-
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Automatic p-version Mesh Generation for Curved DomainsXiao-Juan
Luo, Mark S. Shephard,
Scientific Computation Research Center, Rensselaer Polytechnic
Institute, Troy, NY 1218
Robert M. O’Bara, Rocco Nastasia and Mark W. BeallSimmetrix
Inc., Clifton Park, NY 12065
AbstractTo achieve the exponential rates of convergence possible
with the p-version finite element mrequires properly constructed
meshes. In the case of piecewise smooth domains these
mecharacterized by having large curved elements over smooth
portions of the domain and geocally graded curved elements to
isolate the edge and vertex singularities that are of interespaper
presents a procedure under development for the automatic generation
of such mesgeneral three dimensional domains defined in solid
modeling systems. Two key steps in thedure are the determination of
the singular model edges and vertices and the creation of gecally
graded elements around those entities. The other key step is the
use of generalelement mesh modification procedures to correct any
invalid elements created by the curvmesh entities on the model
boundary which is required to ensure proper geometric approximof
the domain. Example meshes are included to demonstrate the features
of the procedure
Keywords: p-version method, curved meshes, graded meshes
1. Introduction
The performance of the p-version finite element method is
characterized by its ability to aexponential rates of convergence
thus allowing the desired level of accuracy to be obtainemany fewer
degrees of freedom than other methods such as standard h-version
finite emethod. However, to attain the exponential rates of
convergence the meshes must satisfy srequirements in terms of mesh
entity geometric approximation and mesh gradation that asatisfied
by current mesh generation methods developed for h-version meshes
of linear odratic elements.
As the order of the finite element basis functions are
increased, so must the level of geoapproximation of those finite
element faces and edges that lie on curved domain boundarieerwise
the error due to geometric approximation will become the dominate
error thus yieldinfurther increase in finite element order
meaningless [13, 6]. Unlike the case of low order Lagain finite
elements where it is convenient to employ the same functions for
the finite elebasis and geometric shape, the p-version finite basis
functions do not represent a convchoice for defining the geometry
of the element. In this work Bezier polynomials are selectethe
geometric representation of the elements because they provide a
convenient frameworkconstruction of geometric approximations of any
order and they effectively support the geombased operations needed
in a finite element calculation.
To attain the optimal convergence, p-version finite elements
require strongly graded cmeshes. In the neighborhood of
singularities the meshes must be geometrically gradeddirections of
the singular gradients [14, 1, 2, 3, 4]. In the smooth portions of
the domain afrom the singularities, the meshes must be very
coarse.
1
-
to cre-orithmle. Aed on
version
to pro-for p-
h geo-used
thm tolar geo-der toet of
etrichat thetion ofnergyf thateticalever,f con-metric].
Theetric
etry ofrposesl costpriatetiveed [5].edure
ations
esh
The ideal approach to accomplish the mesh generation process for
the p-version method isate curved elements of the size and shape
desired one at a time. However, the lack of algand the high level
of computational effort required of such an approach make it
unfeasibcompromise approach is to combine operations to curve a
mesh initially constructed basstandard linear geometry mesh entity
meshing procedures. The steps in the automatic p-mesh generation
procedure currently under development are:
• Automatically isolate the singular edges and vertices in the
model.• Generate a coarse linear mesh accounting for the isolated
singular features.• Curve the singular feature isolation mesh to
maintain a properly curved mesh isolation.• Curve the remaining
mesh entities classified on the curved boundaries.• Apply local
mesh modifications as needed to ensure a valid mesh.
This paper is organized as follows. Section 2. discusses the use
of Bezier representationsvide an effective framework for defining
the shape of mesh entities to any order as neededversion meshes.
Section 3. reviews the requirements to isolate geometric
singularities witmetrically graded meshes to support optimal
hp-version analysis and indicates the algorithmto isolate them for
proper treatment during mesh generation. Section 4. presents an
algorigenerate coarse geometrically graded meshes of linear
geometry around the isolated singumetric entities. The procedure to
curve an initial graded linear geometry mesh to proper orget a
curved mesh with right gradation for hp-version analysis is given
in Section 5. A sexamples meshes are given in Section 6.
2. Mesh geometryIn the p-version finite element method, the
approximation of the mesh to curved geomdomains must be properly
maintained when the finite element basis is improved to ensure
tmapping error does not become the dominant error. A simple
analysis based on the relaapproximation theory to the convergence
of the error in energy norm indicates that the enorm will converge
so long as the geometric approximation of the mesh is within one
order oused in the finite element basis [6]. As for the other norms
of interest, there is limited theorinformation or numerical studies
about how well the geometric must be approximated. Howa study of a
two dimensional benchmark elasticity problem clearly demonstrates
the loss overgence in the energy norm and pointwise norms of
engineering interest when the geoapproximation is not increased as
the order of the finite element basis is increased [13improvement
of geometric approximation is met by assigning appropriate high
order geomshapes to mesh entities classified1 on curved
boundaries.
The functional interfaces to solid modeling systems can support
the direct use of the geomthe portion of the model face or model
edge that a mesh face or edge is classified on for puof performing
the geometry-based calculation needed [6]. However, the high
computationaof this method makes it undesirable. The alternative is
to assign the mesh entity an approgeometric form with Lagrange and
Bezier polynomial being two of the possibilities. An effecprocedure
for creating quadratic Lagrangian’s finite element meshes has been
developHowever, the extensions of this approach to high order
Lagrangian’s indicates that the procquickly became computationally
demanding and geometrically complex. Bezier represent
1. A mesh face or edge that lies on a model face is “classified
on that model face” and a medge that lies on a model classified on
a model edge is “classified on that model edge”.
2
-
f any
nvex
oreions.
hichape.
shapemap-
is
sentedthethan
ationr sim-
mente thefinite
d toly.
gion. Alldariesively
n inrtex.
meshsed onry at aformly
[8, 9] on the other hand provide an attractive alternative for
defining element geometries oorder. Advantageous properties of
Bezier polynomials include:
• The Convex Hull Property - A Bezier curve, surface, or volume
is contained in the cohull formed by its control points.
• The Variation Diminishing Property - An infinite plane can not
intersect a Bezier curve mtimes than it intersects control polygon
which allows more efficient intersection calculat
• All derivatives and products of Bezier functions are easily
computed Bezier functions.• Computationally efficient algorithms
for degree elevation and subdivision are available w
can be used to refine the shape’s convex hull as well as
adaptively refine the mesh’s sh
One important application of the above Bezier properties is the
developed curved elementvalidity test. In the case of a Bezier
tetrahedron volume, the Jacobian of the geometricping is
(1)
where represents the natural coordinates of the element. The
determinant of Jacobiancomputed as
(2)
, and are the three partial derivatives of which are also Bezier
functions.
Because the product of Bezier functions are also Bezier
functions, the can be repreas a polynomial in Beizer form which is
bounded by its convex hull of control points. If all ofcontrol
points of the are greater than zero then the region’s must be
greaterzero everywhere. Comparing to validity checks that test the
value of the at the integrpoints used in performing the analysis,
the new approach has the following advantages foplex mesh
entities:
• The element geometric shape is determined only by its own
control points. Once the eleis valid, the determinant of Jacobian
should always be greater than zero insidelement independent of the
basis functions, polynomial order, or integration rules of
theelement approximation.
• The relation of the to the control polygon provides insight on
how a region founbe invalid due to some portion having negative can
be made valid most effective
• These tests can be extended to test the mesh faces and edges
bounding the mesh reregions using an invalid mesh edge or face (due
to self-intersection) as part of its bounis also invalid.
Identifying and correcting these lower dimension mesh entities can
effectreduce the number of invalid mesh regions that need to be
corrected.
An example of an invalid tetrahedron region determined by the
above algorithm is showFigure 1. In this case the placement of
control point causes at the mesh ve
The Bezier shape of a mesh entity is described by a set of
control points. In the case ofedges classified on the curved model
boundaries, their control points are determined baBezier curve and
surface interpolation methods resulting from evaluating the model
geometset of discrete locations. Common methods usually assume the
interpolation points are uni
J ξ˜
( )x˜
ξ˜
( )
J ξ˜
( ) x˜∂ξ˜
∂-----=
ξ˜
det J( )
det J( ) x˜∂ξ0∂
---------x˜
∂ξ1∂
---------× x˜
∂ξ2∂
---------•=
x˜
∂ξ0∂
---------x˜
∂ξ1∂
---------x˜
∂ξ2∂
--------- x˜
ξ˜
( )
det J( )
det J( ) det J( )det J( )
det J( )
det J( )det J( )
P1 a b c×( ) 0
-
poorn forcurva-
ture-needs
f thecomesperties,exact
hbor-ongerat thethebset ofd. For
eshesertices
and
odelles, themodellculatededral
s to bege. Todeter-
distributed in the parametric space of the mesh entity. However,
this approach can lead togeometric approximations. An alternative
method that improves the geometric approximatioa given order uses a
chord length method [9]. In cases where there are large changes in
theture in the portion model entity being approximated by the mesh
entity the use of a curvabased procedure for selecting the
interpolating points is appropriate [13]. Specific attentionto be
paid to model entities that contain either parametric degeneracies
and/or periodicity.
3. Singular features isolation
The character of partial differential equations’ exact solution
is determined by the shape odomain, the boundary conditions, the
loading functions, and the material parameters. It besingular when
sharp reentrant corners are presented, or sudden change in material
proloads or boundary condition occurs. The influence of the
domain’s geometric shape on thesolution has been studied in [14, 1,
2, 3, 4, 7] which show the solution is singular in a neighood of a
reentrant model edge whose interior dihedral angle . The larger ,
the strthe singularity. As long as one of its connecting model
edges is singular, the exact solutionvicinity of a model vertex
also has singular behavior [2, 7]. Let indicatemodel vertices and
edges of a problem domain. A procedure is needed to define the
suthese entities that are singular based on the geometric
properties of the faces they bounexample, all of the marked edges
and vertices in Figure 2(a) are singular.
Meshes regarded as optimal should be refined with geometrically
graded cylindrical maround singular model edges and geometrically
graded spherical vertices around singular v[14, 1, 2, 3, 4]. For
example, an appropriate p-version mesh at the neighboring of
vertex
are shown in Figure 2(b).
Given a problem domain, it is possible to preprocess the
geometric domain to mark the medges and vertices at which the
solution will be singular in terms of the interior dihedral angof
the pairs of model faces bounding the model edges. In the case of
two-manifold domainprocess considers each model edge and the two
model faces coming to it. If both of thefaces are planar the
interior dihedral angle is constant along the model edge and can be
caat any location along the edge. If either one of the model faces
is non-planar then the dihangle may no longer be constant.
Therefore, the variation of along the model edge needexamined. The
current procedure does this using a simple sampling algorithm along
the edbe completely effective this procedure needs to be
supplemented with a procedure that will
(a) An invalid Bezier tetrahedron region (b) Invalid tetrahedron
region by moving
Figure 1. Determination of invalid Bezier region.
P1
wi π> wi
Gj{ } j 0 1,=
G160
G01
wi
wi
4
-
ingu-ueriesn. Letandardsed toface
.
ntrant
ngular. Thence to
tities
mine all locations where the angle passes through which will
de-mark singular and non-slar portions. The computation of the
angle, , at the requested locations is performed by qto the solid
modeler that provides face normal and tangent vectors at the
specified locatio
be defined as the angle between two face normal vectors. Since
the calculated by stnormal vector operations is always between 0
and , additional information needs to be udetermine whether the
interior dihedral angle is or . Angles between the twotangent
vectors are also employed. If then elseAn example is shown in
Figure 3.
Figure 4 shows an example solid model in the left image and
indicates the isolated reemodel edges in the right image.
On the assumption that dominate singular behavior is reasonably
local to the isolated siedge, the geometrically graded mesh will be
created within the neighborhood of the edgeheight of the graded
layer mesh for an isolated singular edge is defined as 1/2 of the
dista
(b) Model singular features in a 3D domain (b) Desired mesh for
singular model en
Figure 2. Geometrically singular model features.
,
Figure 3. Computation of dihedral angle in 3D domain.
G160
G150
G170
G180
G190
G200
G210
G01
G11
G21
G41
G31
G160
G01
spherical mesh
cylindricalmesh
T
0.15T0.152T
T
0.152T
0.15T
πwi
θi θiπ
θi 2π θi– βiβ∑ i π> wi 2π θi wi π>,–= wi θi wi π wi 2π θi
wi π>,–=wi
T
5
-
es notstancearies ismin-oint
n of aample
int onn
the closet non-connected model boundary entity. A non-connected
entity is one which doshare any common lower order model entities.
A simple algorithm based on the shortest dibetween a set of
sampling locations to their closest points on non-connected model
boundused to get a useful estimation of . A more advanced algorithm
will use bounding boxed toimize the number of entities checked in
detail and will include the use of more intelligent psampling. The
thickness for each layer is calculated as,
, (3)
Where is geometric gradation factor. By default, is set to 0.15
based on the assumptiostrong singularity [2] and three layers of
elements are generated. Figure 5(a) shows an exwhere of is defined
as 1/2 of the distance between and which is the closest
ponon-connected model faces and . Figure 5(b) shows the layer
definition for with aof 0.5 simply to make it easier to view the
layer construction.
Figure 4. Automatically marked set of reentrant model edges.
Figure 5. Attributes for isolated model edge
reentrant edges marked as darker
T
ti
ti αiT= i 0 1 2…, ,=
α α
T G01 P0 P1
G02 G1
2 G01 α
P0
P1
G01
G02
G12
2T
(a)
T
t0
t1 t2
α 0.5=
G01
(b)
G01
6
-
truc-ly gen-metricment
ualityg thewereains.
model/or trimnteract.
meshportanttures,
l entityan be
es are
maticsingularnd thege iso-
l ele-
4. Geometrically graded meshes around isolated features
A modified version of the Generalized Advancing Layer method
[10] is used to create the stured linear geometry meshes around the
singular edges and vertices. The algorithm reliaberates layered
meshes with the desired structure around selected model entities
for geodomains of arbitrary complexity. The elements can be all
tetrahedra, or a combination of eletypes including hexahedra,
wedges and pyramids.
Starting from a linear geometry surface mesh, the algorithm
efficiently generates high qstructured elements by stepping into
the domain creating a set of advancing layers followingiven
geometric progression. Key generalizations to the advancing layers
methodologyintroduced to deal with the complexities of creating
feature isolation meshes in general domOne new feature is a
transitional blend operation to account for large local changes in
thegeometry due to high curvature. Other features were added to
automatically condense andthe layers as needed with the meshes
being generated for unconnected model entities iEach step in the
application of these processes performs checks to insure the
validity of theand to optimize the quality of the mesh entities
created. These enhancements are quite imto the success of feature
isolation when coarse, with respect to local geometric model
feameshes are needed which is the case in meshes for p-version
analysis.
The structured mesh generation is controlled through a set of
attributes applied to the modebeing isolated, which in the case of
singularity isolation are edges and vertices. Attributes cplaced on
any topological entity to control mesh entity size and layer
gradation. The attributthe following:
• first layer height• total height• gradation factor• number of
layers
If attributes are placed on faces in the model a typical
advancing front style mesh of priswedges or hexahedra is created.
In the current application these attributes are assigned toedges or
vertices of the model. The result in those cases are a cylindrical
like mesh arouedges and a spherical like mesh around vertices.
Figure 6 shows an example of singular edlation for a “penny shaped
crack” on the interior of a domain. In this example hexahedraments
were generated around the isolation edge.
Figure 6. Example of edge singularity isolation around a crack
tip.
7
-
createf thehighpro-
4.) andcurveurve als for
rticesppro-
ate fortions
ws theg of
he fol-
in the
der ofs the, col-lements
5. Process of constructing the curved mesh
Although the ideal approach to accomplish the generation of
desired p-version meshes is tocurved mesh entities with gradation
for isolated singular features and fill the remainder odomain with
optimal elements at a time, the lack of a qualified set of
algorithms and thecomputational cost that would be required
currently makes that approach impractical. A commised approach is
to first generate the graded mesh around the isolated features
(Sectionconstruct a coarse straight-sided mesh in the rest of the
domain and then to incrementallythe mesh entities in an ordered
manner. The local mesh modification based procedure to
cstraight-sided mesh for quadratic mesh geometries given in
reference [5] provides key toothe procedures given here. The
current procedure has been extended to:
• Support higher order mesh entity geometry.• Provide
generalized local mesh modification accounting for curved mesh
entities.• Maintain the structure of the mesh configuration around
isolated singular edges and ve
by carefully ordering and controlling the process of curving
mesh entities to ensure the apriate mesh gradations are
maintained.
All three of these improvements are important to the generation
of curved meshes approprioptimal p-version analysis. The proper
order and control of the mesh modification operaaccounting for the
mesh gradations is a most critical aspect of this process. Figure 7
shodifference in meshes without (left image) and with (right image)
consideration of the orderinmesh curving operations. The mesh
curving procedure orders the mesh entity curving into tlowing two
steps:
• Curve the singular feature isolation mesh to ensure a proper
structure is maintainedcurved mesh.
• Curve the remaining mesh entities classified on curved
boundaries to the required orapproximation. Since curving entities
can lead to mesh invalidates, this step includeapplication of
curved mesh modification operations including entity shape change,
splitslapses and swaps as needed to ensure that a valid mesh of
acceptably shaped curved eis provided.
Figure 7. Meshes without (left) and with (right) considering the
ordering of curving.
8
-
neouslygure 8ociated
dis-d thisat theplishedntrol
nge inrved isassoci-at thethe basesideddiago-ove-
5.1 Curve the singular feature isolation mesh
To ensure a properly curved graded mesh around the singular
features, the process simultacurves the mesh entities on the
boundary and those within the structured layer above it.
Fidemonstrates this process with Figure 8(a) showing the linear
structured layer mesh asswith a portion of a curved model edge.
Figure 8(b) shows how the local gradation would berupted if only
the mesh edge, , classified on the model edge were curved. To
avoiproblem the “parallel” and “diagonal” mesh edges (see Figure
8(a)) are also curved such thentire layer maintains its gradation
and element shapes are smooth. This process is accomby moving the
control points of the parallel and diagonal edges based on how far
the copoints of the edge on the model boundary are moved with
consideration given to the chalength of the edges being curved. The
need to account for the length of the edges being cudemonstrated in
Figure 8(c) where all control points were moved the same distance
as theated control points on the mesh edge classified on the curved
boundary . It is clear thshort edges are curved more than desired.
The method used scales the shape change ofmesh edge accounting for
edge lengths as follows. Let , and be the straight-edge length of
the mesh edge on the curved boundary (the base edge), parallel
andnal edge. The scale factors and for parallel and diagonal edges
applied to their mment with respect to that of the base edge
are:
, (4)
Figure 8 (d) shows a smooth curved isolation mesh by using this
method.
(a) Linear graded mesh (b) Curve classified on
(c) Curve the isolation mesh the same as (b) Scaled curve the
isolation meshFigure 8. Curve the singular feature isolation
mesh.
M01 G0
1
M01
Lbase Lpi Ldiith ith
Spi Sdi ith
Spi
LpiLbase------------= Sdi 0.5 Spi 1– Spi+( )=
G01
M01
paralleledges
0th
1th
2th
3th
1th
2th
3thdiagonaledges
G01M0
1
M01 G0
1
G01M0
1 G01
M01
M01
9
-
ves theto a list
th thisfrome of avalidityns thatity ofm out-
ons aree cur-fied on
eseertexere-
, the
mesh
ndaryer side”pertiese theship ofdue toorrect
t
5.2 Curve mesh entities on the curved boundary
After curving the graded mesh around the isolated mesh edges,
the curving procedure curremaining mesh edges/faces classified on
curved model boundaries. The entities are put inwith the attachment
of a proposed Bezier geometric shape to curve them to the
boundary.
The process of curving the mesh entities traverses the list. If
the movement associated wicurving maintains validity of the
connected mesh entities, the entity is curved and removedthe list.
If any invalidates arise, additional processing is required. Since
changing the shapmesh entity changes the shape of the mesh entities
it bounds, the process of checking theof the mesh must check the
shape validity of the connected mesh entities as well. This meain
addition to verifying no self-intersection of the curved mesh face
or edge, the shape validthe connected high dimensional mesh
entities are checked using the shape check algorithlined in Section
2.
In those cases when a mesh entity does cause an invalidity,
local mesh modification operatiapplied as indicated in Section 5.3.
After the needed mesh modifications are performed thrent mesh
entity is removed from the list and any new mesh entities created
that are classicurved boundaries are added to the list.
Figure 9 shows a simple example of curving the mesh edges on
model edge . Mesh edgand are curved to the boundary without causing
any mesh invalidates. However, facbecomes invalid when curving .
Considering that the bounded angle is already small at v
, although possible, just curving would produce two elements
with small angles. Thfore, swapping provides a better opportunity
for maintaining larger angles. Howeverswapped edge must be curved
as shown in Figure 9(c).
5.3 Curved mesh modification operations to correct invalid
element
The curved local mesh modifications processes build upon a set
of operations that includeentity shape modification, split,
collapse and swap.
Mesh entity shape modification is often successful when curving
a mesh entity on the bouhas caused an intersection with another
mesh entity and there is adequate room on the “othof the
intersected mesh entity to re-shape it such that the intersection
is eliminated. The proof the Bezier control polygon are used in
this process to determine if re-shaping will eliminatintersection
and make the element valid again. For example, one can examine the
relationthe three partial derivative vectors at a mesh vertex where
the Jacobian is negativethe intersections of bounding mesh
entities. Figure 10 presents two possible solutions to c
(a) Initial straight-sided mesh (b) Curve , and (c) Correct
invalid elemencaused by curving
Figure 9. Curving the mesh edges classified on .
G01 M0
1
M11 M0
2
M21
M00 M3
1
M31
M11
M21
M01
G01
M11 M2
1
M01
G01
M02M3
1
M00
M21
M01
G01
M11
M01 M1
1 M21
M21
G01
det J( )
10
-
lying
undary,litducedsh ininintro-lem.
n theirown inparent
urrentto re-nge themore
hows aedge
the invalid region given in Figure 1. An example where valid
mesh is regained by only appinterior mesh entity shape manipulation
is shown in Figure 11.
In those cases where two neighboring mesh entities of a single
element are on a curved bothe curving process will create angles of
180o. The only option in this case is to execute a spoperation so
that a new mesh entity not classified on that same boundary entity
is introbetween those two entities. Figure 12 gives an example in
which when the initial linear meFigure 12(a) is curved there are
180o angles at the four mesh vertices with circles around
themFigure 12(b) as well as along the two edges connecting the two
pairs of them. In this caseducing edge splits of the edges on the
top and bottom of the cylinder will eliminate the probThe process
of splitting takes advantage of the property that the Bezier
geometries maintaishapes under a subdivision process. For example,
a curved face and region split are shFigure 13 where the newly
created curved regions exactly decompose the shape of
theregions.
There are cases where the applying only curving and splitting
180o angles will not produce a validor satisfactory mesh result in
terms of element shape. This typically occurs where the cmesh
configuration is such that there is insufficient room in the
desired direction of motionshape mesh entities. The goal of curved
mesh entity collapse and swap operators is to chamesh topology to
produce sufficient space. Since these operators are
computationallydemanding than the others, they are only considered
when they are needed. Figure 14 scase where the invalidity caused
by curving to is effectively corrected by collapsing
(a) Move and (rotate plane define by) - curve 2 new sides
(b) Move the positive side of planedefined by - curve 1 new
side
Figure 10. Correct invalid region by shape manipulating
(a) Invalid mesh (b) Close up of the invalidity (c) Valid
meshFigure 11. Curve mesh corrected by shape improvement.
P1P2
P0
P2
P1P0
P1 P2b c×
P1c a×
M01 G0
1
11
-
better
ome-those
vertexandroper
gions
(see Figure 14(c)). The collapse produces a better mesh
configuration maintainingangles than just curving edge .
The primary new complexity in collapse and swap operations
introduced by curved mesh getry is to determine appropriate
geometric shapes for the new mesh entities created
duringoperations. Figure 15(a) shows the curved domain defined by
the five regions connected to
that will be deleted when collapsing edge by moving to . New
edgesshown in Figure 15(b) as well as their higher order connected
entities need to have p
(a) Linear mesh (b) Invalid after curving becauseface interior
angles are bigger
than 180o
(b) Valid mesh by edge split
Figure 12. Splitting to correct 180o angles.
(a) Face split - create 3 new curved regions (b) Region split -
create 4 new curved reFigure 13. Curved face and region split
operations.
(a) Linear mesh (b) Curve (c) CollapseFigure 14. Edge collapse
to fix invalid element.
Geometry model
M11
M21
M01 G0
1M00
M11
M01
G01M0
0
M11
M21
M01
G01M0
0
M01 M1
1
M80 M0
1 M80 M7
0 M11
M21
12
-
metrice equal
tenesswablepri-
cted asindi-
te the. For,,adrilat-
arethe
the
elytained
curved shapes assigned to them to make the operation valid.
Compatibility between geoshape and finite element basis indicates
the geometric order of these mesh entities should bto or less than
that used for finite element to satisfy the standard finite element
complerequirement. Thus, the geometric order of a new edge or face
can only elevate up to the alloorder. Within the limits of the
order of geometry allowed, it is important to determine an approate
shape for the new mesh entities. For example, if the edge shape of
were construstraight-sided instead of curved it would intersect the
boundary of the affected region whichcates a mesh invalidity. To
avoid this problem blending operators [11] are applied to
compushape for new edges inside of curved quadrilateral polygons
without interfering with othersexample, the shape of edge is
determined by the curved polygon bounded by ,and , and the shape of
edge determined by curved polygon bounded by ,and (Figure 15(b)).
The general procedure to compute a curved edge shape inside a
queral polygon is as follows:
Let be the shapes of the curved quadrilateral polygon andthe
locations of corresponding vertices , , and . The blend mapping
insidepolygon is,
(5)
If the new edge from to is quadratic, substitute into Eq. 5 to
computecontrol point location,
(6)
If the edge is cubic, and are used for the two control points
respectiv(Figure 16). Since simple blend operations do not ensure
the generated entities will be con
(a) Curved polyhedral for collapsing (b) Mesh after
collapsing
Figure 15. Curved edge collapsing.
M21
M11 M2
0 M30 M8
0
M70 M2
1 M40 M6
0 M70
M80
M20
M30
M40
M60
M70
M10
M80
M01Collapse edge
Deletingvertex
M20
M30
M40
M60
M70
M10
New edges
M11
M21
M01
φ0 φ1 φ2 φ3, , ,( ) P0 P1 P2 P3, , ,( )M0
0 M10 M2
0 M30
x 1 v–( )φ0 uφ1 vφ2 1 u–( )φ3 1 u–( ) 1 v–( )P0– 1 v–( )uP1–
uvP2– 1 u–( )vP3–+ + +=
M00 M2
0 u v 0.5= =
x12--- φ0 φ1 φ2 φ3+ + +( )
14--- P0 P1 P2 P2+ + +( )–=
u v13---= = u v 2
3---= =
13
-
ed if
ions toopri-ompu-.
sifiedpply a
entitymesh
ply
propri-
ll newratio-.
ducere 17(b)of thets. Asneedorderuracy.fine-
anicalwn in
in the bounding entities, the validity checks must still be
performed and locations adjustneeded.
The incremental procedure being developed to select the
appropriate local mesh modificatcorrect invalid elements builds on
the linear geometry algorithm of reference [12] with apprate
extensions to account for the added complexity of curved mesh
entities. Based on the ctational cost and complexity of each
operation, the process works in the following five steps
S1. Determine if the invalidity is caused by pairs of
neighboring mesh faces or edges clason the boundary such that
angles of or greater are created. In these cases asplit operation
that will introduce additional mesh entities to subdivide those
angles.
S2. Check whether there is adequate room on the “other side” of
the intersected meshand its geometric order is sufficient to
re-shape. If these conditions are met, applyentity re-shape to fix
the problem.
S3. Examine the possibility of applying a collapse operation to
eliminate the invalidity. Apthe collapse if possible.
S4. Examine the application of a swap operation to create the
space required to have apately shaped mesh entities in the swapped
configuration to eliminate the invalidity.
S5. If none of the above processes can be applied, apply local
refinement and place amesh entities classified on the boundary into
the list of entities to be processed. Thenal for performing
subdivision is to create more options for applying other
operations
Although the last operation (S5) does typically allow completion
of the process, it does introundesired mesh refinement where the
new elements are often not ideally shaped (see Figuin the examples
section). In such cases an alternative is to inflate the order of
the geometrymesh entity to provide additional shape freedom without
the need to add additional elemenshown in Figure 17(c) this
approach can be successful in obtaining a valid mesh without theof
refining the mesh. Of course, inflating the geometry order can
force the need to inflate theof finite element used at that
location past what may have been required for analysis accHowever,
this is likely to be more cost effective than the additional cost
introduced by the rement process. Additional investigation of these
possibilities is underway.
6. Example resultsCurvilinear mesh examples are presented in
this section. The first example is a biomechmodel used to simulate
blood flow. Linear, quadratic, cubic and quartic meshes are sho
(a) for of quadratic shape (b) for andfor of cubic shape
Figure 16. Blending method to compute geometric shape.
φ0
φ1
φ2
φ3
M00
M10
M20M3
0
P0( )P1( )
P2( )P3( )
P4( )
φ0
φ1
φ2
φ3
M00
M10
M20M3
0
P0( )P1( )
P2( )P3( )
P5( )
P6( )
u v 0.5= = P4 u v 1 3⁄= = P5 u v 2 3⁄= =P6
1800
14
-
andt lin-
of eachon theclearly
devia-from
ct theexam-ementsbenefitpor-e).
Figure 17. The quality of curved mesh with respect to geometry
model is measured bywhich are the normalized maximum distance
deviation to the longest and shortes
ear mesh edge length. is computed as the maximum distance
between sample pointsmesh entity classified on curved model
boundary and their corresponding closest pointsboundary. Statics
for curving the blood vessel meshes are presented in Table 1. The
resultsdemonstrate that geometric approximation error in terms of
normalized maximum distancetion has been improved by increasing the
geometric approximation order especially in goinglinear to cubic
geometry.
Table 1 also includes the statics on which local mesh
modifications were used to correinvalid elements. The shape
manipulation operations are the most frequently applied in thisple.
The number of refinements required has been reduced from 10 for
quadratic shaped elto 3 for cubic shaped elements and 2 for quartic
shaped elements, thus demonstrating theof using high geometric
order to alleviate the need for local refinement to fix invalidity.
Thetion of the domain requiring refinement is circled on curved
meshes shown in Figure 17 (c-
Figure 17. Blood vessel meshes.
Table 1. Statics for curving the blood vessel meshes.
Blood vessel
Linear Quadratic Cubic Quartic
0.1208 0.0702 0.0355 0.0329
1.7028 0.9897 0.5006 0.4648
Collapse 2 1 1
Swap 3 1 1
Split 17 5 5
Recurving 29 15 14
Refinement 10 3 2
∆dmax∆dmin ∆d
∆d
(a) Model (b) Linear mesh
∆dmax∆dmin
15
-
urveding fromble 2.re 19
The next two examples (Figure 18 and Figure 19) are mechanic
components for which cgraded quadratic meshes are generated by
using the procedures discussed above beginnisolating all of the
reentrant model edges. Statics for the examples are presented in
TaRecurving remains the dominated operations to fix invalid
elements. Figure 18 and Figuinclude close-ups of curved graded
meshes around selected singular edges.
Table 2. Statics for curving meshes for models 1 and 2.
Model 1 Model 2
quadratic quadratic
Reentrant model edges 9 64
Collapse 0 7
Swap 0 4
Split 1 10
Recurving 12 57
Refinement 0 4
Figure 17. Blood vessel meshes.
(c) Quadratic mesh (d) Cubic mesh
(e) Quartic mesh
16
-
Figure 18. p-version mesh for model 1.
Figure 19. p-version mesh for model 2.
Reentrant edges marked as lighter
Reentrant edges marked as lighter
17
-
enerals bygularith gra-tainingt of thebound-mesh
uce p-timalthese
DMI-
nics
les
7. ConclusionThis paper discussed an algorithm to automatically
generate curved graded meshes for gthree-dimensional domains
suitable for solving elliptic partial differential equation
problemthe p-version finite element method. The procedure starts
with the automatic isolation of sinreentrant model entities. Linear
geometry elements are generated around those features wdations as
needed for optimal hp-convergence. These elements are then curved
while mainthe appropriate gradation. The procedure next constructs
a linear geometry mesh for the resdomain. The linear mesh entities
classified on curved boundaries are then curved to thosearies.
Whenever those curving operations introduce invalid elements,
generalized curvedmodification operations are applied to eliminate
the problems.
As the results demonstrate, the procedure as currently developed
has the capability to prodversion meshes on general 3-D domains.
Additional improvements to construct more opmesh configurations and
the development of complete hp-adaptive procedures building
onmeshing procedures are underway.
8. AcknowledgmentsThis work was supported by the National
Science Foundation through SBIR grant number0132742.
9. Reference[1] B. Anderson, U. Falk, I. Babuska, T.V.
Petersdorff, “Reliable stress and fracture mecha
analysis of complex components using a h-p version of fem”,Int.
J. Numer. Methods Eng.,38, 2135-2163, 1995.
[2] I. Babuska, M. Suri, “The p and h-p versions of the finite
element method, basic principand properties”,SIAM Review, 36(4)
578-632, 1994
Figure 19. p-version mesh for model 2.
18
-
nd
ing
qua-
p-ver-
p-
ula-
-
ifi-
, “p-
[3] I. Babuska, T.V. Petersdorff, B. Anderson, “Numerical
treatment of vertex singularities aintensity factors for mixed
boundary value problems for the Laplace equation”,SIAM J.Numer.
Anal., 31(5), 1265-1288, 1994.
[4] I. Babuska, B. Anderson, B. Guo, J.M. Melenk, H.S. Oh,
“Finite element method for solvproblems with singular solutions”,J.
Comp. App. Math., 74, 51-70 1996.
[5] S. Dey, R.M. O’Bara, M.S. Shephard, “Towards curvilinear
meshing in 3D: The case ofdratic simplices”,CAD Computer Aided
Design, 33(3), 199-209, 2001.
[6] S. Dey, M.S. Shephard, J.E. Flaherty, “Geometry
representation issues associated withsion finite element
computations”,Comput. Methods Appl. Mech. Engrg.150(1-4),
39-55,1997.
[7] M.R. Dorr, “The approximation of solutions of elliptic
boundary-value problems via the version of the finite element
method”,SIAM, J. Numer. Anal. 23(1), 58-77, 1986.
[8] G.E. Farin,Curves and Surfaces for Computer Aided Geometric
Design - A Practical Guide,3rd Edition. Boston: Academic Press,
1993.
[9] R.T. Farouki and V.T. Rajan, “Algorithms for polynomials in
Bernstein”,Comput. Aid. Geom.Des., 5, 1-26, 1988.
[10]R. Garimella and M.S. Shephard, “Boundary layer mesh
generation for viscous flow simtions in complex geometric
domains”,Int. J. Numer. Meth. Engrg., 49(1-2), 193-218, 2000.
[11]R. Harber, M.S. Shephard, J.F. Abel, R.H. Gallagher, D.P.
Greenberg, “A General Two-Dimensional, Graphical Finite Element
Preprocessor Utilizing Discrete Transfinite Mappings”, Int. J.
Numer. Meth. Engrg. 17, 1015-1044, 1981.
[12]X. Li, M.S. Shephard and M.W. Beall, “3-D Anisotropic Mesh
Adaptation by Mesh Modcations”, submitted toComp. Meth. Appl. Mech.
Engng., 2003.
[13]X.J. Luo, M.S. Shephard, J.F. Remacle, R.M. O’Bara, M.W.
Beall, B.A. Szabo, R. Actisversion mesh generation
issues”,Proceedings of the 11th Meshing Roundtable, SandiaNational
Laboratories, 343-354, 2002.
[14]B.A. Szabo, “Mesh design for the p-version of the finite
element method”,Comp. MethodsAppl. Mech. Eng., 55, 181-197,
1986.
[15]B.A. Szabo and I. Babuska,Finite element analysis, John
Wiley & Sons New York, 1991.
19
Automatic p-version Mesh Generation for Curved DomainsXiao-Juan
Luo, Mark S. Shephard,Scientific Computation Research Center,
Rensselaer Polytechnic Institute, Troy, NY 12180Robert M. O’Bara,
Rocco Nastasia and Mark W. BeallSimmetrix Inc., Clifton Park, NY
120651. Introduction2. Mesh geometry(1)(2)Figure 1. Determination
of invalid Bezier region.
3. Singular features isolationFigure 2. Geometrically singular
model features.Figure 3. Computation of dihedral angle in 3D
domain.Figure 4. Automatically marked set of reentrant model
edges., (3)
Figure 5. Attributes for isolated model edge
4. Geometrically graded meshes around isolated featuresFigure 6.
Example of edge singularity isolation around a crack tip.
5. Process of constructing the curved meshFigure 7. Meshes
without (left) and with (right) considering the ordering of
curving.5.1 Curve the singular feature isolation mesh, (4)Figure 8.
Curve the singular feature isolation mesh.
5.2 Curve mesh entities on the curved boundaryFigure 9. Curving
the mesh edges classified on .
5.3 Curved mesh modification operations to correct invalid
elementFigure 10. Correct invalid region by shape
manipulatingFigure 11. Curve mesh corrected by shape
improvement.Figure 12. Splitting to correct 180o angles.Figure 13.
Curved face and region split operations.Figure 14. Edge collapse to
fix invalid element.Figure 15. Curved edge collapsing.(5)(6)
Figure 16. Blending method to compute geometric shape.S1.
Determine if the invalidity is caused by pairs of neighboring mesh
faces or edges classified ...S2. Check whether there is adequate
room on the “other side” of the intersected mesh entity and i...S3.
Examine the possibility of applying a collapse operation to
eliminate the invalidity. Apply t...S4. Examine the application of
a swap operation to create the space required to have
appropriatel...S5. If none of the above processes can be applied,
apply local refinement and place all new mesh ...
6. Example resultsFigure 17. Blood vessel meshes.Table 1.
Statics for curving the blood vessel meshes.Table 2. Statics for
curving meshes for models 1 and 2.Figure 18. p-version mesh for
model 1.Figure 19. p-version mesh for model 2.
7. Conclusion8. Acknowledgments9. Reference[1] B. Anderson, U.
Falk, I. Babuska, T.V. Petersdorff, “Reliable stress and fracture
mechanics a...[2] I. Babuska, M. Suri, “The p and h-p versions of
the finite element method, basic principles a...[3] I. Babuska,
T.V. Petersdorff, B. Anderson, “Numerical treatment of vertex
singularities and i...[4] I. Babuska, B. Anderson, B. Guo, J.M.
Melenk, H.S. Oh, “Finite element method for solving pro...[5] S.
Dey, R.M. O’Bara, M.S. Shephard, “Towards curvilinear meshing in
3D: The case of quadratic...[6] S. Dey, M.S. Shephard, J.E.
Flaherty, “Geometry representation issues associated with
p-versi...[7] M.R. Dorr, “The approximation of solutions of
elliptic boundary-value problems via the p- ver...[8] G.E. Farin,
Curves and Surfaces for Computer Aided Geometric Design - A
Practical Guide, 3rd ...[9] R.T. Farouki and V.T. Rajan,
“Algorithms for polynomials in Bernstein”, Comput. Aid. Geom.
De...[10] R. Garimella and M.S. Shephard, “Boundary layer mesh
generation for viscous flow simulations...[11] R. Harber, M.S.
Shephard, J.F. Abel, R.H. Gallagher, D.P. Greenberg, “A General
Two- Dimensi...[12] X. Li, M.S. Shephard and M.W. Beall, “3-D
Anisotropic Mesh Adaptation by Mesh Modifications”...[13] X.J. Luo,
M.S. Shephard, J.F. Remacle, R.M. O’Bara, M.W. Beall, B.A. Szabo,
R. Actis, “p- ve...[14] B.A. Szabo, “Mesh design for the p-version
of the finite element method”, Comp. Methods Appl...[15] B.A. Szabo
and I. Babuska, Finite element analysis, John Wiley & Sons New
York, 1991.
