[PPT] - S6465: Physics-Based Modeling of Flexible Tires on Deformable PowerPoint Presentation

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GPU TECHNOLOGY CONFERENCE:

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

Daniel Melanz, Dan Negrut Simulation-Based Engineering Laboratory University of Wisconsin - Madison

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Overview

1)

Motivation & Background

2)

The Tire

3)

The Terrain

4)

Tire-Terrain Interaction

5)

Validation

6)

Conclusions & Future Work

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Energid

Motivation

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The Tire

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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

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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

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( , ) ( )  r e S e x x

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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!

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s e

  Me Q Q

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ANCF – Mass

Getting the mass is easy:
Can be performed as a preprocess

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T

V

dV          



M S S

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ANCF – External Forces

Getting the external force is easy:
Due to gravity:
Due to a concentrated force:
Can be used to apply contact!

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T e 

Q S f

l T e g

A dx   Q S f

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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!

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2 2 11

1 1 ( ) + ( ) 2 2

l l

U EA dx EI dx    



11 11

( ) + ( )

T T l l s

EA dx EI dx                     

 

Q e e

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ANCF Examples

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ANCF – GPU Details (Internal Forces)

1

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2 3 4 5 6 A B Simple mesh:

2 elements

(A&B)

7 nodes (0-6)

Element A: Nodes 0-1-2-3 Element B: Nodes 3-4-5-6 1 1 2 3 3 2 4 4 3 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

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Modeling the Tire

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Modeling the Tire

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The Terrain

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Terrain Models (Terramechanics)

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

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Empirical Methods

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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 = rb

q2 q1

ò s cosq +t sinq

( )dq

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

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Continuum Methods

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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).

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Discrete Methods

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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).

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Tire-Terrain Interaction

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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

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Energid

1) Accounting for contact through complementarity

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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:

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Energid

2) Posing Coulomb’s friction as an optimization problem

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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

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Complementarity Approach: The Math

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Complementarity Approach: The Math

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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.

Complementarity Approach: The Math

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

Complementarity Approach: The Math

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Complementarity Approach: The Math

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Energid

The Optimization Angle

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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

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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 

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Tire-Terrain Interaction

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Tire-Terrain Interaction

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University of Wisconsin - Madison

DEM – GPU Details (Collision Detection)

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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)

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Validation

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Longitudinal Slip Test - Setup

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Source: http://insideracingtechnology.com/

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Longitudinal Slip Test - Results

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Source: http://insideracingtechnology.com/

Slip [-]

1.5
1
0.5

0.5 1

Drawbar Pull Coefficient [-]

0.25
0.2
0.15
0.1
0.05

0.05 0.1 0.15 0.2 0.25

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Single Wheel Test - Setup

Investigates the contact stresses, drawbar pull, wheel torque, and sinkage of a wheel under controlled wheel slip and normal loading

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Measurements were taken for drawbar pull, torque, and sinkage

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Single Wheel Test – Experimental Data

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

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2/2/2016 Energid

Single Wheel Test – DEM Validation

Drawbar Pull vs. Slip Torque vs. Slip Sinkage vs. Slip Normal load = 80 N Normal load = 130 N

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Single Wheel Test - Particle Tracking

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Single Wheel Test - Slip Ratio

Energid

Negative Slip (Towed Wheel) Zero Slip (Perfect Rolling) Positive Slip (Driven Wheel)

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Conclusions & Future Work

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Conclusions

Our goal is to enable Chrono to solve complex, engineering problems

through 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

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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

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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/

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Thank You.

melanz@wisc.edu Simulation Based Engineering Lab Wisconsin Applied Computing Center

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