scholarly journals Computationally Efficient Solution of a 2D Diffusive Wave Equation Used for Flood Inundation Problems

Water ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 2195 ◽  
Author(s):  
Wojciech Artichowicz ◽  
Dariusz Gąsiorowski
Keyword(s):  
Wave Equation ◽  
Parallel Implementation ◽  
Time Integration ◽  
Computational Time ◽  
System Of Nonlinear Equations ◽  
Computationally Efficient ◽  
Time Step ◽  
Finite Element Scheme ◽  
Numerical Errors ◽  
Floodplain Inundation

This paper presents a study dealing with increasing the computational efficiency in modeling floodplain inundation using a two-dimensional diffusive wave equation. To this end, the domain decomposition technique was used. The resulting one-dimensional diffusion equations were approximated in space with the modified finite element scheme, whereas time integration was carried out using the implicit two-level scheme. The proposed algorithm of the solution minimizes the numerical errors and is unconditionally stable. Consequently, it is possible to perform computations with a significantly greater time step than in the case of the explicit scheme. An additional efficiency improvement was achieved using the symmetry of the tridiagonal matrix of the arising system of nonlinear equations, due to the application of the parallelization strategy. The computational experiments showed that the proposed parallel implementation of the implicit scheme is very effective, at about two orders of magnitude with regard to computational time, in comparison with the explicit one.

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.

Newmark-Beta Method in Discrete Elastic Rods Algorithm to Avoid Energy Dissipation

2019 ◽  
Vol 86 (8) ◽  
Author(s):  
Weicheng Huang ◽  
Mohammad Khalid Jawed
Keyword(s):  
Energy Dissipation ◽  
Time Integration ◽  
Integration Scheme ◽  
Integration Step ◽  
Elastic Rods ◽  
Computationally Efficient ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Time Integration Scheme

Discrete elastic rods (DER) algorithm presents a computationally efficient means of simulating the geometrically nonlinear dynamics of elastic rods. However, it can suffer from artificial energy loss during the time integration step. Our approach extends the existing DER technique by using a different time integration scheme—we consider a second-order, implicit Newmark-beta method to avoid energy dissipation. This treatment shows better convergence with time step size, specially when the damping forces are negligible and the structure undergoes vibratory motion. Two demonstrations—a cantilever beam and a helical rod hanging under gravity—are used to show the effectiveness of the modified discrete elastic rods simulator.

Studies on the Accuracy Stability and Efficiency of a New Time Integration Scheme for Ballast Modeling Using the Discrete Element Method

2014 ◽  
Author(s):  
G. F. Mathews ◽  
R. L. Mullen ◽  
D. C. Rizos
Keyword(s):  
Particle Interaction ◽  
Time Integration ◽  
Discrete Element ◽  
Critical Discussion ◽  
Implicit Time ◽  
Integration Scheme ◽  
Computationally Efficient ◽  
Time Step ◽  
Central Difference ◽  
Time Integration Scheme

This paper presents the development of a semi-implicit time integration scheme, originally developed for structural dynamics in the 1970’s, and its implementation for use in Discrete Element Methods (DEM) for rigid particle interaction, and interaction of elastic bodies that are modeled as a cluster of rigid interconnected particles. The method is developed in view of ballast modeling that accounts for the flexibility of aggregates and the arbitrary shape and size of granules. The proposed scheme does not require any matrix inversions and is expressed in an incremental form making it appropriate for non-linear problems. The proposed method focuses on improving the efficiency, stability and accuracy of the solutions, as compared to current practice. A critical discussion of the findings of the studies is presented. Extended verification and assessment studies demonstrate that the proposed algorithm is unconditionally stable and accurate even for large time step sizes. It is demonstrated that the proposed method is at least as computationally efficient as the Central Difference Method. Guidelines for the implementation of the method to ballast modeling are discussed.

Modified precise direct time integration method for the transient response analysis of viscoelastic systems using an internal variable model

2019 ◽  
Vol 26 (3-4) ◽  
pp. 161-174
Author(s):  
Taufeeq Ur Rehman Abbasi ◽  
Hui Zheng
Keyword(s):  
State Space ◽  
Integration Method ◽  
Time Integration ◽  
Internal Variable ◽  
New Method ◽  
Computational Time ◽  
Direct Integration ◽  
Computational Accuracy ◽  
Time Step ◽  
Time Integration Method

Engineering systems for different levels of energy dissipation use internal variable models, which may lead to tremendous problems in accurate analysis. This article aims to provide an alternative direct integration method for the analysis of systems involving an anelastic displacement field model. A new state-space formulation built on an augmented set of anelastic variables for asymmetric systems is developed. Then, a precise time integration method based on state-space matrix formulation is proposed by introducing a Legendre–Gauss quadrature. The new integration method in terms of numerical stability and its implementation is discussed. The effect of sensitivity of the selection of the time-step and computational time on the performance of the new method is investigated by using a multi-degree-of-freedom system. The performance of the new method is also evaluated in terms of both computational accuracy and efficiency at higher degrees of freedom by using a continuum system. It is demonstrated that the computational accuracy and efficiency of the new method on large-scale problems are higher than that of the direct integration linear displacement–velocity method.

Comparisons of Structure-Dependent Explicit Methods for Time Integration

2015 ◽  
Vol 15 (03) ◽  
pp. 1450055 ◽  
Author(s):  
Shuenn-Yih Chang
Keyword(s):  
Time Integration ◽  
Low Frequency ◽  
Unconditional Stability ◽  
Explicit Methods ◽  
Explicit Method ◽  
Computationally Efficient ◽  
Total Response ◽  
Time Step ◽  
Order Accuracy ◽  
Practical Applications

Chang explicit method (CEM)1,2 and CR explicit method3 (CRM) are two structure-dependent explicit methods that have been successfully developed for structural dynamics. The most important property of both integration methods is that they involve no nonlinear iterations in addition to unconditional stability and second-order accuracy. Thus, they are very computationally efficient for solving inertial problems, where the total response is dominated by low frequency modes. However, an unusual overshooting behavior for CR explicit method is identified herein and thus its practical applications might be largely limited although its velocity computing for each time step is much easier than for the CEM.

Second-order time integration of the wave equation with dispersion correction procedures

Geophysics ◽  
2019 ◽  
Vol 84 (4) ◽  
pp. T221-T235
Author(s):  
Rune Mittet
Keyword(s):  
Wave Equation ◽  
Time Domain ◽  
Fourier Transforms ◽  
Time Integration ◽  
Second Order ◽  
Source Distribution ◽  
Time Step ◽  
Combined Application ◽  
Simulation Time ◽  
Spatial Source

Second-order time integration of the wave equation is numerically efficient with time steps close to the limit set by the stability criterion. However, dispersion errors over realistic propagation distances are unacceptable with time steps in this range. Dispersion-free results, using second-order time integration of the wave equation, can be achieved by applying a time-domain prepropagation filter to all source time functions followed by a time-domain postpropagation filter applied to the simulated wavefield at all recording positions. The time-domain implementation of the filtering process is an order of magnitude more effective, in terms of CPU time, compared to an implementation via discrete and fast Fourier transforms. Pre- and postpropagation filters are valid for any simulation time step. The two filters can be calculated once because they are independent of the simulation time step, and they can be applied with any modeling scheme that uses second-order time integration. Second-order time integration results in traveltime and amplitude errors. The amplitude errors depend on the spatial source distribution. The combined application of the two filters removes traveltime-related errors and amplitude-related errors independently of the spatial source distribution.

A Finite-Element Based Framework for Transient Rotor Dynamic Simulations

2020 ◽  
Author(s):  
Anurag Rajagopal ◽  
Dilip K. Mandal
Keyword(s):  
Finite Element ◽  
Time Integration ◽  
Integration Scheme ◽  
Test Case ◽  
Computationally Efficient ◽  
Time Step ◽  
Dynamic Simulations ◽  
Adaptive Time Step ◽  
Time Integration Scheme ◽  
Rotor Dynamic

Abstract Transient simulations play a key role in the analysis and subsequent design of structural components with one or more rotating parts. A framework is proposed to this effect, centered around the finite-element solver OptiStruct, consisting of a time integration scheme built on the Newmark family with an appropriate adaptive time-step control. The process accounts for a computationally efficient handling of nonlinearities that might arise through bearings and casings. This solution is detailed starting from the governing equations for transient rotor dynamics to the nuances of the time marching scheme, and this process is applied to a test case from which conclusions are drawn that might be of interest to practicing engineers. These conclusions include insights into enforced motion, operation at or near critical speeds, rotor damping and contact. This work is aimed at producing a user-friendly and robust tool and process for the practicing engineer to perform complex rotor dynamic analysis.

scholarly journals Hybrid Kinematic-Dynamic Approach to Seismic Wave-Equation Modeling, Imaging, and Tomography

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Alexandr S. Serdyukov ◽  
Anton A. Duchkov
Keyword(s):  
Mechanical Properties ◽  
Wave Equation ◽  
Efficient Implementation ◽  
Equation Modeling ◽  
Computationally Efficient ◽  
Time Step ◽  
Near Surface ◽  
Structure Response ◽  
Seismic Waveforms ◽  
First Arrival

Estimation of the structure response to seismic motion is an important part of structural analysis related to mitigation of seismic risk caused by earthquakes. Many methods of computing structure response require knowledge of mechanical properties of the ground which could be derived from near-surface seismic studies. In this paper we address computationally efficient implementation of the wave-equation tomography. This method allows inverting first-arrival seismic waveforms for updating seismic velocity model which can be further used for estimating mechanical properties. We present computationally efficient hybrid kinematic-dynamic method for finite-difference (FD) modeling of the first-arrival seismic waveforms. At every time step the FD computations are performed only in a moving narrowband following the first-arrival wavefront. In terms of computations we get two advantages from this approach: computation speedup and memory savings when storing computed first-arrival waveforms (it is not necessary to make calculations or store the complete numerical grid). Proposed approach appears to be specifically useful for constructing the so-called sensitivity kernels widely used for tomographic velocity update from seismic data. We then apply the proposed approach for efficient implementation of the wave-equation tomography of the first-arrival seismic waveforms.

AN ITERATIVE TIME-STEPPING METHOD FOR SOLVING FIRST-ORDER TIME DEPENDENT PROBLEMS AND ITS APPLICATION TO THE WAVE EQUATION

2000 ◽  
Vol 08 (01) ◽  
pp. 241-255 ◽  
Author(s):  
GÉZA SERIANI
Keyword(s):  
Wave Equation ◽  
Time Integration ◽  
Spectral Element ◽  
Implicit Time ◽  
Time Step ◽  
Time Stepping ◽  
Point Scheme ◽  
Time Marching ◽  
Matrix Vector ◽  
Time Dependent Problems

Equations describing dynamic problems, after spatial discretization by using the finite element or spectral element method, lead to solve large systems of ODE in time. A family of new time integration algorithms based on an iterative time-stepping (ITS) approach is proposed for solving these systems. The method is developed for first-and second-order differential equations, and applied, in particular, to wave equation. It is an implicit time marching method in which, at each time-step, the solution is computed by a fixed-point scheme. The analysis show that the method is accurate, unconditionally stable and that it allows for efficient and parallel implementations because no matrix inversion is required and only matrix-vector multiplications and vector scaling operations are involved.

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