Vladimir A. Garanzha

Parallel implementation of Newton’s method for solving large-scale linear programs

Parallel versions of a method based on reducing a linear program (LP) to an unconstrained maximization of a concave differentiable piecewise quadratic function are proposed. The maximization problem is solved using the generalized Newton method. The parallel method is implemented in C using the MPI library for interprocessor data exchange. Computations were performed on the parallel cluster MVC-6000IM. Large-scale LPs with several millions of variables and several hundreds of thousands of constraints were solved. Results of uniprocessor and multiprocessor computations are presented.

Barrier variational generation of quasi‐isometric grids

A grid generation method based on the minimization of the discrete barrier functional with feasible set consisting of quasi-isometric grids is suggested. The deviation from isometry for given grid connectivity and prescribed boundary conditions is minimized via the contraction of the feasible set into a small vicinity of the optimal grid. Formulation of functional with given metrics in both physical and logical spaces allows to consider the adaptive grid generation in terms of quasi-isometric grids and cover many practical applications. A fast and reliable grid untangling procedure based on the penalty-like reformulation of barrier functional and the continuation technique is described. Numerical experiments demonstrate that the suggested functional produces high-quality grids with small global condition numbers. Copyright ? 2001 John Wiley & Sons, Ltd.

Generation of three-dimensional delaunay meshes from weakly structured and inconsistent data

A method is proposed for the generation of three-dimensional tetrahedral meshes from incomplete, weakly structured, and inconsistent data describing a geometric model. The method is based on the construction of a piecewise smooth scalar function defining the body so that its boundary is the zero isosurface of the function. Such implicit description of three-dimensional domains can be defined analytically or can be constructed from a cloud of points, a set of cross sections, or a “soup” of individual vertices, edges, and faces. By applying Boolean operations over domains, simple primitives can be combined with reconstruction results to produce complex geometric models without resorting to specialized software. Sharp edges and conical vertices on the domain boundary are reproduced automatically without using special algorithms. Refs. 42. Figs. 25.

Validation of Non-darcy Well Models Using Direct Numerical Simulation

We describe discrete well models for 2-D non-Darcy fluid flow in anisotropic porous media. Attention is mostly paid to the well models and simplified calibration procedures for the control volume mixed finite element methods, including the case of highly distorted grids.

Variational principles in grid generation and geometric modelling: theoretical justifications and open problems

The paper is devoted to analysis of variational principles for construction of mappings with prescribed properties in grid generation and geometric modelling. An attempt is made to formulate general requirements which should be satisfied by the variational principle. Theoretical justification is considered along with review of unsolved problems and fundamental mathematical difficulties. Copyright

Maximum norm optimization of quasi-isometric mappings

A reliable method for maximum norm optimization of spatial mappings is suggested. It is applied to the problem of optimal flattening of surfaces and to precisely controlled surface morphing. Robustness and grid independence of the method are demonstrated on real-life tests. Copyright

Distortion measure of trilinear mapping. Application to 3‐D grid generation

Distortion measures for polylinear mappings are investigated. It is shown that certain distortion measures satisfy the maximum principle which allows us to obtain upper bounds on the distortion measures for hexahedral cells and other types of elements widely used in the finite element method. These estimates allow to apply a maximum-norm optimization technique for spatial mappings in the case of finite element grids consisting of hexahedra. A global hexahedral grid untangling procedure suggested earlier was tested on hard 3-D examples demonstrating its ability to work in a black box mode and its high level of robustness. Copyright

Polyconvex potentials, invertible deformations, and thermodynamically consistent formulation of the nonlinear elasticity equations

It is shown that the nonstationary finite-deformation thermoelasticity equations in Lagrangian and Eulerian coordinates can be written in a thermodynamically consistent Godunov canonical form satisfying the Friedrichs hyperbolicity conditions, provided that the elastic potential is a convex function of entropy and of the minors of the elastic deformation Jacobian matrix. In other words, the elastic potential is assumed to be polyconvex in the sense of Ball. It is well known that Ball’s approach to proving the existence and invertibility of stationary elastic deformations assumes that the elastic potential essentially depends on the second-order minors of the Jacobian matrix (i.e., on the cofactor matrix). However, elastic potentials constructed as approximations of rheological laws for actual materials generally do not satisfy this requirement. Instead, they may depend, for example, only on the first-order minors (i.e., the matrix elements) and on the Jacobian determinant. A method for constructing and regularizing polyconvex elastic potentials is proposed that does not require an explicit dependence on the cofactor matrix. It guarantees that the elastic deformations are quasiisometries and preserves the Lame constants of the elastic material.

Parallel Implementation of Generalized Newton Method for Solving Large-Scale LP Problems

The augmented Lagrangian and Generalized Newton methods are used to simultaneously solve the primal and dual linear programming (LP) problems. We propose parallel implementation of the method to solve the primal linear programming problem with very large number (≈ 2 ·106) of nonnegative variables and a large (≈ 2 ·105) number of equality type constraints.

Highly Accurate Numerical Methods for Incompressible 3D Fluid Flows on Parallel Architectures

We consider an approach to efficient parallel implementation of the high order Control Volume PadE-type Differences (CVPD) applied to spatial time-dependent flow in the mixing tanks. This numerical technology allows to obtain very high quality solutions on the block-structured curvilinear grids with sliding grid capability. In some sense it combines the flexibility of the finite volume methods with the accuracy of the spectral methods. However, the payoff for the high accuracy is that the parallel implementation issues become more complicated as compared to conventional low order approximation methods. Our objective is to demonstrate that reasonable parallel efficiency can be attainedo n the parallel computer platforms without compromising the high accuracy, when the highly accurate non-local discrete operators and implicit time-steppings are usedas the building blocks of the numerical methods. We present numerical results obtained on CRAY C90, CRAY T3D andI BM SP2.