In general, three methodologies stand out in the search for solutions to problems in the engineering field: analytical methods, experimental methods and numerical methods. To understand the numerical method, it is necessary to know the other methodologies used. In this article we will present the concept of the three methods and explore the use of numerical methods in solving engineering problems.
Analytical Method
The Analytical Method is used for less complex problems. It is a simple methodology that provides a direct answer by solving, usually manually, equations based on mathematical formulas, in which input variables are defined to find the result of the output variables.
Analytical calculations cannot be applied to all the challenges encountered in engineering, as solving the equations that describe the phenomenon studied can be too complex to calculate, for example, solving second-order, non-linear partial differential equations. To make the equations solvable, the physical characteristics that describe the phenomenon are idealized, making them much simpler than what is observed in practice. As a result, the product obtained with the result of the equations using the analytical method can deviate considerably from the real product, generating problems of over-dimensioning, which increases the cost of production or implementation of a project.
However, the analytical solutions of simple problems can be used as a basis for understanding the behavior of the system of equations, for developing numerical methods and for verifying computer codes.
To illustrate, a collision between two bodies can be classified as elastic when there is conservation of the system's energy and linear momentum, described by the equation shown in Figure 1. However, most collisions found in nature cannot be classified as elastic, and this assumption is a simplification to facilitate application in specific cases.
Figure 1: Law of Conservation of Motion in an elastic shock
- using the analytical method - SOURCE: Kot Collection.
Experimental Methods
Experimental methods use physical prototypes, which can be full or reduced scale, to study the phenomenon. Tests are carried out to simulate the operating conditions of the product. Classic examples of the application of this methodology are the crash test of a car or the aerodynamic analysis of a formula 1 car in a wind tunnel, as shown in Figure 2 [1].
Figure 2: Formula 1 car simulation for the 2021 season using the "wind tunnel" experimental method - SOURCE: BARRETTO (2019).
Experimental methods provide high quality results and allow a detailed understanding of the event being studied, but require a high level of investment. This investment is not only in relation to the cost of building the prototype, but also in relation to the investment needed to obtain the essential infrastructure to accurately simulate the physical characteristics of operation. Another difficulty encountered when using the experimental method is the time needed to carry out the test, since it may be necessary to carry out several simulations or long-term simulations and build different prototypes until an ideal design is obtained, especially in destructive tests. Finally, some real conditions cannot be reproduced, such as the simulation of oil reservoirs [2].
Numerical method
O método numérico encontra soluções numéricas aproximadas dos mais variados problemas complexos encontrados no mundo real, por meio de aplicações de algoritmos, que possibilitam elaborar e calcular operações matemáticas usando sequências de operações aritméticas mais simples. “A análise numérica idealiza e concebe métodos para aprovar de forma eficiente as soluções de problemas expressados matematicamente” [3].
An algorithm is a finite set of ordered operations that makes it possible to solve a given problem. It consists of a series of specific instructions which, through a continuation of steps, make it possible to find an approximation of the result.
A análise numérica tem como objetivo encontrar uma solução aproximada do valor real por meio de sucessões, utilizando o mínimo de operações elementares possíveis. Apesar de ter seu início antes dos computadores, atualmente é associada à tecnologia da informação. A medida que os computadores ficaram mais acessíveis financeiramente e com maior capacidade de processamento, foi popularizado o uso de métodos e técnicas computacionais para solucionar problemas reais, em que as resoluções manuais podem ser impraticáveis, imprecisas, exigirem um alto investimento financeiro ou um alto tempo de execução.
The numerical method is a methodology that, despite requiring a higher investment than the analytical method due to the need for a hardware infrastructure and software licenses, still requires a lower investment than the experimental method. It is an approach that, even using hypotheses that simplify the phenomenon in relation to that found in reality, still makes it possible to obtain a representation similar to that found in the study of a physical prototype in a shorter time.
Uma aplicação prática dos métodos numéricos é a simulação CFD que a partir das leis da termofluidodinâmica busca resolver o problema delimitado com a utilização de recursos computacionais. No exemplo apresentado na Figura 3, com a aplicação da técnica CFD pode-se determinar a distribuição de pressão em toda a superfície externa de um veículo, determinando-se coeficientes aerodinâmicos de arrasto e downforce, com um custo consideravelmente inferior à técnica de ensaio em túnel de vento.
Figure 3: CFD simulation to define the best aerodynamic profile - SOURCE: Kot Collection.
Numerical solution
To solve a problem using numerical solutions, a few steps must be followed, as shown in Figure 4 [3].
Figure 4: Steps involved in the numerical solution of a physical problem - SOURCE: FRANCO (2011).
The mathematical model must be built from the observation of the phenomenon, using the laws of physics and mathematics. It needs to be built in such a way that it correctly represents the real physical characteristics of the problem.
With the mathematical model, the numerical model is obtained using the approximation method. This methodology is based on discretizing the domain and solving the differential equation at specific points.
The necessary steps in the numerical solution can be summarized as follows:
Physical Model > Mathematical Model > Numerical Model > Numerical Solution > Numerical Results.
Care must be taken at every stage of solving the problem to ensure that the numerical solution reflects the physical phenomenon being reproduced. Otherwise, meaningless solutions can be obtained. "The numerical tool is appropriate and reliable when you have a numerical method that correctly solves the differential equations and a mathematical model that faithfully represents the physical phenomenon." [3].
The numerical solution can present two types of errors when the results found are compared with the actual physical problem. Numerical errors are the result of incorrectly solving differential equations. In this case, it is necessary to compare the result with other analytical or numerical solutions. There are also errors resulting from the use of equations that do not correctly describe the phenomenon being studied.
Conclusion
The numerical method has a number of advantages over other methodologies:
- Low cost compared to the experimental method;
- Shorter problem-solving time;
- Easy to simulate realistic conditions.
However, numerical solutions should not be seen as a substitute for other methods. It is the engineer's responsibility to analyze and determine the best method for solving the problem, taking into account the advantages and limitations of each method.
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References:
[1] BARRETTO, Lawrence. First Look: Formula 1’s 2021 car in the wind tunnel. Formula 1, 21 de agosto de 2019. Disponível em: <https://www.formula1.com/en/latest/article.first-look-formula-1s-2021-car-in-the-wind-tunnel.6ye3S7Pb8NRX1K7PjTBxtS.html> Acesso em: 11 de abril de 2020.
[2] FRANCO, Admilson T. Métodos Numéricos Aplicados à Engenharia: Introdução aos métodos de diferença finitas e de volumes finitos. 2011. 9 f. Mechanical Engineering Course, Academic Department of Mechanics, Federal University of Pará, Curitiba, 2011. Chap. 1.
[3] ARAÚJO, Eduardo. Métodos Numéricos para Simulação na Engenharia. Blog ESS, 30 de novembro de 2017. Disponível em <https://www.esss.co/blog/simulacao-numerica-metodo-analitico-experimental-concorrentes-ou-complementares-na-engenharia/.> Acesso em: 20 de setembro de 2020.
