Adaptive spatial resolution of the optimal transportation meshfree method

Particle-based numerical methods are becoming increasingly popular for the solution of continuum mechanics problems involving large topological changes. These methods do not depend on the sensitive and time-intensive step of mesh generation, unlike more conventional, grid-based numerical techniques. However, particle methods do face several unique challenges which prevent widespread industry adoption, one of which is simulation refinement. Localized refinement of a simulation is required to make many simulations computationally feasible, but such techniques have yet to be adopted in practice. This study focuses on developing an adaptive spatial resolution algorithm for the optimal transportation meshfree method. The study first attempts to automatically determine where simulations most need refinement. Inspiration is drawn from the field of design of experiments, and a stand-alone, gradient-based adaptive sampling method is proposed for design of experiments applications. The new method balances space filling, local refinement, and error minimization objectives while reducing reliance on delicate tuning parameters. Higher-order local maximum entropy approximants are used for metamodelling to confer the approach with intrinsic resistance to data noise and make it more suitable to situations with unreplicated data points. Tests find it performs favourably compared to conventional design of experiments approaches, and investigate the effects of a time-varying dataset on its performance in anticipation of application to particle-based methods. The remainder of the adaptive spatial resolution algorithm is then developed. The novel adaptive design of experiments approach is used to automatically determine the locations in greatest need of refinement and additional nodes are added in the vicinity. Refining the optimal transportation meshfree method requires a novel approach to material point splitting and mass redistribution, which is accomplished through a combination of kernel density estimation and kernel functions. The negative effect of disorder on the optimal transportation meshfree method is also investigated, and motivates the inclusion of particle shifting into the new algorithm. Finally, simple canonical PDEs are investigated using the complete adaptive spatial resolution method, and the solution accuracy is found to increase by up to an order of magnitude for the same number of nodes when the adaptive spatial resolution algorithm is applied.Applied Science, Faculty ofMechanical Engineering, Department ofGraduat