Discrete gradient method in solid mechanics

The discrete gradient method is proposed as a novel numerical tool to perform solid mechanics analysis directly on point-cloud models without converting the models into a finite element mesh. This method does not introduce continuous approximation of the primary unknown field variables; instead, it computes the gradients of the field variables at a node using discrete differentials involving a set of neighboring nodes. The discrete gradients are substituted into Galerkin weak from to derive the algebraic governing equations for further analysis. Therefore, the formulation renders a completely discrete computation that can conduct mechanical analysis on point-cloud representations of patient-specific organs without resorting to finite element method.

Since the method is prone to rank-deficient instability, a stabilized scheme is developed by employing penalty that involves a minor modification to the method. The difference between nodal strain and subcell strain is penalized to prevent the appearance of zero average strain.

This dissertation delineates the theoretical underpins of the method and provides a detailed description of its implementation in two and three-dimensional elasticity problem. Several benchmark numerical tests are presented to demonstrate the accuracy, convergence, and capability of dealing with compressibility and incompressibility constraint without severe locking. An efficient method is also developed to automatically extract point-cloud models from medical images. Two and three-dimensional examples of biomedical applications are presented too.