Last modified: 2022-07-09
Abstract
Abstract:
Groebner bases were introduced, with an algorithm to determine them in 1965 by Bruno Buchberger, an Austrian mathematician. With the increasing capability of symbolic computation in recent decades, considerable progress has been made in the area of advanced computational algebraic geometry. Buchberger’s algorithm for the determination of Groebner bases has been implemented in many mathematical symbolic computational systems, such as the most popular commercial systems: Mathematica and Maple, and because of its availability, the use of the Groebner basis methodology has now become a feasible option for many scientific and engineering applications.
This paper exemplifies the utility of the methodology of Groebner basis in the analysis of geometric nonlinear beam-reinforced thin plates (BRP) for modeling rectangular duct systems under internal pressure. The focus of this study is to illustrate how the methodology can be developed to solve non-linear engineering problems which are usually solved by numerical approaches. In the proposed process for the nonlinear analysis, the governing integro-partial differential equation is derived by applying the principle of minimum potential energy using the exact solution of the BRP plate from linear analysis as a shape function. Appling the variational principle combined with the Ritz method, the governing equation can be transformed into a set of coupled non-linear algebraic equations. With the use of Groebner basis package in Maple, the analytical expressions for the lateral displacements of the BRP plate under internal pressure can be obtained in a fully symbolic form with all the parameters, such as geometric and material properties of the beams and panels. Thus, the displacement expression in terms of pressure, beam stiffness, and panel dimensions can be used as a convenient tool for preliminary design predictions for engineering practice, while it is not always easy to observe the relationships between the parameters using numerical methods. The analytical solutions have been compared with the results using the finite element software, ANSYS. The comparisons’ study indicates that for commonly used duct panels with the aspect ratio less than 1/3, the analytical solutions for displacements have been found to be in close agreement. Finally, the study is found to be a unique alternative, worthy of further investigation, and potentially effective in the analysis of similar problems occurring in a variety of engineering applications.
Keywords: Groebner bases, geometrically nonlinear analysis, beam-reinforce plates, rectangular duct system, computational algebraic geometry