Accurate Real-Time Framework for Complex Pre-defined Cuts in Finite Element Modeling

Abstract

Achieving detailed pre-defined cuts on deformable materials is vitally pivotal for many commercial applications, such as cutting scenes in games and vandalism effects in virtual movies. In these types of applications, the majority of resources are allocated to achieve high-fidelity representations of materials and the virtual environments. In the case of limited computing resources, it is challenging to achieve a convincing cutting effect. On the premise of sacrificing realism effects or computational cost, a considerable amount of research work has been carried out, but the best solution that can be compatible with both cases has not yet been identified. This doctoral dissertation is dedicated to developing a unique framework for representing pre-defined cuts of deformable surface models, which can achieve real-time, detailed cutting while maintaining the realistic physical behaviours. In order to achieve this goal, we have made in-depth explorations from geometric and numerical perspectives. From a geometric perspective, we propose a robust subdivision mechanism that allows users to make arbitrary predetermined cuts on elastic surface models based on the finite element method (FEM). Specifically, after the user separates the elements in an arbitrary manner (i.e., linear or non-linear) on the topological mesh, we then optimise the resulting mesh by regenerating the triangulation within the element based on the Delaunay triangulation principle. The optimisation of regenerated triangles, as a process of refining the ill-shaped elements that have small Aspect Ratio, greatly improves the realism of physical behaviours and guarantees that the refinement process is balanced with real-time requirements. The above subdivision mechanism can improve the visual effect of cutting, but it neglects the fact that elements cannot be perfectly cut through any pre-defined trajectories. The number of ill-shaped elements generated yield a significant impact on the optimisation time: a large number of ill-shaped elements will render the cutting slow or even collapse, and vice versa. Our idea is based on the core observation that the producing of ill-shaped elements is largely attributed to the condition number of the global stiffness matrix. Practically, for a stiffness matrix, a large condition number means that it is almost singular, and the calculation of its inverse or the solution of a system of linear equations are prone to large numerical errors and time-consuming. It motivates us to alleviate the impact of condition number of the global stiffness matrix from the numerical aspects. Specifically, we address this issue in a novel manner by converting the global stiffness matrix into the form of a covariance matrix, in which the number of conditions of the matrix can be reduced by exploiting appropriate matrix normalisation to the eigenvalues. Furthermore, we investigated the efficiency of two different scenarios: an exact square-root normalisation and its approximation based on the Newton-Schulz iteration. Experimental tests of the proposed framework demonstrate that it can successfully reproduce competitive visuals of detailed pre-defined cuts compared with the state-of-the-art method (Manteaux et al. 2015) while obtaining a significant improvement on the FPS, increasing up to 46.49 FPS and 21.93 FPS during and after the cuts, respectively. Also, the new refinement method can stably maintain the average Aspect Ratio of the model mesh after the cuts at less than 3 and the average Area Ratio around 3%. Besides, the proposed two matrix normalisation strategies, including ES-CGM and AS-CGM, have shown the superiority of time efficiency compared with the baseline method (Xin et al. 2018). Specifically, the ES-CGM and AS-CGM methods obtained 5 FPS and 10 FPS higher than the baseline method, respectively. These experimental results strongly support our conclusion which is that this new framework would provide significant benefits when implemented for achieving detailed pre-defined cuts at a real-time rate.