Program package FLUX for the simulation of fundamental and applied problems of fluid dynamics

Based on parallel algorithms of a conservative numerical method, a software package for simulating fundamental and applied fluid dynamics problems in a wide range of parameters is developed. The software is implemented on a cluster computer system. Examples of the numerical simulation of three-dimensional problems in various fields of fluid dynamics are discussed, including problems of external flow around bodies, investigation of aerodynamic characteristics of flying vehicles, flows around a set of objects, flows in nozzles, and flows around underwater constructs.     

Prediction-control problem for some systems of equations of fluid dynamics

By methods of the theory of degenerate semigroups of operators, we investigate inverse problems for some systems of equations of fluid dynamics containing the incompressibility equation ? · v = 0. Namely, we obtain well-posedness criteria for inverse problems for the system of Sobolev equations, which describes the dynamics of small internal motions of a stratified fluid in an equilibrium state under some additional assumptions, and for the system of Oskolkov equations, which, in the linear approximation, describes the dynamics of a viscoelastic incompressible Kelvin-Voigt fluid of order 1.     

Mesh adaptation strategies for problems in fluid dynamics

Mesh adaptation procedures are reviewed and discussed with particular emphasis on methods that are based on mesh movement (r-refinement) and mesh enrichment (h-refinement). Although the need for adaptive refinement is common to many areas of scientific computation, the examples in this survey paper are drawn almost exclusively from the domain of computational fluid dynamics.     

A fast solver for riemann problems

An efficient algorithm is presented for solving the Riemann problem for a polytropic gas. It enables the user to compute the solution for all physically reasonable data. The convergence of the algorithm is proved. The accuracy of the solution is limited only by the accuracy of the computer. There is an a-priori estimation of the required number of iterations. The rate of convergence turns out to be much higher than that of the usual fixed point iteration scheme.     

The application of the FDEM program package with error estimate to industrial problems

The Finite Difference Element Method (FDEM) program package is a robust and efficient black-box solver that solves arbitrary nonlinear systems of elliptic and parabolic partial differential equations under arbitrary nonlinear boundary conditions on arbitrary domains in 2-D and 3-D. FDEM is an unprecedented generalization of the finite difference method on unstructured finite element meshes. From the difference of formulas of different order, we get an easy access to the discretization error. By the knowledge of this error, the mesh may be refined locally to reduce the error to a prescribed relative tolerance. The error estimate is a unique property for such a general black-box. In addition, the FDEMprogram package is efficiently parallelized on distributed memory parallel computers. In this paper we demonstrate the usefulness of the FDEM program package by its application to several industrial problems. This gives completely new results as up to now people have solved these problems blindly, unaware of the error of their solution. The first problem is the numerical simulation of a microreactor where we have one chemical component entering through the main channel and one chemical component entering through a side channel so that there is a reaction of the components. We want to examine the flow field and the behaviour of the chemical components. The second problem is the heat conduction in a high pressure Diesel injection pump. This problem is based on a preceding fluid-structure interaction problem, and we now compute the temperature distribution in the injection pump additionally. Finally, we simulate the distribution of the temperature in a DC/AC converter module with six power-MOSFETs heated with uniform power. At the bottom of the module air cooling is applied. In contrast to the first two elliptic problems, this is a 3-D parabolic problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Preconditioning for boundary control problems in incompressible fluid dynamics

PDE‐constrained optimization problems arise in many physical applications, prominently in incompressible fluid dynamics. In recent research, efficient solvers for optimization problems governed by the Stokes and Navier–Stokes equations have been developed, which are mostly designed for distributed control. Our work closes a gap by showing the effectiveness of an appropriately modified preconditioner to the case of Stokes boundary control. We also discuss the applicability of an analogous preconditioner for Navier–Stokes boundary control and provide some numerical results.     

Numerical algorithm for moving-boundary fluid dynamics problems and its testing

A new Eulerian algorithm is proposed for computing incompressible flows with moving boundaries of arbitrary geometry. The algorithm can be regarded as an extension of Hirt’s classical volume-of-fluid method. The results of its implementation and testing are presented. A comparison of the numerical results with experimental data and computations of other authors reveals good agreement.     

Parallel interior-point solver for structured quadratic programs: Application to financial planning problems

Many practical large-scale optimization problems are not only sparse, but also display some form of block-structure such as primal or dual block angular structure. Often these structures are nested: each block of the coarse top level structure is block-structured itself. Problems with these characteristics appear frequently in stochastic programming but also in other areas such as telecommunication network modelling. We present a linear algebra library tailored for problems with such structure that is used inside an interior point solver for convex quadratic programming problems. Due to its object-oriented design it can be used to exploit virtually any nested block structure arising in practical problems, eliminating the need for highly specialised linear algebra modules needing to be written for every type of problem separately. Through a careful implementation we achieve almost automatic parallelisation of the linear algebra. The efficiency of the approach is illustrated on several problems arising in the financial planning, namely in the asset and liability management. The problems are modelled as multistage decision processes and by nature lead to nested block-structured problems. By taking the variance of the random variables into account the problems become non-separable quadratic programs. A reformulation of the problem is proposed which reduces density of matrices involved and by these means significantly simplifies its solution by an interior point method. The object-oriented parallel solver achieves high efficiency by careful exploitation of the block sparsity of these problems. As a result a problem with over 50 million decision variables is solved in just over 2 hours on a parallel computer with 16 processors. The approach is by nature scalable and the parallel implementation achieves nearly perfect speed-ups on a range of problems. Supported by the Engineering and Physical Sciences Research Council of UK, EPSRC grant GR/R99683/01     

A parallel eigenvalue problem solver for computational nanoelectronics

A new parallel eigenvalue solver for finding the interior eigenvalues of a standard Hermitian eigenvalue problem arising in atomistic simulations in nanoelectronics is presented. It is based on the Tracemin algorithm which finds the p smallest eigenpairs of a generalized Hermitian eigenvalue problem. The original problem is modified using spectrum folding or a quadratic mapping so that the interior eigenvalues are mapped onto the smallest or the largest, respectively. In the latter case the solution of systems in every iteration of Tracemin is avoided and Chebyshev polynomials are used to speedup convergence. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A fast solver for the first biharmonic boundary value problem

Summary This paper provides a fast and storage-saving method for the solution of the first biharmonic boundary value problem (b.v.p.). The b.v.p. is approximated via a special variational finite difference technique suggested earlier by V.G. Korneev. It is shown theoretically that our method produces an approximate solution to the finite difference equations inO(NlnNln^–1) arithmetical operations, whereN is the number of unknowns and (0<<1) denotes the relative accuracy required. The numerical results obtained by our computer code CGMFC decisively substantiate the theoretical estimates given.     

A coupled far-field formulation for time-periodic numerical problems in fluid dynamics

Consider uniform flow past an oscillating body generating a time-periodic motion in an exterior domain, modelled by a numerical fluid dynamics solver in the near field around the body. A far-field formulation, based on the Oseen equations, is presented for coupling onto this domain thereby enabling the whole space to be modelled. In particular, examples for formulations by boundary elements and infinite elements are described.     

Temporal boundary value problems in interfacial fluid dynamics

We study problems in interfacial fluid dynamics which do not have well-posed initial value problems. We prove existence of solutions for these problems by considering instead boundary value problems, where boundary data is specified at two different times. We develop a general framework, for problems on the real line and for problems which are spatially periodic. A variety of boundary conditions are considered, including Dirichlet, Neumann and mixed conditions. The framework is applied to two specific problems from interfacial fluid dynamics: a family of generalizations of the Boussinesq equations developed by Bona, Chen and Saut, and the vortex sheet.     

DrAmpl: a meta solver for optimization problem analysis

Optimization problems modeled in the AMPL modeling language (Fourer et al., in AMPL: a modeling language for mathematical programming, 2002) may be examined by a set of tools found in the AMPL Solver Library (Gay, in Hooking your solver to AMPL, 1997). DrAmpl is a meta solver which, by use of the AMPL Solver Library, dissects such optimization problems, obtains statistics on their data, is able to symbolically prove or numerically disprove convexity of the functions involved and provides aid in the decision for an appropriate solver. A problem is associated with a number of relevant solvers available on the NEOS Server for Optimization (Czyzyk et al., in IEEE J Comput Sci Eng 5:68–75, 1998) by means of a relational database. We describe the need for such a tool, the design of DrAmpl and some of its consequences, and keep in mind that a similar tool could be developed for other algebraic modeling languages.     

On a parallel multilevel solver for linear elasticity problems

In this paper an application of the additive multilevel iteration method to parallel solving of large‐scale linear elasticity problems is considered. The results are derived in the framework of the hierarchical basis finite element discretization defined on a tensor product of one‐dimensional grid and a sequence of nested triangulations . The algorithm was tested on a number of model problems, arising from bridge foundation modeling. Parallel performance of the solver is reported for Cray T3E‐600 and Sun ES/4000 computer systems. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical solution of some nonlinear problems of plate dynamics

We derive the equations of motion of plates for various cases of geometrical nonlinearity and shear strain. A finite-difference numerical solution is given.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 66, pp. 83–89, 1988.     

A direct solver for the Legendre tau approximation for the two-dimensional Poisson problem

A direct solver for the Legendre tau approximation for the two dimensional Poisson problem is proposed. Using the factorization of symmetric eigenvalue problem, the algorithm overcomes the weak points of the Schur decomposition and the conventional diagonalization techniques for the Legendre tau approximation. The convergence of the method is proved and numerical results are presented.     

A fast Fourier transform based direct solver for the Helmholtz problem

This article is devoted to the efficient numerical solution of the Helmholtz equation in a two‐ or three‐dimensional (2D or 3D) rectangular domain with an absorbing boundary condition (ABC). The Helmholtz problem is discretized by standard bilinear and trilinear finite elements on an orthogonal mesh yielding a separable system of linear equations. The main key to high performance is to employ the fast Fourier transform (FFT) within a fast direct solver to solve the large separable systems. The computational complexity of the proposed FFT‐based direct solver is $\mathrm{??}(N\log N)$ operations. Numerical results for both 2D and 3D problems are presented confirming the efficiency of the method discussed.