Mesh Generation
y
Marshall Bern Paul Plassmann
Intro duction
A mesh is a discretization of a geometric domain into small simple shap es such as triangles
or quadrilaterals in two dimensions and tetrahedra or hexahedra in three Meshes nd use
in many application areas In geography and cartography meshes give compact representa
tions of terrain data In computer graphics most ob jects are ultimately reduced to meshes
b efore rendering Finally meshes are almost essential in the numerical solution of partial
di erential equations arising in physical simulation In this chapter we concentrate on al
gorithms for this last application assuming an audience including b oth practitioners such
as engineers and theoreticians such as computational geometers and numerical analysts
Typ es of Geometric Domains
We divide the p ossible inputs rst according to dimension two or three We distinguish
four typ es of planar domains as shown in Figure For us a simple polygon includes b oth
b oundary and interior A polygon with holes is a simple p olygon minus the interiors of some
other simple p olygons its b oundary has more than one connected comp onent A multiple
domain is more general still allowing internal b oundaries in fact such a domain may b e any
planar straight line graph in which the in nite face is b ounded by a simple cycle Multiple
domains mo del ob jects made from more than one material Curved domains allow sides
that are algebraic curves such as splines As in the rst three cases collectively known as
polygonal domains curved domains may or may not include holes and internal b oundaries
Three dimensional inputs have analogous typ es A simple polyhedron is top ologically
equivalent to a ball A general polyhedron may b e multiply connected meaning that it
is top ologically equivalent to a solid torus or some other higher genus solid it may also
have cavities meaning that its b oundary may have more than one connected comp onent
We do assume however that at every p oint on the b oundary of a general p olyhedron
a su ciently small sphere encloses one connected piece of the p olyhedron s interior and
one connected piece of its exterior Finally there are multiple polyhedral domains general
p olyhedra with internal b oundaries and three dimensional curved domains which typically
have b oundaries de ned by spline patches
Construction and mo deling of domain geometry lie outside the scop e of this chapter so
we shall simply assume that domains are given in some sort of b oundary representation
Xerox Palo Alto Research Center Coyote Hill Rd Palo Alto CA b ern parc xerox com
y
Department of Computer Science and Engineering The Pennsylvania State University University Park
PA plassman cse psu edu Supp orted by the Mathematical Information and Computational
Sciences Division subprogram of the O ce of Computational and Technology Research U S Department of
Energy under Contract W Eng
Figure Typ es of two dimensional inputs a simple p olygon b p olygon with holes c multiple
domain and d curved domain
without sp ecifying the exact form of this representation Computational geometers typically
assume exact combinatorial data structures such as linked lists for simple p olygons and
p olygons with holes doubly connected edge lists or quad edge structures for planar
multiple domains and winged edge structures for p olyhedral domains
In practice complicated domains are designed on computer aided design CAD systems
These systems use surface representations designed for visual rendering and then translate
the nal design to another format for input to the mesh generator The Stereolithography
Tessellation Language STL le format originally develop ed for the rapid prototyping of
solid mo dels sp eci es the b oundary as a list of surface p olygons usually triangles and sur
face normals The advantages of the STL input format are that a watertight mo del can
b e ensured and mo del tolerance deviation from the CAD mo del can b e sp eci ed by the
user The Initial Graphics Exchange Sp eci cation IGES format enables a variety of sur
face representations including higher order representations such as B splines and NURBs
Perhaps due to its greater complexity or to sloppy CAD systems or users IGES les often
contain incorrect geometry either gaps or extra material at surface intersections
An alternative approach to format translation is to directly query the CAD system with
say p oint enclosure queries and then construct a new representation based on the answers
to those queries This approach is most advantageous when the translation problem is
di cult as it may b e in the case of implicit surfaces level sets of complicated functions
or constructive solid geometry formulas
With either approach translation or reconstruction by queries the CAD mo del must b e
top ologically correct and su ciently accurate to enable meshing We exp ect to see greater
integration b etween solid mo deling and meshing in the future
Typ es of Meshes
A structured mesh is one in which all interior vertices are top ologically alike In graph
theoretic terms a structured mesh is an induced subgraph of an in nite p erio dic graph
such as a grid An unstructured mesh is one in which vertices may have arbitrarily varying
lo cal neighb orho o ds A block structured or hybrid mesh is formed by a numb er of small
structured meshes combined in an overall unstructured pattern
In general structured meshes o er simplicity and easy data access while unstructured
meshes o er more convenient mesh adaptivity re nement dere nement based on an initial
solution and a b etter t to complicated domains High quality hybrid meshes enjoy the
advantages of b oth approaches but hybrid meshing is not yet fully automatic We shall
discuss unstructured mesh generation at greater length than structured or hybrid mesh
Figure Typ es of meshes a structured b unstructured and c blo ck structured
Figure a Triangulating quadrilaterals b Sub dividing triangles to form quadrilaterals
generation b oth b ecause the unstructured approach seems to b e gaining ground and b ecause
it is more closely connected to computational geometry
The division b etween structured and unstructured meshes usually extends to the shap e
of the elements two dimensional structured meshes typically use quadrilaterals while un
structured meshes use triangles In three dimensions the analogous element shap es are
hexahedra meaning top ological cub es and tetrahedra There is however no essential rea
son for structured and unstructured meshes to use di erent element shap es In fact it
is p ossible to sub divide elements in order to convert b etween triangles and quadrilaterals
Figure and b etween tetrahedra and hexahedra
Organization
Section gives a brief survey of numerical metho ds and their implications for mesh genera
tion Section discusses the in uence of element shap e on accuracy and convergence time
Sections and cover structured and unstructured two dimensional meshes Section
discusses three dimensional hexahedral mesh generation including structured hybrid and
unstructured approaches Finally Section describ es three dimensional unstructured tetra
hedral mesh generation
We shall explain the basic computational geometry results as they arise within a larger
context however Section concludes with a separate theoretical discussion b ecause un
structured planar mesh generation is esp ecially rich in interesting geometric questions
Throughout this article we emphasize practical issues an earlier survey by Bern and Epp
stein emphasized theoretical results Although there is inevitably some overlap b etween
these two surveys we intend them to b e complementary
Mesh generation has a huge literature and we cannot hop e to cover all of it There are
excellent references on numerical metho ds structured mesh generation
and unstructured mesh generation There are also several
nice Web sites with up to date information on mesh generation
Numerical Metho ds
Scienti c computing seeks accurate discrete mo dels for continuous physical phenomena
We can divide the pro cess into three interdep endent steps problem formulation mesh
generation and equation solution In this section we discuss discretization and solution
metho ds and their impact on mesh generation
Discrete Formulation
There are a numb er of approaches to the discrete approximation of partial di erential equa
tions PDEs mo deling a physical system Here we brie y review the standard discretization
metho ds nite di erence nite element and nite volume Although these metho ds result
in linear systems of similar structure the desired characteristics of meshes for these metho ds
di er
The nite di erence method replaces a continuous di erential op erator with a
di erence approximation Consider the partial di erential equation
Lu f
where L is some di erential op erator and u is a function of p osition and p ossibly also of
time We seek an approximate solution of on some geometric domain A standard
nite di erence approach replaces L with a discrete stencil Writing u u x for the
k k
value of u at mesh vertex p osition x the action of the stencil at x can b e approximated
k i
by
X
Lu x A u
i ik k
k adj (x )
i
where adj x is the set of p oints adjacent to x in the mesh and A is a set of weights
i i ik
dep ending only on L and the geometry of the mesh The right hand side of can also b e
discretized yielding a system of linear equations
n
X
A u f
ik k i
k =1
to b e solved for the unknowns u Because the nite di erence stencil gives nonzero weight
k
only to neighb oring vertices this system will b e quite sparse
It is convenient to use the same stencil throughout the mesh this restriction simpli es
not only the software but also the mathematics b ecause the convergence prop erties of a
particular stencil can b e analyzed by Taylor series expansion A nite di erence stencil
gives a more accurate approximation of a continuous op erator when the edges meeting at
vertices are nearly orthogonal For these reasons the nite di erence metho d usually relies
on structured meshes
The nite element method approximates the solution rather than the equation
The essential idea is to replace the continuous function u x with the nite dimensional
P
n
a x where the are basis functions with lo cal supp ort approximationu x
k k k
k =1
These basis functions are typically low order p olynomials so that the action of the dif
ferential op erator L is easy to compute Because the approximation u x is de ned
k
everywhere on the domain an analysis of convergence can b e made in a continuous norm
instead of p ointwise as in the nite di erence metho d
The nite element metho d obtains a discrete approximation by demanding that the
di erential equations b e satis ed for some set of test functions x
i
Z Z
Lu f
i i
An implementation of the nite element metho d involves the computation of co e cients
Z
A L
ik k i
Z
X
L
k i
e
e I
ik
where I denotes the set of elements for which b oth the basis function and the test
ik k
function are nonzero The unknowns a can now b e expressed in terms of the A s by
i k ik
a sparse linear system of equations of the same overall form as
Finite element metho ds are typically no more complicated on unstructured meshes than
on structured meshes Furthermore there is no real advantage to mesh edges meeting
orthogonally For these reasons unstructured meshes nd broad and increasing use in nite
element metho ds
The nite volume method is motivated by physical conservation laws The in nitesimal
version of a conservation law is of the form
d
r
dt
where is the density and is the ux of the conserved quantity In order to maintain the
same physical conservation law on a discrete level the nite volume metho d de nes control
volumes and requires that on each control volume
v
Z Z
d
n
dt
v v
where n is the normal to the surface of the volume The nite volume metho d mo dels
uid dynamics problems esp ecially well b ecause pressure and velo city can b e represented
at centers and vertices of volumes resp ectively
Cel l centered control volumes are usually identical to mesh elements while vertex
centered control volumes form a dual mesh with one volume for each vertex of the original
mesh There are several ways to de ne vertex centered control volumes Two regular grids
may b e overlaid staggered by half an element or in the case of unstructured meshes the
Delaunay triangulation may b e used for the mesh and its dual the Vorono diagram for
the control volumes
Solution Metho ds
The solution of the sparse linear system is usually the most computationally demanding of
the three steps Solution metho ds include direct factorization and preconditioned iterative
metho ds metho ds which can vary dramatically in required storage and computational cost
for di erent problems Moreover the discrete formulation and mesh generation steps greatly
in uence the e cacy of a solution metho d For example if one cho oses to use higher order
basis functions in the nite element metho d one can use a coarser mesh and a smaller but
denser linear system
Direct factorization metho ds such as sparse Cholesky or LU factorization tend b e
slower but more robust than iterative metho ds Luckily the extra computational cost can
b e amortized when the factorization is reused to solve for more than one right hand side
An imp ortant issue in direct metho ds is ordering the vertices to minimize the ll the
numb er of intermediate nonzeros Nested dissection uses graph separators to save an
3 d 1 2
asymptotic factor of O n in the ll Any planar graph admits separators of size O n
2 3
reasonable three dimensional meshes admit separators of size O n
Iterative metho ds have proved e ective in solving the linear systems arising in
physical mo deling Most large problems cannot b e e ectively solved without the use of
preconditioning A p opular preconditioner involves an incomplete factorization Rather
than computing the exact factors for the matrix A LU approximate factors are computed
such that A LU and the preconditioned system
1 1 1
L AU U u L f
is solved iteratively Ideally the incomplete factors should b e easy to compute and require
a mo dest amount of storage and the condition numb er of the preconditioned system should
b e much b etter than the original system
Multigrid metho ds can achieve the ultimate goal of iterative metho ds convergence
in O iterations for certain classes of problems These metho ds use a sequence of meshes
graded from ne to coarse The iterative solution of the ne linear system is accelerated
by solving the coarser systems The cycle rep eatedly pro jects the current residual from a
ner mesh onto the next coarser mesh and interp olates the solution from the coarser mesh
onto the ner One drawback of multigrid is that computing a sequence of meshes may b e
di cult for complicated geometries
Domain decomp osition represents something of a hybrid b etween iterative and
direct approaches This approach divides the domain into p ossibly overlapping small do
mains it typically solves subproblems on the small domains directly but iterates to the
global solution in which neighb oring subproblem solutions agree This approach enjoys
some of the sup erior convergence prop erties of multigrid metho ds while imp osing less of a
burden on the mesh generator In fact the domain is often partitioned so that subproblems
admit structured meshes
Element Shap e
The shap es of elements in a mesh have a pronounced e ect on numerical metho ds For
purp oses of discussion let us de ne the aspect ratio of an element to b e the ratio of its
maximum to its minimum width where width refers to the distance b etween parallel sup
p orting hyp erplanes There are many other roughly equivalent de nitions of asp ect ratio
In general elements of large asp ect ratio are bad Large asp ect ratios lead to p o orly
conditioned matrices worsening the sp eed and accuracy of the linear solver Sp eed degrades
b efore accuracy a triangular mesh with a rather mild sharp est angle say can b e
noticeably slower than a triangular mesh with a minimum angle of
Moreover even assuming that the solver gives an exact answer large asp ect ratios may
give unacceptable interp olation error Here it is useful to distinguish b etween two di erent
typ es of p o or asp ect ratio Early results showed convergence as triangular elements
shrink assuming that all angles are b ounded away from Later analysis however
showed convergence assuming only that angles are b ounded away from a weaker
condition The generalization of this result to three dimensions assumes dihedrals b ounded
away from Bank and Smith recently prop osed a triangle quality measure based on
an analysis of interp olation error the area of a triangle divided by the sum of squared edge
lengths This measure slightly favors sharp triangles over at triangles
Sometimes however elements of large asp ect ratio are go o d If the solution to the
di erential equation is anisotropic meaning that its second derivative varies greatly with
direction then prop erly aligned high asp ect ratio elements give a very e cient mesh The
ideal asp ect ratio of a triangle is the square ro ot of the ratio of the largest to smallest
eigenvalues of the Hessian For triangular meshes it do es not make much di erence
whether long skinny elements have large angles as well as small angles but if the asp ect
ratio exceeds the ideal then large angles are worse than small
Fluid ow problems esp ecially full Navier Stokes simulation that is viscosity included
are strongly anisotropic For example in aero dynamic simulations ideal asp ect ratios may
reach along the surface of the aircraft Quadrilateral and hexahedral meshes have
an advantage in accuracy over triangular and tetrahedral meshes for control volume formu
lations of these problems as they allow faces of elements in the b oundary layer to b e either
almost parallel or almost orthogonal to the surface
Simulations with sho ck fronts for example sup ersonic air ow over a wing are also
strongly anisotropic In this case however the lo cations and directions for high asp ect ratio
elements cannot b e predicted in advance The need for adaptivity remeshing based on an
initial solution now tilts the balance in favor of triangles and tetrahedra Simpson
discusses and surveys the literature on anisotropy
Element shap e a ects another prop erty of the linear system b esides condition numb er
A triangular mesh with well shap ed elements gives a symmetric M matrix p ositive de nite
with negative o diagonal entries for a nite element formulation of an equation with a
Laplacian op erator M matrices are exactly those matrices that satisfy a discrete maximum
principle this desirable prop erty rules out oscillation of the numerical metho d In this case
well shap ed has a precise meaning the two angles opp osite each interior edge of the mesh
should sum to at most This requirement implies that no quadrilaterals are
reversed Section so the triangulation must b e the Delaunay or constrained Delaunay
triangulation Dep ending on the b oundary conditions asso ciated with the di erential equa
tion an M matrix may also require that the single angle opp osite a b oundary edge should
measure at most This requirement go es b eyond Delaunay but it is not hard to sat
isfy this requirement for domains without internal b oundaries simply split outwards facing
obtuse angles by dropping p erp endiculars to the b oundary ip back to a new Delaunay
triangulation and rep eat until there are no reversed quadrilaterals and no outwards facing
obtuse angles
In three dimensions an unstructured tetrahedral mesh gives an M matrix if and only if
P
jej cot is nonnegative where the sum is over all for each edge e in the mesh the sum
e
e
edges e that are opp osite to e in tetrahedra of the mesh and where jej denotes the length
of e and the dihedral angle at e All such sums will b e nonnegative if all dihedrals
e
in the mesh are nonobtuse but this condition is more restrictive than necessary
Finally in a nite volume formulation of Poisson s equation a Delaunay mesh with
Vorono control volumes gives an M matrix even in three dimensions The nite volume
formulation but not the nite element formulation can tolerate Delaunay tetrahedra of
large asp ect ratio so long as all control volumes have go o d asp ect ratios The disparity
b etween the two formulations is surprising b ecause they give the very same matrix in two
dimensions
Structured Two Dimensional Meshes
Structured meshes o er simplicity and e ciency A structured mesh requires signi cantly
less memory say a factor of three less than an unstructured mesh with the same numb er
of elements b ecause array storage can de ne neighb or connectivity implicitly A structured
mesh can also save time to access neighb oring cells when computing a nite di erence
stencil software simply increments or decrements array indices Compilers pro duce quite
e cient co de for these op erations in particular they can optimize the co de for vector
machines
On the other hand it can b e di cult or imp ossible to compute a structured mesh for
a complicated geometric domain Furthermore a structured mesh may require many more
elements than an unstructured mesh for the same problem b ecause elements in a structured
mesh cannot grade in size as rapidly These two di culties can b e solved by the hybrid
structured unstructured approach which decomp oses a complicated domain into blo cks
supp orting structured grids Hybrid approaches however are not yet fully automatic
requiring user guidance in the decomp osition step A complicated three dimensional hybrid
mesh see Section can take weeks or even months of work hence hybrid approaches are
typically used only late in the design cycle
Structured mesh generation can b e roughly classi ed into hand generated and other
elementary approaches algebraic or interp olation metho ds and PDE or variational meth
o ds The PDE approach solves partial di erential equations in order to map
the domain onto another domain with a convenient co ordinate system In this section we
discuss an elliptic PDE approach with a connection to the classical topic of conformal
mapping
A mapping of a region of the complex plane is conformal if it preserves angles in
other words the angle b etween any two curves intersecting at a p oint z is preserved by
the mapping The Riemann mapping theorem states that for any top ological disk there
exists a conformal mapping f that takes the interior of one to one onto the interior of any
other top ological disk such as the unit disk or square There is an obvious connection to
mesh generation a conformal mapping of onto a square grid induces a structured mesh
on with the prop erty that element angles tend towards in the limit of an increasingly
ne discretization
Unfortunately the Riemann mapping theorem only proves the existence of a conformal
mapping it do es not give an algorithm Let us write z x iy and consider the the complex η y M f
Ω R
1 ξ x
Figure The conformal mapping f from the domain de ned by a b oundary discretization to a
rectangle R The inverse of the mapping maps a grid on R onto a structured mesh for
Figure The grid on left was obtained by solving with unit asp ect ratio resulting in a folded over
mesh On the right a more appropriate asp ect ratio has b een chosen
function f z x y i x y If f is analytic as a conformal f will b e assuming
f z then it satis es the Cauchy Riemann equations and Thus
x y y x
2
the functions and must each b e harmonic and satisfy Laplace s equation so that r
2
and r If f is conformal its inverse is as well therefore x and y as functions of
2 2
and are also harmonic and satisfy r x and r y
Consider the regions and R in Figure and assume we already have a discretization
of the b oundary of Finding a suitable b oundary discretization may itself b e a nontrivial
2 2
task The obvious algorithm is to solve r x and r y assuming x and y are
given on the b oundary of R However this approach may not work One may obtain
p o orly shap ed or even inverted elements as shown in Figure a The problem is that the
solutions x and y may b e harmonic but not harmonic conjugate i e satisfy the Cauchy
Riemann equations
The algorithm can b e partially mended by obtaining a b etter estimate for M the
rectangle height implied by the discretization of the b oundary of If we scale the original
co ordinates of the rectangle onto a square with co ordinates with the mapping
and M we obtain the system
2 2
M x x M y y
From the rst order Cauchy Riemann equations we have
2 2 2 2 2
M x y x y
Bar eld obtained reasonable nonoverlapping meshes by estimating the average value of
the right hand side of the ab ove equation and using this value for M One can think of M
as an average asp ect ratio for the original domain if ideal asp ect ratio varies signi cantly
over the domain one can also make this numb er a function of p osition This approach
can b e successful for many physical problems and can b e improved signi cantly if the
generated grid is smo othed as a p ostpro cessing step This approach can also b e extended
to three dimensions where the Riemann mapping theorem no longer holds by the addition
of another average asp ect ratio term
Although the approach just sketched works quite well for some domains it do es not
guarantee the generation of a valid mesh It is interesting that the inverse problem solving
the harmonic equations
xx y y
xx y y
do es guarantee a solution with no inverted elements and a nonvanishing Jacobian
Solving the problem in this form is more di cult b ecause it requires a discretization of the
domain for which we want to nd a grid However the system can b e inverted to form the
nonlinear elliptic system
x x x
y y y
where
2 2
x y
x x y y
2 2
x y
Software designed to solve these systems often includes an additional source term on the
right hand sides of the harmonic systems in to control the lo cal p oint spacing in the
domain
The elliptic metho d just discussed though motivated by conformal mapping do es not
compute true conformal mappings A true conformal mapping induces a structured mesh
with certain advantages for example the Laplacian is the limit of the second order di er
ence on such a grid True conformal mapping however do es not seem to b e widely used
in mesh generation p erhaps b ecause algorithms to compute such mappings are relatively
slow or b ecause they do not allow lo cal control of p oint spacing
In the case that is a simple p olygon the Schwarz Christo el formula o ers an explicit
form for the conformal mappings from the unit disk D to Such a mapping can in
turn b e used to nd conformal mappings from to a square or rectangle Let the p oints
in the complex plane de ning the p olygon in counterclo ckwise order b e z z the
1 n
interior angles at these p oints b e and de ne the normalized angles as
1 n k
Using as the preimages of z z on the edge of the disk the
k 1 n 1 n
Schwarz Christo el formula gives the form of the conformal mapping as
Z
n
Y
k
d f A B
k
0
k =1
There are several programs available to solve for the unknown values SCPACK by
k
Trefethen the SC To olb ox by Driscoll and CRDT by Driscoll and Vavasis
One di culty in the numerical solution is crowding enormous variation in spacing b e
tween the p oints CRDT the latest and apparently b est Schwarz Christo el algorithm
k
overcomes this di culty by rep eatedly remapping so that crowding do es not o ccur near the
p oints b eing evaluated
Unstructured Two Dimensional Meshes
We have already mentioned the advantages of unstructured meshes exibility in tting
complicated domains rapid grading from small to large elements and relatively easy re
nement and dere nement
Unlike structured mesh generation unstructured mesh generation has b een part of main
stream computational geometry for some years and there is a large literature on the sub
ject We consider three principled approaches to unstructured mesh generation in some
detail these approaches use Delaunay triangulation constrained Delaunay triangulation
and quadtrees Then we discuss mesh re nement and improvement In the nal section we
describ e some geometric problems abstracted from unstructured mesh generation
Delaunay Triangulation
Our rst approach to unstructured mesh generation partitions the task into two phases
placement of mesh vertices followed by triangulation Added p oints are called Steiner
points to distinguish them from the domain s original vertices If the placement phase is
smart enough the triangulation phase can b e esp ecially simple considering only the input
vertices and Steiner p oints and ignoring the input edges
The placement phase typically places vertices along the domain b oundary b efore adding
them to the interior The b oundary should b e lined with enough Steiner p oints that the
Delaunay triangulation of all vertices will conform to the domain This requirement inspires
a crisp geometric problem called conforming Delaunay triangulation given a p olygonal
domain add Steiner p oints so that each edge of is a union of edges in the Delaunay
triangulation An algorithm due to Saalfeld lines the edges of with a large numb er
of Steiner p oints uniformly spaced except near the endp oints A more e cient solution
covers the edges of by disks that do not overlap other edges Edelsbrunner and Tan
3
gave the b est theoretical result an algorithm that uses O n Steiner p oints for an n vertex
2
multiple domain They also gave an n lower b ound example
There are several approaches to placing interior Steiner p oints One approach
combines the vertices from a numb er of structured meshes A second approach adds
Steiner p oints in successive layers working in from the domain b oundary as in advancing
front mesh generation Section Figure shows an example A third approach
cho oses interior p oints at random according to some distribution which may b e interp olated
from a coarse quadtree or background triangulation An indep endent random sample is
likely to pro duce some badly shap ed triangles so the generator should oversample
and then lter out p oints to o close to previously chosen p oints Finally there are
deterministic metho ds that achieve essentially the same e ect as random sampling with
ltering these metho ds de ne birth and death rules that dep end up on the density
of neighb oring p oints
All of these metho ds can give anisotropy The rst and second approaches structured
submeshes and advancing front o er lo cal control of element shap es and orientations These
Figure Delaunay triangulation of p oints placed by an advancing front T Barth
e
Figure a Delaunay triangulation b A reversed quadrilateral
approaches may space p oints improp erly where structured meshes or advancing fronts col
lide but this aw can usually b e corrected by ltering p oints and later smo othing the mesh
The third and fourth approaches trade direct control over element shap es for ease of tting
complicated geometries Nevertheless one can achieve anisotropy with these approaches by
computing the Delaunay triangulation within a stretched space For example
Bossen uses a background triangulation to de ne lo cal a ne transformations Delau
nay ips describ ed b elow are then made with resp ect to transformed circles Stretched
Delaunay triangulations have many more large angles than ordinary Delaunay triangula
tions but this should not p ose a problem unless the stretching exceeds the desired amount
Section
The triangulation phase uses the well known Delaunay triangulation The Delaunay
triangulation of a p oint set S fs s s g is de ned by the empty circle condition a
1 2 n
triangle s s s app ears in the Delaunay triangulation DT S if and only if its circumcircle
i j k
encloses no other p oints of S See Figure a There is an exception for p oints in sp ecial
p osition if an empty circle passes through four or more p oints of S we may triangulate
these p oints complete the triangulation arbitrarily So de ned DT S is a triangulation
of the convex hull of S For our purp oses however we can discard all triangles that fall
outside the original domain
There are a numb er of practical Delaunay triangulation algorithms We describ e
only one called the edge ipping algorithm b ecause it is most relevant to our subsequent
2
discussion Its worst case running time of O n is sub optimal but it p erforms quite well
in practice The edge ipping algorithm starts from any triangulation of S and then lo cally
optimizes each edge Let e b e an internal non convex hull edge and Q b e the triangulated
e
quadrilateral formed by the triangles sharing e Quadrilateral Q is reversed if the two angles
e s4 s9
s10
s5 s7 s 2 s8 s1 s6
s3
Figure A sweep line algorithm for computing an initial triangulation
without the diagonal sum to more than or equivalently if each triangle circumcircle
contains the opp osite vertex as in Figure b If Q is reversed we ip it by exchanging
e
e for the other diagonal
compute an initial triangulation of S
place all internal edges onto a queue
while the queue is not empty do
remove the rst edge e
if quadrilateral Q is reversed then
e
ip it and add the outside edges of Q to the queue endif
e
endwhile
An initial triangulation can b e computed by a sweep line algorithm This algorithm adds
the p oints of S by x co ordinate order Up on each addition the algorithm walks around
the convex hull of the already added p oints starting from the rightmost previous p oint and
adding edges until the slop e reverses as shown in Figure The following theorem
guarantees the success of edge ipping a triangulation in which no quadrilateral is reversed
must b e a completion of the Delaunay triangulation
Constrained Delaunay triangulation
There is another way b esides conforming Delaunay triangulation to extend Delaunay tri
angulation to p olygonal domains The constrained Delaunay triangulation of a p ossibly
multiple domain do es not use Steiner p oints but instead rede nes Delaunay triangula
tion in order to force the edges of into the triangulation
A p oint p is visible to a p oint q in if the op en line segment pq lies within and do es
not intersect any edges or vertices of The constrained Delaunay triangulation CDT
contains each triangle not cut by an edge of that has an an empty circumcircle where
empty now means that the circle do es not contain any vertices of visible to p oints inside
the triangle The visibility requirement means that external proximities where wraps
around to nearly touch itself have no e ect Figure provides an example here vertex v
is not visible to any p oint in the interior of triangle abc
The edge ipping algorithm can b e generalized to compute the constrained Delaunay
triangulation only this time we do not allow edges of onto the queue Obtaining an
initial triangulation is somewhat more di cult for p olygonal domains than for p oint sets
The textb o ok by Preparata and Shamos describ es an O n log n time algorithm for a
v
b
c
Figure The constrained Delaunay triangulation of a p olygon with holes
Figure A mesh computed by Rupp ert s algorithm J Rupp ert
computing an initial triangulation This algorithm rst adds edges to to sub divide it into
easy to triangulate monotone faces
Rupp ert building on work of Chew gave a mesh generation algorithm based
on constrained Delaunay triangulation Subsequently Mitchell sharp ened Rupp ert s
analysis and Shewchuk further re ned the algorithm and made an implemen
tation available on the Web Rupp ert s algorithm computes the constrained Delaunay
triangulation at the outset and then adds Steiner p oints to improve the mesh thus uniting
the two phases of the approach describ ed in the last section In cho osing this approach the
user gives up some control over p oint placement but obtains a more e cient mesh with
fewer and rounder triangles
The rst step of Rupp ert s mesh generator cuts o all vertices of the domain at which
the interior angle measures less than The cutting line at such a vertex v should not
intro duce a new small feature to it is b est to cut o an isosceles triangle whose base is
ab out halfway from v to its closest visible neighb or If v has degree greater than two as
might b e the case in a multiple domain then the bases of the isosceles triangles around v
should match up so that no isosceles triangle receives a Steiner p oint on one of its legs
Next the algorithm computes the constrained Delaunay triangulation of the mo di ed
domain The algorithm then go es through the lo op given b elow The last line of the
lo op repairs a constrained Delaunay triangulation after the addition of a new Steiner p oint
c To accomplish this step there is no need to recompute the entire triangulation The
removed old triangles are exactly those with circumcircles containing c which can b e found
by searching outwards from the triangle that contains c and the new triangles that replace
the removed triangles must all b e incident to the new vertex c
while there exists a triangle t with an angle smaller than do
let c b e the center of t s circumcircle
if c lies within the diameter semicircle of a b oundary edge e then
add the midp oint m of e
else add c endif
recompute the constrained Delaunay triangulation
endwhile
The lo op is guaranteed to halt with all angles larger than At this p oint the cut
o isosceles triangles are returned to the domain and the mesh is complete Rupp ert s
algorithm comes with a strong theoretical guarantee all new angles that is angles not
present in the input are greater than and the total numb er of triangles in the mesh
is at most a constant times the minimum numb er of triangles in any such no smal l angle
mesh To prove this e ciency result Rupp ert shows that each triangle in the nal mesh
is within a constant factor of the local feature size at its vertices The lo cal feature size at
p oint p is de ned to b e the radius of the smallest circle centered at p that touches two
nonadjacent edges of the b oundary this is a spacing function intrinsic to the domain
Quadtrees
A quadtree mesh generator starts by enclosing the entire domain inside an
axis aligned square It splits this root square into four congruent squares and continues
splitting squares recursively until each minimal or leaf square intersects in a simple
way Further splits may b e dictated by a user de ned spacing function or balance condition
Quadtree squares are then warp ed and cut to conform to the b oundary A nal triangulation
step gives an unstructured triangular mesh
We now describ e a particular quadtree mesh generator due to Bern Eppstein and
Gilb ert As rst presented the algorithm assumes that is a p olygon with holes
however the algorithm can b e extended to multiple and even to curved domains In fact
the quadtree approach handles curved domains more gracefully than the Delaunay and
constrained Delaunay approaches b ecause the splitting phase can automatically adapt to
the curvature of enclosed b oundary pieces
The algorithm of Bern et al splits squares until each leaf square contains at most one
connected comp onent of s b oundary with at most one vertex Mitchell and Vavasis
improved the splitting phase by cloning squares that intersect in more than one con
nected comp onent so that each copy contains only a single connected comp onent of The
algorithm then splits squares near vertices of two more times so that each vertex lies
within a bu er zone of equal size squares
Next the mesh generator imp oses a balance condition no square should b e adjacent to
one less than one half its size This causes more splits to propagate across the quadtree
increasing the total numb er of leaf squares by a constant factor at most Squares are
Figure A mesh computed by a quadtree based algorithm S Mitchell
then warped to conform to the domain Various warping rules work we give just one
p ossibility In the following pseudo co de jbj denotes the side length of square b
for each vertex v of do
let y b e the closest quadtree vertex to v
move y to v
endfor
for each leaf square b still crossed by an edge e do
move the vertices of b that are closer than jbj to e to their closest p oints on e
endfor
discard faces of the warp ed quadtree that lie outside
Finally the cells of the warp ed quadtree are triangulated so that all angles are b ounded
away from Figure gives a mesh computed by a variant of the quadtree algorithm This
gure shows that cloning ensures appropriate element sizes around holes and almost holes
Notice that a quadtree based mesh exhibits preferred directions horizontal and vertical If
this artifact p oses a problem mesh improvement steps can b e used to redistribute element
orientations The quadtree algorithm enjoys the same e ciency guarantee as Rupp ert s
algorithm In fact the quadtree algorithm was the rst to b e analyzed in this way
Mesh Re nement and Dere nement
Adaptive mesh re nement places more grid p oints in areas where the PDE solution error
is large Lo cal error estimates based on an initial solution are known as a posteriori error
estimates and can b e used to determine which elements should b e re ned For elliptic
problems these estimators asymptotically b ound the true error and can b e computed lo cally
using only the information on an element
One approach to mesh re nement iteratively inserts extra vertices into the triangu
lation typically at edge bisectors or triangle circumcenters as in Section New vertices
along the b oundaries of curved domains should b e computed using the curved b oundary
Figure A triangle divided by a bisection and b regular re nement
rather than the current straight edge thereby giving a truer approximation of the domain as
the mesh re nes Iterative vertex insertion may b e viewed as a mesh improvement step
Section and indeed several generators have combined insertion deletion
ipping and smo othing into a single lo op
Iterative vertex insertion gives a ner mesh but not a nesting or edge conforming re
nement of the original mesh meaning a mesh that includes the b oundaries of the original
triangles Nesting re nements simplify the interp olation step in the multigrid metho d Sec
tion To compute such a re nement we turn to another approach This approach splits
triangles in need of re nement by adding the midp oints of sides The pseudo co de b elow
gives the overall approach
k
solve the di erential equation on the initial mesh T
0
estimate the error on each triangle
while the maximum error on a triangle is larger than the given tolerance do
based on error estimates mark a set of triangles S to re ne
k
divide the triangles in S along with adjacent invalid triangles to get T
k k +1
solve the di erential equation on T
k +1
estimate the error on each triangle
k k
endwhile
There are a numb er of p opular alternatives for step in which the current mesh T is
k
adaptively re ned In regular re nement the midp oints of the sides of a marked
triangle are connected as in Figure b to form four similar triangles Unmarked triangles
that received two or three midp oints are split in the same way Unmarked triangles that
received only one midp oint are bisected by connecting the midp oint to the opp osite vertex
as in Figure a Before the next iteration of bisected triangles are glued back together
and then marked for re nement this precaution guarantees that each triangle in T will
k +1
either b e similar to a triangle in T or b e the bisection of a triangle similar to a triangle
0
in T Thus regular re nement regardless of the numb er of times through the re nement
0
lo op pro duces a mesh with minimum angle at least half the minimum angle in T Hence
0
the angles in T are b ounded away from and
k +1
Rivara prop osed several alternatives for step based on triangle bisec
tion One metho d re nes each marked triangle by cutting from the opp osite vertex to the
midp oint of the longest edge Neighb oring triangles are now invalid meaning that one
side contains an extra vertex these triangles are then bisected in the same way Bisections
Figure The bisection algorithm given in the pseudo co de splits invalid children of re ned triangles
to their sub division p oints rather than to their longest edges
continue until there are no remaining invalid triangles Re nement can propagate quite
far from marked triangles however propagation cannot fall into an in nite lo op b ecause
along a propagation path each bisected edge is longer than its predecessor This approach
like the previous one pro duces only a nite numb er of di erent triangle shap es similarity
classes and the minimum angle is again at least half the smallest angle in T Quite often
0
longest edge re nement actually improves angles
A second Rivara re nement metho d is given in the pseudo co de b elow and illustrated
in Figure This metho d do es not always bisect the longest edge so bisections tend
to propagate less yet the metho d retains the same nal angle b ound as the rst Rivara
metho d
i
Q S f Q denotes marked triangles to b e re ned g
i k
R f R denotes children of re ned triangles g
i
while Q R do
i i
bisect each triangle in Q across its longest edge
i
bisect each triangle in R across its sub divided edge
i
add all invalid children of Q triangles to R
i i+1
add all other invalid triangles to Q
i+1
i i
endwhile
We now discuss the reverse pro cess coarsening or dere nement of a mesh This pro cess
helps reduce the total numb er of elements when tracking solutions to time varying di er
ential equations Coarsening can also b e used to turn a single highly re ned mesh into a
sequence of meshes for use in the multigrid metho d
Figure shows a sequence of meshes computed by a coarsening algorithm due to
Ollivier Go o ch The algorithm marks a set of vertices to delete from the ne mesh elim
inates all marked vertices and then retriangulates the mesh The resulting mesh is node
conforming meaning that every vertex of the coarse mesh app ears in the ne mesh but
not edge conforming One di culty is that the shap es of the triangles degrade as the mesh
is coarsened due to increasing disparity b etween the interior and b oundary p oint densi
ties Meshes pro duced by re nement metho ds are typically easier to coarsen than are less
hierarchical meshes such as Delaunay triangulations Teng Talmor and Miller have re
cently devised an algorithm using Delaunay triangulations of well spaced p oint sets which
pro duces a sequence of b ounded asp ect ratio no de conforming meshes of approximately
minimum depth
Figure A sequence of meshes used by the multigrid metho d for solving the linear systems arising
in mo deling air ow over an airfoil C Ollivier Go o ch
Mesh Improvement
The most common mesh improvement techniques are ipping and smo othing These tech
niques have proved to b e very p owerful in two dimensions and together they can transform
very p o or meshes into very go o d ones so long as the mesh starts with enough vertices
Flipping exchanges the diagonals of a triangulated quadrilateral as in the edge ipping
algorithm for computing Delaunay triangulation Section only the criterion for making
the exchange need not b e the Delaunay empty circle test Flipping can b e used to regularize
vertex degrees minimize the maximum angle or improve almost any other quality measure
of triangles For quality measures optimized by the Delaunay triangulation Section
ipping computes a true global optimum but for other criteria it computes only a lo cal
optimum
Mesh smo othing adjusts the lo cations of mesh vertices in order to improve element
shap es and overall mesh quality In mesh smo othing the top ology of the
mesh remains invariant thus preserving imp ortant features such as the nonzero pattern of
the linear system
Laplacian smo othing is the most commonly used smo othing technique This
metho d sweeps over the entire mesh several times rep eatedly moving each adjustable vertex
to the arithmetic average of the vertices adjacent to it Variations weight each adjacent
vertex by the total area of the elements around it or use the centroid of the incident
elements rather than the centroid of the neighb oring vertices Laplacian smo othing
is computationally inexp ensive and fairly e ective but it do es not guarantee improvement
in element quality In fact Laplacian smo othing can even invert an element unless the
algorithm p erforms an explicit check b efore moving a vertex
Another class of smo othing algorithms uses optimization techniques to determine new
vertex lo cations Both global and lo cal optimization based smo othing o er guaranteed
mesh improvement and validity Global techniques simultaneously adjust all unconstrained
vertices such an approach involves an optimization problem as large as the numb er of
unconstrained vertices and consequently is computationally very exp ensive Lo cal
techniques adjust vertices one by one or an indep endent set of vertices in parallel
resulting in a cost more comparable to Laplacian smo othing Many quality measures
including maximum angle and area divided by sum of squared edge lengths can b e optimized
by techniques related to linear programming
Figure shows the results of a lo cal optimization based smo othing algorithm develop ed
Figure a A mesh resulting from bisection re nement without smo othing b The same mesh
after lo cal optimization based smo othing
by Freitag et al The algorithm was applied to a mesh generated adaptively during the
nite element solution of the linear elasticity equations on a two dimensional rectangular
domain with a hole The mesh on the left was generated using the bisection algorithm for
re nement the edges from the coarse mesh are still evident after many levels of re nement
The mesh on the right was generated by a similar algorithm only with vertex lo cations
optimized after each re nement step Overall the global minimum angle has improved
from to and the average minimum element angle from to
Theoretical Questions
We have mentioned some theoretical results conforming Delaunay triangulation no small
angle triangulation in context In this section we describ e some other theoretical work
related to mesh generation
Optimal Triangulation
Computational geometers have studied a numb er of problems of the following form given
a planar p oint set or p olygonal domain nd a b est triangulation where b est is judged
according to some sp eci c quality measure such as maximum angle minimum angle max
imum edge length or total edge length If the input is a simple p olygon most optimal
3
triangulation problems are solvable by dynamic programming in time O n but if the
input is a p oint set p olygon with holes or multiple domain these problems b ecome much
harder
The Delaunay triangulation constrained Delaunay triangulation in the case of p olygo
nal domains optimizes any quality measure that is improved by ipping a reversed quadri
lateral this statement follows from the theorem that a triangulation without reversed
quadrilaterals must b e Delaunay Thus Delaunay triangulation maximizes the minimum
angle along with optimizing a numb er of more esoteric quality measures such as maximum
circumcircle radius maximum enclosing circle radius and roughness of a piecewise linear
interp olating surface b
Ear
e a
c
Figure Edge insertion retriangulates holes by removing su ciently go o d ears Dotted lines
indicate the old triangulation
As mentioned in Section edge ipping can also b e used as a general optimization
heuristic For example edge ipping works reasonably well for minimizing the maximum
angle but it do es not in general nd a global optimum A more p owerful lo cal im
provement metho d called edge insertion exactly solves the minmax angle problem
as well as several other minmax optimization problems
Edge insertion starts from an arbitrary triangulation and rep eatedly inserts candidate
edges If minimizing the maximum angle is the goal the candidate edge e sub divides the
maximum angle in general the candidate edge is always incident to a worst vertex of
a worst triangle The algorithm then removes the edges that are crossed by e forming
two p olygonal holes alongside e Holes are retriangulated by rep eatedly removing ears
triangles with two sides on the b oundary as shown in Figure with maximum angle
smaller than the old worst angle cab If retriangulation runs to completion then the overall
triangulation improves and edge bc is eliminated as a future candidate If retriangulation
gets stuck then the overall triangulation is returned to its state b efore the insertion of e
and e is eliminated as a future candidate Each candidate insertion takes time O n giving
3
a total running time of O n
n
compute an initial triangulation with all edge slots unmarked
2
while an unmarked edge e cutting the worst vertex a of worst triangle abc do
add e and remove all edges crossed by e
try to retriangulate by removing ears b etter than abc
if retriangulation succeeds then mark bc
else mark e and undo e s insertion endif
endwhile
Edge insertion can compute the minmax eccentricity triangulation or the minmax
3
slop e interp olating surface in time O n By inserting candidate edges in a certain
2
order and saving old partial triangulations the running time can b e improved to O n log n
for minmax angle and maxmin triangle height
We close with some results for two other optimization criteria maximum edge length
and total length Edelsbrunner and Tan showed that a triangulation of a p oint set
that minimizes the maximum edge length must contain the edges of a minimum spanning
tree The tree divides the input into simple p olygons which can b e lled in by dynamic
3 2
programming giving an O n time algorithm improvable to O n Whether a trian
gulation minimizing total edge length minimum weight triangulation can b e solved in
p olynomial time is still op en The most promising approach incrementally computes a
set of edges that must app ear in the triangulation If the required edges form a connected
spanning graph then the triangulation can b e completed with dynamic programming as in
the minmax problem
Steiner Triangulation
The optimal triangulation problems just discussed have limited applicability to mesh gen
eration since they address only triangulation and not Steiner p oint placement Because
exact Steiner triangulation problems app ear to b e intractable typical theoretical results on
Steiner triangulation prove either an approximation b ound such as the guarantees provided
by the mesh generators in Sections and or an order of complexity b ound such as
3
Edelsbrunner and Tan s O n algorithm for conforming Delaunay triangulation
The mesh generators in Sections and give constant factor approximation algo
rithms for what we may call the no smal l angle problem triangulate a domain using a
minimum numb er of triangles such that all new angles are b ounded away from The
provable constants tend to quite large in the hundreds although the actual p erformance
seems to b e much b etter The numb er of triangles in a no small angle triangulation dep ends
on the geometry of the domain not just on the numb er of vertices n an upp er b ound is
given by the sum of the asp ect ratios of triangles in the constrained Delaunay triangulation
We can also consider the no large angle problem triangulate using a minimum num
b er of triangles such that all new angles are b ounded away from The strictest b ound
on large angles that do es not imply a b ound on small angles is nonobtuse triangulation
triangulate a domain such that the maximum angle measures at most Moreover
a nonobtuse mesh has some desirable numerical and geometric prop erties Bern
Mitchell and Rupp ert develop ed a circle based algorithm for nonobtuse triangulation
of p olygons with holes this algorithm gives a triangulation with O n triangles regard
less of the domain geometry Figure shows the steps of this algorithm the domain is
packed with nonoverlapping disks until each uncovered region has either or sides radii
to tangencies are added in order to split the domain into small p olygons and nally small
p olygons are triangulated with right triangles without adding any new sub division p oints
It is currently unknown whether multiple domains admit p olynomial size nonobtuse
triangulations Mitchell however gave an algorithm for triangulating multiple domains
2
using O n log n triangles with maximum angle Tan improved the maximum
2
angle b ound to and the complexity to the optimal O n
Hexahedral Meshes
Mesh generation in three dimensions is not as well develop ed as in two for a numb er
of reasons lack of standard data representations for three dimensional domains greater
software complexity and most relevant to this article some theoretical di culties
This section and the next one survey approaches to three dimensional mesh generation
We have divided this material according to element shap e hexahedral or tetrahedral This
classi cation is not completely strict as many hexahedral mesh generators use triangular
prisms and tetrahedra in a pinch Careful implementations of numerical metho ds can handle
Figure Steps in circle based nonobtuse triangulation
Figure A multiblo ck hexahedral mesh of a submarine showing a blo ck structure and b a
vertical slice through the mesh ICEM CFD
degenerate hexahedra such as prisms In this section we describ e three approaches
to hexahedral mesh generation that vary in their degree of structure and strictness
Multiblo ck Meshes
We start with the approach that pro duces meshes with the most structure and quite often
the highest quality elements A multiblock mesh contains a numb er of small structured
meshes that together form a large unstructured mesh Typically a user must supply the
top ology of the unstructured mesh but the rest of the pro cess is automated Figure shows
a multiblo ck mesh created by ICEM Hexa a system develop ed by ICEM CFD Engineering
In this system the user controls the placement of the blo ck corners and then the mesh
generator pro jects the implied blo ck edges onto domain curves and surfaces automatically
Due to the need for human interaction multiblo ck meshes are not well suited to adaptive
meshing nor to rapidly iterated design and simulation
Figure A two dimensional Cartesian mesh for a biplane wing W Coirier
Cartesian Meshes
We move on to a recently develop ed quick and dirty approach to hexahedral mesh gener
ation The Cartesian approach o ers simple data structures explicit orthogonality of mesh
edges and robust and straightforward mesh generation The disadvantage of this approach
is that it uses non hexahedral elements around the domain b oundary which then require
sp ecial handling
A Cartesian mesh is formed by cutting a rectangular b ox into eight congruent b oxes
each of which is split recursively until each minimal b ox intersects the domain in a simple
way or has reached some small target size This construction is essentially the same as an
o ctree describ ed in Section Requiring neighb oring b oxes to di er in size by at most a
factor of two ensures appropriate mesh grading
Boxes cut by the b oundary are classi ed into a numb er of patterns by determining
which of their vertices lie interior and exterior to Each pattern corresp onds to a di erent
typ e of non hexahedral element Boxes adjacent to ones half their own size can similarly b e
classi ed as non hexahedral elements or alternatively the solution value at their sub division
vertices can b e treated as implicit variables using Lagrange multipliers
Recent uid dynamics simulations have used Cartesian meshes quite successfully in b oth
nite element and nite volume formulations The approach can b e adapted
even to very di cult meshing problems For example Berger and Oliger and Berger
and Colella have develop ed adaptive Cartesian based metho ds for rotational ows and
ows with strong sho cks
Unstructured Hexahedral Meshes
Hexahedral elements retain some advantages over tetrahedral elements even in unstruc
tured meshes Hexahedra t man made ob jects well esp ecially ob jects pro duced by CAD
systems The edge directions in a b ox shap ed hexahedron often have physical signi cance
for example hexahedra show a clear advantage over tetrahedra for a stress analysis of a
b eam The face normals of a b ox meet at the center of the element this prop erty can
b e used to de ne control volumes for nite volume metho ds These advantages are not
inherent to hexahedra but rather are prop erties of b ox shap ed elements which degrade as
the element grows less rectangular Thus it will not su ce to generate an unstructured
hexahedral mesh by transforming a tetrahedral mesh
Armstrong et al are currently developing an unstructured hexahedral mesh generator
based on the medial axis transform The medial axis of a domain is the lo cus of centers of
spheres that touch the b oundary in two or more faces This construction is closely related
to the Vorono diagram of the faces of the domain Srinivasan et al have previously
applied this construction to two dimensional unstructured mesh generation The medial
axis is a natural to ol for mesh generation as advancing fronts meet at the medial axis in
the limit of small equal sized elements By precomputing this lo cus a mesh generator can
more gracefully handle the junctures b etween sections of the mesh
Tautges and Mitchell are developing an all hexahedral mesh generation algorithm
called whisker weaving Whisker weaving is an advancing front approach that xes the
top ology of the mesh b efore the geometry It starts from a quadrilateral surface mesh
which can itself b e generated by an advancing front generator within each face The
algorithm forms the planar dual of the surface mesh and then nds closed lo ops in the
planar dual around the surface of the p olyhedron Each lo op will represent the b oundary
of a layer of hexahedra in the eventual mesh A layer of hexahedra can b e represented by
its dual called a sheet which has one vertex p er hexahedron and edges b etween adjacent
hexahedra As the algorithm runs it lls in sheets from the b oundary inwards
This approach to hexahedral meshing raises an interesting theoretical question which
quadrilateral surface meshes can b e extended to hexahedral volume meshes Mitchell
and Thurston see also Eppstein answered this question in a top ological sense by
showing that any surface mesh on a simple p olyhedron with an even numb er of quadrilaterals
can b e extended to a volume mesh formed by p ossibly curved top ological cub es The
geometric question remains op en
Tetrahedral Meshes
Tetrahedra have several imp ortant advantages over hexahedra unique linear interp olation
from vertices to interior greater exibility in tting complicated domains and ease of
re nement and dere nement In order to realize the last two of these advantages tetrahedral
meshes are almost always unstructured
Most of the approaches to unstructured triangular mesh generation that we surveyed
in Section can b e generalized to tetrahedral mesh generation but not without some new
di culties Before describing Delaunay advancing front and o ctree mesh generators we
3
discuss three theoretical obstacles to unstructured tetrahedral meshing ways in which IR
2
di ers from IR
First not all p olyhedral domains can b e triangulated without Steiner p oints Fig
ure a gives an example of a non tetrahedralizable p olyhedron a twisted triangular
prism in which each rectangular face has b een triangulated so that it b ends in towards the
interior None of the top three vertices is visible through the interior to all three of the b ot
tom vertices hence no tetrahedron formed by the vertices of this p olyhedron can include the
b ottom face Chazelle gave a quantitative bad example shown in Figure b This
p olyhedron includes n gro oves that nearly meet at a doubly ruled curved surface any
2 2
triangulation of this p olyhedron must include n Steiner p oints and n tetrahedra
Bad examples such as these app ear to rule out the p ossibility of generalizing constrained
Delaunay triangulation to three dimensions
Figure a Sch onhardt s twisted prism cannot b e tetrahedralized without Steiner p oints b
2
Chazelle s p olyhedron requires n Steiner p oints
Needle Wedge
Spindle Cap
Sliver
Figure The ve typ es of bad tetrahedra
Second the very same domain may b e tetrahedralized with di erent numb ers of tetra
hedra For example a cub e can b e triangulated with either ve or six tetrahedra As
we shall see b elow the generalization of the edge ip to three dimensions exchanges two
tetrahedra for three or vice versa This variability do es not usually p ose a problem except
3
in the extreme cases For example n p oints in IR can have a Delaunay triangulation with
2
n tetrahedra even though some other triangulation will have only O n
Finally tetrahedra can b e p o orly shap ed in more ways than triangles In two dimensions
there are only two typ es of failure angles close to and angles close to and no failures
of the rst kind implies no failures of the second In three dimensions we can classify p o orly
shap ed tetrahedra according to b oth dihedral and solid angles There are then ve typ es
of bad tetrahedra as shown in Figure A need le p ermits arbitrarily small solid angles
but not large solid angles and neither large nor small dihedral angles A wedge p ermits
b oth small solid and dihedral angles but neither large solid nor large dihedral angles and
so forth Notice that a sliver or a cap can have all face angles b ounded away from b oth
and although the tetrahedron itself may have arbitrarily small solid angles and
interior volume An example is the sliver with vertex co ordinates
and where
Many measures of tetrahedron quality have b een prop osed most of which have
a maximum value for an equilateral tetrahedron and a minimum value for a degenerate
tetrahedron One suitable measure which forbids all ve typ es of bad tetrahedra is the
minimum solid angle A weaker measure which forbids all typ es except slivers is the ratio
of the minimum edge length to the radius of the circumsphere
Figure In three dimensions an edge ip exchanges three tetrahedra sharing an edge for two
tetrahedra sharing a triangle or vice versa
Delaunay triangulation
As in two dimensions p oint placement followed by Delaunay triangulation is a p opular
approach to mesh generation esp ecially in aero dynamics The same p oint placement meth
o ds work fairly well combining structured meshes advancing front and
random scattering with ltering As in two dimensions the placement phase must put
su ciently many p oints on the domain b oundary to ensure that the Delaunay triangulation
will b e conforming Although the three dimensional conforming Delaunay triangulation
problem is not to o hard for most domains of practical interest we do not know of published
solutions
The rst two p oint placement metho ds su er from the same liability in three dimensions
as in two p oints may b e improp erly spaced at junctures b etween fronts or patches All
three metho ds su er from a new sort of problem even a well spaced p oint set may include
sliver tetrahedra in its Delaunay triangulation b ecause a sliver do es not have an unusually
large circumsphere compared to the lengths of its edges For this reason some Delaunay
mesh generators include a sp ecial p ostpro cessing step that nds and removes slivers
Chew p ersonal communication has recently devised an algorithm that removes slivers by
adding Steiner p oints at a random lo cation near their circumcenters
The triangulation phase of mesh generation also b ecomes somewhat more di cult in
three dimensions The generalization of edge ipping exchanges the two p ossible triangula
tions of ve p oints in convex p osition as shown in Figure We call a ip a Delaunay ip
if after the ip the triangulation of the ve p oints satis es the empty sphere condition no
circumsphere encloses a p oint In three dimensions it is no longer true that any tetrahe
dralization can b e transformed into the Delaunay triangulation by a sequence of Delaunay
ips and it is currently unknown whether any tetrahedralization can b e tranformed
into the Delaunay triangulation by arbitrary ips Nevertheless there are provably correct
incremental Delaunay triangulation algorithms based on edge ipping
There are other practical three dimensional Delaunay triangulation algorithms as well
Bowyer and Watson gave incremental algorithms with reasonable exp ected case
p erformance Barb er implemented a randomized algorithm in arbitrary dimension
This algorithm can b e used to compute Delaunay triangulations through a well known
d
reduction which lifts the Delaunay triangulation of p oints in IR to the convex hull
d+1
of p oints in IR
Advancing Front
We have already mentioned an advancing front approach to placing Steiner p oints for De
launay triangulation A pure advancing front mesh generator places the
elements themselves rather than just the Steiner p oints This approach gives more direct
control of element shap es esp ecially near the b oundary which is often a region of sp e
cial interest Advancing front generators seem to b e esp ecially p opular in aero dynamics
simulations
We describ e an advancing front algorithm of L ohner and Parikh as it contains
the essential ideas Desired element size and p erhaps stretching directions are de ned
at the vertices of a coarse background tetrahedralization and interp olated to the rest of
the domain The background mesh can also b e de ned by an o ctree the three dimensional
generalization of a quadtree To get started the b oundaries of the domain are triangulated
the initial front consists of the b oundary faces The algorithm then iteratively cho oses a
face of the front and builds a tetrahedron over that face The algorithm attempts to ll
in clefts left by the last layer of tetrahedra b efore starting the next layer within a layer
the algorithm cho oses small faces rst in order to minimize collisions The fourth vertex of
the tetrahedron will b e either an already existing vertex or a vertex sp ecially created for
the tetrahedron In the latter case the algorithm tries to cho ose a smart lo cation for the
new vertex for example the new vertex may b e placed along a normal to the base face
at a distance determined by asp ect ratios and length functions ultimately derived from the
background triangulation In either case cleft or new vertex the tetrahedron must b e
tested for collisions b efore nal acceptance
Figure shows the surface of a fairly isotropic tetrahedral mesh computed by an
advancing front mesh generator develop ed by ANSYS Inc This generator like the one
just describ ed places elements directly
Marcum and Weatherill have devised an algorithm somewhere b etween pure advanc
ing front and advancing front p oint placement followed by Delaunay triangulation Their
algorithm starts with a coarse mesh and then uses advancing front to place additional
Steiner p oints simply sub dividing the coarse tetrahedra to maintain a triangulation This
mesh is then improved rst by Delaunay and then by minmax solid angle ips Other re
searchers agree that applying ips in this order is more e ective than using either typ e of
ip alone
Octrees
An octree is the natural generalization of a quadtree An initial b ounding cub e is split into
eight congruent cub es each of which is split recursively until each minimal cub e intersects
the domain in a simple way As in two dimensions a balance condition ensures that
no cub e is next to one very much smaller than itself balancing an unbalanced quadtree
or o ctree expands the numb er of b oxes by a constant multiplicative factor The balance
condition need not b e explicit but rather it may b e a consequence of an intrinsic lo cal
spacing function
Shephard and his collab orators have develop ed several o ctree based
mesh generators for p olyhedral domains Their original generator tetrahedralizes leaf
cub es using a collection of prede ned patterns To keep the numb er of patterns manageable
the generator makes the simplifying assumption that each cub e is cut by at most three facets
Figure The surface of a tetrahedral mesh computed by an advancing front generator ANSYS
Inc
Figure The surface of a tetrahedral mesh derived from an o ctree M Yerry and M Shephard
of the input p olyhedron Perucchio et al give a more sophisticated way to conform to
b oundaries Buratynski uses rectangular o ctrees and a hierarchical set of warping rules
The o ctree is re ned so that each domain edge intersects b oxes of only one size Boxes are
warp ed to domain vertices then edges and nally faces
Mitchell and Vavasis generalized the quadtree mesh generator of Bern et al to
three dimensions The generalization is not straightforward primarily b ecause vertices of
p olyhedra may have very complicated lo cal neighb orho o ds This algorithm is guaranteed to
avoid all ve typ es of bad tetrahedra while pro ducing a mesh with only a constant times the
minimum numb er of tetrahedra in any such b ounded asp ect ratio tetrahedralization So far
this is the only three dimensional mesh generation algorithm with such a strong theoretical
guaranty Vavasis has recently released a mo di ed version of the algorithm called
QMG for Quality Mesh Generator including a simple geometric mo deler and equation
solver to b o ot The mo di ed algorithm includes a more systematic set of warping rules in
particular the new warping metho d for an o ctree cub e cut by a single facet generalizes to
any xed dimension
Re nement and Improvement
We discuss improvement b efore re nement b ecause less is known on the sub ject As we
mentioned ab ove edge ipping generalizes to three dimensions and ipping rst by the
Delaunay empty sphere criterion and then by the minmax solid angle criterion seems to
b e fairly e ective Laplacian smo othing also generalizes although exp erimental results
indicate that it is no longer as e ective as in two dimensions Optimization based smo oth
ing app ears to b e more p owerful than simple Laplacian smo othing Freitag and
Ollivier Go o ch recommend combining Delaunay ipping with smo othing for maxmin
dihedral angle or maxmin dihedral angle sine
We now move on to re nement and discuss two di erent re nement algorithms based
up on the natural generalization of bisection to three dimensions To bisect tetrahedron
v v v v across edge v v we add the triangle v v v where v is the midp oint of v v as
0 1 2 3 0 1 01 2 3 01 0 1
shown in Figure This op eration creates two child tetrahedra v v v v and v v v v
0 01 2 3 01 1 2 3
and bisects the faces v v v and v v v which unless they lie on the domain b oundary are
0 1 2 0 1 3
each shared with an adjacent tetrahedron Two tetrahedra that share a face must agree on
how it is to b e bisected otherwise an invalid mesh will b e constructed v v 3 3
v 2 v 2
v v v v 0 1
0 1 v 01
Figure The tetrahedron on the left is bisected to form two new tetrahedra
A single bisection of a tetrahedron can approximately square the minimum solid angle
unlike in two dimensions where the minimum angle of a triangle is decreased by no more
than a factor of two Consider the wedge tetrahedron with vertex co ordinates
and Bisection of the longest edge of this tetrahedron creates a
2
new tetrahedron with minimum solid angle ab out
Rivara and Levin suggested an extension of longest edge Rivara re nement Sec
tion to three dimensions Notice that splitting the longest edge in a tetrahedron also
splits the longest edge on the two sub divided faces and thus the bisection of shared faces
is uniquely de ned Ties can b e broken by vertex lab els Neighb oring invalid tetrahedra
meaning all those sharing the sub divided longest edge are re ned recursively
Rivara and Levin provide exp erimental evidence suggesting that rep eated rounds of
longest edge re nement cannot reduce the minimum solid angle b elow a xed threshold
but this guarantee has not b een proved The guarantee would follow if it could b e shown
that the algorithm generates only a nite numb er of tetrahedron similarity classes v 3 v 13 v 23 v 1
v 2 v 03 v v 01 02
v
0
Figure The rst three levels of longest edge bisection of the canonical tetrahedron Note that
the tetrahedra generated at each level are similar For the nal level of re nement we show only the
four tetrahedra obtained from v v v v Four similar tetrahedra are obtained from v v v v
0 01 2 3 01 1 2 3
A bisection algorithm rst intro duced by B ansch do es indeed generate only a nite
numb er of similarity classes Before describing the algorithm we sketch the argument of
Liu and Jo e which motivates the algorithm The key observation is that there exists
an a ne transformation that maps any tetrahedron to a canonical tetrahedron for which
longest edge bisection generates only a nite numb er of similarity classes Consider the
p
canonical tetrahedron t with co ordinates and In
c
Figure we illustrate the rst three levels of longest edge bisection of t v v v v It can
c 0 1 2 3
b e shown that all the tetrahedra generated at each level of re nement are similar and that
the eight tetrahedra generated after three levels of re nement are similar to t Re nement
c
in the canonical space induces a re nement in the original space with only a nite numb er of
di erent similarity classes A subtlety the similarity classes in the original space corresp ond
to homothets identical up to scaling and translation not similarity classes in the canonical
space Hence the numb er of similarity classes turns out to b e rather than
B ansch Liu and Jo e and Arnold et al give essentially equivalent
algorithms for generating the bisection order we follow B ansch s presentation Each face
in each tetrahedron elects one of its edges to b e its re nement edge so that two conditions
hold the choice for a face is consistent b etween the two tetrahedra that share it and
exactly one edge in each tetrahedon the global re nement edge is chosen by pairs of faces
of the tetrahedron These conditions hold initially if each face picks its longest edge and
ties are broken in any consistent manner for example by vertex or edge lab el order In the
pseudo co de b elow a child face is a triangle like v v v in Figure and a new face is one
01 2 1
like v v v
01 2 3
mark the re nement edge of every face in the current mesh
let T b e the set of marked tetrahedra i
0
while T do
i
bisect each tetrahedron in T across its global re nement edge
i
pick the old edge in each child face as its re nement edge
pick the longest edge in each new face as its re nement edge
i i
let T b e the set of invalid tetrahedra
i
enddo
Conclusions
We have describ ed the current state of the art in mesh generation for nite element meth
o ds Practical issues in mesh generation are roughly in order of imp ortance algorithm
robustness t with underlying physics element quality and mesh e ciency
Unstructured triangular and tetrahedral mesh generation already makes frequent use of
data structures and algorithms familiar in computational geometry We exp ect this trend
to continue We also exp ect and recommend computational geometers to fo cus some
attention on structured meshes and hexahedral meshes
We close with a short list of op en problems of b oth practical and theoretical interest It
is no coincidence that these problems fo cus on three dimensional mesh generation
3
Is the ip graph for a p oint set in IR connected In other words is it p ossible to
convert any triangulation of a p oint set even a p oint set in convex p osition into any
other using only ips Figure
Is there a smo othing algorithm guaranteed to remove slivers A sliver Figure is
the only typ e of bad tetrahedron with well spaced vertices and small circumspheres
3
Is there an algorithm for conforming Delaunay triangulation in IR In other words
place vertices on the b oundary of a p olyhedron so that the Delaunay triangulation
of all vertices original and new contains the p olyhedron
Is there an algorithm for unstructured tetrahedral mesh generation that guarantees
an M matrix for the nite element formulation of Poisson s equation
Give an algorithm for computing the blo cks in a multiblo ck mesh Such an algo
2
rithm should give a small numb er of nicely shap ed blo cks quadrilaterals in IR and
3
hexahedra in IR
Can any quadrilateral surface mesh with an even numb er of quadrilaterals b e extended
to a hexahedral volume mesh
Ackonwledgements
We would like to thank Lori Freitag Paul Heckb ert Scott Mitchell Carl Ollivier Go o ch
Jonathan Shewchuk and Steve Vavasis for help in preparing this survey
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