The Generalized Finite Element Method: An Example ...

The Generalized Finite Element Method: An Example of its Implementation and Illustration of its Performance T. Strouboulis1 , K. Copps1 , and I. Babuška2 , 1 Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843-3141, U.S.A. 2 Texas Institute for Computational and Applied Mathematics, University of Texas at Austin, Austin, TX 78712, U.S.A.

November 11, 1998

Abstract The Generalized Finite Element Method (GFEM) was introduced in [1] as a combination of the standard FEM and the Partition of Unity Method. The standard mapped polynomial finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data (the geometry of the domain, the loads, and the boundary conditions). The special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and are pasted to the existing finite element basis to construct a conforming approximation. The essential boundary conditions can be imposed exactly as in the standard FEM. Adaptive numerical quadrature is used to ensure that the errors in integration do not affect the accuracy of the approximation. This paper gives an example of how the GFEM can be developed for the Laplacian in domains with multiple elliptical voids and illustrates implementation issues and the superior accuracy of the GFEM versus the standard FEM.

1 Introduction In this paper we give an example of the design of the Generalized Finite Element Method (GFEM). As our model problem we employ the Laplacian in domains with multiple inclusions or voids. The GFEM is an extension of the standard Finite Element Method in the sense that it allows us to incorporate into the basis of the approximation any special functions which are known to approximate well the solution locally. These special functions are pasted into the standard FEM basis of mapped polynomials by employing a partition of unity method [2]. For the example case considered here, namely the Laplacian in two dimensions in domains with several elliptical voids, we will employ the special harmonic basis functions corresponding to the problem of the elliptical void in the infinite medium. In [1] we gave examples of the GFEM for domains with reentrant corners and cracks and employed the eigenfunctions for the problem of the infinite wedge as our special functions. The idea of incorporating special functions which reflect the local character of the solution in the approximation is not new. The idea has had a resurgence in the newer Trefftz and “hybrid” methods. For example, see the T-element method of Jirousek and Wrobleski [3], or the commercially available analysis package Procision developed by Apanovitch [4]. One, of course, could include the special functions as global basis functions in the approximation, and then there is no difficulty with ensuring the conformity of the approximation. This global approach, however, destroys the banded structure of the stiffness matrix and for this reason it was never adopted in practical computations. In order to be able to use the special functions only locally, various methods which use formulations different than the standard “displacement” formulation have been employed, e.g., the method of Lagrange multipliers, the Discontinuous Galerkin Method [5], etc. The problem with such approaches is that the question of the stability cannot be easily resolved, and the implementation of such methods requires major changes to existing finite element codes. Recently, there has been increasing interest in the so called meshless methods. The objective of these methods is the construction of an approximation without employing a mesh, in order to facilitate the solution of problems 1 The work of these authors was supported by the U.S. Army Research Office under Grant DAAL03-G-028, by the National Science Foundation under Grant MSS-9025110, by the Texas Advanced Research Program under Grant TARP-71071, and by the U.S. Office of Naval Research under Grants N00014-96-1-0021 and Grant N00014-96-1-1015. 2 The work of this author was supported by the U.S. Office of Naval Research under Grant N00014-90-J-1030 and by the National Science Foundation under Grants DMS-91-20877 and DMS-95-01841.

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in complex domains, or domains with propagating cracks by avoiding the construction of a mesh. The functions which are used to form the basis of the approximation are associated with points which are “sprinkled” at selected locations in the domain. The support of these functions is often circular disks (or spheres in three dimensions) that cover the domain. Let us mention, for example, the RKPM [6], the EFG method [7] and the h–p clouds method [8], [9]. There are two main difficulties with meshless methods. First, essential boundary conditions must be imposed as a constraint to the approximation by employing a penalty method or the method of Lagrange multipliers, see also [10] and this could lead to significant complications including loss of stability. And second, the necessary numerical integration of the approximation functions over circles and spheres and their intersections, may require exorbitant amounts of computer resources to obtain sufficiently accurate values of the integrals. For general 3D problems, the problem of the accurate numerical integration is insurmountable. Further, the adoption of meshless methods means that the existing finite element codes must be replaced by new codes which must be written for the new methods. The GFEM is a more sensible approach for achieving the goals of both the hybrid methods and the meshless methods without much of the associated difficulties. In the GFEM, the pasting of the special functions into the approximation is done by a simple multiplication of the special functions with the vertex “hat” functions (formed by the linear or bilinear finite element shape functions) as in the partition of unity methods of Babuška and Melenk in [11], [12], [13], and [14], also in [15]. The pasting is done without the need of using constraints imposed by either the penalty method or the method of Lagrange multipliers. The GFEM includes the classical FEM as a special case, and the essential boundary conditions can be imposed exactly as in the FEM. Because the GFEM is a direct extension of the standard FEM, its stability (the satisfaction of the inf-sup condition) follows exactly as in the FEM. Further, the GFEM uses the FE mesh for needed numerical integration, but that does not prevent cracks or other features from crossing through element boundaries. Of course, some care must be taken to control the errors in the numerical integration, however, it does not significantly affect either the complexity or the performance of the method. Following this introduction, in section 2, we review the construction of the GFEM approximation. In section 3 we discuss the use of both analytically known functions and numerically constructed special functions for general use in the GFEM. Section 4 details numerical quadrature algorithms for the GFEM.And in section 5, we demonstrate the power of the GFEM for creating highly accurate numerical solutions using minimal computer resources.

2 Construction of the GFEM approximation We will let  ⊂ R2 , be a bounded domain with boundary ∂ = 0 D ∪ 0 N , 0D ∩ 0N = ∅. And we will employ the mixed boundary value problem for the Laplacian as our model problem, namely 1u = 0

on



u=0 ∂u =g ∂n

on

0D

on

0N

We will then employ thenvariational formulation ofo(1), namely, def 1 1 Find uEX ∈ S0D = u ∈ H () u|0 = 0 such that D Z Z gv ∇uEX · ∇v = 0N

(1)

∀ v ∈ S01 D

(2)

where H 1 () is the space of functions with square integrable derivatives in . elem be a finite element mesh, which is the partition of the domain into non-overlapping elements Let 1 = {τj }nj =1 τj , namely =

n[ elem j =1

τj,

and τj ∩ τk = ∅ ∀ j 6 = k

(3)

3

Strouboulis et al. / GFEM: An Example of its Implementation and Illustration of its Performance

In the examples, we will employ regular meshes of curvilinear quadrilateral elements; by regular mesh we mean that the intersection of the boundaries of any two elements, ∂τj ∩ ∂τk , is either empty, a vertex, or an entire element p edge for both τj and τk . We will let S1FEM ⊆ S01 D be the finite element space which corresponds to the mesh 1 and elements of degree pFEM , namely   pFEM def 1 −1 pFEM ˆ = v ∈ S0D v|τ ◦ Fτ ∈ S (4) S1 where Sˆ pFEM is the space of bi–pFEM shape functions over the master square τˆ , and Fτ : τˆ 7→ τ , is the mapping of the master square τˆ onto the element τ , which is constructed using the blending function method [16]. We will p denote by uFEM the standard finite element approximation of uEX , corresponding to S1FEM , namely the solution of the problem: p

Find uFEM ∈ S1FEM such that

Z

Z ∇uFEM · ∇v =

0N

gv

p

∀ v ∈ S1FEM .

(5)

Here we could have also considered meshes with irregular connections and elements of various degrees. Let us now recall the definition of a partition of unity, (see [2]): Definition 2.1 (C0 Partition of Unity Subordinate to the Finite Covering {ωi }) Let  ⊂ R2 and {ωi }ni=1 (n < ∞) be an open covering of , and assume that the functions φ ωi , (i = 1, . . . , n) are such that X φ ωi (x) = 1 ∀ x ∈  (6) φ ωi ∈ C 0 () and supp(φ ωi ) ⊂ ωi ∀ i, with i

and ||φ ωi ||L∞ ≤ C∞ , ||∇φ ωi ||L∞ ≤

CG diam(ωi )

(7)

Then we will say that {φ ωi }ni=1 is a C 0 partition of unity subordinate to the covering {ωi }ni=1 .

y

y' P'

P r

Q(x,y)

Γm

x

x'

Figure 2.1. An elliptical void inside the domain  with contour lines indicating the geometry of the conformal mapping for all points (n) P 0 (x 0 , y 0 ) = Q(P (x, y)). The mapping is used to evaluate the series of handbook solutions in the patch spaces, 9i from equation (15), (n)

on the master circle. The special functions in 9i Neumann boundary condition.

are harmonic in both coordinate systems and satisfy the appropriate imposed Dirichlet or

nfun(ω )

Let {ψjωi }j =1 i be the set of the special functions associated with ωi . We will now introduce the enriched finite element space  n onpatch n onpatch n onfun(ωi )  p , φ ωi ; ψjωi (8) S GFEM = S S1FEM ; ωi i=1

i=1

j =1

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by  S GFEM =

 npatch X nfun(ω X i) p v ∈ S01 D v = w + ai φ ωi ψjωi , w ∈ S1FEM i=1

Below, we will employ (1) def

ωi = ωi

=

[

(9)

j =1

τ,

1 φ ωi = φX i def

and

τ ∈1 Xi ∈∂τ

(10)

1 is the corresponding piecewise bilinear basis. Let us now give an where Xi is the ith vertex of the mesh, and φX i ωi example of how the functions ψj will be chosen in the example of the GFEM considered here. Consider the case that the domain has M internal elliptical voids m , m = 1, . . . , M, each with the boundary 0m = ∂m , as shown in Figure 2.1, and assume that the Neumann condition, (11) ∇u · n = g 0m

or the homogeneous Dirichlet condition,

(n)

is imposed. Let ωi

u

0m

=0

(12)

be the n-layer patch around a vertex Xi , namely [  [  (n) τ0 ωi = (n−1) τ ⊆ωi

(1)

Figure 2.2 show examples of ωi

(13)

τ 0 ∈1 ∂τ ∩∂τ 0 6 =∅

(2)

and ωi .

Xi

Xi

Figure 2.2. The 1–layer and 2–layer patches for the vertex Xi , on a standard finite element mesh. (i)

Let ψj,m be functions such that the expansion uEX = 0m . We will then let (n) 9i

 =

ψ=

M X m=1

(n) δm

pm X j =1

P

(i)

αj,m ψj,m is rapidly convergent in the neighborhood of

(i) αj,m ψj,m

1ψ (i) = 0 in R2 − m j,m

(i) and either ∇ψj,m · n (n) δm

(i) = g or ψj,m = 0; 0m 0  m (n) = 1 if 0m ⊆ ωi (n) else δm =0

(14)

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Here we will let (i)

ψ1,m = am ln rm , (i)

(i)

k + b z−k ), ψ k −k ψ2k,m = R(zm m m 2k+1,m = I(zm + cm zm ),

k = 1, . . . , pm

(15)

  √ where zm = rm cos(θm ) + I sin(θm ) , and I = −1, where (rm , θm ) is the polar coordinate system associated with the mth void, and where the constants (am , bm and cm ) are selected to satisfy either the applied Neumann or homogeneous Dirichlet boundary conditions on 0m . The resulting GFEM approximation reads uGFEM =

npatch X i

φi

 nfun(ω X i) j

(i)

(i)

aj ψj



nFEM

+

X k

bk φ˜ k

(16)

Note that the standard FE functions, φi and φ˜ k , can be constructed using the shape functions on the master element. (i) The special functions ψj , however, are typically expressed in terms of physical coordinate system on the actual domain. The superior performance of the GFEM is based on the following result, nfun(ω Theorem 2.1 X i ) (i) (i) aj ψj ∀ i, such that If there exists a ψi = j =1

X i

||uEX − ψi ||2

then ||uEX −

≤ C2 ε2

(17)

φi ψi ||U() ≤ C ε

(18)

U(ωi )

X i

Here C is a constant which is independent of ε, uEX , 1 and the ψi ’s, but depends on the minimal angle of the elements τk ∈ 1. Theorem 2.1 was first proven in [17], then elaborated on in [2, 11, 12, 13]. The GFEM affords both great flexibility in meshing the domain, and ease of adding patch spaces to selected vertices. Let us assume that the goal is to solve the Laplacian on a domain with three internal voids as shown in Figure 2.3, which illustrates three options for selecting the meshes. Option (a) employs meshes which the inclusions, while options (b) and (c) employ meshes which cover the voids. In (b) and (c), a special quadrature algorithm must be used in the elements which intersect the voids.

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(a)

(b)

(c)

h=1

h = 12

h = 14

Figure 2.3. An example of a square domain with three voids and some of the options provided by the GFEM for constructing the approximation: (a) A traditional FEM mesh which respects the geometry, (b) A regular mesh of squares which intersects the voids, (c) A distorted mesh which intersects the voids.

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3 Handbook Problems The GFEM allows the employment of special functions for each special feature of a domain, like voids, inclusions, cracks, corners, etc. to be added to the approximation without changing the underlying mesh that discretizes the domain, see Fig. 3.1. The special functions which will be also called “handbook” solutions may be known analytically or may be computed numerically. In the standard FEM, the mesh must: (1) Represent the geometry and changes in material properties and (2) be sufficiently refined to deliver the desired accuracy. The GFEM can achieve these goals through the choice of the special functions to be included in the approximation and can easily modify the approximation to account for local changes in the geometry and the material properties. GFEM mesh

FEM mesh

Crack

Void

Figure 3.1. A schematic example which shows the flexibility of the GFEM compared with the standard FEM. The GFEM can represent local changes in the geometry by changing the special functions instead of using local remeshing.

Given a problem, a large database of special functions can be computed and stored in the preprocessing phase of the computation. Each special function can be parameterized and cataloged according to: its geometry, the type of differential equation, and boundary conditions. Figure 3.2 provides a schematic example of the construction (i) of a series of handbook solutions for the case of a set of three voids. For each handbook function ψj to be

desired support of handbook solutions

...

Figure 3.2. An example which illustrates the problem involved in the computation of the special functions for an array of three voids: The special functions are obtained by solving a series of Neumann problems with appropriate boundary conditions.

computed, there is a problem to be solved on a slightly larger domain around the feature with a unique set of Neumann boundary conditions that satisfy the consistency condition.

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4 Element Integration Algorithms In the standard FEM, the approximation is constructed by elementwise mapped polynomial functions and the evaluation of the element integrals is done by a Gauss quadrature of a sufficiently high order (q × q degree product rule in 2D) over each element. In the GFEM, however, the use of Gauss quadrature is not practical for two reasons: (1) The entries of the stiffness matrix and the load vector are integrals over complex geometries, e.g., an element partially intersected by one or several voids, and (2) The integrals involve products of derivatives of special functions which may be rough or singular. If care is not taken to control accuracy of these integrals, the accuracy of the GFEM may degrade, as it will be shown in the examples below. The accuracy of the integrals can be controlled by using two types of adaptive quadrature algorithms which deal with the two types of difficulties mentioned above. In the case of singular functions around cracks and re-entrant corners, which was discussed in [1], other types of quadrature are needed. Let us now discuss the two types of adaptive integration in the context of a model problem. Consider the problem illustrated in Figure 4.1, consisting of the square domain [−1, 1] × [−1, 1], which includes three elliptical voids, with radii rmin , rmax and center c as follows: 1. rmin = 0.05, rmax = 0.5, c = (−0.25, 0.375), 45 degree angle of orientation 2. rmin = rmax = 0.1875, c = (0.25, −0.375) 3. rmin = rmax = 0.09375, c = (0.8125, 0.5625)

Homogeneous Neumann boundary conditions are imposed on the boundary of the voids, and at the outer boundary a non-homogeneous Neumann boundary condition with g = ∇u·n where u(x, y) = x −y is applied. We distinguish

3 1

2

Figure 4.1. The model problem consists of the square domain [−1, 1] × [−1, 1] including 3 elliptical voids with zero flux boundary conditions, and outer boundary loaded by constant unit flux.

two types of elements in a mesh, elements that cover voids, and those that do not. Note that depending on n (the number of layers employed in the approximation) an element without voids may still include special functions for one or several voids. We will use two different adaptive quadrature algorithms in an element depending on the element type. 4.1 Integration for elements covering features Let τ be an element which covers, at least partially, one or several voids and let n onsub τ,sub = ω 1sub τ k k=1

(19)

be a subdivision of τ into subelements which approximate the geometry of the feature, see Figure 4.2, followed by additional refinement of the subelements that contribute the major part of the error in the computed integral. Here we use the “trapezoidal” quadrature in subelements intersecting the feature boundary, and a higher order quadrature, e.g., the cubic “Simpson’s” rule, for all other subelements.

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Strouboulis et al. / GFEM: An Example of its Implementation and Illustration of its Performance

3

4

2

1

mesh

(a)

2

3

4

1 3 2

1

2

mesh 3

(b) Figure 4.2. Example of meshes with elements which cover at least partially one or several voids and the meshes of subelements employed in the numerical quadrature.

Algorithm 4.1 (Globally adaptive quadrature for elements crossing features) Assume that the goal is to compute the integral Z I[f] = f τ

Begin: Initial Subdivision:

Let nsub = 1, ω1τ,sub = τ . do l = 1 to m (where m is a positive integer) For each subelement intersecting the boundary of a void, ωkτ,sub ∩ 0m 6 = ∅, subdivide it into four subelements ωjτ,sub (k) .

τ,sub ) + 1, j = 1, . . . 4. Define level(ωjτ,sub (k) ) = level(ωk

end do Initial Estimate:

For all subelements ωkτ,sub intersecting the void boundary use the “trapezoidal” rule with extrapolation to estimate the value of the integral over the region Iωτ,sub and the error Eωτ,sub k k (2-norm of the vector of errors). For all other ωkτ,sub in the element, use the cubic “Simpson” rule (or a suitable higher order rule) and extrapolation to get Iωτ,sub and the error Eωτ,sub . k k P Compute the estimate of the total integral I = Iωτ,sub . k P Compute the estimate of the error E = Eωτ,sub . k

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E > εrel |I|2 Find the maximum error in all subelements, Emax = max(Eωτ,sub ).

Control:

do while

Process Regions: Update:

Divide the subelement(s) that attain Emax . Recompute value of global integral I and the error E. end do

k

k

4.2 Integration for elements not intersecting features For all elements which support special functions, the DCUHRE algorithm (see [18]) will be used which provides an approximation of the integral of a vector function I[f] over the master element rectangle. This algorithm uses a globally adaptive strategy with the same quadrature rule, of polynomial degree 7–13, for all the subregions and integrands. The algorithm uses directional refinement of the subregions, i.e., it divides selected subregions into two pieces along the coordinate axes where the integrand has the largest fourth divided difference. 4.3 Dependence of global accuracy on the numerical integration

0

10

Error in Approximation, u0 – u GFEM

(Ω)

u0

(Ω)

Let us now demonstrate that accurate integration of entries in the stiffness matrix and force vector is necessary for controlling the accuracy of the approximate solution uGFEM . We considered the model problem shown in Figure 4.1, and the meshes shown in Figure 2.3(b). Figure 4.3 shows the h–convergence of the relative error in the energy norm for tolerances εrel = 0.1, εrel = 0.01, and εrel = 0.001. Figure 4.4 depicts the pointwise error in the modulus of the gradient for the pm = 1, 2-layer GFEM on the h = 21 mesh, for integration tolerances εrel = 0.1, εrel = 0.01, and εrel = 0.001. Note the tolerance εrel = 0.1, εrel = 0.01 are not sufficient for controlling the accuracy of the computed solution. On the other hand, the tolerance εrel = 0.001 is sufficient. Let us note that in these results we employed as exact solution the GFEM approximation with pm = 4, and 2-layers on the same mesh.

–1

10

h=1

h = 12 h=

rel

= 0.1

rel

= 0.01

rel

= 0.001

14

–2

10

h = 18

–3

10

10

100 Degrees of freedom, NDOF

1000

Figure 4.3. A larger integration tolerance, εrel , greatly degrades the accuracy of the approximate solution when the special functions are used. Here the GFEM approximation with pm = 1 and a 2-layer approximation was employed.

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Strouboulis et al. / GFEM: An Example of its Implementation and Illustration of its Performance

(a)

(b)

Relative error in modulus of the gradient (percent of the average modulus of the gradient)

(c)

0%

1%

2%

5%

10%

40%

60%

Figure 4.4. An example which shows the effect of the accuracy of the numerical quadrature on the accuracy of the computed solution. The contours of the error, measured in the relative modulus of the flux for the regular mesh pm = 1, 2-layer GFEM h = 21 and NDOF = 63. (a) εrel = 0.1, (b) εrel = 0.01, and (c) εrel = 0.001. The results obtained for εrel = 0.001 are practically identical with those corresponding to exact integration.

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5 Performance of the GFEM We will now compare the accuracy of a two different versions of the GFEM versus the accuracy of the standard FEM. For the comparison we will employ the model problem shown in Figure 4.1. We will consider the case in which the voids are meshed, and the case in which the mesh covers the voids. 5.1 Meshes constructed around the voids In the first kind of GFEM approximation, the mesh is constructed around the voids as shown in Figure 2.3(a). In this case we can use either the standard FEM or the GFEM which employs the harmonic functions corresponding to the voids in n-layers around them. For purposes of comparison on these meshes, we employed as the exact solution the one shown in Figure 5.1. Figure 5.2 shows the contour of the error in the gradient for four different choices of the approximation. From the results shown in Figure 5.2, we conclude that the GFEM can achieve much higher accuracy for the same number of degrees of freedom than the standard FEM on standard FEM meshes.

modulus of gradient %maximum

0%

4%

6%

8%

10%

15%

30%

100%

Figure 5.1. The gradient of the exact solution uEX which was obtained using the standard FEM with pFEM = 6 on the mesh shown with h = 18 , 768 elements, and 28,078 degrees of freedom.

5.2 Meshes which cover the voids The second kind of GFEM employs meshes which cover the voids as shown in Figures 2.3(b) and (c). Figure 5.3 depicts the error in the modulus of the gradient, for the cases pm = 1, pm = 2, and 2-layer approximation on the regular meshes of squares and the distorted meshes. The exact error was obtained using the GFEM on the same mesh with pm = 4, and 2-layer approximation as uEX . From the results shown in Figure 5.3 we can see that relatively high accuracy can be obtained using a very small number of degrees of freedom when compared with the standard FEM. Note, especially the low error in the gradient near the tips of the elliptical void.

Strouboulis et al. / GFEM: An Example of its Implementation and Illustration of its Performance

(a)

(b)

(c)

(d)

13

Relative error in modulus of the gradient (percent of the average modulus of the gradient)

0%

1%

2%

5%

10%

40%

60%

Figure 5.2. The accuracy of the standard FEM versus the accuracy of the GFEM for meshes constructed around the voids. Contours of the error, measured in the relative modulus of the gradient for (a) standard FEM pFEM = 1, h = 41 , 838 degrees of freedom; (b) standard FEM pFEM = 3, h = 21 , 484 degrees of freedom; (c) GFEM pm = 1, 1–layer approximation, h = 1, 67 degrees of freedom; (d) GFEM pm = 1, 2–layer approximation, h = 21 , 208 degrees of freedom.

Strouboulis et al. / GFEM: An Example of its Implementation and Illustration of its Performance

(a)

14

(b)

(c)

(d) Relative error in modulus of the gradient (percent of the average modulus of the gradient)

0%

1%

2%

5%

10%

40%

60%

Figure 5.3. The accuracy of the GFEM for meshes overlapping the voids. The contours of the error, measured in the relative modulus of the gradient on regular and distorted meshes with the 2-layer approximation. (a) Distorted mesh pm = 1, h = 1 approximation, 49 degrees of freedom; (b) Distorted mesh pm = 2, h = 1 approximation, 91 degrees of freedom; (c) Regular mesh pm = 1, h = 21 approximation, 63 degrees of freedom; (d) Regular mesh pm = 2, h = 21 approximation, 117 degrees of freedom.

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6 Conclusions The GFEM can be designed to achieve the goals of both the “hybrid” type methods and the meshless methods. In particular: (a) The GFEM can easily incorporate special functions that approximate the exact solution well locally. (b) The GFEM can be used on meshes which cover parts of the boundary of the domain. (c) The GFEM is free of the difficulties with numerical integration and the application of Dirichlet boundary conditions associated with the meshless methods, (d) The GFEM can be easily incorporated into the existing FEM codes. Note that the problem of the control of the quadrature error is the major pitfall of the so called meshless methods. In a future paper, we will address the problem of the a posteriori estimation of the error and the numerical generation of the special functions.

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