Discrete Differential Form Subdivision and Vector Field Generation over Volumetric Domain

Author: Huang, Jinghao

Year: 2012

Degree: Dissertation (Ph.D.)

Advisor: Schroeder, Peter

Committee Members: Schroeder, Peter; Aivazis, Michael A. G.; Bruno, Oscar P.; Desbrun, Mathieu

Option: Applied And Computational Mathematics; Materials Science

DOI: 10.7907/FXF2-4447

Abstract

This thesis presents a new method to construct smooth l- and 2-form subdivision schemes over the 3D volumetric domain. Based on the subdivided 1- and 2-form coefficient field, smooth vector fields can be constructed using Whitney forms. To obtain stencils in the regular setting, classical 0-form subdivision and linear 1- and 2-form subdivision over the octet mesh are introduced. Then, convoluting with a smooth operator, smooth 1- and 2-form subdivision schemes in the regular case can be determined up to one free parameter. This parameter can be determined by a novel technique based on spectrum and momentum considerations. However, artifacts exist in boundary regions because of the incomplete regular support and the shrinking feature of the original 0-form subdivision scheme. To address these problems, the projection-scaling method and the expansion method are introduced and compared. The former method projects arbitrary discrete differential forms to a subspace spanned by low-order potential fields. The algorithm subdivides these potential fields and reconstructs the discrete form in the refined level using linear combinations. Scaling is included for elements near the boundary to offset the effect of mesh shrinkage. Alternatively, for the expansion method, a compatible nonshrinking 0-form subdivision scheme is constructed first. Based on the new 0-form subdivision method, extending 1- and 2-forms beyond the boundary becomes natural. In the experiment, no noticeable artifacts, including attenuation, enlarging or undesirable bend, are found in practice.

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