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

Aristizabal, Mauricio; Hernández-Estrada, José L.; Garcia, Manuel; Millwater, Harry

doi:10.1016/j.amc.2023.127981

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

Journal Article · 03 April 2023 · Applied Mathematics and Computation

DOI:https://doi.org/10.1016/j.amc.2023.127981· OSTI ID:2417924

^[1]; Hernández-Estrada, José L. ^[2]; ^[3]; Millwater, Harry ^[4]

Univ. of Texas at San Antonio, TX (United States); Universidad EAFIT, Medellín (Colombia); OSTI
Universidad EAFIT, Medellín (Colombia)
Angelo State Univ., San Angelo, TX (United States)
Univ. of Texas at San Antonio, TX (United States)

Here, 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.

View Accepted Manuscript (DOE)

Research Organization:: Univ. of Texas at San Antonio, TX (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA); US Army Research Office (ARO)

Grant/Contract Number:: NA0003948

OSTI ID:: 2417924

Alternate ID(s):: OSTI ID: 1968911

Journal Information:: Applied Mathematics and Computation, Journal Name: Applied Mathematics and Computation Journal Issue: C Vol. 451; ISSN 0096-3003

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

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Related Subjects

97 MATHEMATICS AND COMPUTING
Catenary
Elastic cable
Hypercomplex numbers
Mathematics
Newton-Raphson
Nonlinear equations

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

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