scholarly journals Explicit integration of equations of motion solved on computer cluster

2019 ◽  
Author(s):  
Václav Rek
Keyword(s):  
Equations Of Motion ◽  
Explicit Integration ◽  
Computer Cluster

A Sequential Integration Method

1988 ◽  
Vol 110 (4) ◽  
pp. 382-388
Author(s):  
Liang-Wey Chang ◽  
James F. Hamilton
Keyword(s):  
Integration Method ◽  
Equations Of Motion ◽  
Slow Motion ◽  
Explicit Integration ◽  
Time Step ◽  
Lumped Model ◽  
Guaranteed Stability ◽  
Fast Motion ◽  
The Stability ◽  
Secular Terms

This paper presents a method for simulating systems with two inertially coupled motions, i.e., a slow motion and a fast motion. The equations of motion are separated into two sets of coupled nonlinear ordinary differential equations. For each time step, the two sets of equations are integrated sequentially rather than simultaneously. Explicit integration methods are used for integrating the slow motion since the stability of the integration is not a problem and the explicit methods are very convenient for nonlinear equations. For the fast motion, the equations are linear and the implicit integrations can be used with guaranteed stability. The size of time step only needs to be chosen to provide accuracy of the solution for the modes that are excited. The interaction between the two types of motion must be treated such that secular terms do not appear due to the sequential integration method. A lumped model of a flexible pendulum will be presented in this paper to illustrate the application of the method. Numerical results for both simultaneous and sequential integration are presented for comparison.

scholarly journals Inverse Nonlinear Finite Element Methods for Surgery Simulation and Image Guidance

2008 ◽  
Author(s):  
Philip Pratt
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Equations Of Motion ◽  
Nonlinear Finite Element ◽  
Weighted Sums ◽  
Global Coordinate System ◽  
Surgery Simulation ◽  
Explicit Integration ◽  
Time Step ◽  
Nonlinear Finite Element Methods

Nonlinear finite element methods are described in which cyclic organ motion is implied from 4D scan data. The equations of motion corresponding to an explicit integration of the total Lagrangian formulation are reversed, such that the sequence of node forces which produces known changes in displacement is recovered. The forces are resolved from the global coordinate system into systems local to each element, and at every simulation time step are expressed as weighted sums of edge vectors. In the presence of large deformations and rotations, this facilitates the combination of external forces, such as tool-tissue interactions, and also positional constraints. Applications in the areas of surgery simulation and minimally invasive robotic interventions are discussed, and the methods are illustrated using CT images of a pneumatically-operated beating heart phantom.

Mutibody Vehicle Dynamics Analysis Using an Explicit-Implicit Integrator With Subsystem Synthesis Method

2011 ◽  
Author(s):  
Sung-Soo Kim ◽  
Wan Hee Jeong ◽  
Junyoun Jo ◽  
Ji-Hyeun Wang
Keyword(s):  
Vehicle Dynamics ◽  
Integration Method ◽  
Equations Of Motion ◽  
Effective Means ◽  
Synthesis Method ◽  
Implicit Integration ◽  
Explicit Integration ◽  
Virtual Reference ◽  
Computational Speed ◽  
Speed Up

This paper proposes an explicit-implicit numerical integration method in order to apply to multibody vehicle dynamics model based on a subsystem synthesis method. The subsystem synthesis method can provide effective means to independently analyze each subsystem with virtual reference body. In the proposed method, the explicit integration is used for solving the equations of motion for a base body, while the implicit integration is utilized for obtaining the solutions of the equations of motion for each subsystem. For the purpose of the application of the implicit formulas easily, a subsystem synthesis method with the Cartesian coordinates is developed. In order to show the application viability and effectiveness of the proposed method, an extensive comparative study has been performed through simulations. Then, the proposed method is compared to conventional implicit integration method applied to an overall system. When simulating the bump run of a multibody vehicle model with compliance effect such as bushing elements, the proposed method achieves about 2 times computational speed-up. Furthermore, the simulation study reveals that the larger the number of the attached subsystems is, the better the computational efficiency of the proposed method is than that of the conventional implicit integration method.

Microplane Constitutive Model and Computational Framework for Blood Vessel Tissue

2005 ◽  
Vol 128 (3) ◽  
pp. 419-427 ◽  
Author(s):  
Ferhun C. Caner ◽  
Ignacio Carol
Keyword(s):  
Finite Element ◽  
Soft Tissue ◽  
Constitutive Modeling ◽  
Arterial Wall ◽  
Convergence Rates ◽  
Equations Of Motion ◽  
Small Time ◽  
Dynamic Relaxation ◽  
Explicit Integration ◽  
Biological Soft Tissue

This paper presents a nonlinearly elastic anisotropic microplane formulation in 3D for computational constitutive modeling of arterial soft tissue in the passive regime. The constitutive modeling of arterial (and other biological) soft tissue is crucial for accurate finite element calculations, which in turn are essential for design of implants, surgical procedures, bioartificial tissue, as well as determination of effect of progressive diseases on tissues and implants. The model presented is defined at a lower scale (mesoscale) than the conventional macroscale and it incorporates the effect of all the (collagen) fibers which are anisotropic structural components distributed in all directions within the tissue material in addition to that of isotropic bulk tissue. It is shown that the proposed model not only reproduces Holzapfel’s recent model but also improves on it by accounting for the actual three-dimensional distribution of fiber orientation in the arterial wall, which endows the model with advanced capabilities in simulation of remodeling of soft tissue. The formulation is flexible so that its parameters could be adjusted to represent the arterial wall either as a single material or a material composed of several layers in finite element analyses of arteries. Explicit algorithms for both the material subroutine and the explicit integration with dynamic relaxation of equations of motion using finite element method are given. To circumvent the slow convergence of the standard dynamic relaxation and small time steps dictated by the stability of the explicit integrator, an adaptive dynamic relaxation technique that ensures stability and fastest possible convergence rates is developed. Incompressibility is enforced using penalty method with an updated penalty parameter. The model is used to simulate experimental data from the literature demonstrating that the model response is in excellent agreement with the data. An experimental procedure to determine the distribution of fiber directions in 3D for biological soft tissue is suggested in accordance with the microplane concept. It is also argued that this microplane formulation could be modified or extended to model many other phenomena of interest in biomechanics.

A Stable Fluid-Structure-Interaction Algorithm: Application to Industrial Problems

2005 ◽  
Vol 128 (4) ◽  
pp. 516-524 ◽  
Author(s):  
D. Abouri ◽  
A. Parry ◽  
A. Hamdouni ◽  
E. Longatte
Keyword(s):  
Tube Bundle ◽  
Equations Of Motion ◽  
Mechanical Analysis ◽  
Industrial Applications ◽  
Explicit Integration ◽  
Flow Meter ◽  
Navier Stokes Equation ◽  
Fluid Structure ◽  
Positive Displacement ◽  
Wide Range

Fluid-structure interactions occur in a wide range of industrial applications, including vibration of pipe-work, flow meters, and positive displacement systems as well as many flow control devices. This paper outlines computational methods for calculating the dynamic interaction between moving parts and the flow in a flow-meter system. Coupling of phenomena is allowed without need for access to the source codes and is thus suitable for use with commercially available codes. Two methods are presented: one with an explicit integration of the equations of motion of the mechanism and the other, with implicit integration. Both methods rely on a Navier-Stokes equation solver for the fluid flow. The more computationally expensive, implicit method is recommended for mathematically stiff mechanisms such as piston movement. Industrial-application examples shown are for positive displacement machines, axial turbines, and steam-generator tube-bundle vibrations. The advances in mesh technology, including deforming meshes with nonconformal sliding interfaces, open up this new field of application of computational fluid dynamics and mechanical analysis in flow meter design.

scholarly journals A Stable Fluid Rigid Body Interaction Algorithm: Application to Industrial Problems

2004 ◽  
Author(s):  
D. Abouri ◽  
A. Parry ◽  
A. Hamdouni
Keyword(s):  
Equations Of Motion ◽  
Mechanical Analysis ◽  
Industrial Applications ◽  
Navier Stokes ◽  
Explicit Integration ◽  
Flow Meter ◽  
Navier Stokes Equation ◽  
Positive Displacement ◽  
Flow Control Devices ◽  
Wide Range

Fluid-mechanism interactions occur in a wide range of flow meter categories including turbine and positive displacement systems as well as many flow control devices. This paper outlines computational methods for calculating the dynamic interaction between moving parts and the flow in a flow meter system. Coupling of phenomena is allowed without need for access to the source codes and is thus suitable for use with commercially available codes. Two methods are presented one with an explicit integration of the equations of motion of the mechanism and the other with implicit integration. Both methods rely on a Navier-Stokes equation solver for the fluid flow. The more computationally expensive implicit method is recommended for mathematically stiff mechanisms such as piston movement. Example industrial applications shown are for positive displacement machines and axial turbines. The advances in mesh technology including deforming meshes with non-conformal sliding interfaces opens up this new field of application for Computational Fluid Dynamics and mechanical analysis in flow meter design.

Conservative Split-Explicit Time Integration Methods for the Compressible Nonhydrostatic Equations

2007 ◽  
Vol 135 (8) ◽  
pp. 2897-2913 ◽  
Author(s):  
J. B. Klemp ◽  
W. C. Skamarock ◽  
J. Dudhia
Keyword(s):  
Equations Of Motion ◽  
Time Integration ◽  
Test Cases ◽  
Explicit Integration ◽  
Linear Dispersion ◽  
Acoustic Gravity Waves ◽  
Wide Range ◽  
Terrain Following ◽  
Integration Techniques ◽  
Time Integration Methods

Abstract Historically, time-split schemes for numerically integrating the nonhydrostatic compressible equations of motion have not formally conserved mass and other first-order flux quantities. In this paper, split-explicit integration techniques are developed that numerically conserve these properties by integrating prognostic equations for conserved quantities represented in flux form. These procedures are presented for both terrain-following height and hydrostatic pressure (mass) vertical coordinates, two potentially attractive frameworks for which the equation sets and integration techniques differ significantly. For each set of equations, the linear dispersion equation for acoustic/gravity waves is derived and analyzed to determine which terms must be solved in the small (acoustic) time steps and how these terms are represented in the time integration to achieve stability. Efficient techniques for including numerical filters for acoustic and external modes are also presented. Simulations for several idealized test cases in both the height and mass coordinates are presented to demonstrate that these integration techniques appear robust over a wide range of scales, from subcloud to synoptic.

Development and implementation of model smoothing method in the framework of absolute nodal coordinate formulation

2021 ◽  
pp. 146441932110217
Author(s):  
Zhigang Zhang ◽  
Hongsheng Mao ◽  
Junjian Hou ◽  
Liangwen Wang ◽  
Gengxiang Wang
Keyword(s):  
High Frequency ◽  
Absolute Nodal Coordinate Formulation ◽  
Equations Of Motion ◽  
Model Development ◽  
Internal Force ◽  
Dynamic Responses ◽  
Time Interval ◽  
Explicit Integration ◽  
Absolute Nodal Coordinate ◽  
Frequency Components

A model smoothing formulation used to filter out the high-frequency components in the dynamical modeling process is developed and implemented in the framework of absolute nodal coordinate formulation (ANCF). Unlike the conventional method which employs stiff ordinary differential equation (ODE) solvers with numerical damping to calculate the dynamic responses of the ANCF model, a new formulation with the controlled high frequency is proposed in this paper. Using the average stress defined in a time interval to replace the instant stress in the virtual power of the internal force, the additional inertial and damping terms are introduced to the equations of motion of the ANCF element. It is proved that the highest frequency magnitude of the modified ANCF model can be adjusted by changing the length parameter of the time interval. As the high-frequency components are filtered out in the process of the model development, the traditional stiff problems of ANCF elements can be solved by using well-developed explicit integration algorithms, particularly in the very stiff and thin structures. Dynamic analysis of the classic examples including the pendulum beam and plate are performed to evaluate the performance of the model smoothing formulation of ANCF elements. Numerical results show that the proposed method can greatly improve the calculation efficiency of ANCF models, and the calculation accuracy can be also guaranteed.

The motion of a lunar orbiter

1966 ◽  
Vol 25 ◽  
pp. 373
Author(s):  
Y. Kozai
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Artificial Satellite ◽  
Equations Of Motion ◽  
Triaxial Ellipsoid ◽  
The Moon ◽  
Secular Motion ◽  
The Earth ◽  
Close Satellite ◽  
The Mean

The motion of an artificial satellite around the Moon is much more complicated than that around the Earth, since the shape of the Moon is a triaxial ellipsoid and the effect of the Earth on the motion is very important even for a very close satellite.The differential equations of motion of the satellite are written in canonical form of three degrees of freedom with time depending Hamiltonian. By eliminating short-periodic terms depending on the mean longitude of the satellite and by assuming that the Earth is moving on the lunar equator, however, the equations are reduced to those of two degrees of freedom with an energy integral.Since the mean motion of the Earth around the Moon is more rapid than the secular motion of the argument of pericentre of the satellite by a factor of one order, the terms depending on the longitude of the Earth can be eliminated, and the degree of freedom is reduced to one.Then the motion can be discussed by drawing equi-energy curves in two-dimensional space. According to these figures satellites with high inclination have large possibilities of falling down to the lunar surface even if the initial eccentricities are very small.The principal properties of the motion are not changed even if plausible values ofJ3andJ4of the Moon are included.This paper has been published in Publ. astr. Soc.Japan15, 301, 1963.

On the motion of comet P/d’Arrest

1974 ◽  
Vol 22 ◽  
pp. 145-148
Author(s):  
W. J. Klepczynski
Keyword(s):  
Equations Of Motion ◽  
Variational Equations ◽  
Partial Derivatives

AbstractThe differences between numerically approximated partial derivatives and partial derivatives obtained by integrating the variational equations are computed for Comet P/d’Arrest. The effect of errors in the IAU adopted system of masses, normally used in the integration of the equations of motion of comets of this type, is investigated. It is concluded that the resulting effects are negligible when compared with the observed discrepancies in the motion of this comet.

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