Parameter-Transfer Finite Element Method for Structural Analysis

A theoretical formulation of a method to solve structural problems is given from a general point of view. The procedure creates a mathematical model of typical elements taking into account the whole behavior of the structure under concern such as static, dynamic, aerodynamic . . . thermal actions. In analogy with other numerical methods like finite elements, a "library" of models, named parameter-transfer finite elements (P-TFE) can be obtained for simple structures. Complex geometries can be approached by a discretization technique assembling the contributions from each element. An application to the panel flutter problem is given in which the proposed method overcomes most difficulties of the usual techniques of solution. To assess the validity of the method, a comparison with results obtained by other authors is made. The present approach of analysis is based on an analytical solution of the differential equations that describe the behav- ior of a single elementary member. The solution is obtained as the series expansion in terms of one parameter to obtain a mathematical model named parameter-transfer finite element (P-TFE). Complex geometries can be modeled by assembling these elements.1'2 A simple mathematical model of a structural problem, which can be solved by standard library routines, can be obtained. In fact, the series expansion solution gives a polynomial and not a transcendental expression of unknowns. From a numerical point of view, the approximate solution, up to the desired term, can be obtained by manipulating a poly- nomial matrix. The present method is very general and can be applied to several physical problems in which the solution involves some parameter. Some cases of interest might be thermal, aeroelas- tic, and stability problems. For each of these cases, it is possible to create a library of parametric finite elements, which takes into account the behavior of the structure in the whole. The proposed approach can be particularly useful in those cases in which the exact transfer function is not known and also in which classical approximate approaches, such as the Galerkin or finite element methods (FEM), are not applicable without strong computational efforts. In the following paragraph the theoretical formulation of the method is given. The method is then validated by an application to the case of monodimensional panel flutter. This problem has been extensively studied by a number of authors using different approaches such as Galerkin's method3'7 or finite elements.8'13 Comparisons with the results obtained with the present method are reported in the numerical examples.