1. INTRODUCTION

High order (p-version) finite element methods, character-ized by the capability of exponential rate of convergence,are gaining popularity in industry. The basic functions of p-version finite elements, their convergence properties andaspects of their computer implementation have receivedextensive consideration in the literature (e.g.[1,3,10,16,17,18]). However, the critical technical issues ofthe appropriate geometric representation of p-version finiteelements for solving partial differential equations over gen-eral three dimensional domains have not received adequateconsideration. This paper first demonstrates that the accu-racy of finite element solutions is strongly influenced byhow well the geometry is approximated. Consideration isthen given to a set of procedures being developed forproper generation of curved elements for p-versionanalyses.

Section 2 outlines the advantages of p-version finite ele-ments assuming the proper choice of meshing and mappingprocedures as required to preserve the superior rates of con-vergence. Section 3 examines the role of the meshgeometric approximation on the accuracy of the resultsobtained in terms of a specific curved domain problem witha known exact solution. This simple example clearly dem-onstrates that the use of quadratic geometricapproximations for p-version finite elements does not leadto satisfactory solution results, in the sense that using p-lev-

els greater than 3 or 4 will produce results that are affectedby the errors in mapping.

The requirements of Section 2 and results of Section 3demonstrate the need for new mesh generation technolo-gies to support p-version finite elements. Section 4overviews current efforts on the development of such amesh generation capability. Central to this new approach isthe use of Bezier basis for the geometric representation ofthe element shapes. This basis allows one to effectivelyincrease the order of geometric appropriation in an effi-cient manner to any order desired. In addition, this basissupports effective methods for the execution of key opera-tions such as determining the validity of curved finiteelements and determining which mesh entities requireshape change to make an invalid element valid.

2. p-VERSION FINITE ELEMENT MESHES

A finite element mesh serves two purposes: First, to allowrepresentation of an arbitrary body by a collection of ele-ments on which piecewise polynomial functions(occasionally augmented by other functions) are defined,and second, to control the error of approximation in termsof the data of interest.

The error of approximation depends on the finite elementmesh and the polynomial degree of elements. In conven-tional FEA codes, the polynomial degree of elements are

p-Version Mesh Generation Issues

Xiao-Juan Luo, Mark S. Shephard, Jean-François RemacleScientific Computation Research Center, Rensselaer Polytechnic Institute, Troy, NY 12810,

xluo@scorec.rpi.edu, shephard@scorec.rpi.edu, remacle@scorec.rpi.edu

Robert M. O’Bara, Mark W. BeallSimmetrix, Inc., Troy, NY 12180

obara@simmetrix.com, mbeall@simmetrix.com

Barna SzabóWashington University, St. Louis, MO 63130

szabo@me.wustl.edu

Ricardo ActisEngineering Software Research and Development, Inc., St. Louis, Missouri

ricardo@esrd.com

ABSTRACT

Higher order (p-version) finite element methods have been shown to be clearly superior to low order finite element methodswhen properly applied. However, realization of the full benefits of p-version finite elements for general 3-D geometries requiresthe careful construction and control of the mesh. A 2-D elasticity problem with curved boundary is used to clearly illustrate theinfluence of the mesh shape geometric approximation order and shape representation method on the accuracy of finite elementsolution in a p-version analysis. Consideration is then given to a new approach for the representation of mesh geometry for p-ver-sion meshes and to the automatic generation of p-version meshes.

Keywords: geometric approximation, Bezier polynomials, curvilinear mesh

fixed at 1 or 2, and the error is controlled by making suffi-ciently fine meshes: The diameter of the largest element isdenoted by h. The errors of approximation are reduced as his reduced. The term h-version refers to this approach.Since the mid-1980’s an alternative, known as the p-ver-sion, matured sufficiently for use in professional practice.In this approach the primary role of the mesh is to repre-sent the topological and geometric description of the objectbeing modeled by a collection of elements and the error iscontrolled by the polynomial degree of elements, denotedby p. The error is reduced as p is increased. The p-versionhas certain advantages, which include faster rates of con-vergence and the ability to produce a sequence of solutions,corresponding to increasing p, automatically and withoutthe need to alter the finite element mesh (so long as themesh provides a satisfactory geometric approximation tothe domain). This allows monitoring the convergence of thedata of interest and estimating the errors of approximation.A large number of papers are available on this subject, see,for example, references [1,10,11,12,13,14].

The p-version poses certain new requirements for meshing.Since the size of elements is much larger than in the h-ver-sion, it is essential to use advanced mapping procedures sothat the domain geometry is properly represented and inte-grated. Various procedures have been developed andimplemented using special collocation points, known as theBabuska points, in connection with blending function tech-niques see, for example, references [2,7]. A typical finiteelement mesh used in the p-version is shown in Figure 1.The solution, representing the von Mises stress contours,was obtained with StressCheck†.

The proper choice of the mesh topological and geometricrepresentation depends on the goals of computation. Insolid mechanics the goals of computation are: (a) to deter-mine stiffness characteristics of a structure, includingnatural frequencies; (b) to determine the strength character-istics, including stress maxima and stress intensity factors,and (c) to determine stability limits (buckling loads). Instiffness and stability computations it is generally possibleto simplify meshing by omitting small features, such as fil-lets, bosses, small holes, etc. In strength computations,

when the goal is to compute the maximal stress, it is neces-sary to include fillets and all relevant features at leastwithin the region where the maximal stress is sought. Thisis known as the region of primary interest. A frequent con-ceptual error in finite element analysis is reporting stressesin regions where the fillets and other small features wereomitted.

The error of approximation has two sources: the local errorand the pollution error. The pollution error, associated withthe region of secondary interest, is controlled by ensuringthat the error in the natural norm of the formulation, usu-ally the energy norm, is sufficiently small. The local error,associated with the region of primary interest, depends onthe local discretization (choice of mesh, mapping and thepolynomial degree). This error is most efficiently con-trolled by the p-version of the finite element method.

3. INFLUENCE OF GEOMETRICAPPROXIMATION

This section discusses the influence of geometric approxi-mation on the solution accuracy of p-version finite elementmethod by using a benchmark two-dimensional elasticityproblem for which analytic expressions for the exact dis-placement and stress field are known.

3.1 Model problem

An infinite plane weakened by an elliptical hole isdeformed by the application of uniform tensile stress in thevertical direction at infinity as shown in Figure 2a. The rel-evant geometric parameters are the major axis and

minor axis of the inner ellipse. These parameters are typ-

ically related to a third parameter, , as

, , (1)

where corresponds to a circle and is asharp crack.

Due to the double symmetry of the problem, only one quar-ter of the sub-domain ABCDE needs to be investigated asshown in Figure 2b. The exact stresses for the infinitedomain problem are known along edges BC and DC andgiven by:

(2)

(3)

(4)

where and , and

are the stress components expressed in elliptical coor-

dinate system [15].

The mapping between Cartesian coordinate system

and elliptical coordinate system is

Figure 1. A typical finite element mesh in the p-version.

† StressCheck® is a trademark of Engineering Soft-ware Research and Development, Inc., St. Louis,Missouri

a

b

m

a 1 m+= b 1 m–= 0 m 1<≤

m 0= m 1=

σx12--- S 1 2θcos–( ) 2τρρ 2θ 2τρθ 2θsin–cos+( )=

σy12--- S 1 2θcos+( ) 2τρρ 2θ 2τρθ 2θsin+cos+( )=

σxy12--- τρρ τθθ–( ) 2θ 2τρθ 2θcos+sin( )=

S ρ4 m2– 2m– 2ρ2 2θcos+= τρρ τρθ

τθθ

ρ θ,( )

x y,( )ρ θ,( )

, ,

Traction (Neumann) boundary conditions are applied onedges BC and CD and symmetric Dirichlet boundary condi-tions are imposed on edges DE and AB.

3.2 Maximum stress and strain energy

The maximum stress of this problem is concen-

trated at vertex A and is a function of ratio only,

being defined as . The finite element

stress is computed directly in this study by computing thestrains from the displacement solution then applying theappropriate isotropic material stress-strain relationship.

Search for the maximum computed stress is con-

ducted over not only Gauss Quadrature points but also thevertices of each element [12]. The relative error in maxi-mum stress defined as

% (5)

is of great engineering interest.

The exact potential energy, , of the sub-domainloaded by traction only without body force can be com-puted as

(6)

=

Where and are normal and tangential trac-

tion components and displacement componentsrespectively [12]. The angle measured from the positive

axis to the normal of boundaries BC and CD are

, so Eq. (6) can be simplified as

(7)

where are stress tensor components and

are displacement components [4]. The exact potentialenergy is obtained by numerically solving Eq. (7) to anaccuracy substantially greater that any of the finite elementsolutions. The finite element potential energy iscomputed by evaluating the product of load vector andfinite element solution over the boundary. The relativeerror in energy norm is defined as

% (8)

3.3 Finite element meshes and geometric approxi-mation shapes

A parameter is selected to construct the first testmodel. An isotropic material with Young’s Module of 1.0and Poisson’s ratio of is used under the assumption ofplane strain. The stress applied at the infinite boundary is1.0. Table 1 provides the exact potential energy (to 7 digitssignificant figures) and the exact stress.

The finite element method used for solving the problemconsists of a double discretization. First, a mesh is intro-duced in order to discretize the geometrical domain. Then,the solution function space is approximated by a finitedimensional function space. Both geometrical and functionspace approximations introduce discretizations errors intothe solution. In this work, we use polynomials for both dis-cretizations where represents the polynomial order for

function space discretization and represents the polyno-mial order for geometrical discretization.

The discretization error of the finite element method can bewritten as the sum of two contributions:

(9)

where is the geometrical error and is the func-

Figure 2. Elliptical hole in an infinite plane under theuniform tensile stress at infinity

a

ρ 1=

x

y

P

b

P

(a)

x

y

A

D

E

B

C

w 4=

l 4=

a

b

(b)

x R ρ mρ----+

θcos= y R ρ mρ----–

θsin= Ra b+

2------------=

σmaxexact

a b⁄σmax

exact 1 2a b⁄+=

σmaxfem

ε∞σmax

fem σmaxexact–

σmaxexact

---------------------------------- 100×=

Πexact

Table 1: Test problem

0.25 1.25 0.75 1.667 -7.8462131

Πexact T nun T tut+( )ds∫°–=

T nun T tut+( ) s T nun T tut+( ) sdC

D

∫+dB

C

∫–

T n T t, un ut,

αx

00 900,

Πexact uxσx uyτxy+( ) y uyσy uxτxy+( ) xdC

D

∫+dB

C

∫–=

σx σy τxy, , ux uy,

Π fem

ε Πexact Π fem–Πexact

----------------------------------- 100×=

m 0.25=

0.3

m a b a b⁄ σmaxexact Πexact

13 3⁄

p

q

E fem E pfem Eq

fem+=

Eqfem E p

fem

tional error. Studies of the convergence and accuracy of thefinite element method for domains where geometrical error,

, is identically zero or carefully controlled to be

small, thus allowing study of the functional error, ,

are common. We investigate the total error when the geo-

metrical discretization error, , contribution can be

significant. In this study a coarse mesh with only one edgeclassified on the ellipse AE (see Figure 3) is used to per-form the analysis and function polynomial orders from

to are used and geometric polynomial

orders from to are used. (A more com-plete study including additional mesh configuration andgeometric approximation details is in preparation for publi-cation [4])

For the mesh edge that is used to geometrically discretizethe ellipse, linear (q=1), quadratic (q=2), cubic (q=3) andquartic (q=4) geometric approximations are selected. Twodifferent fitting methods are applied for the q>1 cases. The

first is a interpolant where the interpolating points areequally spaced in the parametric space of the edge. The

second enforces continuity at the vertices A and E (see[4] for specifics on the construction of those geometricapproximations). Close-ups of the geometric approxima-tions in the vicinity of vertex A are shown in Figure 4.

3.4 Result analysis

Convergence curves for the relative error in energy normwith respect to polynomial order for the various geomet-ric approximation orders are shown in Figure 5. Whenthe polynomial order increases past the geometric approxi-mation order, the error in mapping begins to dominate thesolution which is consistent with the basic theory [3]. Thediscretization error approaches a limit as increases. This

limit is essentially the because for very high we

solve the PDE nearly exactly on an approximated geometri-cal domain. The geometrical error is less when the

geometric approximation order increases.

The performance of the different geometric approxima-tions on the norm of the maximum stress is a bit morecomplex. Figure 6a shows the relative error in maximumstress for the geometric approximations and Figure 6b

shows it for the geometric approximations. For

(which is ) the computed maximum stress is underesti-mated at , but quickly increases past the exact valueto overestimate the exact value by relative error of 122% at

. Such behavior is expected since as increaseswe are moving toward the solution of a problem with asharp corner at point A where the stress theoretically goesto infinity. As is increased for the case the sharp-

(a) Linear mesh (b) quartic mesh

Figure 3. Linear and quartic meshes

(a) and quadratic approximation shapes

Figure 4. Different geometric approximation shapes

Eqfem

E pfem

Eqfem

p 2= p 10=

q 1= q 4=

C0

C1

C0

C0

0.9 1 1.1 1.20

0.1

0.2

0.3

0.4

0.5

x

y

model edgeC0 quadratic shapeC1 quadratic shape

C0 C1

(b) and cubic approximation shapes

(c) and quartic approximation shapes

Figure 4. Different geometric approximation shapes

0.9 1 1.1 1.20

0.1

0.2

0.3

0.4

0.5

x

y

model edgeC0 cubic shapeC1 cubic shape

C0 C1

0.9 1 1.1 1.20

0.1

0.2

0.3

0.4

0.5

x

ymodel edgeC0 quartic shapeC1 quartic shape

C0 C1

pq

p

Eqfem p

Eqfem

L∞

C0

C1 q 1=

C0

p 1=

p 10= p

q C0

ness of the slope discontinuity at vertex A is decreased andthe stress results become more accurate. However, it isinteresting to note that in the case of the quadratic geo-metric approximation the stress is overpredicted by 45%,this is because the curved edge is not perpendicular to thesymmetry plane due to the error in mapping.

A comparison of the and geometric approxima-tions cases indicates that they underestimate the exact valuefor low order . In the case of the geometric approxi-mations the stress becomes overestimated when

continues to increase. In the case of the geometricapproximations the stress is always underestimated for

and while the does slightly overesti-mate the value for high . Figure 7 provides a more direct

comparison of the and cases for the various geo-metric orders.

It should be noted that the geometric approximation errorfor the quadratic geometric approximations, , at

are substantial with an overestimate of 45% for

the case and underestimate of 16% for the case.The cubic geometric approximations, , yield asmaller error at with an overestimate of 7.7% for

the case and underestimate of 5.0% for the case.The quartic geometric approximations, , yield thesmallest error at with an overestimate of 2.8% for

the case and 0.29% for the case. These results areconsistent with those presented in reference [15] where theellipse geometry was approximated using a blending func-tion method.

4. CURVILINEAR MESH GENERATION

Simmetrix Inc. is developing tools to generate meshes withcurved elements given an initial linear mesh. The currentdevelopment efforts are aimed at the effective representa-tion and definition of meshes consisting of mixedgeometric order elements as shown in Figure 8.

4.1 Representing Mesh Shape

In the current work Bezier representations [5,6] are beingused to define the mesh geometry instead of the more stan-dard Lagrange interpolations due to the followingproperties of Bezier polynomials:

(a) Relative error in energy norm for shapes

(b) Relative error in energy norm for shapes

Figure 5. Relative error in energy norm of different

C0

2 3 4 5 6 7 8 9 101.5

2

2.5

3

p order

Rel

ativ

e er

ror

in e

nerg

y no

rm (

%)

q = 1q = 2q = 3q = 4

q − Geometric approximation order

C0

2 3 4 5 6 7 8 9 101.5

2

2.5

3

p order

Rel

ativ

e er

ror i

n en

ergy

nor

m (%

)

q = 2q = 3q = 4

q − Geometric approximation order

C1

p q,

C0 C1

p C0

p

C1

q 2= q 3= q 4=p

C0 C1

q 2=p 10=

C0 C1

q 3=p 10=

(a) interpolation shape

(b) continuous shape

Figure 6. The convergence curve of relative error inmaximum stress for different shapes

2 4 6 8 10−60

−40

−20

0

20

40

p order

Rel

ativ

e er

ror

in m

axim

um s

tres

s (%

)

q = 1q = 2q = 3q = 4

q − Geometric approximation order

C0

2 4 6 8 10−60

−40

−20

0

20

40

Re

lativ

e e

rro

r in

ma

xim

um

str

ess

(%

)

p order

q = 2q = 3q = 4

q − Geometric approximation order

C1

C0 C1

q 4=p 10=

C0 C1

• Convex Hull Property: A Bezier curve, surface, or vol-ume is contained in the convex hull formed by itscontrol points.

• As with other approximating bases, it has good shapepreserving properties.

• Better qualified properties to allow more efficient

intersection calculations.• All derivatives are also Bezier functions and can be

easily computed.• Products of Bezier functions are easily computed and

are a Bezier function.• Computationally efficient algorithms for degree eleva-

tion and subdivision are available. These can be usedto refine the shape’s convex hull as well as adaptivelyrefine the mesh’s shape.

• Variation Diminishing Property: an infinite plane cannot intersect a Bezier curve more times than it inter-sects control polygon.

Two major issues involved with generating curved meshesare determining if an entity’s curved shape is valid, and iftwo or more mesh entities intersect outside of sharedboundaries. In terms of the intersection issue, application ofthe variation diminishing property for mesh edges and theconvex hull property for mesh faces has resulted in compu-tationally efficient tests to determine if entities interferewith each other.

4.2 Determining Shape Validity

In the case of determining the validity of a mesh entity’sshape, previous implementations only tested the validity ofmesh regions at the integration sites that were to be used inperforming the analysis. The method assumes a region isvalid if the Jacobian is greater than zero at the integrationsites. Though this approach is sufficient for analysis inwhich the element shape, order and integration rule arefixed, it suffers from the following drawbacks:

• If the analysis changes the integration rule then theintegration locations will change. As a result a regionthat was considered valid may suddenly becomeinvalid. The only way to use this approach would be totest each region with respect to all the possible inte-gration sites which can be a large number ofevaluations when the polynomial order of the elementcan be increased.

• The test itself does not provide insight on how theregion can be made valid either by changing theregion’s topology or its geometry.

• The test only focused on mesh regions. If a mesh edgeor face were invalid (due to self-intersection) then allregions using it as part of its boundaries would also beinvalid. Identifying and correcting these lower dimen-sion mesh entities would effectively reduce thenumber of invalid mesh regions that need to becorrected.

Instead of checking for local validity of a mesh entity’sshape, Simmetrix has developed an approach based on theabove Bezier properties to test for the global validity of amesh entity. In the case of a region, the global validitycheck function is based on the fact that the Jacobian func-tion can be represented as a box product of partialderivatives. In the case of a Bezier volume, the partialderivatives are also Bezier functions. Combining the 3 par-tials to form the Jacobian also results in a polynomial thatcan be represented in Bezier form. Since a Bezier polyno-mial is bounded by its convex hull of control points, theBezier form of the Jacobian provides bounds on the Jaco-bian itself. If all the control points of the Jacobian aregreater than zero then the region’s Jacobian must be greater

(a) Convergence curve for

(b) Convergence curve for

Figure 7. Convergence curve for and shapes

Figure 8. Example of mesh entities composed ofdifferent polynomial orders

2 4 6 8 10−50

0

50

p order

Rel

ativ

e er

ror

in m

axim

um s

tres

s (%

)

C1, q = 2C0, q = 2C0, q = 3C1, q = 3

q − geometric approximation order

q 2 3,=

2 4 6 8 10−25

−20

−15

−10

−5

0

5

p order

Rel

ativ

e er

ror i

n m

axim

um s

tress

(%)

C1, q = 4C0, q = 4

q − geometric approximation order

q 4=

C0 C1

Quadratic Edge

Quadratic Face

Cubic FaceCubic Edge

Linear Edges w/

Quadratic Shapes

LinearFace

LinearEdges

Linear Edge w/

Cubic Shape

than zero everywhere. The test works for any polynomialorder. The test also provides additional information in thecase of an invalid shape in terms of possible modificationsthat can be used to correct the shape, see Figure 9. The topimage shows four mesh faces made invalid due to the curv-ing of four mesh edges classified on the circular hole. Thecenter image is a close up of one of the invalidities. Onepossible solution to this problem would be to curve each ofthe faces’ remaining linear edges in order to resolve theinvalidities. The result of these shape modifications areshown in the bottom image.

One important piece needed for curvilinear mesh genera-tion that is missing is the aspect of mesh quality of curvedmeshes. Quality metrics are extremely important whenchoosing which mesh modification to apply in order to cor-rect an invalid mesh shape. For example, another solutionto the problem posed in Figure 9 is to relocate the faces’vertices opposite to curved edges.

4.3 Approximating the Model’s Boundary

Another important issue is how to best approximate themodel boundary. As previously discussed, the approxima-tion error between the original geometric model and themesh can have a strong impact on the analysis. Currently

interpolation methods are used to “fit” the mesh boundaryto the model boundary at certain sample points on theboundary; however, using traditional methods, such aschord length interpolation [9], can cause undesirableboundary artifacts. In polynomial surfaces that are beyondquadratic. One of the major issues is to find appropriateparametric locations for the interpolation points that are noton the edges of a triangular surface. Optimization tech-niques that improve the quality of the surface mesh byreducing these artifacts need to be investigated. Simmetrixhas implemented one such technique as shown inFigure 10..

Both meshes use the same interpolation points. However,the top image shows very pronounced “creases” on thesphere’s surface. In the bottom image the creasing is muchless pronounced.

As previously mentioned, in addition to basic interpolation

Figure 9. The effect of curving mesh edges classified onthe planar face in order to correct the original

highlighted invalid mesh faces Figure 10. Comparison of non-optimized mesh faceswith optimized mesh faces

Non-Optimmized

Optimized

approaches, constraints such as the order of geometric con-tinuity between mesh entities need to be taken intoconsideration when meshing the boundary.

4.4 Preliminary Results

Figure 11 shows the result of generating a curved mesh ofmaximum polynomial degree q = 3 from a linear mesh. Theboundary mesh entities’ shapes have been optimized toreduce undesirable artifacts. Figure 12 shows a more com-plex example of curve mesh generation as well as theimpact raising the polynomial degree on the boundarymesh. Figure 13 and Figure 14 show the application of qua-dratic curving on models supplied by ESRD. The initiallinear meshes show the desired coarseness of the meshes.

5. CLOSING REMARK

The p-version finite element method provides an effectivemethod to apply simulation technologies in engineeringdesign. However, as this paper has pointed out, the applica-

Figure 11. The result of curving a straight sided meshclassified on a sphere using a maximum polynomial

degree of 3.

Figure 12. More complex curving example showing howthe polynomial degree affects the shape of the

highlighted mesh edge.

Figure 13. The application of Simmetrix’s quadraticcurving tool on a model supplied by ESRD.

tion of these methods requires the careful construction ofthe meshes and control of their geometric approximation tocurved domains. The brief introduction to the methods cur-rently under development to support p-version meshgeneration indicate the need to address a number of issuesthat do not need to be considered when low order finite ele-ments are to be used.

6. ACKNOWLEDGEMENTS

Aspects of this work are supported by the NationalScience Foundation (DMI-0132742) and the Office ofNaval Research (N00014-99-0725).

7. REFERENCES

[1] I. Babuska and B. Szabó, "Trends and New Problemsin Finite Element Methods", The Mathematics of Finite

Elements and Applications, J.R. Whiteman, Ed, JohnWiley and Sons, Chichester, pp. 1-33 (1997)

[2] Q. Chen and I. Babuska, "Approximate Optimal Pointsfor Polynomial Interpolation of Real Functions in anInterval and in a Triangle", Comput. Methods Appl.Mech. Engrg., Vol. 128, pp. 405-417 (1995).

[3] S. Dey, M.S. Shephard and J.E. Flaherty, “Geometry-based issues associated with p-version finite elementcomputations”, Comput., Methods. Appl. Mech.Engrg., Vol 150, pp. 39-55 (1997).

[4] X.J. Luo, M.S. Shephard and J.F. Remacle, “The effectof geometric approximation for curved domains in p-version finite element method”, in preparation.

[5] G.E. Farin, Curves and Surfaces for Computer AidedGeometric Design - A Practical Guide, 3rd Edition.Boston: Academic Press, (1993).

[6] R.T. Farouki and V.T. Rajan, “Algorithms for polyno-mials in Bernstein”, Comput. Aid. Geom. Des., Vol. 5,pp. 1-26 (1988).

[7] G. Királyfalvi and B. Szabó, "Quasi-Regional Map-ping for the p-Version of the Finite Element Method,"Finite Elements in Analysis and Design, Vol. 27, pp.85-97 (1997).

[8] N.I. Muskhelishvili, Some Basic Problems of theMathematical Theory of Elasticity, Groningen, P.Noordhoff, (1963).

[9] L. Piegl and W. Tiller, The NURBS Book - 2nd Edition,Berlin: Springer-Verlag (1997).

[10] B. Szabó, "The p- and hp-Versions of the Finite Ele-ment Method in Solid Mechanics," Comput. Methods.Appl. Mech. Engrg., Vol. 80, pp. 185-195 (1990).

[11] B. Szabó, "On Reliability in Finite Element Computa-tions", Comp. & Structures, Vol. 39, pp. 729-734(1991).

[12] B. Szabó and I. Babuska, Finite Element Analysis,John Wiley & Sons New York, (1991).

[13] B. Szabó and R. Actis, "Finite Element Analysis inProfessional Practice," Comput. Methods. Appl. Mech.Engrg., Vol. 133, pp. 209-228 (1996).

[14] B. Szabó, "Quality Assurance in the Numerical Simu-lation of Mechanical Systems," ComputationalMechanics for the 21st Century, B.H.W. Topping, Ed.,Saxe-Coburg Pub., Edinburgh pp.51-69, (2000).

[15] Z. Yosibash and B. Szabó, “Convergence of StressMaxima in Finite elent Computations”, Comm. Numer.Methods Engrg. Vol. 10: pp. 683-697, (1994).

[16] I. Babuska and M. Suri, “The p and h-p versions of thefinite element method, basic priciples and properties”,SIAM J. Numer. Anal., Vol. 36(4) pp. 578-631 (1994).

[17] P. Carnevall and R.B. Morris, Y. Tsuji and G. Taylor,“New basis functions and computational procedures ofp-version finite element analysis, Int. J. Numer. Meth-ods. Engrg, Vol. 36 pp. 3759-3779 (1993).

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Figure 14. Another example of the quadratic curvingtool on a model supplied by ESRD.