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The p-version of the nite element method compared to an

adaptive h-version for the deformation theory of plasticity

A. Duster 1, E. Rank *

Fakultat Bauwesen, Lehrstuhl fur Bauinformatik, Technische Universitat Munchen, Arcisstr. 21, D-80290, Munchen, Germany

Abstract

A p-version of the nite element method is applied to the deformation theory of plasticity and the results are compared to a state-of-the-art adaptive h-version. It is demonstrated that even for nonlinear elliptic problems the p-version is a very efcient discretization

strategy. 2001 Elsevier Science B.V. All rights reserved.

Keywords: p-version; Deformation theory of plasticity; Hencky plasticity

1. Introduction

The p-version of the nite element method has been investigated very intensively during the past 15 years

and it has turned out to be superior to the classical h-version in a signicant number of elds of practical

importance. Most applications were restricted to linear elliptic problems like the Poisson equation, the

Lame equations, the ReissnerMindlin plate problem, etc. Recently, it was shown that the p-version can besuccessfully extended to geometrically nonlinear problems [14] and to elastoplastic analysis of two-

dimensional structures [12,28].

The aim of this paper is to compare the p-version of the nite element method for the deformation theory

of plasticity with a state-of-the-art adaptive h-version in order to stress the fact that high-order nite element

methods can compete with adaptive h-versions. The results of the p-version will be compared to the latest

results of the current German research project, `Adaptive nite element methods in applied mechanics' [5,6].

The outline of the paper is as follows: rst we will set up some basic notations concerning the defor-

mation theory of plasticity. In Section 3, a short introduction to the p-version is given. The efciency of the

p-version is demonstrated by two numerical examples in Section 4. Our numerical results are veried by

similar computations having been performed independently with a different p-version code by Actis and

Szabo [30].

2. The deformation theory of plasticity

The deformation theory of plasticity, rst proposed by Hencky [10], is valid for an isotropic material

under radial 2 loading up to strains lower than the damage threshold (see e.g. [15,16,29]). Following the

www.elsevier.com/locate/cmaComput. Methods Appl. Mech. Engrg. 190 (2001) 19251935

* Corresponding author. Tel.: +89-289-23048; fax: +89-289-25051; http://www.inf.bauwesen.tu-muenchen.de/start.htm.

E-mail addresses: [email protected] (A. Du ster), [email protected] (E. Rank).1 Partially supported by the German Science Foundation DFG under contract RA624/2-3.2

A loading is said to be radial or proportional, if during the loading the ratios of the stress components remain constant.

0045-7825/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved.

PII: S 0 0 4 5 - 7 8 2 5 ( 0 0 ) 0 0 2 1 5 - 2


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notation given in [13], the Hencky problem in perfect plasticity for small strains with the von Mises yield

criterion can be stated as follows:

Find the displacement eld u and the stress eld r such that

divr f in XY

r P2leDu j treuI in XY

u 0 on CDY"t rn 0 on CNY

1

where X is a bounded domain in R3,

P2leDu X2leDu if k2leDuk6r0Y

r0keDuk

eDu if k2leDuk b r0Y

&2

with k k X

X p

. Boundary conditions are given at CD and CN, where displacements and tractions are

imposed, respectively. eDu eu 13

treuI is the deviatoric part of the strain tensor

eu 12

grad u

gradT uX 3

The material constants are the shear modulus l, the bulk modulus j and the yield stress ryield

3a2p

r0.

The corresponding weak form of Eq. (1) reads:

Find u P V fvx P H1X 3X v 0 on CDg such that

X

ru X ev dX

X

v fdX

CN

v "t dC Vv P VX 4

The main dierence between the deformation theory and the ow theory of PrandtlReuss [15,16,20] is thecomputation of the plastic strains. In ow theory, the plastic strains ep e ee are given by the rateequation

ep crDX 5

Due to the postulated proportional loading in case of the deformation theory the ow rule (5) can be

formally integrated, leading to an algebraic equation for determining the plastic strains

ep crDX 6

An existence theorem for proportional loading can be found in [15].

3. The p-version

While in the standard h-version of the nite element method the mesh is rened to achieve convergence,

the polynomial degree of the shape functions remains unchanged. Usually low-order approximation of

degree p 1 or p 2 is chosen. The p-version leaves the mesh unchanged and increases the polynomialdegree of the shape functions locally or globally. In most implementations a hierarchical set of shape

functions is applied, providing a simple and consistent facility of implementation in one-, two- or three-

dimensional analysis. Guidelines to construct these meshes a priori can often be given much more easily for

the p-version than for the h-version [26,27]. For linear elliptic problems, it was also proven that a sequence

of meshes can be constructed so that the approximation error only depends on the polynomial degree p and

not on the order of singularities in the exact solution (see [2]).

1926 A. Duster, E. Rank / Comput. Methods Appl. Mech. Engrg. 190 (2001) 19251935


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Following Szabo and Babuska [26], our p-version implementation uses hierarchical basis functions,

which can easily be implemented up to any desired polynomial degree. The ansatz spaceSpYpXqst applied inthis report is the space of polynomials on Xqst 1Y 1 1Y 1 being spanned by the set of all mono-mials n

igjY iYj 0Y 1Y F F F Yp. Considering, e.g. a nite element for plane elasticity problems based on theansatz spaceSpYpXqst with p 8, the size of the n n-stiffness matrix is n 162. The amount of degreesof freedom corresponding to the bubble modes is 98 being about 60% of the total number of degrees of

freedom on element level. As the bubble degrees of freedom are purely local to the element, they can be

condensed using a modied Cholesky decomposition for the element stiffness matrices. This results infurther increase of computation time on the element level but drastic decrease of solution time because the

condition number of the global stiffness matrix is strongly reduced (see e.g. [22,24]). Several authors

[1,17,19] have investigated these observations in detail, interpreting the bubble mode condensation as a

preconditioning procedure.

Another main dierence between h- and p-version nite element methods lies in mapping requirements.

Because in the p-version the element size is not reduced as the polynomial degree is increased, the de-

scription of the geometry has to be independent of the number of elements. This results in the necessity

to construct elements which exactly represent the boundary. The isoparametric mapping, used in stan-

dard nite element formulations, can be seen as a special case of mapping using the blending function

method [9,26]. Following these ideas element boundaries can be implemented as (almost) arbitrarily

curved edges.In addition to the geometric exibility of higher-order elements there are several reasons for their

attractiveness like high accuracy, robustness and computational efciency. High accuracy is due to an

exponential rate of convergence in the case of an analytical exact solution. This exponential rate can

even be obtained for problems with singularities when an increase of the polynomial order is combined

with local mesh renement in an hp-version. The robustness of the p-version allows the use of strongly

distorted elements and prevents from Poisson ratio locking in cases of nearly incompressible materials

and from shear locking in thin plate situations based on ReissnerMindlin theory (see e.g. [11,21]). In

[22,23], the computational efciency of the p-version for ReissnerMindlin problems was compared to

the h-version and it was shown that the p-version is comparable in speed to a h-version computation

with the same amount of degrees of freedom, yet signicantly more accurate. Furthermore, it was

shown that the p-version is superior in parallel efciency as compared to a classical h-version approach

(see [24]).

4. Numerical results for a benchmark problem

Fig. 1 shows a quarter of a square plate with central hole and unit thickness, loaded by uniform

traction of magnitude p fp0 f100. The material is assumed to be elastic-perfectly-plastic and plane-strain conditions are considered. The computations are carried out for the load parameters f 3X0 andf 4X5, with shear modulus l 80193X8, bulk modulus j 164206X0 and yield stress ryield 450. Thissquare plate was dened as a benchmark problem for the current German research project `Adaptive

nite-element-methods in applied mechanics' [5,6]. We will therefore refer to the results documented in

[3,4,6,25].To nd an approximate solution of the weak form (4) for the given benchmark we use the p-version with

polynomial spaces SpYpXqst on quadrilaterals taking advantage of the blending function method. Thenonlinear computations are performed using one load step and applying the NewtonRaphson method

combined with a radial return algorithm (see e.g. [7,8,16]).

4.1. Perforated square plate under uniform tension with load factor f 3X0

First, we will consider the benchmark system with a load factor f 3X0. A reference solution is computedon a ne mesh consisting of 5568 elements and a uniform polynomial degree up to p 7 with a corre-sponding number of 546,755 degrees of freedom (see Fig. 2, left mesh). Results of interest are the dis-

placements ux at node 5, uy at node 4, the stress ryy at node 2 and

A. Duster, E. Rank / Comput. Methods Appl. Mech. Engrg. 190 (2001) 19251935 1927


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W1

2

X

eTrdX 7

with

e exxY eyyY cxy T

Y r rxxYryyYrxy T

X

It is evident from Table 1 that all results are stable to at least 6 digits and that they can be used as a

reference for further investigations. To test now the eciency of the p-version, a second mesh consisting of

Table 1

Reference solution on a mesh with 5568 elements for load factor f 3X0

p Degrees of freedom W Node 2 ryy Node 4 uy Node 5 ux

1 11 292 2044.847134 623.013300 1X402843825E 01 5X089190291E 022 44 856 2045.261162 521.794799 1X403456683E 01 5X086476901E 023 100 692 2045.267638 517.562995 1X403465985E 01 5X086435253E 024 178 800 2045.269309 517.470322 1X403468249E 01 5X086424710E 025 279 180 2045.268311 517.452278 1X403466922E 01 5X086430958E 02

6 401 831 2045.268696 517.453434 1X403467423E 01 5X086428634E 027 546 755 2045.268489 517.453434 1X403467144E 01 5X086429928E 02

Fig. 1. Perforated square plate under uniform tension.

Fig. 2. Two meshes with 5568 and 4 elements.



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only 4 elements was chosen (see Fig. 2). A series of computations varying the polynomial degree uniformly

from p 1 up to p 19 was performed. To integrate the stiffness matrices, an integration rule withn p 1

2Gaussian points per element was applied. In order to get an impression of the error related to

the numerical integration, the computations were also performed for an integration rule with n 402

Fig. 3. Relative errorjuxYrefuxYFEj

uxYref100% at node 5, load factor f 3X0.

Fig. 4. Relative errorjuyYrefuyYFEj

uyYref100% at node 4, load factor f 3X0.



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Gaussian points per element. The results are plotted in the form of relative errors in double logarithmic

scale in Figs. 36. Regarding the computations, where an integration rule with n p 12 Gaussian pointsper element was applied, one observes that both displacements ux and uy at nodes 5 and 4, respectively, are

well within the 0.1% relative error range when using a polynomial degree pP

5. Only 544 degrees of

Fig. 5. Relative errorjryyYrefryyYFEj

ryyYref100% at node 2, load factor f 3X0.

Fig. 6. Relative error g

jWrefWFEjWref

q100%, load factor f 3X0.



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freedom with a corresponding polynomial degree p 8 are needed to compute the stress ryy at node 2(where the most critical stress concentration occurs), with a relative error less than 5%. Considering these

numerical investigations, it becomes obvious that an accuracy being relevant for practical issues can be

easily gained. Further improvement of the quality of the results can be achieved, if a more accurate inte-

gration rule is applied. This is indicated in Figs. 36, where the errors of the results of interest, gained with

an integration rule with n 402 Gaussian points per element are also plotted.

4.2. Perforated square plate under uniform tension with load factor f 4X5

Again, we choose the same benchmark system, yet now with a load factor f 4X5, where the numericalresults for h-version computations are available. Three different meshes with 2, 4 and 10 p-elements are

chosen (see Fig. 7). A series of computations for polynomial degrees p6 17 for the mesh with 2 elements

and p6 9 for the meshes with 4 and 10 elements was carried out. The following results are of interest: uy at

node 4, ux at node 5 and the integral of the displacement uy along the edge being delimited by the nodes 4

and 5. In order to draw a comparison to an adaptive h-version, we refer to the results of Barthold, Schmidt

and Stein [3,4,6,25]. The computations there, were performed with the Q1P0 element differing from the

well-known bilinear quadrilateral element by including an additional elementwise constant pressure degree

of freedom (see [18]). A mesh consisting of 64 Q1P0 elements was rened in 10 steps using the equilibrium

criterion by Babu!ska and Miller, yielding 875 elements with 1816 degrees of freedom (see Fig. 8). In

[3,4,6,25] the results for a sequence of graded meshes and a reference solution obtained with 24,200 Q1P0

elements with a corresponding number of 49,062 degrees of freedom are also given. Comparing the results

of the uniform p-version with those of the h-version based on a sequence of graded meshes, we observe that

the efciency of the p-version is superior (see Figs. 911). The discretization with 4 elements, p 9 and 684degrees of freedom provides an accuracy which cannot be reached by the h-version even when using 4096

Q1P0 elements with 8320 degrees of freedom. Even compared to a h-renement, resulting in an adapted

mesh with 875 Q1P0 elements it can be seen that a uniform p-version is much more accurate. Furthermore,

Fig. 7. Three meshes with 2, 4 and 10 p-elements.

Fig. 8. Initial mesh with 64 Q1P0 elements and adapted mesh with 875 Q1P0 elements (from [3]).



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Fig. 9. Displacement uy at node 4, load factor f 4X5.

Fig. 10. Displacement ux at node 5, load factor f 4X5.



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Fig. 11.n5n4

juyj dx, load factor f 4X5.

Fig. 12. von Mises stress, load factor f 4X5, for meshes with 2, 4 and 10 elements.



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it should be noted that the p-version offers additional advantages. To compute the stiffness matrices for

p 9, 4 102 400 integration points are needed for the mesh consisting of 4 elements. For integration ofthe 875 Q1P0 stiffness matrices 875 22 3500 Gaussian points are required. Comparing these numbers itbecomes evident that the advantages of the p-version increase with the complexity of the numerical com-

putation required at each integration point. If for example, the ow theory of PrandtlReu with nonlinear

isotropic hardening were used, a nonlinear equation to determine the plastic multiplier would have to be

solved at each Gaussian point.

Returning to the Hencky model and considering the von Mises equivalent stress as shown in Fig. 12, it is

observed that results are very similar for all three meshes. Obviously, two p-elements are sufcient to

compute an excellent approximation for the given problem. The calculations were performed with p 17for the 2-element mesh and p 9 for the meshes with 4 and 10 elements.

Fig. 13 shows the regions where yielding occurs, indicated by the corresponding integration points.

Within the accuracy limited by the discrete set of these points (an integration rule with n p 12

Gaussian points was applied), the result is identical for all three discretizations.

5. Conclusions

In this paper, a uniform p-version was compared to a state-of-the-art h-version for the deformation

theory of plasticity. For a given benchmark problem, the p-version turned out to be signicantly more

accurate than the h-version, even when compared to an adaptive h-renement. We expect from our in-

vestigations that the p-version for the deformation theory is superior, as long as the computational domain

can be discretized into a small number of elements. Only when the complexity of a structure itself requires a

ne mesh, the h-version will be advantageous. Yet, as many structures can be described with only a few

p-elements, further investigation of the p-version for more complex material models seems to be very

promising.

Acknowledgements

The authors wish to thank Professor Erwin Stein and Dipl.-Ing. Matthias Schmidt from University of

Hannover for supporting this work. Extensive help has been given for a successful implementation of the

Hencky material model into our p-version code. The generous preparation of results for the benchmark

problem enabled us to draw a direct comparison between h- and p-version.

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Fig. 13. Integration points where yielding occurs, load factor f 4X5.



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