A Logarithmic Complexity Divide-and-Conquer Algorithm for Multi-flexible Articulated Body Dynamics

2006 ◽  
Vol 2 (1) ◽  
pp. 10-21 ◽  
Author(s):  
Rudranarayan M. Mukherjee ◽  
Kurt S. Anderson
Keyword(s):  
Degrees Of Freedom ◽  
Temporal Integration ◽  
Equations Of Motion ◽  
Turnaround Time ◽  
Divide And Conquer ◽  
Dynamics Simulation ◽  
Flexible Body ◽  
Integration Step ◽  
Large Rotations ◽  
Nonlinear Equations Of Motion

This paper presents an efficient algorithm for the dynamics simulation and analysis of multi-flexible-body systems. This algorithm formulates and solves the nonlinear equations of motion for mechanical systems with interconnected flexible bodies subject to the limitations of modal superposition, and body substructuring, with arbitrarily large rotations and translations. The large rotations or translations are modelled as rigid body degrees of freedom associated with the interconnecting kinematic joint degrees of freedom. The elastic deformation of the component bodies is modelled through the use of modal coordinates and associated admissible shape functions. Apart from the approximation associated with the elastic deformations, this algorithm is exact, non-iterative, and applicable to generalized multi-flexible chain and tree topologies. In its basic form, the algorithm is both time and processor optimal in its treatment of the nb joint variables, providing O(log(nb)) turnaround time per temporal integration step, achieved with O(nb) processors. The actual cost associated with the parallel treatment of the nf flexible degrees of freedom depends on the specific parallel method chosen for dealing with the individual coefficient matrices which are associated locally with each flexible body.

A Logarithmic Complexity Divide and Conquer Algorithm for Flexible Multibody Dynamics

2005 ◽  
Author(s):  
Rudranarayan Mukherjee ◽  
Kurt Anderson
Keyword(s):  
Elastic Deformation ◽  
Degrees Of Freedom ◽  
Parallel Implementation ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Flexible Multibody Dynamics ◽  
Divide And Conquer ◽  
Element Analysis ◽  
Large Rotations ◽  
Nonlinear Equations Of Motion

This paper presents an efficient algorithm for parallel implementation of multi-flexible-body dynamics systems simulation and analysis. The effective overall computational cost of the algorithm is logarithmic when implemented with a processor optimal O(n) processors. This algorithm formulates and solves the nonlinear equations of motion for mechanical systems with interconnected flexible bodies subject to small elastic deformation together with large rotations and translations. The large rotations or translations are modeled as rigid body degree of freedom associated with the interconnecting kinematic joint degrees of freedom. The elastic deformation of the component bodies is modeled through the use of admissible shape functions generated using standard finite element analysis software or otherwise. Apart from the approximation associated with the elastic deformations, this algorithm is exact, non-iterative and applicable to generalized multi-flexible chain and free topologies.

Multi-Flexible-Body Simulations Using Interpolating Splines in a Divide-and-Conquer Scheme

2013 ◽  
Author(s):  
Imad M. Khan ◽  
Woojin Ahn ◽  
Kurt Anderson ◽  
Suvranu De
Keyword(s):  
Equations Of Motion ◽  
Linear Equations ◽  
Divide And Conquer ◽  
Flexible Body ◽  
Divide And Conquer Algorithm ◽  
Large Rotations ◽  
Floating Frame ◽  
Interpolating Splines ◽  
Divide And Conquer Scheme ◽  
Logarithmic Complexity

A new method for modeling multi-flexible-body systems is presented that incorporates interpolating splines in a divide-and-conquer scheme. This algorithm uses the floating frame of reference formulation and piece-wise interpolation spline functions to construct and solve the non-linear equations of motion of the multi-flexible-body systems undergoing large rotations and translations. We compare the new algorithm with the flexible divide-and-conquer algorithm (FDCA) that uses the assumed modes method and may resort to sub-structuring in many cases [1]. We demonstrate, through numerical examples, that in such cases the interpolating spline-based approach is comparable in accuracy and superior in efficiency to the FDCA. The algorithm retains the theoretical logarithmic complexity inherent to the divide-and-conquer algorithm when implemented in parallel.

Divide-and-Conquer-Based Large Deformation Formulations for Multi-Flexible Body Systems

2013 ◽  
Author(s):  
Imad M. Khan ◽  
Kurt S. Anderson
Keyword(s):  
Modeling And Simulation ◽  
Role Model ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Divide And Conquer ◽  
Model Complexity ◽  
Flexible Body ◽  
Elastic Bodies ◽  
Large Rotations ◽  
High Computational Cost

In the dynamic modeling and simulation of multi-flexible-body systems, large deformations and rotations has been a focus of keen interest. The reason is a wide variety of application area where highly elastic components play important role. Model complexity and high computational cost of simulations are the factors that contribute to the difficulty associated with these systems. As such, an efficient algorithm for modeling and simulation of systems undergoing large rotations and large deflections may be of great importance. We investigate the use of absolute nodal coordinate formulation (ANCF) for modeling articulated flexible bodies in a divide-and-conquer (DCA) framework. It is demonstrated that the equations of motion for individual finite elements or elastic bodies, as obtained by the ANCF, may be assembled and solved using a DCA type method. The current discussion is limited to planar problems but may easily be extended to spatial applications. Using numerical examples, we show that the present algorithm provides an efficient and robust method to model multibody systems employing highly elastic bodies.

An Emerging O(N) Model and Algorithm for Virtual Prototyping of Dynamics of Molecular Conformation

2008 ◽  
Author(s):  
Andrew Ries ◽  
Shanzhong Shawn Duan
Keyword(s):  
Degrees Of Freedom ◽  
Molecular Conformation ◽  
Molecular System ◽  
Equations Of Motion ◽  
Integration Step ◽  
Computational Costs ◽  
System A ◽  
Hard Constraints ◽  
Solution Of Equations ◽  
Computationally Intensive

Molecular dynamics is effective for nano-scale phenomenon analysis. There are two major computational steps associated with computer simulation of dynamics of molecular conformation and they are the calculation of the interatomic forces and the formation and solution of the equations of motion. Currently, these two computational steps are treated separately, but in this paper an O(N) (order N) procedure is presented for an integration between these computational steps. For computational costs associated with calculating the interatomic forces, an internal coordinate method (ICM) approach is used for determining potentials due to both the bonding and non-bonding interactions. Thus, the potential gradients can be expressed as a combination of the potential in absolute and relative coordinates. For computational costs associated with the formation and solution of the equations of motion for the system, a constraint method that is used in computational multibody dynamics is utilized. This frees some degrees of freedom so that Kane’s method can be applied for the recursive formation and solution of equations of motion for the atomistic molecular system. Because the inclusion of lightly excited high frequency degrees of freedom, such as inter-atomic oscillations and rotation about double bonds would force the use of very small integration step sizes, holonomic constraints are introduced to freeze these “uninteresting” degrees of freedom. By introducing these hard constraints the time scale can be appropriately sized for to provide a less computationally intensive dynamic simulation of molecular conformation. The algorithm developed improves computational speed significantly when compared with any traditional O(N3) procedure.

scholarly journals Open Quantum Dynamics Theory on the Basis of Periodical System-Bath Model for Discrete Wigner Function

2021 ◽  
Author(s):  
Yuki Iwamoto ◽  
Yoshitaka Tanimura
Keyword(s):  
Distribution Function ◽  
Quantum Dynamics ◽  
Degrees Of Freedom ◽  
Wigner Distribution ◽  
Equations Of Motion ◽  
Heat Bath ◽  
Planck Equation ◽  
Mesh Size ◽  
Dynamics Simulation ◽  
Open Quantum

Abstract Discretizing distribution function in a phase space for an efficient quantum dynamics simulation is non-trivial challenge, in particular for a case that a system is further coupled to environmental degrees of freedom. Such open quantum dynamics is described by a reduced equation of motion (REOM) most notably by a quantum Fokker-Planck equation (QFPE) for a Wigner distribution function (WDF). To develop a discretization scheme that is stable for numerical simulations from the REOM approach, we find that a two-dimensional (2D) periodically invariant system-bath (PISB) model with two heat baths is an ideal platform not only for a periodical system but also for a system confined by a potential. We then derive the numerically ''exact'' hierarchical equations of motion (HEOM) for a discrete WDF in terms of periodically invariant operators in both coordinate and momentum spaces. The obtained equations can treat non-Markovian heat-bath in a non-perturbative manner at finite temperatures regardless of the mesh size. The stability of the present scheme is demonstrated in a high-temperature Markovian case by numerically integrating the discrete QFPE with by a coarse mesh for a 2D free rotor and harmonic potential systems for an initial condition that involves singularity.

A State-Time Formulation for Multibody Systems Dynamics Simulation: Part I — Extension to Systems With General Topology

2005 ◽  
Author(s):  
Mojtaba Oghbaei ◽  
Kurt S. Anderson
Keyword(s):  
Data Storage ◽  
Equations Of Motion ◽  
A Priori ◽  
Turnaround Time ◽  
General Topology ◽  
Dynamics Simulation ◽  
Constraint Forces ◽  
Time Dynamic ◽  
State Time ◽  
Massively Parallel Computing

This paper presents an extension of a newly developed multibody system dynamics formulation to systems with general topology. The State-Time dynamic formulation, which has been recently developed by the authors, provides the means to yield significantly reduced simulation turnaround time through its ability to better exploit massively parallel computing resources. The rules provided in this article are useful in automating the generation of system’s equations of motion and in determining the final form of the system tangent matrix arising in this formulation. A priori knowledge of this structure assists one to find a proper ordering for the rows and columns of this matrix such that the final structure is optimized from data storage and solution expense perspectives. Also, the extended formulation enables one to eliminate the constraint forces or to bring only the desirable ones into evidence and as such results in a reduced set of equations and unknowns. Examples are provided to demonstrate application of the given rules.

Divide-and-Conquer Based Adaptive Coarse Grained Simulation of RNA

2010 ◽  
Author(s):  
Mohammad Poursina ◽  
Kishor Bhalerao ◽  
Kurt Anderson
Keyword(s):  
Molecular Modeling ◽  
Degrees Of Freedom ◽  
Traditional Approach ◽  
Equations Of Motion ◽  
Critical Role ◽  
Divide And Conquer ◽  
Coarse Grained ◽  
Biomolecular Systems ◽  
Spatial And Temporal Scales ◽  
O 10

Molecular modeling has gained increasing importance in recent years for predicting important structural properties of large biomolecular systems such as RNA which play a critical role in various biological processes. Given the complexity of biopolymers and their interactions within living organisms, efficient and adaptive multi-scale modeling approaches are necessary if one is to reasonably perform computational studies of interest. These studies nominally involve multiple important physical phenomena occurring at different spatial and temporal scales. These systems are typically characterized by large number of degrees of freedom O(103) – O(107). The temporal domains range from sub-femto seconds (O(10−16)) associated with the small high frequency oscillations of individual tightly bonded atoms to milliseconds (O(10−3)) or greater for the larger scale conformational motion. The traditional approach for molecular modeling involved fully atomistic models which results in fully decoupled equations of motion. The problems with this approach are well documented in literature.

High-Speed Vehicle Models Based on a New Concept of Vehicle/Structure Interaction Component: Part II—Algorithmic Treatment and Results for Multispan Guideways

1993 ◽  
Vol 115 (1) ◽  
pp. 148-155 ◽  
Author(s):  
L. Vu-Quoc ◽  
M. Olsson
Keyword(s):  
High Speed ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Structural Equations ◽  
Component Part ◽  
Vehicle Speed ◽  
Vehicle Structure ◽  
Algorithmic Treatment ◽  
Nonlinear Equations Of Motion ◽  
High Speed Vehicle

The predictor structural equations for the vehicle models developed in Part I are derived here for use with a new class of predictor/corrector algorithms to solve the mildly nonlinear equations of motion of the vehicle/structure models. Having all accelerations of the vehicle component eliminated, and with the aid of further simplifying approximations, the predictor structural equations are linear with respect to the structural degrees of freedom. In the algorithms, the predictor structural equations are different from the corrector structural equations; the proposed algorithmic treatment has been proved (elsewhere) to yield accurate energy balance. Results obtained for both continuous and discontinuous guideways are discussed, and optimal guideway configurations suggested. Effects of high-speed vehicle braking on a flexible guideway are analyzed using the vehicle models and the proposed algorithmic treatment. The influence of the guideway flexibility on the vehicle speed, an important feature of the present formulation, is clearly demonstrated.

scholarly journals Speed Oscillations of a Vehicle Rolling on a Wavy Road

2021 ◽  
Vol 11 (21) ◽  
pp. 10431
Author(s):  
Walter V. Wedig
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Travel Speed ◽  
Differential Algebraic Equations ◽  
Polar Coordinates ◽  
Algebraic Equations ◽  
Traffic Lights ◽  
Analytical Approximations ◽  
Vertical Vibrations ◽  
Nonlinear Equations Of Motion

Every driver knows that his car is slowing down or accelerating when driving up or down, respectively. The same happens on uneven roads with plastic wave deformations, e.g., in front of traffic lights or on nonpaved desert roads. This paper investigates the resulting travel speed oscillations of a quarter car model rolling in contact on a sinusoidal and stochastic road surface. The nonlinear equations of motion of the vehicle road system leads to ill-conditioned differential–algebraic equations. They are solved introducing polar coordinates into the sinusoidal road model. Numerical simulations show the Sommerfeld effect, in which the vehicle becomes stuck before the resonance speed, exhibiting limit cycles of oscillating acceleration and speed, which bifurcate from one-periodic limit cycle to one that is double periodic. Analytical approximations are derived by means of nonlinear Fourier expansions. Extensions to more realistic road models by means of noise perturbation show limit flows as bundles of nonperiodic trajectories with periodic side limits. Vehicles with higher degrees of freedom become stuck before the first speed resonance, as well as in between further resonance speeds with strong vertical vibrations and longitudinal speed oscillations. They need more power supply in order to overcome the resonance peak. For small damping, the speeds after resonance are unstable. They migrate to lower or supercritical speeds of operation. Stability in mean is investigated.

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