Showing papers on "Homotopy analysis method published in 1992"

Homotopy Methods for Solving the Optimal Projection Equations for the H2 Reduced Order Model Problem

Abstract: The optimal projection approach to solving the H2 reduced order model problem produces two coupled, highly nonlinear matrix equations with rank conditions as constraints Due to the resemblance of these equations to standard matrix Lyapunov equations, they are called modified Lyapunov equations The algorithms proposed herein utilize probability-one homotopy theory as the main tool It is shown that there is a family of systems (the homotopy) that make a continuous transformation from some initial system to the final system With a carefully chosen initial problem a theorem guarantees that all the systems along the homotopy path will be asymptotically stable, controllable and observable One method, which solves the equations in their original form, requires a decomposition of the projection matrix using the Drazin inverse of a matrix It is shown that the appropriate inverse is a differentiable function An effective algorithm for computing the derivative of the projection matrix that involves solving a set of Sylvester equations is given Another class of methods considers the equations in a modified form, using a decomposition of the pseudogramians based on a contragredient transformation Some freedom is left in making an exact match between the number of equations and the number of unknowns, thus effectively generating a family of methods

Homotopy-determinant algorithm for solving nonsymmetric eigenvalue problems

Abstract: The eigenvalues of a matix A are the zeros of its characteristic polynomial f[lambda] = det[A - [lambda]I]. With Hyman's method of determinant evaluation, a new homotopy continuation method, homotopy-determinant method, is developed in this paper for finding all eigenvalues of a real upper Hessenberg matrix. In contrast to other homotopy continuation methods, the homotopy-determinant method calculates eigenvalues without computing their corresponding eigenvectors. Like all homotopy methods, our method solves the eigenvalue problem by following eigenvalue paths of a real homotopy whose regularity is established to the extent necessary. The inevitable bifurcation and possible path jumping are handled by effective processes. 18 refs., 4 figs., 1 tab.

Globally optimal rational approximation using homotopy continuation methods

Abstract: Homotopy continuation methods are applied to the nonlinear problem of approximating a higher-order system by a lower-order rational model, such that the mean-square modeling error is minimized. A homotopy function is constructed which creates distinct paths from each of the known solutions of a simple problem to each of the solutions of the desired nonlinear problem. This homotopy function guarantees that the globally optimum rational approximation solution may be determined by finding all the solutions. A simple numerical continuation algorithm is described for following the paths to the optimum solution. A numerical example is included which demonstrates that the globally optimum model will be obtained by applying this homotopy continuation method. >

Preconditioned Iterative Methods for Homotopy Curve Tracking

Abstract: Homotopy algorithms are a class of methods for solving systems of nonlinear equations that are globally convergent with probability one. All homotopy algorithms are based on the construction of an appropriate homotopy map and then the tracking of a curve in the zero set of this homotopy map. The curve-tracking algorithms used here require the solution of a series of very special systems. In particular, each (n + 1) x (n + 1) system is in general nonsymmetric but has a leading symmetric indefinite n x n submatrix (typical of large structural mechanics problems, for example). Furthermore, the last row of each system may by chosen (almost) arbitrarily. The authors seek to take advantage of these special properties. The iterative methods studied here include Craig''s variant of the conjugate gradient algorithm and the SYMMLQ algorithm for symmetric indefinite problems. The effectiveness of various preconditioning strategies in this context are also investigated, and several choices for the last row of the systems to be solved are explored.

Integrated homotopy sweep technique for computer-aided geometric design

Abstract: A homotopy sweep technique is proposed for use in CAGD. The technique integrates the sweep technique and the homotopy model through sharing of a common parameter which controls the sweeping as well as the homotopic deformation of the cross sectional shape. The deformation is determined by two functions, a blending function which controls the smooth transition of one contour to the other, and a scaling function which shapes the outline of the sweep object. The sufficient conditions for the homotopy sweep surface patches to meet with first order geometry continuity are discussed. The technique provides a convenient and intuitive control of the shape of the cross section between two defined contours, and allows generalized cylinders to be modelled with lesser user input.

Homotopy methods for solving decoupled power flow equations

Abstract: Explores the application of probability-one homotopy algorithms for solving the load flow equations under the decoupling assumption. The existence of solutions and convergence properties of homotopy methods are interpreted within the power systems framework, via an analogy between nonlinear resistive circuits in steady state and power systems. These ideas are further related to system stability. The effectiveness and performance of various homotopy functions are presented through simulation results for several small examples of power systems. >

Some computational methods for systems of nonlinear equations and systems of polynomial equations

Abstract: This paper gives a brief survey and assessment of computational methods for finding solutions to systems of nonlinear equations and systems of polynomial equations. Starting from methods which converge locally and which find one solution, we progress to methods which are globally convergent and find an a priori determinable number of solutions. We will concentrate on simplicial algorithms and homotopy methods. Enhancements of published methods are included and further developments are discussed.

A computation of power system characteristic by general homotopy and investigation of its stability

Abstract: Presents an application of a general homotopy method to the computation of the characteristics of a power system such as PV curves. The load flow equations are derived from a set of nonlinear differential and algebraic equations. A homotopy continuation method is used for following the characteristics of the power system. The stability of the load flow solutions is discussed by deriving the variational equation. >

The real homotopy-based method for computing solutions of electric power systems

Abstract: Presents a real homotopy-based method to calculate the roots of any system of polynomial equations in n real variables. This method eliminates some of the computational effort required in the existing homotopy- and imbedding-based continuation methods. The method allows for the computation of real solutions of even large-sized deficient polynomial systems. As an example, the method is applied to the load flow analysis of power systems. >

Parallel Homotopy Algorithm for Large Sparse Generalized Eigenvalue Problems: Application to Hydrodynamic Stability Analysis

Abstract: A parallel homotopy algorithm is presented for finding a few selected eigenvalues (for example those with the largest real part) of Az = λBz with real, large, sparse, and nonsymmetric square matrix A and real, singular, diagonal matrix B. The essence of the homotopy method is that from the eigenpairs of Dz = λBz, we use Euler-Newton continuation to follow the eigenpairs of A(t)z = λBz with A(t) ≡ (1−t)D + tA. Here D is some initial matrix and “time” t is incremented from 0 to 1. This method is, to a large degree, parallel because each eigenpath can be computed independently of the others. The algorithm has been implemented on the Intel hypcrcubc. Experimental results on a 64-nodc Intel iPSC/860 hypercube are presented. It is shown how the parallel homotopy method may be useful in applications like detecting Hopf bifurcations in hydrodynamic stability analysis.

A homotopy method for optimal design using an envelope function

Abstract: Philip Y. Shin and Henry A. Castillo ** Naval Postgraduate School An efficient and easy-to-implement procedure for the globally convergent homotopy method for nonlinear optimization is discussed in this paper. An envelope function is used to augment multiple inequality constraints so that difficult active set strategy can be avoided. The procedure is applied for a simple constrained minimization problem, and its convergency is tested for different values of an envelope function parameter. The results indicates that for a large value of the parameter, fast convergence is achieved, but it could result in large errors in the solution. The paper also demonstrates application to optimal design of one-ring stiffened shells. A sequence of optimal designs has been obtained as a function of external hydrostatic pressure. It is seen from the results that the single ring-stiffener works more effectively at lower levels of pressure. INTRODUCTION columns and stiffened composite plates. The multimodal characteristics which occur at the exact optimum are also discussed in the papers. One of the advantages of the homotopy method over other conventional optimization techniques is that instead of obtaining a simple optimum, which is typical of other methods, the homotopy technique generates, in a single computer execution, an entire family of optimum designs for a given parameter. So the method can be applied to the types of problems in which a designer's interest is in comparing different optimal designs for certain values of a parameter. Also, the method shows guaranteed convergence to a desired optimum design. However, the method is hampered when there are several inequality constraints. When the system of equations, derived from the Lagrange multipliers technique, are solved by a homotopy tracing algorithm, the solution path has several branches due to changes in the active constraint set. At a branching point, the Jacobian matrix looses its full ranks, so numerical ill-conditioning The objective of this paper is to devise a ------------------technique to eliminate this branching point Assistant Professor, Department of Mechanical Engistrategy. Since the homo to^^ optimization has neering, Member A I M , ASME. practical difficulties in dealing with several inequality constraints, special treatments of " Graduate Student, Lieutenant, U.S. Navy. augmenting all the inequalities by using the Kreisselmeier-Steinhauser [7] smooth envelope functions are discussed. This approach is first This paper is declared a work of the U.S. Government and tested for a simple constrained function is not subject to copyright protection in the United States. minimization, then, it is applied for optimal design of stiffened cylindrical shells under hydrostatic pressure.