Optimization Methods to Improve Particle Configurations For Smoothed Particle Hydrodynamics -Research SummaryPresented for LLNL WCI Summer Intern Poster Session, Aug. 2017

1. : Chosen Through Line-Search
Optimization Methods to Improve Particle Configurations for Smoothed Particle Hydrodynamics
1Luke Evans, 2Walt Nissen
1San Francisco State University, 2Lawrence Livermore National Laboratory
Figure 4: (Left) Voronoi Tessellation, (Right) --limited Voronoi Tessellation [2]
[1] Diehl, S. and Rockefeller, G. and Fryer, C.L, and Riethmiller, D., and Statler, T.S.,
“Generating Optimal Initial Conditions for Smoothed Particle Hydrodynamics
Simulations”, Publications of the Astronomical Society of Australia, 32, e048 (2015)
[2] Bullo F., Martinez S., Cortès J., “Motion Control with Distributed Information”, Control
Systems Magazine, Aug. 2007: 75 – 88.
[3] Bullo F., Martinez S., Cortès J., Distributed Control of Robotic Networks, Princeton
University Press, 2009. 117.
[4] “Mesh-Free Computational Fluid Dynamics”, College of Engineering and Informatics:
Biomedical Engineering, NUI Galway, 20 June 2016
Motivation: Authors in [1] cite a locational optimization algorithm in [2] as inspiration
Locational Optimization: Fixed number of facilities, searching for optimal placement of facilities given some
objective and constraints
• Coverage(Area Problem)[2]: points , radii ,
objective function:
The Voronoi Tessellation of points is a
partition of a domain into regions , where
is the set of points closer to than any other ,
denoted the Voronoi cell of . In an -limited Voronoi
Tessellation, the cell for is .
• controls fractional value of displacements, but is a free parameter : no pre-determined value
• Recommendation in [1]: if iterations don’t converge, pick smaller or decrease per iteration,
very ad hoc
• must be chosen for every different simulation so that iterations converge, but convergence
does not guarantee that gives optimal placement
• Is it possible determine within the algorithm that:
1) allows convergence?
2) gives optimal configuration for a given SPH problem?
: A Free Parameter?
Application:
• Redefine , without
• For the relaxation algorithm, it can be shown that approximates
• With , we can frame the relaxation algorithm as a gradient ascent algorithm
Maximizing A(P) for an r-limited Voronoi Tessellation[2]:
• For gradient ascent, each must move in the direction of
• faces the same direction as the averaged normal
vector to the arcs on the boundary of cell
Relaxation Algorithm as Gradient Ascent
Relaxation Algorithm(“WVT”)
In [1], technique relaxes particles through net displacements based
on distances to neighbors and smoothing lengths:
Net displacement of particle for all within
Contribution of neighboring particle inversely proportional to
i.e. : close neighbors displace much more than far ones
To maximize a function with a gradient ascent algorithm, we generate a sequence of inputs
with the property that at every , and with the rule ,
where is a scalar denoted the step-size, chosen to guarantee a sufficient increase for .The
process of choosing is known as a line search.
• Gradient descent for coverage: , corresponds to ,
• Redefine as the step-size chosen from a line search for the above problem
Significance:
1) that admits a convergent setup is now determined within the relaxation algorithm
2) local optimization framework can measure how effective a setup is
3) gradient descent framework can generate closer to optimal configurations
Implementation:
• Tested various modifications of the relaxation algorithm and coverage function
• Implemented in Draco from the PMESH project
• Currently testing accuracy of algorithm with line-search mu in Spheral++
We investigate the results given in [1] concerning particle
configurations for Smoothed Particle Hydrodynamics configurations. In
particular, we broaden the algorithm given for relaxation of initial
particle placements in regards to locational optimization and gradient
ascent methods.
Abstract
• Smoothed Particle Hydrodynamics is a meshless method for fluid flows, where a set
of discrete particles are carried with the flow:
• The fluid properties at any position are interpolated from property of nearby
particles
• Fluid property , position with smoothing length , smoothing kernel 𝑊
• Smoothing length determines the smoothing neighborhood of [1]
• Spatially Uniform v. Spatially Adaptive Distributions:
• Smoothing lengths : identical or varying across domain
• Initial particle setup needs:
• Interpolation Accuracy: Should accurately reproduce a given density field,
perform well with uniform and varying densities
• Isotropy: Setup should not cause preferred directions in simulation
Initial Conditions
Bibliography
This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.
Figure 1: Smoothing kernel acting over the smoothing neighborhood of [4]
Figure 3: Net displacement of
particle
Figure 2: Various setup methods for spherical volumes [1]
Figure 5: visualizing for several points [3]
LLNL-POST-699268