Homotopy Method Research Articles

A new stochastic residual error based homotopy approach for stability analysis of structures with large fluctuation of random parameters

AbstractThe large fluctuation of uncertain parameters introduces a great challenge in the stability analysis of structures. To address this problem, a novel stochastic residual error based homotopy method is proposed in this article. This new method used the concept of homotopy to reconstruct a new governing equation for stochastic elastic buckling analysis, and the closed‐form solutions of the isolated buckling eigenvalues and eigenvectors are obtained by the stochastic homotopy analysis method. On this basis, apth order origin moment of the stochastic residual error with respect to the elastic buckling equation is defined. Then, the optimal form of the homotopy series can be determined automatically by minimizing thepth order origin moment, which overcomes the disadvantage of highly relying on sample values of the existing homotopy stochastic finite element method. Moreover, the proposed method is developed to deal with the stochastic closely spaced buckling eigenvalue problem. Three mathematical examples and three buckling eigenvalue examples, including a variable cross‐section column, a 7‐story frame, and a Kiewitt single‐layer latticed spherical shell, are performed to illustrate the accuracy and effectiveness of the proposed method by comparing with the existing methods when dealing with large fluctuation of random parameters.

Brain Tumor Segmentation Based on Bendlet Transform and Improved Chan-Vese Model.

Automated segmentation of brain tumors is a difficult procedure due to the variability and blurred boundary of the lesions. In this study, we propose an automated model based on Bendlet transform and improved Chan-Vese (CV) model for brain tumor segmentation. Since the Bendlet system is based on the principle of sparse approximation, Bendlet transform is applied to describe the images and map images to the feature space and, thereby, first obtain the feature set. This can help in effectively exploring the mapping relationship between brain lesions and normal tissues, and achieving multi-scale and multi-directional registration. Secondly, the SSIM region detection method is proposed to preliminarily locate the tumor region from three aspects of brightness, structure, and contrast. Finally, the CV model is solved by the Hermite-Shannon-Cosine wavelet homotopy method, and the boundary of the tumor region is more accurately delineated by the wavelet transform coefficient. We randomly selected some cross-sectional images to verify the effectiveness of the proposed algorithm and compared with CV, Ostu, K-FCM, and region growing segmentation methods. The experimental results showed that the proposed algorithm had higher segmentation accuracy and better stability.

Conditional selective inference for robust regression and outlier detection using piecewise-linear homotopy continuation

In this paper, we consider conditional selective inference (SI) for a linear model estimated after outliers are removed from the data. To apply the conditional SI framework, it is necessary to characterize the events of how the robust method identifies outliers. Unfortunately, the existing conditional SIs cannot be directly applied to our problem because they are applicable to the case where the selection events can be represented by linear or quadratic constraints. We propose a conditional SI method for popular robust regressions such as least-absolute-deviation regression and Huber regression by introducing a new computational method using a convex optimization technique called homotopy method. We show that the proposed conditional SI method is applicable to a wide class of robust regression and outlier detection methods and has good empirical performance on both synthetic data and real data experiments.

Numerical solution of density-driven groundwater flows using a generalized finite difference method defined by an unweighted least-squares problem

Density-driven groundwater flows are described by nonlinear coupled differential equations. Due to its importance in engineering and earth science, several linearizations and semi-linearization schemes for approximating their solution have been proposed. Among the more efficient are the combinations of Newtonian iterations for the spatially discretized system obtained by either scalar homotopy methods, fictitious time methods, or meshless generalized finite difference method, with several implicit methods for the time integration. However, when these methods are used, some parameters need to be determined, in some cases, even manually. To overcome this problem, this paper presents a novel generalized finite differences scheme combined with an adaptive step-size method, which can be applied for solving the governing equations of interest on non-rectangular structured and unstructured grids. The proposed method is tested on the Henry and the Elder problems to verify the accuracy and the stability of the proposed numerical scheme.

Machine learning the real discriminant locus

Parameterized systems of polynomial equations arise in many applications in science and engineering with the real solutions describing, for example, equilibria of a dynamical system, linkages satisfying design constraints, and scene reconstruction in computer vision. Since different parameter values can have a different number of real solutions, the parameter space is decomposed into regions whose boundary forms the real discriminant locus. This article views locating the real discriminant locus as a supervised classification problem in machine learning where the goal is to determine classification boundaries over the parameter space, with the classes being the number of real solutions. This article presents a novel sampling method which carefully samples a multidimensional parameter space. At each sample point, homotopy continuation is used to obtain the number of real solutions to the corresponding polynomial system. Machine learning techniques including nearest neighbors, support vector classifiers, and neural networks are used to efficiently approximate the real discriminant locus. One application of having learned the real discriminant locus is to develop a real homotopy method that only tracks real solution paths unlike traditional methods which track all complex solution paths. Examples show that the proposed approach can efficiently approximate complicated solution boundaries such as those arising from the equilibria of the N=4 Kuramoto model which was previously intractable using traditional methods.

Two-stage homotopy method to incorporate discrete control variables into AC-OPF

Alternating-Current Optimal Power Flow (AC-OPF) is an optimization problem critical for planning and operating the power grid. The problem is traditionally formulated using only continuous variables. Typically, control devices with discrete-valued settings, which provide valuable flexibility to the network and improve resilience, are omitted from AC-OPF formulations due to the difficulty of integrality constraints. We propose a two-stage homotopy algorithm to solve the AC-OPF problem with discrete-valued control settings. This method does not rely on prior knowledge of control settings or other initial conditions. The first stage relaxes the discrete settings to continuous variables and solves the optimization using a robust homotopy technique. Once the solution has been obtained using relaxed models, the second homotopy problem gradually transforms the relaxed settings to their nearest feasible discrete values. We test the proposed algorithm on several large networks with switched shunts and adjustable transformers and show it can outperform a similar state-of-the-art solver.

Solution to a Damped Duffing Equation Using He’s Frequency Approach

In this paper, we generalize He's frequency approach for solving the damped Duffing equation by introducing a time varying amplitude. We also solve this equation by means of the homotopy method and the Lindstedt–Poincaré method. High accurate formulas for approximating the Jacobi elliptic function cn are formally derived using Chebyshev and Pade approximation techniques.

Nonlinear homotopy interior-point algorithm for 6-DoF powered landing guidance

The six-degree-of-freedom (6-DoF) powered landing guidance problem usually is difficult to solve for several reasons. First, this problem coupled with low-frequency translational and high-frequency rotational equations of motion is naturally large-scale. Second, the constraints enforced into the problem are highly coupled, nonlinear and non-convex, which makes the problem quite sensitive to the initial value guesses. Third, the initial position, velocity and attitude of the landing vehicle is quite difficult to forecast due to the complex endo-atmospheric environment, thus the traditional guidance methods are difficult to obtain a predetermined trajectory. In this paper, for the first time we proposed a nonlinear homotopy interior-point optimization algorithm (NHOPT) for solving the 6-DoF powered landing guidance problem. Different from general external homotopy strategies, we embed the concept of homotopy methods into the barrier line-search interior-point method. Using vertical landing without position and velocity offset, it is easy to provide initial guesses, which we define as the initial homotopy subproblem. Then a class of related homotopy subproblems is solved along a homotopy path using the algorithm until the original problem is converged. Numerical results demonstrate that the algorithm has high performance in terms of convergence and computational efficiency.

Fixed Point and Homotopy Methods in Cone A-Metric Spaces and Application to the Existence of Solutions to Urysohn Integral Equation

The purpose of this article is to introduce an ordered implicit relation that can be used for the existence of fixed points of new contractions defined in cone A-metric spaces. We investigate a fixed-point method for proving the existence of Urysohn integral equation solutions. We prove an homotopy result by the application of obtained fixed-point theorem. The hypothesis is demonstrated with examples.

Explicit Solutions to Large Deformation of Cantilever Beams by Improved Homotopy Analysis Method I: Rotation Angle

An improved homotopy analysis method (IHAM) is proposed to solve the nonlinear differential equation, especially for the case when nonlinearity is strong in this paper. As an application, the method was used to derive explicit solutions to the rotation angle of a cantilever beam under point load at the free end. Compared with the traditional homotopy method, the derivation includes two steps. A new nonlinear differential equation is firstly constructed based on the current nonlinear differential equation of the rotation angle and the auxiliary quadratic nonlinear differential equation. In the second step, a high-order non-linear iterative homotopy differential equation is established based on the new non-linear differential equation and the auxiliary linear differential equation. The analytical solution to the rotation angle is then derived by solving this high-order homotopy equation. In addition, the convergence range can be extended significantly by the homotopy–Páde approximation. Compared with the traditional homotopy analysis method, the current improved method not only speeds up the convergence of the solution, but also enlarges the convergence range. For the large deflection problem of beams, the new solution for the rotation angle is more approachable to the engineering designers than the implicit exact solution by the Euler–Bernoulli law. It should have significant practical applications in the design of long bridges or high-rise buildings to minimize the potential error due to the extreme length of the beam-like structures.

Residue-regulating homotopy method for strongly nonlinear oscillators

Residue-regulating homotopy method for strongly nonlinear oscillators

A Homotopy Method for the Constrained Inverse Problem in the Multiphase Porous Media Flow

This paper considers the constrained inverse problem based on the nonlinear convection-diffusion equation in the multiphase porous media flow. To solve this problem, a widely convergent homotopy method is introduced and proposed. To evaluate the performance of the mentioned method, two numerical examples are presented. This method turns out to have wide convergence region and strong anti-noise ability.

Application of He's homotopy and perturbation method to solve heat transfer equations: A python approach

In this research article, an attempt has been made to solve the linear/nonlinear and steady/unsteady heat transfer equations using the Homotopy and Perturbation method (HPM). Moreover, the implementation of HPM has been done by using SymPy, a library in python, to solve problems symbolically. Total three problems were dealt viz. steady-state conduction with heat generation, lumped capacitance analysis with a variable specific heat of the material, and heat transfer in uniform rectangular fin with radiation from the surface. In all the cases, the HPM has given excellent results compared to the analytical and numerical. Finally, the execution of SymPy has been explained, and a detailed procedure to implement HPM through python has been presented for all three cases.

Algorithm for solving fractional partial differential equations using homotopy analysis method with Pade approximation

<span>In recent years nonlinear problems have several methods to be solved and utilize a well-known analytic tools such as homotopy analysis method. In general, homotopy analysis method had gain a wide focus and improvement especially in typical nonlinear problem. The aim of this paper is to use homotopy method of analysis to solve partial differential equation in addition to improve method’s efficiency. The method in this paper is to apply approximation to Pade’ approach to obtain sufficient efficiency. As a result, the improvement has been verified by solving two cases beside a mean value comparison of the homotopy analysis method’s squared error with the improved form.</span>

A Novel Collision-Free Homotopy Path Planning for Planar Robotic Arms.

Achieving the smart motion of any autonomous or semi-autonomous robot requires an efficient algorithm to determine a feasible collision-free path. In this paper, a novel collision-free path homotopy-based path-planning algorithm applied to planar robotic arms is presented. The algorithm utilizes homotopy continuation methods (HCMs) to solve the non-linear algebraic equations system (NAES) that models the robot’s workspace. The method was validated with three case studies with robotic arms in different configurations. For the first case, a robot arm with three links must enter a narrow corridor with two obstacles. For the second case, a six-link robot arm with a gripper is required to take an object inside a narrow corridor with two obstacles. For the third case, a twenty-link arm must take an object inside a maze-like environment. These case studies validated, by simulation, the versatility and capacity of the proposed path-planning algorithm. The results show that the CPU time is dozens of milliseconds with a memory consumption less than 4.5 kB for the first two cases. For the third case, the CPU time is around 2.7 s and the memory consumption around 18 kB. Finally, the method’s performance was further validated using the industrial robot arm CRS CataLyst-5 by Thermo Electron.

Experimental investigation and modelling of separation behavior for the Bi–Cu–Pb system in vacuum distillation containing one binary azeotrope

The VLE (vapor-liquid equilibrium) experiments for the Bi–Cu–Pb system containing a sub-binary azeotrope at 1 and 10 Pa were carried out. The VLE data for the sub-binary and ternary systems were correlated using the Wilson equation. Based on the Euler-Newton homotopy method, the azeotropic points of the Bi–Cu–Pb system at different temperatures were predicted. The results showed good agreement with experimental data. The thermodynamic consistency of the experimental values was verified by Van Ness test. This work provides a theoretical basis for vacuum distillation of ternary alloy containing a sub-binary azeotrope. It is helpful to determine suitable experimental conditions for the separation and purification of crude metals in vacuum distillation.

Parameter estimation with the multigrid-homotopy method for a nonlinear diffusion equation

This paper is concerned with the numerical identification of the piecewise constant permeability function for a nonlinear diffusion equation within the two-phase porous media flow. A novel iteration scheme based on the multigrid idea and homotopy method, named as the multigrid-homotopy method, has been developed for solving this problem. Numerical results indicate that this method can overcome the numerical instability and be robust to data noise. In addition, compared with the multigrid method, the multigrid-homotopy method expands the convergence domain; compared with the homotopy method, this method improves the convergence rate. The results are promising.

Large Scale Multi-Period Optimal Power Flow With Energy Storage Systems Using Differential Dynamic Programming

With theincreased penetration of renewable sources, power gridsare becoming stressed due to fluctuating generation. To alleviate stress from inconsistent sources, utilities employ energy storage systems alongside renewable sources and rely on dispatching synchronous generators. However, optimal dispatch of such devices is limited by traditional AC optimal power flow methods that do not account for time-dependent constraints, such as the state of charge of energy storage systems and generator ramping constraints. Multi-period optimal power flow is proposed as a large non-convex non-linear problem to optimally dispatch and control generators and energy storage elements across multiple time periods. In this paper, we introduce a scalable, robust framework to solve multi-period optimal power flow using a differential dynamic programming scheme that makes it capable of scaling to large systems containing energy storage devices. We demonstrate the efficacy of this solution by optimizing the SyntheticUSA testcase over a set of time periods. A robust homotopy method is applied to achieve fast simulation times that can be parallelized for further improvements.

Regularization homotopy method for solving distance equations

Regularization homotopy method for solving distance equations

Rayleigh–Bloch waves above the cutoff

Extensions of Rayleigh–Bloch waves above the cutoff frequency are studied via the discrete spectrum of a transfer operator for a channel containing a single cylinder with quasi-periodic side-wall conditions. Above the cutoff, the Rayleigh–Bloch wavenumber becomes complex valued and an additional wavenumber appears. For small- to intermediate-radius values, the extended Rayleigh–Bloch waves are shown to connect the Neumann and Dirichlet trapped modes before embedding in the continuous spectrum. A homotopy method involving an artificial damping term is proposed to identify the discrete spectrum close to the embedding. Moreover, Rayleigh–Bloch waves vanish beyond some frequency but reappear at higher frequencies for small and large cylinders. The existence and properties of the Rayleigh–Bloch waves are connected with finite-array resonances.