Harmonic study and reduction of 3D finite element structures subject to geometric nonlinearities

Simulation of structural dynamics involving global nonlinearities, such as large amplitude motion, is generally dealt with over-simplified models, like beams or plates, which need strong and relatively restrictive hypotheses to be truth worthy. The main reason is that the simulation cost may become really problematic if the number of degrees of freedom is large as all the unknowns of the problem are directly impacted by those nonlinear effects. The use of a 3D analysis allows those limiting assumptions to be overridden but results in a very large model. The work presented in this paper deals with this matter and is devoted to the study and reduction of 3D finite element models subject to geometric nonlinearities. As a working example, we propose to study a simple beam structure meshed with 3D elements that can undergo large displacements. The methodology can however be applied to more complex 3D meshed structures. The forced periodic solution is found using the Harmonic Balance Method (HBM), which we compare against an Euler-Bernoulli beam model with von Karman assumptions. A reduction method adapted for large displacements nonlinearity is also applied on the 3D model. The methodology relies on creating an extended reduction basis for a Galerkin procedure. As the projection on only linear modes fails to capture nonlinear effects, we extend the linear basis with modal derivatives [1]. These modal derivatives are calculated by differentiating two eigenvalue problems, one on the normal structure, and one on the structure deformed along particular eigenvectors. The reduction basis used is therefore formed with classical linear modes and modal derivatives. The projection of the nonlinear forces on this basis may however still be cumbersome as nonlinear effects occur on every degree of freedom. A way around this matter is to evaluate the nonlinear reduced stiffnesses using the STEP (STiffness Evaluation Procedure) method [2]. These stiffnesses are calculated by a combination of appropriate reduction basis vectors. Simulations are computed on a clamped-clamped beam subject to large displacements around its first bending mode for a 1D (Euler-Bernoulli theory) and a 3D models, with or without reduction. They are validated by comparison with results given in the literature [3]. The reduction basis used is composed with the first linear bending mode and its associated modal derivative. Such simple basis gives accurate results on both transverse and longitudinal displacements. Results from the 3D model are consistent with the Euler-Bernoulli formulation, exhibiting only small differences due to simplifying assumptions on the 1D model. The resolution time shows the great capacities of the methodology to deal with the main issue of 3D model by reducing drastically its computational cost. These promising results validate the methodology for 3D nonlinear elements. Further work will focus on the main difficulty in the use of modal derivatives which is to find robust criteria to choose which modal derivative must be included in the enhanced reduction basis. We will also aim at using more complicated geometries going towards industrial applications such as the simulation of turbofan blades undergoing large displacement. [1] P.M. Slaat, J. de Jongh, A.A.H.J. Sauren. Computers and Structures, 54(6), 1155-1171, 1995. [2] A.A. Muravyov, S.A. Rizzi. Computers and Structures, 81, 1513-1523, 2003. [3] A. Grolet, F. Thouverez. CanCNSM, Montreal, 2013