Showing papers on "Homotopy analysis method published in 2001"

Convergent regions of the Newton homotopy method for nonlinear systems: theory and computational applications

Abstract: This paper introduces the concept of the convergent region of a solution of a general nonlinear equation using the Newton homotopy method. The question of whether an initial guess converges to the solution of our interest using the Newton homotopy method is investigated. It is shown that convergent regions of the Newton homotopy method are equal to stability regions of a corresponding Newton dynamic system. A necessary and sufficient condition for the adjacency of two solutions using the Newton homotopy method is derived. An algebraic characterization of a convergent region and its boundary for a large class of nonlinear systems is derived. This characterization is explicit and computationally feasible. A numerical method to determine the convergent region and to establish simple criteria to avoid revisits of the same solutions from different initial guesses is developed. It is shown that for general nonlinear systems or gradient systems, it is computationally infeasible to construct a set of initial guesses which converge to the set of all type-one equilibrium points on the stability boundary of a stable equilibrium point x/sub s/ from a finite number of function values and derivatives near x/sub s/ using the Newton homotopy method. Several examples are applied to illustrate the theoretical developments.

Constructive homotopy methods for finding all or multiple DC operating points of nonlinear circuits and systems

Abstract: This paper develops a constructive homotopy-based methodology for finding all or multiple dc operating points of nonlinear circuits and systems. Several sufficient conditions for the connectivity of all the solutions along a single homotopy path are derived. These conditions offer criteria to determine an initial guess from which all the solutions can be obtained by following the single homotopy path. For the class of nonlinear circuits and systems in which all of the solutions lie on several homotopy paths, a new systematic method to explicitly construct a starting point for each homotopy path is developed. From a practical viewpoint, the constructive method developed does not require the difficult task of finding a good initial guess and is applicable to general nonlinear circuits and systems. From a methodological viewpoint, the constructive method developed is applicable to general homotopy methods with different homotopy functions. One particularly important feature of the developed constructive method is that when this method is applied to a dc problem with box constraints, it is very computationally efficient because it locates only feasible solutions satisfying the given box constraints. The proposed method is applied to several test examples with promising results.

Using homotopy to invert geophysical data

Abstract: Homotopy is a powerful tool for solving nonlinear equations. It is used here to solve small‐dimensional geophysical inverse problems by locating the solutions of the governing normal equations. An Euler‐Newton numerical continuation scheme is used to map trajectories in model space that start from a prescribed solution to a trivial set of equations and terminate at a solution to the inverse problem. The trajectories often map out a continuum of equivalent solutions that are caused by model equivalences or overparameterization. This allows exploration of the solution space topology. The homotopy method, in this application, is relatively insensitive to the choice of starting model. Several examples based on synthetic controlled‐source electromagnetic (CSEM) responses are shown to illustrate the method. An inversion of actual CSEM data from the Canadian Shield is also provided.

Linear iterative solvers based on perturbation techniques

Abstract: In this paper, we propose a new class of algorithms to solve large scale linear algebraic equations. This method is based on homotopy, perturbation technique and Pade approximants.

A multigrid approach for steady state laminar viscous flows

Abstract: In this paper, we use the laminar viscous flow in a lid-driven cavity as an example to describe and verify a numerical scheme for non-linear partial differential equations. The proposed scheme combines a new analytical method for strongly non-linear problems, namely the homotopy analysis method, with the multigrid techniques. A family of formulas at different orders is given. At the lowest order, the current approach is the same as the traditional multigrid methods. However, our high-order scheme needs a fewer number of iterations and less CPU time than the classical ones. Copyright © 2001 John Wiley & Sons, Ltd.

On the Homotopy Classification of Elliptic Boundary Value Problems

Abstract: The present paper deals with the homotopy classification problem of boundary value problems for elliptic operators. We start with classical boundary value problems. The ellipticity condition allows us to reduce classical problems to the Dirichlet problem for the Laplace operator and also to obtain the homotopy classification. We then study general case of operators, that do not necessarily satisfy the Atiyah-Bott condition. The boundary value problem reduces then to the so-called spectral boundary value problem.

Forward displacement analyses of symmetrical stewart platform mechanisms based on homotopy method

Abstract: This paper deals with the problems of forward displacement analyses of symmetrical Stewart platform mechanisms based on the homotopy method The mathematical model is set upA new type coefficient homotopy function is built upThe problem of path intersection in the process of iteration is discussed The principle of selecting the complex constant parameter,which has an influence on the path trace, is givenIt can be used efficiently in homogeneous mechanisms with different structure parameters and the same mechanism with different link variablesA numerical example is given