Research Activities

Jonas T. Holdeman, Jr.

I have been studying interpolation functions for divergence-free and irrotational vector fields. These have application to computation of the flow of an (idealized) incompressible fluid and to computational electromagnetics using the finite element method (FEM). I have derived strongly divergence-free bases on quadrilateral, triangular, and hexahedral meshes in two and three dimensions for application to this problem.

I have been working on a book titled A Handbook of New Finite Elements and Methods for Incompressible Flow and Electromagnetics. This is a compilation of my research into finite elements for incompressible fluid flow. Some chapters are,

• Part 1, Function Approximation

  • Cartesian Elements for Solenoidal Fields in Two Dimensions

  • Finite Elements for Solenoidal Fields in Two-dimensional Curvilinear Coordinates

  • Cartesian Elements for Irrotational Fields in Two Dimensions

  • Finite Elements for Irrotational Fields in Two-dimensional Curvilinear Coordinates

  • Three-dimensional Elements for Solenoidal Fields

  • Three-dimensional Elements for Irrotational Fields

  • Simplexical and Prismoidal Finite Elements

  • Geometric Mapping

• Part 2. Applications

  • Incompressible Flow Equations

• Appendix

Compatible pairs are identified for mixing quadrilateral and triangular elements.

Several years ago I wrote two papers on the subject. Copies of these papers are available for viewing in the portable document format (PDF) using the Adobe Acrobat reader. Adobe Acrobat Reader is available as a free download from Adobe.

I. Some Lagrange Interpolation Functions for Solenoidal and Irrotational Vector Fields (276KB) (view abstract)

II. Some Hermite Interpolation Functions for Solenoidal and Irrotational Vector Fields (276KB) (view abstract)

Governing Equation for Incompressible Flow; Revisiting the Navier-Stokes Equation (168KB) (view abstract)

Can the classical flow problems with analytic solutions be derived in the context of the pressureless governing equation in simple mathematical terms suitable for undergraduate instruction? Here are some of my thoughts on the subject in a rough-draft form.

Revisiting Some Analytical Solutions to Incompressible Flow Using Projection by Inspection (152KB)

Here are a few recent oral conference presentations.

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

“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. (view abstract)

“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. (view abstract)

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

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

“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. (view abstract)

These and other papers along with abstracts can be found in my list of publications.

Here is an unpublished paper:

“Incompressible Flow - Dynamic or Kinematic?,”

The Navier-Stokes equations for incompressible flow are orthogonally decomposed into an equation for fluid flow that does not contain pressure, and an equation for pressure as a function of flow. This shows that incompressible flow is a kinematic problem, with the incompressibility serving as an underlying conservation law.