An Algorithm for the Automatisation of Pseudo Reductions of PDE Systems Arising from the Uniform-approximation Technique

• Patrick Schneider
• Reinhold Kienzler
Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 15)

Abstract

One way to develop theories for the elastic deformation of two- or one-dimensional structures (like, e.g., shells and beams) under a given load is the uniform-approximation technique (see [2] for an introduction). This technique derives lower-dimensional theories from the general three-dimensional boundary value problem of linear elasticity by the use of series-expansions. It leads to a set of power series in one or two characteristic parameters, which are truncated after a given power, defining the order of the approximating theory. Finally, a so-called pseudo reduction of the resulting PDE system in the unknown displacement coefficients is performed, as the last step of the derivation of a consistent theory. The aim is to find a main differential equation system (at best a single PDE) in a few main variables (at best only one) and a set of reduction differential equations, which express all other unknown variables in terms of the variables of the main differential equation system, so that the original PDE system is identically solved by inserting the reduction equations, if the main variables are a solution of the main differential equation system. To find a valid pseudo reduction by inserting the PDEs of the original system into each other is a complicated and very time-consuming task for higher-order theories. Therefore, an structured algorithm seeking all possibilities of valid pseudo reductions (to a given number of PDEs in a given number of variables) is presented. The key idea is to reduce the problem to finding a solution of a linear equation system, by treating each product of different powers of characteristic parameters with the same variable as formally independent variables. To this end, all necessary equations, which can be build from the original PDE system, have to be identified and added to the system a-priori.

keywords

Uniform-approximation technique. Pseudo reduction. Partial differential equations. Higher-order theory. Linear elasticity.

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