Evaluation of Time Integration Schemes for the Generalized Interpolation Material Point Method Philip Wallstedt

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Evaluation of Time Integration Schemes for the
Generalized Interpolation Material Point Method
Philip Wallstedt philip.wallstedt_at_utah.eduJim
Guilkey james.guilkey_at_utah.edu
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Recurring Themes in Time Integration

• Close coupling between space and time
• Low order is better
• Linear error theories not applicable
• Behavior drastically different for large
  deformation
• So whats really going on?
• Manufactured solutions measure effect of time
  integration on
• Accuracy
• Stability

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Compare USF and USL
Update Stress Last
Update Stress First
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Compare Centered-Difference and USL
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Initialization to a Negative Half Time Step
If you know the answer
Use data at time 0
The easy way if first time step then
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Axis-Aligned Displacement in a Unit Square
Functions of coordinate directions only Corners
and edges of GIMP particles remain
aligned Sliding boundaries zero normal velocity
at surface.
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Axis-Aligned Displacement cont.
Diagonal terms only
Stress
Momentum
Solve for body force from momentum
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Axis-Aligned Displacement
Von Mises Stress Straight rows and columns only
with the right answer
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Definition of Error at a Particle
For a smooth problem in space and time check all
particles and all time steps
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Spatial Convergence
CD-GIMP is 2nd order the initialization
shortcut works USL changes to 1st order effect
of half step initialization Minor change to UGIMP
causes large error USF and MPM visually OK but
poor accuracy
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Temporal Convergence
Most methods display zero temporal convergence
until stability is lost, even though CD-GIMP is
formally 2nd order in time. USL loses one spatial
order, and gains one temporal sum of 2? We
conclude that spatial error dominates temporal
error such that reduced CFL has no benefit.
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GIMP Convergence Order

• Hypothesis
• GIMP is second order in space when
• Problem is smooth in space and time
• Particle edges are aligned
• Material boundary is accurately represented
• Therefore the 2D code is verified.

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Expanding Ring
Free surfaces with implied zero normal
stress GIMP particle edges not aligned More
general and representative
Stress Displacement
radius
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Expanding Ring Displacement
Radial Symmetry
Capital R is radius in reference configuration.
X and Y Displacement Components
Stress Displacement
radius
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Expanding Ring Cartesian Coordinates
Gradient Operators in terms of R and ?
Stress with zero Poissons ratio
Now we let Maple do the hard part . . .
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Stress Matrix
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Find c1, c2, and c3 by rotating the stress matrix
Rotation Matrix Q
Maple finds a simple answer
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Solve for Body Force
Maple generates C-compatible code for b
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Expanding Ring Results
Normal Stress
Hoop Stress
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Ring Spatial Convergence
The miss-alignment of particles and the
stair-stepped surface now dominate the
error. UGIMP gives up only for the highest
resolutions.
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Ring Temporal Convergence
UGIMP and GIMP nearly same accuracy Same trends
otherwise little temporal convergence
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Conclusions
The Method of Manufactured solutions allows order
of accuracy to be demonstrated for realistic
large deformation problems. The CD-GIMP
combination is significantly better than choices
involving UGIMP, MPM, USF, and USL. Formal
temporal orders of accuracy are usually not
observed in real solutions because spatial error
dominates temporal. CD-GIMP can be 2nd order if
problem is smooth, particle edges are aligned,
and surface is well-represented. Convergence
drops to 1st order for non-aligned
particles. Thanks to Mike Steffen, Mike Kirby ,
LeThuy Tran, and Martin Berzins, as well as DOE
grant W-7405-ENG-48.
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Expanding Disk C code from Maple
t1 pipi t3 1/rho t4 t1Et3 t5 RR t6 t5R t8 c2t5 t9 c1R t11 T(c3t6t8t9) t12 cos(H) t17 TT t18 t17T t19 c2c2 t20 t19t19 t27 c3c3 t31 Tc2 t34 t17c2 t35 c1c1 t39 t35c1 t42 t27t27 t44 t5t5 t48 t19c2 t52 t27c3 t54 t44R t57 t18t27 t61 t18t52 t66 t18c3 t77 t5c1 t80 12.0t18t20t619.0Tt19R68.0Tt27t612.0t31c19.0t34t353.0t18c2t3972.0t18t42t44t624.0t17t48t5120.0t17t52t54191.0t57 t54t19195.0t61t44t5c28.0t66Rt3956.0t57t6t3580.0t66t44t4824.0 t18t48t77 t82 Rt35 t88 t17c3 t89 t6t19 t94 t17t27 t98 t44c2 t101 t17t19 t104 Tc3 t124 15.0t18t19t82120.0t61t54c1130.0t88t8924.0t88t82112.0t94t6c1221.0t94t9830.0t101t932.0t104t976.0t104t86.0c2122.0 t88t8c161.0t66t8t35130.0t66t89c1221.0t57t98c116.0c3R t125 t80t124 t127 t104t5 t129 t31R t131 Tc1 t151 1/(3.0t1272.0t1291.0t131)/(1.03.0t94t445.0t88t6c24.0 t88t773.0t34t9t17t352.0t1312.0t101t53.0t1294.0t127) t156 sin(H) b0 -t4t11t12-t3ETt12t125t151/2.0 b1 -t4t11t156-t3t125ETt156t151/2.0 b2 0.0