PNNL-13501 - Pacific Northwest National Laboratory

Graphics, Inc., workstation. The complexity of this
algorithm is conjectured to be linear in the number of
pairs of overlapping elements.
Seconds
100000
10000
1000
100
10
1
Summary and Conclusions
During the last 3 years, we have researched and applied
the mathematical theory of combinatorial algebraicgeometric
and topological invariants to remap, reconnect,
and renormalize multiscale, multidimensional, adaptive,
moving hybrid grids. In each of these cases, scalars,
vectors, and tensors are remapped by sets of coordinatefree
transformations between logical and real physical
n-space. This emerging computational technology, based
upon multiscale, multidimensional Free-Lagrangian
numerical methods, implies that the boundaries (external
and internal) of the computational domain must be
canonically based on the invariance of the solution(s) to
the partial differential equations that model the physical
problem domain. These may include, but are not limited
to material interfaces, liquid-gas interfaces, contact
discontinuities, and shock fronts.
References
Remaping Time
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Problem
Brute Force
Centroid Distance
Bridge
Figure 1. Tre-remapping 3-D hydro-transport calculation
Leutoff D, KG Roesner, et al. 1992. “Numerical
investigation of 3-dimensional shock focusing effects by
an invariant difference scheme.” Fluid Dynamics
Research 10(4-6):469.
134 FY 2000 Laboratory Directed Research and Development Annual Report
Vorodnitsyn V. 1994. “Finite difference models entirely
inheriting continuous symmetry of ordinary differential
equations.” Int. J. of Modern Phys., C 5(4):723.
Publications and Presentations
Oliveira and Trease. September 2000. “A symmetry
invariant first-order remapping methodology.” 7 th
International Conference on Numerical Grid Generation
in Computational Field Simulations.
Oliveira JS and Bailey. “Another universal axiomatization
of cubes.” Algebra Universalis (submitted).
Oliveira JS and Bailey. “The theory of cubic lattices.”
Mathematical Association of America (submitted).
Oliveira and Bailey. “Automorphisms of cubic
algebras-I.” Journal of Algebraic Combinatorics
(submitted).
Oliveira and Bailey. “Free cubic algebras.” Advances in
Mathematics (submitted).
Oliveira and Bailey. “Automorphisms of cubic
algebras-II.” (in internal peer review).
Oliveira and Bailey. “Automorphisms of cubic
algebras-IV.” (in internal peer review).
Oliveira and Bailey. “Some remarks on the algebra of
filters of a cubic algebra.” (in internal peer review).
Oliveira and Bailey. “Cube-like structures generated by
filters.” (in internal peer review).
Oliveira and Bailey. “MV-Algebras and cubic algebras.”
(in internal peer review).
Oliveira and Bailey “The face lattice of a multicube.” (in
internal peer review).
Oliveira and Bailey. “Congruencies and homomorphisms
of cubic algebras.” (in internal peer review).