Efficient and Accurate Mixed-Mode Fracture and Shape Sensitivity Analysis Using the Hypercomplex Finite Element Method

The hypercomplex-variable finite element method (ZFEM) is a numerical method that extends the classical finite element method (FEM) to calculate highly accurate arbitrary-order sensitivities of the response variables. ZFEM can compute sensitivities with respect to shape parameters, material properties, and loads. This dissertation addressed previous limitations of ZFEM regarding its ease of implementation, memory usage, and efficiency. First, the hypercomplex algebra library "MultiZ" was developed. MultiZ allows its users to easily convert real-variable numerical analysis code to multicomplex- or multidual-variable for sensitivity analysis. Second, the block forward substitution (BFS) considerably reduced the memory usage and solution time of ZFEM's hypercomplex system of equations compared to the previously used Cauchy-Riemann matrix approach. Furthermore, two accurate and efficient ZFEM-based methods were developed for two-dimensional linear elastic fracture mechanics problems: i) a local ZFEM strategy to compute the strain energy release rate. Local ZFEM is as accurate as the state-of-the-art J-integral method but exhibits superior computational efficiency. ii) a complex interaction integral method that accurately computes the rate of the mixed-mode stress intensity factors.