressure gradient than by the necessary energy to vaporize water within foliage. Transpiration and soil water drainage were very sensitive to changes in air and dew point temperature and to solar radiation under low atmospheric vapor-pressure deficit conditions associated with reduced air temperature. Leaf water potential and stomatal conductance were reduced under conditions having high evapotranspiration. Representative air and dew point temperature input data for a particular application are necessary for satisfactory results, whereas irradiation may be less well characterized for applications with high atmospheric vapor-pressure deficit. The effects of a general rise in atmospheric temperature on forest water budgets are discussed.

“A Triangular Finite Element Mesh Generator for Fluid Dynamic Systems of Arbitrary Geometry,” C. Kleinstreuer and Jonas T. Holdman, International Journal for Numerical Methods in Engineering 15 (9) 1325-1334 (1980).

ABSTRACT. A two-dimensional finite element mesh generator capable of discretizing arbitrarily-shaped flow regions is described. This economical, interactive computer code, written in FORTRAN IV and employing PLOT 10 software together with TEKTRONIX graphics terminals (4000 series or similar devices), generates a triangular network of variable element density according to the geometry and local kinematic flow patterns of a given fluid flow problem. The algorithm is an easy-to-use preprocessor and mesh generator delivering, for a particular source program, suitable input data which contribute to a high degree of accuracy, efficiency and flexibility of the numerical scheme.
The code was tested for the triangularization of the confluence region of the Ohio and Tennessee rivers with two islands forming subchannels of different sizes.

“Magnetic Torque on a Shielded Superconducting Gyroscope,” L. B. Holdeman and J. T. Holdeman, Jr., Journal of Applied Physics 47 4936-4943 (1976).

ABSTRACT. The torque on a superconducting sphere rotating in an arbitrary magnetic field is calculated. The result is expressed in terms of the coefficients of the expansion of the magnetic field in spherical harmonic functions, but in general a boundary-value problem must be solved to obtain these coefficients. The boundary-value problem is solved and the torque calculated for configurations pertinent to the gyroscope relativity experiment, a proposed satellite experiment which is to test various theories of gravitation. Typical numerical results for these torques are given and compared with the predicted relativistic effects.

“Ambiguity in the Nucleon-Nucleon Phase Parameters Near 330 MeV,” J. T. Holdeman, Jr., and P. Signell, Physical Review Letters 27 1393-1396 (1971).

ABSTRACT. The two-nucleon data in the energy range 290-350 MeV are found to contain an additional solution to the one recently reported by MacGregor et al. Potential models are much closer to this additional solution. Experiments are proposed for deciding between the two solutions.

“Discrepancies in Low Energy Scattering Data,” J. T. Holdeman, P. Signell and M. Sher, Physical Review Letters 24 243-245 (1970).

ABSTRACT. A strong disagreement for predictions near 10 MeV is found between two previous multienergy phase-shift analyses. The data also contain contradictions, but are found to definitely favor one of the analyses over the other. Unresolved normalization discrepancies make further experimental work in the energy range 1-10 MeV mandatory.

“Legendre Polynomial Expansions of Hypergeometric Functions,” J. T. Holdeman, Journal of Mathematical Physics 11 114-117 (1970).

ABSTRACT. The expansion of a class of hypergeometric functions in a series of Legendre polynomials is derived. The range of validity and meaning to be attached to the sums is investigated. Several applications to the problem of the scattering of charged particles are presented.

“A Method for the Approximation of Functions Defined by Formal Series Expansions in Orthogonal Polynomials,” Jonas T. Holdeman, Jr., Mathematics of Computation 23 275-287 (1969).

ABSTRACT. An algorithm is described for numerically evaluating functions defined by formal (and possible divergent) series as well as convergent series of orthogonal functions which are, apart from a factor, orthogonal polynomials. When the orthogonal functions are polynomials, the approximations are rational functions. The algorithm is similar in some respects to the method of Pade approximantes. A rational approximation involving Tchebychev polynomials due to H. Machley and described by Kogbetliantz [1] is a special case of the algorithm.

“Remarks on Charged Particle Scattering,” J. T. Holdeman and R. M. Thaler, Physical Review 139 (Series II) B1186-1192 (1965).

ABSTRACT. The scattering of charged particles from a shielded Coulomb potential is reviewed. The limit as the shielding radius becomes infinite is discussed. A method of determining reaction cross sections, recently introduced by the authors is treated in detail and applied to the scattering of protons from He⁴ and H² at 40 MeV.

“Optical theorem as Applied to Charged-Particle Scattering to Yield Total Reaction Cross-Sections From Elastic-Scattering Data,” J. T. Holdeman and R. M. Thaler, Physical Review Letters 14 81-82 (1965).

SUMMARY. Total reaction cross-sections for charged particle scattering may be obtained by a careful examination of the elastic-scattering data. The procedure for doing so arises from the optical theorem. The feasibility of the procedure was checked by using an optical-model code to generate “experimental” data. Reaction cross-sections were also obtained in a few cases where sufficient data exists.