Stable and Unstable Axisymmetric Solutions for Membranes of Revolution

1989 ◽  
Vol 42 (11S) ◽  
pp. S289-S294 ◽  
Author(s):  
Hubertus J. Weinitschke
Keyword(s):  
Numerical Methods ◽  
Boundary Value Problems ◽  
Boundary Data ◽  
Boundary Value ◽  
Elastic Membrane ◽  
Membrane Theory ◽  
Principal Stresses ◽  
Radial Displacements ◽  
Axisymmetric Solutions ◽  
Nonlinear Elastic

Axisymmetric boundary value problems in nonlinear elastic membrane theory are studied, with prescribed tensile stresses or radial displacements at the edge(s). The membrane of revolution is subjected to a load parallel to the axis or normal to the deformed surface. Analytical and numerical methods are presented to determine the range of boundary data for which the solutions are tensile and stable in the sense that the principal stresses are nonnegative everywhere in the membrane.

Numerical methods for the solution of special and general sixth-order boundary-value problems, with applications to Bénard layer eigenvalue problems

1990 ◽  
Vol 431 (1883) ◽  
pp. 433-450 ◽  
Keyword(s):  
Numerical Methods ◽  
Boundary Value Problems ◽  
Eigenvalue Problems ◽  
Boundary Value ◽  
Second Order ◽  
Sixth Order ◽  
Asymptotic Estimates ◽  
Eighth Order ◽  
The Family ◽  
Convergent Method

A family of numerical methods is developed for the solution of special nonlinear sixth-order boundary-value problems. Methods with second-, fourth-, sixth- and eighth-order convergence are contained in the family. The problem is also solved by writing the sixth-order differential equation as a system of three second-order differential equations. A family of second- and fourth-order convergent methods is then used to obtain the solution. A second-order convergent method is discussed for the numerical solution of general nonlinear sixth-order boundary-value problems. This method, with modifications where necessary, is applied to the sixth-order eigenvalue problems associated with the onset of instability in a Bénard layer. Numerical results are compared with asymptotic estimates appearing in the literature.

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