Proceedings of the IMPLAST 2010 Conference
October 12-14 2010 Providence, Rhode Island USA
© 2010 Society for Experimental Mechanics, Inc.
The Optimal Transportation Meshfree (OTM) method for
ballistic limit impact simulations
B. Li, A. Kidane, G. Ravichandran and M. Ortiz∗
∗
Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA, U.S.A.,[email protected]
Abstract
We develop an Optimal Transportation Meshfree (OTM) method for simulating extremely large deformation
and ballistic limit impact problems. The method combines concepts from Optimal Transportation theory with
material-point sampling and local max-ent meshfree approximation. The proposed OTM method generalizes
the Benamou-Brenier differential formulation of optimal mass transportation problems to problems including
arbitrary geometries, essential boundary conditions, constitutive behavior, and multibody contacts. An energybased material-point erosion algorithm is also proposed for simulating discontinuous material failure phenomena.
Three dimensional OTM simulations with material-point erosion algorithm of impact of metallic plates by spherical
particles over a range of impact velocity and thicknesses of plates have been performed to study the performance
and efficiency of the OTM method. In-house experiments with the same configurations have been conducted to
validate the OTM method.
1
Introduction
It’s well known there are many complexities associated with the ballistic limit impact. Experiments may be very
expensive and time consuming, and it’s possible to obtain very few data by the limitation of current laboratory
facilities. Numerical methods, on the other hand, have become very useful and important tools to predict the
ballistic limit and look into the details of the complex process. However the accurate simulation of terminal ballistics
is a grand challenge in scientific computing that places exacting demands on physics models, solvers and computing
resources. In order to meet these challenges, we develop an Optimal Transportation Meshfree (OTM) method for
simulating extremely large deformation and ballistic limit impact problems.
The proposed OTM method is a meshfree updated-lagrangian methodology which combines concepts from Optimal Transportation theory with material-point sampling and local max-ent meshfree approximation, and overcomes
the essential difficulties in grid-based numerical methods like Lagrangian and Eulerian finite element method. The
rationale behind the approach is as follows. We resort the Benamou-Brenier [3] differential formulation of optimal
mass transportation problems and its connection to the Wasserstein distance [5] to discretize the inertial action in
space and time within a strictly variational framework. The resulting discretization may be regarded as the result
of restricting the inertial action to mass measures concentrated on material points undergoing piecewise rectilinear
motions. The density of such mass measures and the constrained minimization structure of the problem may be expected to confer the discretization robust convergence properties. The optimal transportation variational framework
also results in: proper mass matrices and inertia forces in the presence of continuously varying spatial interpolation;
geometrically exact mass transport and satisfaction of the continuity equation; and exact linear and angular momentum conservation. Finally, fields requiring differentiation, such as deformation and velocity fields, are interpolated
from nodal values using max-ent shape functions. These shape functions are reconstructed continuously from the
nodal set and have the key property of possessing a Kronecker-delta property at the boundary, which enables the
direct imposition of displacement boundary conditions. An energy-based material-point erosion algorithm is also
proposed for simulating fracture and fragmentation phenomena. The convergence of the material-point erosion algorithm to Griffith-fracture solutions is ensured by means of Γ-convergence [7]. Three dimensional OTM simulations
with material-point erosion algorithm of impact of metallic plates by spherical particles have been conducted. The
accuracy and performance of the method in presence is illustrated in this paper by comparing with the in-house
experiments.
Following a brief description of the methodology and material-point erosion algorithm in section 2, the main
numerical results using the OTM method with the comparison to the in-house experiment results are presented in
section 3. In section 4 the main conclusion afforded by the analysis is provided.
2
2.1
Methodology
The Optimal Transportation Meshfree method
We begin by briefly summarizing the OTM method of spatial and temporal discretization. The aim is to approximate
a flow at discrete times 𝑡0 , 𝑡1 , . . . , 𝑡𝑘 , 𝑡𝑘+1 , . . . . To this end, we consider the semi-discrete action sum
𝑆𝑑 (𝜑1 , . . . , 𝜑𝑁 −1 ) =
}
𝑁
−1 {
∑
1 𝑑2𝑊 (𝜌𝑘 , 𝜌𝑘+1 ) 1
−
[𝑈
(𝜑
)
+
𝑈
(𝜑
)]
(𝑡𝑘+1 − 𝑡𝑘 )
𝑘
𝑘+1
2 (𝑡𝑘+1 − 𝑡𝑘 )2
2
(1)
𝑘=0
where 𝜑𝑘 : 𝐵 ⊂ ℝ𝑛 → ℝ𝑛 is the deformation mapping at time 𝑡𝑘 , 𝜌𝑘 is the corresponding mass density at time 𝑡𝑘 ,
∫
𝑈 (𝜑) =
𝑓 (∇𝜑) 𝑑𝑥
(2)
𝐵
is the free energy of the solid, 𝑓 (∇𝜑) is the local free-energy density per unit mass, and
∫
𝑑2𝑊 (𝜌𝑎 , 𝜌𝑏 ) =
inf
∣𝑇 (𝑥) − 𝑥∣2 𝜌𝑎 (𝑥) 𝑑𝑥
𝑇 :𝐵→𝐵
𝜌𝑎 =𝜌𝑏 det(∇𝑇 )
(3)
is the Wasserstein distance between two mass densities 𝜌𝑎 and 𝜌𝑏 over 𝐵. The term 21 𝑑2𝑊 (𝜌𝑘 , 𝜌𝑘+1 )(𝑡𝑘+1 − 𝑡𝑘 ) in
(1) supplies a measure of the inertial action between times 𝑡𝑘 and 𝑡𝑘+1 . We also note that in writing (1) we have
restricted attention to elastic behavior and unforced systems for simplicity. Extensions accounting for forcing, e. g.,
in the form of body forces, boundary tractions, and extensions to inelasticity may be found in [6].
In order to render the governing equations fully discrete, we begin by approximating mass densities by point
masses, namely,
𝑀
∑
(
)
𝜌ℎ,𝑘 (𝑥) =
𝑚𝑝 𝛿 𝑥 − 𝑥𝑝,𝑘 ,
(4)
𝑝=1
(
)
where 𝑥𝑝,𝑘 represents the position at time 𝑡𝑘 of a material point of constant mass 𝑚𝑝 and 𝛿 𝑥−𝑥𝑝,𝑘 is the Dirac-delta
distribution centered at 𝑥𝑝,𝑘 , Fig. 1. Li et al. [6] have shown that the constancy of the material point masses 𝑚𝑝
is indeed equivalent to the weak satisfaction of the continuity equation. To complete the spatial discretization, we
approximate the incremental deformation mapping as
𝜑ℎ,𝑘→𝑘+1 (𝑥) =
𝑁
∑
𝑥𝑎,𝑘+1 𝑁𝑎,𝑘 (𝑥),
(5)
𝑎=1
where {𝑥𝑎,𝑘+1 , 𝑎 = 1, . . . , 𝑁 } is an array of nodal coordinates at time 𝑡𝑘+1 and 𝑁𝑎,𝑘 (𝑥) are conforming shape
functions defined over the configuration at time 𝑡𝑘 . In calculations, we specifically use max-ent shape functions [1]
computed from the array {𝑥𝑎,𝑘 , 𝑎 = 1, . . . , 𝑁 } of nodal coordinates at time 𝑡𝑘 . Since max-ent shape functions are
strongly localized, the interpolation at a material point 𝑥𝑝,𝑘 depends solely on the nodes contained in a small local
neighborhood 𝒩𝑝,𝑘 of the material point, Fig. 1. In calculations, the local neighborhoods are continuously updated
using range searches [4] to account for the relative motion between material points and nodes.
Inserting these approximations into (1) we obtain the fully-discrete action
𝑁
−1 ∑
𝑀 {
∑
𝑚𝑝 ∣𝑥𝑝,𝑘+1 − 𝑥𝑝,𝑘 ∣2
2 (𝑡𝑘+1 − 𝑡𝑘 )2
𝑘=0 𝑝=1
]}
[
(
)
(
)
1
(𝑡𝑘+1 − 𝑡𝑘 ),
− 𝑚𝑝 𝑓 ∇𝜑ℎ,𝑘 (𝑥𝑝,𝑘 ) + 𝑚𝑝 𝑓 ∇𝜑ℎ,𝑘+1 (𝑥𝑝,𝑘+1 )
2
𝑆ℎ (𝜑ℎ,1 , . . . , 𝜑ℎ,𝑁 −1 ) =
(6)
Np,k
Figure 1: The optimal transportation meshfree methodology.
Algorithm 1 OTM time step
Require: Initial and final times for time step, 𝑡𝑘 , 𝑡𝑘+1 .
Require: Initial nodal coordinates, 𝑥𝑘 = {𝑥𝑎,𝑘 , 𝑎 = 1, . . . , 𝑁 }.
Require: Initial material-point coordinates, {𝑥𝑝,𝑘 , 𝑝 = 1, . . . , 𝑀 }.
Require: Initial shape functions {{𝑁𝑎,𝑘 (𝑥𝑝,𝑘 ), 𝑎 ∈ 𝒩𝑝,𝑘 }, 𝑝 = 1, . . . , 𝑀 }.
Require: Ibid gradients {{∇𝑁𝑎,𝑘 (𝑥𝑝,𝑘 ), 𝑎 ∈ 𝒩𝑝,𝑘 }, 𝑝 = 1, . . . , 𝑀 }.
Require: Initial deformation gradients, {𝐹𝑝,𝑘 , 𝑝 = 1, . . . , 𝑀 }.
Require: Initial material state at all material points.
−1
1: Compute mass matrix 𝑀𝑘 , nodal forces 𝑓𝑘 , accelerations 𝑎𝑘 = 𝑀𝑘 𝑓𝑘 .
2: Update nodal coordinates: 𝑥𝑘+1 = 𝑥𝑘 + (𝑡𝑘+1 − 𝑡𝑘 )(𝑣𝑘 + 12 (𝑡𝑘+1 − 𝑡𝑘−1 )𝑎𝑘 ).
3: Update nodal velocities: 𝑣𝑘+1 = (𝑥𝑘+1 − 𝑥𝑘 )/(𝑡𝑘+1 − 𝑡𝑘 ).
4: Update material-point coordinates: 𝑥𝑝,𝑘+1 = 𝜑𝑘→𝑘+1 (𝑥𝑝,𝑘 ), 𝑝 = 1, . . . , 𝑀 .
5: Update deformation gradients: 𝐹𝑝,𝑘+1 = ∇𝜑𝑘→𝑘+1 (𝑥𝑝,𝑘 )𝐹𝑝,𝑘 , 𝑝 = 1, . . . , 𝑀 .
6: Effect constitutive updates at material points.
7: Update local material-point neighborhoods 𝒩𝑝,𝑘+1 , 𝑝 = 1, . . . , 𝑀 .
8: Compute shape functions {𝑁𝑎,𝑘+1 (𝑥𝑝,𝑘+1 ), 𝑎 ∈ 𝒩𝑝,𝑘+1 }, 𝑝 = 1, . . . , 𝑀 .
9: Ibid gradients {∇𝑁𝑎,𝑘+1 (𝑥𝑝,𝑘+1 ), 𝑎 ∈ 𝒩𝑝,𝑘+1 }, 𝑝 = 1, . . . , 𝑀 .
where we again consider the unforced elastic case for simplicity. The discrete trajectories now follow from the discrete
Hamilton’s principle
𝛿𝑆ℎ = 0
(7)
of stationary action.
The algorithm resulting from the preceding scheme is listed in Algorithm 1. We see from Fig. 1 that the OTM
scheme can be solved forward explicitly. This forward solution has the usual structure of explicit time-integration
and updated-Lagrangian schemes. In particular, all the finite kinematics of the motion, including the mass density
and volume updates, are geometrically exact. In addition, the continuous reconstruction of the local material-point
neighborhoods and shape functions has the effect of automatically reconnecting the material points and the nodal set,
at no cost of remapping the local states carried by the material points. This property of the method is particularly
convenient for inelastic materials whose local material state often includes additional internal variable information. A
particularly convenient feature of OTM, which is common to other material-point based methods [2], is that seizing
contact is automatically accounted for. This is so because of the cancelation of linear momentum that naturally
occurs when colliding nodes come within the local neighborhoods of material points.
2.2
The Material-Point Erosion Algorithm
Finally, we briefly outline the material-point erosion algorithm proposed to simulate the fracture and fragmentation
phenomena in terminal ballistics calculations. Within the context of OTM calculations, fracture can be modeled
simply by eroding or failing material points according to an energy-release criterion. Following Fraternali et al. [7]
we compute the energy-release rate attendant to the removal of material point 𝑝 as
𝐺𝑝,𝑘+1 =
1
𝜖2
∑
𝑚𝑞 𝑓𝑘 (𝐹𝑞,𝑘+1 ),
(8)
𝑥𝑞,𝑘+1 ∈𝐵𝜖 (𝑥𝑝,𝑘+1 )
where 𝐵𝜖 (𝑥𝑝,𝑘+1 ) is the ball of radius 𝜖 centered at 𝑥𝑝,𝑘+1 . The radius 𝜖 defines a length scale intermediate between
the discretization size and the macroscopic size of the bodies. The material point is removed when
𝐺𝑝,𝑘+1 ≥ 𝐺𝑐 ,
(9)
where 𝐺𝑐 is a critical energy release rate that measures the material-specific energy required to create a fracture surface of unit area. For linear elasticity, Fraternali et al. have shown that criterion (8) and (9) result in approximations
that converge to Griffith fracture in the limit of an infinitely fine discretization.
3
3.1
Results
Experimental Setup
A series of in-house ballistics experiments on aluminum alloy 6061-T6 plates by s2 tool hardened steel spherical
particles are performed. These experiments provide a severe test to validate the OTM method. The projectile with
diameter 0.07” is glued on to a light-weight styrofoam sabot 1” in diameter and 2” in length and propelled by a gas
gun. A general view of the gas gun used in the tests is shown in Fig. 2. The gas gun is filled with helium with the
Figure 2: A general view of the gas gun.
pressure at the desired value of 20 to 80 psi, depending on the required velocity in the range of 100 to 400 m/s. The
target plate with thickness in {0.032”, 0.04”, 0.05”, 0.063”} is positioned at the end of the barrel and attached to
the 1” wide steel frame. The two vertical edges of the plate are free and the other two edges (top and bottom) are
clamped to the steel frame, which resulted in a 6” x 4” target area. Near the impact end of the barrel, light emitting
diodes are placed in two orifices on the periphery of the barrel and two fast-response photo detectors are placed in
orifices diametrically opposite to the diodes. When the projectile passes through these orifices it generates a ramp
pulse whose duration is recorded using a high-speed WaveSurfer 24Xs oscilloscope (LeCroy, Chestnut Ridge, NY),
which has a bandwidth of 200 MHz and a sampling rate of 2.5 Giga samples per second. The projectile velocity is
then calculated from the time of travel of the projectile across the two detectors. The calculated projectile velocity
is verified by calculating the pulse duration corresponding to the projectile crossing a single detector. The velocity
of the projectile is calibrated against gas pressure and can be controlled within 2 to 5 m/s. The perforation area is
measured using high-precision gage pins and verified using a surface profile scanning technique. The surface profile
scanning technique employs a ConoScan 3000 (Optimet, North Andover, MA) which uses a conoscopic holography
technology and has a precision of 10 m over a scan area of up to 120 x 120 mm2 . A typical surface profile is shown
in Fig. 3.
3.2
Numerical Results
The OTM simulations employ the same configurations including the exact essential boundary conditions, but simulate
target plates with thicknesses continuously in the range of 0.032” to 0.063”. In our tests all material are described
by means of engineering 𝐽2 -viscoplasticity models with power-law hardening, rate-sensitivity and thermal softening.
The material parameters have been chosen according to the related literature [8]. Fig. 4 shows a typical perforation
process for the impact of a 0.063” Al6061-T6 plate by the steel spherical particle at 370 m/s.
A parametric study of the ballistics behavior of the material system in the thickness range [0.032”, 0.063”]
and velocity range [100, 400]m/s is performed by the OTM method. A comparison of the numerical results and
experimental results is shown in Fig. 5, which illustrates the fact that the present method is in principle capable to
capture three dimensional ballistics phenomena and matches the reality very well. As it may be seen, the numerical
results overestimate the experimental data by as much as 15 m/s. However due to the weight of the sabot, the
ballistic limits measured from experiments might be underestimated. Fig. 6(a) shows the predicted perforation
area of the 0.05” thick aluminum alloy target impacted by steel spherical projectile at different velocities. The
experimental measurements are also shown for comparison and validation. Again the agreement between simulations
and experiments is excellent. The computed variation of the residual velocity 𝑉𝑟 with the incident velocity 𝑉𝑖 for the
same configuration is illustrates in Fig. 6(b).
4
Conclusions
We have presented an Optimal Transportation Meshfree (OTM) method which enables the prediction of terminal
ballistics. The theoretical basis of the OTM method guarantees the exact conservation of mass, linear and angular
momentum. The proposed material point erosion algorithm greatly extends the applicability of the OTM method.
The performance and accuracy of the proposed method has been tested by three dimensional ballistic limit impact
simulations and in-house experiments. The agreement between numerical and experimental results is excellent.
Acknowledgements
The authors gratefully acknowledge the support of the Department of Energy National Nuclear Security Administration under Award Number DE-FC52-08NA28613 through Caltech’s ASC/PSAAP Center for the Predictive Modeling
and Simulation of High Energy Density Dynamic Response of Materials.
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Figure 3: A typical 3D surface profile of the perforated aluminum target measured by conoscope.
(a) 𝑡 = 2𝜇𝑠
(b) 𝑡 = 10𝜇𝑠
(c) 𝑡 = 20𝜇𝑠
(d) 𝑡 = 32𝜇𝑠
Figure 4: Perforation of 0.063” thick aluminum plate at impact velocity 370 m/s.
experiments with perforation
experiments without perforation
Velocity(m/s)
interface between regions with and without
perforation by the OTM method
Region with perforation
Region without perforation
Thickness (thousands of an inch)
Figure 5: A parametric study of the aluminum plate with different thicknesses in the range [0.032”, 0.063”] impacted
by steel spherical projectile with different velocities in the range of [100, 400]m/s.
without plate
Vr [m/s]
0.05” OTM
Vi [m/s]
(a)
(b)
Figure 6: The 0.05” thick aluminum alloy target impacted by steel spherical projectile.(a)Perforation area at different
velocities;(b)Incident velocity 𝑉𝑖 vs. residual velocity 𝑉𝑟 .