A block forward substitution method for solving the hypercomplex finite element system of equations

The hypercomplex finite element method, ZFEM, allows the analyst to compute highly-accurate arbitrary-order shape, material property, and loading derivatives by augmenting the traditional finite element method with multiple imaginary degrees of freedom. In ZFEM, the real variables are converted to hypercomplex variables such as multicomplex, multidual, or quaternions. By uplifting the real variables to hypercomplex, derivatives are computed in an automated fashion using a standard finite element formulation. The use of multicomplex or multidual numbers provides higher-order derivatives. The drawback of ZFEM is that it increases the number of degrees of freedom of the real variable system by a factor 2n, where n is the order of the required derivative. In consequence, ZFEM increases the memory consumption and the solution time of the system of equations compared to the real variable system. The block forward substitution method (BFS), proposed in this work, addresses the memory and runtime issues. This new method solves the original real-valued FEM system once. Then, the derivatives are computed using pseudo-loads with the original system of equations. In contrast with the conventional solution method of ZFEM, BFS computes the hypercomplex contributions to the stiffness matrix element-wise, and it never assembles nor solves the full hypercomplex system of equations. In effect, the BFS method generalizes the first-order semi-analytical complex variable method to any order derivative. The BFS method provides the capability to allow a combination of realvariable and hypercomplex-variable elements within the same model. The numerical results indicate that a first-order derivative can be obtained with 1% to 8% additional computational time of the real-variable analysis. This allows the computation of multiple first-order derivatives by post-processing of a single FEM analysis. Additionally, it was shown that fourth-order shape sensitivities can be computed in less than 5% additional runtime of the real-variable FEM analysis. Published by Elsevier B.V.