A Simple Explicit–Implicit Time-Marching Technique for Wave Propagation Analysis

2018 ◽  
Vol 16 (01) ◽  
pp. 1850082 ◽  
Author(s):  
Delfim Soares
Keyword(s):  
Wave Propagation ◽  
Time Integration ◽  
Critical Time ◽  
Implicit Time ◽  
Time Step ◽  
Stiffness Matrices ◽  
Time Integrators ◽  
Propagation Analysis ◽  
Time Marching ◽  
Critical Time Step

A new explicit–implicit time integration technique is proposed here for wave propagation analysis. In the present formulation, the time integrators of the model are selected at the element level, allowing each element to be considered as explicit or implicit. In the implicit elements, controllable algorithm dissipation is provided, enabling an [Formula: see text]-stable formulation. In the explicit elements, null amplitude decay is considered, enabling maximal critical time-step values. The new methodology renders a very simple and effective time-marching algorithm. Here, just displacement–velocity relations are considered, and no computation of accelerations is required. Moreover, explicit/implicit analyses can be carried out just by the tuning of local effective matrices, inputting or not stiffness matrices into their computations. At the end of the paper, numerical results are presented, illustrating the performance and potentialities of the new method.

scholarly journals A Weighted Residual Parabolic Acceleration Time Integration Method for Problems in Structural Dynamics

2007 ◽  
Vol 7 (3) ◽  
pp. 227-238 ◽  
Author(s):  
S.H. Razavi ◽  
A. Abolmaali ◽  
M. Ghassemieh
Keyword(s):  
Fourth Order ◽  
Initial Conditions ◽  
Time Integration ◽  
Critical Time ◽  
Linear Acceleration ◽  
Time Step ◽  
Time Integration Method ◽  
Acceleration Method ◽  
The Stability ◽  
Critical Time Step

AbstractIn the proposed method, the variation of displacement in each time step is assumed to be a fourth order polynomial in time and its five unknown coefficients are calculated based on: two initial conditions from the previous time step; satisfying the equation of motion at both ends of the time step; and the zero weighted residual within the time step. This method is non-dissipative and its dispersion is considerably less than in other popular methods. The stability of the method shows that the critical time step is more than twice of that for the linear acceleration method and its convergence is of fourth order.

scholarly journals The effects of element shape on the critical time step in explicit time integrators for elasto-dynamics

2014 ◽  
Vol 101 (11) ◽  
pp. 809-824 ◽  
Author(s):  
Harm Askes ◽  
Antonio Rodríguez-Ferran ◽  
Jack Hetherington
Keyword(s):  
Critical Time ◽  
Time Step ◽  
Time Integrators ◽  
Critical Time Step ◽  
Element Shape

scholarly journals Bipenalty method for time domain computational dynamics

2009 ◽  
Vol 466 (2117) ◽  
pp. 1389-1408 ◽  
Author(s):  
Harm Askes ◽  
Miguel Caramés-Saddler ◽  
Antonio Rodríguez-Ferran
Keyword(s):  
Time Integration ◽  
Critical Time ◽  
Time Step ◽  
Beam Elements ◽  
Computational Dynamics ◽  
Integration Schemes ◽  
Time Integration Schemes ◽  
The Common ◽  
Critical Time Step ◽  
Better Than

Penalty functions can be used to add constraints to systems of equations. In computational mechanics, stiffness-type penalties are the common choice. However, in dynamic applications stiffness penalties have the disadvantage that they tend to decrease the critical time step in conditionally stable time integration schemes, leading to increased CPU times for simulations. In contrast, inertia penalties increase the critical time step. In this paper, we suggest the simultaneous use of stiffness penalties and inertia penalties, which is denoted as the bipenalty method . We demonstrate that the accuracy of the bipenalty method is at least as good as (and usually better than) using either stiffness penalties or inertia penalties. Furthermore, for a number of finite elements (bar elements, beam elements and square plane stress/plane strain elements) we have derived ratios of the two penalty parameters such that their combined effect on the critical time step is neutral. The bipenalty method is thus superior to using stiffness penalties, because the decrease in critical time step can be avoided. The bipenalty method is also superior to using inertia penalties, since the constraints are realized with higher accuracy.

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.

scholarly journals A novel semi-implicit scheme for elastodynamics and wave propagation in nearly and truly incompressible solids

2021 ◽  
Author(s):  
Chennakesava Kadapa
Keyword(s):  
Wave Propagation ◽  
Wave Speed ◽  
Critical Time ◽  
Incompressible Material ◽  
Implicit Scheme ◽  
Bulk Wave ◽  
Lumped Mass ◽  
Time Step ◽  
Material Models ◽  
Shear Wave Speed

AbstractThis paper presents a novel semi-implicit scheme for elastodynamics and wave propagation problems in nearly and truly incompressible material models. The proposed methodology is based on the efficient computation of the Schur complement for the mixed displacement-pressure formulation using a lumped mass matrix for the displacement field. By treating the deviatoric stress explicitly and the pressure field implicitly, the critical time step is made to be limited by shear wave speed rather than the bulk wave speed. The convergence of the proposed scheme is demonstrated by computing error norms for the recently proposed LBB-stable BT2/BT1 element. Using the numerical examples modelled with nearly and truly incompressible Neo-Hookean and Ogden material models, it is demonstrated that the proposed semi-implicit scheme yields significant computational benefits over the fully explicit and the fully implicit schemes for finite strain elastodynamics simulations involving incompressible materials. Finally, the applicability of the proposed scheme for wave propagation problems in nearly and truly incompressible material models is illustrated.

scholarly journals Investigation of Air Pocket Behavior in Pipelines Using Rigid Column Model and Contributions of Time Integration Schemes

Water ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 785
Author(s):  
Arman Rokhzadi ◽  
Musandji Fuamba
Keyword(s):  
Transient Flow ◽  
Time Integration ◽  
Driving Pressure ◽  
Implicit Time ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Time Step ◽  
Column Model ◽  
Integration Schemes ◽  
Backward Euler

This paper studies the air pressurization problem caused by a partially pressurized transient flow in a reservoir-pipe system. The purpose of this study is to analyze the performance of the rigid column model in predicting the attenuation of the air pressure distribution. In this regard, an analytic formula for the amplitude and frequency will be derived, in which the influential parameters, particularly, the driving pressure and the air and water lengths, on the damping can be seen. The direct effect of the driving pressure and inverse effect of the product of the air and water lengths on the damping will be numerically examined. In addition, these numerical observations will be examined by solving different test cases and by comparing to available experimental data to show that the rigid column model is able to predict the damping. However, due to simplified assumptions associated with the rigid column model, the energy dissipation, as well as the damping, is underestimated. In this regard, using the backward Euler implicit time integration scheme, instead of the classical fourth order explicit Runge–Kutta scheme, will be proposed so that the numerical dissipation of the backward Euler implicit scheme represents the physical dissipation. In addition, a formula will be derived to calculate the appropriate time step size, by which the dissipation of the heat transfer can be compensated.

scholarly journals A Semi-Implicit Version of the MPAS-Atmosphere Dynamical Core

2015 ◽  
Vol 143 (9) ◽  
pp. 3838-3855 ◽  
Author(s):  
Steven Sandbach ◽  
John Thuburn ◽  
Danail Vassilev ◽  
Michael G. Duda
Keyword(s):  
Time Integration ◽  
Standard Test ◽  
Newton Iteration ◽  
Atmospheric Modeling ◽  
Implicit Time ◽  
Integration Scheme ◽  
Baroclinic Wave ◽  
Time Step ◽  
Integration Schemes ◽  
Helmholtz Problem

Abstract An important question for atmospheric modeling is the viability of semi-implicit time integration schemes on massively parallel computing architectures. Semi-implicit schemes can provide increased stability and accuracy. However, they require the solution of an elliptic problem at each time step, creating concerns about their parallel efficiency and scalability. Here, a semi-implicit (SI) version of the Model for Prediction Across Scales (MPAS) is developed and compared with the original model version, which uses a split Runge–Kutta (SRK3) time integration scheme. The SI scheme is based on a quasi-Newton iteration toward a Crank–Nicolson scheme. Each Newton iteration requires the solution of a Helmholtz problem; here, the Helmholtz problem is derived, and its solution using a geometric multigrid method is described. On two standard test cases, a midlatitude baroclinic wave and a small-planet nonhydrostatic gravity wave, the SI and SRK3 versions produce almost identical results. On the baroclinic wave test, the SI version can use somewhat larger time steps (about 60%) than the SRK3 version before losing stability. The SI version costs 10%–20% more per step than the SRK3 version, and the weak and strong scalability characteristics of the two versions are very similar for the processor configurations the authors have been able to test (up to 1920 processors). Because of the spatial discretization of the pressure gradient in the lowest model layer, the SI version becomes unstable in the presence of realistic orography. Some further work will be needed to demonstrate the viability of the SI scheme in this case.

A Smoothing Scheme for Seismic Wave Propagation Simulation with a Mixed FEM of Diffusionized Wave Equation

2021 ◽  
pp. 2150061
Author(s):  
Ryuta Imai ◽  
Naoki Kasui ◽  
Masayuki Yamada ◽  
Koji Hada ◽  
Hiroyuki Fujiwara
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Seismic Wave ◽  
Time Integration ◽  
Coupled System ◽  
Seismic Wave Propagation ◽  
Implicit Time ◽  
Integration Scheme ◽  
Weak Formulation ◽  
Mixed Fem

In this paper, we propose a smoothing scheme for seismic wave propagation simulation. The proposed scheme is based on a diffusionized wave equation with the fourth-order spatial derivative term. So, the solution requires higher regularity in the usual weak formulation. Reducing the diffusionized wave equation to a coupled system of diffusion equations yields a mixed FEM to ease the regularity. We mathematically explain how our scheme works for smoothing. We construct a semi-implicit time integration scheme and apply it to the wave equation. This numerical experiment reveals that our scheme is effective for filtering short wavelength components in seismic wave propagation simulation.

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