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

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

Overview Motivation & Background 1) The Tire 2) The Terrain 3) Tire-Terrain Interaction 4) Validation 5) Conclusions & Future Work 6) 4/7/2016 2 University of Wisconsin

Motivation 2/2/2016 3 Energid

The Tire 4/7/2016 4 University of Wisconsin

ANCF – What is it? • A bsolute N odal C oordinate F ormulation • 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 4/7/2016 5 University of Wisconsin

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  x x r ( , ) e S ( ) e 4/7/2016 6 University of Wisconsin

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:   Me Q Q s e • We can use this to determine how the element moves over time! 4/7/2016 7 University of Wisconsin

ANCF – Mass • Getting the mass is easy:        T M S S dV o     V o • Can be performed as a preprocess 4/7/2016 8 University of Wisconsin

ANCF – External Forces • Getting the external force is easy: • Due to gravity: l   T Q A S f dx e g 0 • Due to a concentrated force: e  T Q S f • Can be used to apply contact! 4/7/2016 9 University of Wisconsin

ANCF – Internal Forces • Getting the internal force is hard : • Using the equation for strain energy: l l   1 1    2 2 U EA ( ) dx + EI ( ) dx 11 2 2 0 0 • We take the derivative of the strain energy with respect to the nodal coordinates   T   T l   l        11     Q EA ( ) dx + EI ( ) dx   s 11     e e 0 0 • Bad News: Must be performed at every time step • Good News: Can be performed in parallel! 4/7/2016 10 University of Wisconsin

ANCF Examples 4/7/2016 11 University of Wisconsin

ANCF – GPU Details (Internal Forces) 2 1 Memory representation: Nodal information A 5 3 4 0 2 1 6 3 4 0 Memory representation: Internal force information Simple mesh: B 3 3 5 4 1 2 0 6 - 2 elements (A&B) 6 - 7 nodes (0-6) Problem: Node overlap results in race conditions! 5 Solution: Internal forces are calculated on a per Element A: Nodes 0-1-2-3 element basis, a parallel reduce-by-key is used transform the element data into nodal data Element B: Nodes 3-4-5-6 4/7/2016 12 University of Wisconsin

Modeling the Tire 4/7/2016 13 University of Wisconsin

Modeling the Tire 4/7/2016 14 University of Wisconsin

The Terrain 4/7/2016 15 University of Wisconsin

Terrain Models (Terramechanics) • There are three main techniques that are used to study terramechanics: 2/2/2016 16 Energid

Empirical Methods • A force balance in the vertical direction yields an equation for the weight, W , of the tire: q 1 ( ) d q ò s cos q + t sin q W = rb q 2 � • Once the limits of the contact patch are determined, the drawbar pull and torque can be calculated by integrating the stresses over the wheel Forces, torques, and stresses on a driven, rigid wheel. Dynamic Bekker implementation. 2/2/2016 17 Energid

Continuum Methods • 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 (above). Continuum model (behind). 2/2/2016 18 Energid

Discrete Methods • 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). 2/2/2016 19 Energid

Tire-Terrain Interaction 4/7/2016 20 University of Wisconsin

The Complementarity Approach • Two important concepts Accounting for contact through complementarity 1) Posing Coulomb’s friction as an optimization problem 2) 2/2/2016 21 Energid

1) Accounting for contact through complementarity • 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: 2/2/2016 22 Energid

2) Posing Coulomb’s friction as an optimization problem • Actors in the Friction Force play, at a contact i : • Normal force  n • Friction coefficient µ • Relative slip velocity v S at the contact point • Two orthogonal directions d u and d w spanning the contact tangent plane • Components of friction force,  u and  w , found as solution of small optimization problem 2/2/2016 23 Energid

Complementarity Approach: The Math 4/7/2016 24

Complementarity Approach: The Math 4/7/2016 25

Complementarity Approach: The Math 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 26

Complementarity Approach: The Math M. Anitescu, Optimization-based Simulation of Nonsmooth Rigid Multibody Dynamics, Math. Program. 105 (1)(2006) 113-143 4/7/2016 27

Complementarity Approach: The Math 4/7/2016 28

The Optimization Angle 2/2/2016 29 Energid

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 4/7/2016 30

Complementarity Approach: Putting Things in Perspective • 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  2/2/2016 31 Energid

Tire-Terrain Interaction 4/7/2016 32 University of Wisconsin

Tire-Terrain Interaction 4/7/2016 33 University of Wisconsin

DEM – GPU Details (Collision Detection) • 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) University of Wisconsin - 6/29/2015 34 Madison

Validation 4/7/2016 35 University of Wisconsin

Longitudinal Slip Test - Setup Source: http://insideracingtechnology.com/ 4/7/2016 36 University of Wisconsin

Longitudinal Slip Test - Results 0.25 0.2 0.15 Drawbar Pull Coefficient [-] 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 -1.5 -1 -0.5 0 0.5 1 Slip [-] Source: http://insideracingtechnology.com/ 4/7/2016 37 University of Wisconsin

Single Wheel Test - Setup Investigates the contact stresses, drawbar pull, wheel torque, and sinkage of a wheel under controlled wheel slip and normal loading 2/2/2016 38 Energid

Single Wheel Test – Experimental Data • Measurements were taken for drawbar pull, torque, and sinkage Drawbar Pull vs. Slip Torque vs. Slip Sinkage vs. Slip 2/2/2016 39 Energid

Single Wheel Test – DEM Validation Drawbar Pull vs. Slip Torque vs. Slip Sinkage vs. Slip Normal load = 80 N Normal load = 130 N 2/2/2016 40 Energid

Single Wheel Test - Particle Tracking 2/2/2016 41 Energid

Single Wheel Test - Slip Ratio Negative Slip (Towed Wheel) Zero Slip (Perfect Rolling) Positive Slip (Driven Wheel) 2/2/2016 42 Energid

Conclusions & Future Work 4/7/2016 43 University of Wisconsin