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No / We present an explicit polynomial solution method for surface generation. In this case the surface in question is characterized by some boundary configuration whereby the resulting surface conforms to a fourth order linear elliptic Partial Differential Equation, the Euler–Lagrange equation of a quadratic functional defined by a norm. In particular, the paper deals with surfaces generated as explicit Bézier polynomial solutions for the chosen Partial Differential Equation. To present the explicit solution methodologies adopted here we divide the Partial Differential Equations into two groups namely the orthogonal and the non-orthogonal cases. In order to demonstrate our methodology we discuss a series of examples which utilize the explicit solutions to generate smooth surfaces that interpolate a given boundary configuration. We compare the speed of our explicit solution scheme with the solution arising from directly solving the associated linear system.

Membrane structures are commonly used in many elds. The studies of thesestructures are of increasing interest. The two projects focus on the evaluations ofequilibrium states for uid-pressurized membranes under dierent loading conditions,and the corresponding instability behavior.The rst part of the current work discusses the instability behavior of a thin,planar, circular and initially horizontal membrane subjected to downwards or upwards uid pressure. The membrane structures exhibit large deformations under uid pressure. Various instability behaviors have been observed for dierent loadingparameters. Limit and bifurcation points have been detected for dierent loadingconditions. Dierent loading parameters have been used to interpret the instabilitybehavior. The eects on instability of parameters, the initial states of the membrane,and the chosen mesh have been discussed.The second part of the current work discusses instability behavior of a thin,spherical and closed membrane containing gas and uid placed on a horizontal rigidand non-friction plane. A multi-parametric loading has been described. By addingthe practically relevant controlling equations, the complex equilibrium paths werefollowed using the generalized path following algorithm, and the stability conclusionswere made dierently, according to the considered load parameters and theconstraints. A generalized eigenvalue analysis was used to evaluate the stabilitybehavior including the constraint eects. Fold line evaluations were performed toanalyze the parametric dependence of the instability behavior. A solution surfaceapproach was used to visualize the mechanical response under this multi-parametricsetting. / <p>QC 20151029</p>