Spectral solution to a problem on the axisymmetric nonlinear deformation of a cylindrical membrane shell due to pressure and edges convergence

A geometrically and physically nonlinear model of a membrane cylindrical shell, which has been built and tested, describes the behavior of a airbag made of fabric material. Based on the geometrically accurate relations of "strain-displacement", it has been shown that the equilibrium equations of the shell, written in terms of Biot stresses, together with boundary conditions acquire a natural physical meaning and are the consequences of the principle of virtual work. The physical properties of the shell were described by Fung’s hyper-elastic biological material because its behavior is similar to that of textiles. For comparison, simpler hyper-elastic non-compressible Varga and Neo-Hookean materials, the zero-, first-, and second-order materials were also considered. The shell was loaded with internal pressure and convergence of edges. The approximate solution was constructed by an spectral method; the exponential convergence and high accuracy of the equilibrium equations inherent in this method have been demonstrated. Since the error does not exceed 1 % when keeping ten terms in the approximations of displacement functions, the solution can be considered almost accurate. Similar calculations were performed using a finite element method implemented in ANSYS WB in order to verify the results. Differences in determining the displacements have been shown to not exceed 0.2 %, stresses – 4 %. The study result has established that the use of Fung, Varga, Neo-Hookean materials, as well as a zero-order material, lead to similar values of displacements and stresses, from which displacements of shells from the materials of the first and second orders significantly differ. This finding makes it possible, instead of the Fung material whose setting requires a significant amount of experimental data, to use simpler ones – a zero-order material and the Varga material

Novel Expressions for the Derivatives of Sixth Kind Chebyshev Polynomials: Spectral Solution of the Non-Linear One-Dimensional Burgers’ Equation

This paper is concerned with establishing novel expressions that express the derivative of any order of the orthogonal polynomials, namely, Chebyshev polynomials of the sixth kind in terms of Chebyshev polynomials themselves. We will prove that these expressions involve certain terminating hypergeometric functions of the type 4F3(1) that can be reduced in some specific cases. The derived expressions along with the linearization formula of Chebyshev polynomials of the sixth kind serve in obtaining a numerical solution of the non-linear one-dimensional Burgers’ equation based on the application of the spectral tau method. Convergence analysis of the proposed double shifted Chebyshev expansion of the sixth kind is investigated. Numerical results are displayed aiming to show the efficiency and applicability of the proposed algorithm.

The influence of hyporheic fluxes on regional groundwater discharge zones

The importance of hyporheic water fluxes induced by hydromorphologic processes at the streambed scale and their consequential effects on stream ecohydrology have recently received much attention. However, the role of hyporheic water fluxes in regional groundwater discharge is still not entirely understood. Streambed-induced flows not only affect mass and heat transport in streams but are also important for the retention of solute contamination originating from deep in the subsurface, such as naturally occurring solutes as well as leakage from the future geological disposal of nuclear waste. Here, we applied a multiscale modeling approach to investigate the effect of hyporheic fluxes on regional groundwater discharge in the Krycklan catchment, located in a boreal landscape in Sweden. Regional groundwater modeling was conducted using COMSOL Multiphysics constrained by observed or modeled representations of the catchment infiltration and geological properties, reflecting heterogeneities within the subsurface domain. Furthermore, streambed-scale modeling was performed using an exact spectral solution of the hydraulic head applicable to streaming water over a fluctuating streambed topography. By comparing the flow fields of watershed-scale groundwater discharge with and without consideration of streambed-induced hyporheic flows, we found that the flow trajectories and the distribution of the travel times of groundwater were substantially influenced by the presence of hyporheic fluxes near the streambed surface. One implication of hyporheic flows is that the groundwater flow paths contract near the streambed interface, thus fragmenting the coherent areas of groundwater upwelling and resulting in narrow “pinholes” of groundwater discharge points.

Spectra of elliptic potentials and supersymmetric gauge theories

We study a relation between asymptotic spectra of the quantum mechanics problem with a four components elliptic function potential, the Darboux-Treibich-Verdier (DTV) potential, and the Omega background deformed N=2 supersymmetric SU(2) QCD models with four massive flavors in the Nekrasov-Shatashvili limit. The weak coupling spectral solution of the DTV potential is related to the instanton partition function of supersymmetric QCD with surface operator. There are two strong coupling spectral solutions of the DTV potential, they are related to the strong coupling expansions of gauge theory prepotential at the magnetic and dyonic points in the moduli space. A set of duality transformations relate the two strong coupling expansions for spectral solution, and for gauge theory prepotential.