Parallel Mesh Generation

In this poster we focus and present our preliminary results pertinent to the integration of multiple parallel Delaunay mesh generation methods into a coherent hierarchical framework. The goal of this project is to study our telescopic approach and to develop Delaunay-based methods to explore concurrency at all hardware layers using abstractions at (a) medium-grain level for many cores within a single chip and (b) coarse-grain level, i.e., sub-domain level using proper error metricand application-specific continuous decomposition methods. © 2015 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of organizing committee of the 24th International Meshing Roundtable (IMR24).

Numerical simulation is an important tool used in various fields of computational science and engineering. The models to be solved by simulation are predominantly based on equations which require the discretization of a spatial domain. The accuracy of the simulation results is heavily influenced by the properties of the underlying spatial discretization, the mesh. Thus, adapting a mesh to meet certain criteria is an integral step to achieve a desired accuracy and high computational performance. With the trend of more and more cores on a compute node, it is essential to efficiently exploit this available on-node parallelism of modern multi-core systems. Our work introduces a flexible shared-memory parallelized mesh adaptation framework. We show the integrability of available serial mesh adaptation algorithms and applicability to multi-region meshes. The first step of the framework is the partitioning of the initial mesh, where all the resulting partitions are subsequently assigned to independent sets using graph coloring algorithms. These sets are then processed in parallel using two different adaptation algorithms: A template-based algorithm and a Delaunay-based algorithm provided by the TetGen software. We perform benchmarks using a cube geometry with different mesh resolutions as well as a model of a microelectronic transistor device structure to demonstrate the scalability of our approach and investigate the resulting element quality. The obtained speedups for this inherently memory-bound problem and for constant problem sizes range between 4.6 and 8.6 using 16 threads.

Technological Advances in Medical Imaging have enabled the acquisition of images accurately describing biological tissues. Finite Element (FE) methods on these images provide the means to simulate biological phenomena such as brain shift registration, respiratory organ motion, blood flow pressure in vessels, etc. FE methods require the domain of tissues be discretized by simpler geometric elements, such as triangles in two dimensions, tetrahedra in three, and pentatopes in four. This exact discretization is called a mesh. The accuracy and speed of FE methods depend on the quality and fidelity of the mesh used to describe the biological object. Elements with bad quality introduce numerical errors and slower solver convergence. Also, analysis based on poor fidelity meshes do not yield accurate results specially near the surface. In this dissertation, we present the theory and the implementation of both a sequential and a parallel Delaunay meshing technique for 3D and —for the first ti...

h i g h l i g h t s • We present a parallel Delaunay method for 3D mesh generation and refinement. • We show that our method achieves unprecedented scalability for up to 144 cores. • We show that our single-threaded performance is faster than CGAL and TetGen. • We show that quality and fidelity is comparable to CGAL and TetGen.

In this paper, we present a Delaunay refinement algorithm for 4dimensional (3D+t) segmented images. The output mesh is proved to consist of sliver-free simplices. Assuming that the hyper-surface is a closed smooth manifold, we also guarantee faithful geometric and topological approximation. We implement and demonstrate the effectiveness of our method on publicly available segmented cardiac images.

h i g h l i g h t s • We present a parallel Delaunay method for 3D mesh generation and refinement. • We show that our method achieves unprecedented scalability for up to 144 cores. • We show that our single-threaded performance is faster than CGAL and TetGen. • We show that quality and fidelity is comparable to CGAL and TetGen.

In this paper, we present a scalable three dimensional hybrid parallel Delaunay image-to-mesh conversion algorithm (PDR.PODM) for distributed shared memory architectures. PDR.PODM is able to explore parallelism early in the mesh generation process because of the aggressive speculative approach employed by the Parallel Optimistic Delaunay Mesh generation algorithm (PODM). In addition, it decreases the communication overhead and improves data locality by making use of a data partitioning scheme offered by the Parallel Delaunay Refinement algorithm (PDR). PDR.PODM utilizes an octree structure to decompose the initial mesh and to distribute the bad elements to different octree leaves (subregions). A set of independent subregions are selected and refined in parallel without any synchronization among them. In each subregion, a group of threads is assigned to insert or delete multiple points based on the refinement rules offered by PODM. We tested PDR.PODM on Blacklight, a distributed shared memory (DSM) machine in the Pittsburgh Supercomputing Center, and observed a weak scaling speedup of 163.8 and above for up to 256 cores as opposed to PODM whose weak scaling speedup is only 44.7 on 256 cores. The end result is that we can generate 18 million elements per second as opposed to 14 million per second in our earlier work. To the best of our knowledge, PDR.PODM exhibits the best scalability among parallel guaranteed quality Delaunay mesh generation algorithms running on DSM supercomputers.

Meshes composed of well-centered simplices have nice orthogonal dual meshes (the dual Voronoi diagram). This is useful for certain numerical algorithms that prefer such primal-dual mesh pairs. We prove that well-centered meshes also have optimality properties and relationships to Delaunay and minmax angle triangulations. We present an iterative algorithm that seeks to transform a given triangulation in two or three dimensions into a well-centered one by minimizing a cost function and moving the interior vertices while keeping the mesh connectivity and boundary vertices fixed. The cost function is a direct result of a new characterization of well-centeredness in arbitrary dimensions that we present. Ours is the first optimization-based heuristic for well-centeredness, and the first one that applies in both two and three dimensions. We show the results of applying our algorithm to small and large two-dimensional meshes, some with a complex boundary, and obtain a well-centered tetrahed...

We introduce a parametrized notion of genericity for Delaunay triangulations which, in particular, implies that the Delaunay simplices of δ-generic point sets are thick. Equipped with this notion, we study the stability of Delaunay triangulations under perturbations of the metric and of the vertex positions. We quantify the magnitude of the perturbations under which the Delaunay triangulation remains unchanged.

Well-centered meshes (meshes composed of well-centered simplices) have the advantage of having nice orthogonal dual meshes (the dual Voronoi diagram), which is useful for certain numerical algorithms that require or prefer such primal-dual mesh pairs. We present a characterization of a well-centered n-simplex and introduce a cost function that quantifies well-centeredness of a simplicial mesh. We investigate some properties of the cost function and describe an iterative algorithm for optimizing the cost function. The algorithm can transform a given triangulation into a well-centered one by moving the interior vertices while keeping the mesh connectivity and boundary vertices fixed. We show the results of applying our algorithm to small, large, and graded two-dimensional meshes as well as one tiny three-dimensional mesh and a small tetrahedralization of the cube. Also, we prove for planar meshes that the optimal triangulation with respect to the cost function is the minmax angle tria...

Well-centered meshes (meshes composed of well-centered simplices) have the advantage of having nice orthogonal dual meshes (the dual Voronoi diagram), which is useful for certain numerical algorithms that require or prefer such primal-dual mesh pairs. We present a characterization of a well-centered n-simplex and introduce a cost function that quantifies well-centeredness of a simplicial mesh. We investigate some properties of the cost function and describe an iterative algorithm for optimizing the cost function. The algorithm can transform a given triangulation into a well-centered one by moving the interior vertices while keeping the mesh connectivity and boundary vertices fixed. We show the results of applying our algorithm to small, large, and graded two-dimensional meshes as well as one tiny three-dimensional mesh and a small tetrahedralization of the cube. Also, we prove for planar meshes that the optimal triangulation with respect to the cost function is the minmax angle tria...

Meshes composed of well-centered simplices have nice orthogonal dual meshes (the dual Voronoi diagram). This is useful for certain numerical algorithms that prefer such primal-dual mesh pairs. We prove that well-centered meshes also have optimality properties and relationships to Delaunay and minmax angle triangulations. We present an iterative algorithm that seeks to transform a given triangulation in two or three dimensions into a well-centered one by minimizing a cost function and moving the interior vertices while keeping the mesh connectivity and boundary vertices fixed. The cost function is a direct result of a new characterization of well-centeredness in arbitrary dimensions that we present. Ours is the first optimization-based heuristic for well-centeredness, and the first one that applies in both two and three dimensions. We show the results of applying our algorithm to small and large two-dimensional meshes, some with a complex boundary, and obtain a well-centered tetrahed...

In this paper, we propose a consistent parallel unstructured mesh generator based on a multi-phase SPH method. A set of physics-motivated modeling equations are developed to achieve the targets of domain decomposition, communication volume optimization and high-quality unstructured mesh generation simultaneously. A unified density field is defined as the target function for both partitioning the geometry and distributing the mesh-vertexes. A multi-phase Smoothing Particle Hydrodynamics (SPH) method is employed to solve the governing equations. All the optimization targets are achieved implicitly and consistently by the particle relaxation procedure without constructing triangulation/tetrahedralization explicitly. The target of communication reduction is achieved by introducing a surface tension model between distinct partitioning sub-domains, which are characterized by colored SPH particles. The resulting partitioning diagram features physically localized sub-domains and optimized interface communication. The target of optimiz

In adjoint-based sensitivity simulations based on the continuous adjoint equations on industrial geometries two main problems arise. Firstly, the formulation of the continuous adjoint equation set must be consistent with the physics of the problem in order to reliably capture the shape gradient for further use in optimisation. Giannakoglou and co-workers have recently shown that for Reynolds Averaged Navier-Stokes equations this should involve the solution of the adjoint turbulence model, since the established practice of “frozen turbulence” in adjoint equations leads to inaccurate shape derivative. Secondly, for accurate shape derivative both the forward and adjoint equation set need to be solved to a high tolerance and typically require meshes of very high quality: a local numerical error or lack of convergence in parts of domain may seriously pollute the adjoint model solution and the shape derivative. To make the problem more difficult, adjoint equation sets, have proven difficu...

In the report a parallel mesh generator in 3d/4d is presented. The mesh generator was developed as a part of the research project on space-time discretizations for partial differential equations in the least-squares setting. The generator is capable of constructing meshes for space-time cylinders built on an arbitrary 3d space mesh in parallel. The parallel implementation was created in the form of an extension of the finite element software MFEM. The code is publicly available in the Github repository [1].

This paper deals with the parallelization of Delaunay triangulation, a widely used space partitioning technique. Two parallel implementations of a three{dimensional incremental construction algorithm are presented. The rst is based on the decomposition of the spatial domain, while the second relies on the master{slaves approach. Both parallelization strategies are evaluated, stressing practical issues rather than theoretical complexity. We report on the exploitation of two di erent parallel environments: a tightly{ coupled distributed memory MIMD architecture and a network of workstations cooperating under the Linda environment. Then, a third hybrid solution is proposed, speci cally addressed to the exploitation of higher parallelism. It combines the other two solutions by grouping the processing nodes of the multicomputer into clusters and by exploiting parallelism at two different levels.

The paper deals with the parallelization of Delaunay triangulation, a widely used space partitioning technique. Two parallel implementations of a three-dimensional incremental construction algorithm are presented. The first is based on the decomposition of the spatial domain, while the second relies on the master-slaves approach. Both parallelization strategies are evaluated, stressing practical issues rather than theoretical complexity. We report on the exploitation of two different parallel environments: a tightly ...

h i g h l i g h t s • We present a parallel Delaunay method for 3D mesh generation and refinement. • We show that our method achieves unprecedented scalability for up to 144 cores. • We show that our single-threaded performance is faster than CGAL and TetGen. • We show that quality and fidelity is comparable to CGAL and TetGen.

This paper deals with the parallelization of Delaunay triangulation, a widely used space partitioning technique. Two parallel implementations of a three{dimensional incremental construction algorithm are presented. The rst is based on the decomposition of the spatial domain, while the second relies on the master{slaves approach. Both parallelization strategies are evaluated, stressing practical issues rather than theoretical complexity. We report on the exploitation of two di erent parallel environments: a tightly{ coupled distributed memory MIMD architecture and a network of workstations cooperating under the Linda environment. Then, a third hybrid solution is proposed, speci cally addressed to the exploitation of higher parallelism. It combines the other two solutions by grouping the processing nodes of the multicomputer into clusters and by exploiting parallelism at two different levels.

The paper deals with the parallelization of Delaunay triangulation, a widely used space partitioning technique. Two parallel implementations of a three-dimensional incremental construction algorithm are presented. The first is based on the decomposition of the spatial domain, while the second relies on the master-slaves approach. Both parallelization strategies are evaluated, stressing practical issues rather than theoretical complexity. We report on the exploitation of two different parallel environments: a tightly ...

The paper deals with the parallelization of Delaunay triangulation, a widely used space partitioning technique. Two parallel implementations of a three-dimensional incremental construction algorithm are presented. The first is based on the decomposition of the spatial domain, while the second relies on the master-slaves approach. Both parallelization strategies are evaluated, stressing practical issues rather than theoretical complexity. We report on the exploitation of two different parallel environments: a tightly ...

The paper deals with the parallelization of Delaunay triangulation, a widely used space partitioning technique. Two parallel implementations of a three-dimensional incremental construction algorithm are presented. The first is based on the decomposition of the spatial domain, while the second relies on the master-slaves approach. Both parallelization strategies are evaluated, stressing practical issues rather than theoretical complexity. We report on the exploitation of two different parallel environments: a tightly ...

Delaunay meshing is a popular technique for mesh generation. Usually , the mesh has to be refined so that certain fidelity and quality criteria are met. Delaunay refinement is achieved by dynamically inserting and removing points in/from a Delaunay triangulation. In this paper, we present a robust parallel algorithm for computing Delaunay triangulations in three dimensions. Our triangulator offers fully dynamic parallel insertions and removals of points and is thus suitable for mesh refinement. As far as we know, ours is the first method that parallelizes point removals, an operation that significantly slows refinement down. Our shared memory implementation makes use of a custom memory manager and lightweight locks which greatly reduce the communication and synchronization cost. We also employ a contention policy which is able to accelerate the execution times even in the presence of high number of rollbacks. Evaluation on synthetic and real data shows the effectiveness of our method on widely used multi-core SMPs.

In this paper, we present a parallel Image-to-Mesh Conversion (I2M) algorithm with quality and
fidelity guarantees achieved by dynamic point insertions and removals. Starting directly from an image,
its implementation is capable of recovering the isosurface and meshing the volume with tetrahedra
of good shape. Our tightly-coupled shared-memory parallel speculative execution paradigm employs
carefully designed contention managers, load balancing, synchronization and optimizations schemes.
These techniques are shown to boost not only the parallel but also the single-threaded efficiency of
our code. Specifically, our single-threaded performance is faster than both CGAL and TetGen, the state
of the art sequential open source meshing tools we are aware of. The effectiveness of our method is
demonstrated on Blacklight, the Pittsburgh Supercomputing Center’s cache-coherentNUMAmachine.We
observe a more than 82% strong scaling efficiency for up to 64 cores, and a more than 82% weak scaling
efficiency for up to 144 cores, reaching a rate of more than 14.3 million elements per second. This is the
fastest 3D Delaunay mesh generation and refinement algorithm, to the best of our knowledge.

In this paper, we present a Delaunay refinement algorithm for 4-dimensional (3D+t) segmented images. The output mesh is proved to consist of sliver-free simplices. Assuming that the hyper-surface is a closed smooth manifold, we also guarantee faithful geometric and topological approximation. We implement and demonstrate the effectiveness of our method on publicly available segmented cardiac images. Finally, we devise a tightly-coupled parallelization technique to boost the performance of our 4-dimensional mesher, thereby taking advantage of the multi-core and many-core platforms already available in the market.

Delaunay triangulation (DT) is a well known method for object reconstruction. There are at least three algorithms to construct triangulation, i.e. in-line (incremental insertion), incremental construction, and divide and conquer. Each of them has trade-off in the implementation on parallel algorithms.