The dual eigenvalue problems of the conformable fractional Sturm–Liouville problems

In this paper, we are concerned with the eigenvalue gap and eigenvalue ratio of the Dirichlet conformable fractional Sturm–Liouville problems. We show that this kind of differential equation satisfies the Sturm–Liouville property by the Prüfer substitution. That is, the nth eigenfunction has $n-1$ n − 1 zero in $( 0,\pi ) $ ( 0 , π ) for $n\in \mathbb{N}$ n ∈ N . Then, using the homotopy argument, we find the minimum of the first eigenvalue gap under the class of single-well potential functions and the first eigenvalue ratio under the class of single-barrier density functions. The result of the eigenvalue gap is different from the classical Sturm–Liouville problem.

Inverse Infiltration Modeling of Dike Covers Made of Dredged Material Using PEST and AMALGAM

A variety of studies recently proved the applicability of different dried, fine-grained dredged materials as replacement material for erosion-resistant sea dike covers. In Rostock, Germany, a large-scale field experiment was conducted, in which different dredged materials were tested with regard to installation technology, stability, turf development, infiltration, and erosion resistance. The infiltration experiments to study the development of a seepage line in the dike body showed unexpected measurement results. Due to the high complexity of the problem, standard geo-hydraulic models proved to be unable to analyze these results. Therefore, different methods of inverse infiltration modeling were applied, such as the parameter estimation tool (PEST) and the AMALGAM algorithm. In the paper, the two approaches are compared and discussed. A sensitivity analysis proved the presumption of a non-linear model behavior for the infiltration problem and the Eigenvalue ratio indicates that the dike infiltration is an ill-posed problem. Although this complicates the inverse modeling (e.g., termination in local minima), parameter sets close to an optimum were found with both the PEST and the AMALGAM algorithms. Together with the field measurement data, this information supports the rating of the effective material properties of the applied dredged materials used as dike cover material.

Damage identification in a complex truss structure using modal characteristics correlation method and sensitivity-weighted search space

Damage identification for complex structures is a challenging task due to the large amount of structural elements, limited number of measured modes and uncertainties in referenced numerical models. This article presents a study on enhancing the effectiveness of modal characteristics correlation methods for damage identification of complex structures. First, a correlation method using change in the ratio of modal strain energy to eigenvalue is introduced. Damage information is determined via a forward approach by optimizing the correlation level between the patterns of the analytical and measured changes in the ratio of modal strain energy to eigenvalue. Different from traditional optimization-based forward methods that require accurate numerical models, damage sensitivity coefficients of the ratio of modal strain energy to eigenvalue are directly estimated from the experimental modal information. To enhance the damage identification capability, both the elemental modal strain energy–eigenvalue ratio and the total modal strain energy–eigenvalue ratio components are examined in the correlation function. Second, a sensitivity-weighted search space scheme incorporated with genetic algorithm is developed to overcome the ill-posed problem that causes false detection errors. Finally, the correlation method and the enhanced technique are experimentally tested on a complex truss model with nearly 100 elements. To deal with the huge number of degrees of freedom in this structure, a multi-layout roving test with the adoption of redundant channels is designed, and a three-criterion strategy is used for the selection of modes. Results demonstrate the effectiveness of the proposed damage assessment framework to locate and estimate damage in complex truss structures.

Eigenvalue Ratios of Non-Negatively Curved Graphs

We derive an optimal eigenvalue ratio estimate for finite weighted graphs satisfying the curvature-dimension inequalityCD(0, ∞). This estimate is independent of the size of the graph and provides a general method to obtain higher-order spectral estimates. The operation of taking Cartesian products is shown to be an efficient way for constructing new weighted graphs satisfyingCD(0, ∞). We also discuss a higher-order Cheeger constant-ratio estimate and related topics about expanders.