scholarly journals Bifurcations of the von Kármán equations with Robin boundary conditions

1999 ◽  
Vol 38 (3-4) ◽  
pp. 85-112 ◽  
Author(s):  
C.S. Chien ◽  
Y.J. Kuo ◽  
Z. Mei
Keyword(s):  
Boundary Conditions ◽  
Robin Boundary Conditions ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Von Kármán ◽  
Robin Boundary

Solubility of the von Karman equations with nonhomogeneous boundary conditions in nonsmooth domains

1990 ◽  
Vol 50 (2) ◽  
pp. 1491-1497
Author(s):  
O. John ◽  
V. A. Kondratiev ◽  
D. M. Lekveishvili ◽  
J. Necas ◽  
O. A. Oleinik
Keyword(s):  
Boundary Conditions ◽  
Nonhomogeneous Boundary ◽  
Von Karman ◽  
Nonsmooth Domains ◽  
Von Kármán Equations ◽  
Nonhomogeneous Boundary Conditions ◽  
Von Kármán

On the swirling flow between rotating coaxial disks, asymptotic behaviour, II

1981 ◽  
Vol 90 (3-4) ◽  
pp. 317-346 ◽  
Author(s):  
Heinz Otto Kreiss ◽  
Seymour V. Parter
Keyword(s):  
Boundary Conditions ◽  
Asymptotic Behaviour ◽  
Swirling Flow ◽  
Simple Zeros ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Image Position ◽  
Von Kármán

SynopsisConsider solutions 〈H(x, ε), G(x, ε)〉 of the von Kármán equations for the swirling flow between two rotating coaxial disksandWe assume that |H(x, ε)| + |Hʹ(x, ε)| + |G(x, ε)|≦B. This work considers shapes and asymptotic behaviour as ε→0+. We consider the type of limit functions 〈H(x), G(x)〉 that are permissible. In particular, if 〈H(x, ε), G(x, ε)〉 also satisfy the boundary conditions H(0, ε)=H(1, ε)=0, Hʹ(0, ε)=Hʹ(1, ε)=0 then H(x) has no simple zeros. That is, there does not exist a point Z ε [0, 1] such that H(x)=0, Hʹ(z)≠0. Moreover, the case of “cells” which oscillate is studied in detail.

scholarly journals A Review on Analytical Modeling for Collapse Mode Capacitive Micromachined Ultrasonic Transducer of the Collapse Voltage and the Static Membrane Deflections

Micromachines ◽  
2021 ◽  
Vol 12 (6) ◽  
pp. 714
Author(s):  
Jiujiang Wang ◽  
Xin Liu ◽  
Yuanyu Yu ◽  
Yao Li ◽  
Ching-Hsiang Cheng ◽  
...  
Keyword(s):  
Analytical Modeling ◽  
Ultrasonic Transducer ◽  
Governing Equations ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Collapse Mode ◽  
Capacitive Micromachined Ultrasonic Transducer ◽  
Gap Height ◽  
Von Kármán ◽  
Analytical Expressions

Analytical modeling of capacitive micromachined ultrasonic transducer (CMUT) is one of the commonly used modeling methods and has the advantages of intuitive understanding of the physics of CMUTs and convergent when modeling of collapse mode CMUT. This review article summarizes analytical modeling of the collapse voltage and shows that the collapse voltage of a CMUT correlates with the effective gap height and the electrode area. There are analytical expressions for the collapse voltage. Modeling of the membrane deflections are characterized by governing equations from Timoshenko, von Kármán equations and the 2D plate equation, and solved by various methods such as Galerkin’s method and perturbation method. Analytical expressions from Timoshenko’s equation can be used for small deflections, while analytical expression from von Kármán equations can be used for both small and large deflections.

scholarly journals An efficient adaptive grid method for a system of singularly perturbed convection-diffusion problems with Robin boundary conditions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Li-Bin Liu ◽  
Ying Liang ◽  
Xiaobing Bao ◽  
Honglin Fang
Keyword(s):  
Boundary Conditions ◽  
Grid Method ◽  
Posteriori Error ◽  
Adaptive Grid ◽  
Singularly Perturbed ◽  
Robin Boundary Conditions ◽  
Euler Formula ◽  
Convection Diffusion ◽  
Backward Euler ◽  
Robin Boundary

AbstractA system of singularly perturbed convection-diffusion equations with Robin boundary conditions is considered on the interval $[0,1]$ [ 0 , 1 ] . It is shown that any solution of such a problem can be expressed to a system of first-order singularly perturbed initial value problem, which is discretized by the backward Euler formula on an arbitrary nonuniform mesh. An a posteriori error estimation in maximum norm is derived to design an adaptive grid generation algorithm. Besides, in order to establish the initial values of the original problems, we construct a nonlinear optimization problem, which is solved by the Nelder–Mead simplex method. Numerical results are given to demonstrate the performance of the presented method.

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