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

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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)

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.

Suggested Citation

  • Manuel Conesa & Juan Francisco Sánchez-Pérez & Gonzalo García-Ros & Enrique Castro & Julio Valenzuela, 2023. "Normalization Method as a Potent Tool for Grasping Linear and Nonlinear Systems in Physics and Soil Mechanics," Mathematics, MDPI, vol. 11(20), pages 1-19, October.
  • Handle: RePEc:gam:jmathe:v:11:y:2023:i:20:p:4321-:d:1261495
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    References listed on IDEAS

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    1. I. Alhama & M. Cánovas & F. Alhama, 2014. "On the Nondimensionalization Process in Complex Problems: Application to Natural Convection in Anisotropic Porous Media," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-13, June.
    2. Juan Francisco Sánchez-Pérez & Gloria Motos-Cascales & Manuel Conesa & Francisco Moral-Moreno & Enrique Castro & Gonzalo García-Ros, 2022. "Design of a Thermal Measurement System with Vandal Protection Used for the Characterization of New Asphalt Pavements through Discriminated Dimensionless Analysis," Mathematics, MDPI, vol. 10(11), pages 1-18, June.
    3. Juan Francisco Sánchez-Pérez & Gonzalo García-Ros & Manuel Conesa & Enrique Castro & Manuel Cánovas, 2023. "Methodology to Obtain Universal Solutions for Systems of Coupled Ordinary Differential Equations: Examples of a Continuous Flow Chemical Reactor and a Coupled Oscillator," Mathematics, MDPI, vol. 11(10), pages 1-28, May.
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