A boundary element solution to the soap bubble problem

Abstract

The boundary element method (BEM) is applied to the soap bubble problem, that is to the problem of determining the surface that a soap bubble constrained by bounding contours assumes under the action of molecular forces. This is also the shape of a uniformly stretched membrane bounded by one or more nonintersecting curves. As the slopes of the membrane surface are ®nite, their square can not be neglected and the resulting governing equation is non-linear. The problem is solved using the analogue equation method (AEM). According to this method the non-linear membrane is substituted by a linear one subjected to a ®ctitious transverse load. The ®ctitious load is established using the BEM. Numerical examples are presented which illustrate the method and demonstrate its accuracy. This application of the BEM to non-linear problems shows that BEM is a versatile computational method for allpurpose use in engineering analysis. The solution of the problem at hand is very important in engineering, since the soap bubble surface can be used as the best initial form for membrane roofs.

Large deflection analysis of elastic space membranes

“…Inserting Equations (2a) and (7) into the above equation we have A = e + z, e 3 + u , e + w, e 3 = ( + u , )e + (w, +z, )e 3 (10) and the components of the metric surface tensor of the deformed membrane are obtained as…”

“…The use of Cartesian co-ordinates simplifies the application of the AEM as it yields simple analogue equations that can be effectively treated on the projected domain. The solution is complete, in the sense it includes the establishment of the reference shape (minimal surface) following the procedure described in Reference [10], the deformed shape under combined prestress and self-weight, and the final deformed shape under the in-service three-dimensional loading. In each stage, the three displacement components as well as their derivatives are computed from their integral representations of the substitute problems, which are used as mathematical formulae.…”

A boundary-only meshless method for numerical solution of the Eikonal equation

“…According to developed AEM (MAEM), the nonlinear governing equation is replaced with equivalent nonhomogeneous linear one (analog equation) with known fundamental solution and under the same boundary conditions. The MAEM has been successfully used for a wide variety of PDEs such as: soap bubble problem [31], heat flows in bodies with nonlinear material properties, determination of a surface with constant mean curvature or with constant Gaussian curvature and the problem of minimal surface [30], nonlinear static and dynamic problems in general bodies [32], nonlinear dynamic analysis of heterogeneous orthotropic members [33], 2D elastostatic problem in inhomogeneous anisotropic bodies [34], for some type of elliptic problems [35] and etc.…”

Application of hybrid Trefftz finite element method to non‐linear problems of minimal surface

“…When the slopes are sufficiently small, their squares and products can be neglected and Equation (2) can reduce to the classical Laplace equation [8] …”