Geometry-aware partitioning of complex domains for parallel quad meshing

Highlights

• A new data partitioning algorithm for large-scale quad mesh generation.

• A parallel computing framework for distributed geometric processing and meshing.

• A quad mesh generator for efficient scientific simulations of large geometric data.

Abstract

We develop a partitioning algorithm to decompose complex 2D data into small and simple subregions suitable for effective distributed and parallel quadrilateral mesh generation. To support high-quality quad mesh generation, the partitioning reduces to solving an integer quadratic optimization problem with linear constraints. Directly solving this problem is expensive for large-scale data. Hence, we also suggest a more efficient two-step algorithm to obtain an approximate solution. First, we partition the region into a set of cells using $L_{\infty }$ 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 mesh generation. Through comparisons on the quality of the final meshes and the performance of simulations run on these meshes, we show that our decomposition algorithm outperforms existing partitioning approaches by offering more simulation-friendly regular meshes.

Introduction

Mesh generation is a fundamental and critical problem in geometric data modeling and processing. In most scientific and engineering tasks that involve numerical computations or simulations (e.g. solving partial differential equations (PDE) using finite difference methods (FDM), finite element methods (FEM), or finite volume methods (FVM)) on geometric regions or objects, the first step of the procedure is always to discretize the geometric data using polygonal or polyhedral meshes. The quality of this tessellation often significantly affects the subsequent computation accuracy, efficiency, and numerical stability. Compared with commonly used irregular tessellations (such as triangular meshes in 2D and tetrahedral meshes in 3D), the regular tessellations (such as quadrilateral meshes in 2D and hexahedral meshes in 3D) are sometimes preferred in finite element analysis or isogeometric analysis in structural mechanics, fluid dynamics simulations, and several other fields, because they demonstrate more efficient and accurate computing [1] and more economic storage.

In recent years, the acquisition or generation of large high-resolution geometric datasets pose new challenges to the design of effective processing algorithms. These big and complex data are expensive to model and analyze using existing sequential algorithms, as the limited CPU and memory are often insufficient to handle billions of elements efficiently. Parallel algorithms utilizing high-performance computers make it possible to solve large and complex problems efficiently on hundreds or thousands of computing clusters and is therefore desirable.

Divide-and-conquer is a natural and effective strategy in parallel mesh generation for large geometric data. The given region or object is first decomposed into a set of solvable and simplified subparts; then each subpart is distributed to a working processor for mesh construction; finally, individually generated meshes are merged to get the final result. Parallel mesh generation strategies such as Delaunay-based methods and advancing front techniques have been used for both triangulation [2], [3], [4] or tetrahedralization  [5], [6]. Following this general paradigm, in this work we aim to develop a partitioning algorithm on complex and large-scale 2D regions for parallel quadrilateral mesh generation.

In geometric processing through divide-and-conquer, the partitioning of data often directly dictates the algorithm’s efficiency and quality. We formulate three main criteria as follows: (1) The areas of the subregions should be similar; (2) The boundary length of each subregion should be small compared with its area; and (3) Each subregion should have desirable geometric property, and more specifically for quad meshing, it should have corner angles close to $k\pi /2,k\in \{0,1,2,3\}$. The first two criteria determine the parallel performance: the balance of work load on different processors, and the communication cost among processors. Efficient parallelization should balance the workload distributed to different processors and should minimize interprocessor communication to reduce synchronization and waiting. The third criterion on the subregion’s geometry affects the quality of the final quad tessellation. For example, on a square subregion one can generate a quad mesh where every element is uniform and not sheared; but on a triangle subregion, the smallest angle of the resultant quad mesh will inevitably be smaller than the smallest boundary angle of this triangle patch. Hence, for effective quad mesh generation, it is desirable to partition the geometry into subregions whose boundary angles are near $k\pi /2$ to construct less-sheared quad elements. To incorporate the geometric constraint in data partitioning, however, is sometimes prohibitively expensive. We will discuss this issue in Section  3 and propose a more efficient strategy to tackle it. The three criteria are shown in Fig. 1.

After data partitioning, we distribute subregions to different working processors for mesh generation. In our implementation, we use consistent boundary sampling and advancing front for parallel meshing on each subregion. The sub-meshes are finally merged together then a relaxation is performed to get the final result. The main contributions of this paper include:

• We study geometry-aware data partitioning for effective parallel mesh generation, and suggest new models to partition large and complex 2D regions into subparts with desirable geometric shapes for quad meshing. Compared with existing partitioning algorithms, our approach could lead to more efficient parallel processing and higher-quality meshing results.

• We develop a parallel computing framework for quad mesh generation of large-scale 2D regions. It can effectively utilize parallel computational resources to handle big geometric data. We demonstrate that in coastal modeling the massive-size coastal terrains and oceans can be discretized effectively.

• We demonstrate that our framework is simulation-friendly. We perform two simulations to show that the meshes generated by our framework can reach higher accuracy than others.