A Method of Improving Time Integration Algorithm Accuracy for Long-Term Dynamic Simulation

2020 ◽  
Vol 20 (07) ◽  
pp. 2050079
Author(s):  
Shi Li ◽  
Leibo Qin ◽  
Hongchao Guo ◽  
Dixiong Yang
Keyword(s):  
Mass Matrix ◽  
Time Integration ◽  
Numerical Dispersion ◽  
Scaling Method ◽  
Integration Algorithm ◽  
Original Algorithm ◽  
Mass Scaling ◽  
Scaling Matrix ◽  
Period Elongation

This paper aims to improve the accuracy of time integration algorithms (TIAs) for long-term simulation of structural dynamics. To this end, a new method of reducing the period elongation (numerical dispersion) was proposed by utilizing the mass scaling matrix. Firstly, the period elongation of explicit Gui-[Formula: see text] algorithm as a representative was analyzed, and the strategy of enhancing the algorithm accuracy was investigated. Subsequently, the period elongation was reduced by introducing a parameter to change the mass matrix, which is weighted by the original mass matrix and stiffness matrix. The bisection method is utilized to determine the parameter according to the formulation of period elongation. Since just the mass matrix of original algorithm is changed slightly, the convergence rate of original TIA remains unchanged and the proposed mass scaling method imposes little influence on the stability condition of original algorithm. Moreover, this method has few modifications to the computer program of TIA and is easy to implement. Finally, both linear and nonlinear long-term dynamic response analyses for multiple-degree-of-freedom systems indicated that the proposed mass scaling method is effective and convenient to reduce the period elongation for several typical TIAs.

Symplectic interior penalty discontinuous Galerkin method for solving the seismic scalar wave equation

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. T133-T145 ◽  
Author(s):  
Xijun He ◽  
Dinghui Yang ◽  
Xiao Ma ◽  
Yanjie Zhou
Keyword(s):  
Wave Equation ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Time Integration ◽  
High Accuracy ◽  
High Order ◽  
Numerical Dispersion ◽  
Scalar Wave ◽  
Long Time

To improve the computational accuracy and efficiency of long-time wavefield simulations, we have developed a so-called symplectic interior penalty discontinuous Galerkin (IPDG) method for 2D acoustic equation. For the symplectic IPDG method, the scalar wave equation is first transformed into a Hamiltonian system. Then, the high-order IPDG formulations are introduced for spatial discretization because of their high accuracy and ease of dealing with computational domains with complex boundaries. The time integration is performed using an explicit third-order symplectic partitioned Runge-Kutta scheme so that it preserves the Hamiltonian structure of the wave equation in long-term simulations. Consequently, the symplectic IPDG method combines the advantages of discontinuous Galerkin method and the symplectic time integration. We investigate the properties of the method in detail for high-order spatial basis functions, including the stability criteria, numerical dispersion and dissipation relationships, and numerical errors. The analyses indicate that the symplectic SIPG method is nondissipative and retains low numerical dispersion. We also find that different symplectic IPDG methods have different convergence behaviors. It is indicated that using coarse meshes with a high-order method produces smaller errors and retains high accuracy. We have applied our method to simulate the scalar wavefields for different models, including layered models, a rough topography model, and the Marmousi model. The numerical results show that the symplectic IPDG method can suppress numerical dispersion effectively and provide accurate information on the wavefields. We also conduct a long-term experiment that verifies the capability of symplectic IPDG method for long-time simulations.

scholarly journals Modeling Structural Dynamics Using FE-Meshfree QUAD4 Element with Radial-Polynomial Basis Functions

Materials ◽  
2021 ◽  
Vol 14 (9) ◽  
pp. 2288
Author(s):  
Hongming Luo ◽  
Guanhua Sun
Keyword(s):  
Structural Dynamics ◽  
Interpolation Method ◽  
Mass Matrix ◽  
Time Integration ◽  
Partition Of Unity ◽  
Implicit Time ◽  
Lumped Mass ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
Consistent Mass Matrix

The PU (partition-of-unity) based FE-RPIM QUAD4 (4-node quadrilateral) element was proposed for statics problems. In this element, hybrid shape functions are constructed through multiplying QUAD4 shape function with radial point interpolation method (RPIM). In the present work, the FE-RPIM QUAD4 element is further applied for structural dynamics. Numerical examples regarding to free and forced vibration analyses are presented. The numerical results show that: (1) If CMM (consistent mass matrix) is employed, the FE-RPIM QUAD4 element has better performance than QUAD4 element under both regular and distorted meshes; (2) The DLMM (diagonally lumped mass matrix) can supersede the CMM in the context of the FE-RPIM QUAD4 element even for the scheme of implicit time integration.

Direct Sensitivity Analysis of Multibody Systems: A Vehicle Dynamics Benchmark

2019 ◽  
Vol 14 (2) ◽  
Author(s):  
Alfonso Callejo ◽  
Daniel Dopico
Keyword(s):  
Sensitivity Analysis ◽  
Multibody Systems ◽  
Automatic Differentiation ◽  
Time Integration ◽  
General Purpose ◽  
Contact Forces ◽  
Design Parameters ◽  
Computational Techniques ◽  
Computationally Efficient ◽  
Integration Algorithm

Algorithms for the sensitivity analysis of multibody systems are quickly maturing as computational and software resources grow. Indeed, the area has made substantial progress since the first academic methods and examples were developed. Today, sensitivity analysis tools aimed at gradient-based design optimization are required to be as computationally efficient and scalable as possible. This paper presents extensive verification of one of the most popular sensitivity analysis techniques, namely the direct differentiation method (DDM). Usage of such method is recommended when the number of design parameters relative to the number of outputs is small and when the time integration algorithm is sensitive to accumulation errors. Verification is hereby accomplished through two radically different computational techniques, namely manual differentiation and automatic differentiation, which are used to compute the necessary partial derivatives. Experiments are conducted on an 18-degree-of-freedom, 366-dependent-coordinate bus model with realistic geometry and tire contact forces, which constitutes an unusually large system within general-purpose sensitivity analysis of multibody systems. The results are in good agreement; the manual technique provides shorter runtimes, whereas the automatic differentiation technique is easier to implement. The presented results highlight the potential of manual and automatic differentiation approaches within general-purpose simulation packages, and the importance of formulation benchmarking.

scholarly journals A Hybrid Analytics Paradigm Combining Physics-Based Modeling and Data-Driven Modeling to Accelerate Incompressible Flow Solvers

Fluids ◽  
2018 ◽  
Vol 3 (3) ◽  
pp. 50 ◽  
Author(s):  
Sk. Rahman ◽  
Adil Rasheed ◽  
Omer San
Keyword(s):  
Poisson Equation ◽  
Incompressible Flow ◽  
Data Exchange ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
Time Integration ◽  
Data Driven ◽  
Integration Algorithm ◽  
Reduced Order ◽  
Advection Diffusion Equation

Numerical solution of the incompressible Navier–Stokes equations poses a significant computational challenge due to the solenoidal velocity field constraint. In most computational modeling frameworks, this divergence-free constraint requires the solution of a Poisson equation at every step of the underlying time integration algorithm, which constitutes the major component of the computational expense. In this study, we propose a hybrid analytics procedure combining a data-driven approach with a physics-based simulation technique to accelerate the computation of incompressible flows. In our approach, proper orthogonal basis functions are generated to be used in solving the Poisson equation in a reduced order space. Since the time integration of the advection–diffusion equation part of the physics-based model is computationally inexpensive in a typical incompressible flow solver, it is retained in the full order space to represent the dynamics more accurately. Encoder and decoder interface conditions are provided by incorporating the elliptic constraint along with the data exchange between the full order and reduced order spaces. We investigate the feasibility of the proposed method by solving the Taylor–Green vortex decaying problem, and it is found that a remarkable speed-up can be achieved while retaining a similar accuracy with respect to the full order model.

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.

Impact Dynamic Analysis of Horizontally Vibrated Conveyor with Coulomb Friction Modeling

2012 ◽  
Vol 619 ◽  
pp. 26-29
Author(s):  
Chao Sheng Song ◽  
Qi Ming Huang ◽  
Zhan Gao ◽  
Jie Xu
Keyword(s):  
Coulomb Friction ◽  
Impact Analysis ◽  
Time Integration ◽  
Impact Excitation ◽  
Friction Modeling ◽  
Integration Algorithm ◽  
Conveyor System ◽  
Time Integration Algorithm ◽  
And Control ◽  
The Impact

This paper introduces dynamic impact analysis as an effective technique for studying the response of horizontal vibrated conveyor with time-varying impact excitation by the falling of the scrap. A two degree-of-freedoms impact dynamic model is formulated considering the static and dynamic coulomb friction between the scrap and chute. Then the time integration algorithm was applied in the program to solve the dynamic equations. Using the proposed method, the impact effects of ideal single scrap and multiple scraps on the dynamic response of the conveyor were analyzed. Computational results reveal numerous interesting dynamic characteristics which can be used to forecast and control the vibration of the scrap and conveyor system.

Energy-Momentum Conserving Time Integration of Modally Reduced Flexible Multibody Systems

2013 ◽  
Author(s):  
Alexander Humer ◽  
Johannes Gerstmayr
Keyword(s):  
Angular Momentum ◽  
Multibody Systems ◽  
Mass Matrix ◽  
Time Integration ◽  
Flexible Multibody Dynamics ◽  
Integration Scheme ◽  
Integration Schemes ◽  
Flexible Multibody ◽  
Floating Frame ◽  
Energy And Momentum

Many conventional time integration schemes frequently adopted in flexible multibody dynamics fail to retain the fundamental conservation laws of energy and momentum of the continuous time domain. Lack of conservation, however, in particular of angular momentum, may give rise to unexpected, unphysical results. To avoid such problems, a scheme for the consistent integration of modally reduced multibody systems subjected to holonomic constraints is developed in the present paper. As opposed to the conventional approach, in which the floating frame of reference formulation is combined with component mode synthesis for approximating the flexible deformation, an alternative, recently proposed formulation based on absolute coordinates is adopted in the analysis. Owing to the linear relationship between the generalized coordinates and the absolute displacement, the inertia terms in the equations of motion attain a very simple structure. The mass matrix remains independent of the current state of deformation and the velocity dependent term known from the floating frame approach vanishes due to the absence of relative coordinates. These advantageous properties facilitate the construction of an energy and momentum consistent integration scheme. By the mid-point rule, algorithmic conservation of both linear and angular momentum is achieved. In order to consistently integrate the total energy of the system, the discrete derivative needs to be adopted when evaluating the strain energy gradient and the derivative of the algebraic constraint equations.

scholarly journals Time Integration and Assessment of a Model for Shape Memory Alloys Considering Multiaxial Nonproportional Loading Cases

2014 ◽  
Author(s):  
Wael Zaki ◽  
Xiaojun Gu ◽  
Claire Morin ◽  
Ziad Moumni ◽  
Weihong Zhang
Keyword(s):  
Shape Memory Alloys ◽  
Shape Memory ◽  
Time Integration ◽  
Implicit Integration ◽  
Proportional Loading ◽  
Simulation Data ◽  
Integration Algorithm ◽  
Implicit Integration Algorithm ◽  
User Material Subroutine ◽  
Martensite Reorientation

The paper presents a numerical implementation of the ZM model for shape memory alloys that fully accounts for non-proportional loading and its influence on martensite reorientation and phase transformation. Derivation of the time-discrete implicit integration algorithm is provided. The algorithm is used for finite element simulations using Abaqus, in which the model is implemented by means of a user material subroutine. The simulations are shown to agree with experimental and numerical simulation data taken from the literature.

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