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 Examples
Linear Triangular Mesh, Unstructured
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Mesh Examples
Linear Quadrilateral Mesh, Structured
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Moving Meshes

• To simulate blood flow our mesh needs to be able
  to keep track of cell boundaries and fluid as
  they move in time.
• Essentially, two approaches
• Eulerian
• Lagrangian

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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 More work to solve equations each
  time step.

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Lagrangian Framework

• Elements themselves move over time, boundaries
  are real and exist in the geometry

• Advantages
• Since boundaries lie in geometry, they are more
  accurate
• Less equations to solve

• Problems
• As mesh moves elements deform
• Elements may be added and removed over time

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Moving Mesh Example Our PrototypeUnstructured
Lagrangian Moving Mesh, with Quadratic Bezier
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 of degree n is determined by n1
  control values, p0 pn1
• A Bezier curve of degree n can be represented as
  a linear combination of n1 basis polynomials
• For Quadratic Bezier curves this takes the form
• B(t) (1-t)2p0 2t(1-t)p1 t2p2

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Why Use Bezier Curves?

• 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
• Use Quadratic BSplines to represent boundaries in
  the mesh

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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 well sized and well shaped to minimize
  error

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Mesh Hierarchy
Curved Bezier Mesh
Control Mesh
Logical Mesh
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Delaunay Triangulation

• Examine circumcenter of each triangle
• Delaunay if circles center is within triangle
• Delaunay Meshes maximize the minimum angle
• Small angles are bad because they increase
  interpolation error

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Mesh Quality

• Mesh size (Number of Elements)
• Mesh grading (Avoid drastic element size changes)
• Element Quality (Avoid large interpolation
  errors)

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Bezier Triangle Quality

• Linear triangles must not have small angles
• Delaunay property keeps quality logical mesh
• Higher-Order Quality
• Quality of triangles in Control Mesh affect
  quality of curved triangle.
• Edge Smoothing keeps quality control mesh

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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)
• Edge flips alone can ensure Delaunay

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Cleaning Process Step 2Edge Smoothing for
High-Order Quality

• Identify overly curved triangles, smooth edge and
  re interpolate
• Keeps control mesh well shaped
• Also 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
• Identify poor logical triangles
• Cases where edge flips produce too much
  interpolation error
• Use Rupert Refinement to insert circumcenters of
  logical triangles

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Overview

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