We investigate bifurcation scenario of the von Karman equations with partially clamped boundary conditions defined in rectangular domains. First, we study how the (preconditioned) Block GMRES method can be used in the context of continuation methods for tracing solution curves of large systems of nonlinear equations. Next, we discuss linear stabilities of the von Karman equations with partially clamped boundary conditions. In particular, we calculate the first seven eigenvalues and its associated eigenfunctions of the linearized von Karman equations via computer algebra. The Block GMRES method is used to solve linear systems and to detect singularity along solution paths of the discrete problem. Sample numerical results are reported. (C) 2001 Elsevier Science B.V. All rights reserved
This study presents a new diffusion model named the gradient-dependent model. It is based on the strain-dependent model, a modification of Fick’s diffusion model, and is employed to solve penetrant diffusion in a 2D-infinite plate with finite thickness. An immersion test is performed and analyzed for studying moisture diffusing through the surfaces of a polymer matrix composite laminate into the material and the induced in-plane expansion is measured. The hygric expansion is proportional to the moisture concentration, which can be derived from the gradient-dependent model involving parameters standing for the hygric property of the material. By fitting the theoretical solution of the hygric expansion to the data, the parameters for locating the penetrant front can be obtained and they are very important in describing the hygric behavior of the material.
We describe a special Gauss–Newton method for tracing solution manifolds with singularities of multiparameter systems. First we choose one of the parameters as the continuation parameter, and fix the others. Then we trace one-dimensional solution curves by using continuation methods. Singularities such as folds, simple and multiple bifurcations on each solution curve can be easily detected. Next, we choose an interval for the second continuation parameter, and trace one-dimensional solution curves for certain values in this interval. This constitutes a two-dimensional solution surface. The procedure can be generalized to trace a k-dimensional solution manifold. Numerical results in 1D, 2D and 3D second-order semilinear elliptic eigenvalue problems given by Lions [1982] are reported.
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