William I. Thacker

Algorithm 905: SHEPPACK: Modified Shepard Algorithm for Interpolation of Scattered Multivariate Data

Scattered data interpolation problems arise in many applications. Shepard’s method for constructing a global interpolant by blending local interpolants using local-support weight functions usually creates reasonable approximations. SHEPPACK is a Fortran 95 package containing five versions of the modified Shepard algorithm: quadratic (Fortran 95 translations of Algorithms 660, 661, and 798), cubic (Fortran 95 translation of Algorithm 791), and linear variations of the original Shepard algorithm. An option to the linear Shepard code is a statistically robust fit, intended to be used when the data is known to contain outliers. SHEPPACK also includes a hybrid robust piecewise linear estimation algorithm RIPPLE (residual initiated polynomial-time piecewise linear estimation) intended for data from piecewise linear functions in arbitrary dimension m. The main goal of SHEPPACK is to provide users with a single consistent package containing most existing polynomial variations of Shepard’s algorithm. The algorithms target data of different dimensions. The linear Shepard algorithm, robust linear Shepard algorithm, and RIPPLE are the only algorithms in the package that are applicable to arbitrary dimensional data.

Magnetohydrodynamic flow and heat transfer about a rotating disk with suction and injection at the disk

This paper studies the effects of a partial magnetic field on the flow and heat transfer about a porous rotating disk. Using modem quasi-Newton and globally convergent homotopy methods, numerical solutions are obtained for a wide range of magnetic field strengths and injection and suction velocities. Results are presented graphically in terms of three nondimensional parameters. There is excellent agreement with previous work and asymptotic formulas.

Solving the canonical dual of box-and integer-constrained nonconvex quadratic programs via a deterministic direct search algorithm

This paper presents a massively parallel global deterministic direct search method (VTDIRECT) for solving nonconvex quadratic minimization problems with either box or±1 integer constraints. Using the canonical dual transformation, these well-known NP-hard problems can be reformulated as perfect dual stationary problems (with zero duality gap). Under certain conditions, these dual problems are equivalent to smooth concave maximization over a convex feasible space. Based on a perturbation method proposed by Gao, the integer programming problem is shown to be equivalent to a continuous unconstrained Lipschitzian global optimization problem. The parallel algorithm VTDIRECT is then applied to solve these dual problems to obtain global minimizers. Parallel performance results for several nonconvex quadratic integer programming problems are reported.

Magnetohydrodynamic Flow Between a Solid Rotating Disk and a Porous Stationary Disk

In this paper we examine the flow of a conducting fluid between a solid rotating disk and a stationary porous disk with uniform section of fluid through the porous disk in the presence of a magnetic fiels. The equations of motion are solved using least change secant update quasi-Newton and modern root finding techniques. The fluid motion depends on the cross-flow Reynolds number, rotation Reynolds number and Hartmann number. The effects of the parameters on the flow field are presented graphically.

Magnetohydroynamic Free Convection from a Disk Rotating in a Vertical Plane

The non-axisymmetric motion (produced by a buoyancy induced cross flow) of a fluid in contact with a rotating disk and in the presence of a magnetic field normal to the disk is studied. Using modern quasi- Newton techniques, B-splines, and a Galerkin approximation to the fluid motion equations, numerical solutions are obtained for a wide range of magnetic field strengths and Prandtl numbers (ratio of kinematic viscosity to thermal conductivity). Results are presented both in tabular and graphical form in terms of two non-dimensional parameters. There is excellent agreement with previous work.

The Nonlinear Stability of a Heavy Rigid Plate Supported by Flexible Columns

A heavy rigid platform is supported by thin elastic legs. The governing equations for large deformations are formulated and solved numerically by homotopy and quasi-Newton methods. Nonlinear phenomena such as non-uniqueness, catastrophe and hysteresis are found. A global critical load for nonlinear stability is introduced.

Stability and Postbuckling of a Platform with Flexible Legs Resting on a Slippery Surface

A rigid platform is supported by thin elastic legs. The legs are able to slide on the ground as they deform. The governing equations for large deformations are formulated and solved numerically by homotopy and quasi-Newton methods. Nonlinear phenomena such as nonuniqueness are found. A global critical load for nonlinear stability is presented.

Micro programming rational arithmetic operations

For many years, real number arithmetic, whether accomplished in hardware, software, or firmware, has utilized a floating point scheme. This paper summarizes the results of a study assessing rational representation as a possible replacement for floating point representation. The Hewlett-Packard 2100A microprogrammable computer was used as the test vehicle. Both accuracy and speed comparisons were used to judge the usefulness of the substitution.