The Rensselaer Polytechnic Institute Computational Dynamics Laboratory

04/22/23 Rensselaer Computational Dynamics Laboratory 1

The Rensselaer Polytechnic InstituteComputational Dynamics Laboratory


Who are We?

• FacultyProfessor Kurt S. Anderson

• Graduate StudentsRudranarayan MukherjeeKishor BhaleraoMohammad Poursina


• BS & MS from Berkeley• Ph.D. from Stanford University• TRW Space and Technology• Visiting scholar, lecturer, and research

fellow at the Technische Hochscule-Darmstadt

• Professor at The Ohio State University • RPI faculty member since August 1995

– American Academy of Mechanics (AAM)–American Institute of Aeronautics and Astronautics (AIAA)–American Society of Automotive Engineers (SAE)–American Society of Mechanical Engineers (ASME)–US Association of Computational Mechanics (USACM)–American Society of Engineering Education (ASEE)–Associate Editor of Journal of Guidance Control and Dynamics, AIAA–Tau Beta Pi, Pi Tau Sigma, Phi Beta Kappa, and Sigma Xi.

Professor Kurt S. Anderson


Current Researchers

• Michael Sadowski, PhD Student• Focus: Unilateral Constraints via R.C.R.

• Mojtaba , PhD Student• Focus: State-Time Dynamic Formulation

Massively Parallel Dynamic Simulation

• James Critchley, PhD• Focus: Parallel Implementation of R.C.R.

Developing reduced order models formolecular dynamicsDeveloping an object oriented multibodydynamics software package

Current Researchers


• Mojtaba Oghbaei• Focus: State-Time Dynamic Formulation




Current Researchers


• Mojtaba , PhD Student• Focus: State-Time Dynamic Formulation




Current Researchers


• Mojtaba Oghbaei• Focus: State-Time Dynamic Formulation





• Rudranarayan Mukherjee, PhD Student• Focus: Evaluation of parallel algorithms for

applicability to protein folding and macro molecular dynamics

• Past Researchers– Shanzhong Duan, Ph.D.– YuHung Hsu, Ph.D.– Omer Gundogdu, Ph.D.– Jason Rosner, MS– Philip Stephanou, MS

• Rudranarayan Mukherjee, PhD Student• Focus: Evaluation of parallel algorithms for

applicability to protein folding and macro molecular dynamics

• Past Researchers– Shanzhong Duan, Ph.D.– YuHung Hsu, Ph.D.– Omer Gundogdu, Ph.D.– Jason Rosner, MS– Philip Stephanou, MS


Selected Publications• K.S. Anderson and S. Duan, "Parallel Implementation of a Low Order

Algorithm for Dynamics of Multibody Systems on a Distributed Memory Computing System", journal Engineering with Computers, Vol. 16, No. 2, 2000, pp 96-108

• Y.H. Hsu and K.S. Anderson, "Recursive Sensitivity Analysis for Constrained Multi-rigid-body Dynamic Systems Design Optimization", Structural and Multidisciplinary Optimization, Vol. 24, No. 4, pp. 312-324, October 2002

• K.S. Anderson and M.J. Sadowski, “An Efficient Method for Contact/Impact Problems in Multibody Systems: Topologies with Many Loops”, Fourth Symposium on Multibody Dynamics and Vibration ASME DETC 2003

• J.H. Critchley and K.S. Anderson , "A Generalized Recursive Coordinate Reduction Method for Multibody Dynamic Systems", Journal of MultiscaleComputational Engineering, accepted for publication

• J. H. Critchley and K . S. Anderson, "A Parallel Logarithmic Order Algorithm for General Multi-body System Dynamics", Accepted for publication in Multibody System Dynamics Journal, 2004

Selected Publications• K.S. Anderson and S. Duan, "Parallel Implementation of a Low Order

Algorithm for Dynamics of Multibody Systems on a Distributed Memory Computing System", journal Engineering with Computers, Vol. 16, No. 2, 2000, pp 96-108

• Y.H. Hsu and K.S. Anderson, "Recursive Sensitivity Analysis for Constrained Multi-rigid-body Dynamic Systems Design Optimization", Structural and Multidisciplinary Optimization, Vol. 24, No. 4, pp. 312-324, October 2002

• K.S. Anderson and M.J. Sadowski, “An Efficient Method for Contact/Impact Problems in Multibody Systems: Topologies with Many Loops”, Fourth Symposium on Multibody Dynamics and Vibration ASME DETC 2003

• J.H. Critchley and K.S. Anderson , "A Generalized Recursive Coordinate Reduction Method for Multibody Dynamic Systems", Journal of MultiscaleComputational Engineering, accepted for publication

• J. H. Critchley and K . S. Anderson, "A Parallel Logarithmic Order Algorithm for General Multi-body System Dynamics", Accepted for publication in Multibody System Dynamics Journal, 2004


What Do We Do?

• A Unified ApproachBridging the Gap Between Dynamics, Computer Science, and Numerics

• Recursive Coordinate ReductionRCR Parallelism and Application to Unilateral Constraints

• State-Time Dynamic FormulationState-of-the-Art Dynamic Formulation with the

Aim of Massively Parallel Computing


A Unified Approach

Numerics

ComputerScience

Dynamics NumericsNumerics

ComputerScience

ComputerScience

DynamicsDynamics

Decompose & solve equationsas they are formed

Decompose & solve equationsas they are formed

Formulate equations of motionto efficiently exploit available(parallel) computing resources

Formulate equations of motionto efficiently exploit available(parallel) computing resources

Develop numerics for minimummessage passing and optimalload balance

Develop numerics for minimummessage passing and optimalload balance


Associated Applications

Large Space Structures

n~2000, m~100

NASA

Tracked VehicleSimulation

n~952, m~88

A.R.O.

Massively RedundantMEMS devices

n~10000, m~2000

Zyvex, NIST

Multi-Scalemodeling ofNano-composites

n~105, m~ n

NSF NIRT

Automotive HandlingDurability, and Safetyn~24+, m~15+Ford

M1 AbramsInternational Space Station

Carbon Nano-tubeMicro-Gripper and Manipulator

Molecular Dynamics/Protein Folding

N~107, m~104

Sandia, NSF

Note: n= Number of System Generalized Coordinates, m = Number of System Constraints


Applications of Efficient Methods• Comprehensive design

optimization• Operator-in-the-loop

simulation• Hardware-in-the-loop

simulation• Model based predictive

control• Interactive design iteration• Virtual prototyping


A Need for EfficiencyRealizable simulations of large systems

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

7

8

9

10x 10

4flo

ps

degrees of freedom (n)

)(nO

)( 3nO

)(log 2 nO

*requiresa parallelcomputer

*


Multi-ScaleMultibody Dynamics

Discrete(fine scale)

Efficient Force CalculationsEfficient Force Calculations Efficient Multibody Efficient Multibody Dynamics AlgorithmsDynamics Algorithms

Adaptive Resolution ControlAdaptive Resolution Control-Generalized Momentum FormulationGeneralized Momentum Formulation

Adaptive Resolution Change :Adaptive Resolution Change :discrete, rigid and flexible modelsdiscrete, rigid and flexible models

Adaptive Domain Change:Adaptive Domain Change:H and P type refinementH and P type refinement

Articulated Rigid BodyArticulated Rigid BodyModel – Coarse grainedModel – Coarse grained

Articulated Flexible BodyArticulated Flexible BodyModel – Coarse grainedModel – Coarse grained

Multi-time Step Integration SchemesMulti-time Step Integration Schemes

Hierarchic Multi-resolution Substructured ModelHierarchic Multi-resolution Substructured Model

Better Fidelity and Faster SimulationsBetter Fidelity and Faster Simulations


Recursive Coordinate Reduction• A new low order kinematic

loop solution• Velocity level constraint

enforcement results in superior numerical stability

0 5 10 15 20 2510

-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

disp

lace

men

t er

ror

(met

ers)

time (seconds)

Acceleration ConstrainedVelocity Constrained

Four-bar linkage example

KinematicLoop


RCR ParallelismA new divide and conquer parallel algorithm

0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

8

9

10x 10

4

Number of Processors

Eff

ectiv

e F

lops

O(n)RCRPRCRP balanced

1-1001-100

. . .. . .

100

104.4

104.5

104.6

104.7

104.8

104.9

Number of Processors

Eff

ectiv

e Fl

ops

O(n)RCRPRCRP balanced

5050 5026-49

74-5150

26-49

74-51

Performance for a 128 Body Chain System


• Quarter Rotor Stage– Intellisuite– AUTOLEV– O(n+m)

• Parallel Pin Assembly

Bilateral Constraintsn=44, m=38, c=1

(n=173, m=149, c=4)

UnilateralConstraint

Unilateral Constraints via RCR


Efficient Design Sensitivity Determination for Multibody Systems

Design optimization of multibody systems (MBS) is time-consuming and complex tasks.

Optimization techniques with fast convergence (e.g., gradient-based) are often beneficial within this context.

Goals

Modeling

Analysis

Validation

Simulation


Sensitivity Analysis

Sensitivity analysis plays an important role in gradient-based optimization techniques and modern engineering applications.

Sensitivity analysis is also an asset to:

Assessment of design trend

Control algorithm developments

Determination of coupling strength in multidisciplinary design optimization (MDO)


Methods Developed Here OfferConsiderable Computational Savings

1600

0

800

400

1200

2000

O(n) Scale

O(n4) Empirical Data

O(n ) Empirical Data

Best Fit Quartic

Best Fit Linear

2 4 6 8 10 12

O(n4) Scale

0.1

0.2

0.3

0.4

0.5

Number of Degrees of Freedom nSi

mul

atio

n Ti

me

(sec

onds

)

•Traditional “Exact” Sensitivity Methods O(n4) [Cost Quartic in n]

•“Exact” Senstitivity Methods Developed hereO(n+m) [Cost Linear in n & m]

Examples:Simple Automobile Model: n=24, Collections of MEMS Devices: n~10000Detailed M1 Abrams: n=952, Detailed Nano-Structure: n~105

Space Station: n>2000 Future Needs: n>???


Outcomes:• Dynamic Simulation cost O(n+m) overall [Traditionally O(n3+nm2+m3) ]

• Design Sensitivity Analysis cost O(n+m) overall [Traditionally O(n4+n2m2+m3) ]

Research Spawned out of this Work (Funding Agency)•Efficient molecular dynamic modeling ( NSF NIRT†, Sandia†)•Multi-scale, multi-physics composite material modeling (NSF†, Sandia†)•Efficient track and drive chain modeling (A.R.O. †, MDI‡)•Virtual prototyping (Ford‡)•Distributed modeling/control of heavily redundant MEMS systems (NYSCAT‡, Zyvex‡) •Advanced computing aerospace system modeling (NASA)

† Proposal submitted or soon to be submitted‡ Collaboration or funding already established

Methods Developed Here OfferConsiderable Computational Savings


State-Time (ST) Formulation• State-Time vs. Traditional State Dynamic Formulations

– Sequential bottlenecks of current MBS dynamic algorithms– Parallel Scalability in Time

• Need for State-Time Approach– Full exploitation of future massively parallel computing

resources (>106 processors)– Growing interest and the need for simulation and analysis of

large-scale systems & even modest-size systems

• Using State-Time as an Implicit Integration Scheme in Sequential Applications– Minimization of the error over the current time step in an

‘average sense’ instead of ‘local’ minimization


– Sequential bottlenecks of current MBS dynamic algorithms–







– Sequential bottlenecks of current MBS dynamic algorithms– Parallel Scalability in Time







– Sequential bottlenecks of current MBS dynamic algorithms–







Full State-Time Implementation• Dynamic Analysis of Double Pendulum

x

y

1q

2q

111 ,, lIm

222 ,, lIm1q

2q

yf1

xf1

xf2

yf2

gm1

gm2

Parameters (SI):

12.1

15.0

118.0

1

02

02

2

2

01

01

1

1

qlm

qlm


x

y

1q

2q

111 ,, lIm

222 ,, lIm1q

2q

yf1

xf1

xf2

yf2

gm1

gm2

Parameters (SI):


x

y

1q

2q

111 ,, lIm

222 ,, lIm1q

2q

yf1

xf1

xf2

yf2

gm1

gm2

x

y

1q

2q

111 ,, lIm

222 ,, lIm1q

2q

yf1

xf1

xf2

yf2

gm1

gm2

Parameters (SI):

12.1

15.0

118.0

1

02

02

2

2

01

01

1

1

qlm

qlm

12.1

15.0

118.0

1

02

02

2

2

01

01

1

1

qlm

qlm


x

y

1q

2q

111 ,, lIm

222 ,, lIm1q

2q

yf1

xf1

xf2

yf2

gm1

gm2

x

y

1q

2q

111 ,, lIm

222 ,, lIm1q

2q

yf1

xf1

xf2

yf2

gm1

gm2

Parameters (SI):


Selected Simulation Results

Angular displacement, q1 vs. t

Position of the mass center, y2 vs. t

Constraint force, F1y vs. t

Sparse structure of the Tangent Matrix

0.000864

0.002432

0.006216

0.02728

0.12734

0.35022

0.90381

n

0.000864

0.002432

0.006216

0.02728

0.12734

0.35022

0.90381

n2

Error


Comparison Note• Using the State-Space Approach :

– # of equations and variables (spatial variables) per integration step : 4

– Parallelizability : Spatial domain

• Using the State-Time Approach :– # of equations and variables (spatial variables) per node

of a temporal element : 14– Parallelizability : Spatial & Temporal domain

• What methodology performs better in case of …– simulating even a modest-size system where millions of

integration steps are required for a desirable simulation?– having stiff modes/sharp gradients in some variables?– having a massive parallel computing machine?

Comparison Note• Using the State-Space Approach :

– # of equations and variables (spatial variables) per integration step : 4

– Parallelizability : Spatial domain

• Using the State-Time Approach :– # of equations and variables (spatial variables) per node

of a temporal element : 14– Parallelizability : Spatial & Temporal domain

• What methodology performs better in case of …– simulating even a modest-size system where millions of

integration steps are required for a desirable simulation?– having stiff modes/sharp gradients in some variables?– having a massive parallel computing machine?


ST as an Implicit Integration Scheme• Dynamic Analysis of a Stiff Double Pendulum

2.443.113.24Order of Accuracy

126754729# of steps

2.443.113.24Order of Accuracy

126754729# of steps

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