Browsing by Author "Glowinski, Roland"

Now showing 1 - 6 of 6
  • Thumbnail Image
    Item
    A Domain Decomposition Method for the Acoustic Wave Equation Allowing for Discontinuous Coefficients and Grid Change
    (1994-01) Bamberger, Alain; Glowinski, Roland; Tran, Quang Huy
    A domain decomposition technique is proposed for the computation of the acoustic wave equation, in which the bulk modulus and density fields are allowed to be discontinuous at the interfaces. Inside each subdomain, the method presented coincides with the second order finite difference schemes traditionally used in geophysical modelling. However, the possibility of assigning to each subdomain its own space-step makes numerical simulations much less expensive. Another interest of the method lies in the fact that its hybrid variational formulation naturally leads to exact equations for gridpoints on the interfaces. Transposing Babuska-Brezzi's formalism on mixed and hybrid finite elements provides a suitable functional framework for this domain decomposition formulation and shows that the inf-sup condition remains the basic requirement for convergence to occur.
  • Thumbnail Image
    Item
    A Quadratically Constrained Minimization Problem Arising from PDE of Monge-Ampére Type
    (2008-05) Sorensen, D.C.; Glowinski, Roland
    This note develops theory and a solution technique for a quadratically constrained eigenvalue minimization problem. This class of problems arises in the numerical solution of fully-nonlinear boundary value problems of Monge-Ampére type. Though it is most important in the three dimensional case, the solution method is directly applicable to systems of arbitrary dimension. The focus here is on solving the minimization subproblem which is part of a method to numerically solve a Monge-Ampére type equation. These subproblems must be evaluated many times in this numerical solution technique and thus efficiency is of utmost importance. A novelty of this minimization algorithm is that it is finite of complexity O(N^3) with the exception of solving a very simple rational function of one variable. This function is essentially the same for any dimension. This result is quite surprising given the nature of the minimization problem.
  • Thumbnail Image
    Item
    Numerical Simulation and Optimal Shape for Viscous Flow by a Fictitious Domain Method
    (1994-08) Glowinski, Roland; Kearsley, Anthony J.; Pan, Tsorng-Whay; Periaux, Jacques
    In this article we discuss the fictitious domain solution of the Navier-Stokes equations modelling unsteady incompressible viscous flow. The method is based on a Lagrange multiplier treatment of the boundary conditions to be satisfied and is particularly well suited to the treatment of no-slip boundary conditions. This approach allows the use of structured meshes and fast specialized solvers for problems on complicated geometries. Another interesting feature of the fictitious domain approach is that it allows the solution of optimal shape problems without regriding. The resulting methodology is applied to the solution of flow problems including external incompressible viscous flow modelled by the Navier-Stokes equations and then to an optimal shape problem for Stokes and Navier-Stokes flow.
  • Thumbnail Image
    Item
    On the Simulation and Control of Some Friction Constrained Motions
    (1993-05) Glowinski, Roland; Kearsley, Anthony J.
    In this paper, some issues involved with numerical simulation and control of some elasto-dynamic systems are discussed. The motivation is the simulation of dry or Coulomb friction in the joints that link together remote manipulator systems used in aerospace operations (for example, space shuttle remote manipulator systems). The goal here is to develop numerical techniques to simulate and control these systems, while properly modeling the Coulomb friction. The numerical procedure described employs a finite difference time discretization in conjunction with a vector of multipliers that predicts the friction effect for all time. In addition to this discrete multiplier technique an associated regularization procedure that greatly improves the behavior of these multipliers is also presented. Numerical examples conclude the paper.
  • Thumbnail Image
    Item
    Supercomputing and the Finite Element Approximation of the Navier-Stokes Equations for Incompressible Viscous Fluids
    (1988-06) Glowinski, Roland
    We discuss in this paper the numerical simulation of unsteady incompressible flows modeled by the Navier-Stokes equations, concentrating most particularly on flows at Reynold number of the order of 10^3 to 10^4. The numerical methodology described here is of modular type and well suited to super computers; it is based on time discretization by operator splitting, and space discretization by low order finite element approximations, leading to highly sparse matrices. The Stokes subproblems originating from the splitting are treated by an efficient Stokes solver, particularly efficient for flow at high Reynold numbers; the nonlinear subproblems associated with the advection are solved by a least squares/preconditioned conjugate gradient method. The methodology discussed here is then applied to the simulation of jets in a cavity, using a CRAY X-MP supercomputer. Various visualizations of the numerical results are presented, in order to show the vortex dynamics taking place in the cavity.
  • Thumbnail Image
    Item
    The use of optimization techniques in the solution of partial differential equations from science and engineering
    (1996) Kearsley, Anthony Jose; Glowinski, Roland; Tapia, Richard A.
    Optimal Control of systems governed by Partial Differential Equations is an applications-driven area of mathematics involving the formulation and solution of minimization problems. Given a physical phenomenon described by a differential equation, the Optimal Control Problem (OCP) seeks to force state variables to behave in a particular, desired way. This manipulation of state variables is achieved through the control variables. Many problems arising in applications of science and engineering can fruitfully be viewed and formulated as OCP problems. From the OCP point of view, one sees the structure underlying the optimization problem. In this thesis we will propose and analyze algorithms for the solution of Nonlinear Programming Problems (NLP) designed to exploit the OCP structure.