Homotopy Continuation Technique Research Articles

A continuation-based method for finding laminated composite stacking sequences

A method of recovering laminate ply stacking sequences from a set of up to twelve lamination parameters using polynomial homotopy continuation techniques is presented. The ply angles are treated as continuous variables, and are allowed to take any value between -90° and +90°. The individual plies are assumed to be orthotropic and have constant stiffness. The method is fully deterministic, and does not rely on an optimisation process to establish the stacking sequence. Polyhedral continuation methods are used to limit the solution space in which the stacking sequences are sought. The method can reliably find every stacking sequence solution that exists to achieve a precisely specified set of lamination parameter “targets”, with the number of real solutions to a feasible combination of target properties found to vary from 1 to over 100. The same method is also demonstrated to be able to find stacking sequences to satisfy a set of specified ABD stiffness matrix terms, as might be required following a direct-stiffness modelling design process.

Spatial pattern formation in reaction-diffusion models: a computational approach.

Reaction-diffusion equations have been widely used to describe biological pattern formation. Nonuniform steady states of reaction-diffusion models correspond to stationary spatial patterns supported by these models. Frequently these steady states are not unique and correspond to various spatial patterns observed in biology. Traditionally, time-marching methods or steady state solvers based on Newton's method were used to compute such solutions. However, the solutions that these methods converge to highly depend on the initial conditions or guesses. In this paper, we present a systematic method to compute multiple nonuniform steady states for reaction-diffusion models and determine their dependence on model parameters. The method is based on homotopy continuation techniques and involves mesh refinement, which significantly reduces computational cost. The method generates one-parameter steady state bifurcation diagrams that may contain multiple unconnected components, as well as two-parameter solution maps that divide the parameter space into different regions according to the number of steady states. We applied the method to two classic reaction-diffusion models and compared our results with available theoretical analysis in the literature. The first is the Schnakenberg model which has been used to describe biological pattern formation due to diffusion-driven instability. The second is the Gray-Scott model which was proposed in the 1980s to describe autocatalytic glycolysis reactions. In each case, the method uncovers many, if not all, nonuniform steady states and their stabilities.

Homotopy Continuation for Spatial Interference Alignment in Arbitrary MIMO X Networks

In this paper, we propose an algorithm to design interference alignment (IA) precoding and decoding matrices for arbitrary MIMO X networks. The proposed algorithm is rooted in the homotopy continuation techniques commonly used to solve systems of nonlinear equations. Homotopy methods find the solution of a target system by smoothly deforming the solution of a start system which can be trivially solved. Unlike previously proposed IA algorithms, the homotopy continuation technique allows us to solve the IA problem for both unstructured (i.e., generic) and structured channels such as those that arise when time or frequency symbol extensions are jointly employed with the spatial dimension. To this end, we consider an extended system of bilinear equations that include the standard alignment equations to cancel the interference, and a new set of bilinear equations that preserve the desired dimensionality of the signal spaces at the intended receivers. We propose a simple method to obtain the start system by randomly choosing a set of precoders and decoders, and then finding a set of channels satisfying the system equations, which is a linear problem. Once the start system is available, standard prediction and correction techniques are applied to track the solution all the way to the target system. We analyze the convergence of the proposed algorithm and prove that, for many feasible systems and a sufficiently small continuation parameter, the algorithm converges with probability one to a perfect IA solution. The simulation results show that the proposed algorithm is able to consistently find solutions achieving the maximum number of degrees of freedom in a variety of MIMO X networks with or without symbol extensions. Further, the algorithm provides insights into the feasibility of IA in MIMO X networks for which theoretical results are scarce.

Hard-Spring Bistability and Effect of System Parameters in a Two-Degree-of-Freedom Vibration System with Damping Modeled by a Fractional Derivative

The harmonic balance coupled with the continuation algorithm is a well-known technique to follow the periodic response of dynamical system when a control parameter is varied. However, deriving the algebraic system containing the Fourier coefficients can be a highly cumbersome procedure, therefore this paper introduces polynomial homotopy continuation technique to investigate the steady state bifurcation of a two-degree-of-freedom system including quadratic and cubic nonlinearities subjected to external and parametric excitation forces under a nonlinear absorber. The fractional derivative damping is considered to examine the effects of different fractional order, linear and nonlinear damping coefficients on the steady response. By means of polynomial homotopy continuation, all the possible steady state solutions are derived analytically, i.e. without numerical integration. Coexisting periodic solutions, saddle-node bifurcation and various effects of fractional damping on the steady state response are found and investigated. It is shown that the fractional derivative order and damping coefficient change the bifurcating curves qualitatively and eliminate the saddle-node bifurcation during resonance. Moreover, the system response depicts bigger and bigger region of hard-spring bistability with increasing fractional derivative order, but the region of hard-spring bistability of steady response becomes gradually small and then disappears when we increase the linear and nonlinear damping coefficients. In addition, the analytical results are verified by comparison with the numerical integration ones, it can be found that the present approximate resonance responses are in good agreement with numerical ones.

An efficient computational algorithm for the initial value problem associated with the universal Y functions

AbstractIn the present paper, an efficient iterative method of arbitrary integer order of convergent ≥2 based on the homotopy continuation techniques for the solution of the initial value problem of space dynamics using the universal Y functions is presented. The method is of dynamic nature in the sense that, ongoing from one iterative scheme to the subsequent one, only additional instruction is needed. Most importantly, the method does not need any prior knowledge of the initial guess. This is a property which avoids the critical situations between divergent to very slow convergent solutions that may exist in other numerical methods which depend on initial guesses. A computational package for digital implementation of the method is given, together with numerical applications for elliptic, hyperbolic, and parabolic orbits. The accuracy of the results for all orbits is O(10–16). (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A combined adaptive control parametrization and homotopy continuation technique for the numerical solution of bang–bang optimal control problems

A combined adaptive control parametrization and homotopy continuation technique for the numerical solution of bang–bang optimal control problems

A COMBINED ADAPTIVE CONTROL PARAMETRIZATION AND HOMOTOPY CONTINUATION TECHNIQUE FOR THE NUMERICAL SOLUTION OF BANG–BANG OPTIMAL CONTROL PROBLEMS

AbstractWe present an efficient computational procedure for the solution of bang–bang optimal control problems. The method is based on a well-known adaptive control parametrization method, which is one of the direct methods for numerical solution of optimal control problems. First, the adaptive control parametrization method is reviewed and then its advantages and disadvantages are illustrated. In order to resolve the need for a priori knowledge about the structure of optimal control and for resolving the sensitivity to an initial guess, a homotopy continuation technique is combined with the adaptive control parametrization method. The present combined method does not require any assumptions on the control structure and the number of switching points. In addition, the switching points are captured accurately; also, efficiency of the method is reported through illustrative examples.

NEIMARK BIFURCATIONS OF A GENERALIZED DUFFING–VAN DER POL OSCILLATOR WITH NONLINEAR FRACTIONAL ORDER DAMPING

A generalized Duffing–van der Pol oscillator with nonlinear fractional order damping is introduced and investigated by the residue harmonic homotopy. The cubic displacement involved in fractional operator is used to describe the higher-order viscoelastic behavior of materials and of aerodynamic damping. The residue harmonic balance method is employed to analytically generate higher-order approximations for the steady state responses of an autonomous system. Nonlinear dynamic behaviors of the harmonically forced oscillator are further explored by the harmonic balance method along with the polynomial homotopy continuation technique. A parametric investigation is carried out to analyze the effects of fractional order of damping and the effect of the magnitude of imposed excitation on the system using amplitude-frequency curves. Jump avoidance conditions are addressed. Neimark bifurcations are captured to delineate regions of instability. The existence of even harmonics in the Fourier expansions implies symmetry-breaking bifurcation in certain combinations of system parameters. Numerical simulations are given by comparing with analytical solutions for validation purpose. We find that all Neimark bifurcation points in the response diagram always exist along a straight line.

A homotopy continuation approach for analysing finite-time singularities in thin liquid films

We consider self-similar solutions related to rupture of thin liquid films on a solid substrate that evolve solely under the stabilizing influence of surface tension and the destabilizing influence of effective van der Waals interactions between the free surface of the film and the substrate. Such solutions have been previously analysed in the literature and various numerical approaches to obtain such solutions have been proposed. Such approaches are based either on shooting or finite-difference schemes and require well-chosen initial guesses for solutions. We propose an alternative numerical method, which is based on a homotopy approach and continuation techniques and allows one to reach self-similar solutions from analytically known small-amplitude steady solutions of the related thin-film equation. We argue that this method is more robust than previously proposed methods and does not require initial guesses to obtain solutions. Although the present study focuses on the particular case of self-similar solutions related to planar rupture that have square-root far-field behaviour, our approach can also be used to obtain planar solutions having a different far-field behaviour and radially symmetric self-similar solutions for the considered thin-film equation. We expect the approach to be also valid for other equations of similar type that show a subcritical primary bifurcation and finite-time singularities.

RESONANCE RESPONSE OF A SIMPLY SUPPORTED ROTOR-MAGNETIC BEARING SYSTEM BY HARMONIC BALANCE

Both the primary and superharmonic resonance responses of a rigid rotor supported by active magnetic bearings are investigated by means of the total harmonic balance method that does not linearize the nonlinear terms so that all solution branches can be studied. Two sets of second order ordinary differential equations governing the modulation of the amplitudes of vibration in the two orthogonal directions normal to the shaft axis are derived. Primary resonance is considered by six equations and superharmonic by eight equations. These equations are solved using the polynomial homotopy continuation technique to obtain all the steady state solutions whose stability is determined by the eigenvalues of the Jacobian matrix. It is found that different shapes of frequency-response and forcing amplitude-response curves can exist. Multiple-valued solutions, jump phenomenon, saddle-node, pitchfork and Hopf bifurcations are observed analytically and verified numerically. The new contributions include the foolproof multiple solutions of the strongly nonlinear system by means of the total harmonic balance. Some predicted frequency varying amplitudes could not be obtained by the multiple scales method.

Dynamics of a Duffing-van der Pol oscillator with time delayed position feedback

The two main objectives of this article are (1) to determine the bifurcating periodic responses accurately and to study the effects of time delay and feedback gain on the steady state response in autonomous oscillators by means of the residue harmonic balance method; and (2) to study the dynamics of both the autonomous and non-autonomous Duffing-van der Pol oscillators with delayed position feedback as examples. The results of the steady state response to frequency and amplitude excitation versus the time delay, positive and negative feedback gains are studied. Both saddle-node and subcritical Hopf bifurcations are found and discussed. The obtained second-order solutions are in excellent agreement with numerical integration results. By combining both harmonic balance and polynomial homotopy continuation techniques, the superharmonic resonance responses of the non-autonomous oscillator are investigated in order to obtain all possible steady state solutions. The effects of the feedback gains and time delay ...

New progress in real and complex polynomial root-finding

Matrix methods are increasingly popular for polynomial root-finding. The idea is to approximate the roots as the eigenvalues of the companion or generalized companion matrix associated with an input polynomial. The algorithms also solve secular equation. QR algorithm is the most customary method for eigen-solving, but we explore the inverse Rayleigh quotient iteration instead, which turns out to be competitive with the most popular root-finders because of its excellence in exploiting matrix structure. To advance the iteration we preprocess the matrix and incorporate Newton’s linearization, repeated squaring, homotopy continuation techniques, and some heuristics. The resulting algorithms accelerate the known numerical root-finders for univariate polynomial and secular equations, and are particularly well suited for the acceleration by using parallel processing. Furthermore, even on serial computers the acceleration is dramatic for numerical approximation of the real roots in the typical case where they are much less numerous than all complex roots.

Computational developments for distance determination of stellar groups

In this paper, we consider a statistical method for distance determination of stellar groups. The method depends on the assumption that the members of the group scatter around a mean absolute magnitude in Gaussian distribution. The mean apparent magnitude of the members is then expressed by frequency function, so as to correct for observational incompleteness at the faint end. The problem reduces to the solution of a highly transcendental equation for a given magnitude parameter α. For the computational developments of the problem, continued fraction by the Top-Down algorithm was developed and applied for the evaluation of the error function erf(z). The distance equation Λ(y) = 0 was solved by an iterative method of second order of convergence using homotopy continuation technique. This technique does not need any prior knowledge of the initial guess, a property which avoids the critical situations between divergent and very slow convergent solutions, that may exist in the applications of other iterative methods depending on initial guess. Finally, we apply the method for the nearby main sequence late type stars assuming that the stars of each group of the same spectral type scatter around a mean absolute magnitude in a Gaussian distribution. The accuracies of the numerical results are satisfactory, in that, the percentage errors between r and the mean values are respectively: (2.4%, 1.6%, 0.72%, 0.66%, 3.5%, 2.4%, 2%, 2.5%, 0.9%) for the stars of spectral types: (F5V, F6V, F7V, F8V, F9V, G0V, G2V, G5V, G8V).

The secant–homotopy continuation method

Abstract A new secant–homotopy continuation formula is presented. In this paper, the traditional Newton–Raphson method is modified by the secant theory, and the homotopy continuation technique is applied to a new secant–homotopy continuation formula. By means of adjusting the auxiliary function, we can solve the nonlinear equations and guarantee the solutions exactly without divergence rather than the traditional numerical methods such as the Newton–Raphson method.

Computations of the cosmic distance equation

In this paper, an iterative method of second order of convergence for solving the distance equation is developed using homotopy continuation technique. The method does not need any priori knowledge of the initial guess, a property which avoids the critical situations between divergent to very slow convergent solutions, that may exist in the application of other numerical methods depending on initial guess. Moreover, this method uses the most efficient technique of the continued fraction expansions for the evaluation of the basic functions involved. The accuracy of the method is at least of O(10 −8), a sufficient accuracy that secures very accurate distance determination.

A modified formula of ancient Chinese algorithm by the homotopy continuation technique

A modified formula, iteration representation, of ancient Chinese algorithm by combining the homotopy continuation technique is presented. Considering the convergence, the new iteration formula and the technique for the choice of the auxiliary homotopy function are developed. Once we have the condition f( x 1) = f( x 2) or f( x) is an even function, we will diverge in traditional ancient Chinese algorithm. The paper provides some useful rules to overcome this defect.

Matrix-valued nevanlinna-pick interpolation with complexity constraint: an optimization approach

Over the last several years, a new theory of Nevanlinna-Pick interpolation with complexity constraint has been developed for scalar interpolants. In this paper we generalize this theory to the matrix-valued case, also allowing for multiple interpolation points. We parameterize a class of consisting of most interpolants of no higher degree than the central solution in terms of spectral zeros. This is a complete parameterization, and for each choice of interpolant we provide a convex optimization problem for determining it. This is derived in the context of duality theory of mathematical programming. To solve the convex optimization problem, we employ a homotopy continuation technique previously developed for the scalar case. These results can be applied to many classes of engineering problems, and, to illustrate this, we provide some examples. In particular, we apply our method to a benchmark problem in multivariate robust control. By constructing a controller satisfying all design specifications but having only half the McMillan degree of conventional H/sup /spl infin// controllers, we demonstrate the advantage of the proposed method.

Homotopy continuation method for calculating critical loci of binary mixtures

This work describes an algorithm that uses the Heidemann and Khalil (1980, A.I.Ch.E. J., 26, 769–779) critical point criteria and a homotopy continuation technique for calculating critical loci in binary mixtures. This approach is able to track smoothly transition from G/L critical points to L/L critical points without the need of changing initial estimates. Such transitions marked by the existence of a turning point in the critical locus are commonly found in type III and IV mixtures. The tangent plane global stability tests are included to locate UCEPs and LCEPs which are intersections of the critical locus and the LLG line. The continuation method is as efficient as a gradient-based approach like Heidemann and Khalil’s method and as robust as the Hicks and Young’s grid-search approach. It also provides physical insight to the change in phase behavior as the composition and the equation of states parameters change.

Application of New Homotopy Continuation Techniques to Variable Geometry Trusses

A VGT, or Variable Geometry Truss, can be thought of as a statically determinate truss that has been modified to contain some number of variable length members. These extensible members allow the truss to vary its configuration in a controlled manner. Some of the typical applications envisioned for VGTs are as booms to position equipment in space, as supports for space antennae, and as berthing devices. Recently, they have also been proposed as parallel-actuated, long chain, high dexterity manipulators. This paper will demonstrate the use of homotopy continuation in solving the kinematics of relatively complex variable geometry trusses (VGTs) including the octahedron and the decahedron. The procedural aspects are described in detail with the help of examples.