A Bezier Based Approach to Unstructured Moving Meshes

1. A Bezier Based Approach to Unstructured Moving Meshes ALADDIN and Sangria Gary Miller David Cardoze Todd Phillips Noel Walkington Mark Olah Miklos Bergou

2. The Sangria Project • Goal: Simulation of blood flow on a microscopic level • Need to solve Navier-Stokes fluid dynamics equations • Challenges • Cells have a non-linear boundary that changes over time • Discontinuities across boundaries

3. Problem: Need to keep track of various functions over our domain (Pressure, Temperature, Velocity, etc.) Need to deal with dynamic curved domain Must represent these functions in a small amount of space on a computer Representation must be accurate Representation must be efficient for numerically solving PDE’s Solution: Use a Mesh Divide domain into simple geometric elements Define a finite set of basis functions on these elements Approximate function as a linear combination of basis functions Only need to store scalar coefficients on nodes to represent function Motivation for Meshing

4. Mesh Example Linear Triangular Mesh, Unstructured

5. Eulerian Framework • Domain is statically meshed and used throughout the simulation • Boundaries and blood cell locations are simply functions defined on the domain • Advantage: Geometry is simple, static mesh does not move or deform • Disadvantage: Boundaries are only approximations • Disadvantage: Many elements required for accuracy

6. Lagrangian Framework • Elements themselves move over time, boundaries are “real” and exist in the geometry • Problems • As mesh moves elements deform • Elements may be added and removed over time • Advantages • Since boundaries lie in geometry, they are more accurate • Requires fewer elements

7. What Type of Elements? • Linear Triangles? • Very easy to represent (set of 3 points) • Very easy to deal with geometrically • Quality metrics well understood • No small angles implies good linear element • NOT good at approximating curved boundaries or domains • NOT good at approximating non-linear motion

8. Advantages of Curved Elements • Better approximation of curved domain boundaries and curved boundaries within the mesh • Better approximation of non-linear motion in a moving mesh

9. Bezier Curves • A Bezier Curve is a polynomial curve, parameterized over u=[0,1] • A Bezier curve of degree n is determined by n+1 control values, p0 …pn+1 • We use Quadratic Bezier Curves (3 control points) • Benefits: • Easy evaluation, since they are polynomials • Easy subdivision via the deCasteljau algorithm • End point values are interpolated along curves • Curve lies within the convex hull of its control points

10. Bezier Triangles • Triangle made from 3 Bezier edges • Defined by set of 6 control points • Consists of 4 underlying linear triangles, called the control mesh

11. BSplines for Boundaries • BSplines are piecewise Bezier curves • They maintain an additional condition of continuity along the curve • We use Quadratic BSplines which interface naturally with our Quadratic Bezier Edges

12. Mesh Implementation • Unstructured mesh where elements consist of Bezier Edges and Bezier Triangles • BSplines used for boundaries • Uses Lagrangian framework, so elements of mesh move over time • Mesh moves in discrete time steps based on velocity field given by Navier-Stokes • As mesh moves, elements will become deformed, areas in need of detail will change as well • Apply cleaning operations at each time step to keep mesh, of high quality; well sized and well shaped • Remember that functions are only approximations, so we must constantly strive to keep errors small

13. Delaunay Triangulation • Examine circumcircle of each triangle • Triangle is Delaunay if no other points are within circle • Delaunay Meshes maximize the minimum angle • Small angles are bad because they increase interpolation error Edge Flip

14. Mesh Quality • High Quality mesh ensures low interpolation error • Mesh size (Number of Elements) • Mesh grading (Avoid drastic element size changes) • Element Quality (Avoid large interpolation errors) • Linear triangles must not have small angles • Higher-Order Quality via edge smoothing

15. Cleaning Process: Step 1Edge Flips to Maintain Delaunay • A quadratic edge flip is implemented as four edge flips in the control mesh • Localized operation (2 Triangles)

16. Cleaning Process: Step 2Edge Smoothing for High-Order Quality • Identify overly curved triangles, smooth edge and reinterpolate • Keeps control mesh well shaped • Also a localized operation (2 Triangles)

17. Cleaning Process: Step 3Mesh Coarsening • Given a sizing function on mesh, determine areas that have too many small triangles • Coarsen mesh by using edge flips and vertex removal • Keeps the number of mesh elements low • Local operation (expected num. of triangles <=6)

18. Cleaning Process: Step 4Mesh Refinement • Identify poorly sized triangles – Too big • Identify poor logical triangles – Small angles • Use Rupert Refinement; insert circumcenters of logical triangles, then flip out to Delaunay • Expected constant number of edge flips

19. Overview of Operation • Project functions onto mesh’s basis • Move mesh discretely according to velocity function • Clean mesh • Edge Flips • Smoothing • Coarsening • Refinement • Send functions to solver, receive new functions, and repeat

20. Moving Mesh Example: Our Prototype

21. My Research: Optimizing Implementation • Original mesh implementation done in Ruby • Advantages of Ruby • Ruby is flexible like Perl, object oriented like Java, and can be used functionally like ML • Easy to evaluate new techniques and algorithms • Garbage collection • Disadvantages of Ruby • Garbage collection! • Difficult to control heap usage • Slow control structures (for_each) • Primitives are large (float ~ 24 bytes)

22. Solution: Implement Mesh in C++ • Achieve speedups and small memory footprint by: • Efficiently managing heap allocation • Separating data from mesh topology using dictionaries • Reducing dependence on large hash structures • Developing mutable data structures (B-Splines) • Utilize more efficient algorithms • Fast linear system solver for spline movement • Fibonacci heaps for coarsening

23. Project Organization C++ RUBY SOLVER SOLVER CLEANER CLEANER MESH OPERATORS MESH OPERATORS TOPOLOGY TOPOLOGY AND DATA DATA COMMON FILE FORMAT

24. Program Organization C SOLVER RUBY SOLVER RUBY DEBUGGER RUBY WRAPPERS SOLVER API SIMULATION CLEANER BEZIER MESH BOUNDARY MESH MESH I/O CELL COMPLEX V E F DATA PTS CONTROL PTS

25. Results • Achieved memory savings by a factor of 100. • Able to practically operate on meshes with more than 300,000 degrees of freedom • Achieved move/clean speeds that approach those of production-quality moving meshes • Smaller meshes move/clean in “real time” • New implementation is flexible and provides easy integration with other programs and languages

26. 28,000 Node Mesh 110,000 Node Mesh