Homotopy approach for random eigenvalue problem

Abstract

A novel approach, referred to as the homotopy stochastic finite element method, is proposed to solve the eigenvalue problem of a structure associated with some amount of uncertainty based on the homotopy analysis method.For this approach, an infinite multivariate series of the involved random variables is proposed to express the random eigenvalue or even a random eigenvector. The coefficients of the multivariate series are determined using the homotopy analysis method. The convergence domain of the derived series is greatly expanded compared with the Taylor series due to the use of an approach function of the parameter h. Therefore, the proposed method is not limited to random parameters with small fluctuation. However, in practice, only singlevariable and double-variable approximations are employed to simplify the calculation. The numerical examples show that with a suitable choice of the auxiliary parameter h, the suggested approximations can produce very accurate results and require reduced or similar computational efforts compared with the existing methods.KEYWORDS homotopy analysis method, perturbation method, random eigenvalue problem, stochastic finite element method, Taylor series 1 | INTRODUCTION Algebraic eigenvalue problems are a class of basic and significant problems in various fields, such as structural dynamics and structural stability. Currently, the computation of eigenvalues and eigenvectors is well comprehended for deterministic problems. 1,2 In many practical cases, however, the physical properties of the structural systems are not deterministic. For instance, the stiffness of a beam can be affected by material imperfections such that the stiffness distribution along the beam is irregular and difficult to measure and the boundary constraints of beam structures are usually uncertain. 3,4 Therefore, it is extremely necessary to use random variables to more realistically describe the uncertain characteristics that exist in eigenvalue problems in engineering.Due to the randomness of the input parameters, such as the modulus of elasticity, of a physical problem, the desired output or eigenvalues will also be random. The methods for computing these random outputs are generally composed of 2 categories. The first category includes simulation-based methods. All orders of statistical moments and all probability density functions of the eigenvalues can be determined by repeatedly carrying out computations of the deterministic problem in the simulation-based methods. Direct Monte-Carlo (DMC) simulation is the most important and fundamental simulation-based method, 5-9 but it requires considerable computational effort, especially for large systems. Even so, DMC simulation can be conducted and is considered a closed-form solution for evaluating approximation methods. The second category for random analysis, stochastic finite element methods (SFEM), 10-13

An efficient reduced‐order method for stochastic eigenvalue analysis

“…In this section, we simply discuss the method for solving the deterministic eigenproblem Equation (20). For each vector d, the single vector iteration method is enough for our purpose.…”

“…By adopting the power iteration to compute the maximum eigenvalue of Equation (20), a new solution d (j) is computed based on a known approximation d (j−1) ,…”

“…k of the deterministic eigenproblem Equation (20). The middle loop from step 2 to 14 corresponds to computing the kth couple {𝜆 k (𝜃), d k }.…”

A Novel Stochastic Approach for Static Damage Identification of Beam Structures Using Homotopy Analysis Algorithm

“…Many stochastic finite element methods [34][35][36][37][38][39][40], such as the generalized spectral stochastic finite element method [36] and stochastic reduced basis methods [37], may be used to solve Equation (14). For a balance between efficiency and accuracy, it is recommended in this paper to utilize a new proposed homotopy analysis algorithm to determine the solution of the stochastic damage identification equation in Equation (14).…”

“…For a balance between efficiency and accuracy, it is recommended in this paper to utilize a new proposed homotopy analysis algorithm to determine the solution of the stochastic damage identification equation in Equation (14). The homotopy analysis algorithm in the literature [38][39][40] can be used to establish a homotopy relationship between the deterministic damage index and the stochastic damage index.…”