Bimodal optimal design of vibrating plates using theory and methods of nondifferentiable optimization

This paper is concerned with an optimal design problem of vibrating plates. The optimization problem consists in maximizing the smallest eigenvalue of the elliptic eigenvalue problem describing the free plate vibration. The thickness of the plate is the variable subject to optimization. The volume of the plate is constant and the thickness of the plate is bounded.In this paper, we consider the case where the smallest eigenvalue is multiple. This implies that the optimization problem is nondifferentiable. A necessary optimality condition is formulated. The finite-element method is employed as an approximation method. A nonsmooth optimization method is used to solve this optimization problem. Numerical examples are provided.This work was supported by the Polish Academy of Sciences and the Education Ministry of Japan. Lemarechal's implementation of his method was used for numerical computations.on leave from Systems Research Institute, Warsaw, Poland.     

On a non-homogeneous eigenvalue problem involving a potential: An Orlicz–Sobolev space setting

In this paper we study a non-homogeneous eigenvalue problem involving variable growth conditions and a potential V. The problem is analyzed in the context of Orlicz–Sobolev spaces. Connected with this problem we also study the optimization problem for the particular eigenvalue given by the infimum of the Rayleigh quotient associated to the problem with respect to the potential V when V lies in a bounded, closed and convex subset of a certain variable exponent Lebesgue space.     

Concentration phenomena in nonlinear eigenvalue problems with variable exponents and sign-changing potential

In this paper we establish the concentration of the spectrum in an unbounded interval for a class of eigenvalue problems involving variable growth conditions and a sign-changing potential. We also study the optimization problem for the particular eigenvalue given by the infimum of the associated Rayleigh quotient when the variable potential lies in a bounded, closed and convex subset of a certain variable exponent Lebesgue space.     

Non-linear eigenvalue problems for <Emphasis Type="Italic">p</Emphasis>-Laplacian with variable domain

In this work we consider some eigenvalue problems for p-Laplacian with variable domain. Eigenvalues of this operator are taken as a functional of the domain. We calculate the first variation of this functional, using the obtained formula investigate behavior of the eigenvalues when the domain varies. Then we consider one shape optimization problem for the first eigenvalue, prove the necessary condition of optimality relatively domain, offer an algorithm for the numerical solution of this problem.     

Existence of an extremal ground state energy of a nanostructured quantum dot

This paper is concerned with two rearrangement optimization problems. These problems are motivated by two eigenvalue problems which depend nonlinearly on the eigenvalues. We consider a rational and a quadratic eigenvalue problem with Dirichlet’s boundary condition and investigate two related optimization problems where the goal function is the corresponding first eigenvalue. The first eigenvalue in the rational eigenvalue problem represents the ground state energy of a nanostructured quantum dot. In both the problems, the admissible set is a rearrangement class of a given function.     

A nonlinear eigenvalue problem arising in a nanostructured quantum dot

In this paper we investigate a minimization problem related to the principal eigenvalue of the s-wave Schrödinger operator. The operator depends nonlinearly on the eigenparameter. We prove the existence of a solution for the optimization problem and the uniqueness will be addressed when the domain is a ball. The optimized solution can be applied to design new electronic and photonic devices based on the quantum dots.     

Inverse Eigenvalue Problem in Structural Dynamics Design

1 Introduction Structural dynamics design is to design a structure subject to the dynamic characteristics re- quirement, i.e., determine physical and geometrical parameters such that the structure has the given frequencies and (or) mode shapes. This problem often arises in engineering connected with vibration. Recently, Joseph [1], Li et al. [2,3] converted the structural dynamics design to the following inverse eigenvalue problem. GIEP Let x = (x1, , xm)T , and let A(x) and B(x) be real n…     

Duality Theory in Nonlinear Buckling Analysis for von Kármán Equations

The nonlinear eigenvalue problem in buckling analysis is studied for von Kármán plates. By using the general duality theory developed by Gao-Strang [1, 2] it is proved that the stability criterion for the bifurcated state depends on a reduced complementary gap function. The duality theory is established for nonlinear bifurcation problems. This theory shows that the nonlinear eigenvalue problem is eventually equivalent to a coupled quadratic dual optimization problem. A series of equivalent variational principles are constructed and a lower bound theorem for the first eigenvalue of the buckling factor is proved.     

Approximating eigenvectors and eigenvalues across a wide range of design

This paper concerns optimization of the design of large structures in which the solution to a finite-element-based vibration problem is part of the objective function. A method is presented where a spline-fit is used as an approximation of the eigenvalues and eigenvectors across a given design range. subroutines facilitate nonlinear sensitivity calculations which provide the eigenvalue and eigenvector derivatives used in the approximation. This paper presents an example that illustrates good approximations to frequency and mode shapes valid across a wide range of design changes. The method provides the potential for changing mode shapes interactively, and using that information as part of the optimization process.     

A method based on Rayleigh quotient gradient flow for extreme and interior eigenvalue problems

Recently, a continuous method has been proposed by Golub and Liao as an alternative way to solve the minimum and interior eigenvalue problems. According to their numerical results, their method seems promising. This article is an extension along this line. In this article, firstly, we convert an eigenvalue problem to an equivalent constrained optimization problem. Secondly, using the Karush-Kuhn-Tucker conditions of this equivalent optimization problem, we obtain a variant of the Rayleigh quotient gradient flow, which is formulated by a system of differential-algebraic equations. Thirdly, based on the Rayleigh quotient gradient flow, we give a practical numerical method for the minimum and interior eigenvalue problems. Finally, we also give some numerical experiments of our method, the Golub and Liao method, and EIGS (a Matlab implementation for computing eigenvalues using restarted Arnoldi’s method) for some typical eigenvalue problems. Our numerical experiments indicate that our method seems promising for most test problems.     

基于形状优化框架下的Steklov特征值问题研究

<正>1引言特征值问题在应用数学分支和工程中,尤其是在最优设计问题中,有很多的应用,所以特征值问题的最优化已经有了较为深入的研究,见在我们的研究当中,最优设计问题常常以一种指定载荷的设计下、能量的极小化问题的形式出现.在大多数关于最优设计的文章里面,我们更重视在一个固定载荷下条件下结构的最     

The Eigenvalue Problem for a One-Dimensional Differential Operator with a Variable Coefficient and Nonlocal Integral Conditions*

We consider conditions for the existence of the eigenvalue λ = 0 in the eigenvalue problem for a differential operator with a variable coefficient and integral conditions. We analyze how these conditions depend on such properties of the coefficient p(x) as monotonicity and symmetry and observe some other properties of the spectrum of the eigenvalue problem. Particularly, we show by a numerical experiment that the fundamental theorem on the increase of the eigenvalues in the case of increasing coefficient p(x) is not valid for the eigenvalue problem with nonlocal conditions.     

Global optimization for the Biaffine Matrix Inequality problem

It has recently been shown that an extremely wide array of robust controller design problems may be reduced to the problem of finding a feasible point under a Biaffine Matrix Inequality (BMI) constraint. The BMI feasibility problem is the bilinear version of the Linear (Affine) Matrix Inequality (LMI) feasibility problem, and may also be viewed as a bilinear extension to the Semidefinite Programming (SDP) problem. The BMI problem may be approached as a biconvex global optimization problem of minimizing the maximum eigenvalue of a biaffine combination of symmetric matrices. This paper presents a branch and bound global optimization algorithm for the BMI. A simple numerical example is included. The robust control problem, i.e., the synthesis of a controller for a dynamic physical system which guarantees stability and performance in the face of significant modelling error and worst-case disturbance inputs, is frequently encountered in a variety of complex engineering applications including the design of aircraft, satellites, chemical plants, and other precision positioning and tracking systems.     

Topology and material orientation optimization based on evolution equations

Many modern high-performance materials have inherent anisotropic elastic properties and its local material orientation can be considered to be an additional design variable for the topology optimization [1–3]. We extend our previous model for topology optimization with variational controlled growth [4–6] for linear elastic anisotropic materials, for which the material orientation is introduced as an additional design variable. We solve the optimization problem purely with the principles of thermodynamics by minimizing the Gibbs energy. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Tensor eigenvalue complementarity problems

This paper studies tensor eigenvalue complementarity problems. Basic properties of standard and complementarity tensor eigenvalues are discussed. We formulate tensor eigenvalue complementarity problems as constrained polynomial optimization. When one tensor is strictly copositive, the complementarity eigenvalues can be computed by solving polynomial optimization with normalization by strict copositivity. When no tensor is strictly copositive, we formulate the tensor eigenvalue complementarity problem equivalently as polynomial optimization by a randomization process. The complementarity eigenvalues can be computed sequentially. The formulated polynomial optimization can be solved by Lasserre’s hierarchy of semidefinite relaxations. We show that it has finite convergence for generic tensors. Numerical experiments are presented to show the efficiency of proposed methods.     

Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming

Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, for a tensor with nonnegative entries (which we refer it as a nonnegative tensor), efficient numerical schemes have been proposed to calculate its maximum eigenvalue based on a Perron–Frobenius-type theorem. In this paper, we consider a new class of tensors called essentially nonnegative tensors, which extends the concept of nonnegative tensors, and examine the maximum eigenvalue of an essentially nonnegative tensor using the polynomial optimization techniques. We first establish that finding the maximum eigenvalue of an essentially nonnegative symmetric tensor is equivalent to solving a sum of squares of polynomials (SOS) optimization problem, which, in its turn, can be equivalently rewritten as a semi-definite programming problem. Then, using this sum of squares programming problem, we also provide upper and lower estimates for the maximum eigenvalue of general symmetric tensors. These upper and lower estimates can be calculated in terms of the entries of the tensor. Numerical examples are also presented to illustrate the significance of the results.     

Subset simulation for unconstrained global optimization

Global optimization problem is known to be challenging, for which it is difficult to have an algorithm that performs uniformly efficient for all problems. Stochastic optimization algorithms are suitable for these problems, which are inspired by natural phenomena, such as metal annealing, social behavior of animals, etc. In this paper, subset simulation, which is originally a reliability analysis method, is modified to solve unconstrained global optimization problems by introducing artificial probabilistic assumptions on design variables. The basic idea is to deal with the global optimization problems in the context of reliability analysis. By randomizing the design variables, the objective function maps the multi-dimensional design variable space into a one-dimensional random variable. Although the objective function itself may have many local optima, its cumulative distribution function has only one maximum at its tail, as it is a monotonic, non-decreasing, right-continuous function. It turns out that the searching process of optimal solution(s) of a global optimization problem is equivalent to exploring the process of the tail distribution in a reliability problem. The proposed algorithm is illustrated by two groups of benchmark test problems. The first group is carried out for parametric study and the second group focuses on the statistical performance.     

Recurrent neural dynamic models for equilibrium and eigenvalue problems

Neural networks (NN) have been used in a number of interesting applications. In this paper, two neural dynamic models which belong to the class of recurrent neural networks (RNN) have been formulated for the solution of equilibrium and eigenvalue problems. The RNN is comprised of two layers, namely, variable layer and constraint layer, which correspond to the number of design variables in the problem. In addition, the recurrent connections and feed forward connections are used to represent the incremental values in the design parameters. The stability of the neural dynamic model for the equilibrium problem has been guaranteed using Lyapunov's function. Illustrative examples and results of the computer simulation of the neural dynamic model have also been presented.     

A space decomposition scheme for maximum eigenvalue functions and its applications

In this paper, we study nonlinear optimization problems involving eigenvalues of symmetric matrices. One of the difficulties in solving these problems is that the eigenvalue functions are not differentiable when the multiplicity of the function is not one. We apply the \({\mathcal {U}}\)-Lagrangian theory to analyze the largest eigenvalue function of a convex matrix-valued mapping which extends the corresponding results for linear mapping in the literature. We also provides the formula of first-and second-order derivatives of the \({\mathcal {U}}\)-Lagrangian under mild assumptions. These theoretical results provide us new second-order information about the largest eigenvalue function along a suitable smooth manifold, and leads to a new algorithmic framework for analyzing the underlying optimization problem.     

A dynamical method of DAEs for the smallest eigenvalue problem

This article gives a new method based on the dynamical system of differential-algebraic equations for the smallest eigenvalue problem of a symmetric matrix. First, the smallest eigenvalue problem is converted into an equivalent constrained optimization problem. Second, from the Karush–Kuhn–Tucker conditions for this special equality-constrained problem, a special continuous dynamical system of differential-algebraic equations is obtained. Lastly, based on the implicit Euler method and an analogous trust-region technique, we obtain a prediction-correction method to compute a steady-state solution of this special system of differential-algebraic equations, and consequently obtain the smallest eigenvalue of the original problem. We also analyze the local superlinear property for this new method, and present the promising numerical results, in comparison with other methods.