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Solid Geometry Processing on Deconstructed Domains
Author(s) -
Sellán Silvia,
Cheng Herng Yi,
Ma Yuming,
Dembowski Mitchell,
Jacobson Alec
Publication year - 2019
Publication title -
computer graphics forum
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.578
H-Index - 120
eISSN - 1467-8659
pISSN - 0167-7055
DOI - 10.1111/cgf.13592
Subject(s) - geometry processing , tetrahedron , discretization , geometry , solid geometry , computer science , boundary (topology) , complex geometry , convergence (economics) , differential geometry , variety (cybernetics) , finite element method , boundary representation , mathematics , mathematical analysis , polygon mesh , artificial intelligence , physics , economics , thermodynamics , economic growth
Many tasks in geometry processing are modelled as variational problems solved numerically using the finite element method. For solid shapes, this requires a volumetric discretization, such as a boundary conforming tetrahedral mesh. Unfortunately, tetrahedral meshing remains an open challenge and existing methods either struggle to conform to complex boundary surfaces or require manual intervention to prevent failure. Rather than create a single volumetric mesh for the entire shape, we advocate for solid geometry processing on deconstructed domains , where a large and complex shape is composed of overlapping solid subdomains. As each smaller and simpler part is now easier to tetrahedralize, the question becomes how to account for overlaps during problem modelling and how to couple solutions on each subdomain together algebraically . We explore how and why previous coupling methods fail, and propose a method that couples solid domains only along their boundary surfaces. We demonstrate the superiority of this method through empirical convergence tests and qualitative applications to solid geometry processing on a variety of popular second‐order and fourth‐order partial differential equations.