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 3D

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

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

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

Step 7

Step 8

Step 9

Step 10

Step 11

Step 12

Step 13

Step 14

Step 15

Step 16

Step 17

Step 18

Step 19

Step 20

Step 21

Step 22

Step 23

Step 24

Step 25

Step 26

Step 27

Step 28

Step 29

Step 30

Step 31

Step 32

Parallelization – 5 Equal parts

Conformity of Locally Refined Grids No hanging Nodes Maintaining Conformity in any Locally Refined Grid –Consider Triangles, Tetrahedrons or N- Simplices 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

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

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

3D 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 3D is currently subject to research

Questions? ??? Thank You!