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.

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 Thetis coastal ocean model: discontinuous Galerkin discretization for the three-dimensional hydrostatic equations

2018 ◽  
Author(s):  
Tuomas Kärnä ◽  
Stephan C. Kramer ◽  
Lawrence Mitchell ◽  
David A. Ham ◽  
Matthew D. Piggott ◽  
...  
Keyword(s):  
Discontinuous Galerkin ◽  
Unstructured Grid ◽  
Splitting Method ◽  
Time Integration ◽  
River Estuary ◽  
Coastal Ocean ◽  
Second Order ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Element Discretization

Abstract. Unstructured grid ocean models are advantageous for simulating the coastal ocean and river-estuary-plume systems. However, unstructured grid models tend to be diffusive and/or computationally expensive which limits their applicability to real life problems. In this paper, we describe a novel discontinuous Galerkin (DG) finite element discretization for the hydrostatic equations. The formulation is fully conservative and second-order accurate in space and time. Monotonicity of the advection scheme is ensured by using a strong stability preserving time integration method and slope limiters. Compared to previous DG models advantages include a more accurate mode splitting method, revised viscosity formulation, and new second-order time integration scheme. We demonstrate that the model is capable of simulating baroclinic flows in the eddying regime with a suite of test cases. Numerical dissipation is well-controlled, being comparable or lower than in existing state-of-the-art structured grid models.

A Novel Sub-Stepping Method with Numerical Dissipation Control for Time Integration of Highly Flexible Structures

2018 ◽  
Vol 10 (10) ◽  
pp. 1850106 ◽  
Author(s):  
Saeed Mohammadzadeh ◽  
Mehdi Ghassemieh
Keyword(s):  
Equilibrium Points ◽  
Flexible Structures ◽  
Time Integration ◽  
Single Step ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Step Method ◽  
Time Step ◽  
Time Increment ◽  
Time Integration Scheme

Sub-stepping time integration methods attempt to march each time step with multiple sub-steps. Generally, for the first sub-step, a single-step method is applied and the following sub-steps are solved using a method that utilizes the data obtained from two or three previous equilibrium points. Despite the robust stability in problems, control of numerical dissipation in sub-stepping schemes is a tough task due to applying different algorithms on a time increment. In order to overcome this insufficiency, a new sub-stepping time integration scheme, which uses two sub-steps in each time increment, is proposed. Newmark and quadratic acceleration methods are applied on the first and second sub-steps, respectively. Both methods utilize constant parameters that enable the control of numerical dissipation in the analysis. For the proposed method, the stability analysis revealed the unconditional stability region for the pertinent parameters. Additionally, the precision investigation disclosed an advantage of the proposed method with the presence of minor period elongation error. Finally, the application of the proposed method is illuminated via several numerical examples. In addition to numerical dissipation control, the proposed method proved to have an outstanding advantage over other methods in solving highly flexible structures more efficiently and more accurately.

UNSTEADY FLOW COMPUTATIONS OVER MOVING BODY USING DYNAMIC MESHES

2004 ◽  
Vol 01 (03) ◽  
pp. 507-518
Author(s):  
J. C. MANDAL ◽  
J. BALLMANN
Keyword(s):  
Compressible Flows ◽  
Time Integration ◽  
Time Discretization ◽  
Second Order ◽  
Integration Scheme ◽  
Arbitrary Lagrangian Eulerian ◽  
Time Step ◽  
Naca0012 Airfoil ◽  
Moving Body ◽  
Time Integration Scheme

An efficient implicit unstructured grid algorithm for solving unsteady inviscid compressible flows over moving body employing an Arbitrary Lagrangian Eulerian formulation is presented. In the present formulation, the time discretization is performed using a second-order accurate 3-point time integration scheme and the upwind-biased space discretization using second-order accurate finite volume formulation with Venkatakrishnan limiter. The face-velocities of the control volumes are computed using Geometric Conservation Laws. The nonlinear system arising from the implicit formulation is solved using an ILU preconditioned Newton–Krylov iteration at every time step. The computed results for two test cases involving harmonically oscillating NACA0012 airfoil are presented in order to demonstrate the efficacy of the present solver.

scholarly journals One-Parameter Controlled Non-Dissipative Unconditionally Stable Explicit Structure-Dependent Integration Methods with No Overshoot

2021 ◽  
Vol 11 (24) ◽  
pp. 12109
Author(s):  
Veerarajan Selvakumar ◽  
Shuenn-Yih Chang
Keyword(s):  
Time Integration ◽  
Unconditional Stability ◽  
Second Order ◽  
Numerical Dissipation ◽  
Implicit Methods ◽  
Time Step ◽  
Order Accuracy ◽  
Structural Dynamic ◽  
Integration Methods ◽  
New Family

Although many families of integration methods have been successfully developed with desired numerical properties, such as second order accuracy, unconditional stability and numerical dissipation, they are generally implicit methods. Thus, an iterative procedure is often involved for each time step in conducting time integration. Many computational efforts will be consumed by implicit methods when compared to explicit methods. In general, the structure-dependent integration methods (SDIMs) are very computationally efficient for solving a general structural dynamic problem. A new family of SDIM is proposed. It exhibits the desired numerical properties of second order accuracy, unconditional stability, explicit formulation and no overshoot. The numerical properties are controlled by a single free parameter. The proposed family method generally has no adverse disadvantage of unusual overshoot in high frequency transient responses that have been found in the currently available implicit integration methods, such as the WBZ-α method, HHT-α method and generalized-α method. Although this family method has unconditional stability for the linear elastic and stiffness softening systems, it becomes conditionally stable for stiffness hardening systems. This can be controlled by a stability amplification factor and its unconditional stability is successfully extended to stiffness hardening systems. The computational efficiency of the proposed method proves that engineers can do the accurate nonlinear analysis very quickly.

scholarly journals Thetis coastal ocean model: discontinuous Galerkin discretization for the three-dimensional hydrostatic equations

2018 ◽  
Vol 11 (11) ◽  
pp. 4359-4382 ◽  
Author(s):  
Tuomas Kärnä ◽  
Stephan C. Kramer ◽  
Lawrence Mitchell ◽  
David A. Ham ◽  
Matthew D. Piggott ◽  
...  
Keyword(s):  
Discontinuous Galerkin ◽  
Unstructured Grid ◽  
Splitting Method ◽  
Time Integration ◽  
River Estuary ◽  
Coastal Ocean ◽  
Second Order ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Element Discretization

Abstract. Unstructured grid ocean models are advantageous for simulating the coastal ocean and river–estuary–plume systems. However, unstructured grid models tend to be diffusive and/or computationally expensive, which limits their applicability to real-life problems. In this paper, we describe a novel discontinuous Galerkin (DG) finite element discretization for the hydrostatic equations. The formulation is fully conservative and second-order accurate in space and time. Monotonicity of the advection scheme is ensured by using a strong stability-preserving time integration method and slope limiters. Compared to previous DG models, advantages include a more accurate mode splitting method, revised viscosity formulation, and new second-order time integration scheme. We demonstrate that the model is capable of simulating baroclinic flows in the eddying regime with a suite of test cases. Numerical dissipation is well-controlled, being comparable or lower than in existing state-of-the-art structured grid models.

scholarly journals Performance of Composite Implicit Time Integration Scheme for Nonlinear Dynamic Analysis

2008 ◽  
Vol 2008 ◽  
pp. 1-16 ◽  
Author(s):  
William Taylor Matias Silva ◽  
Luciano Mendes Bezerra
Keyword(s):  
Transient Response ◽  
Time Integration ◽  
Nonlinear Dynamic Analysis ◽  
Implicit Time ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Time Step ◽  
Structural Dynamic ◽  
Implicit Time Integration ◽  
Time Integration Scheme

This paper presents a simple implicit time integration scheme for transient response solution of structures under large deformations and long-time durations. The authors focus on a practical method using implicit time integration scheme applied to structural dynamic analyses in which the widely used Newmark time integration procedure is unstable, and not energy-momentum conserving. In this integration scheme, the time step is divided in two substeps. For too large time steps, the method is stable but shows excessive numerical dissipation. The influence of different substep sizes on the numerical dissipation of the method is studied throughout three practical examples. The method shows good performance and may be considered good for nonlinear transient response of structures.

On the Friction Coefficient in the Numerical Simulation of Tidal Wave Propagation

2010 ◽  
Author(s):  
Z. Y. Song ◽  
C. Cheng ◽  
F. M. Xu ◽  
J. Kong
Keyword(s):  
Numerical Simulation ◽  
Wave Propagation ◽  
Friction Factor ◽  
Tidal Wave ◽  
Computational Time ◽  
Numerical Dissipation ◽  
Courant Number ◽  
Time Step ◽  
Large Time Step ◽  
The Relationship

Based on the analytical solution of one-dimensional simplified equation of damping tidal wave and Heuristic stability analysis, the precision of numerical solution, computational time and the relationship between the numerical dissipation and the friction dissipation are discussed with different numerical schemes in this paper. The results show that (1) when Courant number is less than unity, the explicit solution of tidal wave propagation has higher precision and requires less computational time than the implicit one; (2) large time step is allowed in the implicit scheme in order to reduce the computational time, but the precision of the solution also reduce and the calculation precision should be guaranteed by reducing the friction factor: (3) the friction factor in the implicit solution is related to Courant number, presented as the determined friction factor is smaller than the natural value when Courant number is larger than unity, and their relationship formula is given from the theoretical analysis and the numerical experiments. These results have important application value for the numerical simulation of the tidal wave.

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.

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.

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