Parallel Two-Dimensional Unstructured Anisotropic Delaunay Mesh Generation of

Complex Domains for Aerospace Applications

Juliette Pardue Dr. Andrey Chernikov

Computer Science Department of Old Dominion University

NIA CFD Seminar September 27, 2016

Need for Parallel Mesh Generation

• Detailed geometry with small features require high density of elements • Hours to generate mesh in serial • Mesh may not fit in one CPU’s memory

• Parallel PDE solver software matured faster than parallel mesh generation software

• Mesh generation now a bottleneck for finite element analysis

2

Major Contributions• Adaptation of isotropic

decomposition [1] to anisotropic • 368 speedup on 512 processes

3[1] Blelloch, Miller, and Talmor, 12th Annual Symposium on Computational Geometry, 1996

Major Contributions• Adaptation of isotropic

decomposition [1] to anisotropic • 368 speedup on 512 processes

3[1] Blelloch, Miller, and Talmor, 12th Annual Symposium on Computational Geometry, 1996

• Computational savings • Isotropic vs anisotropic • Same simulation parameters • Same triangle sizing function

and same input geometry • Isotropic mesh has 14 times

more triangles and took twice as many iterations to converge

Anisotropic Boundary Layer• Boundary layers in fluid mechanics are anisotropic in nature • Isotropic cannot accurately and efficiently represent domain • Discretize mesh to efficiently capture these flow velocities

4[2] Aubry et al., 53rd AIAA CFD Conference, 2015

Anisotropic Boundary Layer• Boundary layers in fluid mechanics are anisotropic in nature • Isotropic cannot accurately and efficiently represent domain • Discretize mesh to efficiently capture these flow velocities

4[2] Aubry et al., 53rd AIAA CFD Conference, 2015

• Points inserted along normals where strong gradients exist

• Yields substantial CPU savings without compromising accuracy

Boundary Layer Refinement

• Surface slope discontinuities • Creates ill-suited elements • Affects local mesh density • Causes interpolation errors

when computing PDE solution

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Boundary Layer Refinement

• Surface slope discontinuities • Creates ill-suited elements • Affects local mesh density • Causes interpolation errors

when computing PDE solution

5

• Fan of rays added at sharp trailing edges

Boundary Layer Refinement

• Surface slope discontinuities • Creates ill-suited elements • Affects local mesh density • Causes interpolation errors

when computing PDE solution

5

• Fan of rays added at sharp trailing edges

• Laplacian direction vector smoothing used at blunt trailing edges

Boundary Layer Intersections

• Search domain pruned with 
axis-aligned bounding boxes

• Alternating digital tree (ADT) [3] to check each element’s 
rays for intersections

6[3] Bonet and Peraire, International Journal for Numerical Methods in Engineering, 1991

Boundary Layer Intersections

• Search domain pruned with 
axis-aligned bounding boxes

• Alternating digital tree (ADT) [3] to check each element’s 
rays for intersections

6[3] Bonet and Peraire, International Journal for Numerical Methods in Engineering, 1991

• other rays from same element • surface of geometry • other elements’ outer border

• Candidate intersections from ADT checked with 2D line-orientation test

Boundary Layer Intersections

• Search domain pruned with 
axis-aligned bounding boxes

• Alternating digital tree (ADT) [3] to check each element’s 
rays for intersections

6[3] Bonet and Peraire, International Journal for Numerical Methods in Engineering, 1991

• other rays from same element • surface of geometry • other elements’ outer border

• Candidate intersections from ADT checked with 2D line-orientation test

Resolving Boundary Layer Intersections

• Laplacian direction vector smoothing for contiguous self-intersections

• Clipping for multi-element intersections and non-contiguous self-intersections

Resolving Boundary Layer Intersections

• Laplacian direction vector smoothing for contiguous self-intersections

• Clipping for multi-element intersections and non-contiguous self-intersections

Gradation Control• Provides mathematically good transition to isotropic region • Poor gradation causes truncation errors in flow solution • ADT used to ensure gradation does not cause intersections

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Gradation Control

Parallel Triangulation of Boundary Layer

• Duality between 2D Delaunay triangulation and 3D lower convex hull of paraboloid

• 3D lower convex hull faces correspond to Delaunay triangles

10[1] Blelloch, Miller, and Talmor, 12th Annual Symposium on Computational Geometry, 1996

Parallel Triangulation of Boundary Layer

• Duality between 2D Delaunay triangulation and 3D lower convex hull of paraboloid

• 3D lower convex hull faces correspond to Delaunay triangles

• Paraboloid flattened to vertical plane • Each segment of 2D lower convex

hull corresponds to edge of a Delaunay triangle

• Each side of dividing path can be triangulated independently

10[1] Blelloch, Miller, and Talmor, 12th Annual Symposium on Computational Geometry, 1996

Decomposed Delaunay Subdomains

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Graded Isotropic Inviscid Region

• Sizing function controls gradation • Discretize borders between

subdomains to guarantee Delaunay criteria • Maximum circumradius-to-

shortest-edge ratio of √2 • Maximum triangle area bound

by sizing function • Interior of decoupled

subdomains refined concurrently

12[4] Linardakis and Chrisochoides, SIAM Journal on Scientific Computing, 2008

Load Balancing• Idle processes waste CPU resources • Process’ work load is total number of estimated triangles of it’s

remaining subdomains • Global memory window on root process • If a process is low on work

1. Fetch global memory window 2. Compute which process has the most work 3. Request work from this max process

13

14(a) (b)

14(c) (d)

14(e) (f)

Flow Solution

15

Performance• Compared performance to fastest

sequential algorithm, Triangle • Speedup: ratio of execution time

of fastest sequential algorithm to execution time of parallel algorithm

• Efficiency: ratio of speedup to number of processes used

• Strong scalability: performance when total work is kept fixed

• Weak scalability: performance when total work increases proportionally to processes

16[5] Shewchuk, Applied Computational Geometry: Towards Geometric Engineering, 1996

Performance• Compared performance to fastest

sequential algorithm, Triangle • Speedup: ratio of execution time

of fastest sequential algorithm to execution time of parallel algorithm

• Efficiency: ratio of speedup to number of processes used

• Strong scalability: performance when total work is kept fixed

• Weak scalability: performance when total work increases proportionally to processes

16[5] Shewchuk, Applied Computational Geometry: Towards Geometric Engineering, 1996

Conclusion• Parallel meshing has not matured as fast as parallel solvers • Adaptation of isotropic decomposition to anisotropic domains • Anisotropic meshes contain fewer elements and converge quicker • Strong scaling speedup of 368 on 512 processes • Weak scaling efficiency of 79% on 1024 processes • Push-button mesh generator • High-fidelity anisotropic boundary layer • Globally Delaunay, graded isotropic inviscid region

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References[1] G.E. Blelloch, G.L. Miller, and D. Talmor, “Developing a Practical Projection-

Based Parallel Delaunay Algorithm,” Proc. 12th Annual Symposium on Computational Geometry, 1996, pp. 186-195.

[2] R. Aubry, K. Karamete, E. Mestreau, D. Gayman, and S. Dey, “Ensuring a Smooth Transition from Semi-Structured Surface Boundary Layer Mesh to Fully Unstructured Anisotropic Surface Mesh,” Proc. 53rd AIAA Computational Fluid Dynamics Conference, 2015.

[3] J. Bonet and J. Peraire, “An Alternating Digital Tree (ADT) Algorithm for 3D Geometric Searching and Intersection Problems,” International Journal for Numerical Methods in Engineering, vol. 31, 1991, pp. 1–17.

[4] L. Linardakis and N. Chrisochoides, “Graded Delaunay Decoupling Method for Parallel Guaranteed Quality Planar Mesh Generation,” SIAM Journal on Scientific Computing, vol. 30, 2008, pp. 1875-1891.

[5] J.R. Shewchuk, “Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator,” Applied Computational Geometry: Towards Geometric Engineering vol. 1148, 1996, pp. 203-222.