Normalization Method as a Potent Tool for Grasping Linear and Nonlinear Systems in Physics and Soil Mechanics

Manuel Conesa (), Juan Francisco Sánchez-Pérez, Gonzalo García-Ros, Enrique Castro and Julio Valenzuela
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Manuel Conesa: Department of Applied Physics and Naval Technology, Universidad Politécnica de Cartagena (UPCT), Campus Alfonso XIII, 30202 Cartagena, Spain
Juan Francisco Sánchez-Pérez: Department of Applied Physics and Naval Technology, Universidad Politécnica de Cartagena (UPCT), Campus Alfonso XIII, 30202 Cartagena, Spain
Gonzalo García-Ros: Department of Mining and Civil Engineering, Universidad Politécnica de Cartagena (UPCT), Campus Alfonso XIII, 30202 Cartagena, Spain
Enrique Castro: Department of Applied Physics and Naval Technology, Universidad Politécnica de Cartagena (UPCT), Campus Alfonso XIII, 30202 Cartagena, Spain
Julio Valenzuela: Metallurgical and Mining Engineering Department, Universidad Católica del Norte, Antofagasta 1240000, Chile

Mathematics, 2023, vol. 11, issue 20, 1-19

Abstract: To address physical problems that require solving differential equations, both linear and nonlinear analytical methods are preferred when possible, but numerical methods are utilized when necessary. In this study, the normalization technique is established, which is a simple mathematical approach that requires only basic manipulation of the governing equations to obtain valuable information about the solution. The methodology of this technique involves adopting appropriate references to obtain the dimensionless form of the governing equation, after which the terms of the equation are balanced, obtaining the dimensionless monomials governing the solution. Thorough knowledge of the physical processes involved is necessary to find the best references. The main advantages of this technique are the simplicity of the methodology, the acquisition of valuable information about the solution without the need for complex mathematical calculations, and its applicability to nonlinear problems. However, it is important to consider the difficulty in selecting appropriate references in more complex scenarios. This study applies this normalization methodology to different scenarios, showing how choosing appropriate references lead to the independent dimensionless monomials. Once obtained, it was possible to identify different situations concerning the value of monomials. It will be when they are close to unity, and therefore normalized, when they fundamentally affect the solution of the problem. Finally, we present two cases, one linear and one complex, about the application of normalization to the challenging problem of soil consolidation in ground engineering, illustrating how the technique was used to obtain the solution and its many advantages.

Keywords: linear or nonlinear ODE’s; normalization; education in science and engineering; network simulation method; forced oscillator; nondimensionalization; universal solution; soil constitutive relationships (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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