A finite volume based formulation of geometrically exact Simo-Reissner beams and their contact interaction

The thesis explores the application of the finite volume method to the numerical discretisation of nonlinear Simo-Reissner beam formulations, focusing on modelling flexible slender rods and cables. These structures find utility in diverse fields such as aerospace engineering, marine sectors, textile industries, and biomechanics. Traditional methods for simulating large deformations in such slender structures involve nonlinear beam formulations coupled with finite element-based numerical techniques. This thesis presents a novel approach by adapting the finite volume method to the well-established nonlinear Simo-Reissner beam formulation. Initially developed for fluid dynamics, the finite volume method has evolved to encompass solid mechanics and multi-physics scenarios in recent years. This thesis extends the finite volume method to the geometrically exact beam formulation, with the numerical implementations carried out in open-source software OpenFOAM. The work begins by formulating the governing equations in a strong integral form, upon which a cell-centred version of finite volume spatial discretisation is applied. Various aspects, such as the choice of rotation parametrisation, mathematical formulation of the beam kinematics, conjugate strain measures and the linearisation of the strong integral form of governing equations, are described. The spatial discretisation of the computational beam domain and the equation discretisation for each computational volume are presented. A block-coupled Newton-Raphson procedure is employed to solve the discretised equilibrium equations. The efficacy of the proposed methodology is demonstrated by comparing the simulated numerical results with classic benchmark test cases available in the literature. The objectivity of strain measures for the current formulation and mesh convergence studies for initially straight and curved beam configurations are also discussed. The quasi-static finite volume nonlinear beam solver is extended to include the inertial contributions. An implicit Newmark finite time integration scheme is used to discretise temporal terms. Aspects of energy stability and conservation during the evolution of dynamics are studied for different time step values using benchmark dynamic beam test cases from the literature. Finally, to model multiple interactions between beams, frictionless and frictional beam-to-beam contact algorithms developed in the literature using the finite element methods are adapted towards finite volume implementations of the nonlinear Simo-Reissner beams. A point-to-point frictionless contact algorithm for large contact angles between the beams and a line-to-line frictionless contact interaction for almost parallel beams is implemented using penalty and augmented Lagrangian contact constraint methods. The $C^1$-continuous Hermite spline polynomials are used for smooth contact detection for both algorithms. Details on the contact detection techniques, effects of spatial domain discretisation on contact formulation, and convergence tolerances used for the contact formulation are discussed. This is followed by numerical analysis of series of benchmark contact test cases where the potential of the finite volume beam solver is investigated. Tangential contact forces due to friction between beam surfaces are also considered for point-to-point interaction of beams, where a linear Coulomb friction model and linear penalty constraint are used to compute the slip (sliding) and stick states of frictional motion. The thesis ends with a description of the coding implementations of the finite volume beam solver in the open-source OpenFOAM environment. A broad, high-level perspective of the beam solver, the coding structure, the step-by-step simulation procedure, and meshing considerations are discussed, reinforcing the understanding of finite volume discretisation within the context of beam domains.