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     

Stability Analysis of A Floating Caliper Disc Brake

In this paper, the onset of the dynamic instability in a .oating caliper disc brake system is studied. The disc is modelled as a .exible rotating plate, the pad is taken as a small mass and a distributed sti.ness, and the motion of the caliper is also taken into consideration. The linearized equations of motion about the equilibrium position are derived assuming a constant braking force. The equations are subsequently discretized with the help of appropriate shape functions. The eigenvalue problem is then solved to detect the onset of instability. The effects of damping on the stability of motion are investigated as well.     

Spike‐Type Solutions to One Dimensional Gierer–Meinhardt Model with Lévy Flights

The Gierer–Meinhardt model with Lévy flights is shown to give rise to patterns of spikes with algebraically decaying tails. The spike shape is given by a solution to a fractional differential equation. Near an equilibrium formation the spikes drift according to the differential equations of the form known for Fickian diffusion, but with a new homoclinic. A nonlocal eigenvalue problem of a new type is formulated and studied. The system is less stable due to the Lévy flights, though the behavior of eigenvalues is changed mainly quantitatively.     

The stress driven instability in elastic crystals: Mathematical models and physical manifestations

Summary Equilibrium equations and stability conditions for the simple deformable elastic body are derived by means of considering a minimum of the static energy principle. The energy is supposed to be sum of the volume (elastic) and the surface terms. The ability to change relative positions of different material particles is taken into account, and appropriate natural definitions of the first and second variations of the energy are introduced and calculated explicitly. Considering the case of negligible magnitude of the surface tension, we establish that an equilibrium state of a nonhydrostatically stressed simple elastic body (of any physically reasonable elastic energy potential and of any symmetry) possessing any small smooth part of free surface is always unstable with respect to relative transfer of the material particles along the surface. Surface tension suppresses the mentioned instability with respect to sufficiently short disturbances of the boundary surface and thus can probably provide local smoothness of the equilibrium shape of the crystal. We derive explicit formulas for critical wavelength for the simplest models of the internal and surface energies and for the simplest equilibrium configurations. We also formulate the simplest problem of mathematical physics, revealing peculiarities and difficulties of the problem of equilibrium shape of elastic crystals, and discuss possible manifestations of the above-mentioned instability in the problems of crystal growth, materials science, fracture, physical chemistry, and low-temperature physics.     

Relative equilibrium configurations of point vortices on a sphere

The problem of constructing and classifying equilibrium and relative equilibrium configurations of point vortices on a sphere is studied. A method which enables one to find any such configuration is presented. Configurations formed by the vortices placed at the vertices of Platonic solids are considered without making the assumption that the vortices possess equal in absolute value circulations. Several new configurations are obtained.     

A model for surface diffusion of trans-membrane proteins on lipid bilayers

The equilibrium theory of lipid membranes is modified to include the effects of a continuous distribution of trans-membrane proteins. These influence membrane shape and evolve in accordance with a diffusive balance law. The model is purely mechanical in the absence of the proteins. Conditions ensuring energy dissipation in the presence of diffusion are given and an example constitutive function is used to simulate the coupled inertia-less interplay between membrane shape and protein distribution. The work extends an earlier continuum theory of equilibrium configurations of composite lipid-protein membranes to accommodate surface diffusion.     

Thermal Buckling Analysis of FGM Sandwich Circular Plates Under Transverse Nonuniform Temperature Field Actions北大核心CSCD

Based on the von Kármán geometric nonlinear plate theory, the displacement⁃type geometric nonlinear governing equations for FGM sandwich circular plates under transverse nonlinear temperature field actions were derived. With the immovable clamped boundary condition, the analytical formula for dimensional critical buckling temperature differences of the system was obtained from the solution of the linear eigenvalue problem. Moreover, the 2⁃point boundary value problem of ordinary differential equations was solved with the shooting method. The effects of geometric parameters, constituent material properties, gradient indexes, temperature field parameters and layer⁃thickness ratios on the critical buckling temperature differences, the thermal postbuckling equilibrium paths, and the buckling equilibrium configurations of FGM sandwich circular plates, were investigated. The results show that, with the increases of the thickness⁃radius ratio, the relative thickness of the FGM layer and the gradient index, the FGM sandwich circular plate's critical buckling temperature difference will increase monotonically. Given a fixed radius and a fixed total thickness, the postbuckling deformation of the FGM sandwich circular plate will decrease significantly with the relative thickness of the FGM layer. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Equilibrium problem for the eigenvalues of banded block Toeplitz matrices

We consider banded block Toeplitz matrices T_n with n block rows and columns. We show that under certain technical assumptions, the normalized eigenvalue counting measure of T_n for n → ∞ weakly converges to one component of the unique vector of measures that minimizes a certain energy functional. In this way we generalize a recent result of Duits and Kuijlaars for the scalar case. Along the way we also obtain an equilibrium problem associated to an arbitrary algebraic curve, not necessarily related to a block Toeplitz matrix. For banded block Toeplitz matrices, there are several new phenomena that do not occur in the scalar case: (i) The total masses of the equilibrium measures do not necessarily form a simple arithmetic series but in general are obtained through a combinatorial rule; (ii) The limiting eigenvalue distribution may contain point masses, and there may be attracting point sources in the equilibrium problem; (iii) More seriously, there are examples where the connection between the limiting eigenvalue distribution of T_n and the solution to the equilibrium problem breaks down. We provide sufficient conditions guaranteeing that no such breakdown occurs; in particular we show this if T_n is a Hessenberg matrix.     

The inverse scattering problem for a partially coated penetrable cavity with interior measurements

In this paper we consider the inverse scattering problem for a cavity that is bounded by a partially coated penetrable inhomogeneous medium of compact support and recover the shape of the cavity and the surface conductivity from a knowledge of measured scattered waves due to point sources located on a curve or surface inside the cavity. First, we prove that both the shape of the cavity and the surface conductivity on the coated part can be uniquely determined from a knowledge of the measured data. Next, we establish a linear sampling method for determining both the shape of the cavity and the surface conductivity. A central role in our justification is played by an eigenvalue problem which we call the exterior transmission eigenvalue problem. Finally, we present some numerical examples to illustrate the validity of our method.     

密封容器组合壳自由振动的精确解

给出了一类密封容器组合壳自由振动问题的精确解,基于Love经典薄壳理论,导出了具有任意经线形状的旋转壳体在轴对称振动时的基本方程,组合壳结构中球壳与柱壳的连接条件是通过连接处的变形连续性和内力平衡关系得出的。问题的数学模型被归结为常微分方程组在球壳和 壳两个区间上的特征值问题。振动模态函数是由Legendre和三角函数构造出来,并且得到了精确的频率方程。所有的计算都是在Maple程序下运行的,无论是精确的符号运算还是具有所需有效数学精度的数值计算,都表明该文所编译的Maple程序是简单而有效的。固有频率的数值结果同文献中有限元法和其它数值方法的结果作了比较。作为一个标准,该文给出的精确解对于检验各种近似方法的精密度是有价值的。     

滑动支撑的弹性梁静态方程的可解性

给出了滑动支撑的弹性梁静态方程可解的一组两参数非共振条件。     

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.     

On the Optimal Insulation of Conductors

We coat a conductor with an insulator and equate the effectiveness of this procedure with the rate at which the body dissipates heat when immersed in an ice bath. In the limit, as the thickness and conductivity of the insulator approach zero, the dissipation rate approaches the first eigenvalue of a Robin problem with a coefficient determined by the shape of the insulator. Fixing the mean of the shape function, we search for the shape with the least associated Robin eigenvalue. We offer exact solutions for balls; for general domains, we establish existence and necessary conditions and report on the results of a numerical method.     

A nonlinear eigenvalue problem governing the eversion of elastic spherical caps

Summary This paper deals with a nonlinear eigenvalue problem of variational type which describes the everted equilibrium shapes of a thin elastic spherical cap. The everted solutions of the equilibrium problem do exist under certain conditions involving the relative magnitudes of the geometrical parameters characterizing the cap. In the mathematical formulation the everted solution and the significant parameters respectively correspond to the nontrivial zeros and the eigenvalues of a nonlinear completely continuous operator. The present paper characterizes the spectrum as well as the correspondent path of solutions.     

The stability and small oscillations of an unduloidal fluid bridge between two coaxial discs under conditions of weightlessness

A fluid bridge between two identical coaxial discs is considered which, in equilibrium, has the form of a convex unduloid (that is, a wave-like surface). It is shown that the stability of the equilibrium and the existence of small oscillations of the fluid depend on the coercivity of the bilinear form associated with the operator arising in the problem which is determined by the potential of the surface tension forces. The problem reduces to an operator equation in which one of the operators is associated, by virtue of Laplace's law, with the mean curvature of the perturbed free surface. The problem of coercivity reduces to an auxiliary eigenvalue problem. The conditions of stability are found to be satisfied if all of the eigenvalues of the problem are strictly greater than unity. Sufficient conditions for stability are obtained using arguments based on the theory of elliptic functions. The existence of natural frequencies is proved using functional analysis methods.     

Asymptotic behavior for the principal eigenvalue of a reinforcement problem

In this paper, we consider the asymptotic behavior for the principal eigenvalue of an elliptic operator with piecewise constant coefficients. This problem was first studied by Friedman in 1980. We show how the geometric shape of the interface affects the asymptotic behavior for the principal eigenvalue. This is a refinement of the result by Friedman.     

Construction of Exact Solutions for Equilibrium Configurations of the Boundary of a Conducting Liquid Deformed By an External Electric Field

In a two-dimensional plane-symmetric formulation, we consider the problem of the equilibrium configurations of the free surface of a conducting capillary liquid placed in an external electric field. We find a one-parameter family of exact solutions of the problem according to which the fluid takes the shape of a blade. Such a configuration provides formally unlimited local field amplification: the field strength is maximum at the edge of the blade and drops to zero at the periphery. For a given potential difference between the liquid and the flat electrode located above it, we find threshold values of the electric field strength at the edge of the liquid blade, the radius of curvature of the edge, and the distance from the edge to the electrode limiting the region of existence of the solutions.     

Universality of the double scaling limit in random matrix models

We study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support. We establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit. The limiting kernels are constructed out of functions associated with the second Painlevé equation. This extends a result of Bleher and Its for the special case of a critical quartic potential. The two main tools we use are equilibrium measures and Riemann‐Hilbert problems. In our treatment of equilibrium measures we allow a negative density near the critical point, which enables us to treat all cases simultaneously. The asymptotic analysis of the Riemann‐Hilbert problem is done with the Deift‐Zhou steepest‐descent analysis. For the construction of a local parametrix at the critical point we introduce a modification of the approach of Baik, Deift, and Johansson so that we are able to satisfy the required jump properties exactly. © 2005 Wiley Periodicals, Inc.     

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 spectral projection method for transmission eigenvalues

We consider a nonlinear integral eigenvalue problem, which is a reformulation of the transmission eigenvalue problem arising in the inverse scattering theory. The boundary element method is employed for discretization, which leads to a generalized matrix eigenvalue problem. We propose a novel method based on the spectral projection. The method probes a given region on the complex plane using contour integrals and decides whether the region contains eigenvalue(s) or not. It is particularly suitable to test whether zero is an eigenvalue of the generalized eigenvalue problem, which in turn implies that the associated wavenumber is a transmission eigenvalue. Effectiveness and efficiency of the new method are demonstrated by numerical examples.