Parallel Mesh Generation Research Articles

A Hybrid Parallel Delaunay Image-to-mesh Conversion Algorithm Scalable on Distributed-memory Clusters

In this paper, we present a scalable three dimensional hybrid MPI+Threads parallel Delaunay image-to-mesh conversion algorithm. A nested master-worker communication model for parallel mesh generation is implemented which simultaneously explores process-level parallelization and thread-level parallelization: inter-node communication using MPI and inter-core communication inside one node using threads. In order to overlap the communication (task request and data movement) and computation (parallel mesh refinement), the inter-node MPI communication and intra-node local mesh refinement is separated. The master thread that initializes the MPI environment is in charge of the inter-node MPI communication while the worker threads of each process are only responsible for the local mesh refinement within the node. We conducted a set of experiments to test the performance of the algorithm on Turing, a distributed memory cluster at Old Dominion University High Performance Computing Center and observed that the granularity of coarse level data decomposition, which affects the coarse level concurrency, has a significant influence on the performance of the algorithm. With the proper value of granularity, the algorithm expresses impressive performance potential and is scalable to 30 distributed memory compute nodes with 20 cores each (the maximum number of nodes available for us in the experiments).

Using sequential NETGEN as a component for a parallel mesh generator

A parallel tetrahedral mesh generator is developed using the existing sequential NETGEN mesh generator. Mesh generation algorithms developed decompose the geometry into multiple sub-geometries sequentially on a master node and then mesh each sub-geometry in parallel on multiple processors. Two methods are implemented. The first decomposes the geometry and produces conforming surface sub-meshes from which volume meshes can be generated in parallel. A second refinement based method also makes use of the CAD geometry information. A scalable mesh migration algorithm that utilizes “owner updates” rule is implemented. Results show that using the refinement based method, a mesh with a billion elements can be generated in about a minute.

A Geometry-aware Data Partitioning Algorithm for Parallel Quad Mesh Generation on Large-scale 2D Regions

We develop a partitioning algorithm to decompose complex 2D data into small simple subregions for effective parallel quad meshing. We formulate the partitioning problem for effective parallel quad meshing as a quadratic integer optimization problem with linear constraints. Directly solving this problem is expensive for large-scale data partitioning. Hence, we suggest a more efficient two-step algorithm to obtain an approximate solution. First, we partition the region into a set of cells using L∞ Centroidal Voronoi Tessellation (CVT), then we solve a graph partitioning on the dual graph of this CVT to minimize the total partitioning boundary length, while enforcing the load balancing and each subregion's connectivity. With this decomposition, subregions are distributed to multiple processors for parallel quadrilateral mesh generation. We demonstrate that our decomposition algorithm outperforms existing approaches by offering a higher-quality partitioning, and therefore, improved performance and quality in mesh generation.

Scalable 3D Hybrid Parallel Delaunay Image-to-Mesh Conversion Algorithm for Distributed Shared Memory Architectures

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.

Generating Multibillion Element Unstructured Meshes on Distributed Memory Parallel Machines

We present a parallel mesh generator called PMSH that is developed as a wrapper code around the open source sequential Netgen mesh generator. Parallelization of the mesh generator is carried out in five stages: (i) generation of a coarse volume mesh; (ii) partitioning of the coarse mesh; (iii) refinement of coarse surface mesh to produce fine surface submeshes; (iv) remeshing of each fine surface submesh to get a final fine mesh; (v) matching of partition boundary vertices followed by global vertex numbering. A new integer based barycentric coordinate method is developed for matching distributed partition boundary vertices. This method does not have precision related problems of floating point coordinate based vertex matching. Test results obtained on an SGI Altix ICE X system with 8192 cores confirm that our approach does indeed enable us to generate multibillion element meshes in a scalable way.

Boundary recovery for 3D Delaunay triangulation

New ideas are presented in this paper for the boundary recovery of 3D Delaunay triangulation. Fully constrained Delaunay triangulations in terms of geometrical and topological integrities on all boundary edges and facets are required in many applications, such as meshing by components, fluid–structure interactions, parallel mesh generation, local remeshing and interface problems. The geometry of boundary edges and facets can be recovered by the introduction of Steiner points. However, for a fully constrained Delaunay triangulation, these Steiner points have to be removed or repositioned towards the interior of the domain to restore the topological integrity of the boundary edges and the facets. It is found that Steiner points on edges could be removed more systematically following a specific sequence in an alternative manner rather than a random selection commonly adopted in practice; whereas for Steiner points on a facet, a weight on the Steiner point adjacency would lead to an optimal order to facilitate their removal. A linear programming technique is also employed to determine the feasible region for the relocation of Steiner points in the interior of the domain. Work examples and industrial applications with details in the boundary recovery are presented to illustrate how the algorithm works on objects with difficult boundary conditions.

Parallel multigrid on hierarchical hybrid grids: a performance study on current high performance computing clusters

SUMMARYThis article studies the performance and scalability of a geometric multigrid solver implemented within the hierarchical hybrid grids (HHG) software package on current high performance computing clusters up to nearly 300,000 cores. HHG is based on unstructured tetrahedral finite elements that are regularly refined to obtain a block‐structured computational grid. One challenge is the parallel mesh generation from an unstructured input grid that roughly approximates a human head within a 3D magnetic resonance imaging data set. This grid is then regularly refined to create the HHG grid hierarchy. As test platforms, a BlueGene/P cluster located at Jülich supercomputing center and an Intel Xeon 5650 cluster located at the local computing center in Erlangen are chosen. To estimate the quality of our implementation and to predict runtime for the multigrid solver, a detailed performance and communication model is developed and used to evaluate the measured single node performance, as well as weak and strong scaling experiments on both clusters. Thus, for a given problem size, one can predict the number of compute nodes that minimize the overall runtime of the multigrid solver. Overall, HHG scales up to the full machines, where the biggest linear system solved on Jugene had more than one trillion unknowns. Copyright © 2012 John Wiley & Sons, Ltd.

Improvements in the reliability and element quality of parallel tetrahedral mesh generation

SUMMARY The paper presents a parallel tetrahedral mesh generation approach based on recursive bidivisions using triangular surfaces. Research was conducted for addressing issues concerning mesh generation reliability and element quality. A novel procedure employing local modification techniques is proposed for repairing the intersecting interdomain mesh instead of directly repeating the bidivision procedure, which improves the robustness of the complete meshing procedure significantly. In addition, a new parallel quality improvement scheme is suggested for optimizing the distributed volume meshes. The scheme is free of any communication cost and highly efficient. Finally, mesh experiments of hundreds of millions of elements are performed to demonstrate the reliability, effectiveness and efficiency of the proposed method and its potential applications to large-scale simulations of complex aerodynamics models. Copyright © 2012 John Wiley & Sons, Ltd.

Parallel Node-based Local Tetrahedral Mesh Generation

Parallel Node-based Local Tetrahedral Mesh Generation

Effective out-of-core parallel delaunay mesh refinement using off-the-shelf software

We present three related out-of-core parallel mesh generation algorithms and their implementations for small size computational clusters. Computing out-of-core permits to solve larger problems than otherwise possible on the same hardware setup. Also, when using shared computing resources with high demand, a problem can take longer to compute in terms of wall-clock time when using an in-core algorithm on many nodes instead of using an out-of-core algorithm on few nodes. The difference is due to wait-in-queue delays that can grow exponentially to the number of requested nodes. In one specific case, using our best method and only 16 nodes it can take several times less wall-clock time to generate a 2 billion element mesh than to generate the same size mesh in-core with 121 nodes. Although our best out-of-core method exhibits unavoidable overheads (could be as low as 19% in some cases) over the corresponding in-core method (for mesh sizes that fit completely in-core), this is a modest and expected performance penalty. We evaluated our methods on traditional clusters of workstations as well as presented preliminary performance evaluation on emerging BlueWaters supercomputer.

MDEC: MeTiS-based Domain Decomposition for Parallel 2D Mesh Generation

Abstract Domain decomposition methods are commonly employed within the context of parallel numerical algorithms. Most often, the domain decomposition is performed before the main computation begins. Within the context of mesh generation, parallel mesh generation is desired when the goal is to mesh a very large geometric domain or if very high accuracy is required. In this paper, we propose a novel technique, which we call the MeTiS-based Domain Decomposition (MDEC) technique, for the decomposition of geometric domains into subdomains for use in parallel 2D mesh generation. Our technique is based upon discrete domain decomposition [1]. The algorithm proceeds by first constructing a background mesh which satisfies a minimum angle constraint of 30 degrees and second partitioning this initial coarse mesh or background mesh into subdomains. Finally, adjustments are applied to the triangles with small boundary angles so that all subdomains in the final decomposition contain boundary angles no smaller than 60 degrees which is a guaranteed property of the domain decomposition algorithm. We prove this guarantee for the boundary angles of the MDEC domain decomposition. Our results show that, in comparison to the medial axis domain decomposition (MADD) algorithm [2], our method provides a better balance of subdomain areas, better boundary angles, and a faster decomposition time. In addition, when the MDEC and MADD subdomains are used in conjunction with a parallel constained Delaunay mesh generation technique (PCDM) [3], the meshes are generated in approximately the same time and have very similar element quality.

A template for developing next generation parallel Delaunay refinement methods

We describe a complete solution for both sequential and parallel construction of guaranteed quality Delaunay meshes for general two-dimensional geometries. We generalize the existing sequential point placement strategies for guaranteed quality mesh refinement: instead of a specific position for a new point, we derive two types of two-dimensional regions which we call selection disks. Both types of selection disks are inside the circumdisk of a poor quality triangle, with the Type I disk always inside the Type II disk. We prove that any point placement algorithm which inserts Steiner points inside selection disks of Type I terminates, and any algorithm which inserts Steiner points inside selection disks of Type II produces an asymptotically size-optimal mesh. In the area of parallel Delaunay mesh refinement, we develop a new theoretical framework for the construction of graded meshes on parallel architectures, i.e., for parallel mesh generation with element size controlled by a user-defined criterion. Our sufficient conditions of point Delaunay-independence allow to select points for concurrent insertion in such a way that the standard sequential guaranteed quality Delaunay refinement procedures can be applied in parallel to attain the required element quality constraints. Finally, we present a novel parallel algorithm which can be used in conjunction with any sequential point placement strategy that chooses points within the selection disks. We implemented our algorithm for shared memory multi-core architectures and present the experimental results. Our data show that even on workstations with a few cores, which are now in common use, our implementation is significantly faster than the best sequential counterpart.

A document driven methodology for developing a high quality Parallel Mesh Generation Toolbox

This paper motivates the value of using a document driven methodology to improve the quality of scientific computing applications by illustrating the design and documentation of a Parallel Mesh Generation Toolbox (PMGT). Formal mathematical specification is promoted for writing unambiguous requirements, which can be used to judge the correctness and thus the reliability of PMGT. Mathematics is also shown to improve understandability, reusability and maintainability through modelling software modules as finite state machines. The proposed methodology includes explicit traceability between requirements, design, implementation and test cases. Traceability improves the verification of completeness and consistency and it allows for proper change management. To improve the reliability of PMGT, given the challenge that the correct solution is unknown a priori, an automated testing approach is adopted to verify the known properties of a correct solution, such as conformality and counterclockwise vertex numbering.

Algorithm 870

We present a geometric domain decomposition method and its implementation, which produces good domain decompositions in terms of three basic criteria: (1) The boundary of the subdomains create good angles, that is, angles no smaller than a given tolerance Φ 0 , where the value of Φ 0 is determined by the application which will use the domain decomposition. (2) The size of the separator should be relatively small compared to the area of the subdomains. (3) The maximum area of the subdomains should be close to the average subdomain area. The domain decomposition method uses an approximation of a Medial Axis as an auxiliary structure for constructing the boundary of the subdomains (separators). The N -way decomposition is based on the “divide and conquer” algorithmic paradigm and on a smoothing procedure that eliminates the creation of any new artifacts in the subdomains. This approach produces well shaped uniform and graded domain decompositions, which are suitable for parallel mesh generation.

CFD investigation of helical wire-wrapped 7-pin fuel bundle and the challenges in modeling full scale 217 pin bundle

Flow and temperature distributions of sodium in a heat generating fuel pin bundle with helically wound spacer wire have been predicted from basic principles by solving the three-dimensional conservation equations of mass, momentum and energy, for a wide range of Reynolds number. Turbulence has been modeled using the k– ɛ turbulence model. The geometry details of the bundle and heat flux from the fuel pin are similar to that of the Indian Prototype Fast Breeder Reactor (PFBR) that is currently under construction. The focus of the study is to assess the effect of transverse flow in promoting flow and temperature uniformity. It is seen that the ratio of maximum transverse velocity to the maximum axial velocity is nearly equal to the tangent of the rolling up angle of the spacer wire. Due to the wire wrap, the difference in bulk sodium temperature between the peripheral and central sub-channels is reduced to by a factor of 4 when compared to that without spacer wire. The film drop at the junction between wire and the pin is found to be only ∼70 °C. The predicted results are found to be in close agreement with that of the experimental results reported in literature. The present study considers a 7-pin bundle assembly of one helical pitch. The computational time and memory required for a 217 pin with 15 pitches assembly is ascertained to be 500 times that required for the current study. Hence, research activities have been directed towards developing a parallel CFD code and structural mesh generation software.

Parallel unstructured mesh generation by an advancing front method

Mesh generation is a critical step in high fidelity computational simulations. High-quality and high-density meshes are required to accurately capture the complex physical phenomena. A robust approach for a parallel framework has been developed to generate large-scale meshes in a short period of time. A coarse tetrahedral mesh is generated first to provide the basis of block interfaces and then is partitioned into a number of sub-domains using METIS partitioning algorithms. A volume mesh is generated on each sub-domain in parallel using an advancing front method. Dynamic load balancing is achieved by evenly distributing work among the processors. All the sub-domains are combined to create a single volume mesh. The combined volume mesh can be smoothed to remove the artifacts in the interfaces between sub-domains. A void region is defined inside each sub-domain to reduce the data points during the smoothing operation. The scalability of the parallel mesh generation is evaluated to quantify the improvement on shared- and distributed-memory computer systems.

Delaunay Decoupling Method for Parallel Guaranteed Quality Planar Mesh Refinement

Creating in parallel guaranteed quality large unstructured meshes is a challenging problem. Parallel mesh generation procedures decompose the original mesh generation problem into smaller subproblems that can be solved in parallel. The subproblems can be treated as either completely or partially coupled, or they can be treated as completely decoupled. In this paper we present a parallel guaranteed quality Delaunay method for 2-dimensional domains which is based on the complete decoupling of the subproblems. As a result the method eliminates the communication and the synchronization during the meshing of the subproblems. Moreover, it achieves 100% code reuse of existing, fine-tuned, and well-tested sequential mesh generators. The approach we describe in this paper presents for the first time an effective way to create in parallel guaranteed quality meshes with billions of elements in a few hundreds of seconds, and at the same time demonstrates that these meshes can be generated in an efficient and scalable way. Our performance data indicate superlinear speedups.

On control of adaptation in parallel mesh generation

A black-box methodology of a parallel generation of adaptive anisotropic meshes is described. A control of mesh adaptation, important for robustness and flexibility, may effect parallel properties of the methodology. This is demonstrated with a 3D example.

Guaranteed-quality parallel Delaunay refinement for restricted polyhedral domains

We describe a distributed memory parallel Delaunay refinement algorithm for simple polyhedral domains whose constituent bounding edges and surfaces are separated by angles between 90° to 270° inclusive. With these constraints, our algorithm can generate meshes containing tetrahedra with circumradius to shortest edge ratio less than 2, and can tolerate more than 80% of the communication latency caused by unpredictable and variable remote gather operations. Our experiments show that the algorithm is efficient in practice, even for certain domains whose boundaries do not conform to the theoretical limits imposed by the algorithm. The algorithm we describe is the first step in the development of much more sophisticated guaranteed-quality parallel mesh generation algorithms.

Guaranteed-quality parallel Delaunay refinement for restricted polyhedral domains

We describe a distributed memory parallel Delaunay refinement algorithm for simple polyhedral domains whose constituent bounding edges and surfaces are separated by angles between 90° to 270° inclusive. With these constraints, our algorithm can generate meshes containing tetrahedra with circumradius to shortest edge ratio less than 2, and can tolerate more than 80% of the communication latency caused by unpredictable and variable remote gather operations. Our experiments show that the algorithm is efficient in practice, even for certain domains whose boundaries do not conform to the theoretical limits imposed by the algorithm. The algorithm we describe is the first step in the development of much more sophisticated guaranteed-quality parallel mesh generation algorithms.