This is a survey article about using non-conforming finite elements in solving eigenvalue problems of elliptic operators, with emphasis on obtaining lower bounds. In addition, this article also contains some new materials for eigenvalue approximations of the Laplace operator, which include: 1) the proof of the fact that the non-conforming Crouzeix-Raviart element approximates eigenvalues associated with smooth eigenfunctions from below; 2) the proof of the fact that the non-conforming EQ rot 1 element approximates eigenvalues from below on polygonal domains that can be decomposed into rectangular elements; 3) the explanation of the phenomena that numerical eigenvalues λ 1,h and λ 3,h of the non-conforming Q rot 1 element approximate the true eigenvalues from below for the L-shaped domain. Finally, we list several unsolved problems.
Keywordsnon-conforming element, eigenvalue, lower bound MSC(2000): 65N25, 65N30, 35P15, 65N15
Citation:Yang Y D, Zhang Z M, Lin F B. Eigenvalue approximation from below using non-conforming finite elements.
A lithium
(Li) metal anode is required to achieve a high-energy-density
battery, but because of an undesirable growth of Li dendrites, it
still has safety and cyclability issues. In this study, we have developed
a microsphere-protected (MSP) Li metal anode to suppress the growth
of Li dendrites. Microspheres could guide Li ions to selective areas
and pressurize dendrites during their growth. Interconnections between
microspheres improved the pressurization. By using an MSP Li metal
anode in a 200 mAh pouch-type Li/NCA full cell at 4.2 V, dendrite-free
Li deposits with a density of 0.4 g/cm3, which is 3 times
greater than that in the case of bare Li metal, were obtained after
charging at 2.9 mAh/cm2. The MSP Li metal enhanced the
cyclability to 190 cycles with a criterion of 90% capacity retention
of the initial discharge capacity at a current density of 1.45 mA/cm2.
This paper describes the extension of a wave and finite element (WFE) method to the prediction of noise transmission through, and radiation from, infinite panels. The WFE method starts with a conventional finite element model of a small segment of the panel. For a given frequency, the mass and stiffness matrices of the segment are used to form the structural dynamic stiffness matrix. The acoustic responses of the fluids surrounding the structure are modelled analytically. The dynamic stiffness matrix of the segment is post-processed using periodic structure theory, and coupled with those of the fluids. The total dynamic stiffness matrix is used to obtain the response of the medium to an incident acoustic pressure. Excitation of the structure by oblique plane waves and a diffuse sound field are considered. The response to structural excitation and the consequent radiation are determined. Since the size of the WFE model is small, computational times are small. Various example applications are presented to illustrate the approach, including a thin isotropic panel, an antisymmetric, cross-ply sandwich panel and a symmetric panel with an orthotropic core.
Stable optical properties of high transmittance and low yellow index, which are required for a polyimide film as a flexible display substrate could be affected by thermal imidization even in oxidative-stable fluorinated polyimides.
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