Quaternion and octonion-based finite element analysis methods for computing multiple first order derivatives

The complex Taylor series expansion method for computing accurate first order derivatives is extended in this work to quaternion, octonion and any order Cayley-Dickson algebra. The advantage of this new approach is that highly accurate multiple first order derivatives can be obtained in a single analysis. Quaternion and octonion-based finite element analysis methods were developed in order to compute up to three (quaternion) and up to seven (octonion) first order derivatives of shape, material properties, and/or loading conditions in a single analysis. The traditional finite element formulation was modified such that each degree-of-freedom was augmented with three or seven additional imaginary nodes. The quaternion and octonion-based methods were integrated within the Abaqus commercial finite element code through a user element subroutine. Numerical examples are presented for thermal conductivity and linear elasticity; however, the methodology is general. The results indicate that the quaternion and octonion-based methods provide derivatives of the same high accuracy as the complex finite element method but are significantly more efficient. A Fortran code to solve a simple seven variable quaternion example is given in the Appendix.