Efficient Storage and Processing of Adaptive Triangular Grids using Sierpinski Curves Csaba Attila Vigh Department of Informatics, TU München JASS 2006,

Efficient Storage and Processing of Adaptive Triangular Grids using

Sierpinski Curves

Csaba Attila VighDepartment 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 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 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


• 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

1,0 21,0


• 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

n22


Hilbert’s Mapping and three Iterations


Six Iterations

Peano’s Space-Filling Curve

• Partitioning in 9 Subintervals and Subsquares

• Subintervals mapped to Subsquaresn23 n23

3 4 92 5 81 6 7

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


• 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



• 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


Recursive Construction of the Grid on a Triangular Domain


• 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


• 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)


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


• 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


• 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


Red (Circles), Green (Boxes)


• 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


• 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


• 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


• 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


• 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


• Determining Color of the Nodes1. 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)


• 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


•Recursive Propagation of Edge Parameters

•Knowing Scenario for the Cell also know Scenarios for Subcells


• 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

4321 ,,, xxxx 34321 ,,, xxxx

21 , xx


• Bisection of Tetrahedron along Tagged Edge

,

• Sierpinski Curve Approximated by Polygonal Line of the Tagged Edges

542354314321 ,,,,,,,,,, xxxxxxxxxxxx 215 , xxx

233121 ,,,, xxxxxx


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!

Documents

Efficient Storage and Processing of Adaptive Triangular Grids using Sierpinski Curves Csaba Attila Vigh Department of Informatics, TU München JASS 2006,