Advanced numerical methods for complex scientific and engineering problems: Editorial introduction

The paper presents the comparative study on numerical methods of Euler method, Improved Euler method and fourth-order Runge-Kutta method for solving the engineering problems and applications. The three proposed methods are quite efficient and practically well suited for solving the unknown engineering problems. This paper aims to enhance the teaching and learning quality of teachers and students for various levels. At each point of the interval, the value of y is calculated and compared with its exact value at that point. The next interesting point is the observation of error from those methods. Error in the value of y is the difference between calculated and exact value. A mathematical equation which relates various functions with its derivatives is known as a differential equation. It is a popular field of mathematics because of its application to real-world problems. To calculate the exact values, the approximate values and the errors, the numerical tool such as MATLAB is appropriate for observing the results. This paper mainly concentrates on identifying the method which provides more accurate results. Then the analytical results and calculates their corresponding error were compared in details. The minimum error directly reflected to realize the best method from different numerical methods. According to the analyses from those three approaches, we observed that only the error is nominal for the fourth-order Runge-Kutta method.

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Convergence Stability of Through-Flow Analyses via the Implementation of the Secant and Bisection Iteration Schemes

Volume 4: Cycle Innovations; Industrial and Cogeneration; Manufacturing Materials and Metallurgy; Marine ◽

10.1115/gt2009-59760 ◽

2009 ◽

Author(s):

Vassilios Pachidis ◽

Ioannis Templalexis ◽

Pericles Pilidis

Keyword(s):

Numerical Methods ◽

Numerical Solution ◽

Numerical Scheme ◽

Linear Equations ◽

Original Form ◽

Through Flow ◽

Engineering Problems ◽

Non Linear ◽

Newton Raphson ◽

Non Linear Equations

One of the most frequently encountered problems in engineering is dealing with non-linear equations. For example, the solution of the full Radial Equilibrium Equation (REE) in Streamline Curvature (SLC) through-flow methods is a typical case of a scientific analysis associated with a complex mathematical problem that can not be handled analytically. Various schemes are used routinely in scientific studies for the numerical solution of mathematical problems. In simple cases, these methods can be applied in their original form with success. The Newton-Raphson for example is one such scheme, commonly employed in simple engineering problems that require an iterative solution. Frequently however, the analysis of more complex phenomena may fall beyond the range of applicability of ‘textbook’ numerical methods, and may demand the design of more dedicated algorithms for the mathematical solution of a specific problem. These algorithms can be empirical in nature, developed from scratch, or the combination of previously established techniques. In terms of robustness and efficiency, all these different schemes would have their own merits and shortcomings. The success or failure of the numerical scheme applied depends also on the limitations imposed by the physical characteristics of the computational platform used, as well as by the nature of the problem itself. The effects of these constraints need to be assessed and taken into account, so that they can be anticipated and controlled. This manuscript discusses the development, validation and deployment of a convergence algorithm for the fast, accurate and robust numerical solution of the non-linear equations of motion for two-dimensional flow fields. The algorithm is based on a hybrid scheme, combining the Secant and Bisection iteration methods. Although it was specifically developed to address the computational challenges presented by SLC-type of analyses, it can also be used in other engineering problems. The algorithm was developed to provide a mid-of-the-range option between the very efficient but notoriously unstable Newton-Raphson scheme and other more robust, but less efficient schemes, usually employing some sort of Dynamic Convergence Control (DCC). It was also developed to eliminate the large user intervention, usually required by standard numerical methods. This new numerical scheme was integrated into a compressor SLC software and was tested rigorously, particularly at compressor operating regimes traditionally exhibiting convergence difficulties (i.e. part-speed performance). The analysis showed that the algorithm could successfully reach a converged solution, equally robustly but much more efficiently compared to a hybrid Newton-Raphson scheme employing DCC. The performance of these two schemes, in terms of speed of execution, is presented here. Typical error histories and comparisons of simulated results against experimental are also presented in this manuscript for a particular case-study.

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Workshop on numerical methods for system engineering problems

Mathematical Programming ◽

10.1007/bf01588335 ◽

1980 ◽

Vol 18 (1) ◽

pp. 358-358

Keyword(s):

Numerical Methods ◽

System Engineering ◽

Engineering Problems

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Present and future of the numerical methods in buildings and infrastructures areas of biosystems engineering

Journal of Agricultural Engineering ◽

10.4081/jae.2015.436 ◽

2015 ◽

Vol 46 (1) ◽

pp. 1 ◽

Cited By ~ 3

Author(s):

Francisco Ayuga

Keyword(s):

Finite Element Method ◽

Numerical Methods ◽

The Finite Element Method ◽

Practical Applications ◽

Development Status ◽

Agricultural Engineering ◽

The Past ◽

Engineering Problems ◽

The Status ◽

Element Method

Biosystem engineering is a discipline resulting from the evolution of the traditional agricultural engineering to include new engineering challenges related with biological systems, from the cell to the environment. Modern buildings and infrastructures are needed to satisfy crop and animal production demands. In this paper a review on the status of numerical methods applied to solve engineering problems in the field of buildings and infrastructures in biosystem engineering is presented. The history and basic background of the finite element method is presented. This is the first numerical method implemented and also the more developed one. The history and background of other two more recent methods, with practical applications, the computer fluids dynamics and the discrete element method are also presented. Besides, a review on the scientific and professional applications on the field of buildings and infrastructures for biosystem engineering needs is presented. Today we can simulate engineering problems with solids, engineering problems with fluids and engineering problems with particles and get to practical solutions faster and cheaper than in the past. The paper encourages young engineers and researchers to make progress these tools and their engineering applications. The capacities of all numerical methods in their present development status go beyond the present practical applications. There is a broad field to work on it.

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A Comparison of One-Way and Two-Way Coupling Methods for Numerical Analysis of Fluid-Structure Interactions

Journal of Applied Mathematics ◽

10.1155/2011/853560 ◽

2011 ◽

Vol 2011 ◽

pp. 1-16 ◽

Cited By ~ 56

Author(s):

Friedrich-Karl Benra ◽

Hans Josef Dohmen ◽

Ji Pei ◽

Sebastian Schuster ◽

Bo Wan

Keyword(s):

Numerical Analysis ◽

Numerical Methods ◽

Computational Effort ◽

Complex Geometries ◽

Fluid Structure Interactions ◽

Solution Strategy ◽

Fluid Structure ◽

Coupling Methods ◽

Wide Range ◽

Engineering Problems

The interaction between fluid and structure occurs in a wide range of engineering problems. The solution for such problems is based on the relations of continuum mechanics and is mostly solved with numerical methods. It is a computational challenge to solve such problems because of the complex geometries, intricate physics of fluids, and complicated fluid-structure interactions. The way in which the interaction between fluid and solid is described gives the largest opportunity for reducing the computational effort. One possibility for reducing the computational effort of fluid-structure simulations is the use of one-way coupled simulations. In this paper, different problems are investigated with one-way and two-way coupled methods. After an explanation of the solution strategy for both models, a closer look at the differences between these methods will be provided, and it will be shown under what conditions a one-way coupling solution gives plausible results.

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