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Efficient Algorithms for Solving Static HamiltonJacobi Equations
, 2003
"... Consider the eikonal equation, = 1. If the initial condition is u = 0 on a manifold, then the solution u is the distance to the manifold. We present a new algorithm for solving this problem. More precisely, we present an algorithm for computing the closest point transform to an explicitly described ..."
Abstract

Cited by 48 (6 self)
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Consider the eikonal equation, = 1. If the initial condition is u = 0 on a manifold, then the solution u is the distance to the manifold. We present a new algorithm for solving this problem. More precisely, we present an algorithm for computing the closest point transform to an explicitly described manifold on a rectilinear grid in low dimensional spaces. The closest point transform finds the closest point on a manifold and the Euclidean distance to a manifold for all the points in a grid (or the grid points within a specified distance of the manifold). We consider manifolds composed of simple geometric shapes, such as, a set of points, piecewise linear curves or triangle meshes. The algorithm computes the closest point on and distance to the manifold by solving the eikonal equation = 1 by the method of characteristics. The method of characteristics is implemented efficiently with the aid of computational geometry and polygon/polyhedron scan conversion. Thus the method is named the characteristic/scan conversion algorithm. The computed distance is accurate to within machine precision. The computational complexity of the algorithm is linear in both the number of grid points and the complexity of the manifold. Thus it has optimal computational complexity. The algorithm is easily adapted to sharedmemory and distributedmemory concurrent algorithms. Many query problems...
EXPERIMENTAL STUDY OF THE DEVELOPMENT OF PLASTIC SOLITARY WAVES IN ONEDIMENSIONAL GRANULAR MEDIA BY
"... In this work, a modified split Hopkinson pressure bar (SHPB) is used to impact onedimensional granular chain of spheres. These homogeneous chains used here comprise of brass, aluminum, or stainless steel spherical beads, ranging from a single sphere to a chain of fourteen, and are of interest becaus ..."
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In this work, a modified split Hopkinson pressure bar (SHPB) is used to impact onedimensional granular chain of spheres. These homogeneous chains used here comprise of brass, aluminum, or stainless steel spherical beads, ranging from a single sphere to a chain of fourteen, and are of interest because of their unique wave propagation characteristics, as seen in earlier efforts. Loading magnitudes spanning from 9 kN to 40 kN – considerably higher than most previous works on these systems which have been conducted in the elastic regime – cause the granular chains to deform plastically. These conditions allow for solitary waves, which propagate in the onedimensional array of elastic spheres, to be studied in the plastic regime. The propagating pulse assumes a distinctive shape after travelling through five beads, and can consequently be realized as a plastic solitary wave. The wave speed of this pulse was seen to depend on maximum force, as in the elastic as, although it was measured to be less than the elastic wave speed. In addition, the plastic velocity varied with the 1/9 th power instead of the 1/6 th power for the elastic speed. For the case of brass, the plastic wave propagates at 50 % to 80 % the speed of the elastic wave, depending on whether the incident or transmitted force is compared. It was found that there is also decreasing plasticity along the chain length except at the end beads in contact with the SHPB, which rebounds into the bar and are hit again. This research is the first to investigate in detail the development and evolution of a plastic solitary wave and will form the basis of future work in this area. ii Acknowledgements The author would like to express his utmost appreciation to Professor John Lambros for his assistance and guidance for the past two years. In addition, special thanks to Erheng Wang for his helpful contributions, Jodi Gritten for ordering research materials, and the machinists who manufactured parts needed to conduct the experiments: Greg Milner and Dustin Burns. Special
ACTA UNIVERSITATIS APULENSIS No 15/2008 PROPAGATION OF STRESS WAVES IN ELASTIC SOLIDS
"... Abstract. The propagation of waves in elastic media under dynamic loads (stress waves) are investigated. The nature of deformation, stress, stress–strain relations, and equation of motion are some objectives of investigation. General decompositions of elastic waves are studied. Two planar waves in a ..."
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Abstract. The propagation of waves in elastic media under dynamic loads (stress waves) are investigated. The nature of deformation, stress, stress–strain relations, and equation of motion are some objectives of investigation. General decompositions of elastic waves are studied. Two planar waves in an infinite isotropic elastic medium, and then time–harmonic solutions of the wave equations are analyzed. Spherically symmetric waves in three–dimensional space from a point source, radially symmetric waves in a solid infinite cylinder of radius a, and waves propagated over the surface of an elastic body are studied. Finally, particular solutions of the Navier equations are given.