Measuring characteristic length scales of eigenfunctions of Sturm–Liouville equations in one and two dimensions

The spatial resolution of eigenfunctions of Sturm–Liouville equations in one-dimension is frequently measured by examining the minimum distance between their roots. For example, it is well known that the roots of polynomials on finite domains cluster like O(1/N ²) near the boundaries. This technique works well in one dimension, and in higher dimensions that are tensor products of one-dimensional eigenfunctions. However, for non-tensor-product eigenfunctions, finding good interpolation points is much more complicated than finding the roots of eigenfunctions. In fact, in some cases, even quasi-optimal interpolation points are unknown. In this work an alternative measure, ℓ, is proposed for estimating the characteristic length scale of eigenfunctions of Sturm–Liouville equations that does not rely on knowledge of the roots. It is first shown that ℓ is a reasonable measure for evaluating the eigenfunctions since in one dimension it recovers known results. Then results are presented in higher dimensions. It is shown that for tensor products of one-dimensional eigenfunctions in the square the results reduce trivially to the one-dimensional result. For the non-tensor product Proriol polynomials, there are quasi-optimal interpolation points (Fekete points). Comparing the minimum distance between Fekete points to ℓ shows that ℓ is a reasonably good measure of the characteristic length scale in two dimensions as well. The measure is finally applied to the non-tensor product generalized eigenfunctions in the triangle proposed by Taylor MA, Wingate BA [(2006) J Engng Math, accepted] where optimal interpolation points are unknown. While some of the eigenfunctions have larger characteristic length scales than the Proriol polynomials, others show little improvement.     

Computation of Single Eigenfrequencies and Eigenfunctions of Plate and Shell Structures Using an H-adaptive FE-method

The accuracy of the computation of eigenfrequencies and eigenfunctions with FE-methods can be substantially improved with efficient adaptive procedures. For such an adaptive analysis of plate and shell structures a simple a-posteriori error estimator or indicator for the error in the energy norm and L ₂-norm of the eigenfunction u _h for shell and plate structures is proposed.Both indicators hite shall represent the correct convergence of the estimated error. The estimator for the error in the energy norm is used to enlarge adaptively the dimension of the finite element subspace. On the basis of numerical examples the efficiency and the quality of the improved solution is discussed. In order to validate the quantity of the estimated error Aitken’s extrapolation technique is applied.     

Heat propagation in H-bridge smart power chips under switching conditions

A method for a time-dependent solution of the heat conduction equation for semiconductor power devices will be presented. The mathematical approach to the solution is taken with the help of the eigenfunctions and eigenvalues theory, a suitable approach for these types of problems. Examples for the application of this solution for smart power devices under switching conditions will be presented.     

Revealing stable and unstable modes of denoisers through nonlinear eigenvalue analysis

In this paper, we propose to analyze stable and unstable modes of black-box image denoisers through nonlinear eigenvalue analysis. We aim to find input images for which the denoiser output is proportional to the input. We treat this as a generalized nonlinear eigenproblem. Potential implications are wide, as most image processing algorithms can be viewed as black-box operators. We introduce a generalized nonlinear power-method to solve eigenproblems for such operators. This allows us to reveal stable modes of the denoiser: optimal inputs, achieving superior PSNR in noise removal. Analogously to the linear case, such stable modes show coarse structures and correspond to large eigenvalues. We also provide a method to generate unstable modes, which the denoiser suppresses strongly, which are textural with small eigenvalues. We validate the method using total-variation (TV) and demonstrate it on the EPLL (Zoran–Weiss) and the Non-local means denoisers. Finally, we suggest an encryption–decryption application.     

On the dynamic response of rectangular plate, with moving mass

In this article, the dynamics of plates under influence of relatively large masses, moving along an arbitrary trajectory on the plate surface is considered. The method consists of transformation of the governing equation into a series of eigenfunctions, which satisfy the boundary conditions of the plate. The method presented in this investigation is general and can be applied to general moving mass and moving force systems as well. Furthermore, the article shows that the response of structures due to moving mass, which has often been neglected in the past, must be properly taken into account because it often differs significantly from the moving force model.     

A method of the analysis of the thermal field dynamics in a core and insulation of a DC lead with convectional heat abstraction

A method of the analysis of dynamics of the thermal field in a DC lead is presented in this article. The cooling of the system is modeled by the convectional boundary condition. Field functions are determined analytically by the method of states superposition and the separation of variables. The coefficients of field functions and eigenvalues of the boundary–initial problem are computed by the numerical method. The coefficients are the solution of the respective systems of equations. Those systems are the result of scalar products of the functions non-orthogonal in the region of the core and insulation. On the other hand, the eigenvalues are determined with the aid of the original algorithm, which, among others, takes advantage of field properties and the method of golden partition. Consequentially, the spatial–temporal distribution of a temperature is computed, as well as the averaged time constant of the system and the admissible long-lasting ampacity. The influence of the heat transfer coefficient and thickness of insulation is discussed. The obtained results have been verified with the aid of a finite element method.     

Ambrosetti类型半线性波动方程的唯一性定理

研究了满足Dirichlet边界条件及关于变量t满足周期条件的Ambrosetti半线性波动方程.利用巴拿赫空间中的特征理论,证明了解的唯一性定理.     

Structure-aligned guidance estimation in surface parameterization using eigenfunction-based cross field

In this paper, we present a structure-aligned approach for surface parameterization using eigenfunctions from the Laplace–Beltrami operator. Several methods are designed to combine multiple eigenfunctions using isocontours or characteristic values of the eigenfunctions. The combined gradient information of eigenfunctions is then used as a guidance for the cross field construction. Finally, a global parameterization is computed on the surface, with an anisotropy enabled by adapting the cross field to non-uniform parametric line spacings. By combining the gradient information from different eigenfunctions, the generated parametric lines are automatically aligned with the structural features at various scales, and they are insensitive to local detailed features on the surface when low-mode eigenfunctions are used.     

Degradation of a stratified thermocline in a solar storage tank

The thermal stratification behaviour in a solar storage tank is simulated and analysed using a theoretical model based on an integral transform technique. A comparison with available experimental and theoretical data is used to validate the present theoretical results. The accuracy of the model in simulating the thermal behaviour of stratification is reasonably good, especially when consideration is given to the complexity of the physical mechanisms involved, and the relative simplicity of the model. The effect of the heat loss parameter is investigated and it is found that initially it is strongly spatially and temporally dependent. Therefore, a functional form that accurately represents the heat loss parameter is needed for closer agreement with experimental results. However, after a relatively long time, the assumption of a constant heat loss parameter is adequate to produce acceptable predictions.