An Implicit Time-Stepping Method for Quasi-Rigid Multibody Systems With Intermittent Contact

2007 ◽  
Author(s):  
Nilanjan Chakraborty ◽  
Stephen Berard ◽  
Srinivas Akella ◽  
Jeff Trinkle
Keyword(s):  
Complementarity Problem ◽  
The Body ◽  
Contact Forces ◽  
Implicit Time ◽  
Discrete Problem ◽  
Time Step ◽  
Normal Contact ◽  
Time Stepping ◽  
Intermittent Contact ◽  
Compliant Layer

We recently developed a time-stepping method for simulating rigid multi-body systems with intermittent contact that is implicit in the geometric information [1]. In this paper, we extend this formulation to quasi-rigid or locally compliant objects, i.e., objects with a rigid core surrounded by a compliant layer, similar to Song et al. [2]. The difference in our compliance model from existing quasi-rigid models is that, based on physical motivations, we assume the compliant layer has a maximum possible normal deflection beyond which it acts as a rigid body. Therefore, we use an extension of the Kelvin-Voigt (i.e. linear spring-damper) model for obtaining the normal contact forces by incorporating the thickness of the compliant layer explicitly in the contact model. We use the Kelvin-Voigt model for the tangential forces and assume that the contact forces and moment satisfy an ellipsoidal friction law. We model each object as an intersection of convex inequalities and write the contact constraint as a complementarity constraint between the contact force and a distance function dependent on the closest points and the local deformation of the body. The closest points satisfy a system of nonlinear algebraic equations and the resultant continuous model is a Differential Complementarity Problem (DCP). This enables us to formulate a geometrically implicit time-stepping scheme for solving the DCP which is more accurate than a geometrically explicit scheme. The discrete problem to be solved at each time-step is a mixed nonlinear complementarity problem.

Numerical simulation of higher-order diffusion-wave equations of variable coefficients using the meshless spectral method

2020 ◽  
pp. 2150010
Author(s):  
Manzoor Hussain ◽  
Sirajul Haq
Keyword(s):  
Wave Equations ◽  
Interpolation Method ◽  
Variable Coefficients ◽  
Higher Order ◽  
Implicit Time ◽  
Test Problems ◽  
Time Step ◽  
Diffusion Wave ◽  
Time Step Size ◽  
Time Stepping

In this paper, meshless spectral interpolation technique using implicit time stepping scheme is proposed for the numerical simulations of time-fractional higher-order diffusion wave equations (TFHODWEs) of variable coefficients. Meshless shape functions, obtained from radial basis functions (RBFs) and point interpolation method (PIM), are used for spatial approximation. Central differences coupled with quadrature rule of [Formula: see text] are employed for fractional temporal approximation. For advancement of solution, an implicit time stepping scheme is then invoked. Simulations performed for different benchmark test problems feature good agreement with exact solutions. Stability analysis of the proposed method is theoretically discussed and computationally validated to support the analysis. Accuracy and efficiency of the proposed method are assessed via [Formula: see text], [Formula: see text] and [Formula: see text] error norms as well as number of nodes [Formula: see text] and time step-size [Formula: see text].

A Recursive Hybrid Time-Stepping Scheme for Intermittent Contact in Multi-Rigid-Body Dynamics

2009 ◽  
Vol 4 (4) ◽  
Author(s):  
Kishor D. Bhalerao ◽  
Kurt S. Anderson ◽  
Jeffrey C. Trinkle
Keyword(s):  
Rigid Body ◽  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
Linear Complementarity ◽  
Rigid Body Dynamics ◽  
Divide And Conquer ◽  
Double Pendulum ◽  
Time Stepping ◽  
Bilateral Constraints ◽  
Intermittent Contact

This paper describes a novel method for the modeling of intermittent contact in multi-rigid-body problems. We use a complementarity based time-stepping scheme in Featherstone’s divide and conquer framework to efficiently model the unilateral and bilateral constraints in the system. The time-stepping scheme relies on impulse-based equations and does not require explicit collision detection. A set of complementarity conditions is used to model the interpenetration constraint and a linearized friction cone is used to yield a linear complementarity problem. The divide and conquer framework ensures that the size of the resulting mixed linear complementarity problem is independent of the number of bilateral constraints in the system. This makes the proposed method especially efficient for systems where the number of bilateral constraints is much greater than the number of unilateral constraints. The method is demonstrated by applying it to a falling 3D double pendulum.

scholarly journals Comparison of Terrain-Following and Cut-Cell Grids Using a Nonhydrostatic Model

2016 ◽  
Vol 144 (6) ◽  
pp. 2085-2099 ◽  
Author(s):  
James Shaw ◽  
Hilary Weller
Keyword(s):  
Velocity Field ◽  
Gravity Waves ◽  
Initial Conditions ◽  
Implicit Time ◽  
Test Results ◽  
Time Step ◽  
Induced Gravity ◽  
Time Stepping ◽  
Cut Cell ◽  
Terrain Following

Abstract Terrain-following coordinates are widely used in operational models but the cut-cell method has been proposed as an alternative that can more accurately represent atmospheric dynamics over steep orography. Because the type of grid is usually chosen during model implementation, it becomes necessary to use different models to compare the accuracy of different grids. In contrast, here a C-grid finite-volume model enables a like-for-like comparison of terrain-following and cut-cell grids. A series of standard two-dimensional tests using idealized terrain are performed: tracer advection in a prescribed horizontal velocity field, a test starting from resting initial conditions, and orographically induced gravity waves described by nonhydrostatic dynamics. In addition, three new tests are formulated: a more challenging resting atmosphere case, and two new advection tests having a velocity field that is everywhere tangential to the terrain-following coordinate surfaces. These new tests present a challenge on cut-cell grids. The results of the advection tests demonstrate that accuracy depends primarily upon alignment of the flow with the grid rather than grid orthogonality. A resting atmosphere is well maintained on all grids. In the gravity waves test, results on all grids are in good agreement with existing results from the literature, although terrain-following velocity fields lead to errors on cut-cell grids. Because of semi-implicit time stepping and an upwind-biased, explicit advection scheme, there are no time step restrictions associated with small cut cells. In contradiction to other studies, no significant advantages of cut cells or smoothed coordinates are found.

scholarly journals Efficient extrapolation methods for electro- and magnetoquasistatic field simulations

2003 ◽  
Vol 1 ◽  
pp. 81-86 ◽  
Author(s):  
M. Clemens ◽  
M. Wilke ◽  
T. Weiland
Keyword(s):  
Taylor Series Expansion ◽  
Iterative Solvers ◽  
Gradient Algorithm ◽  
Implicit Time ◽  
Preconditioned Conjugate Gradient ◽  
Test Problems ◽  
Time Step ◽  
Time Stepping ◽  
Multi Stage ◽  
Runge Kutta

Abstract. In magneto- and electroquasi-static time domain simulations with implicit time stepping schemes the iterative solvers applied to the large sparse (non-)linear systems of equations are observed to converge faster if more accurate start solutions are available. Different extrapolation techniques for such new time step solutions are compared in combination with the preconditioned conjugate gradient algorithm. Simple extrapolation schemes based on Taylor series expansion are used as well as schemes derived especially for multi-stage implicit Runge-Kutta time stepping methods. With several initial guesses available, a new subspace projection extrapolation technique is proven to produce an optimal initial value vector. Numerical tests show the resulting improvements in terms of computational efficiency for several test problems. In quasistatischen elektromagnetischen Zeitbereichsimulationen mit impliziten Zeitschrittverfahren zeigt sich, dass die iterativen Lösungsverfahren für die großen dünnbesetzten (nicht-)linearen Gleichungssysteme schneller konvergieren, wenn genauere Startlösungen vorgegeben werden. Verschiedene Extrapolationstechniken werden für jeweils neue Zeitschrittlösungen in Verbindung mit dem präkonditionierten Konjugierte Gradientenverfahren vorgestellt. Einfache Extrapolationsverfahren basierend auf Taylorreihenentwicklungen werden ebenso benutzt wie speziell für mehrstufige implizite Runge-Kutta-Verfahren entwickelte Verfahren. Sind verschiedene Startlösungen verfügbar, so erlaubt ein neues Unterraum-Projektion- Extrapolationsverfahren die Konstruktion eines optimalen neuen Startvektors. Numerische Tests zeigen die aus diesen Verfahren resultierenden Verbesserungen der numerischen Effizienz.

Railhead Corrugation Growth Explained by Dynamic Interaction between Track and Bogie Wheelsets

1996 ◽  
Vol 210 (1) ◽  
pp. 11-20 ◽  
Author(s):  
A Igeland
Keyword(s):  
Contact Stiffness ◽  
Empirical Relationship ◽  
Short Wave ◽  
Contact Forces ◽  
Rolling Contact ◽  
Normal Contact ◽  
Time Stepping ◽  
Vertical Contact ◽  
Semi Empirical ◽  
Locomotive Wheels

Short-wave railhead corrugation growth on tangent tracks due to wear from driven locomotive wheels is investigated. A track model with a given initial random railhead irregularity is used. The track parameters and the railhead irregularities are estimated from measurements performed on a newly built track. The corrugation calculations are based on rolling contact mechanics; a semi-empirical relationship between creep, friction force and normal contact force is employed. Only longitudinal creep and creep forces are taken into account. The vertical contact forces between the moving wheels and the rail are calculated using a time-stepping method, allowing for a non-linear Hertzian wheel/rail contact stiffness. Numerical examples are given. It is found that the two wheelsets of a travelling bogie interact and that both of them should be considered in the same calculation. The bogie wheelbase turns out to be an important parameter.

Modeling and Detection of Localized Tooth Defects in Geared Systems

1999 ◽  
Author(s):  
M. El Badaoui ◽  
V. Cahouet ◽  
F. Guillet ◽  
J. Daniere ◽  
P. Velex
Keyword(s):  
Numerical Simulations ◽  
Gear Tooth ◽  
Equations Of Motion ◽  
Vibration Monitoring ◽  
Detection Methods ◽  
Implicit Time ◽  
Integration Scheme ◽  
Time Step ◽  
Normal Contact ◽  
Detection Techniques

Abstract The early detection of failures in geared systems is an important industrial problem which has still to be addressed from both an experimental and theoretical viewpoint. The proposed paper combines some extensive numerical simulations of a single stage geared unit with localized tooth faults and the use of several detection techniques whose performances are compared and critically assessed. A model aimed at simulating the contributions of local tooth defects such as spalling to the gear dynamic behavior is set up. The pinion and the gear of a pair are modeled as to two rigid cylinders with all six degrees of freedom connected by a series of springs which represent gear body and gear tooth compliances on the base plane. Classical shaft finite elements including torsional, flexural and axial displacements can be superimposed on the gear element together with some lumped stiffnesses, masses, inertias, ... which account for the load machines, bearings and couplings. Tooth defects are modeled by a distribution of normal deviations over a zone which can be located anywhere on the active tooth flanks. Among the numerous available signal processing techniques used in vibration monitoring, cepstrum analysis is sensitive, reliable and it can be adapted to a complex geared system with several meshes. From an analytical analysis of the equations of motion, two complementary detection techniques based upon the acceleration power cesptrum are proposed. The equations of motion and the contact problem between mating flanks are simultaneously solved by coupling an implicit time-step integration scheme and a unilateral normal contact algorithm. The results of the numerical simulations are used as a data base for the proposed detection techniques. The combined influence of the defect location, depth and extent is analyzed for two examples of spur and helical gears with various profile modifications and the effectiveness of the two complementary detection methods is discussed before some conclusions are drawn.

A Recursive Hybrid Time-Stepping Scheme for Intermittent Contact in Multi-Rigid-Body Dynamics

2009 ◽  
Author(s):  
Kishor D. Bhalerao ◽  
Kurt S. Anderson ◽  
Jeffery C. Trinkle
Keyword(s):  
Rigid Body ◽  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
Linear Complementarity ◽  
Rigid Body Dynamics ◽  
Divide And Conquer ◽  
Double Pendulum ◽  
Time Stepping ◽  
Bilateral Constraints ◽  
Intermittent Contact

This paper describes a novel method for the modeling of intermittent contact in multi-rigid-body problems. We use a complementarity based time-stepping scheme in Featherstone’s Divide and Conquer framework to efficiently model the unilateral and bilateral constraints in the system. The time-stepping scheme relies on impulse-based equations and does not require explicit collision detection. A set of complementarity conditions is used to model the interpenetration constraint and a linearized friction cone is used to yield a linear complementarity problem. The Divide and Conquer framework ensures that the size of the resulting mixed linear complementarity problem is independent of the number of bilateral constraints in the system. This makes the proposed method especially efficient for systems where the number of bilateral constraints are much greater than the number of unilateral constraints. The method is demonstrated by applying it to a falling 3D double pendulum.

Modeling and Detection of Localized Tooth Defects in Geared Systems

1999 ◽  
Vol 123 (3) ◽  
pp. 422-430 ◽  
Author(s):  
M. El Badaoui ◽  
V. Cahouet ◽  
F. Guillet ◽  
J. Danie`re ◽  
P. Velex
Keyword(s):  
Numerical Simulations ◽  
Gear Tooth ◽  
Equations Of Motion ◽  
Vibration Monitoring ◽  
Detection Methods ◽  
Implicit Time ◽  
Integration Scheme ◽  
Time Step ◽  
Normal Contact ◽  
Detection Techniques

The early detection of failures in geared systems is an important industrial problem which has still to be addressed from both an experimental and theoretical viewpoint. The proposed paper combines some extensive numerical simulations of a single stage geared unit with localized tooth faults and the use of several detection techniques whose performances are compared and critically assessed. A model aimed at simulating the contributions of local tooth defects such as spalling to the gear dynamic behavior is set up. The pinion and the gear of a pair are assimilated to two rigid cylinders with all six degrees of freedom connected by a series of springs which represent gear body and gear tooth compliances on the base plane. Classical shaft finite elements including torsional, flexural and axial displacements can be superimposed to the gear element together with some lumped stiffnesses, masses, inertias, … which account for the load machines, bearings and couplings. Tooth defects are modeled by a distribution of normal deviations over a zone which can be located anywhere on the active tooth flanks. Among the numerous available signal processing techniques used in vibration monitoring, cepstrum analysis is sensitive, reliable and it can be adapted to complex geared system with several meshes. From an analytical analysis of the equations of motion, two complementary detection techniques based upon acceleration power cepstrum are proposed. The equations of motion and the contact problem between mating flanks are simultaneously solved by coupling an implicit time-step integration scheme and a unilateral normal contact algorithm. The results of the numerical simulations are used as a data base for the proposed detection techniques. The combined influence of the defect location, depth and extent is analyzed for two examples of spur and helical gears with various profile modifications and the effectiveness of the two complementary detection methods is discussed before some conclusions are drawn.

scholarly journals Stability and Conservation Properties of Collocated Constraints in Immersogeometric Fluid-Thin Structure Interaction Analysis

2015 ◽  
Vol 18 (4) ◽  
pp. 1147-1180 ◽  
Author(s):  
David Kamensky ◽  
John A. Evans ◽  
Ming-Chen Hsu
Keyword(s):  
Incompressible Flows ◽  
Fluid Structure Interaction ◽  
Time Integration ◽  
Implicit Time ◽  
Integration Scheme ◽  
Discrete Problem ◽  
Time Step ◽  
Integration Rule ◽  
Fluid Structure ◽  
Structure Interaction

AbstractThe purpose of this study is to enhance the stability properties of our recently-developed numerical method [D. Kamensky, M.-C. Hsu, D. Schillinger, J.A. Evans, A. Aggarwal, Y. Bazilevs, M.S. Sacks, T.J.R. Hughes, “An immersogeometric variational framework for fluid-structure interaction: Application to bioprosthetic heart valves”, Comput. Methods Appl. Mech. Engrg., 284 (2015) 1005–1053] for immersing spline-based representations of shell structures into unsteady viscous incompressible flows. In the cited work, we formulated the fluid-structure interaction (FSI) problem using an augmented Lagrangian to enforce kinematic constraints. We discretized this Lagrangian as a set of collocated constraints, at quadrature points of the surface integration rule for the immersed interface. Because the density of quadrature points is not controlled relative to the fluid discretization, the resulting semi-discrete problem may be over-constrained. Semi-implicit time integration circumvents this difficulty in the fully-discrete scheme. If this time-stepping algorithm is applied to fluid-structure systems that approach steady solutions, though, we find that spatially-oscillating modes of the Lagrange multiplier field can grow over time. In the present work, we stabilize the semi-implicit integration scheme to prevent potential divergence of the multiplier field as time goes to infinity. This stabilized time integration may also be applied in pseudo-time within each time step, giving rise to a fully implicit solution method. We discuss the theoretical implications of this stabilization scheme for several simplified model problems, then demonstrate its practical efficacy through numerical examples.

scholarly journals A Numerical Method Based on Leapfrog and a Fourth-Order Implicit Time Filter

2014 ◽  
Vol 142 (7) ◽  
pp. 2545-2560 ◽  
Author(s):  
Mohamed Moustaoui ◽  
Alex Mahalov ◽  
Eric J. Kostelich
Keyword(s):  
Fourth Order ◽  
Climate Modeling ◽  
Implicit Time ◽  
Time Step ◽  
Third Order ◽  
Time Stepping ◽  
Formal Stability ◽  
Numerical Instabilities ◽  
Potential Alternative ◽  
Leapfrog Scheme

Abstract A time-stepping scheme is proposed. It is based on the leapfrog method and a fourth-order time filter. The scheme requires only one evaluation per time step and uses an implicit filter, but the effort needed to implement it in an explicit manner is trivial. Comparative tests demonstrate that the proposed scheme produces numerical approximations that are more stable and highly accurate compared to the standard Robert–Asselin (RA) and the Robert–Asselin–Williams (RAW) filtered leapfrog scheme, even though both methods use filter coefficients that are tuned such that the 2Δt modes are damped at the same rate. Formal stability analysis demonstrates that the proposed method generates amplitude errors of O[(Δt)4], implying third-order accuracy. This contrasts with the O[(Δt)2] errors produced by the standard RA and RAW filtered leapfrog. A second scheme that produces amplitude errors of O[(Δt)6] is also presented. The proposed scheme is found to do well at controlling numerical instabilities arising in the diffusion equation and in nonlinear computations using Lorenz’s system and the global shallow-water spectral model. In addition to noticeably improving the resolution of the physical modes, the proposed method is simple to implement and has a wider region of stability compared to the existing time-filtered leapfrog schemes. This makes the proposed method a potential alternative for use in atmospheric, oceanic, and climate modeling.

Sign in / Sign up

Export Citation Format

Share Document