固体变温曲线拟合的新方法——特征值方法

本文提出了小温升条件下固体热传导系统变温过程拟合的新方法。结合机理研究和实验分析,辨识热系统特征值,进而把变温过程多元非线性拟合问题转化为多元线性回归问题。     

On relaxed nested factorization and combination preconditioning

The efficient solution of block tridiagonal linear systems arising from the discretization of convection–diffusion problem is considered in this paper. Starting with the classical nested factorization, we propose a relaxed nested factorization preconditioner. Then, several combination preconditioners are developed based on relaxed nested factorization and a tangential filtering preconditioner. Influence of the relaxation parameter is numerically studied, the results indicate that the optimal relaxation parameter should be close to but less than 1. The number of iteration counts exhibit an extremely sensitive behaviour. This phenomena resembles the behaviour of relaxed ILU preconditioner. For symmetric positive-definite coefficient matrix, we also show that the proposed combination preconditioner is convergent. Finally, numerous test cases are carried out with both additive and multiplicative combinations to verify the robustness of the proposed preconditioners.     

Efficiently preconditioned inexact Newton methods for large symmetric eigenvalue problems

In this paper, we propose an efficiently preconditioned Newton method for the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices. A sequence of preconditioners based on the BFGS update formula is proposed, for the preconditioned conjugate gradient solution of the linearized Newton system to solve Au=q(u) u, q(u) being the Rayleigh quotient. We give theoretical evidence that the sequence of preconditioned Jacobians remains close to the identity matrix if the initial preconditioned Jacobian is so. Numerical results onto matrices arising from various realistic problems with size up to one million unknowns account for the efficiency of the proposed algorithm which reveals competitive with the Jacobi–Davidson method on all the test problems.     

Canonical form of para‐Hermitian pencils,generalized spectral factorization,and optimal control over frequency region

We generalize the J‐spectral factorization of para‐Hermitian proper rational matrix in the case when it has no constant inertia on the imaginary axis. This result and the presented numerical algorithm are based on a canonical form of para‐Hermitian matrix pencils. We apply the new spectral factorization to the optimal control of the proper plant by dynamic measurement feedback over a frequency region. Copyright © 2012 John Wiley & Sons, Ltd.     

An efficient method for computing leading eigenvalues and eigenvectors of large asymmetric matrices

An efficient method for computing a given number of leading eigenvalues (i.e., having largest real parts) and the corresponding eigenvectors of a large asymmetric matrixM is presented. The method consists of three main steps. The first is a filtering process in which the equationx = Mx is solved for an arbitrary initial conditionx(0) yielding:x(t)=e ^Mt x(0). The second step is the construction of (n+1) linearly independent vectorsv _m =M ^m x, 0mn orv _m =e ^mMt x ( being a short time interval). By construction, the vectorsv _m are combinations of only a small number of leading eigenvectors ofM. The third step consists of an analysis of the vectors {v _m } that yields the eigenvalues and eigenvectors. The proposed method has been successfully tested on several systems. Here we present results pertaining to the Orr-Sommerfeld equation. The method should be useful for many computations in which present methods are too slow or necessitate excessive memory. In particular, we believe it is well suited for hydrodynamic and mechanical stability investigations.     

一维结构系统重频特征问题的摄动传递矩阵法

基于传递矩阵法，给出一维结构系统重频振动特征问题的二阶摄动计算方法，该方法适用于一般的一维结构系统的实数和复数重频振动特征问题的摄动分析，并且可避免模态截断误差的影响。只要知道某阶重频的特征值和它的特征向量，用该方法便可准确地计算出其特征值和特征向量的一阶与二阶摄动。算例对弹性支承梁的复数重频振动特征问题进行摄动分析，摄动计算结果和精确计算结果吻合良好。     

Distributed estimation of Laplacian eigenvalues via constrained consensus optimization problems

From the recent literature, we know that some consecutive measurements of the consensus protocol can be used to compute the exact average of the initial condition. In this paper, we show that these measurements can also be used for estimating the Laplacian eigenvalues of the graph representing the network. As recently shown in the literature, by solving the factorization of the averaging matrix, the Laplacian eigenvalues can be inferred. Herein, the problem is posed as a constrained consensus problem formulated two-fold. The first formulation (direct approach) yields a non-convex optimization problem solved in a distributed way by means of the method of Lagrange multipliers. The second formulation (indirect approach) is obtained after an adequate re-parameterization. The problem is then convex and is solved by using the distributed subgradient algorithm and the alternating direction method of multipliers (ADMM). The proposed algorithms allow estimating the actual Laplacian eigenvalues with high accuracy.     

Discrete Eigenvalues of Neutron Transport Equation with Quadratically Anisotropic Scattering

The general properties of discrete eigenvalues of the one-speed neutron transport equation with quadratically anisotropic scattering are studied analytically. We investigate the asymptotic behaviors of boundary curves of the complex discrete eigenvalues carefully and plot the distribution of the discrete eigenvalues in parameter space. Our predictions agree with numerical calculation     

On the maximization of the fundamental eigenvalue in topology optimization

The paper considers a classic formulation of the topology optimization problem of discrete or discretized structures. The objective function to be maximized is the smallest natural frequency of the structure. We develop non-heuristic mathematical models paying special attention to the situation when some design variables take zero values. These models take into account multiple load conditions, equilibrium of forces, constraints on compliance and volume, and the effect of possible non-structural mass. We discuss serious obstacles for a successful numerical treatment of this formulation such as non-Lipschitzean behavior and even discontinuity of the objective function. As a cure, we present an equivalent reformulation as a bilinear semidefinite programming problem without the pitfalls of the original problem. An algorithm is presented for finding an approximation of a globally optimal solution up to a user-defined accuracy. The key ingredient of this algorithm is the treatment of a sequence of linear semidefinite programs. Numerical examples are provided for truss structures. Examples of both academic and larger size illustrate the theoretical results achieved and demonstrate the practical use of this approach. We conclude with an extension on multiple non-structural mass conditions.     

Practical stabilization of time-delay fractional-order systems by parametric controllers

This paper stabilization of time-delayed fractional-order systems by unlimited controllers is considered. To achieve the best controller so that the system be stable, the parameters of the feedback matrices are determinate with the minimum norm. Various constraints applied by the designer to obtain the desired performance criteria. We use the partial eigenvalue assignment (PEVA) method to decrease the constraints and ranks of matrices. The presented method is implemented in two numerical examples.