A Bezier Based Approach to Unstructured Moving Meshes

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A Bezier Based Approach to Unstructured Moving
Meshes

• ALADDIN and Sangria
• Gary Miller
• David Cardoze
• Todd Phillips
• Noel Walkington
• Mark Olah
• Miklos Bergou

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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

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Motivation for Meshing

• 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 PDEs

• 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

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Mesh Example
Linear Triangular Mesh, Unstructured
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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

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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

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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

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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

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Bezier Curves

• A Bezier Curve is a polynomial curve,
  parameterized over u0,1
• A Bezier curve of degree n is determined by n1
  control values, p0 pn1
• 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

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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

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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

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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

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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
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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

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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)

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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)

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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 lt6)

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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

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Overview of Operation

• Project functions onto meshs basis
• Move mesh discretely according to velocity
  function
• Clean mesh
• Edge Flips
• Smoothing
• Coarsening
• Refinement
• Send functions to solver, receive new functions,
  and repeat

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Moving Mesh Example Our Prototype
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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)

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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

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Project Organization
C
RUBY
SOLVER
SOLVER
CLEANER
CLEANER
MESH OPERATORS
MESH OPERATORS
TOPOLOGY
TOPOLOGY AND DATA
DATA
COMMON FILE FORMAT
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Program Organization
C SOLVER
RUBY SOLVER
RUBY DEBUGGER
RUBY WRAPPERS
SOLVER API
SIMULATION
CLEANER
BEZIER MESH
MESH I/O
BOUNDARY MESH
CELL COMPLEX
V
E
F
DATA PTS
CONTROL PTS
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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

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28,000 Node Mesh
110,000 Node Mesh