John C. Bruch

A survey of free boundary value problems in the theory of fluid flow through porous media: variational inequality approach — Part II☆

Abstract A survey is presented of work done using variational inequalities in solving free-boundary problems arising in the filtration of liquids through porous materials. In 1971 Baiocchi proposed this new method for solving such problems. This method proved effective not only from the purely theoretical point of view, that is, for proving the existence and uniqueness of a solution, but also from the point of view of yielding new, simple and efficient numerical solution schemes. This new method was systematically and extensively developed by mathematicians at the Laboratory of Numerical Analysis at the National Research Centre in collaboration with the Institute of Hydraulics of the University of Pavia. The aim of this survey is to present the basic ideas of this method, its applications, and other problems to which various researchers have extended the method. Also, the paper is written in such a manner as to be understood by a more general audience other than mathematicians, who are the ones who have done the majority of work in this area.

Seepage from a trapezoidal and a rectangular channel using variational inequalities

Abstract A variational inequality method has been applied to the problem of two-dimensional seepage from trapezoidal channels and a rectangular channel into permeable soil underlain at a finite depth by a drain. Location of the free surface, shape of the seepage region and seepage flow rate all appear as solutions through the utilization of this method. The numerical portion of the problem is solved using successive overrelaxation (SOR) methods with projection. Both finite-difference and finite-element methods are used. A discussion of the computer approach is given along with comparisons of the results.

Displacement and Velocity Feedback Control of a Beam with a Deadband Region

ABSTRACT The problem of controlling the dynamic response of a beam by means of displacement and velocity feedback is solved. The objective of the control is to prevent the maximum deflection and/or velocity of a beam from exceeding given upper and lower bounds. The control is The theory is illustrated by two numerical examples that involve displacement and velocity feedback control. An assessment is made of the effectiveness of the proposed control by defining a performance measure. It is observed that the dynamic response of the beam can be kept within specified bounds by applying a large enough control force, which also depends on the extent of the deadband region. A measure of force spent in the control process is defined and plotted against the dynamic response, which is observed to decrease rapidly as the bounds approach the values set by the uncontrolled beam.

On the solution of transient free-surface flow problems in porous media by a fixed-domain method

Abstract The numerical model presented here is the solution to the exact initial-boundary-value problems arising in recharge fluid flow through porous media having a free surface. Since the problem is nonlinear and includes a moving boundary, a numerical solution is obtained by using a fixed-domain approach. The Baiocchi transformation and method are used to develop a boundary-value problem which is then solved by an iterative method of successive over-relaxation type. Transient free-surface seepage through a two-dimensional dam with accretion is presented as an example problem. The effects of hydraulic conductivity, specific storativity and accretion on the seepage flow are studied. Two modified cases are satisfactorily compared with published results.

A maximum principle approach to optimal control for one-dimensional hyperbolic systems with several state variables

Abstract The maximum principle developed by Sloss et al. [Optimal control of structural dynamic systems in one space dimension using a maximum principle, J. Vibr. Control 11 (2005) 245–261] is used to determine the optimal control functions for a class of one-dimensional distributed parameter structures. The distributed parameter structures are governed by systems of fourth order hyperbolic equations with constant coefficients. A quadratic performance index is formulated as the cost functional of the problem and can be used to represent the energy of the structure and the force spent in the control process. The developed maximum principle establishes a theoretical foundation for the solution of the optimal control problem and relates the optimal control vector to an adjoint variable vector. The method of solution is outlined which involves reducing the original problem to a system of ordinary differential equations. The solution of the general problem is given and a structural control problem is solved to illustrate the solution procedure. The effectiveness of the proposed control solution is shown by comparing the behavior of controlled and uncontrolled systems.

Boundary optimal control of structural vibrations in an annular plate

A maximum principle is formulated and validated for the vibration control of an annulus plate with the control forces acting on the boundary. In addition, the maximum principle can be applied to plates with multiply connected domains. The performance index is specified as a quadratic functional of displacement and velocity along with a suitable penalty term involving the control forces. Using this index an explicit control law is derived with the help of an adjoint variable satisfying the adjoint differential equation and certain terminal conditions together with the proposed maximum principle. The implementation of the theory is presented and the effectiveness of the boundary control is investigated by a numerical example.

Movement of pollutants in a two-dimensional seepage flowfield

Abstract Hydrodynamic dispersion in seepage from a triangular channel is considered. The finite-element method using isoparametric elements is employed to solve this two-dimensional anisotropic convection-dispersion problem. Included are comparisons between finite-elements and finite-difference solutions to this problem.

Optimal sizing of piezo-actuators for minimum-deflection design of frames under uncertain loads

Optimal piezo-actuator sizes and locations for a frame structure under bending load uncertainty are determined with the objective of minimizing the deflection under worst case of loading. The bending moments generated by the piezo-actuators are used for deflection control, i.e., to minimize the maximum deflection. The frame is subjected to a tip load which lies in an uncertainty domain with regard to its magnitude and direction. The specific uncertain loading studied in the present paper involves a load of given magnitude but of unknown direction, which should be determined to produce the highest deflection. The worst case of loading depends on the size and location of the actuators leading to a nested problem of optimization (design) and anti-optimization (uncertainty problem). Results are given for deterministic and uncertain loading conditions.

Seepage streams out of canals and ditches overlying shallow water tables

Abstract When there is seepage from a canal or ditch overlying a shallow water table, water-logging of the soil at distant points is a distinct possibility. Thus, for given flowfield parameters it is desirous to be able to predict the limit of the drainage capacity which can be handled without producing this condition of water-logging at distant points. A simple numerical scheme is presented for the solution of such problems. This numerical scheme based upon the Baiocchi method and transformation gives a quick, simple, and accurate means of obtaining the discharge and the free-surface location knowing the physical parameters of the particular canal or ditch problem. Thus, the data necessary to investigate the water-logging phenomenon for a particular flow situation can readily be obtained from the application of this numerical approach. The numerical results obtained for specific applications are compared with other published results to similar problems. Also, a discussion is presented concerning the numerical results and the performance of the numerical scheme.

Solution of a free-surface boundary value problem using an inverse formulation and the finite element method

Abstract Two-dimensional free surface problems, examples of which are the seepage from triangular and trapezoidal ditches through porous media, have been solved by the finite element method using rectangular elements. The inverse formulation, in which the x- and y-coordinates are considered dependent variables, and the streamfunction and potential function are the independent variables, has been used in the solution. Included are comparisons between finite element, finite difference, and analytical solutions which show the increased accuracy and computational efficiency of the finite element method.