Fractional Newton-Raphson Method Accelerated with Aitken's Method

The Newton-Raphson (N-R) method is characterized by the fact that generating a divergent sequence can lead to the creation of a fractal, on the other hand the order of the fractional derivatives seems to be closely related to the fractal dimension, based on the above, a method was developed that makes use of the N-R method and the fractional derivative of Riemann-Liouville (R-L) that has been named as the Fractional Newton-Raphson (F N-R) method. In the following work we present a way to obtain the convergence of the F N-R method, which seems to be at least linearly convergent for the case where the order $\alpha$ of the derivative is different from one, a simplified way to construct the fractional derivative and fractional integral operators of R-L is presented, an introduction to the Aitken's method is made and it is explained why it has the capacity to accelerate the convergence of iterative methods to finally present the results that were obtained when implementing the Aitken's method in F N-R method.Comment: Newton-Raphson Method, Fractional Calculus, Fractional Derivative of Riemann-Liouville, Method of Aitken. arXiv admin note: substantial text overlap with arXiv:1710.0763

The scattering of SH waves by a finite crack with a superposition based diffraction technique

The problem of diffraction of cylindrical and plane SH waves by a finite crack is revisited -- We construct an approximate solution by the addition of independent diffracted terms -- We start with the derivation of the fundamental case of a semi-infinite crack obtained as a degenerate case of generalized wedge -- This building block is then used to compute the diffraction of the main incident waves -- The interaction between the opposite edges of the crack is then considered one term at a time until a desired tolerance is reached -- We propose a recipe to determine the number of required interactions as a function of frequency -- The solution derived with the superposition technique can be applied at low and high frequencie

Seismic response of three-dimensional rockfill dams using the Indirect Boundary Element Method

The Indirect Boundary Element Method (IBEM) is used to compute the seismic response of a three-dimensional rockfill dam model. The IBEM is based on a single layer integral representation of elastic fields in terms of the full-space Green function, or fundamental solution of the equations of dynamic elasticity, and the associated force densities along the boundaries. The method has been applied to simulate the ground motion in several configurations of surface geology. Moreover, the IBEM has been used as benchmark to test other procedures. We compute the seismic response of a three-dimensional rockfill dam model placed within a canyon that constitutes an irregularity on the surface of an elastic half-space. The rockfill is also assumed elastic with hysteretic damping to account for energy dissipation. Various types of incident waves are considered to analyze the physical characteristics of the response: symmetries, amplifications, impulse response and the like. Computations are performed in the frequency domain and lead to time response using Fourier analysis. In the present implementation a symmetrical model is used to test symmetries. The boundaries of each region are discretized into boundary elements whose size depends on the shortest wavelength, typically, six boundary segments per wavelength. Usually, the seismic response of rockfill dams is simulated using either finite elements (FEM) or finite differences (FDM). In most applications, commercial tools that combine features of these methods are used to assess the seismic response of the system for a given motion at the base of model. However, in order to consider realistic excitation of seismic waves with different incidence angles and azimuth we explore the IBEM. © 2010 IOP Publishing Ltd

Analysis of the role of diffraction in topographic site effects using boundary element techniques

The role played by the diffraction field on the problem of seismic site effects is studied. For that purpose we solve and analyze simple scattering problems under P and SV in-plane wave assumptions, using two well known direct boundary-element-based numerical methods. After establishing the difference between scattered and diffracted motions, and introducing the concept of artificious and physically based incoming fields, we obtain the amplitude of the Fourier spectra for the diffracted part of the response: this is achieved after establishing the connection between the spatial distribution of the transfer function over the studied simple topographies and the diffracted field. From the numerical simulations it is observed that this diffracted part of the response is responsible for the amplification of the surface ground motions due to the geometric effect. Furthermore, it is also found that the diffraction field sets in a fingerprint of the topographic effect in the total ground motions. These conclusions are further supported by observations in the time-domain in terms of snapshots of the propagation patterns over the complete computational model. In this sense the geometric singularities are clearly identified as sources of diffraction and for the considered range of dimensionless frequencies it is evident that larger amplifications are obtained for the geometries containing a larger number of diffraction sources thus resulting in a stronger topographic effect. The need for closed-form solutions of canonical problems to construct a robust analysis method based on the diffraction field is identified