S6465: Physics-Based Modeling of Flexible Tires on...

GPU TECHNOLOGY CONFERENCE:

S6465: Physics-Based Modeling of Flexible Tires on Deformable Terrain with the GPU

Daniel Melanz, Dan Negrut

Simulation-Based Engineering LaboratoryUniversity of Wisconsin - Madison

Overview

1) Motivation & Background

2) The Tire

3) The Terrain

4) Tire-Terrain Interaction

5) Validation

6) Conclusions & Future Work

4/7/2016 2University of Wisconsin

Energid

Motivation

2/2/2016 3

The Tire

4/7/2016 University of Wisconsin 4

ANCF – What is it?

• Absolute Nodal Coordinate Formulation

• Used for the dynamics analysis of flexible bodies that undergo large deformation

• It is consistent with the nonlinear theory of continuum mechanics

• It is computationally efficient:• Constant mass matrix

• Zero Coriolis and centrifugal effects

• Several opportunities for parallelism


ANCF – How is it defined?

• A single ANCF element is defined by a series of nodes

• Each of these nodes are comprised of degrees of freedom that describe:• The position of the node in space

• Vectors that describe the slope of the element at that point

• A shape function is used to translate the nodal coordinates into Cartesian coordinates


( , ) ( )r e S ex x

ANCF – How does it work?

• Now that we can describe a particular element, we can do useful things with it

• Using the Principle of Virtual Work for the continuum, the following governing differential equation is obtained:

• We can use this to determine how the element moves over time!


s e Me Q Q

ANCF – Mass

• Getting the mass is easy:

• Can be performed as a preprocess


T

o

o

V

dV

M S S

ANCF – External Forces

• Getting the external force is easy:• Due to gravity:

• Due to a concentrated force:

• Can be used to apply contact!


T

e Q S f

0

l

T

e gA dx Q S f

ANCF – Internal Forces

• Getting the internal force is hard:• Using the equation for strain energy:

• We take the derivative of the strain energy with respect to the nodal coordinates

• Bad News: Must be performed at every time step

• Good News: Can be performed in parallel!


2 2

11

0 0

1 1( ) + ( )

2 2

l l

U EA dx EI dx

1111

0 0

( ) + ( )

T Tl l

s EA dx EI dx

Qe e

ANCF Examples


ANCF – GPU Details (Internal Forces)

1


2

3

04

56

A

BSimple mesh:

- 2 elements (A&B)

- 7 nodes (0-6)

Element A: Nodes 0-1-2-3

Element B: Nodes 3-4-5-6

0

0

1

1

2 3

32

4

43

5

5

6

6

Memory representation: Nodal information

Memory representation: Internal force information

Problem: Node overlap results in race conditions!

Solution: Internal forces are calculated on a per element basis, a parallel reduce-by-key is used transform the element data into nodal data

Modeling the Tire


The Terrain


Terrain Models (Terramechanics)

2/2/2016 Energid 16

• There are three main techniques that are used to study terramechanics:

Empirical Methods

2/2/2016 Energid 17

• A force balance in the vertical direction yields an equation for the weight, W, of the tire:

• Once the limits of the contact patch are determined, the drawbar pull and torque can be calculated by integrating the stresses over the wheel

W = rbq2

q1

ò s cosq +t sinq( )dq

Forces, torques, and stresses on a driven, rigid wheel. Dynamic Bekker implementation.

Continuum Methods

2/2/2016 Energid 18

• Continuum methods assume matter to be homogeneous and continuous

• Uses a set of partial differential equations (PDE) with boundary conditions

• Meshes are adopted to approximate the solution

• Examples: FDM, FVM, FEM

Continuum model (behind). Continuum model (above).

Discrete Methods

2/2/2016 Energid 19

• The discrete element method (DEM) represents soil as a collection of many three-dimensional bodies

• When elements collide forces and torques are generated using explicit equations

• By modeling soil using individual bodies, DEM can model the soil much more accurately

Bodies with polyhedral geometry.Particle image velocimetry (MIT).

Tire-Terrain Interaction


Energid

The Complementarity Approach

2/2/2016

• Two important concepts

1) Accounting for contact through complementarity

2) Posing Coulomb’s friction as an optimization problem

21

Energid

1) Accounting for contact through complementarity

2/2/2016

• Two possible scenarios• The distance (gap) between bodies is

greater than zero, therefore the contact force n is zero

Or,

• The gap between bodies is zero, therefore the contact force n is non-zero

• One complementarity conditions captures both scenarios:

22

Energid

2) Posing Coulomb’s friction as an optimization problem

2/2/2016

• Actors in the Friction Force play, at a contact i:• Normal force n• Friction coefficient µ

• Relative slip velocity vS at the contact point

• Two orthogonal directions du and dw spanning the contact tangent plane

• Components of friction force, u and w , found as solution of small optimization problem

23

4/7/2016 24

Complementarity Approach: The Math

4/7/2016 25


4/7/2016 26

D.E. Stewart and J.C. Trinkle. An implicit time-stepping scheme for rigid body dynamics with inelastic collisions and coulomb friction. IJNME, 39:2673-2691, 1996.


4/7/2016 27

M. Anitescu, Optimization-based Simulation of Nonsmooth Rigid Multibody Dynamics, Math. Program. 105 (1)(2006) 113-143


4/7/2016 28


Energid

The Optimization Angle

2/2/2016 29

4/7/2016 30

Time Integration• Life is good once the frictional contact forces at the interface

between shapes are available

• Velocity at new time step l+1 computed as

• Once velocity available, the new set of generalized coordinates computed as

Energid

Complementarity Approach: Putting Things in Perspective

2/2/2016

• Complementarity conditions employed to link distance between shapes and normal force

• Friction posed as an optimization problem

• Equations of motion became equilibrium constraints, an appendix to optimization problem

• DVI discretized to lead to nonlinear complementarity problem

• Relaxation yields CCP, which was solved via a QP with conic constraints to compute

31

University of Wisconsin -Madison

DEM – GPU Details (Collision Detection)

6/29/2015 34

• Generate pair-wise geometrical information

• Efficient implementations• Broad phase

• Narrow phase

• Example: 2D collision detection, bins are squares• Body 4 touches bins A4, A5, B4, B5

• Body 7 touches bins A3, A4, A5, B3, B4, B5, C3, C4, C5

• In proposed algorithm, bodies 4 and 7 will be checked for collision by three threads (associated with bin A4, A5, B4)

Validation


Longitudinal Slip Test - Setup


Source: http://insideracingtechnology.com/

Longitudinal Slip Test - Results


Source: http://insideracingtechnology.com/

Slip [-]-1.5 -1 -0.5 0 0.5 1

Dra

wbar

Pull

Coeff

icie

nt [-

]

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Single Wheel Test - SetupInvestigates the contact stresses, drawbar pull, wheel torque, and sinkage of a wheel under controlled wheel slip and normal loading

Energid2/2/2016 38

• Measurements were taken for drawbar pull, torque, and sinkage

2/2/2016 Energid

Single Wheel Test – Experimental Data

Drawbar Pull vs. Slip Torque vs. Slip Sinkage vs. Slip

39

2/2/2016 Energid

Single Wheel Test – DEM ValidationDrawbar Pull vs. Slip Torque vs. Slip Sinkage vs. Slip

No

rma

l lo

ad

= 8

0 N

No

rma

l lo

ad

= 1

30 N

40

Single Wheel Test - Particle Tracking

Energid2/2/2016 41

Single Wheel Test - Slip Ratio

Energid

Negative Slip (Towed Wheel)

Zero Slip (Perfect Rolling)

Positive Slip (Driven Wheel)

2/2/2016 42

Conclusions & Future Work


Conclusions

• Our goal is to enable Chrono to solve complex, engineering problemsthrough the use of novel algorithms implemented on state-of-the-art hardware

• The discrete element method of terramechanics using contact through complementarity has been validated against both analytical and experimental results

• Flexible bodies(Absolute Nodal Coordinate Formulation) are combined with deformable terrain to model complex off-road vehicle dynamics


Future Work

• Use a more robust ANCF plate element for tire modeling

• Perform lateral validation tests

• Incorporate distributed normal force on element face to simulate tire pressure

• Attach the tire to a full vehicle


Thank you.

• Source available for download under BSD-3 http://spikegpu.sbel.org/

• For all of our animations, please visit https://vimeo.com/uwsbel

• For more information about the Simulation-Based Engineering Laboratory, please visit http://sbel.wisc.edu/

University of Wisconsin 464/7/2016

http://spikegpu.sbel.org/

https://vimeo.com/uwsbel

Thank You.

[email protected]

Simulation Based Engineering Lab

Wisconsin Applied Computing Center


mailto:[email protected]

S6465: Physics-Based Modeling of Flexible Tires on...

Documents

Transcript of S6465: Physics-Based Modeling of Flexible Tires on...