Multigrid methods for bifurcation problems: The self adjoint case

This paper deals with multigrid methods for computational problems that arise in the theory of bifurcation and is restricted to the self adjoint case. The basic problem is to solve for arcs of solutions, a task that is done successfully with an arc length continuation method. Other important issues are, for example, detecting and locating singular points as part of the continuation process, switching branches at bifurcation points, etc. Multigrid methods have been applied to continuation problems. These methods work well at regular points and at limit points, while they may encounter difficulties in the vicinity of bifurcation points. A new continuation method that is very efficient also near bifurcation points is presented here. The other issues mentioned above are also treated very efficiently with appropriate multigrid algorithms. For example, it is shown that limit points and bifurcation points can be solved for directly by a multigrid algorithm. Moreover, the algorithms presented here solve the corresponding problems in just a few work units (about 10 or less), where a work unit is the work involved in one local relaxation on the finest grid

One shot methods for optimal control of distributed parameter systems 1: Finite dimensional control

The efficient numerical treatment of optimal control problems governed by elliptic partial differential equations (PDEs) and systems of elliptic PDEs, where the control is finite dimensional is discussed. Distributed control as well as boundary control cases are discussed. The main characteristic of the new methods is that they are designed to solve the full optimization problem directly, rather than accelerating a descent method by an efficient multigrid solver for the equations involved. The methods use the adjoint state in order to achieve efficient smoother and a robust coarsening strategy. The main idea is the treatment of the control variables on appropriate scales, i.e., control variables that correspond to smooth functions are solved for on coarse grids depending on the smoothness of these functions. Solution of the control problems is achieved with the cost of solving the constraint equations about two to three times (by a multigrid solver). Numerical examples demonstrate the effectiveness of the method proposed in distributed control case, pointwise control and boundary control problems

Cumulative reports and publications through December 31, 1988

This document contains a complete list of ICASE Reports. Since ICASE Reports are intended to be preprints of articles that will appear in journals or conference proceedings, the published reference is included when it is available

Research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis and computer science

This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis, and computer science during the period April l, 1988 through September 30, 1988

Cumulative reports and publications through December 31, 1990

This document contains a complete list of ICASE reports. Since ICASE reports are intended to be preprints of articles that will appear in journals or conference proceedings, the published reference is included when it is available

Some aspects of the numerical solution of equilibrium problems in finite elasticity

Bibliography: pages 173-187.Analytical and computational aspects of solution paths for nonlinear equations are examined, with emphasis on problems in which there are many parameters. The solution to problems of this type is described by an equilibrium hypersurface and methods are presented which allow for the determination of the various features of this surface. These include methods for following numerically any curve on the primary surface, and for determining on such a curve all the singular points (both limit and bifurcation points). Further methods are then presented which allow branching onto secondary paths (subsets of secondary surfaces) from bifurcation points in order to trace out these paths and so determine the bifurcation behaviour of the problem considered. To complete the analysis of the equilibrium surface methods are developed to trace the loci of singular points. The locus of a bifurcation point determines the intersection of the primary and secondary equilibrium surfaces while the loci of limit points allow for the determination of regions of stable and unstable behaviour on the equilibrium surface. These methods are applicable to any system of nonlinear equations but the particular application here is to systems of equations obtained from the finite element approximation of boundary-value problems in elasticity. Attention is restricted to plane boundary-value problems involving incompressible hyperelastic materials. The strain-energy function used to characterise these materials is based on a symmetric function of the principal stretches. All of the above ideas are investigated numerically for the problem of a pressurised rubber cylinder subjected to axial extension. This problem contains two identifiable loading parameters and exhibits complex limit and bifurcation behaviour, which is studied in some detail

The State of the Art in Flow Visualization: Partition-Based Techniques

Flow visualization has been a very active subfield of scientific visualization in recent years. From the resulting large variety of methods this paper discusses partition-based techniques. The aim of these approaches is to partition the flow in areas of common structure. Based on this partitioning, subsequent visualization techniques can be applied. A classification is suggested and advantages/disadvantages of the different techniques are discussed as well

Semiannual report

This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, fluid mechanics, and computer science during the period 1 Oct. 1994 - 31 Mar. 1995

Mathematical and computational modelling for the design of pipe bends and compliant systems

This thesis is divided into three parts. In part I some theoretical and numerical processes are considered which arise when modelling the flow of a fluid through a pipe bend or deflector nozzle. These numerical processes include a new form of numerical integration and a finite element formulation which, it is suggested, could readily be extended to handle further realistic problems based on the pseudo three dimensional model chosen here. An introduction to nonlinear dynamics is included in part II leading towards a classification of bifurcational events in the light of recent advances in dynamics research. Most of the dynamical systems considered are dissipative such that the dynamic behaviour of the system decays onto a final steady state motion which may be modelled by a low order system of equations. In this way any resulting instability will adequately be described, qualitatively at least, by the low order bifurcation classified in part II. In part III the application of the geometrical theory of dynamical systems is used to study the wave driven motions of specified compliant offshore facilities with real data provided from structures currently in use in the offshore industry. In particular predictions are sought of any incipient jumps to resonance of the systems which might lead to potentially dangerous loads in the mooring lines or excessive displacements. Throughout the dynamics work stable steady state paths are closely followed and monitored so that any resulting bifurcation, including the possibility of chaotic behaviour, can be analysed with a view to its subsequent prediction

Research in progress in applied mathematics, numerical analysis, fluid mechanics, and computer science

This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, fluid mechanics, and computer science during the period October 1, 1993 through March 31, 1994. The major categories of the current ICASE research program are: (1) applied and numerical mathematics, including numerical analysis and algorithm development; (2) theoretical and computational research in fluid mechanics in selected areas of interest to LaRC, including acoustics and combustion; (3) experimental research in transition and turbulence and aerodynamics involving LaRC facilities and scientists; and (4) computer science