New and Extended Applications of the Divide-and-Conquer Algorithm for Multibody Dynamics

2014 ◽  
Vol 9 (4) ◽  
Author(s):  
Jeremy J. Laflin ◽  
Kurt S. Anderson ◽  
Imad M. Khan ◽  
Mohammad Poursina
Keyword(s):  
Multibody Dynamics ◽  
Large Deformations ◽  
Multibody System ◽  
Divide And Conquer ◽  
Flexible Body ◽  
Ongoing Research ◽  
Flexible Bodies ◽  
Computational Burden ◽  
Divide And Conquer Algorithm ◽  
Model Fidelity

This work presents a survey of the current and ongoing research by the authors who use the divide-and-conquer algorithm (DCA) to reduce the computational burden associated with various aspects of multibody dynamics. This work provides a brief discussion of various topics that are extensions of previous DCA-based algorithms or novel uses of this algorithm in the multibody dynamics context. These topics include constraint error stabilization, spline-based modeling of flexible bodies, model fidelity transitions for flexible-body systems, and large deformations of flexible bodies. It is assumed that the reader is familiar with the “Advances in the Application of the DCA to Multibody System Dynamics” text as the notation used in this work is explained therein and provides a summary of how the DCA has been used previously.

Flexible Multibody Dynamics Formulation by Hamilton’s Equations

2010 ◽  
Author(s):  
Martin M. Tong
Keyword(s):  
Multibody Dynamics ◽  
Multibody System ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Flexible Multibody Dynamics ◽  
Flexible Body ◽  
Flexible Multibody System ◽  
Generalized Coordinates ◽  
Flexible Multibody ◽  
The Matrix

Numerical solution of the dynamics equations of a flexible multibody system as represented by Hamilton’s canonical equations requires that its generalized velocities q˙ be solved from the generalized momenta p. The relation between them is p = J(q)q˙, where J is the system mass matrix and q is the generalized coordinates. This paper presents the dynamics equations for a generic flexible multibody system as represented by p˙ and gives emphasis to a systematic way of constructing the matrix J for solving q˙. The mass matrix is shown to be separable into four submatrices Jrr, Jrf, Jfr and Jff relating the joint momenta and flexible body mementa to the joint coordinate rates and the flexible body deformation coordinate rates. Explicit formulas are given for these submatrices. The equations of motion presented here lend insight to the structure of the flexible multibody dynamics equations. They are also a versatile alternative to the acceleration-based dynamics equations for modeling mechanical systems.

An Efficient Multibody Divide and Conquer Algorithm

2005 ◽  
Author(s):  
James H. Critchley
Keyword(s):  
Multibody Dynamics ◽  
Time Complexity ◽  
Future Generation ◽  
Parallel Computers ◽  
Parallel Processors ◽  
Divide And Conquer ◽  
Parallel Computer ◽  
Divide And Conquer Algorithm ◽  
Computer Resources ◽  
Theoretical Minimum

A new and efficient form of Featherstone’s multibody Divide and Conquer Algorithm (DCA) is presented. The DCA was the first algorithm to achieve theoretically optimal logarithmic time complexity with a theoretical minimum of parallel computer resources for general problems of multibody dynamics, however the DCA is extremely inefficient in the presence of small to modest parallel computers. The new efficient DCA approach (DCAe) demonstrates that large DCA subsystems can be constructed using fast sequential techniques and realize substantial speed increases in the presence of as few as two parallel processors. Previously the DCA was a tool intended for a future generation of parallel computers, this enhanced version promises practical and competitive performance with the parallel computers of today.

Efficient Large Deformation Simulations of Multi-Flexible-Body Systems Using Absolute Nodal Coordinate Beam and Plate Elements

2014 ◽  
Author(s):  
Imad M. Khan ◽  
Kurt S. Anderson
Keyword(s):  
Absolute Nodal Coordinate Formulation ◽  
Divide And Conquer ◽  
Flexible Body ◽  
Elastic Deformations ◽  
Absolute Nodal Coordinate ◽  
Numerical Examples ◽  
Divide And Conquer Algorithm ◽  
Plate Elements ◽  
The Absolute ◽  
The Individual

In this paper, we investigate the absolute nodal coordinate finite element (FE) formulations for modeling multi-flexible-body systems in a divide-and-conquer framework. Large elastic deformations in the individual components (beams and plates) are modeled using the absolute nodal coordinate formulation (ANCF). The divide-and-conquer algorithm (DCA) is utilized to model the constraints arising due to kinematic joints between the flexible components. We develop necessary equations of the new algorithm and present numerical examples to test and validate the method.

On Parallel Methods of Multibody Dynamics

2003 ◽  
Author(s):  
James H. Critchley ◽  
Kurt S. Anderson
Keyword(s):  
Time Complexity ◽  
Multibody System ◽  
Practical Importance ◽  
Divide And Conquer ◽  
Optimal Time ◽  
Worst Case ◽  
Parallel Methods ◽  
Divide And Conquer Algorithm ◽  
Coordinate Reduction ◽  
Parallel Arrays

Optimal time efficient parallel computation methods for large multibody system dynamics are defined and investigated in detail. Comparative observations are made which demonstrate significant deficiencies in operating regions of practical importance and a new parallel algorithm is generated to address them. The new method of Recursive Coordinate Reduction Parallelism (RCRP) outperforms or directly reduces to the fastest general multibody algorithms available for small parallel resources and obtains “O(logk(n))” time complexity in the presence of larger parallel arrays. Performance of this method relative to the Divide and Conquer Algorithm is illustrated with an operations count for the worst case of a multibody chain system.

An Efficient Multibody Divide and Conquer Algorithm and Implementation

2009 ◽  
Vol 4 (2) ◽  
Author(s):  
James H. Critchley ◽  
Kurt S. Anderson ◽  
Adarsh Binani
Keyword(s):  
Multibody Dynamics ◽  
Time Complexity ◽  
Future Generation ◽  
Parallel Computers ◽  
Divide And Conquer ◽  
Parallel Computer ◽  
Recursive Method ◽  
Divide And Conquer Algorithm ◽  
Computer Resources ◽  
Theoretical Minimum

A new and efficient form of Featherstone’s multibody divide and conquer algorithm (DCA) is presented and evaluated. The DCA was the first algorithm to achieve theoretically the optimal logarithmic time complexity with a theoretical minimum of parallel computer resources for general problems of multibody dynamics; however, the DCA is extremely inefficient in the presence of small to modest parallel computers. This alternative efficient DCA (DCAe) approach demonstrates that large DCA subsystems can be constructed using fast sequential techniques to realize a substantial increase in speed. The usefulness of the DCAe is directly demonstrated in an application to a four processor workstation and compared with the results from the original DCA and a fast sequential recursive method. Previously the DCA was a tool intended for a future generation of parallel computers; this enhanced version delivers practical and competitive performance with the parallel computers of today.

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.

Advances in the Application of the Divide-and-Conquer Algorithm to Multibody System Dynamics

2014 ◽  
Vol 9 (4) ◽  
Author(s):  
Jeremy J. Laflin ◽  
Kurt S. Anderson ◽  
Imad M. Khan ◽  
Mohammad Poursina
Keyword(s):  
Computational Complexity ◽  
Task Scheduling ◽  
Constrained Systems ◽  
Divide And Conquer ◽  
Mathematical Framework ◽  
Complete Understanding ◽  
Basic Algorithm ◽  
Flexible Bodies ◽  
Divide And Conquer Algorithm ◽  
Serial Task

This paper summarizes the various recent advancements achieved by utilizing the divide-and-conquer algorithm (DCA) to reduce the computational burden associated with many aspects of modeling, designing, and simulating articulated multibody systems. This basic algorithm provides a framework to realize O(n) computational complexity for serial task scheduling. Furthermore, the framework of this algorithm easily accommodates parallel task scheduling, which results in coarse-grain O(log n) computational complexity. This is a significant increase in efficiency over forming and solving the Newton–Euler equations directly. A survey of the notable previous work accomplished, though not all inclusive, is provided to give a more complete understanding of how this algorithm has been used in this context. These advances include applying the DCA to constrained systems, flexible bodies, sensitivity analysis, contact, and hybridization with other methods. This work reproduces the basic mathematical framework for applying the DCA in each of these applications. The reader is referred to the original work for the details of the discussed methods.

Model Transitions and Optimization Problem in Multi-Flexible-Body Modeling of Biopolymers

2011 ◽  
Author(s):  
Mohammad Poursina ◽  
Imad Khan ◽  
Kurt S. Anderson
Keyword(s):  
Optimization Problem ◽  
Computational Cost ◽  
Divide And Conquer ◽  
Flexible Body ◽  
Basic Form ◽  
Divide And Conquer Algorithm ◽  
Time Optimal ◽  
Logarithmic Complexity ◽  
Algorithm Implementation ◽  
Fidelity Model

This paper presents an efficient algorithm for the simulation of multi-flexible-body systems undergoing discontinuous changes in model definition. The equations governing the dynamics of the transitions from a higher to a lower fidelity model and vice versa are formulated through imposing/removing certain constraints on/from the system. Furthermore, the issue of the non-uniqueness of the results associated with the transition from a lower to a higher fidelity model is dealt with as an optimization problem. This optimization problem is subjected to the satisfaction of the impulse-momentum equations. The divide and conquer algorithm (DCA) is applied to formulate the dynamics of the transition. The DCA formulation in its basic form is time optimal and results in linear and logarithmic complexity when implemented in serial and parallel, respectively. As such, it reduces the computational cost of formulating and solving the optimization problem in the transitions to the finer models. Necessary mathematics for the algorithm implementation is developed and a numerical example is given to validate the method.

Sign in / Sign up

Export Citation Format

Share Document