A Literature Survery on Surface Quadranlugations

Introduction

Quadrilateral surface meshing is important in many areas of numerical simulations, graphics and visualization. Despite its great importance, it is surprising that there are nopublic domain software for 3D surface quadrangulation. Quadrangulation, in general, is far more complex than triangulation and practical algorithms are often based on some sort of heuristics, therefore, has no robust software implementation as Jonathan Shewchuk's Triangle Software.

The goal of this literature survey is to study various approaches used in quadmesh generation, and implement open source general purpose quadmesh algorithms. Since it is unlikely that a single approach will work in all cases, various algorithms must be implemented and distribute as a part of MeshKit.

* Please note that  the reference list is neither exhaustive nor final. 

Approach
 Advantages
 Disadvantges
 Notes
Direct
  1. Probably very high quality mesh.
  2. Controlled regular mesh generated.
  3. Feature preservation is easy to implement.
  1. Difficult to implement correctly.
  2. Suffer from numerical robustness isssues. May need exact arithmatic calculation for intersection tests.
  3. Complex geometric data structures often needed.
  4. Final front collision is difficult to handle and the final mesh quality may depend on the quality mesh generation in the last stage.
Indirect
  1. Base Delunay triangulation on the complex geometric model is backed by mathematical theories and well studied problem. 
  2. Perhaps easiest to implement and has the least space and time computational complexity.
    1. Many heuristics are required for all quad mesh.
    2. Mesh complexity ( #number of quads ) generated is difficult to predict even for simple geometries.
    3. Quality depends on the order in which triangles are merged.

    Parameterization
    1. Well studied problem in computer graphics and many solutions are available for arbitrary geometric models.
    2. All regular quads generated.
    1. Computationally expensive as it require large sparse matrix solver. Even with preconditioners, efficiency is low.
    2. Difficult to preserve features in the CAD models ( such as sharp boundaries ).
    3. No apriori idea about number of final quads patches generated.
    Spectral
    1. Based on differential gemetric operators and eigenvaluoes.
    2. Well studied problem and have sound theoretical foundations.
    3. Guaranteed to produce all quad mesh.
    1. Computationally extremely expensive. Require multiscale second eigenvalue and eigenvector determination. Even with ARPACK solver both time and space computational complexity are too high to be considered this a practical approach.
    2. Preserving feature boundaries is non trivial task.

    Considering all the options for now, it is clear that choosing indirect approach is both practical and less risky.  No matter what approach we choose, mesh optimization may always be required in the last stage, therefore both parametric and spectral approach, although guaranteed to generate all quad mesh without user any heuristics or parameters selection, is less favorable at this time.


    Plans for implementing Indirect All-Quad Meshing Algorithm:

    Base Surface Triangulation : 

    Generating high quality surface mesh is well studied problem by professor Tamal Dey and public domain software Tight Cocone software ( only binaries available )is available from him. For surface mesh triangle improvement, developing "Intrinsic Delaunay Triangulation" is easy. Complexity of triangulation i.e. number of triangles can be increased by refinement methods or reduced by standard Mesh decimation methods.  Therefore, we have all that is needed for surface triangulations and is available in public domain and therefore no efforts will be made in this direction.

    Triangle-to-Quad Transformation:

    To start with, we will follow two papers and try to implement their ideas. Q-morph provides basic foundation on the quad meshing and it has
    mostly based on heuristic. Paper from Suneeta etc. gives new insights into theoretical complexities and provides some new dual based quad meshing algorithms. After the first quick implementation of these two paper, we will focus on mesh optimizations, some heuristics ( if they are needed) and mesh quality improvement method.

    Objectives of the Quad Meshing Source code development:

    1. Develop a quick prototype software for all quad meshing algorithm which works for large number of cases, if not all.
    2. Develop software that must be integrated into MeshKit and eventually put as public domain software.
    3. Use the quad mesh for future hex meshing algorithms.

    References:

    1. Direct Approaches:

      1. Quadrilateral Meshing by Circle Packing: Marshall Bern and David Eppstein
      2. Feature aligned quad dominant remeshing using iterative local updates: Yu-Kun Lai, Leif Kobbelt, Shi-Min Hu
      3. A Robust Two Step Procedure for Quad-Dominant Remeshing: Martin Marinov and Leif Kobbelt
      4. Quadrilateral surface meshes without self-intersecting dual cycles for hexahedral mesh generation: Matthias Muller Hannemann
      5. Meshing by Successive Superelements decomposition (MSD)- A new approach to quadrilateral mesh generation: C.S. Krishnamoorthy, B. Raphael and S.  Mukherjee
      6. Generation of quadrilateral mesh over analytical curved surfaces: T.S. lau, S.H. Lo, C.K. Lee

    2. Indirect Approaches:

      1. Q-Morph: An Indirect Approach to Advancing Front Quad Meshing: S.J.Owen, M.L. Staten, A. Canann and S. Saigal
      2. Converting Triangulation to Quadrangulation: Suneeta Ramaswami, Pedro Ramos, and Godfried Toussaint
      3. Quadrilateral Decomposition by Two-Ear Property Resulting in CAD Segmentation: Maharavo Randrianarivony

    3. Parameterization Approaches:

      1. Optimally Cutting a Surface into a disk:  Jeff Erickson, Sariel Har-Peled
      2. Mixed-Integer Quadrangulation: David Bommes, Henrik Zimmer, Leif Kobbelt
      3. Discrete Conformal Mapping via Circle Patterns: Liliya Kharevych, Boris Springborn, Peter Schroder
      4. Discrete Surface Ricci Flow: Theory and Applications: Miao Jin, Junho Kim and Xianfeng David Gu

    4. Spectral Approaches:

      1. Spectral Surface Quadrangulation: Shen Dong, Peer-Timo Bremer, Michael Garland, Valerio Pascucci, and John C. Hart.
      2. Spectral Quadrangulation with Orientation and Alignment Control.  Jin Huang, Muyang, Jin Ma, Xinguo Liu, Leif Kobbelt, Hujan Bao