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Finite Difference Methods
Published in Matthew N.O. Sadiku, Computational Electromagnetics with MATLAB®, 2018
Numerical integration (also called numerical quadrature) is used in science and engineering whenever a function cannot easily be integrated in closed form or when the function is described in the form of discrete data. Integration is a more stable and reliable process than differentiation. The term quadrature or integration rule will be used to indicate any formula that yields an integral approximation. Several integration rules have been developed over the years. The common ones include Euler's rule,Trapezoidal rule,Simpson's rule,Newton–Cotes rules, andGaussian (quadrature) rules.
Response to General Loading
Published in Hector Estrada, Luke S. Lee, Introduction to Earthquake Engineering, 2017
To obtain shock spectra, we can solve Duhamel's integral analytically or numerically, both of which were introduced in the previous section. Numerical integration can also be performed using one of the standard numerical integration techniques such as Euler (forward rectangular), Trapezoidal (linear), or Simpson's (parabolic) methods, all of which can be found in a standard calculus textbook. In this book, we use the numerical integration operators available in MATLAB; for Duhamel's integral, we apply the conv function used in Example 4.3. Following are some examples used to generate the shock spectra using both analytical solutions and convolution numerical integration. Also, we demonstrate the use of these spectra in determining the maximum response of structural systems.
Differentiation and Integration
Published in Bilal M. Ayyub, Richard H. Mccuen, Numerical Analysis for Engineers, 2015
Bilal M. Ayyub, Richard H. McCuen
Numerical integration can result in errors where a numerical error is defined as the difference between the true value and the estimated value. The true value is unknown; otherwise, it would not be necessary to use the trapezoidal rule. An approximate upper bound on the error can be determined based on the second term of a Taylor series expansion: xi+1−xi2f′′(x)
An Online Database of Benchmark Problems for Verification of Inverse Problems Computer Codes
Published in Heat Transfer Engineering, 2023
The Fredholm integral of the 1st kind is an extremely versatile mathematical model, with applications in geophysics, tomography, image processing, heat/mass transfer, among others [23]. It is a continuous linear model, given by Eq. (20), to be discretized via numerical integration techniques, herein taken to be the Trapezoidal rule. The exact profile for is given by Eq. (21) (with and ) and shown in Figure 13. In this test case, a Gaussian kernel is assumed (cf. Eq. (22) with and The independent variables are discretized in a uniform grid with respectively.
A new recursive Simpson integral algorithm in vibration testing
Published in Australian Journal of Mechanical Engineering, 2021
Jingbo Xu, Xiaohong Xu, Xiaomeng Cui
In the whole measurement process, a key issue is numerical integration. Because the vibration acceleration measured in practice is a non-deterministic function, the integral curve can only be obtained by numerical integration method. The numerical integration methods in common use are rectangular formula method and trapezoidal formula method. They actually use the rectangle and trapezoidal between adjacent points to approximate the integration area. Their characteristic is simple calculation, but the approximation accuracy is poor. Simpson integral replaces rectangular or trapezoidal integral formula by quadratic curve approximation based on Newton-Cotes formula. It has higher approximation accuracy and simple calculation formula and is a practical method in engineering application as referred to (Hong, Kim, and Lee 2010; STIROS 2008). Furthermore, Simpson integral formula can be divided into 1/3 and 3/8 types, as shown.
Investigation into evaluation of wearing bagging behavior for woven garments
Published in The Journal of The Textile Institute, 2021
Xiaoping Zheng, Chengxia Liu, Wenda Wei
The fitted surface was defined as a function f (x, y), the divided a × b region was defined as a bounded closed region D of the surface in the XY datum plane, and the bagging volume VR was the double integral of the function f (x, y) in the closed region D. The method of trapezoidal numerical integration and Simpson numerical integration were usually used to calculate the numerical double integral. Simpson numerical integration was a method to solve the numerical solution of definite integral by using parabola approximate function curve, and it was more precise than trapezoidal numerical integration (Ujević, 2007). In this paper, the Simpson numerical integration method was used to calculate the double integral of the surface in the closed region of D, and the VR was obtained.