Session: 01-07: Nonlinear Problems in Aerospace Structures

Paper Number: 136286

136286 - Fenicsx-Based Geometrically Nonlinear Analysis of Reissner—mindlin Shells for Composite Structures 

Abstract:

Buckling—specifically nonlinear buckling—is a key design driver for composite wind turbine blades, hence the need for a geometrically nonlinear shell finite element code.

In this talk, we present a geometrically nonlinear shell solver for simulating large deformations in composite structures. This work is an extension of our prior research on linear shells [1], with the source code available at https://github.com/RuruX/shell_analysis_fenicsx.git. The initial software implemented a geometrically-linear, quadrilateral-element variant of the shell formulation by Campello et al. [2]. The original shell solver leverages FEniCSx [3], an open-source tool for solving PDEs, for automated finite element method, specifically through its abstraction of mathematical modeling and automatic code generation. This current extension introduces geometrically nonlinearity into the linear shell solver by incorporating nonlinear formulation of kinematics and strains based on the work by Dornisch et al. [4].

To address shear locking in thin structures with quadrilateral elements, we employ biquadratic Lagrange interpolation for the displacement field and bilinear Lagrange interpolation for the rotation field. Additionally, we introduce a drilling stabilization term to the potential energy to compensate the singularity caused by the lack of torsional stiffness in the third rotation degree of freedom. This term is carefully scaled, especially in models with composite materials, to maintain the accuracy and reliability of the simulation results. A set of benchmark problems will be discussed in this talk to validate our formulation. Furthermore, the application of this solver in practical scenarios is demonstrated through a research blade model, with results for nonlinear static analysis compared with Ansys.

The geometrically nonlinear shell solver can be used to solve buckling problems by formulating a generalized eigenvalue problem where the deformed stiffness matrix K_g (also referred to as the “geometric stiffness”) and the undeformed stiffness matrix K are used to compute the buckling load factor (BLF). Leveraging FEniCSx's automated differentiation capability, this shell solver can also provide analytical derivatives effortlessly. This capability is particularly beneficial for gradient-based structural-sizing optimization of composite wind turbine blades, whose design space can easily climb into hundreds of DoFs.

References:

[1] Herrama, A., Xiang, R., Kamensky, D., and Mullings, J., “Towards FEniCSx as an engine for advanced industrial composite shell design.” FEniCS 2022 conference, 11-14 August 2022, University of California, San Diego.

[2] Campello, E. M. B., Pimenta, P., and Wriggers, P., “A triangular finite shell element based on a fully nonlinear shell formulation.” Computational Mechanics, Vol. 31, 2003, pp. 505-518. https://doi.org/10.1007/s00466-003-0458-8.

[3] Scroggs, M. W., Dokken, J. S., Richardson, C. N., and Wells, G. N., “Construction of Arbitrary Order Finite Element Degree-of-Freedom Maps on Polygonal and Polyhedral Cell Meshes,” ACM Trans. Math. Softw., Vol. 48, No. 2, 2022. https://doi.org/10.1145/3524456, URL https://doi.org/10.1145/3524456.

[4] Dornisch, W., Klinkel, S., and Simeon, B., “Isogeometric Reissner–Mindlin shell analysis with exactly calculated director vectors,” Computer Methods in Applied Mechanics and Engineering, Vol. 253, 2013, pp. 491–504. https://doi.org/10.1016/j.cma.2012.09.010.

Presenting Author: Ru Xiang University of California San Diego

Presenting Author Biography: As a PhD candidate in Mechanical Engineering at UC San Diego, I specialize in high-fidelity structural analysis and gradient-based optimization within Prof. John Hwang's Large-Scale Design and Optimization (LSDO) Lab. My research focuses on enhancing the design and performance of electric vertical take-off and landing (eVTOL) aircraft and wind turbine blades. Through advanced computational techniques, I aim to contribute significantly to sustainable transportation and renewable energy technologies.