Speaker 1: Navamita Ray​, Post-doc, MCS​
Title: Array-Based Hierarchical Mesh Generation in Parallel

Abstract:
In the numerical solution of complex partial differential equations (PDE’s) using finite element methods for unstructured meshes, the two most computationally intensive steps are mesh generation and linear solvers. An initial coarse mesh representing the computational domain might not be of sufficient resolution to get the meaningful results out of the discretizations for physical scales that might be present. As a result, the capability to refine a mesh is an essential part ofany simulation process. On the other hand, it is well known that multi-level methods such as geometric multigrid methods (GMG) can theoretically deliver optimal time complexity for solving sparse linear systems from PDE discretizations. Thus, it would be advantageous to use nested multi-level i.e. hierarchical meshes to achieve accuracy and computational efficiency, especially in the context of large-scale parallel computing, as both the number of processors and the mesh resolution increase.

In this talk, we describe an array-based hierarchical mesh generation capability through uniform refinement of unstructured meshes for efficient solution of PDE’s using finite element methods and multigrid solvers.  A multi-degree, multi-dimensional and multi-level framework is designed to generate the nested hierarchies from an initial mesh that can be used for a number of purposes such as multi-level methods to generating large meshes. The capability is developed under the parallel mesh framework “Mesh Oriented dAtaBase” a.k.a MOAB. We describe the underlying data structures and algorithms to generate such hierarchies and present numerical results for computational efficiency and mesh quality. We also present results to demonstrate the applicability of the developed capability to a multigrid finite-element solver.

Speaker ​2: Xinglin Zhao​, PhD Candidate, SUNY (Advisor: Xiangmin Jiao)​
Title: Reconstructing High-order Surface for Meshing in MOAB

Abstract:
Surface meshes and their manipulations are critical for meshing, numerical simulations, and many other related problems. Though polyhedral mesh is widely used to interpolate the geometry, it may be not enough since piecewise linear or bilinear surface generally produce insufficient approximation to the essential geometry. It is demanding to get more accurate estimation of the geometry, especially when mesh adaptation and high-order numerical scheme like hp-FEM are applied on such mesh. When a continuous CAD model or parametric finite elements are not available, one effective approach to reconstruct the geometry is local polynomial fitting.

In this talk, we will explore one high order surface reconstruction algorithm based on local fittings and demonstrate its effectiveness in mesh adaptation. The capability is being developed under parallel mesh framework “Mesh Oriented dAtaBase” a.k.a MOAB. We describe the design and its applicability to reconstruct high order surface for meshing.

Speaker ​3: Cao Lu​, PhD Candidate, SUNY (Advisor: Xiangmin Jiao)
Title: 2-D Parallel PDE Solvers for Structured and Unstructured Meshes ​with DMMoab

Abstract:
DMMoab is an extension of PETSc's DM framework to provide support for unstructured meshes. The meshes are described by the MOAB database, which provides a wide range of tools for mesh queries and manipulations. In this talk, we will showcase some implementations of 2-D PDE solvers based on DMMoab. The examples include the reaction-diffusion equation, the Bratu model, elliptic equations with discontinuous coefficients ​using geometric multigrid​ solvers​. We support the finite element method (FEM) and the generalised finite difference method (GFD) for spatial discretizations. DMMoab is utilized to provide data management and uniform refinement for structured and unstructured meshes. We will present some preliminary results for the numerical solutions and the parallel performance.