Some Remarks on Generalized Finite Element Methods (GFEM) in

WCCM VFifth World Congress on

Computational MechanicsJuly 7–12, 2002, Vienna, Austria

Eds.: H.A. Mang, F.G. Rammerstorfer,J. Eberhardsteiner

Some Remarks on Generalized Finite Element Methods(GFEM) in Solid Mechanics

Kai G. Schwebke∗, Stefan M. Holzer

Institute of Mathematics and Information Science in Civil EngineeringUniversity of the Federal Armed Forces, Werner-Heisenberg-Weg 39, D–85577 Neubiberg, Germany

e-mail: [email protected]

Key words: GFEM, higher order methods

AbstractAfter a short introduction to Generalized Finite Element Methods (GFEM) for two-dimensional trian-gular elements, a technique is presented to impose Dirichlet-type boundary conditions to global higherorder GFEM ansatz spaces. The convergence rates ofh-, p-, hp- and enrichedp-versions of the GFEMare discussed with respect to singularities. The method is applied to a two dimensional Poisson modelproblem and the global errors measured in energy norm are compared.

Kai G. Schwebke, Stefan M. Holzer

1 Introduction

1.1 GFEM

The generalized finite element method (GFEM) was first introduced in [1] and [2]. It combines desirablefeatures of the standard finite element method and meshless methods.

The key difference of the GFEM compared to the traditional FEM is the construction of the ansatz space.Each node of the finite element mesh carries a number of ansatz functions, expressed in terms of theglobal coordinate system. Those ansatz functions are multiplied by a partition of unity and serve aselement ansatz functions in the patch constituted by the elements incident at the node.

The partition of unity can be formed by the linear ansatz functions also used in theh-version of the FEM.For two-dimensional triangular elements this leads to the Hat functions in figure 1.

Figure 1: Hat functions

The multiplication of the ansatz functions with the partition of unity fades the influence of the ansatzfunctions to zero at the patch boundaries while retainingC0 continuity. At the same time the influenceof the neighbors’ ansatz functions rises to one effectively, smoothly blending the ansatz spaces into eachother.

The minimal set of ansatz functions is the constant trial function at each node. The result is the classiclinear ansatz space of theh-version of the FEM.

To improve the approximation quality of the ansatz space, higher order polynomials may be used. Thisleads to thep-version of the GFEM.

If a-priori knowledge about the problem is present (e.g. analytical solutions or pre-calculated high qualitysolutions), it can be also included in the ansatz as an enrichment.

Integration and element assembling remain unchanged compared to the classical FEM. Because theinfluence of ansatz functions vanishes outside the patch boundaries, the equation system remains banded,but may contain linear dependent entries and needs therefore appropriate solvers.

1.2 Higher Order GFEM

Higher order methods achieve optimal, exponential convergence rate on smooth problems. On the pres-ence of singularities, however, only twice the linear convergence of theh-version can be achieved.

One possibility to circumvent this is the use of mesh refinements around the singularities. This leads tothehp-version of the FEM.

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WCCM V, July 7–12, 2002, Vienna, Austria

The GFEM allows for another solution: The ansatz space is enriched with an analytically derived orpre-calculated function for the singular part. The remaining problem is smooth and can therefore beoptimally approximated with a higher order ansatz on an unchanged, coarse mesh. Using this strategy noremeshing is needed.

In contrast to [3], [4] and [5], where a conventional FEM discretization is enriched with GFEM ansatzfunctions, we use a pure GFEM discretization for the entire ansatz. Compared to a standardp-versionthe data structures needed to represent the discretization are much less complicated. Independent of theproblem’s dimension only the topologicalelement↔noderelation is required. This relation is coveredby a node list and an element list. E.g. standardp-version in three dimensions would also requireele-ment↔edgeandelement↔facerelations. A drawback of this approach are linear dependencies betweenthe ansatz functions which lead to a rank deficient global stiffness matrix. Special solving techniques,which cannot be discussed here, are required.

2 Essential Boundary Conditions

To achieve highly accurate results on a coarse mesh using a higher order method, exact representationof the geometry and boundary conditions is necessary. In [6] it is shown that the approximation qualitydeteriorates considerably when imposing inexactly represented Dirichlet type boundary conditions.

Using only a minimal set of low order ansatz functions (e.g. constant at each node, which resemblesthe classicalh-version), it is easy to impose Dirichlet type boundary conditions — one simply omitsall ansatz functions related to nodes lying on a Dirichlet boundary. This treatment cannot be applied tohigher order ansatz spaces, because the resulting ansatz is no longer minimally conforming. The ansatzhas finite strain energy, but is not the largest space possible — some linear combinations of omittedfunctions would also conform.

2.1 Dirichlet Boundary along an axis

To impose a homogeneous Dirichlet boundary condition along the borderΓ of the domainΩ in figure 2using a higher order monomial ansatz atP , one omits theshadedfunctions in the ansatz space (1).

Ω

Γ x

y

P

Figure 2: Dirichlet boundary aligned with global axis

3


1x y

x2 xy y2

x3 x2y xy2 y3

· · ·

(1)

Note that this subspace is exact and minimal conforming, which means it is the largest possible spacewith finite strain energy.

A different, and more flexible approach to impose the same restriction on the ansatz space is the multi-plication of every ansatz function by the linear functionϕ(x, y) = y as in (2), effectively increasing thedegree of every ansatz by one and dropping the lastp+ 1 ansatz functions.

y · 1y · x y · y

y · x2 y · xy y · y2

· · ·

(2)

2.2 Arbitrary Dirichlet Boundary Shapes

We propose this idea ofmultiplicative restriction of the ansatz space to be generalized to arbitrarilyshaped boundaries with every suitable complete polynomial space.

P2

P1

P3 P4

P5

Ω

Γ

P2

P1

P3 P4

P5

Ω

Γ

Figure 3: Linear and blended mesh

Figure 3 shows two examples of a domainΩ with a Dirichlet boundaryΓ. The first domain is bounded bypiecewise linear segments, the second one by arc segments forming a circle. The domains are discretizedinto 4 triangles between the nodesP1 . . . P5. Each node carries a polynomial tensor product space (3)constructed from Legendre polynomials (4).

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L0

L1(x) · L0(y) L0(x) · L1(y)L2(x) · L0(y) L1(x) · L1(y) L0(x) · L2(y)

L3(x) · L0(y) L2(x) · L1(y) L1(x) · L2(y) L0(x) · L3(y)· · ·

(3)

with

L0(x) = 1L1(x) = x

(n+ 1)Ln+1(x) = (2n+ 1)xLn(x)− nLn−1(x)(4)

To impose the essential boundary condition, the functions of the ansatz spaces at the nodesP1,P2,P4 andP5 are multiplied with the linear Hat function composed of the corresponding element shape functionsand the blended linear Hat function respectively ((5), (6), (7), figure 4 and figure 5). The ansatz spacesof nodes which do not lie on a Dirichlet boundary remain unchanged.

φ1(x, y) = s1

φ2(x, y) = s2

φ3(x, y) = s3

(5)

with(xy

)= s1

(x1

y1

)+ s2

(x2

y2

)+ s3

(x3

y3

)+~v1(t1) · (1− s3) + ~v2(t2) · (1− s1) + ~v3(t3) · (1− s2)

(6)

and

t1 = 2 · s2s1+s2

− 1t2 = 2 · s3

s2+s3− 1

t3 = 2 · s1s1+s3

− 1(7)

The function used to impose a Dirichlet boundary condition should have the following properties to avoiddegradation of the ansatz space:

• C0 continuous between elements

• smooth everywhere else

• (piecewise) linear

The unblended Hat functions meet these criteria and lead to an exact and minimal conforming ansatz.They are therefore a natural choice to impose Dirichlet boundary conditions on edges resolved in themesh. Our experiments show that also linear blended (and therefore nonlinear) Hat functions could beused for this purpose. Other functions (e.g. cone function) could even be used to impose a boundarycondition independent of the mesh. In this case non-smooth features (e.g. center peak of a cone) of

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1

3

2

v(t )11

(x / y)1 1

(x / y)3 3

(x / y)2 2

(s /s /s )321

Figure 4: Barycentric coordinates and linear blending

Figure 5: Linear and blended Hat

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-1.000e-01 1.000e-01 -1.000e-04 1.000e-04

Figure 6: Absolute local error with unaligned and aligned multiplicator function

the function should be aligned with the mesh to avoid decreased approximation quality of the resultingansatz.

Figure 6 shows a numerical example using an unaligned cone function (8) and an aligned linear blendedfunction to restrict the ansatz space. The middle node of the mesh is shifted out of the center to avoiderror cancellation due to symmetry. Both calculations where performed with 7 degrees of freedom. Theabsolute local error in the aligned case is over four orders of magnitude smaller and equally distributed.In the unaligned case, the error is concentrated at the peak of the cone, which is not resolved in theunderlying mesh.

φcone =√r − (x− x0)2 − (y − y0)2 (8)

3 Examples

The following examples solves as a model problem the Poisson equation (9) in two dimensions withhomogeneous Dirichlet boundary condition under a constant load ofc = 1.

−(∂2u

∂x2+∂2u

∂y2

)= c on Ω

u = 0 on ΓD(9)

The transformation of Poisson’s equation to the weak form leads to:

∫Ω

∂u

∂x· ∂v∂x

+∂u

∂y· ∂v∂y

dΩ =∫

Ωvp dΩ (10)

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3.1 Curved domain boundary

In this example the model problem is solved on a domain bounded by linear and circular edges. Figure 7shows the mesh used for GFEM discretization.

Figure 7: GFEM p-version mesh

The performance of a standardp-version (table 1) and a GFEMp-version (table 2) is compared in figure 8.

Both methods perform equally well with respect to global error in energy norm.

Table 1: Domain with curved boundary, standardp-version

DOF energy relative error β

12 1.942324 29.6804%76 2.745970 0.5853% 2.13

192 2.761660 0.0173% 3.80360 2.762089 0.0018% 3.62580 2.762130 0.0003% 3.83

Table 2: Domain with curved boundary, GFEMp-version


6 2.046840 25.8965%26 2.638836 4.4640% 1.2062 2.758780 0.1216% 4.15

114 2.761475 0.0240% 2.66182 2.762040 0.0035% 4.09

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0.0001

0.001

0.01

0.1

1

10

100

1 10 100 1000

Err

or in

Ene

rgy

Nor

m [%

]

Degrees of Freedom

GFEM with curved boundary

standard pGFEM p

Figure 8: Error in energy norm with curved boundary

3.2 Analytical Enrichment

The following example solves the model problem on an L-shaped region.

The performance of theh-, p-, hp- and analytically enrichedp-version of the GFEM is compared.

Figure 9 shows the analytical solution in the area around the singularity at the reentrant corner and theresulting base function after multiplication with the Hat function. This base function is used to enrichthe ansatz space of the patch around the node at the singular point in the enrichedp-version calculation.Note that the analytical solution fulfills the Dirichlet boundary condition by construction.

Figure 9: Analytical base function and resulting ansatz

Figure 10 shows the different meshes used for the discretization. Thep-version mesh is used twice, with

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and without enrichment of the ansatz with the analytical solution for the singularity.

Figure 10:h-, hp- andp-version mesh

The global error analysis (figure 11) shows the performance of the various methods with respect to energynorm.

1e-05

0.0001

0.001

0.01

0.1

1

10

10 100 1000 10000

Err

or in

Ene

rgy

Nor

m [%

]

Degrees of Freedom

GFEM on L-shaped Domain

hp

hpp enriched

Figure 11: Error in energy norm for L-shaped domain

Theh- andp-versions achieve algebraic convergence, though thep-convergence on the uniform mesh isfaster then theh-convergence. Using ahp-approach on a graded mesh we get even faster convergence atthe cost of remeshing.

Thep-version with enrichment clearly exhibits exponential convergence and performs even better thanthehp-version.

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Table 3: L-shaped domain,h-version


8 59.495192 11.0667%21 63.680628 4.8103% 0.8665 63.909366 4.4684% 0.07

225 65.566258 1.9917% 0.65833 66.427336 0.7046% 0.79

3201 66.744192 0.2309% 0.8312545 66.848280 0.0753% 0.82

Table 4: L-shaped domain,p-version


21 63.680628 4.8103%63 66.128845 1.1507% 1.30

126 66.589322 0.4624% 1.32210 66.719740 0.2675% 1.07315 66.804980 0.1401% 1.60441 66.835929 0.0938% 1.19588 66.854478 0.0661% 1.22756 66.865210 0.0500% 1.11945 66.872080 0.0397% 1.03

Table 5: L-shaped domain,hp-version


64 66.584786 0.4692%134 66.831690 0.1001% 2.09239 66.886361 0.0184% 2.93386 66.896148 0.0038% 3.31582 66.897953 0.0011% 3.06834 66.898392 0.0004% 2.62

Table 6: L-shaped domain, enrichedp-version


22 65.602828 1.9370%64 66.720910 0.2657% 1.86

127 66.878978 0.0294% 3.21211 66.897338 0.0020% 5.30316 66.898308 0.0005% 3.22442 66.898586 0.0001% 4.30589 66.898646 0.0000% 4.17

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4 Conclusion

We presented a method of imposing Dirichlet type boundary conditions to global higher order pureGFEM ansatz spaces. Using this type of polynomial ansatz together with analytical enrichment to coversingularities we achieved exponential convergence of the error in energy norm on an unchanged, coarsemesh with geometrically unresolved singularities.

References

[1] J. M. Melenk,On generalized finite element methods, Ph.D. dissertation, University of Maryland,College Park, MD, (1995)

[2] I. Babuska, J. M. Melenk,The partition of unity finite element method: Basic theory and applica-tions, Comp. Meth. Appl. Mech. Engrg., 139, (1996), 289–314

[3] N. Moes, J. Dolbow, T. Belytschko,A finite element method for crack growth without remeshing,Int. J. Numer. Methods Engrg., 46, (1999), 131–150

[4] I. Babuska, T. Strouboulis, K. Copps,The design and analysis of the generalized finite elementmethod, Comp. Meth. Appl. Mech. Engrg., 181, (2000), 43–69

[5] T. Strouboulis, K. Copps, I. Babuska,The generalized finite element method: an example of its im-plementation and illustration of its performance, Int. J. Numer. Methods Engrg, 47, (2000), 1401–1417

[6] T. Strouboulis, K. Copps, I. Babuska,The generalized finite element method, Comp. Meth. Appl.Mech. Engrg., 190, (2001), 4108–4113

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Some Remarks on Generalized Finite Element Methods (GFEM) in