scholarly journals NUMERICAL WAVE FIELDS QUASISTATIC MODELING IN FLUID-FILLED POROELASTIC MEDIA

2021 ◽  
Vol 2 (2) ◽  
pp. 298-311
Author(s):  
Sergey A. Solovyev ◽  
Vadim V. Lisitsa
Keyword(s):  
Temporal Frequency ◽  
Low Frequency ◽  
Wide Frequency Range ◽  
Algebraic Equations ◽  
Static State ◽  
Frequency Space ◽  
Poroelastic Media ◽  
Poroelastic Materials ◽  
Linear Algebraic Equations ◽  
Dependent Strain

This paper presents a numerical algorithm to simulate low-frequency loading of fluid-filled poroelastic materials and estimate the effective frequency-dependent strain-stress relations for such media. The algorithm solves Biot equation in quasi-static state in the frequency space. As a result a system of linear algebraic equations have to be solved for each temporal frequency. We use the direct solver, based on the $LU$ decomposition to resolve the SLAE. According to the presented numerical examples the suggested algorithm allows reconstructing the stiffness tensor within a wide Frequency range.

Wavelet Domain Solution of CSRBF SLAE for Image Interpolation Using Iterative Methods

2004 ◽  
Author(s):  
Luis A. Diago ◽  
Masaki Kitago ◽  
Ichiro Hagiwara
Keyword(s):  
Radial Basis Functions ◽  
Low Frequency ◽  
Direct Methods ◽  
Image Interpolation ◽  
Iterative Solution ◽  
Basis Functions ◽  
Wavelet Domain ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Radial Basis

Radial Basis Functions (RBF) are popular for interpolating scattered data. In this context, the solution of the system of linear algebraic equations (SLAE) is the most time-consuming operation. Techniques fail with large point sets consisting of more than several thousands of points when direct methods and global support are used. In this paper we demostrate that the solution of the SLAE in the wavelet domain is suitable for the problem of image interpolation by means of Compactly-Supported Radial Basis Functions (CSRBF). The iterative solution of SLAE with highly irregular matrices cannot be accelerated by wavelet transformation and subsequent sparcification if the transformed matrix is still highly irregular. To solve the SLAE in the wavelet domain, the ordering of the samples defines the spacial relationship and the energy of the coefficients in the low frequency domain. Two sorting algorithms for the wavelet domain solution are tested and compared with the spacial solution of the SLAE. Examples of image interpolation by means of CSRBF demostrate the superiority of the solution in the wavelet domain using GMRES iterative method.

LOW-FREQUENCY ACOUSTIC SCATTERING OF A PLANE WAVE FROM A SOFT AND HARD TORUS

2007 ◽  
Vol 15 (02) ◽  
pp. 181-197 ◽  
Author(s):  
GEORGE VENKOV
Keyword(s):  
Separation Of Variables ◽  
Near Field ◽  
Scattering Problem ◽  
Acoustic Scattering ◽  
Low Frequency ◽  
Diagonal Form ◽  
Legendre Functions ◽  
Algebraic Equations ◽  
Analytical Technique ◽  
Linear Algebraic Equations

A plane acoustic wave is scattered by either a soft or a hard small torus. The incident wave has a wavelength which is much larger than the characteristic dimension of the scatterer and thus the low-frequency approximation method is applicable to the scattering problem. It is shown that there exists exactly one toroidal coordinate system that fits the given geometry. The R-separation of variables is utilized to obtain the series expansion of the fields in terms of toroidal harmonics (half-integer Legendre functions of first and second kind). The scattering problem for the soft torus is solved analytically for the near field, governing the leading two low-frequency coefficients, as well as for the far field, where both the amplitude and the cross-section are evaluated. The scattering problem for the hard torus appears to be much more complicated in calculations. The Neumann boundary condition on the surface of the torus leads to a three-term recurrence relation for the series coefficients corresponding to the scattered fields. Thus, the potential boundary-value problem for the leading low-frequency approximations is reduced to infinite systems of linear algebraic equations with three-diagonal matrices. An analytical technique for solving systems of diagonal form is developed.

3D numerical simulation of elastic waves with a frequency-domain iterative solver

Geophysics ◽  
2018 ◽  
Vol 83 (6) ◽  
pp. T333-T344 ◽  
Author(s):  
Mikhail Belonosov ◽  
Victor Kostin ◽  
Dmitry Neklyudov ◽  
Vladimir Tcheverda
Keyword(s):  
Wave Propagation ◽  
Frequency Domain ◽  
Krylov Subspace ◽  
Waveform Inversion ◽  
Low Frequency ◽  
Forward Modeling ◽  
Inversion Method ◽  
Algebraic Equations ◽  
Iterative Solver ◽  
Linear Algebraic Equations

The efficiency of any inversion method for estimating the medium parameters from seismic data strongly depends on simulation of the wave propagation, i.e., forward modeling. The requirements are that it should be accurate, fast, and computationally efficient. When the inversion is carried out in the frequency domain (FD), e.g., FD full-waveform inversion, only a few monochromatic components are involved in the computations. In this situation, FD forward modeling is an appealing potential alternative to conventional time-domain solvers. Iterative FD solvers, based on a Krylov subspace iterative method, are of interest due to their moderate memory requirements compared with direct solvers. A huge issue preventing their successful use is a very slow convergence. We have developed an iterative solver for the elastic wave propagation in 3D isotropic heterogeneous land models. Its main ingredient is a novel preconditioner, which provides the convergence of the iteration. We have developed and justified a method to invert our preconditioner effectively on the base of the 2D fast Fourier transform and solving a system of linear algebraic equations with a banded matrix. In addition, we determine how to parallelize our solver using the conventional hybrid parallelization (MPI in conjunction with OpenMP) and demonstrate the good scalability for the widespread 3D SEG/EAGE overthrust model. We find that our method has a high potential for low-frequency simulations in land models with moderate lateral variations and arbitrary vertical variations.

On Improved Approximate Solution of the Fredholm Integral Equation

2006 ◽  
Vol 6 (3) ◽  
pp. 264-268
Author(s):  
G. Berikelashvili ◽  
G. Karkarashvili
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Fredholm Integral Equation ◽  
Algebraic Equations ◽  
Order Convergence ◽  
One Dimensional ◽  
Averaging Operator ◽  
Linear Algebraic Equations ◽  
Convergence Solution

AbstractA method of approximate solution of the linear one-dimensional Fredholm integral equation of the second kind is constructed. With the help of the Steklov averaging operator the integral equation is approximated by a system of linear algebraic equations. On the basis of the approximation used an increased order convergence solution has been obtained.

scholarly journals Exact statistical solution for the hopping transport of trapped charge via finite Markov jump processes

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Andrey A. Pil’nik ◽  
Andrey A. Chernov ◽  
Damir R. Islamov
Keyword(s):  
Charge Transport ◽  
Thin Layers ◽  
Jump Processes ◽  
Dielectric Films ◽  
Algebraic Equations ◽  
Markov Jump ◽  
Statistical Solution ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
Thin Dielectric Films

AbstractIn this study, we developed a discrete theory of the charge transport in thin dielectric films by trapped electrons or holes, that is applicable both for the case of countable and a large number of traps. It was shown that Shockley–Read–Hall-like transport equations, which describe the 1D transport through dielectric layers, might incorrectly describe the charge flow through ultra-thin layers with a countable number of traps, taking into account the injection from and extraction to electrodes (contacts). A comparison with other theoretical models shows a good agreement. The developed model can be applied to one-, two- and three-dimensional systems. The model, formulated in a system of linear algebraic equations, can be implemented in the computational code using different optimized libraries. We demonstrated that analytical solutions can be found for stationary cases for any trap distribution and for the dynamics of system evolution for special cases. These solutions can be used to test the code and for studying the charge transport properties of thin dielectric films.

scholarly journals Two matrix methods for solution of nonlinear and linear Lane-Emden type equations with mixed condition by operational matrix

2015 ◽  
Vol 4 (3) ◽  
pp. 420 ◽  
Author(s):  
Behrooz Basirat ◽  
Mohammad Amin Shahdadi
Keyword(s):  
Bernstein Polynomials ◽  
Operational Matrix ◽  
Numerical Procedure ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Mixed Condition ◽  
Matrix Methods ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
The Given

<p>The aim of this article is to present an efficient numerical procedure for solving Lane-Emden type equations. We present two practical matrix method for solving Lane-Emden type equations with mixed conditions by Bernstein polynomials operational matrices (BPOMs) on interval [<em>a; b</em>]. This methods transforms Lane-Emden type equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equations. We also give some numerical examples to demonstrate the efficiency and validity of the operational matrices for solving Lane-Emden type equations (LEEs).</p>

scholarly journals Optimal Random Packing of Spheres and Extremal Effective Conductivity

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1063
Author(s):  
Vladimir Mityushev ◽  
Zhanat Zhunussova
Keyword(s):  
Effective Conductivity ◽  
Dimensional Space ◽  
Elementary Function ◽  
Voronoi Tessellation ◽  
Random Packing ◽  
Periodicity Cell ◽  
Algebraic Equations ◽  
Optimal Packing ◽  
Linear Algebraic Equations ◽  
Initial Location

A close relation between the optimal packing of spheres in Rd and minimal energy E (effective conductivity) of composites with ideally conducting spherical inclusions is established. The location of inclusions of the optimal-design problem yields the optimal packing of inclusions. The geometrical-packing and physical-conductivity problems are stated in a periodic toroidal d-dimensional space with an arbitrarily fixed number n of nonoverlapping spheres per periodicity cell. Energy E depends on Voronoi tessellation (Delaunay graph) associated with the centers of spheres ak (k=1,2,…,n). All Delaunay graphs are divided into classes of isomorphic periodic graphs. For any fixed n, the number of such classes is finite. Energy E is estimated in the framework of structural approximations and reduced to the study of an elementary function of n variables. The minimum of E over locations of spheres is attained at the optimal packing within a fixed class of graphs. The optimal-packing location is unique within a fixed class up to translations and can be found from linear algebraic equations. Such an approach is useful for random optimal packing where an initial location of balls is randomly chosen; hence, a class of graphs is fixed and can dynamically change following prescribed packing rules. A finite algorithm for any fixed n is constructed to determine the optimal random packing of spheres in Rd.

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