scholarly journals Lagrangian meshfree finite difference particle method with variable smoothing length for solving wave equations

2018 ◽  
Vol 10 (7) ◽  
pp. 168781401878924 ◽  
Author(s):  
Sheng Wang ◽  
Yong Ou Zhang ◽  
Jing Ping Wu
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Time Integration ◽  
Simple Wave ◽  
Domain Size ◽  
Particle Method ◽  
Computational Time ◽  
Integration Scheme ◽  
Smoothed Particle ◽  
Support Domain

In a Lagrangian meshfree particle-based method, the smoothing length determines the size of the support domain for each particle. Since the particle distribution is irregular and uneven in most cases, a fixed smoothing length sometime brings too much or insufficient neighbor particles for the weight function which reduces the numerical accuracy. In this work, a Lagrangian meshfree finite difference particle method with variable smoothing length is proposed for solving different wave equations. This pure Lagrangian method combines the generalized finite difference scheme for spatial resolution and the time integration scheme for time resolution. The new method is tested via the simple wave equation and the Burgers’ equation in Lagrangian form. These wave equations are widely used in analyzing acoustic and hydrodynamic waves. In addition, comparison with a modified smoothed particle hydrodynamics method named the corrective smoothed particle method is also presented. Numerical experiments show that two kinds of Lagrangian wave equations can be solved well. The variable smoothing length updates the support domain size appropriately and allows the finite difference particle method results to be more accurate than the constant smoothing length. To obtain the same level of accuracy, the corrective smoothed particle method needs more particles in the computation which requires more computational time than the finite difference particle method.

A New Unconditionally Stable Time Integration Method for Analysis of Nonlinear Structural Dynamics

2013 ◽  
Vol 80 (2) ◽  
Author(s):  
Ali Akbar Gholampour ◽  
Mehdi Ghassemieh ◽  
Mahdi Karimi-Rad
Keyword(s):  
Time Integration ◽  
Second Order ◽  
Numerical Dispersion ◽  
Computational Time ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Time Step ◽  
Similar Time ◽  
Unconditionally Stable ◽  
Average Acceleration

A new time integration scheme is presented for solving the differential equation of motion with nonlinear stiffness. In this new implicit method, it is assumed that the acceleration varies quadratically within each time step. By increasing the order of acceleration, more terms of the Taylor series are used, which are expected to have responses with better accuracy than the classical methods. By considering this assumption and employing two parameters δ and α, a new family of unconditionally stable schemes is obtained. The order of accuracy, numerical dissipation, and numerical dispersion are used to measure the accuracy of the proposed method. Second order accuracy is achieved for all values of δ and α. The proposed method presents less dissipation at the lower modes in comparison with Newmark's average acceleration, Wilson-θ, and generalized-α methods. Moreover, this second order accurate method can control numerical damping in the higher modes. The numerical dispersion of the proposed method is compared with three unconditionally stable methods, namely, Newmark's average acceleration, Wilson-θ, and generalized-α methods. Furthermore, the overshooting effect of the proposed method is compared with these methods. By evaluating the computational time for analysis with similar time step duration, the proposed method is shown to be faster in comparison with the other methods.

Time-space-domain implicit finite-difference methods for modeling acoustic wave equations

Geophysics ◽  
2018 ◽  
Vol 83 (4) ◽  
pp. T175-T193 ◽  
Author(s):  
Enjiang Wang ◽  
Jing Ba ◽  
Yang Liu
Keyword(s):  
Finite Difference ◽  
Acoustic Wave ◽  
Wave Equations ◽  
Finite Difference Methods ◽  
Second Order ◽  
Computational Time ◽  
Time Step ◽  
Space Domain ◽  
Time Space ◽  
Temporal Dispersion

It has been proved that the implicit spatial finite-difference (FD) method can obtain higher accuracy than explicit FD by using an even smaller operator length. However, when only second-order FD in time is used, the combined FD scheme is prone to temporal dispersion and easily becomes unstable when a relatively large time step is used. The time-space domain FD can suppress the temporal dispersion. However, because the spatial derivatives are solved explicitly, the method suffers from spatial dispersion and a large spatial operator length has to be adopted. We have developed two effective time-space-domain implicit FD methods for modeling 2D and 3D acoustic wave equations. First, the high-order FD is incorporated into the discretization for the second-order temporal derivative, and it is combined with the implicit spatial FD. The plane-wave analysis method is used to derive the time-space-domain dispersion relations, and two novel methods are proposed to determine the spatial and temporal FD coefficients in the joint time-space domain. First, we fix the implicit spatial FD coefficients and derive the quadratic convex objective function with respect to temporal FD coefficients. The optimal temporal FD coefficients are obtained by using the linear least-squares method. After obtaining the temporal FD coefficients, the SolvOpt nonlinear algorithm is applied to solve the nonquadratic optimization problem and obtain the optimized temporal and spatial FD coefficients simultaneously. The dispersion analysis, stability analysis, and modeling examples validate that the proposed schemes effectively increase the modeling accuracy and improve the stability conditions of the traditional implicit schemes. The computational efficiency is increased because the schemes can adopt larger time steps with little loss of spatial accuracy. To reduce the memory requirement and computational time for storing and calculating the FD coefficients, we have developed the representative velocity strategy, which only computes and stores the FD coefficients at several selected velocities. The modeling result of the 2D complicated model proves that the representative velocity strategy effectively reduces the memory requirements and computational time without decreasing the accuracy significantly when a proper velocity interval is used.

Fatigue Life Prediction Due to Slug Flow in Extra Long Submarine Gas Pipelines Using Fourier Expansion Series

2009 ◽  
Author(s):  
Euro Casanova ◽  
Orlando Pelliccioni ◽  
Armando Blanco
Keyword(s):  
Fatigue Life ◽  
Dynamic Response ◽  
Slug Flow ◽  
Fourier Expansion ◽  
Time Integration ◽  
Computational Cost ◽  
Gas Production ◽  
Computational Time ◽  
Integration Scheme ◽  
Expansion Series

Some offshore gas production fields require transporting of production fluids through very long submarines pipelines, without a previous separation process. In these cases, a slug flow pattern may develop for some production conditions. Condensate slugs traveling in the pipeline, act as moving loads for the piping structure, especially for the unsupported pipe spans which can be of even hundreds of meters long, due to irregular sea bottom, therefore producing a dynamic response of the pipeline that in some cases may significantly reduce its fatigue life. In this work a previously presented model [1], which combines fluid equations for predicting slug characteristics and a structural finite element model of horizontal pipelines transporting slugs, is modified for reducing computational cost and to adapt fatigue life calculations to the case of submarine piping. In order to calculate maximum amplitudes of the dynamic response without a time integration scheme, it is considered that traveling slugs produce periodical loads in time for every spatial point of the pipeline, and consequently these loads may be expressed by means of Fourier expansion series. With these assumptions, a more realistic fatigue calculation for a diversity of pipelines conditions is obtained. Results show that for this improved model computational time is dramatically reduced, without a lost in precision, when compared to the previous model requiring a time integration process.

Elimination of temporal dispersion from the finite-difference solutions of wave equations in elastic and anelastic models

Geophysics ◽  
2019 ◽  
Vol 84 (2) ◽  
pp. T47-T58 ◽  
Author(s):  
Lasse Amundsen ◽  
Ørjan Pedersen
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Fourier Transform ◽  
Finite Difference ◽  
Wave Equations ◽  
Time Integration ◽  
Sinc Function ◽  
Partial Differential ◽  
Time Frequency ◽  
Temporal Dispersion

Time integration of wave equations can be carried out with explicit time stepping using a finite-difference (FD) approximation. The wave equation is the partial differential equation that governs the wavefield that is solved for. The FD approximation gives another partial differential equation — the one solved in the computer for the FD wavefield. This approximation to time integration in numerical modeling produces a wavefield contaminated with temporal dispersion, particularly at high frequencies. We find how the Fourier transform can be used to relate the two partial differential equations and their solutions. Each of the two wavefields is then a time-frequency transformation of the other. First, this transformation allows temporal dispersion to be eliminated from the FD wavefield, and second, it allows temporal dispersion to be added to the exact wavefield. The two transforms are band-limited inverse operations. The transforms can be implemented by using time-step independent, noncausal time-varying digital filters that can be precomputed exactly from sums over Bessel functions. Their product becomes the symmetric Toeplitz matrix with the elements defined through the cardinal sine (sinc) function. For anelastic materials, the effect of numerical time dispersion in a wavefield propagating in a medium needs special treatment. Dispersion can be removed by using the time-frequency transform when the FD wavefield is modeled in a medium with the frequency-modified modulus relative to the physical modulus of interest. In the rheological model of the generalized Maxwell body, the frequency-modified modulus is written as a power series, which allows a term-by-term Fourier transform to the time domain. In a low-frequency approximation, the modified modulus obtains the same form as the physical modulus, and it can be implemented as changes in the unrelaxed modulus and shifts of the relaxation frequencies and their strengths of the physical modulus.

Numerical Method

2013 ◽  
Author(s):  
Gary A. Glatzmaier
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Numerical Method ◽  
Fluid Velocity ◽  
Time Integration ◽  
Spectral Space ◽  
Integration Scheme ◽  
Time Step ◽  
Time Integration Scheme ◽  
Solving Equations

This chapter describes a numerical method for solving equations of thermal convection on a computer. It begins by introducing the vorticity-streamfunction formulation as a means of conserving mass. The approach involves updating for the vorticity first and then solving for the fluid velocity each time step. The chapter continues with a discussion of two very different spatial discretizations, whereby the vertical derivatives are approximated with a finite-difference method and the horizontal derivatives with a spectral method. The nonlinear terms are computed in spectral space. The chapter also considers the Adams-Bashforth time integration scheme and explains how the Poisson equation can be solved at each time step for the updated streamfunction given the updated vorticity.

scholarly journals An Improved Implementation of Time Domain Elastodynamic BIEM in 3D for Large Scale Problems and its Application to Ultrasonic NDE

2007 ◽  
Vol 1 (2) ◽  
Author(s):  
Hitoshi Yoshikawa ◽  
Naoshi Nishimura
Keyword(s):  
Time Domain ◽  
Large Scale ◽  
Scattering Problem ◽  
Time Integration ◽  
Simple Wave ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Computational Time ◽  
Integration Algorithm ◽  
Ultrasonic Nde

This paper discusses a three dimensional implementation of boundary integral equation method (BIEM) for large scale time domain elastodynamic problems and its application to ultrasonic nondestructive evaluation (NDE). We improve the time integration algorithm of the BIEM in order to reduce the required computational time. We show the e±ciency of the proposed method by applying it to a simple wave scattering problem and to a more realistic crack determination problem related to ultrasonic NDE.

Weighted Finite Difference and Boundary Element Methods Applied to Groundwater Pollution Problems

1991 ◽  
Vol 23 (1-3) ◽  
pp. 517-524
Author(s):  
M. Kanoh ◽  
T. Kuroki ◽  
K. Fujino ◽  
T. Ueda
Keyword(s):  
Boundary Element Method ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Boundary Element ◽  
Difference Method ◽  
Groundwater Pollution ◽  
Small Region ◽  
Boundary Element Methods ◽  
Computational Time ◽  
Element Method

The purpose of the paper is to apply two methods to groundwater pollution in porous media. The methods are the weighted finite difference method and the boundary element method, which were proposed or developed by Kanoh et al. (1986,1988) for advective diffusion problems. Numerical modeling of groundwater pollution is also investigated in this paper. By subdividing the domain into subdomains, the nonlinearity is localized to a small region. Computational time for groundwater pollution problems can be saved by the boundary element method; accurate numerical results can be obtained by the weighted finite difference method. The computational solutions to the problem of seawater intrusion into coastal aquifers are compared with experimental results.

scholarly journals Approximate Solutions of Time Fractional Diffusion Wave Models

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 923 ◽  
Author(s):  
Abdul Ghafoor ◽  
Sirajul Haq ◽  
Manzoor Hussain ◽  
Poom Kumam ◽  
Muhammad Asif Jan
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Quadrature Rule ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Haar Wavelets ◽  
Test Problems ◽  
Algebraic Equations ◽  
Diffusion Wave ◽  
Wave Models

In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. One and two dimensional Haar wavelets are used for the discretization of a spatial operator while time fractional derivative is approximated using second order finite difference and quadrature rule. The scheme has an excellent feature that converts a time fractional partial differential equation to a system of algebraic equations which can be solved easily. The suggested technique is applied to solve some test problems. The obtained results have been compared with existing results in the literature. Also, the accuracy of the scheme has been checked by computing L 2 and L ∞ error norms. Computations validate that the proposed method produces good results, which are comparable with exact solutions and those presented before.

scholarly journals Parametric instability in a free-evolving warped protoplanetary disc

2020 ◽  
Vol 500 (3) ◽  
pp. 4248-4256
Author(s):  
Hongping Deng ◽  
Gordon I Ogilvie ◽  
Lucio Mayer
Keyword(s):  
Wave Equations ◽  
Hydrodynamic Instability ◽  
Parametric Instability ◽  
Low Viscosity ◽  
Crossing Time ◽  
Particle Hydrodynamics ◽  
Smoothed Particle ◽  
First Time ◽  
Bending Wave ◽  
Grid Based

ABSTRACT Warped accretion discs of low viscosity are prone to hydrodynamic instability due to parametric resonance of inertial waves as confirmed by local simulations. Global simulations of warped discs, using either smoothed particle hydrodynamics or grid-based codes, are ubiquitous but no such instability has been seen. Here, we utilize a hybrid Godunov-type Lagrangian method to study parametric instability in global simulations of warped Keplerian discs at unprecedentedly high resolution (up to 120 million particles). In the global simulations, the propagation of the warp is well described by the linear bending-wave equations before the instability sets in. The ensuing turbulence, captured for the first time in a global simulation, damps relative orbital inclinations and leads to a decrease in the angular momentum deficit. As a result, the warp undergoes significant damping within one bending-wave crossing time. Observed protoplanetary disc warps are likely maintained by companions or aftermath of disc breaking.

Sign in / Sign up

Export Citation Format

Share Document