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Nonlinear Optimization
Published in Jeffery J. Leader, Numerical Analysis and Scientific Computation, 2022
However, there are a number of difficulties with this method. Even if the local minimization algorithm converges each time and furthermore converges to a new minimum each time (as opposed to one previously located, which is possible if it is not a descent method), the root-finding algorithm will often find the current minimum, not a new point. (Deflation may be helpful here.) Additionally, if the root-finding algorithm fails to find a root, it is not clear whether it is because no other root exists, or because the method is simply not finding a new one. There is no general fix for this problem.
Ground response curve of deep circular tunnel in rock mass exhibiting Hoek–Brown strain-softening behaviour considering the dead weight loading
Published in European Journal of Environmental and Civil Engineering, 2021
A closed-form expression for cannot be obtained. Therefore, must be calculated numerically by use of suitable root-finding algorithms such as Newton–Raphson method (Press, Teukolsky, Vetterling, & Flannery, 1992). It should be noted that Equation (47) can also be used for defining by applying instead of It should be notified that the plastic region around the circular opening is formed only when the internal support pressure is lower than the value obtained from Equation (47)
A Generalized Eigenvalue Formulation for Core-Design Applications
Published in Nuclear Science and Engineering, 2023
Nicolo’ Abrate, Sandra Dulla, Piero Ravetto, Paolo Saracco
Due to its great flexibility, can be used to deal with more complex problems, for example, the determination of the moderation ratio for a heterogeneous arrangement of fuel and moderator layers surrounded by a reflector. In this case, the iterative search of the critical moderation ratio is more expensive due to the larger number of spatial nodes needed to resolve the spatial gradients. Of course, the design is usually carried out using effective, nonlinear root-finding algorithms, like the Newton-Raphson method,25 to minimize the number of static calculations required. However, the presence of more solutions, like in this case, can have a detrimental effect on such techniques, limiting their effectiveness. This aspect is even more important when the design calculation includes other parameters, like the thermohydraulic quantities.26 As is visible from Fig. 4, which again has been obtained by iterating the calculations, three critical configurations could be devised for a lattice composed of 17 3-cm thick sheets of uranium oxide enriched at 3%, and 16 3-cm thick layers of water surrounded on both sides by 20 cm of water used as a reflector. It is important to remark here that despite the fact that the curve seems to approach a fourth zero around , it was not possible to find a solution to the static problem around these values of due to the ill conditioning of the moderator cross sections, which assume very large, unphysical values.
Asphalt concrete master curve using dynamic backcalculation
Published in International Journal of Pavement Engineering, 2022
Gabriel Bazi, Tatiana Bou Assi
In numerical analysis, the Newton's method – also known as the Newton-Raphson method – is a root-finding algorithm which produces iteratively better approximations to the roots of a function. In optimisation, the Newton’s method is used to determine the stationary points of a function (minima or maxima) by finding the roots of the function’s first derivative. The Newton’s method uses the first few terms of the Taylor series (equation 9) of a real-valued function at .For finding the roots of a function, the linear approximation (first two terms) of the Taylor series are used and is set equal to zero (equation 10). The offset needed to improve the root for every iteration is shown in equation 11.For finding the stationary points (minima or maxima), the function’s first derivative is determined using the quadratic approximation (first three terms) of the Taylor series and set equal to zero. The offset , shown in equation 12, is calculated using the first and second derivatives.The process starts with an initial guess and a better approximation of the root is obtained after every iteration. This process is repeated as many times as necessary to get the desired accuracy. Figure 4 provides an illustration of how the root of a function f(x) = x3 + x−1 is better approximated after every iteration starting with an initial guess x0 = −0.6 and using equations 10 or 11 (Sauer 2012).