Browsing by Author "Gunzburger, Max D."

Now showing 1 - 20 of 28
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    An alternating direction search algorithm for low dimensional optimization: an application to power flow
    Burrell, Tinal R. (Virginia Tech, 1993-05-05)
    Presented in this paper is a scheme for minimizing the cost function of a three-source technique to arrive at an approximation point (I,J) that is within one unit of the true minimum. The Line-Step algorithm is applied to several systems and is also compared to other minimization techniques, including the Equal Incremental Loss Algorithm. Variations are made on the Line-Step Algorithm for faster convergence and also to handle inequality constraints.
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    Analysis and approximation of the velocity tracking problem for Navier-Stokes flows with distributed control
    Gunzburger, Max D.; Manservisi, S. (Siam Publications, 2000-05)
    We consider the mathematical formulation, analysis, and the numerical solution of a time-dependent optimal control problem associated with the tracking of the velocity of a Navier-Stokes ow in a bounded two-dimensional domain through the adjustment of a distributed control. The existence of optimal solutions is proved and the first-order necessary conditions for optimality are used to derive an optimality system of partial differential equations whose solutions provide optimal states and controls. Semidiscrete-in-time and fully discrete space-time approximations are defined and their convergence to the exact optimal solutions is shown. A gradient method for the solution of the fully discrete equations is examined, as are its convergence properties. Finally, the results of some illustrative computational experiments are presented.
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    Analysis and finite element approximation of an optimal shape control problem for the steady-state Navier-Stokes equations
    Kim, Hongchul (Virginia Tech, 1993-12-05)
    An optimal shape control problem for the steady-state Navier-Stokes equations is considered from an analytical point of view. We examine a rather specific model problem dealing with 2-dimensional channel flow of incompressible viscous fluid: we wish to determine the shape of a bump on a part of the boundary in order to minimize the energy dissipation. To formulate the problem in a comprehensive manner, we study some properties of the Navier-Stokes equations. The penalty method is applied to relax the difficulty of dealing with incompressibility in conjunction with domain perturbations and regularity requirements for the solutions. The existence of optimal solutions for the penalized problem is presented. The computation of the shape gradient and its treatment plays central role in the shape sensitivity analysis. To describe the domain perturbation and to derive the shape gradient, we study the material derivative method and related shape calculus. The shape sensitivity analysis using the material derivative method and Lagrange multiplier technique is presented. The use of Lagrange multiplier techniques,from which an optimality system is derived, is justified by applying a method from functional analysis. Finite element discretizations for the domain and discretized description of the problem are given. We study finite element approximations for the weak penalized optimality system. To deal with inhomogeneous essential boundary condition, the framework of a Lagrange multiplier technique is applied. The split formulation decoupling the traction force from the velocity is proposed in conjunction with the penalized optimality system and optimal error estimates are derived.
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    Analysis and numerical approximations of exact controllability problems for systems governed by parabolic differential equations
    Cao, Yanzhao (Virginia Tech, 1996-06-16)
    The exact controllability problems for systems modeled by linear parabolic differential equations and the Burger's equations are considered. A condition on the exact controllability of linear parabolic equations is obtained using the optimal control approach. We also prove that the exact control is the limit of appropriate optimal controls. A numerical scheme of computing exact controls for linear parabolic equations is constructed based on this result. To obtain numerical approximation of the exact control for the Burger's equation, we first construct another numerical scheme of computing exact controls for linear parabolic equations by reducing the problem to a hypoelliptic equation problem. A numerical scheme for the exact zero control of the Burger's equation is then constructed, based on the simple iteration of the corresponding linearized problem. The efficiency of the computational methods are illustrated by a variety of numerical experiments.
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    Analysis, finite element approximation, and computation of optimal and feedback flow control problems
    Lee, Hyung-Chun (Virginia Tech, 1994-07-05)
    The analysis, finite element approximation, and numerical simulation of some control problems associated with fluid flows are considered. First, we consider a coupled solid/fluid temperature control problem. This optimization problem is motivated by the desire to remove temperature peaks, i.e., "hot spots", along the bounding surface of containers of fluid flows. The heat equation of the solid container is coupled to the energy equation for the fluid. Control is effected by adjustments to the temperature of the fluid at the inflow boundary. We give a precise statement of the mathematical model, prove the existence and uniqueness of optimal solutions, and derive an optimality system. We study a finite element approximation and provide rigorous error estimates for the error in the approximate solution of the optimality system. We then develop and implement an iterative algorithm to compute the approximate solution. Second, a computational study of the feedback control of the magnitude of the lift in flow around a cylinder is presented. The uncontrolled flow exhibits an unsymmetric Karman vortex street and a periodic lift coefficient. The size of the oscillations in the lift is reduced through an active feedback control system. The control used is the injection and suction of fluid through orifices on the cylinder; the amount of fluid injected or sucked is determined, through a simple feedback law, from pressure measurements at stations along the surface of the cylinder. The results of some computational experiments are given; these indicate that the simple feedback law used is effective in reducing the size of the oscillations in the lift. Finally, some boundary value problems which arise from a feedback control problem are considered. We give a precise statement of the mathematical problems and then prove the existence and uniqueness of solutions to the boundary value problems for the Laplace and Stokes equations by studying the boundary integral equation method.
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    Compensator design for a system of two connected beams
    Huang, Wei (Virginia Tech, 1994-08-05)
    The goal of this paper is to study the LQG problem for a class of infinite dimensional systems. We investigate the convergence of compensator gains for such systems when standard finite element schemes are used to discretize the problem. We are particularly interested in the analysis of the uniformly exponential stability of the corresponding closed - loop systems resulting from the finite dimensional compensators. A specific multiple component flexible structure is used to focus the analysis and to test problem in numerical simulations. An abstract framework for analysis and approximation of the corresponding dynamics system is developed and used to design finite - dimensional compensators. Linear semigroup theory is used to establish that the systems are well posed and to prove the convergence of generic approximation schemes. Approximate solutions of the optimal regulator and optimal observer are constructed via Galerkin - type approximations. Convergence of the scheme is established and numerical results are presented to illustrate the method
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    Computational simulation of type-II superconductivity including pinning phenomena
    Du, Q.; Gunzburger, Max D.; Peterson, Janet S. (American Physical Society, 1995-06-15)
    A flexible tool, based on the finite-element method, for the computational simulation of vortex phenomena in type-II superconductors has been developed. These simulations use refined or newly developed phenomenological models including a time-dependent Ginzburg-Landau model, a variable-thickness thin-film model, simplified models valid for high values of the Ginzburg-Landau parameter, models that account for normal inclusions and Josephson effects, and the Lawrence-Doniach model for layered superconductors. Here, sample results are provided for the case of constant applied magnetic fields. Included in the results are cases of flux pinning by impurities and by thin regions in films.
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    A finite element analysis of high kappa, high field Ginzburg-Landau type model of superconductivity
    Karamikhova, Rossitza (Virginia Tech, 1995-08-05)
    This work is concerned with the formulation and analysis of a simplified GinzburgLandau type model of superconductivity which is valid for large K and large magnetic field strengths. This model, referred to as the High kappa model, is derived via formal asymptotic expansion of the full, time-dependent Ginzburg-Landau equations. The model accounts for the effects of both applied magnetic fields and currents of constant magnitude. A notable feature of our model is that the systems for the leading order terms for the magnetic potential and the order parameter are decoupled. Finite element approximations of the High kappa model are introduced using standard Galerkin discretization in space and Backward-Euler and Crank-Nicolson discretization schemes in time. We establish existence and uniqueness results for the fully-discrete equations as well as optimal L2 and HI error estimates for the Backward-Euler-Galerkin and the Crank-Nicolson-Galerkin problems. Computational experiments are performed with several combinations of spatial and time discretizations of the High kappa model equations. Among other things our numerical approximations show good agreement for rates of convergence in space and time with the corresponding theoretical values. Finally, some well known steady-state and dynamic phenomena valid for type II superconductors are illustrated numerically.
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    Hysteresis phenomena of ferromagnetic bodies using the nonlocal exchange energy model
    Keane, Michael K. (Virginia Tech, 1993-06-05)
    We examine the relaxed minimization problem for ferromagnetic bodies using the nonlocal exchange energy model. We show that the model possesses a wide range of phenomena including hysteresis, hysteresis subloops, Barkhausen effect, and demagnetization. The results are in three parts. First, we examine analytically the problem of a unit sphere of ferromagnetic material. We show that when the exchange energy is zero we duplicate De Simone's model which has a wide range of measure-valued minimizers. As the exchange energy grows our model stabilizes at the saturated solutions of the Stoner-Wohlfarth model. Here, the measure-valued minimizers are eliminated. Next, we examine numerically the problem of a body composed of several unit spheres of ferromagnetic material. We show that a constrained problem that focuses on the resultant field energy produces results similar to the unconstrained problem with considerable savings in time and resources. Finally, we examine numerically the constrained problem on a moderately large body. It is shown that the constrained problem contains all the hysteresis phenomena mentioned above.
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    Issues related to least-squares finite element methods for the stokes equations
    Deang, Jennifer M.; Gunzburger, Max D. (Siam Publications, 1998-10)
    Least-squares finite element methods have become increasingly popular for the approximate solution of first-order systems of partial differential equations. Here, after a brief review of some existing theories, a number of issues connected with the use of such methods for the velocity-vorticity-pressure formulation of the Stokes equations in two dimensions in realistic settings are studied through a series of computational experiments. Finite element spaces that are not covered by existing theories are considered; included in these are piecewise linear approximations for the velocity. Mixed boundary conditions, which are also not covered by existing theories, are also considered, as is enhancing mass conservation. Next, problems in nonconvex polygonal regions and the resulting nonsmooth solutions are considered with a view toward seeing how accuracy can be improved. A conclusion that can be drawn from this series of computational experiments is that the use of appropriate mesh-dependent weights in the least-squares functional almost always improves the accuracy of the approximations. Concluding remarks concerning three-dimensional problems, the nonlinear Navier-Stokes equations, and the conditioning of the discrete systems are provided.
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    Least squares finite element methods for the Stokes and Navier-Stokes equations
    Bochev, Pavel B. (Virginia Tech, 1994-07-18)
    The central goal of this work is to define and analyze least squares finite element methods for the Stokes and Navier-Stokes equations that are practical and optimal in a systematic and rigorous way. To accomplish this task we begin by developing the least squares theory for the linear Stokes equations. We introduce least squares methods based on the minimization of functionals that involve residuals of the equations of an equivalent first order formulation for the Stokes problem. We show that for the Stokes equations there are two general types of boundary conditions. For the first type, practical least squares methods can be defined and analyzed in a fairly standard way, based on application of the Agmon, Douglis and Nirenberg a priori estimates. For the second type of boundary conditions this task is more difficult and involves mesh dependent (weighted) least squares functionals. Among the main results are the optimal error estimates for the weighted least squares method in two and three space dimensions. Then, we formulate two least squares methods for the nonlinear Navier-Stokes equations written as a first order system. We consider the first method as a conforming discretization of an abstract nonlinear problem and the second weighted one, which is more practical, as a nonconforming discretization of the same abstract problem. As a result, the analysis of the first method fits into the framework of the approximation theory of Brezzi, Rappaz and Raviart and the analysis of the second does not. Thus, we develop an abstract approximation theory that is suitable for nonconforming discretizations of the abstract problem. The central result is based on the application of our abstract theory to the weighted least squares method. We prove that this method results in optimally accurate approximations for the Navier-Stokes equations. We believe that these error analyses of Chapter are the first treatment of a least squares formulation for a nonlinear problem in the current literature. We then discuss various implementation issues, including theoretical and numerical estimates of condition numbers and the presentation of numerical examples. In particular, we study the numerical convergence rates of various implementations of least squares methods and demonstrate that the weights are necessary for the optimal rates to hold. Finally, we compare numerical results for the driven cavity flow problem with some benchmark results reported in the literature.
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    Least-squares finite element approximations to solutions of interface problems
    Cao, Y. Z.; Gunzburger, Max D. (Siam Publications, 1998-02)
    A least-squares finite element method for second-order elliptic boundary value problems having interfaces due to discontinuous media properties is proposed and analyzed. Both Dirichlet and Neumann boundary data are treated. The boundary value problems are recast into a firstorder formulation to which a suitable least-squares principle is applied. Among the advantages of the method are that nonconforming, with respect to the interface, approximating subspaces may be used. Moreover, the grids used on each side of an interface need not coincide along the interface. Error estimates are derived that improve on other treatments of interface problems and a numerical example is provided to illustrate the method and the analyses.
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    Nucleation of superconductivity in finite anisotropic superconductors and the evolution of surface superconductivity toward the bulk mixed state
    Kogan, V. G.; Clem, J. R.; Deang, Jennifer M.; Gunzburger, Max D. (American Physical Society, 2002-02-14)
    In anisotropic superconductors having an arbitrary orientation of the sample surface relative to the crystal principal axes, the surface critical field H-c3 is less than 1.695H(c2) unless the field is situated along one of the principal crystal planes. Below H-c3 in the vicinity of nucleation, the order parameter scales as rootH(c3)-H. Computational studies for infinite cylinders having rectangular cross sections are presented which show that, due to corners and a finite cross section, the surface superconductivity state persists for fields above the theoretically predicted value for semi-infinite samples. They also show that vortices exist within the Surface superconductivity sheath above the bulk critical field.
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    Numerical approximation and identification problems for singular neutral equations
    Cerezo, Graciela M. (Virginia Tech, 1994-04-25)
    A collocation technique in non-polynomial spline space is presented to approximate solutions of singular neutral functional differential equations (SNFDEs). Using solution representations and general well-posedness results for SNFDEs convergence of the method is shown for a large class of initial data including the case of discontinuous initial function. Using this technique, an identification problem is solved for a particular SNFDE. The technique is also applied to other different examples. Even for the special case in which the initial data is a discontinuous function the identification problem is successfully solved.
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    On the Lawrence-Doniach and Anisotropic Ginzburg-Landau models for layered superconductors
    Chapman, S. Jonathan; Du, Qiang; Gunzburger, Max D. (Siam Publications, 1995-02)
    The authors consider two models, the Lawrence-Doniach and the anisotropic Ginzburg-Landau models for layered superconductors such as the recently discovered high-temperature superconductors. A mathematical description of both models is given and existence results for their solution are derived. The authors then relate the two models in the sense that they show that as the layer spacing tends to zero, the Lawrence-Doniach model reduces to the anisotropic Ginzburg-Landau model. Finally, simplified versions of the models are derived that can be used to accurately simulate high-temperature superconductors.
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    Optimal Boundary and Distributed Controls for the Velocity Tracking Problem for Navier-Stokes Flows
    Sandro, Manservisi (Virginia Tech, 1997-05-05)
    The velocity tracking problem is motivated by the desire to match a desired target flow with a flow which can be controlled through time dependent distributed forces or time dependent boundary conditions. The flow model is the Navier-Stokes equations for a viscous incompressible fluid and different kinds of controls are studied. Optimal distributed and boundary controls minimizing a quadratic functional and an optimal bounded distributed control are investigated. The distributed optimal and the bounded control are compared with a linear feedback control. Here, a unified mathematical formulation, covering several specific classes of meaningful control problems in bounded domains, is presented with a complete and detailed analysis of all these time dependent optimal control velocity tracking problems. We concentrate not only on questions such as existence and necessary first order conditions but also on discretization and computational aspects. The first order necessary conditions are derived in the continuous, in the semidiscrete time approximation and in the fully finite element discrete case. This derivation is needed to obtain an accurate meaningful numerical algorithm with a satisfactory convergence rate. The gradient algorithm is used and several numerical computations are performed to compare and understand the limits imposed by the theory. Some computational aspects are discussed without which problems of any realistic size would remain intractable.
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    Optimization Based Domain Decomposition Methods for Linear and Nonlinear Problems
    Lee, Hyesuk Kwon (Virginia Tech, 1997-06-27)
    Optimization based domain decomposition methods for the solution of partial differential equations are considered. The crux of the method is a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the common boundaries between subdomains; the constraints are the partial differential equations. First, we consider a linear constraint. The existence of optimal solutions for the optimization problem is shown as is its convergence to the exact solution of the given problem. We then derive an optimality system of partial differential equations from which solutions of the domain decomposition problem may be determined. Finite element approximations to solutions of the optimality system are defined and analyzed as is an eminently parallelizable gradient method for solving the optimality system. The linear constraint minimization problem is also recast as a linear least squares problem and is solved by a conjugate gradient method. The domain decomposition method can be extended to nonlinear problems such as the Navier-Stokes equations. This results from the fact that the objective functional for the minimization problem involves the jump in dependent variables across the interfaces between subdomains. Thus, the method does not require that the partial differential equations themselves be derivable through an extremal problem. An optimality system is derived by applying a Lagrange multiplier rule to a constrained optimization problem. Error estimates for finite element approximations are presented as is a gradient method to solve the optimality system. We also use a Gauss-Newton method to solve the minimization problem with the nonlinear constraint.
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    Parameter identification in linear and nonlinear parabolic partial differential equations
    Zhang, Lan (Virginia Tech, 1995)
    The research presented in this dissertation is carried out in two parts; the first, which is the main work of this dissertation, involves development of continuous differentiability of the solution with respect to the unknown parameters. For linear parabolic partial differential equations, only mild conditions are assumed on the admissible parameter space. The nonlinear partial differential equation we consider is a generalized Burgers’ equation, for which we establish the well-posedness and the smoothness properties of the solution with respect to the parameters. In the second part, we consider parameter identification problems for these two parameter dependent systems. The identification scheme which we use here is the quasilinearization method. Based on the results in the first part of this work, we obtain existence and local convergence of the algorithm. We also present some numerical examples which demonstrate the performance of the quasilinearization scheme.
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    Parameter identification in parabolic partial differential equations using quasilinearization
    Hammer, Patricia W. (Virginia Tech, 1990-07-15)
    We develop a technique for identifying unknown coefficients in parabolic partial differential equations. The identification scheme is based on quasilinearization and is applied to both linear and nonlinear equations where the unknown coefficients may be spatially varying. Our investigation includes derivation, convergence, and numerical testing of the quasilinearization based identification scheme
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    Sensitivity analysis and computational shape optimization for incompressible flows
    Burkhardt, John (Virginia Tech, 1995-05-08)
    We consider the optimization of a cost functional defined for a fluid flowing through a channeL Parameters control the shape of an obstruction in the flow, and the strength of the inflow. The problem is discretized using finite elements. Optimization algorithms are considered which use either finite differences or sensitivities to estimate the gradient of the cost functional. Problems of scaling, local minimization, and cost functional regularization are considered. Methods of improving the efficiency of the algorithm are proposed.