Solution and sensitivity analysis of nonlinear equations using a hypercomplex-variable Newton-Raphson method

• Published in: Applied mathematics and computation Vol. 451; no. C; p. 127981
• Main Authors: Aristizabal, Mauricio, Hernández-Estrada, José L., Garcia, Manuel, Millwater, Harry
• Format: Journal Article
• Language: English
• Published: United States Elsevier Inc 15.08.2023; Elsevier
• Subjects: Catenary; Elastic cable; Hypercomplex numbers; Newton-Raphson; Nonlinear equations; Elastic cable; Catenary; Nonlinear equations; Hypercomplex numbers; Newton-Raphson
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2023.127981

•The traditional Newton-Raphson method to solve systems of nonlinear equations is enhanced to compute the required Jacobian with machine precision by applying hypercomplex perturbations to the degrees of freedom of the problem.•Arbitrary-order derivatives of the solution to the system of equations with respect to design parameters are computed by applying additional hypercomplex perturbations to the design parameters of interest.•Derivatives of the solution to the non-linear equations are calculated with respect to any parameter within the system of equations.•Multi dual algebra is used to compute the Jacobian and the system derivatives.•A reduced order model is constructed using the high-order derivatives.•The catenary of an elastic cable subject to its own weight and a vertical point load was evaluated with the methodology and derivatives of up to 10th order with respect to material, loading, and geometrical parameters and were used to construct reduced order models. The classical Newton-Raphson (NR) method for solving nonlinear equations is enhanced in two ways through the use of hypercomplex variables and algebra. In particular, i) the Jacobian is computed in a highly accurate and automated way, and ii) the derivative of the solution to the nonlinear equations is computed with respect to any parameter contained within the system of equations. These advances provide two significant enhancements in that it is straightforward to provide an accurate Jacobian and to construct a reduced order model (ROM) of arbitrary order with respect to any parameter of the system. The ROM can then be used to approximate the solution for other parameter values without requiring additional solutions of the nonlinear equations. Several case studies are presented including 1D and 2D academic examples with fully functioning Python code provided. Additionally, a case of study of the catenary of an elastic cable subject to its own weight and a vertical point load. Derivatives up to 10th order were computed with respect to material, loading, and geometrical parameters. The derivatives were used to generate reduced order models of the cable deformation and reaction forces at its ends with respect to multiple input parameters. Results show that from a single hypercomplex evaluation of the cable under a single vertical point load, it is possible to generate an accurate reduced order model capable of predicting the cable deformation with 1.5 times the load in the opposite direction and with 3.5 times the load in the same direction without resolving the system of equations.