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

Highlights: • Quaternion- and octonion-based finite element methods were developed. • Multiple accurate first order derivatives computed in a single run. • Extends the complex Taylor series expansion method to any order Cayley-Dickson algebra. • The method was implemented through the addition of imaginary nodes/degrees of freedom. • The method was implemented within the commercial finite element code Abaqus. -- Abstract: 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.