AbstractParticle tracking can serve as a useful tool in engineering analysis, visualization, and is an essential component of many Eulerian-Lagrangian (EL) transport schemes. In this work, we consider particle tracking (PT) techniques that are robust and efficient for complex, transient velocity fields on unstructured computational meshes. Several test problems from common 2-D rotational and 3-D helical examples to a remediation scenario in a large-scale, 3-D groundwater system are used to evaluate the proposed PT schemes.

Introduction• Transport processes are central to many of the Army’s

contemporary scientific and engineering challenges.• Accurate characterization of the evolution and

migration of quantities of interest is essential. • Inaccurate, overly diffused or oscillatory transport

simulation results can be quite misleading.• EL transport schemes combine a Lagrangian

representation for advection with Eulerian descriptions for sources/sinks and other physical processes. When they work well, EL methods provide significantly better resolution of advection using low-order approximations in time and space.

• EL formulations invariably contain a core PT component.

• PT quality dictates much of the accuracy of the whole EL approximation as well as efficiency on serial and parallel platforms.

• In the past, most research has focused on how well various EL schemes solve transport with PT error avoided or minimized by using velocity fields that allow analytical tracking.

• Here, we consider PT techniques that are robust and efficient for complex, transient velocity fields on unstructured computational meshes.

• Specifically, we formulate semi-analytical approximations for relevant, low-order velocity representations and develop general purpose techniques based on adaptive, variable-order Runge-Kutta (RK) integration with error control that are suitable for generic velocity approximations.

Particle Tracking

• Governing equation:

• Adaptive element-by-element tracking

Multistage RK:

Adaptive time steps:

• Semi-analytical tracking

RT0 velocity:

€

dx

dt= v(x, t)

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x(t) = x0 +v t

vxt

e1

2vxtΔt 2 +vxΔt

−1 ⎛

⎝ ⎜

⎞

⎠ ⎟+ v0 −

vxv t

vxt

⎛

⎝ ⎜

⎞

⎠ ⎟⋅ e

1

2vxt Δt +

vx

vxt

⎛

⎝ ⎜

⎞

⎠ ⎟2

π

2vxt

⋅ erfcvxt

2

vx

vxt

⎛

⎝ ⎜

⎞

⎠ ⎟− erfc

vxt

2Δt +

vx

vxt

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥, for vxt > 0

x(t) = x0 +v t

vxt

e1

2vxtΔt 2 +vxΔt

−1 ⎛

⎝ ⎜

⎞

⎠ ⎟+ v0 −

vxv t

vxt

⎛

⎝ ⎜

⎞

⎠ ⎟⋅ e

1

2vxt Δt +

vx

vxt

⎛

⎝ ⎜

⎞

⎠ ⎟2

2

−vxt

⋅ es2

−vxt

2

vx

vxt

−vxt

2Δt +

vx

vxt

⎛

⎝ ⎜

⎞

⎠ ⎟

∫ ds, for vxt < 0

x(t) = x0 + v0

evxΔt −1

vx

+ v t

evxΔt − 1+ Δt( )

vx( )2 , for vxt = 0, vx ≠ 0

x(t) = x0 + v0Δt +1

2v t Δt( )

2, for vxt = 0, vx = 0

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Δt* = Δt1⋅ s f ⋅1

D

⎛

⎝ ⎜

⎞

⎠ ⎟1

P

, where D = maxj =1, d

E j

ε a +ε r ⋅max x j , x j*

( ); E j = x j − x j

*

Δt* = Δt1⋅xb − x0

x1 − x0

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v(x, t) = a e (t) + be (t)x, where ae ∈Rd ; be ∈R

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k1 = v x0, t0( ); k2 = v x0 + Δt1a2,1k1, t0 + c 2Δt1( ); ...

kS = v x0 + Δt1⋅ (aS,1k1 + ...+ aS,S −1kS −1), t0 + c SΔt1( )

⇒ x1 = x0 + Δt1⋅ b1k1 + ...+ bSkS( )

Semi-analytical solution:

€

k1€

k2

€

k3

€

k4

€

x0

€

x1

€

x0

€

x1

€

x1*

€

x0 €

x1

€

xb

Particle Tracking

Error 21 x 21 41 x 41 81 x 81 e1 0.266 0.00888 0.00348

e∞ 0.128 0.0328 0.0152

Example 2, Semi-analytical tracking error

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v = cos(πt /3)(2π (x −1/2), - 2π (y −1/2),3/2)

Helical velocity fieldIn the first example, we considersemi-analytical tracking (SA) in ahelical velocity field given by

on the unit cube. The maximum error associated with projecting onto RT0 was 0.0360 on a coarse 11x11x11 mesh was 0.036

Level set vortex example

Next, we consider the SA tracking forlevel set propagation,

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vx = cos(πt /8)sin(2πy)sin2(πx), vy = -cos(πt /8)sin(2πx)sin2(πx)

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∂u

∂t+

r v • ∇u = 0

and compare SA tracking on a 41x41 mesh with a stabilized FEM solution from a 161x161 mesh.

Groundwater remediation

Here, we consider efficiency of the RK schemes for an RDX cleanup scenario

Method TS nt,ax10-6 nt,tx10-6 e∞

RK2 0.01 195. 195 7.94

RK2 0.01 19.7 20.3 17.6

RK2 1.0 2.38 3.31 79.4

RK2 Cr = 1 1.67 5.39 175.

RK4 0.1 19.7 20.3 0.766

RK4 1.0 2.37 3.29 2.23

RK4 Cr = 1 0.751 1.97 5.29

RK45 (10-6) 0.1 4.53 5.28 -

RK45 (10-6) 1.0 2.99 4.10 0.389

RK45 (10-6) Cr = 1 2.86 5.13 0.633

Conclusions• The semi-analytical approach can be extended to unstructured

simplicial meshes in R2,3. The resulting scheme is accurate and may be a viable computational alternative for problems where RT0 velocities are available.

• Increasing the approximation order from two to four in our numerical element-by-element tracking scheme improved computational efficiency significantly. The adaptive RK45 further improved efficiency and provided formal error control to meet user-defined accuracy requirements.

• Steady-state rotation:

• Accelerated vortex:

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vx = 0.002πy, vy = −0.002πx; Ω = [−2000,2000] × [−2000,2000]

Method TS nt,ax10-6 nt,tx10-6 e1 e∞

RK2 5 0.976 1.23 5.60 16.8

RK2 20 0.363 6.80 78.8 224.

RK2 200 0.290 0.747 539. 1300.

RK2 Cr = 1 0.290 0.748 538. 1300.

RK4 2 2.28 2.51 0.0361 0.282

RK4 20 0.359 0.667 1.16 3.48

RK4 100 0.253 0.636 13.4 41.8

RK4 200 0.251 0.640 28.0 236.

RK4 500 0.252 0.648 33.5 317.

RK4 1,000 0.252 0.647 33.6 317.

RK4 Cr = 1 0.250 0.639 36.6 322.

RK45 (10-7) 100 14.8 15.9 0.0221 0.124

RK45 (10-7) 1,000 14.8 15.9 0.0222 0.124

RK45 (10-7) Cr = 1 14.8 15.9 0.0222 0.124

RK45 (10-6) 100 1.38 2.18 0.0646 0.273

RK45 (10-6) 1,000 1.39 2.19 0.0625 0.273

RK45 (10-6) Cr = 1 1.39 2.19 0.0626 0.273

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vx = 5cos(πt /8)sin(2πy)sin2(πx)

vy = −5cos(πt /8)sin(2πx)sin2(πy)