China
Africa
Explore chapters and articles related to this topic
Ⓡ
Published in Yeong Koo Yeo, Chemical Engineering Computation with MATLAB®, 2020
Sometimes we need to represent a function based on knowledge about its behavior at a set of discrete points. Interpolation is a method that produces a function that best matches the given data while also providing a good approximation to the unknown values at intermediate points. Suppose that values of a function f(xi) are known at a set of independent variables xi, as shown in Table 2.12.
Engineering Mechanics and Mechanical Behavior of Materials
Published in Ashutosh Kumar Dubey, Amartya Mukhopadhyay, Bikramjit Basu, Interdisciplinary Engineering Sciences, 2020
Ashutosh Kumar Dubey, Amartya Mukhopadhyay, Bikramjit Basu
This involves mathematical representation of the problem with suitable approximations and assumptions. This method can be used for real-life complex problems containing 3D geometries and varying loading conditions. The resulting solution is obtained by solving the governing equations with suitable approximations. As it uses approximations, the results need to be validated using experimental methods or analytical calculations. This method can solve the complex problem in considerable time and involves a minimal cost. Compared to analytical methods, the results obtained from numerical methods are approximate. A few numerical techniques are as follows: Finite-element method (FEM)Finite difference method (FDM)Finite volume method (FVM)Boundary element method (BEM)
Modeling of Thermal Systems
Published in Yogesh Jaluria, Design and Optimization of Thermal Systems, 2019
The procedure of obtaining a best fit to a given data set is often known as regression. Let us first consider fitting a straight line to a data set. This curve fitting is known as linear regression and is important in a wide variety of engineering applications because linear approximations are often desirable and also because many nonlinear variations such as exponential and power-law forms can be reduced to a linear best fit, as seen later. Let us take the equation of the straight line for curve fitting as f(x)=a+bx
Study on indoor thermal comfort of different age groups in winter in a rural area of China’s hot-summer and cold-winter region
Published in Science and Technology for the Built Environment, 2022
Jiahao Wan, Qinli Deng, Zeng Zhou, Zhigang Ren, Xiaofang Shan
Interpolation is an effective method of discrete function approximation. When the position at some nodes but not the specific expression of the function is given, the algebraic interpolation method is used to give the approximate form of the function. The approximate values at other points can be estimated through the values at finite points of the function. There are some commonly used interpolation formulas, such as Lagrange interpolation, Newton interpolation, Hermite interpolation, Spline interpolation, etc. The Newton interpolation formula is a common representation of Hermite interpolating polynomials at a sequence of nodes (Carnicer, Khiar, and Peña 2019). It introduces the concept of difference quotient, which makes it easier to calculate when the interpolation nodes increase. The Newton interpolation formula is given by Equation 7 and it can be expressed by Equation 8: where is the difference quotient of zero order at is the difference quotient of order at is the truncation error of Newton interpolation formula.
Passivity and passivity indices of nonlinear systems under operational limitations using approximations
Published in International Journal of Control, 2021
Hasan Zakeri, Panos J. Antsaklis
The most common form of approximation is linearisation, which gives us a very tractable model with many analysis and synthesis tools available. The relation between passivity of a nonlinear system and passivity of its approximation is studied in Xia, Antsaklis, Gupta, and McCourt (2015) and Xia, Antsaklis, Gupta, and Zhu (2017), where the authors show that when the linearised model is simultaneously strictly passive and strictly input passive, the nonlinear system is passive as well, within a neighbourhood of the equilibrium point around which the linearisation is done. However, in general, the linearisation is only valid within a limited neighbourhood, and the approximation error can be high. The relation between approximation error, the neighbourhood of study, and passivity/dissipativity are not evident in linearisation. In Topcu and Packard (2009b), the relation between linearisation and optimisation based study of nonlinear systems is presented, and conditions are presented based on linearisation for the feasibility of the optimisation problem.
Response surface optimization for a nonlinearly constrained irregular experimental design space
Published in Engineering Optimization, 2019
The outer approximation concept is applied to convert a nonlinearly constrained experimental design into a linearized experimental design space for obtaining optimal interior design points. For the nonlinear functions associated with the nonlinear constraints in Equation (6) on the design space, linearization of the nonlinear constraints is used in this article. Linearization is a linear approximation of a nonlinear function in a small region around anchor points. As shown in Figure 1(a), the inner linearization of is done first and the inner linear function will move in parallel towards the nonlinear function until it touches any point on the nonlinear function. In this article, the touching point will be referred to as an anchor point. Around the anchor point at a = (0.707, 0.707), three outer linear functions are created in the design space. Owing to the nature of outer linear functions, infeasible design spaces are often created and computer-generated optimal design points may fall in those regions. To reduce the number of infeasible spaces, imposing additional outer linear functions on the design space is recommended. As shown in Figure 1(b, c), the nonlinear function is well approximated by imposing additional outer linear functions on the design space. The potential question is how many piecewise outer linear functions are needed. This is an issue of feasibility conditions, and the proposed exchange algorithm outlined in Section 5 defines the required number of outer linear functions. Outer linear functions can be obtained as follows: where is the rth anchor point which touches the nonlinear function.