scholarly journals Finite difference approximation of eigenvibrations of a bar with oscillator

2020 ◽  
Vol 329 ◽  
pp. 03030
Author(s):  
D. M. Korosteleva ◽  
L. N. Koronova ◽  
K. O. Levinskaya ◽  
S. I. Solov’ev
Keyword(s):  
Finite Difference ◽  
Spectral Problem ◽  
Model Problem ◽  
Second Order ◽  
Harmonic Oscillators ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Eigenvalues And Eigenfunctions ◽  
Constant Coefficients ◽  
Theoretical Results

The second-order ordinary differential spectral problem governing eigenvibrations of a bar with attached harmonic oscillator is investigated. We study existence and properties of eigensolutions of formulated bar-oscillator spectral problem. The original second-order ordinary differential spectral problem is approximated by the finite difference mesh scheme. Theoretical error estimates for approximate eigenvalues and eigenfunctions of this mesh scheme are established. Obtained theoretical results are illustrated by computations for a model problem with constant coefficients. Theoretical and experimental results of this paper can be developed and generalized for the problems on eigenvibrations of complex mechanical constructions with systems of harmonic oscillators.

scholarly journals Eigenvibrations of a beam with two mechanical resonators attached to the ends

2020 ◽  
Vol 329 ◽  
pp. 03009
Author(s):  
L. N. Koronova ◽  
D. M. Korosteleva ◽  
K. O. Levinskaya ◽  
S. I. Solov’ev
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Spectral Problem ◽  
Fourth Order ◽  
Model Problem ◽  
Mechanical Resonators ◽  
Eigenvalues And Eigenfunctions ◽  
Constant Coefficients ◽  
The Finite Difference Method ◽  
Theoretical Results

The fourth-order ordinary differential spectral problem describing vertical eigenvibrations of a beam with two mechanical resonators attached to the ends is studied. This problem has positive simple eigenvalues and corresponding eigenfunctions. We define limit differential spectral problem and establish the convergence of the eigenvalues and eigenfunctions of the original spectral problem to the eigenvalues and eigenfunctions of the limit spectral problem as parameters of the attached resonators tending to infinity. The initial fourth-order ordinary differential spectral problem is approximated by the finite difference method. Theoretical error estimates for approximate eigenvalues and eigenfunctions are derived. Obtained theoretical results are illustrated by computations for model problem with constant coefficients. Theoretical and experimental results of this paper can be developed for the problems on eigenvibrations of complex mechanical constructions with systems of resonators.

scholarly journals Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation

2012 ◽  
Vol 12 (1) ◽  
pp. 193-225 ◽  
Author(s):  
N. Anders Petersson ◽  
Björn Sjögreen
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Half Space ◽  
Material Model ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Space Problem ◽  
Standard Linear Solid ◽  
Standard Linear

AbstractWe develop a stable finite difference approximation of the three-dimensional viscoelastic wave equation. The material model is a super-imposition of N standard linear solid mechanisms, which commonly is used in seismology to model a material with constant quality factor Q. The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables, making it significantly more memory efficient than the commonly used first order velocity-stress formulation. The new scheme is a generalization of our energy conserving finite difference scheme for the elastic wave equation in second order formulation [SIAM J. Numer. Anal., 45 (2007), pp. 1902-1936]. Our main result is a proof that the proposed discretization is energy stable, even in the case of variable material properties. The proof relies on the summation-by-parts property of the discretization. The new scheme is implemented with grid refinement with hanging nodes on the interface. Numerical experiments verify the accuracy and stability of the new scheme. Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are used to demonstrate how the number of viscoelastic mechanisms and the grid resolution influence the accuracy. We find that three standard linear solid mechanisms usually are sufficient to make the modeling error smaller than the discretization error.

scholarly journals An Efficient Compact Difference Method for Temporal Fractional Subdiffusion Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Lei Ren ◽  
Lei Liu
Keyword(s):  
Finite Difference ◽  
Fourth Order ◽  
Difference Method ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Approximation Formula ◽  
Discrete Energy ◽  
Compact Finite Difference ◽  
Second Order In Time

In this paper, a high-order compact finite difference method is proposed for a class of temporal fractional subdiffusion equation. A numerical scheme for the equation has been derived to obtain 2-α in time and fourth-order in space. We improve the results by constructing a compact scheme of second-order in time while keeping fourth-order in space. Based on the L2-1σ approximation formula and a fourth-order compact finite difference approximation, the stability of the constructed scheme and its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method. Applications using two model problems demonstrate the theoretical results.

Finite differences on minimal grids

Geophysics ◽  
1994 ◽  
Vol 59 (9) ◽  
pp. 1435-1443 ◽  
Author(s):  
Sophie‐Adélade Magnier ◽  
Peter Mora ◽  
Albert Tarantola
Keyword(s):  
Finite Difference ◽  
Conventional Method ◽  
Finite Differences ◽  
Quantitative Estimation ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Elastic Media ◽  
Numerical Tests ◽  
Propagation Of Waves

Conventional approximations to space derivatives by finite differences use orthogonal grids. To compute second‐order space derivatives in a given direction, two points are used. Thus, 2N points are required in a space of dimension N; however, a centered finite‐difference approximation to a second‐order derivative may be obtained using only three points in 2-D (the vertices of a triangle), four points in 3-D (the vertices of a tetrahedron), and in general, N + 1 points in a space of dimension N. A grid using N + 1 points to compute derivatives is called minimal. The use of minimal grids does not introduce any complication in programming and suppresses some artifacts of the nonminimal grids. For instance, the well‐known decoupling between different subgrids for isotropic elastic media does not happen when using minimal grids because all the components of a given tensor (e.g., displacement or stress) are known at the same points. Some numerical tests in 2-D show that the propagation of waves is as accurate as when performed with conventional grids. Although this method may have less intrinsic anisotropies than the conventional method, no attempt has yet been made to obtain a quantitative estimation.

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