
10th International Meshing Roundtable
Panel Session
Panelists 
NAFEMS 
Topic #1: Mesh Quality 
Topic #2: Facets v. NURBs
Panelists
 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
 coauthor 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 WisconsinMadison, 1990, Nuclear Engineering
 Led CUBIT project from 19961998.
 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
 coauthor 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.
NAFEMS Aims
 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 10303107 (E)
ISO/CD 1030351 (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.
Armstrong
 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
Knupp
What makes a good quality metric:
 This is a minimal set of requirements of a good mesh quality
metric, very inclusive
 Intended to be noncontroversial (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
nonplanar quads
A proper finite element quality metric is a function f
from R^{dN} to R with the following properties:
 Any orientationpreserving permutation of the indices of the
vertices yields the same value of f
 f is unitless
 The value of f is invariant to translation and/or rotation of
the coordinate system
 The domain D of f is clearly specified
 f is referenced to an ideal element or set of elements
that describes the desired geometric configuration of the physical
element
 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
 f, as a function of node positions, is continuous everywhere
on D.
Vavasis
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/minangle in 2D, 1/minsolidangle 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 productofsidelengths
 criteria involving dihedralangle
 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 CiarletRaviart
conditions
 these involve the condition number of the Jacobian as well as higher
derivatives of the mapping functions
Discussion
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 hrefinement, 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 45 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 inlight of the ongoing 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.
Armstrong
 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
Ferguson
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 n_{f} for each triangle and
flat x_{mf} for each edge
 Use Hermite polynomial to place midside node x_{m}
 For each edge
 IF adjacent triangle vertex normals are equal
 Candidate x_{mc} is set to x_{m}
 ELSE
 Candidate x_{mc} is computed from intersection of
adjacent triangles
 Let d_{m} = x_{mc}  x_{mf} be the vector
from flat midside to candidate midside
 Inflate geometry x_{ms} = x_{mf} + s*d_{m}
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 biparabolic 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
Tautges
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/nonlinear, etc.) easy to perform just by substituting in
appropriate tool
 allows analyst to use best tool for each stage of the process
Schröder
NURBS
 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!
Vavasis
 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 highorder elements or for problems that depend on
C^{1} behavior
 not so good for interfacing simulations on two sides of the boundary
Discussion
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.

