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Capacitance-Voltage and Drive-Level–Capacitance Profiling
Published in Jian V. Li, Giorgio Ferrari, Capacitance Spectroscopy of Semiconductors, 2018
Typical experiments have historically used ac bias with rms values from about 20–300 mV, with 10 or so data points. This gives a rather lengthy data collection time, especially at lower frequencies, which may be problematic for some devices that exhibit metastable changes under bias stress. Truncation of the expansion of Eq. 4.21 is justified when δV ≪ C0/C1, and this criterion can help determine the number of data points, and order of the polynomial, needed for a good curve fit. Normally, a second-order polynomial curve fit is sufficient, and even though the coefficient C2 is unnecessary for finding NDL it is often required for an accurate fit. In some cases, only a few data points and a first-order polynomial are sufficient for a reasonable extraction of C0 and C1.
Numerical Methods
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2022
The main idea of the Runge–Kutta method (which is actually a family of many algorithms including Euler's methods) was proposed in 1895 by Carle Runge29 who intended to extend Simpson's quadrature formula to ODEs. Later Martin Kutta30, Karl Heun, then John Butcher [7], and others generalized these approximations and gave a theoretical background. These methods effectively approximate the true slope of SA on each step interval [xn,xn+1] by the weighted average of slopes evaluated at some sampling points within this interval. Mathematically this means that the truncation of the Taylor series expansion yn+1=yn+hyn′+h22yn′′+h36yn′′′+⋯
Framework for Biomedical Algorithm Designs
Published in Pietro Salvo, Miguel Hernandez-Silveira, Krzysztof Iniewski, Wireless Medical Systems and Algorithms, 2017
Su-Shin Ang, Miguel Hernandez-Silveira
For each value of λ, the solution space, J, can be described by means of a polytope, where each data point is computed for unique values of q and t. The boundaries of the polytope are defined by its vertices. As the algorithm progresses, the polytope gradually “migrates” to a region where the global minimum occurs, and shrinks in size, as it converges around the optimum solution. At each step of the optimization process, the vertices are ranked according to its cost, and the vertex with the lowest cost is returned at the end of the optimization process. There are a few phases in the optimization algorithm, as follows: Reflection: The vertex with the highest cost is reflected about the centroid of the polytope and replaced if the new vertex has a lower cost than the old vertex.Expansion: With reference to the centroid, the polytope is expanded in the direction of the vertex with the lowest cost, on the basis of the heuristic that the “good” solutions are likely to occur in proximity.Contraction: The polytope is shrunk during more mature stages of the optimization process.Reduction: The polytope is shrunk around the vertex with the lowest cost, as the optimization process converges rapidly around the optimum solution.
A study on the vibration characteristics of functionally graded cylindrical beam in a thermal environment using the Carrera unified formulation
Published in Mechanics of Advanced Materials and Structures, 2023
Congshuai He, Junchao Zhu, Yuting Hua, Dakuan Xin, Hongxing Hua
The displacement component can be expressed using the Taylor expansion in the following form where and represent the order of the Taylor expression, the -th term of the Taylor expansion, the sum of the Taylor expansion terms, and the axial displacement variable of the structure, respectively. The expression for the axial displacement variable is given by where and denote the expansion coefficients, represents the truncation index, and can be expressed as
An Investigation of Uncertainty Propagation in Non-equilibrium Flows
Published in International Journal of Computational Fluid Dynamics, 2022
For the second-order truncation, the particle distribution in gPC expansion becomes Substituting the above solution into Eq. (7), we come to where Note that in the deterministic limit, the derivatives of Maxwellian can be evaluated with and Equation (29) reduces to deterministic Navier–Stokes equations The stress tensor and heat flux are related to particle transport phenomena with non-vanishing mean free path, i.e. where p is the thermodynamic pressure, is the identity tensor, and k is the Boltzmann constant.
Dynamics of some more invariant solutions of (3 + 1)-Burgers system
Published in International Journal for Computational Methods in Engineering Science and Mechanics, 2021
R. Kumar, M. Kumar, A. K. Tiwari
On the other hand, J. Zhou et al. [13] could attain soliton, complexion and multi-solitary wave type solutions of Eq. (1) with the help of Painlevé truncation expansion and extended Riccati equation method. The approach used by Dai et al. [14] was slightly different. They have applied the exp-function method to derive an exact solution of the Riccati equation generating them from Eq. (1). Furthermore, Kong and Dai [15] constructed localized excitation based variable separation solutions while Li et al. [16] carried out the periodic wave, solitary wave and rational function solutions of the BS by applying extended mapping approach and a linear variable separation method, whereas, Ju and Yue [17] explored inverse strong symmetries and strong symmetries of another form of a (3 + 1)-BS and got the analytical solution employing the multilinear variable separation approach. Christou et al. [18] generated optimal subalgebra for BS and listed the similarity reductions to solve the system analytically. Lv et al. [19] used the Lie symmetry method to derive symmetry reductions and exact solutions of the Burgers equation. A Painlevé analysis of another form of (3 + 1)-BS is performed by Webb and Zank [20] reduced into a linear heat-like equation. A. M. Wazwaz [21] applied Bäcklund transformation via Hirota’s bilinear form to the Burgers equation and derived some kink wave solutions. Later, these systems were analytically studied by employing the tanh-coth method. In another works [22]–[24] of Wazwaz, he contributed to solve different forms of BS.