The P-Version of the Finite Element Method_ch5_ecm003g

Chapter 5The p-version of the Finite Element Method

Barna Szabo1, Alexander Duster2 and Ernst Rank2

1 Washington University, St. Louis, MO, USA2 Lehrstuhl fur Bauinformatik, Technische Universitat Munchen, Munich, Germany

1 Introduction 119

2 Implementation 120

3 Convergence Characteristics 126

4 Performance Characteristics 131

5 Applications to Nonlinear Problems 133

6 Outlook 136

Acknowledgments 137

Notes 137

References 137

Further Reading 139

1 INTRODUCTION

The p-version of the finite element method (FEM) ispresented as a method for obtaining approximate solutionsto generalized formulations of the form

‘Find u ∈ X such that B(u, v) = F(v) for all v ∈ Y ’ (1)

where u and v are scalar or vector functions in one, two, orthree dimensions. In the displacement formulation of solidmechanics problems, for example, u is the displacementfunction, X is the space of admissible displacement func-tions, v is the virtual displacement function, Y is the space

Encyclopedia of Computational Mechanics, Edited by ErwinStein, Rene de Borst and Thomas J.R. Hughes. Volume 1: Funda-mentals. 2004 John Wiley & Sons, Ltd. ISBN: 0-470-84699-2.

of admissible virtual displacement functions, B(u, v) is thevirtual work of internal stresses, and F(v) is the virtualwork of external forces.

More generally, u (resp. X) is called the trial function(resp. trial space) and v (resp. Y ) is called the test function(resp. test space), B(u, v) is a bilinear form defined on X ×Y and F(v) is a linear functional defined on Y . Associatedwith the spaces X and Y are the norms ‖ · ‖X and ‖ · ‖Y . Thedefinitive properties of bilinear forms and linear functionalsare listed, for example, in Szabo and Babuska (1991),Schwab (1998), and Babuska and Strouboulis (2001).

The definitions for B(u, v), F(v), X, and Y depend onthe choice of the generalized formulation and the boundaryconditions. The solution domain will be denoted by andthe set of functions u that satisfy the condition B(u, u) ≤C < ∞ on will be called the energy space and denotedby E(). The exact solution will be denoted by uEX . Theenergy norm defined by

‖u‖E() :=√

12B(u, u) (2)

will be associated with the spaces X ⊂ E() and Y ⊂E(). It can be shown that this formulation is equiva-lent to the minimization of the potential energy functionaldefined by

(u) := 12B(u, u) − F(u) (3)

The exact solution uEX of equation (1) is the minimizer of(u) on the space X ⊂ E().

In the finite element method, finite dimensional sub-spaces S ⊂ X and V ⊂ Y are constructed. These spacesare characterized by the finite element mesh, the polyno-mial degrees assigned to the elements, and the mapping

120 The p-version of the Finite Element Method

functions. Details are given in Section 2. An approxima-tion to uEX , denoted by uFE , is obtained by solving thefinite dimensional problem:

‘Find uFE ∈ S such that B(uFE , v) = F(v) for all v ∈ V ’(4)

The dimension of V is the number of degrees of freedom,denoted by N .

A key theorem states that the finite element solution uFEminimizes the error in energy norm

‖uEX − uFE ‖E() = minu∈S

‖uEX − u‖E() (5)

It is seen that the error in energy norm depends on thechoice of S. Proper choice of S depends on the regularityof uEX , the objectives of computation, and the desired levelof precision.

Another important theorem establishes the followingrelationship between the error measured in energy normand the potential energy:

‖uEX − uFE ‖2E() = (uFE ) − (uEX ) (6)

Proofs are available in Szabo and Babuska (1991). In thep-version, this theorem is used in a posteriori estimation oferror in energy norm.

The data of interest are functionals of uEX : 1(uEX ),

2(uEX ), . . . approximated by 1(uFE ), 2(uFE ), . . . . Animportant objective of finite element computations is toestablish that the relative errors in the data of interest aresmall. Therefore, it is necessary to show that

|i(uEX ) − i(uFE )| ≤ τi |i(uEX )| i = 1, 2, . . . (7)

where τi are prescribed tolerances. Of course, i(uEX )

is generally unknown; however, i(uEX ) is known to beindependent of the choice of the space S. Therefore, ifwe compute a sequence of finite element solutions corre-sponding to a hierarchic sequence of finite element spacesS1 ⊂ S2 ⊂ S3 · · · then i(uFE ) → i(uEX ). The limitingvalue of i(uFE ) and hence τi can be estimated. The p-version of the finite element method is well suited for thecreation of hierarchic finite element spaces and hence theestimation and control of errors in terms of the data ofinterest.

2 IMPLEMENTATION

From the theoretical point of view, the quality of approxi-mation is completely determined by the finite element spacecharacterized by the finite element mesh , the polynomial

degrees of elements p, and the mapping functions Q (seeSection 2.4). Specifically, the finite element space S is aset of functions constructed from polynomials defined onstandard elements that are mapped onto the elements of thefinite element mesh, subject to the appropriate continuityrequirements to ensure that it is a subset of the energy space

S := u|u ∈ E(), u(Qk) ∈ Spk , k = 1, 2, . . . ,M()

where Qk is the mapping function for the kth element, Spk

is the polynomial space of degree pk associated with thekth element, and M() is the number of elements. Differ-ent sets of basis functions, called shape functions, can bechosen to define the same finite element space; however,there are some important considerations:

1. For a wide range of mapping parameters, the round-offerror accumulation with respect to increasing polyno-mial degree should be as small as possible. (Ideally,the element-level stiffness matrices should be perfectlydiagonal, but it is neither necessary nor practical tochoose the shape functions in that way in two and threedimensions.)

2. The shape functions should permit computation of thestiffness matrices and load vectors as efficiently aspossible.

3. The shape functions should permit efficient enforce-ment of exact and minimal continuity.

4. The choice of the shape functions affects the per-formance of iterative solution procedures. For largeproblems, this can be the dominant consideration.

The first three points suggest that shape functions shouldbe constructed from polynomial functions that have certainorthogonality properties; should be hierarchic, that is, theset of shape functions of polynomial degree p should be inthe set of shape functions of polynomial degree p + 1, andthe number of shape functions that do not vanish at vertices,edges, and faces should be the smallest possible. Some ofthe shape functions used in various implementations of thep-version are described in the following.

2.1 Hierarchic shape functions forone-dimensional problems

The classical finite element nodal basis functions in onedimension on the standard element st = (−1, 1) are illus-trated on the left-hand side of Figure 1.

The standard shape functions are defined by the set ofLagrange polynomials

Np

i (ξ) =p+1∏

j=1,j =i

ξ − ξj

ξi − ξj

(8)

The p-version of the Finite Element Method 121

p = 1

p = 2

p = 3

Figure 1. Set of one-dimensional standard and hierarchic shapefunctions for p = 1, 2, 3. (Reproduced by permission of JohnWiley & Sons, Ltd from A. Duster, H. Broker and E. Rank, Int.J. Numer. Meth. Eng., 52, 673–703 (2001).)

The points ξj where

Np

i (ξj ) = δij =

0 if i = j

1 if i = j(9)

are called nodes. There are certain advantages in selectingthe nodes to be the Gauss–Lobatto points as done in thespectral element method, which is also addressed in thisencyclopedia (see Chapter 3, Volume 3). This approachhas been modified to suit the p-version of the finite elementmethod in Melenk, Gerdes and Schwab (2001). Note thatthe sum of all Lagrange polynomials for a given polynomialdegree p equals unity:

p+1∑i=1

Np

i (ξ) = 1 (10)

Every function that can be represented as a linear combi-nation of this standard basis can be represented also by theset of hierarchic basis functions (see the right-hand sideof Figure 1). A principal difference between the two basesis that in the hierarchic case all lower-order shape func-tions are contained in the higher-order basis. The set ofone-dimensional hierarchic shape functions, introduced bySzabo and Babuska (1991), is given by

N1(ξ) = 12 (1 − ξ) (11)

N2(ξ) = 12 (1 + ξ) (12)

Ni(ξ) = φi−1(ξ), i = 3, 4, . . . , p + 1 (13)

with

φj (ξ) =√

2j − 1

2

∫ ξ

−1Lj−1(x) dx

= 1√4j − 2

(Lj (ξ) − Lj−2(ξ)), j = 2, 3, . . . (14)

where Lj(ξ) are the Legendre polynomials. The first twoshape functions N1(ξ), N2(ξ) are called nodal shape func-tions or nodal modes. Because

Ni(−1) = Ni(1) = 0, i ≥ 3 (15)

p = 1

p = 2

p = 3

p = 1

p = 2

p = 3

Figure 2. Hierarchic structure of stiffness matrix and load vectorwith p = 3. (Reproduced by permission of John Wiley & Sons,Ltd from E. Stein (Editor), Error-controlled Adaptive Finite Ele-ments in Solid Mechanics, 263–307 (2002).)

the functions Ni(ξ), i = 3, 4, . . . are called internal shapefunctions, internal modes, or bubble modes. The orthogo-nality property of Legendre polynomials implies

∫ 1

−1

dNi

dξ

dNj

dξdξ = δij , i ≥ 3 and j ≥ 1

or i ≥ 1 and j ≥ 3 (16)

If equations are ordered in such a way that all linearmodes are numbered from 1 to n1, all quadratic modes arenumbered from n1 + 1 to n2 and so on, stiffness matricescorresponding to polynomial order 1 to p − 1 are subma-trices of the stiffness matrix corresponding to polynomialorder p. Figure 2 depicts the structure of a stiffness matrixand a load vector corresponding to polynomial degree ofp = 3 schematically.

2.2 Hierarchic shape functions for quadrilaterals

The standard quadrilateral finite element is shown inFigure 4. Two types of standard polynomial spaces, thetrunk space Spξ,pη

ts (qst) and the tensor product space

Spξ,pη

ps (qst), are discussed in the following. The tensor

product space Spξ,pη

ps (qst) consists of all polynomials on

qst = [(−1, 1) × (−1, 1)] spanned by the set of monomials

ξiηj where i = 0, 1, . . . , pξ, j = 0, 1, . . . , pη, whereas thetrunk space Spξ,pη

ts (qst) on

qst = [(−1, 1) × (−1, 1)] is

spanned by the set of all monomials

• ξiηj with i = 0, . . . , pξ, j = 0, . . . , pη, i + j =0, . . . , maxpξ, pη

• ξη for pξ = pη = 1• ξpξη for pξ ≥ 2• ξηpη for pη ≥ 2.


1

ξ2 η2

η3

ξ η

ξ3 ξ2η ξ η2

η4ξ3η ξ2η2 ξ η3

ξ5 ξ4η

ξ6 ξ η5 η6ξ5η ξ3η3 ξ2η4ξ4η2

ξ4

ξ η

ξ3η2 ξ2η3 ξ η4 η5

3,3(Ωqst)ts

3,3(Ωqst)ps

Figure 3. The trunk space S3,3ts (

qst) and the tensor product space

S3,3ps (

qst). (Reproduced by permission of John Wiley & Sons, Ltd

from E. Stein (Editor), Error-controlled Adaptive Finite Elementsin Solid Mechanics, 263–307 (2002).)

The difference between the two standard polynomialspaces can be readily visualized when considering thespanning sets in Pascal’s triangle. The set of monomials forpξ = pη = 3 for both the trunk and the tensor product spaceis shown in Figure 3. All monomials inside the dashedline span the trunk space S3,3

ts (qst), whereas the monomials

bordered by the solid line are essential for the tensor productspace S3,3

ps (qst).

Two-dimensional shape functions can be classified intothree groups: nodal, edge, and internal shape functions.Using the numbering convention shown in Figure 4, theseshape functions are described in the following.

N4

N1 E1 N2

E3

E4E2

N3

η, pη

ξ, pξ

Figure 4. Standard quadrilateral element qst: definition of nodes,

edges, and polynomial degree.

1. Nodal or vertex modes: The nodal modes

NNi

1,1(ξ, η) = 14 (1 + ξiξ)(1 + ηiη),

i = 1, . . . , 4 (17)

are the standard bilinear shape functions, well knownfrom the isoparametric four-noded quadrilateral ele-ment. (ξi ,ηi) denote the local coordinates of the ithnode.

2. Edge or side modes: These modes are definedseparately for each individual edge, they vanish at allother edges. The corresponding modes for edge E1read:

NE1i,1(ξ, η) = 1

2 (1 − η)φi (ξ), i ≥ 2 (18)

3. Internal modes: The internal modes

N inti,j (ξ, η) = φi (ξ)φj (η), i, j ≥ 2 (19)

45

Side modes

Vertex modes

Internal modes

1 2 3 4

1

2

3

4

5

6

7

8

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16 17

18 19 20 21 22 23

24 25 26 27 28 29 30

31 32 33 34 35 35 37 38

39 40 41 42 43 44 46 47

Vertex/side number

Pol

ynom

ial d

egre

e

Figure 5. Hierarchic shape functions for quadrilateral elements. Trunk space, p = 1 to p = 8. (From Finite Element Analysis; B. Szaboand I. Babuska; Copyright (1991) John Wiley & Sons, Inc. This material is used by permission of John Wiley & Sons, Inc.)


are purely local and vanish at the edges of thequadrilateral element.

As already indicated, the indices i, j of the shape functionsdenote the polynomial degrees in the local directions ξ, η.In Figure 5, all hierarchic shape functions that span thetrunk space are plotted up to order p = 8.

2.3 Hierarchic shape functions for hexahedrals

The implementation of high-order finite elements in threedimensions can be based on a hexahedral element formula-tion (see Figure 6), again using the hierarchic shape func-tions introduced by Szabo and Babuska (1991). High-orderhexahedral elements are suited for solid, ‘thick’ structuresand for thin-walled structures alike. In the case of plate- orshell-like structures, one local variable can be identified tocorrespond with the thickness direction and it is possibleto choose the polynomial degree in the thickness directiondifferently from those in the in-plane direction; see Duster,Broker and Rank (2001). Generalizing the two-dimensionalconcept, three-dimensional shape functions can be classifiedinto four groups:

1. Nodal or vertex modes: The nodal modes

NNi

1,1,1(ξ, η, ζ) = 1

8(1 + ξiξ)(1 + ηiη)(1 + ζiζ),

i = 1, . . . , 8 (20)

are the standard trilinear shape functions, well knownfrom the isoparametric eight-noded brick element.(ξi , ηi , ζi) are the local coordinates of the ith node.

2. Edge modes: These modes are defined separately foreach edge. If we consider, for example, edge E1 (seeFigure 6), the corresponding edge modes read:

NE1i,1,1(ξ, η, ζ) = 1

4 (1 − η)(1 − ζ)φi (ξ), i ≥ 2

(21)

3. Face modes: These modes are defined separately foreach individual face. If we consider, for example, faceF1, the corresponding face modes read:

NF1i,j,1(ξ, η, ζ) = 1

2 (1 − ζ)φi (ξ)φj (η), i, j ≥ 2

(22)

4. Internal modes: The internal modes

N inti,j,k(ξ, η, ζ) = φi (ξ)φj (η)φk(ζ), i, j, k ≥ 2

(23)

are purely local and vanish at the faces of the hexahe-dral element.

N6

N2

N3

N4

N8

E9 E12

E8E6

E2E3

E7

N1E1 E4

E11

E10

E5

F1

F3

F4

F5

F6F2

N5

N7

η

ζ

ξpξ, p

pζ, q

pη, p

Figure 6. Standard hexahedral element hst: definition of nodes,

edges, faces, and polynomial degree.

The indices i, j, k of the shape functions denote thepolynomial degrees in the local directions ξ, η, ζ.

Three different types of trial spaces can be readilydefined: the trunk space Spξ,pη,pζ

ts (hst), the tensor prod-

uct space Spξ,pη,pζ

ps (hst), and an anisotropic tensor product

space Sp,p,q(hst). A detailed description of these trial

spaces can be found in Szabo and Babuska (1991). Thepolynomial degree for the trial spaces Spξ,pη,pζ

ts (hst) and

Spξ,pη,pζ

ps (hst) can be varied separately in each local direc-

tion (see Figure 6). Differences between the trunk andproduct spaces occur in the face modes and internal modesonly. For explanation, we first consider the face modes,for example, the modes for face 1. Indices i, j denote thepolynomial degrees of the face modes in ξ and η direction,respectively.Face modes (face F1): N

F1i,j,1(ξ, η, ζ) = 1/2(1 − ζ)φi (ξ)

φj (η)

trunk space tensor product space

i = 2, . . . , pξ − 2 i = 2, . . . , pξ

j = 2, . . . , pη − 2 j = 2, . . . , pη

i + j = 4, . . . , maxpξ, pηThe definition of the set of internal modes is very similar.

Indices i, j, k now denote the polynomial degrees in thethree local directions ξ, η, and ζ.Internal modes: N int

i,j,k(ξ, η, ζ) = φi(ξ)φj (η)φk(ζ)

trunk space tensor productspace

i = 2, . . . , pξ − 4 i = 2, . . . , pξ

j = 2, . . . , pη − 4 j = 2, . . . , pη

k = 2, . . . , pζ − 4 k = 2, . . . , pζ

i + j + k = 6, . . . , maxpξ, pη, pζ


The space Sp,p,q(hst) defines an anisotropic set of shape

functions determined by two polynomial degrees p and q

(see Figure 6). All shape functions of higher order in ξ

and η direction are associated with the polynomial degreep. These shape functions correspond to the edges 1, 2, 3,4, 9, 10, 11, 12, to the faces 1 and 6 and to all internalmodes. Shape functions for faces 1 and 6 are equal tothose of the trunk space Spξ,pη,pζ

ts (hst) with pξ = pη = p.

q defines the degree of all shape functions of higher orderin ζ-direction that are associated with the edges 5, 6, 7,8, with the faces 2, 3, 4, 5, and with all internal modes.The modes corresponding to the faces 2, 3, 4, 5, are equalto those of the tensor product space Spξ,pη,pζ

ps (hst) with

p = pξ = pη and q = pζ. Considering a polynomial degreep = q = pξ = pη = pζ, one observes that the number ofinternal modes of Sp,p,q(h

st) is larger than that of thetrunk space Spξ,pη,pζ

ts (hst) but smaller than that of the tensor

product space Spξ,pη,pζ

ps (hst).

Owing to the built-in anisotropic behavior of the trialspace Sp,p,q(h

st), it is important to consider the orientationof the local coordinates of a hexahedral element. Figure 7shows how hexahedral elements should be oriented whenthree-dimensional, thin-walled structures are discretized.The local coordinate ζ of the hexahedral element corre-sponds to the thickness direction. If the orientation of allelements is the same then it is possible to construct dis-cretizations where the polynomial degree for the in-planeand thickness directions of thin-walled structures can betreated differently.

2.4 Mapping

In low-order finite element analysis (FEA), the most fre-quently used mapping technique for the geometric descrip-tion of the domain of computation is the application ofisoparametric elements where the standard shape functionsare used for the geometric description of elements. Thesame shape functions are used for the approximation ofthe unknown solution and for the shape of the elements.

x

z

y

ξζ

η

ξ

ζ

η~

~

~

Qe

Figure 7. Modelling thin-walled structures with hexahedralelements.

Using elements of order p = 1 or p = 2, the boundary ofthe domain is approximated by a polygonal or by a piece-wise parabolic curve, respectively. As the mesh is refined,the boundary of the domain is approximated more and moreaccurately. When using the p-version, on the other hand, themesh remains fixed. It is therefore important to model thegeometry of the structure accurately with the fixed numberof elements. This calls for a method that is able to describecomplex geometries using only a few elements. Gordonand Hall (1973a,b) proposed the blending function methodthat is usually applied when describing curved boundariesof p-version finite elements; see, for example, Szabo andBabuska (1991) and Duster, Broker and Rank (2001). Afterintroducing blending function mapping, an example willcompare polynomial interpolation versus exact blendingmapping and demonstrate the necessity of a precise descrip-tion of geometry.

2.4.1 The blending function method

Consider a quadrilateral element as shown in Figure 8where edge E2 is assumed to be part of a curved bound-ary. The shape of edge E2 is assumed to be defined bya parametric function E2 = [E2x(η), E2y(η)]T, where η isthe local coordinate of the element. The transformationof the local coordinates ξ = [ξ,η]T into the global coor-dinates x = [x, y]T = Qe = [Qe

x(ξ, η), Qey(ξ, η)]T can be

formulated by the two functions

x = Qex(ξ, η) =

4∑i=1

NNi

1,1(ξ, η)Xi

+(E2x(η) −

(1 − η

2X2 + 1 + η

2X3

))1 + ξ

2

y = Qey(ξ, η) =

4∑i=1

NNi

1,1(ξ, η)Yi

+(E2y(η) −

(1 − η

2Y2 + 1 + η

2Y3

))1 + ξ

2(24)

N4

N1 N2

E3

E4

E1

E2

N3

η

ξ(X1, Y1)

(X4, Y4)

(X2, Y2)

(X3, Y3)

y

x

E2x (η)E2y(η)

Qe

Figure 8. Blending function method for quadrilateral elements.


where the first term corresponds to the standard bilinearmapping that is familiar from the isoparametric conceptfor quadrilateral elements with p = 1. The second termtakes the curved edge E2 into account. Therefore, thebilinear mapping is augmented by the blended differencebetween the curve E2 = [E2x(η), E2y(η)]T and the straightline connecting the nodes N2 and N3. The blending term(1 + ξ)/2 ensures that the opposite edge E4 – where (1 +ξ)/2 = 0 – is not affected by the curvilinear description ofedge E2.

If a quadrilateral in which all edges are curved is to beconsidered, the blending function method can be expandedsuch that the mapping reads

x = Qex(ξ, η) = 1

2(1 − η)E1x(ξ) + 1

2(1 + ξ)E2x(η)

+ 1

2(1 + η)E3x(ξ) + 1

2(1 − ξ)E4x(η)

−4∑

i=1

NNi

1,1(ξ, η)Xi

y = Qey(ξ, η) = 1

2(1 − η)E1y(ξ) + 1

2(1 + ξ)E2y(η)

+ 1

2(1 + η)E3y(ξ) + 1

2(1 − ξ)E4y(η)

−4∑

i=1

NNi

1,1(ξ, η)Yi (25)

where

Eix(ξ), Eiy(ξ), for i = 1, 3

Eix(η), Eiy(η), for i = 2, 4 (26)

are parametric functions describing the shape of the edgesEi , i = 1, 2, 3, 4. Therefore the blending function methodallows arbitrary parametric descriptions of the edges ofelements.

2.4.2 Accuracy of mapping versus polynomialinterpolation

The following numerical example demonstrates the impor-tance of accurate representation of geometry when a p-extension is to be applied in order to find a finite elementapproximation. A quarter of a linear elastic square platewith a central circular hole and unit thickness (1 mm)is loaded by a traction Tn = 100 MPa (see Figure 9).The dimensions are chosen to be b = h = 100 mm andR = 10 mm. At the lower and right side of the plate,symmetry conditions are imposed. The isotropic linear elas-tic material behavior is characterized by Young’s modulusE = 206 900 MPa, Poisson’s ratio ν = 0.29, and plane

h

y

x

bR

Tn

5

1 7 2 63

98

45 4

1 23

Figure 9. Perforated square plate under uniform tension.

stress assumptions. The strain energy of the plate – obtainedby an ‘overkill’ finite element approximation – amounts to247.521396 Nmm. The plate is discretized by four quadri-lateral elements and the circle with radius R = 10 mm isrepresented by applying

1. exact blending: that is, the exact parametric descrip-tion of a circle is applied;

2. parabolic description: two parabolas are used tointerpolate the circle with a corresponding relative error(|R − R|/R)100(%) < 0.0725(%), where R denotesthe radius of the interpolated circle.

A p-extension based on the tensor product space Sp,pps (

qst),

p = 1, . . . , 8 is performed and the relative error in energynorm for both the exact blending and the parabolic bound-ary interpolation is plotted versus the degrees of freedomon a log–log scale in Figure 10. Owing to the smooth-ness of the exact solution of the problem, the p-extensionin conjunction with the exact blending shows exponentialrate of convergence (see equation (29) in Section 3.2). Inthe case of the parabolic boundary interpolation, the con-vergence rate of the p-extension deteriorates for p ≥ 3and the strain energy finally converges to an incorrectvalue. Consider the stresses, for instance, stress compo-nent σyy at point 2; we observe that the p-extension withp = 1, . . . , 20 and exact blending converges rapidly whilethe stress obtained with parabolic boundary interpolationdiverges (see Figure 11).

Although the relative error of the parabolic geometricinterpolation seems to be very small, it has a stronginfluence on the accuracy of the p-extension. The strainenergy of the approximation converges to an incorrectvalue and the stress component σyy at point 2 evendiverges. The reason for this is that an artificial stresssingularity is introduced. Considering the first derivativesat the interelement node 6, and at symmetry nodes 2


0.001

0.01

0.1

1

10

Rel

ativ

e er

ror

in e

nerg

y no

rm (

%)

10 100 1000Degrees of freedom (N)

Exact blendingParabolic interpolation

Figure 10. Influence of the blending on the relative error inenergy norm.

200

220

240

260

280

300

320

Poi

nt 2

: σyy

500 1000 1500 2000 2500 3000 35000Degrees of freedom (N)

Exact blendingParabolic interpolation

Figure 11. Influence of the blending on the stress component σyy

at point 2.

and 3, discontinuities are observed. They lead to stresssingularities similar to stress concentrations at corners. Oneway of avoiding these stress singularities is to use theexact blending or to apply the so-called quasi-regionalmapping described in Kiralyfalvi and Szabo (1997). Theidea of the quasi-regional mapping is to combine theblending function method with a polynomial interpolationof geometry, using optimal collocation points; see Chenand Babuska (1995, 1996). An example of the effectivenessof this kind of mapping is given in connection with ageometrically nonlinear problem in Section 5.2. A detailedcomparison of exact and polynomial blending is given byBroker (2001).

3 CONVERGENCE CHARACTERISTICS

In this section, some key theoretical results that establishrelationships between the error in energy norm and the

number of degrees of freedom associated with hierarchicsequences of finite element spaces: S1 ⊂ S2 ⊂ · · · arepresented.

In the early implementations of the finite element me-thod, the polynomial degrees were restricted to p = 1 orp = 2 only. Finite element spaces were enlarged by meshrefinement, that is, by reducing the diameter of the largestelement, denoted by h. Subsequently, this limitation wasremoved, allowing enlargement of finite element spaces byincreasing the polynomial degree of elements, denoted byp, while keeping the mesh fixed. To distinguish betweenthe two approaches, the terms ‘h-version’ and ‘p-version’gained currency. We will consider three strategies forconstructing finite element spaces:

(a) h-Extension: The polynomial degree of elements isfixed, typically at some low number, such as p = 1or p = 2, and the number of elements is increasedsuch that h is progressively reduced.

(b) p-Extension: The mesh is fixed and the polynomialdegree of elements is increased.

(c) hp-Extension: The mesh is refined and the polynomialdegrees of elements are concurrently increased.

A fourth strategy, not considered here, introduces basisfunctions, other than the mapped polynomial basis functionsdescribed in Section 2, to represent some local character-istics of the exact solution. This is variously known as thespace enrichment method, partition of unity method, andmeshless method.

It is of considerable practical interest to know how thefirst space S1 should be constructed and when and howh-extension, p-extension, or hp-extension should be used.The underlying principles and practical considerations aresummarized in the following.

3.1 Classification

It is useful to establish a simple classification for the exactsolution based on a priori information available concerningits regularity. The exact solution, denoted by uEX in thefollowing, may be a scalar function or a vector function.

Category A: uEX is analytic everywhere on the solutiondomain including the boundaries. By definition, afunction is analytic in a point if it can be expandedinto a Taylor series about that point. The solutionis in category A also when analytical continuation isapplicable.

Category B: uEX is analytic everywhere on the solutiondomain including the boundaries, with the exceptionof a finite number of points (or in 3D, a finite num-ber of points and edges). The locations where the


exact solution is not analytic are called singular pointsor singular edges. The great majority of practicalproblems in solid mechanics belong in this category.Problems in category B are characterized by piecewiseanalytic data, that is, the domain is bounded by piece-wise analytic functions and/or the boundary conditionsare piecewise analytic.

Category C: uEX is neither in category A nor in cate-gory B.

At corner singularities and at intersections of materialinterfaces in two-dimensional problems, the exact solutiontypically can be written in the form

uEX =∞∑i=1

Airλi Fi (θ), r < ρ, λmin > 0 (27)

where r , θ are polar coordinates centered on the singularpoint, Ai , λi are real numbers, Fi is an analytic (orpiecewise analytic) vector function, and ρ is the radius ofconvergence. Additional details can be found in Grisvard(1985). This is known as an asymptotic expansion of thesolution in the neighborhood of a singular point. Analogousexpressions can be written for one and three dimensionswith λmin > 1 − d/2 where d is the number of spatialdimensions. The minimum value of λi corresponding toa nonzero coefficient Ai characterizes the regularity (alsocalled ‘smoothness’) of the exact solution. In the followingsection, the key theorems concerning the asymptotic ratesof convergence of the various extension processes aresummarized.

3.2 A priori estimates

A priori estimates of the rates of convergence are availablefor solutions in categories A, B, and C. Convergence iseither algebraic or exponential. The algebraic estimate is ofthe form

‖uEX − uFE ‖E() ≤ k

Nβ(28)

and the exponential estimate is of the form

‖uEX − uFE ‖E() ≤ k

exp(γN θ)(29)

These estimates should be understood to mean that thereexists some positive constant k, and a positive constant β

(resp. γ and θ) that depend on uEX , such that the error willbe bounded by the algebraic (resp. exponential) estimate asthe number of degrees of freedom N is increased. Theseestimates are sharp for sufficiently large N .

The asymptotic rates of convergence for two-dimensionalproblems are summarized in Table 1 and for three-dimen-sional problems in Table 2. In these tables, p (resp. λ)represents the minimum polynomial degree assigned tothe elements of a finite element mesh (resp. λmin inequation (27)) (see Chapter 4, this Volume).

3.3 The choice of finite element spaces

The theoretical results described in Section 3.2 provide animportant conceptual framework for the construction offinite element spaces (see Chapter 3, this Volume).

3.3.1 Problems in category A

Referring to Tables 1 and 2, it is seen that for problems incategory A, exponential rates of convergence are possiblethrough p- and hp-extensions. These convergence rates canbe realized provided that all singular points lie on elementvertices and edges. For both the p- and hp-extensions, theoptimal mesh consists of the smallest number of elementsrequired to partition the solution domain into triangular

Table 1. Asymptotic rates of convergence in two dimensions.

Category Type of extension

h p hp

A Algebraic Exponential Exponentialβ = p/2 θ ≥ 1/2 θ ≥ 1/2

B Algebraic Note 1 Algebraic Exponentialβ = (1/2) min(p,λ) β = λ θ ≥ 1/3

C Algebraic Algebraic Note 2β > 0 β > 0

Note 1: Uniform or quasi-uniform mesh refinement is assumed. In thecase of optimal mesh refinement, βmax = p/2.Note 2: When uEX has a recognizable structure, then it is possibleto achieve faster than algebraic rates of convergence with hp-adaptivemethods.

Table 2. Asymptotic rates of convergence in three dimensions.

Category Type of extension

h p hp

A Algebraic Exponential Exponentialβ = p/3 θ ≥ 1/3 θ ≥ 1/3

B Note 3 Exponentialθ ≥ 1/5

C Algebraic Algebraic Note 2β > 0 β > 0

Note 3: In three dimensions, uEX cannot be characterized by a singleparameter. Nevertheless, the rate of p-convergence is at least twice therate of h-convergence.


a

a

Detail A

xa a

y

Detail A

0.15a

0.15

a0.

15a

0.15a

0.0225a

Figure 12. Example of a geometric mesh (detail).

and quadrilateral elements in two dimensions; tetrahedral,pentahedral, and hexahedral elements in three dimensions.

When h-extensions are used, the optimal rate of conver-gence in 2D is algebraic with β = p/2. The optimal meshgrading depends on both p and the exact solution.

3.3.2 Problems in category B

When the exact solution can be written in the form ofequation (27), there is an optimal design of the discretiza-tion in the neighborhood of the singular point. The finiteelements should be laid out so that the sizes of elementsdecrease in geometric progression toward the singular point(located at x = 0) and the polynomial degrees of elementsincrease away from the singular point. The optimal grad-ing is q = (

√2 − 1)2 ≈ 0.17 that is independent of λmin.

In practice, q = 0.15 is used. These are called geometricmeshes. An example of a geometric mesh in two dimensionsis given in Figure 12.

The ideal distribution of polynomial degrees is that thelowest polynomial degree is associated with the small-est element and the polynomial degrees increase linearlyaway from the singular points. This is because the errorsin the vicinity of singular points depend primarily onthe size of elements, whereas errors associated with ele-ments farther from singular points, where the solution issmooth, depend mainly on the polynomial degree of ele-ments. In practice, uniform p-distribution is used, whichyields very nearly optimal results in the sense that con-vergence is exponential, and the work penalty associatedwith using uniform polynomial degree distribution is notsubstantial.

3.4 A simple 1D model problem

In this section, we will consider an axially loaded linearelastic bar as depicted in Figure 13.

Although the solution of the underlying simple modelproblem (30)–(32) can be stated in a closed form, it

f (x)F

xL

EA

Figure 13. Linear elastic bar.

is worth studying because it implies many of the fea-tures that also appear in more complex models. Fur-thermore, the general concept of the p-version can bereadily represented when considering the simple one-dimensional model problem. A tutorial program for a one-dimensional h-, p-, and hp-version of the finite elementmethod, where the following problems can be investi-gated in detail, has been implemented in Maple [1]. Thesolution u(x)(length) of the ordinary differential equa-tion (30) describes the displacement of the bar in x-direction, being loaded by a traction f (x)(force/length) anda load F(force). E(force/length2) denotes Young’s modu-lus, A(length2) the cross-sectional area, and L(length) thelength of the bar.

−(EAu′(x))′ = f (x) on = [x|0 ≤ x ≤ L] (30)

u = 0 at x = 0 (31)

EAu′ = F at x = L (32)

For the sake of simplicity, it is assumed that the dis-placement u(x) and strain ε = du/dx are small and thatthe bar exhibits a linear elastic stress–strain relation-ship, that is, σ = Eε with σ being uniformly distributedover the cross-sectional area A. Equation (31) definesa Dirichlet boundary condition at x = 0 and equation(32), a Neumann boundary condition at x = L. For adetailed study of this model problem, see Szabo andBabuska (1991). The variational or weak formulationof the model problem (30)–(32), which is the basisfor a finite element approximation can be stated asfollows:

Find u ∈ X satisfying (homogeneous) Dirichlet boundaryconditions, such that

B(u, v) = F(v) for all v ∈ Y (33)

where

B(u, v) =∫ L

0EAu′v′ dx

and F(v) =∫ L

0f v dx + Fv(L) (34)


3.4.1 A numerical example with a smooth solution

Figure 13 shows an elastic bar where it is assumedthat EA = L = 1, f (x) = −sin(8x) and F = 0. The p-version discretizations consist of one element with p =1, 2, 3, . . . , 8, whereas the h-version is based on a uni-formly refined mesh with up to eight linear (p = 1)elements.

First, we will consider the p-version discretization. Theexact solution uEX (x) = −(1/64)sin(8x) + (1/8)cos(8)x

of the problem (33)–(34) is approximated by a polynomialexpression on the basis of the hierarchic shape functions(11)–(13)

uFE (ξ) = N1(ξ)U1 + N2(ξ)U2 +pmax∑p=2

Np+1(ξ)ap+1 (35)

where pmax = 8. U1 and U2 denote the nodal displacements,whereas a3, . . . , a9 are coefficients determining the higher-order terms of the approximation uFE (ξ). Owing tothe orthonormality property (16) of the higher-ordershape functions, the element stiffness matrix, Ke

ij = (2/L)∫ 1−1 EA(dNi(ξ)/dξ)(dNj (ξ)/dξ) dξ, i, j = 1, 2, 3, . . . , 9, is

almost perfectly diagonal:

Ke =

1 −1 0 0 · · · 0−1 1 0 0 · · · 0

0 0 2 0 · · · 00 0 0 2 · · · 0...

......

.... . .

...

0 0 0 0 · · · 2

(36)

Computing the element load vector, F ei = (L/2)

∫ 1−1 Ni(ξ)

f (x(ξ)) dξ, i = 1, 2, 3, . . . , 9, one finds

Fe = [−0.1095,−0.0336, −0.0269, −0.0714, 0.0811,

0.0433, −0.0230, −0.0073, 0.0026]T (37)

Because of the homogenous Dirichlet boundary condition(u(0) = uFE (0) = 0 → U1 = 0), the solution of the result-ing diagonal equation system is trivial in this case. InFigure 14, the p-version approximation uFE (x) for p =1, 2, 3, . . . , 8 is plotted together with the exact solutionof the problem. For a first comparison of the accuracy,the same problem is solved by applying the h-version withp = 1 based on a uniformly refined mesh with decreasingelement size hi = 1/i, i = 1, . . . , 8. Again, the approxima-tion and the exact solution is drawn (see Figure 15).

In Figure 16, the relative error in energy norm

(er)E() = ‖uEX − uFE ‖E()

‖uEX ‖E()

(38)

p = 1 p = 2 p = 3 p = 4

p = 5 p = 6 p = 7 p = 8

Figure 14. p-version solution uFE (x) based on one element withp = 1, 2, 3, . . . , 8. A color version of this image is available athttp://www.mrw.interscience.wiley.com/ecm

h = 1 h = 1/2 h = 1/3 h = 1/4

h = 1/5 h = 1/6 h = 1/ 7 h = 1/8

Figure 15. h-version solution uFE (x) based on a uniform refinedmesh with p = 1. A color version of this image is available athttp://www.mrw.interscience.wiley.com/ecm

is plotted versus the number of degrees of freedom ina double logarithmic style. By the classification given inSection 3.2, this problem is in category A, where the p-version exhibits exponential convergence (29), whereasthe asymptotic rate of convergence of the h-extension isalgebraic (28). For category A problems in one dimension,the parameter β in equation (28) is β = p. Since in this casep = 1, the asymptotic rate of convergence is 1, as shownin Figure 16.

le-12

le-10

le-08

le-06

le-04

0.01

1

100

Rel

ativ

e er

ror

in e

nerg

y no

rm (

%)

1 10

Degrees of freedom (N)

1.01.0

Uniform h-extensionUniform p-extension

Figure 16. Comparison of the h- and p-version: relative error inenergy norm.


3.4.2 A numerical example with a nonsmoothsolution

In the following example, we will again consider the weakformulation (33)–(34) of the model problem (30)–(32)where f (x) is now chosen such that the exact solution isnonsmooth. We define f (x) = λ(λ − 1)xλ−2, F = 0 andEA = L = 1, resulting in an exact solution uEX = −xλ +λx, where λ is the parameter controlling the smoothnessof the solution. If λ < 1.0, then the first derivative of theexact solution will exhibit a singularity at x = 0 and thegiven problem will be in category B. Note that λ > 1/2 isa necessary condition for obtaining a finite strain energy ofthe exact solution. For the following numerical example, λ

is chosen to be 0.65.In Figure 17, the relative error in energy norm (38) is

plotted versus the number of degrees of freedom on alog–log scale. p-Extension was performed on one ele-ment with p = 1, . . . , 50, whereas the h-extension wasperformed on meshes with equal sized elements h =1, . . . , 1/50 with p = 1. Since the given problem is in cate-gory B, both extensions show algebraic convergence of type(28). The asymptotic rate of convergence of the h-extensionis given by

β = min(p, λ − 1

2

) = 0.15 (39)

and can be clearly observed in Figure 17. The rate ofconvergence of the uniform p-extension is twice the rateof the uniform h-extension. This is due to the fact that thepoint where the exact solution exhibits singular behaviorcoincides with a node.

When combining mesh refinement with an increasein polynomial degree, exponential convergence in energynorm (29) can be achieved with an hp-extension, even when

10

100

Rel

ativ

e er

ror

in e

nerg

y no

rm (

%)

1 10 100


1.00.15

0.31.0

Uniform h-extensionUniform p-extension

Figure 17. Comparison of the h- and p-version: relative error inenergy norm.

the exact solution uEX has singularities. The mesh is refinedtowards the singular points by geometric progression usingthe common factor q = 0.15. The location of the nodalpoints Xi is given by

Xi =

0 for i = 0Lqnel−i for i = 1, 2, . . . , nel

(40)

A polynomial degree pmin = 1 is assigned to the elementat the singularity, and increases linearly away from thesingular point to the maximum degree

pmax = (2λ − 1)(nel − 1) (41)

where λ denotes the smoothness of the solution and nel, thetotal number of elements of the corresponding mesh. Withthis hp-extension, one obtains an exponential convergencein energy norm as shown in Figure 18 (hp-version, q =0.15, λ = 0.65). Using about 100 degrees of freedom, theerror is by several orders of magnitude smaller than thatof a uniform p-version with one element or of a uniformh-version with p = 1.

Figure 18 also shows the results of uniform p-extensionsobtained on geometrically refined meshes with q = 0.15.These extensions are performed on meshes with nel =4, 8, 12, 16, 20, 24 elements with p being uniformly increa-sed from 1 to 8. In the preasymptotic range, the p-extension on fixed, geometrically graded meshes showsan exponential convergence rate. In the asymptotic range,the exponential convergence decreases to an algebraic rate,being limited by the smoothness λ of the exact solution. Ifproper meshes are used, that is, if the number of refinementscorresponds to the polynomial degree, then any requiredaccuracy is readily obtained.

0.01

0.1

1

10

100

Rel

ativ

e er

ror

in e

nerg

y no

rm (

%)

1 10 100


Uniform h-extension, p = 1

4 elements, q = 0.15, p = 1, ..., 88 elements, q = 0.15, p = 1, ..., 8

12 elements, q = 0.15, p = 1, ..., 816 elements, q = 0.15, p = 1, ..., 820 elements, q = 0.15, p = 1, ..., 824 elements, q = 0.15, p = 1, ..., 8

hp-extension, q = 0.15, λ = 0.65

Uniform p-extension

Figure 18. Comparison of the h-, p-, and hp-version: relativeerror in energy norm.


3.5 Model problem: The L-shaped domain

In order to illustrate the convergence characteristics of theh-, p-, and hp-extensions for category B problems, weconsider an L-shaped domain in two-dimensional elasticity,under the assumption of plane strain conditions usingPoisson’s ratio 0.3. The reentrant edges are stress-free. Inthe xy coordinates system shown in Figure 12, the exactsolution (up to rigid body displacement and rotation terms)corresponding to the first term of the asymptotic expansionis

ux = A1

2Grλ1 [(κ − Q1(λ1 + 1)) cos λ1θ − λ1 cos(λ1 − 2)θ]

(42)

uy = A1

2Grλ1 [(κ + Q1(λ1 + 1)) sin λ1θ + λ1 sin(λ1 − 2)θ]

(43)

where G is the shear modulus, λ1 = 0.544483737, Q1 =0.543075579, and κ = 1.8. The coefficient A1 is calleda generalized stress intensity factor. Details are availablein Szabo and Babuska (1991). The corresponding stresscomponents are

σx = A1λ1rλ1−1[(2 − Q1(λ1 + 1)) cos(λ1 − 1)θ

− (λ1 − 1) cos(λ1 − 3)θ] (44)

σy = A1λ1rλ1−1[(2 + Q1(λ1 + 1)) cos(λ1 − 1)θ

+ (λ1 − 1) cos(λ1 − 3)θ] (45)

τxy = A1λ1rλ1−1[(λ1 − 1) sin(λ1 − 3)θ

+ Q1(λ1 + 1) sin(λ1 − 1)θ] (46)

This model problem is representative of an important classof problems. The reentrant edges are stress-free, the otherboundaries are loaded by the tractions that correspond tothe exact stress distribution given by equations (44) to(46). Since the exact solution is known, it is possibleto compute the exact value of the potential energy fromthe definition of (uEX ) given by equation (3) and usingB(uEX , uEX ) = F(uEX ) from equation (1):

(uEX ) = −1

2

∮[ux(σxnx +τxyny) + uy(τxynx + σyny)] ds

= −4.15454423A1a

2λ1

E(47)

where nx , ny are the components of the unit normal tothe boundary and a is the dimension shown in the inset inFigure 12. The convergence paths for h- and p-extensionsare shown in Figure 19.

0.1

1.0

10.0

100.0

Rel

ativ

e er

ror

in e

nerg

y no

rm (

%)

h = a/ 2

h = a/4

h = a/2

p = 8

p = 8

h = a/6h = a/8

h = a/10 h = a/2

h = a/10

p = 2

10.2721

0.544

10.544

1

2

3

4

56

7 Uniform mesh

h − extensionp − extension

a

a

a a

102 103 104 105 106 107 10810


Figure 19. Convergence paths for the L-shaped domain. (FromFinite-Element Analysis; B. Szabo and I. Babuska; Copyright(1991) John Wiley & Sons, Inc. This material is used bypermission of John Wiley & Sons, Inc.)

It is seen that the asymptotic rates of convergence areexactly as predicted by the estimate (28). However, whenp-extension is used on a geometric mesh, the preasymptoticrate is exponential or nearly so. This can be explained byobserving that the geometric mesh shown in Figure 12 isoverrefined for low polynomial degrees, hence the dominantsource of the error is that part of the domain where the exactsolution is smooth and hence the rate of convergence isexponential, as predicted by the estimate (29). Convergenceslows to the algebraic rate for small errors, where thedominant source of error is the immediate vicinity of thesingular point.

The error estimate frequently used in conjunction withp-extensions is based on the equation (6) and the use ofRichardson extrapolation utilizing the a priori estimate (28).When hp-adaptivity has to be considered, local-based errorestimators have to be applied; see, for example, Ainsworthand Oden (2000) and Melenk and Wohlmuth (2001). Bydefinition, the effectivity index θ is the estimated errordivided by the true error. The estimated and true errors andthe effectivity indices are shown in Table 3. The parameterβ is the same as that in equation (28).

4 PERFORMANCE CHARACTERISTICS

We have seen in Figure 19 that for a fixed accuracy (say1%) there is a very substantial reduction in the numberof degrees of freedom when p-extension is performed onproperly designed meshes. From a practical point of view,the important consideration is the cost of computational


Table 3. L-shaped domain. Geometric mesh, 18 elements, trunkspace. Plane strain, ν = 0.3. Estimated and true relative errors inenergy norm and effectivity index θ.

p N (uFE )E

A21a

2λ1 tz

β (er )E (%) θ

Est.’d True Est.’d True

1 41 −3.886332 — — 25.42 25.41 1.002 119 −4.124867 1.03 1.03 8.44 8.46 1.003 209 −4.148121 1.37 1.36 3.91 3.93 0.994 335 −4.152651 1.33 1.30 2.09 2.14 0.985 497 −4.153636 0.99 0.94 1.42 1.48 0.966 695 −4.153975 0.78 0.68 1.09 1.17 0.937 929 −4.154139 0.69 0.60 0.89 0.99 0.898 1199 −4.154238 0.69 0.56 0.75 0.86 0.87

∞ ∞ −4.154470 0.54 0

resources rather than the number of degrees of freedom.The proper basis for comparing the performance character-istics of various implementations of the h- and p-versionsof the finite-element method is the cost of computation.The cost has to be evaluated with respect to representativemodel problems, such as the L-shaped domain problem dis-cussed in Section 3.5, given specific goals of computation,the required accuracies, and the requirement that a reason-ably close estimate of the accuracy of the computed dataof interest be provided. It is essential that comparisons ofperformance include a verification process, that is, a pro-cess by which it is ascertained that the relative errors inthe data of interest are within prespecified error tolerances.Verification is understood in relation to the exact solutionof the mathematical model, not in relation to some physicalreality that the model is supposed to represent. The conse-quences of wrong engineering decisions based on erroneousinformation usually far outweigh the costs of verification.

Comparative performance characteristics of the h- andp-versions were first addressed in Babuska and Scapolla(1987) and Babuska and Elman (1989) through analyses ofcomputational complexity and theoretical error estimates aswell as computer timing of specific benchmark problems.The main conclusions are summarized as follows:

1. Only for the uncommon cases of very low accuracyrequirements and very irregular exact solutions are low-order elements preferable to high-order elements. High-order elements typically require smaller computationaleffort for the same level of accuracy.

2. High-order elements are more robust than low-orderelements. This point is discussed further in Section 4.1below.

3. The most effective error control procedures combineproper mesh design coupled with progressive increasein p. For details, we refer to Rank and Babuska (1987),Babuska and Suri (1990), and Rank (1992).

4. Accuracies normally required in engineering computa-tion can be achieved with elements of degree 8 or lessfor most practical problems.

5. Computation of a sequence of solutions correspond-ing to a hierarchic sequence of finite element spacesS1 ⊂ S2 ⊂ · · · provides for simple and effective estima-tion and control of error for all data of interest, basedon various types of extrapolation and extraction pro-cedures; see, for example, Szabo and Babuska (1988),Szabo (1990), and Yosibash and Szabo (1994).

As a general rule, for problems in categories A and B(defined in Section 3.1), which include the vast majorityof practical problems in solid mechanics, p-extension onproperly designed meshes is the most efficient general solu-tion strategy. The performance of p-extensions in solvingproblems on category C is discussed in Section 5.1.1.

In the p-version, the elemental matrices are large andtheir computation is time-consuming. On the other hand,these operations lend themselves to parallel computation;see, for example, Rank et al. (2001). Furthermore, it hasbeen shown that a substantial reduction in time can beachieved if special integration techniques are used (seeNubel, Duster and Rank, 2001), or if the hierarchic structureis sacrificed (see Melenk, Gerdes and Schwab, 2001).

4.1 Robustness

A numerical method is said to be robust when it performswell for a broad class of admissible data. For example,in the displacement formulation of linear elasticity, lettingPoisson’s ratio ν approach 1/2 causes the volumetric strain(div u) to approach zero. This introduces constraints amongthe variables, effectively reducing the number of degreesof freedom, and hence causing the rate of convergence inenergy norm to decrease, in some cases, very substantially.This phenomenon is called locking. Locking also causesproblems in the recovery of the first stress invariantfrom the finite element solution. A similar situation existswhen the thickness approaches zero in plate models basedon the Reissner formulation. For a precise definition ofrobustness, we refer to Babuska and Suri (1992). It wasshown in Vogelius (1983) that the rate of convergencein p-extensions is not influenced by ν → 1/2 on straightsided triangles. It is also known that the h-version usingstraight triangles does not exhibit locking, provided thatp ≥ 4. For curvilinear elements, the rate of p-convergenceis slower, and for the h-version the locking problem isgenerally much more severe. Although the p-version isaffected by membrane locking, in the range of typical plateand shell thicknesses that occur in practical engineeringproblems, locking effects are generally not substantial. For


an investigation of membrane locking in cylindrical shells,we refer to Pitkaranta (1992).

4.2 Example

The following example is representative of shell intersec-tion problems. Specifically, the intersection of two cylindri-cal shells is considered. Referring to Figure 20, the outsideradius of shell A is RA = 140 mm, the outside radius ofshell B is RB = 70 mm. The wall thickness of shell A (resp.shell B) is tA = 8.5 mm; (resp. tB = 7.5 mm). The axes ofthe shells intersect at α = 65. The length of shell A is 800mm, the length of shell B, measured from the point of inter-section of the axes of the shells, is 300 mm. The modulusof elasticity is E = 72.4 MPa, Poisson’s ratio is ν = 0.3.

The intersection of the outer surfaces of the shellsis filleted by a ‘rolling ball fillet’, that is, the filletsurface is generated as if rolling a sphere of radius rf =10.0 mm along the intersection line. The mesh consistsof 34 hexahedral elements. The shell intersection region,comprised of 16 elements, is the darker region shown inFigure 20. The complement is the shell region. Quasi-regional mapping utilizing 6 × 6 collocation points percurved face was employed.

The inside surface is loaded by a pressure p. In orderto establish equilibrium, a normal traction Tn is applied onthe surface SB , which is the surface of intersection betweenshell B and a plane perpendicular to its axis:

Tn = p(RB − tB)2

tB(2RB − tB)(48)

The yz plane is a plane of symmetry. The other surfacesare traction-free. Appropriate rigid body constraints wereimposed in order to prevent motion parallel to the plane ofsymmetry.

The objective is to estimate the magnitude of the maximalvon Mises stress to within 5% relative error. In the

Shell A

Shell B

Surface SB

yz

x

Figure 20. Example: Shell intersection problem. The darkerregion, comprised of 16 elements, is the shell intersection region.

20

40

60

80

100

120

140

160

Nor

mal

ized

max

. von

mis

ses

stre

ss Space p,p,p(Ωhst)ts

Space p,p,1(Ωhst)ts

Space p,p,3(Ωhst)ts

p = 1

100 1000 10 000


23

4 5 6 7 8

6

7

8

Figure 21. Example: Shell intersection problem. Convergence ofthe maximum von Mises stress normalized with respect to theapplied pressure p. The estimated limit is 64.7. The maximumoccurs in the shell intersection region.

shell intersection region, the solution varies substantiallyover distances comparable to the thickness. Therefore,dimensional reduction cannot be justified for this region.Fully three-dimensional elements, that is, elements basedon the trunk spaces Sp,p,p

ts (hst) with p = 1, 2, . . . , 8 were

used in the shell intersection region, whereas the anisotropicspaces Sp,p,q

ts (hst) were used in the shell region. The

computations were performed with StressCheck [2].The results are shown in Figure 21. Very strong con-

vergence to the estimated limit value of 64.7 is observedwhen the isotropic spaces Sp,p,p

ts (hst) are employed. This

is true also for the anisotropic spaces Sp,p,qts (h

st) for q ≥ 2but not for q = 1. The reason is that q = 1 implies kine-matic assumptions similar to those of the Naghdi shelltheory. This introduces an artificial singularity along thefaces where q changes abruptly from 8 to 1. Essentially, thisis a modeling error in the sense it pertains to the question ofwhether and where a particular shell model is applicable,given that the goal is to approximate some functional ofthe exact solution of the underlying fully three-dimensionalproblem. Some aspects of this problem have been addressedin Schwab (1996) and Actis, Szabo and Schwab (1999).This example illustrates the importance of convergence testson the data of interest, including tests on the choice ofdimensionally reduced models.

5 APPLICATIONS TO NONLINEARPROBLEMS

5.1 Elastoplasticity

The p- and hp-versions of the finite element method havebeen widely accepted as efficient, accurate, and flexible


methods for analyzing linear problems in computationalmechanics. On the other hand, applications of the p- andhp-versions to nonlinear problems are relatively recentand hence less well known. Considering for instance, theJ2 flow theory of elastoplasticity, a loss of regularity isexpected along the boundary of the plastic zone. Followingthe classification of Section 3.1, this problem is of Class C,that is, it has an unknown line (in 2D) or surface (in 3D) ofsingular behavior in the interior of the domain. Therefore,only an algebraic rate of convergence can be expected.However, this asymptotic rate does not give informationon the preasymptotic behavior, that is, on the accuracy of ap-extension for a finite number of degrees of freedom, andespecially on the question of computational investment fora desired accuracy of quantities of engineering interest.

To shed some light on this question, we will investigatethe deformation theory of plasticity, first proposed byHencky (1924), as a very simple model problem forelastoplasticity. For a detailed description and numericalinvestigation of this model problem, see Szabo, Actis andHolzer (1995) and Duster and Rank (2001). We refer toHolzer and Yosibash (1996), Duster and Rank (2002), andDuster et al. (2002) for a study of the more complex andphysically more realistic flow theory of plasticity, whereeach load integration step in an incremental analysis can beconsidered equivalent to the model problem investigated inthe following section.

5.1.1 A benchmark problem

As a numerical example, we again use the structure ofFigure 9 in Section 2.4.2 showing a quarter of a squareplate with central hole and unit thickness, loaded nowby a uniform tension of magnitude Tn = 450 MPa. Thedimensions of the plate are chosen to be b = h = 10 mmand the radius is set to R = 1 mm. The material isnow assumed to be elastic, perfectly plastic and planestrain conditions are assumed. The shear modulus is µ =80193.8 MPa, the bulk modulus is κ = 164206.0 MPa,and the yield stress is σ0 = 450.0 MPa. This problem wasdefined by Stein (2002) as a benchmark problem for theGerman research project ‘Adaptive finite-element methodsin applied mechanics’.

To find an approximate solution for the given bench-mark, we use the p-version based on the tensor productspace Sp,p

ps (qst) taking advantage of the blending function

method. Three different meshes with 2, 4 and 10 p-elementshave been chosen (see Figure 22). A series of computa-tions for polynomial degrees p ≤ 17 for the mesh with 2elements and p ≤ 9 for the meshes with 4 and 10 ele-ments was performed. In order to make a comparison withan adaptive h-version, we refer to the results of Barthold,

Figure 22. Three meshes with 2, 4, and 10 p-elements. (Reprintedfrom Comput. Methods Appl. Mech. Engng., 190, A. Duster andE. Rank, The p-version of the finite-element method is comparedto an adaptive h-version for the deformation theory of plasticity,1925–1935, Copyright (2001), with permission from Elsevier.)

Schmidt and Stein (1997, 1998) and Stein et al. (1997). Thecomputations there were performed with the Q1-P0 ele-ment differing from the well known bilinear quadrilateralelement by including an additional, elementwise constantpressure degree of freedom. A mesh consisting of 64 Q1-P0 elements was refined in 10 steps using the equilibriumcriterion, yielding 875 elements with 1816 degrees of free-dom (see Figure 23). In Barthold, Schmidt and Stein (1997,1998) and Stein et al. (1997), the results of a sequenceof graded meshes and a reference solution obtained with24 200 Q1-P0 elements with a corresponding number of49 062 degrees of freedom are also given. Comparing theresults of the uniform p-version with those of the h-versionbased on a sequence of graded meshes, we observe thatthe efficiency of the p-version is superior (see Figures 24,25). The discretization with 4 elements, p = 9, and 684degrees of freedom provides an accuracy that cannot bereached by the h-version, even when using 4096 Q1-P0elements with 8320 degrees of freedom. Even comparedto an h-refinement, resulting in an adapted mesh with 875Q1-P0 elements, it can be seen that a uniform p-versionis much more accurate. Although the p-version is signifi-cantly more elaborate than the h-version, when comparingthe computational effort per degree of freedom, investiga-tions on the computational cost to obtain highly accurateresults have clearly shown a superiority of high-order ele-ments. For further information, including three-dimensionalexamples of the J2 flow theory with nonlinear isotropichardening, we refer to Duster and Rank (2001, 2002),Duster (2002), Duster et al. (2002), Rank et al. (2002), andStein (2002).

5.1.2 An industrial application

The following example is concerned with a structuralcomponent of moderate complexity, called a dragbracefitting, shown in Figure 26. This part is representative ofstructural components used in the aerospace sector in thatrelatively thin plate-like regions are reinforced by integrallymachined stiffeners. The overall dimensions are length L =219.6 mm and width w = 115 mm. The material is typicallyaluminum or titanium, which exhibit strain hardening. For


Figure 23. Initial mesh with 64 Q1-P0 elements and adaptedmesh with 875 Q1-P0 elements (see Barthold, Schmidt andStein, 1997). (Reprinted from Comput. Methods Appl. Mech.Engng., 190, A. Duster and E. Rank, The p-version of thefinite-element method compared to an adaptive h-version for thedeformation theory of plasticity, 1925–1935, Copyright (2001),with permission from Elsevier.)

0.2

0.21

0.22

0.23

0.24

0.25

Dis

plac

emen

t uy

at n

ode

4

10 100 1000 10 000


Reference solution2 elements, p = 1, 2, 3, ...,15,17

4 elements, p = 1, 2, 3, ..., 7, 910 elements, p = 1, 2, 3, ..., 7, 9

Graded Q1-P0 elements875 adapted Q1-P0 elements

Figure 24. Displacement uy at node 4. (Reprinted from Comput.Methods Appl. Mech. Engng., 190, A. Duster and E. Rank, Thep-version of the finite-element method compared to an adaptiveh-version for the deformation theory of plasticity, 1925–1935,Copyright (2001), with permission from Elsevier.)

the purposes of this example, an elastic–perfectly plasticmaterial was chosen because it poses a more challengingproblem from the numerical point of view. The materialproperties are those of an ASTM A-36 steel; the yield pointis 248 MPa, the modulus of elasticity is E = 200 GPa, andPoisson’s ratio is ν = 0.295. The mathematical model isbased on the deformation theory of plasticity.

The lugs A and B are fully constrained and sinusoidallydistributed normal tractions are applied through lugs C

and D. The resultants of the tractions are F and 2F

respectively, acting in the negative x direction as the darkregion shown schematically in Figure 26. The goal of thecomputation is to determine the extent of the plastic zone,given that F = 5.5 kN. The mesh consists of 2 tetrahedralelements, 22 pentahedral elements, and 182 hexahedralelements.

0.06

0.065

0.07

0.075

0.08

Dis

plac

emen

t ux

at n

ode

5

10 100 1000 10 000


Reference solution2 elements, p = 1, 2, 3, ...,15, 17

4 elements, p = 1, 2, 3, ..., 7, 910 elements, p = 1, 2, 3, ..., 7, 9

Graded Q1-P0 elements875 adapted Q1-P0 elements

Figure 25. Displacement ux at node 5. (Reprinted from Comput.Methods Appl. Mech. Engng., 190, A. Duster and E. Rank, Thep-version of the finite-element method compared to an adaptiveh-version for the deformation theory of plasticity, 1925–1935,Copyright (2001), with permission from Elsevier.)

A

B

D

C

2FF

z

x

y

Figure 26. Example: Dragbrace fitting. Elastic-plastic solution,p = 7, trunk space, N = 49 894. In the dark region, the equivalentstrain exceeds the yield strain.

The region of primary interest is the neighborhood of theloaded lugs. The results of linear analysis indicate that themaximal von Mises stress in this region is 1040 MPa, thatis, 4.2 times the yield stress. Therefore, nonlinear analysishas to be performed. The region where the equivalentstrain exceeds the yield strain is shown in Figure 26. Thecomputations were performed with StressCheck.

5.2 Geometric nonlinearity

The following example illustrates an application of thep-version to a geometrically nonlinear problem. In geomet-rically nonlinear problems, equilibrium is satisfied in the


deformed configuration. The constitutive laws establish arelationship either between the Piola–Kirchhoff stress ten-sor and the Euler–Lagrange strain tensor or the Cauchystress tensor and the Almansi strain tensor. The formula-tion in this example is based on the Cauchy stress and theAlmansi strain; see Noel and Szabo (1997). The mappingfunctions given by equation (25) are updated iteratively bythe displacement vector components. For example, at theith iteration, the x-coordinate is mapped by

x(i) = Qex(ξ, η, ζ) + u(i)

x (ξ, η, ζ) (49)

It is known that when a thin elastic strip is subjected topure bending, it deforms so that the curvature is constantand proportional to the bending moment:

1

ρ= M

EI(50)

where ρ is the radius of curvature, M is the bendingmoment, E is the modulus of elasticity, and I is the momentof inertia. Poisson’s ratio ν is zero. In this example, a thinstrip of length L = 100 mm, thickness t = 0.5 mm, andwidth b = 5 mm is subjected to normal tractions on Face A

shown in Figure 27, which correspond to M chosen so thatρ = L/2π:

Tn = −2πE

Ly (51)

where y is measured from the mid surface in the directionof the normal in the current configuration. The three dis-placement vector components are set to zero on Face B.Three hexahedral elements were used. The anisotropicspace Sp,p,1

ts (hst) described in Section 2.3 was used. Map-

ping was by the blending function method using 6 × 6collocation points in the quasi-regional mapping proceduredescribed by Kiralyfalvi and Szabo (1997). The computa-tions were performed with StressCheck. The load Tn wasapplied in 20 equal increments. The final deformed con-figuration, a nearly perfect cylindrical body, is shown inFigure 27. The exact solution of a perfectly cylindrical mid-dle surface (the elastica) is the limiting case with respectto the thickness approaching zero.

This example illustrates the following: (a) In the p-version, very large aspect ratios can be used. (b) Quasi-regional mapping, which is an extension of isoparametricmapping combined with the blending function method, iscapable of providing a highly accurate representation ofthe geometrical description with very few elements overlarge deformations. In this example, Face A was rotated360 degrees relative to its reference position.

L

y

xz

b

Face B

Face A

Figure 27. Example: Thin elastic strip. Geometrically nonlinearsolution. Three hexahedral elements, anisotropic space S 8,8,1

ts (hst),

N = 684.

6 OUTLOOK

Although implementations of the p-version are available ina number of commercial finite element computer codes,widespread applications of the p-version in professionalpractice have been limited by three factors:

1. The infrastructure of the most widely used FEA soft-ware products was designed for the h-version, andcannot be readily adapted to meet the technical require-ments of the p-version.

2. In typical industrial problems, finite element meshes aregenerated by automatic mesh generators that producevery large numbers of tetrahedral elements mappedby low-order (linear or quadratic) polynomial map-ping functions. When the mapping functions are oflow degree, the use of high-order elements is gen-erally not justified. This point was illustrated inSection 2.4.2. Nevertheless, numerous computationalexperiments have shown that p-extension performedon tetrahedral meshes up to p = 4 or p = 5 providesefficient means of verification for the computed datawhen the mappings are proper, that is, the Jacobiandeterminant is positive over every element. Experi-ence has shown that many commercial mesh gener-ators produce improperly mapped elements. As meshgenerators improve and produce fewer elements andmore accurate mappings, this obstacle will be graduallyremoved.

3. The demand for verified information in industrial appli-cations of FEMs has been generally weak; however,as computed information is becoming an increas-ingly important part of the engineering decision-making process, the demand for verified data, andhence the importance of the p-version, is likely toincrease.


At present, the p-version is employed in industrial appli-cations mainly where it provides unique technical capa-bilities. Some examples are: (a) Analysis of mechanicaland structural components comprised of plate- and shell-like regions where dimensional reduction is applicable,and solid regions where fully three-dimensional represen-tation is necessary. An example of this kind of domainis shown in Figure 26 where it would not be feasibleto employ fully automatic mesh generators because thefillets would cause the creation of an excessive numberof tetrahedral elements. On the other hand, if the filletswere omitted, then the stresses could not be determined inthe most critical regions. (b) Two- and three-dimensionallinear elastic fracture mechanics where p-extensions ongeometric meshes, combined with advanced extraction pro-cedures, provide verified data very efficiently; see, forexample, Szabo and Babuska (1988) and Andersson, Falkand Babuska (1990). (c) Plate and shell models where therobustness of the p-version and its ability to resolve bound-ary layer effects are important; see, for example, Babuska,Szabo and Actis (1992), Actis, Szabo and Schwab (1999),and Rank, Krause and Preusch (1998). (d) Analysis ofstructural components made of composite materials wherespecial care must be exercised in choosing the mathemati-cal model; large aspect ratios must be used and geometricas well as material nonlinear effects may have to be con-sidered; see Engelstad and Actis (2003). (e) Interpretationof experimental data where strict control of the errorsof discretization (as well as the experimental errors) isessential for proper interpretation of the results of physicalexperiments.

The p-version continues to be a subject of research aimedat broadening its application to new areas. Only a fewof the many important recent and ongoing research activ-ities can be mentioned here. Application of the p- andhp-versions to mechanical contact is discussed in Paczeltand Szabo (2002) and the references listed therein. Theproblem of hp-adaptivity was addressed in the papersDemkowicz, Oden and Rachowicz (1989), Oden et al.(1989), Rachowicz, Oden and Demkowicz (1989), andDemkowicz, Rachowicz and Devloo (2002). The designof p-adaptive methods for elliptic problems was addressedin Bertoti and Szabo (1998). The problem of combin-ing p- and hp-methods with boundary element meth-ods (BEMs) for the solution of elastic scattering prob-lems is discussed in Demkowicz and Oden (1996). Fur-ther information on coupling of FEM and BEM can befound in this encyclopedia (see Chapter 13, this Vol-ume). Application of hp-adaptive methods to Maxwellequations was reported in Rachowicz and Demkowicz(2002).

ACKNOWLEDGMENTS

The writers wish to thank Dr Ricardo Actis of EngineeringSoftware Research and Development, Inc., St Louis Mis-souri, USA for assistance provided in connection with theexamples computed with StressCheck and Professor IstvanPaczelt of the University of Miskolc, Hungary, for helpfulcomments on the manuscript.

NOTES

[1] Waterloo Maple Inc., 57 Erb Street West, Waterloo,Ontarlo, Canada (www.maplesoft.com). The worksheetcan be obtained from the Lehrstuhl fur Bauinformatik,Technische Universitat Munchen, Germany (www.inf.bv.tum.de/∼duester).

[2] StressCheck is a trademark of Engineering SoftwareResearch and Development, Inc., St. Louis, Missouri,USA (www.esrd.com).

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FURTHER READING

Simo JC and Hughes TJR. Computational Inelasticity. Springer-Verlag: New York, 1998.

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