Efficient Storage and Processing of Adaptive Triangular Grids

Efficient Storage and Processing of Adaptive Triangular Grids using Sierpinski Curves Csaba Attila Vigh Department of Informatics, TU München JASS 2006, course 2: Numerical Simulation: From Models to Visualizations

Outline • Adaptive Grids – Introduction and basic ideas • Space-Filling curves – Geometric generation – Hilbert’s, Peano’s, Sierpinski’s curve • Adaptive Triangular Grids – Generation and Efficient Processing • Extension to 3 D

Adaptive Grids – Basics • Why do we need Adaptive grids? • Modeling and Simulation – PDE – mathematical model – Discretization – Solution with Finite Elements or similar methods – Demand for Adaptive Refinement – very often

Adaptive Grids – Basics • Adaptive Refinement – Trade-off between Memory Requirements and Computing Time – Need to obtain Neighbor Relationships between Grid Cells – Storing Relationships Explicitly leads to: • Arbitrary Unstructured Grids • Considerable Memory Overhead

Adaptive Grids – Basics • Adaptive Refinement - want to save memory? – Use a Strongly Structured Grid – Use Recursive Splitting of Cells (Triangles) – Neighbor Relations must be computed – Computing Time should be small

Adaptive Grids – Basics • Processing of Recursively Refined (Triangular) Grid – Linearize Access to the Cells using Space. Filling Curves • For Triangles – Sierpinski Curve – Use a Stack System for Cache-Efficiency – Parallelization Strategies using Space-Filling Curves are readily available

Space-Filling Curves • 1878, Cantor – Any two Finite-Dimensional Manifolds have same Cardinality – [0, 1] can be Mapped Bijectively onto the Square [0, 1]x[0, 1], or onto the Cube • 1879, Netto – such a Mapping is necessarily Discontinuous

Space-Filling Curves • Is then possible to obtain a Surjective Continuous Mapping? or • Is there a Curve that passes through every Point of a Two-Dimensional Region? • 1890, Peano constructed the first one

Hilbert’s Space-Filling Curve • Hilbert’s Geometric Generating Process – If Interval I ( ) can be mapped continuously onto the square Q ( ) • Partition I into Four Congruent Subintervals • Partition Q into Four Congruent Subsquares – Then each Subinterval can be Mapped Continuously onto one of the Subsquares – Next continue the Partitioning Process on the Subintervals and Subsquares

Hilbert’s Space-Filling Curve • Hilbert’s Geometric Generating Process – After n Partitioning Steps I and Q are split into Congruent Replicas – Subsquares can be arranged such that • Adjacent Subintervals correspond to Adjacent Subsquares with an Edge in common • Inclusion Relationships are preserved

Hilbert’s Space-Filling Curve Hilbert’s Mapping and three Iterations

Hilbert’s Space-Filling Curve Six Iterations

Peano’s Space-Filling Curve • Partitioning in 9 Subintervals and Subsquares • Subintervals mapped to Subsquares Peano’s Mapping

Peano’s Space-Filling Curve Three Iterations of the Peano Curve

Sierpinski’s Space-Filling Curve Four Iterations of the Sierpinski Curve • Slicing the Square into half by its Diagonal • Half of the Curve lies on one Triangle • Other half lies on the other Triangle

Sierpinski’s Space-Filling Curve • Curve may be viewed as a Map from Unit Interval I onto a Right Isosceles Triangle T • T with Vertices at (0, 0), (2, 0), (1, 1) • Hilbert’s Generating Principle – Partition I into two Congruent Subintervals – Partition T into two Congruent Subtriangles – Order of Subtriangles shown in the next picture

Sierpinski’s Space-Filling Curve

Sierpinski’s Space-Filling Curve • Curve starts from (0, 0), ends at (2, 0) • Exit Point from each Subtriangle coincides with Entry Point of the next one • Requirement on Orientation in Subtriangles shown in picture below

Recursively Structured Triangular Grids and Sierpinski Curves – Computational Domain • Right Isosceles Triangle – Starting Cell – Grid constructed recursively • Split each Triangle Cell into 2 Congruent Subcells • Splitting Repeated until Desired Resolution is Reached • Grid may be Adaptive – Local Splitting

Recursively Structured Triangular Grids and Sierpinski Curves Recursive Construction of the Grid on a Triangular Domain

Recursively Structured Triangular Grids and Sierpinski Curves • Cells are in Linear Order on the Sierpinski Curve • Corresponds to Depth-First Traversal of the Substructuring Tree • Additional Memory 1 bit per Cell indicating whether – Cell is a Leave, or – Cell is Adaptively Refined

Recursively Structured Triangular Grids and Sierpinski Curves • Extensions for Flexibility – Several Initial Triangles may be used – Arbitrary Triangles may be used if • Structure of Recursive Subdivision preserved • One Leg is defined as Tagged Edge and will take the role of the Hypotenuse – Tagged Edge can be replaced by a Linear Interpolation of the Boundary (see next picture)

Recursively Structured Triangular Grids and Sierpinski Curves Subdividing Triangles at Boundaries

Discretization of the PDE • A Discretization with Linear FE – Generates • Element Stiffness Matrices • Right Hand Sides – Accumulates them into Global System of Equations for the Unknowns on the Nodes • We consider it to be too Memory Consuming

Discretization of the PDE • Assumption – Stiffness Matrix Computation possible on the fly, or – Hardcode it into the Software • Typical for Iterative Solvers – Contain Matrix-Vector Product between Stiffness Matrix and Unknowns • Memory used only for storing Grid Structure

Discretization of the PDE • Classical Node-Oriented Processing – Loop over Unknowns (Nodes on Grid) – Requires Access to all neighbor Nodes – Difficult in a Recursively Structured Grid – Neighbor could be on a Different Subtree • Our Approach: Cell-Oriented Processing

Cache Efficient Processing of the Computational Grid • Cell-Oriented Processing – Need Access to Unknowns for each Cell – Process Elements along the Sierpinski Curve • Sierpinski Curve Divides Unknowns into two halves – Left of the Curve: Red Nodes – Right of the Curve: Green Nodes – See picture next

Cache Efficient Processing of the Computational Grid Red (Circles), Green (Boxes)

Cache Efficient Processing of the Computational Grid • Access to Unknowns is like Access to a Stack • Consider Unknowns 5 to 10 – During Processing Cells to the Left – Access in Ascending Order – During Processing Cells to the Right – Access in Descending Order • Nodes 8, 9, 10 Placed in turn on Top of the Stack

Cache Efficient Processing of the Computational Grid • System of Four Stacks – to Organize Access to Unknowns – Read Stack holds Initial Value of Unknowns – Two Helper Stacks – Red and Green – hold Intermediate Values of Unknowns of respective Color – Write Stack stores Updated Values of Unknowns

Cache Efficient Processing of the Computational Grid • When Moving from one Cell to the other – 2 Unknowns Adjacent to Common Edge can always be reused – 2 Unknowns opposite to Common Edge must be processed: • One from Exited Cell • One in the New Cell

Cache Efficient Processing of the Computational Grid • Unknown from Exited Cell – Put onto Write Stack – if processing complete – Put onto Helper Stack of respective Color – if needed by other Cells • Unknown in the New Cell – Take from Read Stack – if never used it before – Take from Helper Stack of respective Color – if already used it before

Cache Efficient Processing of the Computational Grid • Unknown from Exited Cell – Count number of Accesses – Determine whether Processing is Complete or not – Determine the Color – Left or Right side of the Sierpinski Curve ? – Curve Enters and Exits at the 2 Nodes adjacent to the Hypotenuse – Only 3 possible Scenarios

Cache Efficient Processing of the Computational Grid • Determining Color of the Nodes 1. Curve Enters through Hypotenuse – Exits across Opposite Leg 2. Curve Enters through Adjacent Leg – Exits through Hypotenuse 3. Curve Enters and Exits across the Opposite Legs Red (circles), Green (boxes)

Cache Efficient Processing of the Computational Grid • Unknown in the New Cell – Determine Color as above – Determine whether New or Old • Consider the 3 Triangle Cells adjacent to “This Cell” • One is Old – where the Curve entered • One is New – where the Curve exits • Third Cell may be Old or New – check Adjacent Edges – Both New Third Cell is New Unknown is New – Unknown is Old otherwise

Cache Efficient Processing of the Computational Grid • Recursive Propagation of Edge Parameters • Knowing Scenario for the Cell also know Scenarios for Subcells

Cache Efficient Processing of the Computational Grid • Processing of the Grid is managed by a set of 6 Recursive Procedures • On the Leaves the Discretization-Level Operations are performed • Example from Maple worksheet is next

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Parallelization – 5 Equal parts

Conformity of Locally Refined Grids • No hanging Nodes • Maintaining Conformity in any Locally Refined Grid – Consider Triangles, Tetrahedrons or NSimplices Refined with Recursive Bisections – Need only Finite Number of Additional Bisections for Completion – Locality of Refinement is preserved – Grid will not become Globally Uniformly Refined

3 D Sierpinski Curves • • 2 D Sierpinski Curve fills a Triangle 3 D Curve expected to fill a Tetrahedron How to subdivide a Tetrahedron? Tetrahedron with a Tagged Edge: – 4 -Tuple – Edge with is • Directed • Tagged • Takes the role of the Hypotenuse

3 D Sierpinski Curves • Bisection of Tetrahedron along Tagged Edge , • Sierpinski Curve Approximated by Polygonal Line of the Tagged Edges

3 D Sierpinski Curves Bisection of a Tagged Tetrahedron. Red Arrows approximate the Sierpinski Curve.

Conclusion • Algorithm Efficiently generates and processes Adaptive Triangular Grids • Memory Requirement is minimal • Hope to achieve Computational Speed competitive with Algorithms based on Regular Grids • Extension to 3 D is currently subject to research

Questions? ? Thank You!