10th International Meshing Roundtable
Topic #1: Mesh Quality |
Topic #2: Facets v. NURBs
- Cecil Armstrong,
- 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,
- President, Visual Kinematics, Inc., a supplier of CAE software
- Prior CAE experience at Ardent Computer and Lockheed Missiles
- MS in Physics and Math, Univ. of Minnesota
- Pat Knupp,
- 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,
- Associate Professor of Computer Science, Caltech
- PhD in Computer Science, 1994, Princeton
- Expert on wavelet based methods for computer graphics
- Tim Tautges,
- 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,
- 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
- 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
- Many companies are obtaining benefits from implementing PDM systems
- These typically enable the Design and Configuration Management (CM)
- International Standards are being developed for both PDM and
- 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
- 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
- Describes the International Standards being developed in this area
- Suggest how the reader may proceed
ISO/CD 10303-107 (E)
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
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
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].
- lots of valuable technology and insights
- archiving mechanism
- slow acceptance
- long gestation period
- minimal benefits to users so far
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
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
- Provides a solution of desired accuracy
- 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
- 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
A proper finite element quality metric is a function f
from RdN to R with the following properties:
- Any orientation-preserving 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
- 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
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
- 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
- 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.
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.
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
Scott Mitchell: What is reference element? Average size of element
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
- Users may need to have knowledge of shape (can’t drill elliptical
- 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
- Candidate xmc is computed from intersection of
- 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
- 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
- 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:
- 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.
- 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)
- integration or ability to use different tools to perform the
same function at some point in the analysis process
- different types of analyses (e.g. different physics,
linear/non-linear, etc.) easy to perform just by substituting in
- allows analyst to use best tool for each stage of the process
- high level control (control points)
- compact representation
- multiresolution structure
- difficult to maintain and manage
- large models very slow
- very general
- direct hardware implementation
- 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
- Mechanical response
- thinshell equations
- 4th order PDE
- subdivision ideal!
- analysis, optimization, etc...
Hesitations / Need Elements
- direct evaluation
- variety of schemes
- multigrid/wavelet connection
- boundary conditions / constraints
- Many advantages
- arbitrary topology
- wavelet connection
- easy to implement
- 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
- 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
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
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?