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An upwind cell centred Finite Volume Method for large strain explicit solid dynamics in OpenFOAM
Abstract and Figures
In practical engineering applications involving extremely complex geometries, meshing typically constitutes a large portion of the overall design and analysis time. In the computational mechanics community, the ability to perform calculations on tetrahedral meshes has become increasingly important. For these reasons, automated tetrahedral mesh generation by means of Delaunay and advancing front techniques has recently received increasing attention in a number of applications, namely: crash simulations, cardiovascular modelling, blast and fracture modelling. Unfortunately, modern industry codes in solid mechanics (e.g. LS-DYNA, ANSYS AUTODYN, ABAQUS/Explicit, Altair Hypercrash) typically rely on the use of traditional displacement based Finite Element formulations which possess several distinct disadvantages, namely: (1) reduced order of convergence for strains and stresses in comparison to displacements; (2) high frequency noise in the vicinity of shocks; and (3) numerical instabilities associated with shear locking, volumetric locking and pressure checker-boarding. In order to address the above mentioned shortcomings, a new mixed-based set of equations for solid dynamics formulated in a system of first order hyperbolic conservation laws was introduced. The new set of conservation laws has a similar structure to that of the well known Euler equations in the context of Computational Fluid Dynamics (CFD). This enables us to borrow some of the available CFD technologies and to adapt the method in the context of solid dynamics. This thesis builds on the work carried out in Lee et. al. 2013 by further developing the upwind cell centred finite volume framework for the numerical analysis of large strain explicit solid dynamics and its tailor-made implementation within the open source code OpenFOAM, extensively used in industrial and academic environments. The object oriented nature of OpenFOAM implementation provides a very efficient platform for future development. In this computational framework, the primary unknown variables are linear momentum and deformation gradient tensor of the system. Moreover, the formulation is further extended for an additional set of geometric strain measures comprising of the co-factor of deformation gradient tensor and the Jacobian of deformation, in order to simulate polyconvex constitutive models ensuring material stability. The domain is spatially discretised using a standard Godunov-type cell centred framework where second order accuracy is achieved by employing a linear reconstruction procedure in conjunction with a slope limiter. This leads to discontinuities in variables at the cell interface which motivate the use of a Riemann solver by introducing an upwind bias into the evaluation of numerical contact fluxes. The acoustic Riemann solver presented is further developed by applying preconditioned dissipation to improve its performance in the near incompressibility regime and extending its range to contact applications. Moreover, two evolutionary frameworks are proposed in this study to satisfy the underlying involutions (or compatibility conditions) of the system. Additionally, the spatial discretisation is also represented through a node-based cell centred finite volume framework for comparison purposes. From a temporal discretisation point of view, a two stage Total Variation Diminishing Runge-Kutta time integrator is employed to ensure second order accuracy. Additionally, inclusion of a posteriori global angular momentum projection procedure enables preservation of angular momentum of the system. Finally, benchmark numerical examples are simulated to demonstrate mesh convergence, momentum preservation and the locking-free nature of the formulation. Moreover, the robustness and accuracy of the computational framework has been thoroughly examined through a series of challenging numerical examples involving contact scenarios and complex computational domains.
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