10IMR Panel

10th International Meshing Roundtable
Panel Session

Panelists | NAFEMS | Topic #1: Mesh Quality | Topic #2: Facets v. NURBs


Cecil Armstrong, c.armstrong@qub.ac.uk
  • Head of School, Mechanical and Manufacturing Engineering, Queen's University of Belfast
  • Research interests include Finite Element Modeling
  • Member of NAFEMS CAD/FEA Committee

Gordon Ferguson, gordy@vki.com
  • President, Visual Kinematics, Inc., a supplier of CAE software component libraries
  • Prior CAE experience at Ardent Computer and Lockheed Missiles & Space
  • MS in Physics and Math, Univ. of Minnesota

Pat Knupp, pknupp@sandia.gov
  • Principle Member of Technical Staff, Sandia National Laboratories
  • Mesh quality and sweeping research as part of CUBIT project
  • PhD, Applied Mathematics, University of New Mexico, with Dr. Stanly Steinberg
  • co-author of "Fundamentals of Grid Generation"

Peter Schröder, ps@cs.caltech.edu
  • Associate Professor of Computer Science, Caltech
  • PhD in Computer Science, 1994, Princeton
  • Expert on wavelet based methods for computer graphics

Tim Tautges, tjtautg@sandia.gov
  • Principal Member of Technical Staff, Sandia National Laboratories
  • Adjunct Professor of Engineering Physics, University of Wisconsin - Madison
  • PhD, University of Wisconsin-Madison, 1990, Nuclear Engineering
  • Led CUBIT project from 1996-1998.

Steve Vavasis, vavasis@cs.cornell.edu
  • Associate Professor of Computer Science, Cornell University
  • PhD, 1989, Stanford
  • Previous research positions at Sandia, Argonne, Bell Labs, Xerox PARC, NASA
  • co-author of the QMG meshing software package

Special Introduction to NAFEMS

Prof. Cecil Armstrong began the panel with a brief overview of NAFEMS, the International Association for the Engineering Analysis Community. Prof. Armstrong is a member of NAFEMS' CAD/CAE Integration committee.


  • Creating Awareness
    • By providing best practice advice documents
    • By developing mechanisms for sharing knowledge
  • Delivering Education and Training
    • By developing a seminar and conference programme
    • By providing training courses and materials
  • Stimulating Standards
    • For sharing of analysis data and results
    • For highly qualified engineering analysts

Product Data Management (PDM) and The Engineering Analysis Environment

  • Many companies are obtaining benefits from implementing PDM systems
    • These typically enable the Design and Configuration Management (CM) tasks
  • International Standards are being developed for both PDM and Engineering Analysis
  • PDM systems are starting to be applied to Engineering Analysis tasks
  • There is a need to educate Analysis Professionals about PDM

PDM & the Engineering Analysis Environment

  • A publication by NAFEMS aimed at Analysis Professionals
  • Produced by the CAD/FE Integration Working Group
  • Summary
    • Aimed at people who don't know about PDM
    • Outlines the fundamentals of PDM and CM
    • Describes the benefits of applying PDM ot the Engineering Analysis task
    • Describes the International Standards being developed in this area
    • Suggest how the reader may proceed

ISO/CD 10303-107 (E)

  • Product data representation and exchange: Integrated application resource: Finite element analysis definition relationships
  • by Keith Hunten and David Leal
  • This document specifies the association between idealised concepts within a finite element analysis and the design definitions of those concepts. The concepts that are idealised within a finite element analysis include the action that is simulated, the states of the product, and the section properties.

    This document also specifies the association between a representation of a field that is calculated by a finite element analysis and a property of a defined state of a product.

ISO/CD 10303-51 (E)

  • Product data representation and exchange: Integrated generic resource: Mathematical description
  • by Derek Pashley and David Leal
  • This document specifies the use of a mathematical values for identification of properties, products, states or activities; the use of mathematical spaces as identification schemes for spaces or set of properties, products, states or activities; and the use of mathematical functions to describe property avariation within a set of space of products, states or activities.
  • EXAMPLE 2 - The volume of geometric space within 'my duct' is a product which is has air flowing through it. there is a set of planes within this volume, such that each is approximately normal to the direction of flow. Each of these planes can be regarded as a feature of the volume.

    There are properties for each of the planes in the volume, such as average pressure, average velocity and average temperature. Hence there is a varaiation of average pressue (say) with respect to the set of planes.

    In order to describe a variation of average pressure with respect to the set of planes within the volume, the set of planes is parametrized by the unit real interval [0.0, 1.0].

Standards Development

  • Benefits
    • lots of valuable technology and insights
    • archiving mechanism
  • Problems
    • slow acceptance
    • long gestation period
    • minimal benefits to users so far

Further Information


Topic #1: "What Makes a Mesh Good?"

The question of what makes a mesh "good" continues to vex users of computational analysis software. From a strictly practical standpoint, a mesh is good if the solution computed on it is accurate to the level desired by the analyst. This implies that mesh adaption may potentially remove the burden of mesh quality assessment from the analyst.

But mesh adaption is far from achieving universal acceptance and implementation, leaving a need for a priori mesh quality assessment. This involves computation of a mesh's inherent properties (metrics) and comparison of these metrics against acceptability criteria, hopefully provided by the analysis software vendors. What makes a good metric and how it should be computed are of considerable interest to analysts and software developers alike.

Therefore, Panelists are asked for their opinions on grid quality metrics and to take questions from the audience on the same.


  • Provides a solution of desired accuracy
  • Needs
    • Required density & orientation as f(space, time)
    • Discretisation error estimates
    • Modelling error estimates
    • Is physics, level of detail, dimensionality etc appropriate?
    • Capture of evolving physical phenomena & geometry


What makes a good quality metric:

  • This is a minimal set of requirements of a good mesh quality metric, very inclusive
  • Intended to be non-controversial (better have a good reason not to satisfy these)
  • List of requirements raises awareness of deficiences in current metrics
  • If a metric satisfies the requirements it does not mean (i) that it need not satisfy other requirements or (ii) it is useful
  • Urge you to check your metrics, fix those that do not satisfy, and recalibrate
  • Example of usefulness of this list: Cubit Robinson metrics for non-planar quads

A proper finite element quality metric is a function f from RdN to R with the following properties:

  1. Any orientation-preserving permutation of the indices of the vertices yields the same value of f
  2. f is unitless
  3. The value of f is invariant to translation and/or rotation of the coordinate system
  4. The domain D of f is clearly specified
  5. f is referenced to an ideal element or set of elements that describes the desired geometric configuration of the physical element
  6. 0 ≤ f ≤ 1, with f=1 if and only if the physical element attains the ideal node configuration and f=0 if and only if the physical element is degenerate
  7. f, as a function of node positions, is continuous everywhere on D.


Aspect ratio is the metric of most importance. The following are all equivalent for triangles and tetrahedra, up to constant factors:

  • longest side divided by min altitude
  • radius of smallest containing sphere divided by radius of inscribed sphere
  • 1/min-angle in 2D, 1/min-solid-angle in 3D
  • condition number of affine mapping from reference triangle/tetrahedron (any norm)

Bounded by aspect ratio

  • The following are bounded above and below by the aspect ratio, but not within a constant factor:
    • radius of circumsphere divided by radius of inscribed sphere
    • cube of longest edge over volume

Not bounded by aspect ratio

  • The following are not equivalent to aspect ratio:
    • longest edge over shortest edge
    • volume over product-of-side-lengths
    • criteria involving dihedral-angle
    • crieria involving max angle or max solid angle
    • radius of containing sphere divided by edge length

Higher order elements

  • for classical finite element methods, need Ciarlet-Raviart conditions
  • these involve the condition number of the Jacobian as well as higher derivatives of the mapping functions


John Chawner: Confucius said that a main with two clocks never knows what time it really is. Can someone provide a definitive list of formulas for computing common mesh quality metrics?

Ted Blacker: Users need to develop a sense of familiarity with a metric and changes should only be introduced for good, easily explained reasons. It is important to use a single consistent measure.

Mark Shephard: Discussing a priori metrics is a waste of time. What's needed is work in the area of a posteriori measures to eliminate truncation and discretization errors adaptively.

David White: But error metrics are solver sensitive making a posteriori metric development difficult.

Steve Vavasis: Error estimates are sensitive to skew so adaption may not work well.

Gordon Ferguson: We should define practical bounds on metrics.

Ted Blacker: Adaptivity is not always reliable - a priori metric work is needed in the near term.

Tim Baker: Certain "aesthetic" aspects of mesh quality should be independent of solver and developed a priori. If you consider the path from mesh to solution to be a road, the issue is whether we go to the trouble of building a road without potholes (i.e. a priori mesh quality) or build a road with many small potholes and require everyone to drive an SUV (i.e. a posteriori metrics).

unknown: When considering the road's quality should we look at every pothole or just the biggest? (i.e. is it better to measure mesh quality in an average sense or to consider only the worst element in the mesh).

Gordon Ferguson: We need to define not only what a good mesh is but also what a "safe" mesh is.

Steve Vavasis: If you start with a mesh that has a priori aspect ratio bounds, and then you perform adaptive h-refinement, then it is guaranteed (according to certain recent results) that the refined meshes will continue to satisfy an aspect ratio bound. The theoretical guarantees assume that the domain has polyhedral boundaries.

Cecil Armstrong: The target mesh metric may be a highly stretched anisotropic element. The recent book "Mesh Generation: application to finite elements" by Frey and George is a useful starting point.

Jonathon Shewchuck: Disagrees with the notion that aspect ratio is the best metric due to interpolation error issues. Also disagrees with the requirement for unitless measures. What about two adjacent isosocles triangles but one is 10x larger than the other?

Pat Knupp: (in reply to Shewchuck) Compare element size to reference element.

Scott Mitchell: What is reference element? Average size of element across mesh?

John Chawner: What if mesh has length scales that vary by 4-5 orders of magnitude? Is the concept of average really relevant? Will it be able to find local deviations of factor of 2? 5? 10?

Cecil Armstrong: Another interesting question is how do analysts know when their solution is good?

John Chawner: "Budgetary" convergence (when time runs out). Also from accumulated corporate practice, benchmark cases, and local gurus. Users need and want mesh guidelines now.  

Topic #2: "What's the Future of Geometry: NURBs or Faceted Models?"

Recent years have seen an upward trend in use of Faceted Models (e.g. STL files) for geometry definition relative to the use of Solid Models based on NURBs. One reason given for this trend is the relative simplicity of Facets over NURBs, especially in-light of the on-going problems with CAD data interoperability. There are also practical issues: legacy data (an old FE model) may be the only representation available for a given object.

On the other hand, Faceted Geometry has problems of its own. Faceted models can be as sloppy as NURB models. Faceted models may contain a very large number of small triangles in order to accurately capture an object's shape - this bulk can slow down meshing software. Finally, the faceted nature of the geometry may be reflected in the mesh and therefore the analysis.

Panelists are asked for their opinions on the future of geometry for meshing and to take questions from the audience on the same.


  • Mesh generation users shouldn’t need to know representation of underlying shape
  • Users may need to have knowledge of shape (can’t drill elliptical holes)
  • Only linear geometry is robust
  • Subdivision surfaces generating lots of interest in CAGD community


Geometry Reconstruction from Facets

  • Let geometry engine tessellate geometry
  • Triangles and vertex normals are available
    • Traditionally used for rendering
  • Associations with original geometry entities are available
  • Triangle vertex normals are key
    • Disambiguate surface discontinuities
    • Useful to reconstruct higher order geometry
    • Accurate computation of curvature
  • Tessellation techniques continue to improve
  • Trend for downstream applications to use tessellated geometry

Geometry Reconstruction Steps

  • Compute facet normal nf for each triangle and flat xmf for each edge
  • Use Hermite polynomial to place midside node xm
  • For each edge
    • IF adjacent triangle vertex normals are equal
      • Candidate xmc is set to xm
    • ELSE
      • Candidate xmc is computed from intersection of adjacent triangles
  • Let dm = xmc - xmf be the vector from flat midside to candidate midside
  • Inflate geometry xms = xmf + s*dm as s goes from 0 to 1
  • Check triangle vertex normals of inflated triangle with flat normal
  • Stop inflation process at an edge if normal inversion occurs
  • Reconstructed geometry represented as 6 node bi-parabolic triangles

Faceted Geometry

  • Advantages
    • Ease of integration to CAD geometry
    • Fast projection of new points to geometry
    • No geometry callbacks - CAD geometry engine free for other tasks
    • Mesh existing FE meshes, mesh STL style geometry
    • Mesh generation process is encapsulated
    • Proof of existence of valid mesh
    • Indirect methods ensure valid mesh at all times
    • Free of CAD geometry topological constraints
  • Disadvantages
    • Point not on geometry
    • No explicit curvature tolerances
    • Tessellation failure
    • Non manifold triangulation
    • Self intersection of facetted geometry and reconstruction
    • Improved geometry reconstruction not guaranteed


Integration of geometry (and other preprocessing tools) into the analysis process:

Vertical integration:

  • Definition
    • integration of tools which each perform a different preprocessing/analysis function into an integrated process, so that information from any given tool is available to other tools in the process.
  • Benefits
    • data propagation through CAE process for persistence (Design to Analysis iterations)
    • use of data/tools anywhere in the process (e.g. using geometry for adaptivity or for smooth boundary conditions)
    • enables new combinations of tools for new capability (e.g. optimization, adaptivity)

Horizontal integration:

  • Definition
    • integration or ability to use different tools to perform the same function at some point in the analysis process
  • Benefits
    • different types of analyses (e.g. different physics, linear/non-linear, etc.) easy to perform just by substituting in appropriate tool
    • allows analyst to use best tool for each stage of the process



  • Advantages
    • high level control (control points)
    • compact representation
    • multiresolution structure
  • Disadvantages
    • difficult to maintain and manage
    • large models very slow

Faceted Models

  • Advantages
    • very general
    • direct hardware implementation
  • Disadvantages
    • heavy weight representation
    • good editing semantics difficult
    • limited multiresolution structure

  • What is subdivision?
    • Smooth surfaces as the limit of a sequence of refinements

  • What is a Loop Scheme?
    • Generalizes quartic box splines
    • very simple rules

Integrated Design

  • Mechanical response
    • thinshell equations
      • 4th order PDE
      • subdivision ideal!
    • analysis, optimization, etc...

Hesitations / Need Elements

  • direct evaluation
  • theory
  • variety of schemes
  • multigrid/wavelet connection
  • booleans
  • boundary conditions / constraints

Why Subdivision?

  • Many advantages
    • arbitrary topology
    • scalable
    • wavelet connection
    • easy to implement
    • efficient
  • From meshes to surfaces!


  • OK for octree mesh generators, e.g., QMG 2.0, at least in some cases. Not so good for Delaunay or advancing front.
  • Not so good for high-order elements or for problems that depend on C1 behavior
  • not so good for interfacing simulations on two sides of the boundary


Steve Vavasis: Facets are compatible with with some mesh algorithms.

Tim Tautges: Facets can be as sloppy as NURBS so they shouldn't be considered a panacea for sloppy CAD.

Peter Schröder: Both patches (NURBS) and facets can be used with corresponding advantages and disadvantages. Facets can be "heavy". Subdivision surfaces gives best of both worlds.

Gordon Freguson: You can reconstruct higher order geometry from facets based on facet vertex normals. Facets also provide for faster projections

Cecil Armstrong: The "geometry fundamentalists" believe that lines and planes are the only reliable geometric types for computations, hence favoring facets.

Tim Tautges: Use facets to analyze model but then use NURBS for final computations.

Alla Sheffer: Facets are already a mesh and allow for synergistic use of existing technology.

John Chawner: NURBS are still needed for high order surfaces that can't be recovered from faceted models.

Steve Owen: Nurbs are consistent across software packages due to standards like STEP and IGES. There currently isn't such a standard for facets now.

last modified 07 November 2001
Misquoted? Misrepresented? Misspelled?
Contact jrc@pointwise.com.