Meshfree Modelling of Hyperelastic Mechanics
Large deformation mechanics faces numerous challenges when simulated using mesh-based methods, notably the finite element method. Firstly, the computational modeler is required to fit a mesh to the body of interest, which is a tedious and time-consuming process. Secondly, these meshes are typically simplex shapes and therefore can resist deformation artificially as their deformation increases, and numerical instability can occur as these simplexes/polygons invert for deformation beyond their resolution. Thirdly, these meshes can be resolved further into smaller pieces, a process called refinement, but the process of robust automatic refinement is still another challenge face by modelers. Therefore, in this thesis, we are motivated to use a meshfree method to simulate large deformations of hyperelastic mechanics without the dependency of a body mesh. The thesis then explores some of the challenges faced by meshfree methods currently and further developing and maturing its strengths of solving PDEs with only random nodal distributions. We found that these meshfree methods often depends on a background mesh for quadrature process, hence currently many meshfree methods are still not as “mesh-independent” as we wish them to be. Therefore we attempted to develop a truly meshfree method is in this thesis, where the quadrature is performed in overlapping support domains of a set of functions that form partition of unity of the body. We manage to improve on the Partition of Unity Quadrature scheme, where polynomial consistency can be restored for the first time in overlapping quadrature schemes, while still conserving energy according to the work conjugacy theory. The methods fall short that the quadrature accuracy must be increased with refinement of particle distributions, a common observation among truly meshfree methods. Another issue that meshfree methods faces is the ability to enforce incompressibility in solid mechanics. Hyperelasticity does not only undergo large deformations, they also exhibit nearly- to-truly incompressible behaviour. Such incompressibility constraints may be enforced using mixed formulations, but this requires the user to define the distribution of particles representing primal and dual variables in a certain fashion, such that a stability criterion known as the inf-sup condition is satisfied. This criterion is, however, normally satisfied through the use of a mesh. A more natural approach would have been to set the distributions to be the same set for meshfree method, such that we may avoid the need of tessellation to satisfy the inf-sup condition. Therefore in this thesis, we further propose a stabilised mixed meshfree formulation based on a stabilisation technique called the Polynomial Pressure Projection (PPP) to penalise the violation of the inf-sup condition caused from using the same distributions of primal and dual variables. Similar to the previous investigation, we continue to improve on the quadrature of our stabilised mixed formulation such that the so-called integration constraint may be satisfied, by means of correcting the classical gradient field. Some of the associated properties of the quadrature scheme and the corrected gradients are given in the appendices, where it theoretically explained on why the use of a background mesh is convergent and the resulting bilinear form using corrected gradient remains stable. The stabilised mixed formulation is tested to be effective in linear elasticity mechanics and non- convective fluids, motivating us to further extend it to large deformation mechanics of hyperelasticity, albeit using a more strengthen form of the PPP. We also unified the gradient correction/quadrature scheme with the stabilised mixed formulation using the Veubeke-Hu- Washizu (VHW) approach. We ran some simple test cases and found that the stabilised mixed formulation continues to converge optimally and being locking free. A simple comparison between stabilised meshfree and finite element methods, however, highlighted that meshfree methods consumes more computational power and yield solution with larger errors, an observation that is consistent with the current literature, mainly due to meshfree basis functions are more costly to be computed, stored and integrated in weak form. Next, we recast the formulation in Eulerian setting to enable us to compute undeformed geometry when given a deformed geometry. We continue to use the enhanced/strengthen PPP stabilisation method in the VHW framework, so that the gradient correction scheme is encompassed naturally. We also showed that our Eulerian formulation can recover the standard Cauchy’s law of equilibrium, with some mesh-dependent conditions that diminish with refinement. When we tested against simple test cases, we observed that the formulation also yields solutions with optimal convergence, besides being free from spurious pressure modes. Besides that, when we re-paramerise the undeformed geometry as unknown variables in the stabilised Lagrangian mixed formulation, we also can demonstrate that the unloaded configuration can be computed, using an example of inflated and extended cylinder. Lastly, a summary of our endeavour in solving some challenges in meshfree method is given in the last chapter.
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