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We study the optimization of an energy function used by the meshing community to measure and improve mesh quality. This energy is non-traditional because it is dependent on both the primal triangulation and its dual Voronoi (power) diagram. The energy is a measure of the mesh's quality for usage in Discrete Exterior Calculus (DEC), a method for numerically solving PDEs. In DEC, the PDE domain is triangulated and this mesh is used to obtain discrete approximations of the continuous operators in the PDE. The energy of a mesh gives an upper bound on the error of the discrete diagonal approximation of the Hodge star operator. In practice, one begins with an initial mesh and then makes adjustments to produce a mesh of lower energy. However, we have discovered several shortcomings in directly optimizing this energy, e.g. its non-convexity, and we show that the search for an optimized mesh may lead to mesh inversion (malformed triangles). We propose a new energy function to address some of these issues.

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TYPE Other Report YEAR 2017

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ECP Milestone Report WBS 2.3.4.13 ECP/VTK-m FY18Q1 [MS-18/01-03] Multiblock / Gradients / Release STDA05-5

Moreland, Kenneth D.; Pugmire, David; Geveci, Berk

The FY18Q1 milestone of the ECP/VTK-m project includes the implementation of a multiblock data set, the completion of a gradients filtering operation, and the release of version 1.1 of the VTK-m software. With the completion of this milestone, the new multiblock data set allows us to iteratively schedule algorithms on composite data structures such as assemblies or hierarchies like AMR. The new gradient algorithms approximate derivatives of fields in 3D structures with finite differences. Finally, the release of VTK-m version 1.1 tags a stable release of the software that can more easily be incorporated into external projects.

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TYPE Other Report YEAR 2017

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Statistical Inference for Porous Materials using Persistent Homology

Moon, Chul; Heath, Jason E.; Mitchell, Scott A.

We propose a porous materials analysis pipeline using persistent homology. We rst compute persistent homology of binarized 3D images of sampled material subvolumes. For each image we compute sets of homology intervals, which are represented as summary graphics called persistence diagrams. We convert persistence diagrams into image vectors in order to analyze the similarity of the homology of the material images using the mature tools for image analysis. Each image is treated as a vector and we compute its principal components to extract features. We t a statistical model using the loadings of principal components to estimate material porosity, permeability, anisotropy, and tortuosity. We also propose an adaptive version of the structural similarity index (SSIM), a similarity metric for images, as a measure to determine the statistical representative elementary volumes (sREV) for persistence homology. Thus we provide a capability for making a statistical inference of the uid ow and transport properties of porous materials based on their geometry and connectivity.

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TYPE Other Report YEAR 2017

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