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

Kai Schwebke, Stefan Holzer

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Kai Schwebke, Stefan Holzer. Some Remarks on Generalized Finite Element Methods (GFEM) in Solid Mechanics. Fifth World Congress on Computational Mechanics, Jul 2002, Vienne, Austria. [hal-01371094](https://hal.science/hal-01371094)

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# 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 Engineering

University of the Federal Armed Forces, Werner-Heisenberg-Weg 39, D-85577 Neubiberg, Germany

e-mail: kai.schwebke@unibw-muenchen.de

**Key words:** GFEM, higher order methods

**Abstract**

After a short introduction to Generalized Finite Element Methods (GFEM) for two-dimensional triangular elements, a technique is presented to impose Dirichlet-type boundary conditions to global higher order GFEM ansatz spaces. The convergence rates of *h*-, *p*-, *hp*- and enriched *p*-versions of the GFEM are discussed with respect to singularities. The method is applied to a two dimensional Poisson model problem and the global errors measured in energy norm are compared.
# 1 Introduction

## 1.1 GFEM

The generalized finite element method (GFEM) was first introduced in [1] and [2]. It combines desirable features 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 the global coordinate system. Those ansatz functions are multiplied by a partition of unity and serve as element 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 the *h*-version of the FEM. For two-dimensional triangular elements this leads to the Hat functions in figure 1.

<figure><img src="image_2.png" /><figcaption>Figure 1: Hat functions</figcaption></figure>

The multiplication of the ansatz functions with the partition of unity fades the influence of the ansatz functions to zero at the patch boundaries while retaining $C_0$ continuity. At the same time the influence of the neighbors' ansatz functions rises to one effectively, smoothly blending the ansatz spaces into each other.

The minimal set of ansatz functions is the constant trial function at each node. The result is the classic linear ansatz space of the *h*-version of the FEM.

To improve the approximation quality of the ansatz space, higher order polynomials may be used. This leads to the *p*-version of the GFEM.

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

Integration and element assembling remain unchanged compared to the classical FEM. Because the influence 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 presence of singularities, however, only twice the linear convergence of the *h*-version can be achieved.

One possibility to circumvent this is the use of mesh refinements around the singularities. This leads to the *hp*-version of the FEM.
The GFEM allows for another solution: The ansatz space is enriched with an analytically derived or pre-calculated function for the singular part. The remaining problem is smooth and can therefore be optimally approximated with a higher order ansatz on an unchanged, coarse mesh. Using this strategy no remeshing is needed.

In contrast to [3], [4] and [5], where a conventional FEM discretization is enriched with GFEM ansatz functions, we use a pure GFEM discretization for the entire ansatz. Compared to a standard *p*-version the data structures needed to represent the discretization are much less complicated. Independent of the problem's dimension only the topological *element*↔*node* relation is required. This relation is covered by a node list and an element list. E.g. standard *p*-version in three dimensions would also require *element*↔*edge* and *element*↔*face* relations. A drawback of this approach are linear dependencies between the 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 representation of the geometry and boundary conditions is necessary. In [6] it is shown that the approximation quality deteriorates considerably when imposing in exactly represented Dirichlet type boundary conditions.

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

### 2.1 Dirichlet Boundary along an axis

To impose a homogeneous Dirichlet boundary condition along the border $\Gamma$ of the domain $\Omega$ in figure 2 using a higher order monomial ansatz at $P$, one omits the shaded functions in the ansatz space (1).

<figure><img src="image_3.png" /><figcaption>Figure 2: Dirichlet boundary aligned with global axis</figcaption></figure>
$$ \begin{array}{ccccccc} & & & & 1 & & \\ & x & & y & & & \\ x^2 & xy & & y^2 & & & \\ x^3 & x^2y & & xy^2 & & y^3 & \\ & \vdots & & & & & \\ & & & & & & \end{array} \qquad (1) $$

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

A different, and more flexible approach to impose the same restriction on the ansatz space is the multiplication of every ansatz function by the linear function $\varphi(x, y) = y$ as in (2), effectively increasing the degree of every ansatz by one and dropping the last $p+1$ ansatz functions.

$$ \begin{array}{ccccccc} & & & & y \cdot 1 & & \\ & y \cdot x & & y \cdot y & & \\ y \cdot x^2 & y \cdot xy & & y \cdot y^2 & & \\ & \vdots & & & & & \end{array} \qquad (2) $$

## 2.2 Arbitrary Dirichlet Boundary Shapes

We propose this idea of *multiplicative restriction* of the ansatz space to be generalized to arbitrarily shaped boundaries with every suitable complete polynomial space.

<figure><img src="image_4.png" /><figcaption>Figure 3: Linear and blended mesh</figcaption></figure>

Figure 3 shows two examples of a domain $\Omega$ with a Dirichlet boundary $\Gamma$. The first domain is bounded by piecewise linear segments, the second one by arc segments forming a circle. The domains are discretized into 4 triangles between the nodes $P_1...P_5$. Each node carries a polynomial tensor product space (3) constructed from Legendre polynomials (4).
$$ 
\begin{array}{l}
L_0 \\
L_1(x) \cdot L_0(y) \qquad L_0(x) \cdot L_1(y) \\
L_2(x) \cdot L_0(y) \qquad L_1(x) \cdot L_1(y) \qquad L_0(x) \cdot L_2(y) \\
L_3(x) \cdot L_0(y) \qquad L_2(x) \cdot L_1(y) \qquad L_1(x) \cdot L_2(y) \qquad L_0(x) \cdot L_3(y) \\
\dots
\end{array} 
\qquad (3) $$

with

$$ 
\begin{array}{l}
L_0(x) = 1 \\
L_1(x) = x \\
(n+1)L_{n+1}(x) = (2n+1)xL_n(x) - nL_{n-1}(x)
\end{array} 
\qquad (4) $$

To impose the essential boundary condition, the functions of the ansatz spaces at the nodes $P_1, P_2, P_4$ and $P_5$ are multiplied with the linear Hat function composed of the corresponding element shape functions and the blended linear Hat function respectively ((5), (6), (7), figure 4 and figure 5). The ansatz spaces of nodes which do not lie on a Dirichlet boundary remain unchanged.

$$ 
\begin{array}{l}
\phi_1(x, y) = s_1 \\
\phi_2(x, y) = s_2 \\
\phi_3(x, y) = s_3
\end{array} 
\qquad (5) $$

with

$$ 
\begin{array}{l}
\binom{x}{y} = s_1 \binom{x_1}{y_1} + s_2 \binom{x_2}{y_2} + s_3 \binom{x_3}{y_3} \\
\qquad + \vec{v}_1(t_1) \cdot (1 - s_3) + \vec{v}_2(t_2) \cdot (1 - s_1) + \vec{v}_3(t_3) \cdot (1 - s_2)
\end{array} 
\qquad (6) $$

and

$$ 
\begin{array}{l}
t_1 = 2 \cdot \frac{s_2}{s_1+s_2} - 1 \\
t_2 = 2 \cdot \frac{s_3}{s_2+s_3} - 1 \\
t_3 = 2 \cdot \frac{s_1}{s_1+s_3} - 1
\end{array} 
\qquad (7) $$

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

* $C_0$ 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 the mesh. Our experiments show that also linear blended (and therefore nonlinear) Hat functions could be used for this purpose. Other functions (e.g. cone function) could even be used to impose a boundary condition independent of the mesh. In this case non-smooth features (e.g. center peak of a cone) of
<figure><img src="image_5.png" /><figcaption>Figure 4: Barycentric coordinates and linear blending</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 5: Linear and blended Hat</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 6: Absolute local error with unaligned and aligned multiplicative function</figcaption></figure>

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

Figure 6 shows a numerical example using an unaligned cone function (8) and an aligned linear blended function to restrict the ansatz space. The middle node of the mesh is shifted out of the center to avoid error cancellation due to symmetry. Both calculations where performed with 7 degrees of freedom. The absolute 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 the underlying mesh.

$$ \phi_{\text{cone}} = \sqrt{r - (x - x_0)^2 - (y - y_0)^2} \qquad (8) $$

### 3 Examples

The following examples solves as a model problem the Poisson equation (9) in two dimensions with homogeneous Dirichlet boundary condition under a constant load of $c = 1$.

$$ \begin{array}{l} - \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) = c \quad \text{on } \Omega \\ u = 0 \quad \text{on } \Gamma_D \end{array} \qquad (9) $$

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

$$ \int_{\Omega} \frac{\partial u}{\partial x} \cdot \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \cdot \frac{\partial v}{\partial y} d\Omega = \int_{\Omega} v p d\Omega \qquad (10) $$
## 3.1 Curved domain boundary

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

<figure><img src="image_8.png" /><figcaption>Figure 7: GFEM p-version mesh</figcaption></figure>

The performance of a standard *p*-version (table 1) and a GFEM *p*-version (table 2) is compared in figure 8. Both methods perform equally well with respect to global error in energy norm.

<table><caption>Table 1: Domain with curved boundary, standard *p*-version</caption><thead><tr><td>DOF</td><td>energy</td><td>relative error</td><td>β</td></tr></thead><tbody><tr><td>12</td><td>1.942324</td><td>29.6804%</td><td></td></tr><tr><td>76</td><td>2.745970</td><td>0.5853%</td><td>2.13</td></tr><tr><td>192</td><td>2.761660</td><td>0.0173%</td><td>3.80</td></tr><tr><td>360</td><td>2.762089</td><td>0.0018%</td><td>3.62</td></tr><tr><td>580</td><td>2.762130</td><td>0.0003%</td><td>3.83</td></tr></tbody></table>

<table><caption>Table 2: Domain with curved boundary, GFEM *p*-version</caption><thead><tr><td>DOF</td><td>energy</td><td>relative error</td><td>β</td></tr></thead><tbody><tr><td>6</td><td>2.046840</td><td>25.8965%</td><td></td></tr><tr><td>26</td><td>2.638836</td><td>4.4640%</td><td>1.20</td></tr><tr><td>62</td><td>2.758780</td><td>0.1216%</td><td>4.15</td></tr><tr><td>114</td><td>2.761475</td><td>0.0240%</td><td>2.66</td></tr><tr><td>182</td><td>2.762040</td><td>0.0035%</td><td>4.09</td></tr></tbody></table>
<figure><img src="image_9.png" /><figcaption>Figure 8: Error in energy norm with curved boundary</figcaption></figure>

## 3.2 Analytical Enrichment

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

The performance of the *h*-, *p*-, *hp*- and analytically enriched *p*-version of the GFEM is compared.

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

<figure><img src="image_10.png" /><figcaption>Figure 9: Analytical base function and resulting ansatz</figcaption></figure>

Figure 10 shows the different meshes used for the discretization. The *p*-version mesh is used twice, with
and without enrichment of the ansatz with the analytical solution for the singularity.

<figure><img src="image_11.png" /><figcaption>Figure 10: h-, hp- and p-version mesh</figcaption></figure>

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

<figure><img src="image_12.png" /><figcaption>Figure 11: Error in energy norm for L-shaped domain</figcaption></figure>

The h- and p-versions achieve algebraic convergence, though the p-convergence on the uniform mesh is faster than the h-convergence. Using a hp-approach on a graded mesh we get even faster convergence at the cost of remeshing.

The p-version with enrichment clearly exhibits exponential convergence and performs even better than the hp-version.
<table><caption>Table 3: L-shaped domain, *h*-version</caption><thead><tr><th>DOF</th><th>energy</th><th>relative error</th><th>β</th></tr></thead><tbody><tr><td>8</td><td>59.495192</td><td>11.0667%</td><td></td></tr><tr><td>21</td><td>63.680628</td><td>4.8103%</td><td>0.86</td></tr><tr><td>65</td><td>63.909366</td><td>4.4684%</td><td>0.07</td></tr><tr><td>225</td><td>65.566258</td><td>1.9917%</td><td>0.65</td></tr><tr><td>833</td><td>66.427336</td><td>0.7046%</td><td>0.79</td></tr><tr><td>3201</td><td>66.744192</td><td>0.2309%</td><td>0.83</td></tr><tr><td>12545</td><td>66.848280</td><td>0.0753%</td><td>0.82</td></tr></tbody></table>

<table><caption>Table 4: L-shaped domain, *p*-version</caption><thead><tr><th>DOF</th><th>energy</th><th>relative error</th><th>β</th></tr></thead><tbody><tr><td>21</td><td>63.680628</td><td>4.8103%</td><td></td></tr><tr><td>63</td><td>66.128845</td><td>1.1507%</td><td>1.30</td></tr><tr><td>126</td><td>66.589322</td><td>0.4624%</td><td>1.32</td></tr><tr><td>210</td><td>66.719740</td><td>0.2675%</td><td>1.07</td></tr><tr><td>315</td><td>66.804980</td><td>0.1401%</td><td>1.60</td></tr><tr><td>441</td><td>66.835929</td><td>0.0938%</td><td>1.19</td></tr><tr><td>588</td><td>66.854478</td><td>0.0661%</td><td>1.22</td></tr><tr><td>756</td><td>66.865210</td><td>0.0500%</td><td>1.11</td></tr><tr><td>945</td><td>66.872080</td><td>0.0397%</td><td>1.03</td></tr></tbody></table>

<table><caption>Table 5: L-shaped domain, *hp*-version</caption><thead><tr><th>DOF</th><th>energy</th><th>relative error</th><th>β</th></tr></thead><tbody><tr><td>64</td><td>66.584786</td><td>0.4692%</td><td></td></tr><tr><td>134</td><td>66.831690</td><td>0.1001%</td><td>2.09</td></tr><tr><td>239</td><td>66.886361</td><td>0.0184%</td><td>2.93</td></tr><tr><td>386</td><td>66.896148</td><td>0.0038%</td><td>3.31</td></tr><tr><td>582</td><td>66.897953</td><td>0.0011%</td><td>3.06</td></tr><tr><td>834</td><td>66.898392</td><td>0.0004%</td><td>2.62</td></tr></tbody></table>

<table><caption>Table 6: L-shaped domain, enriched *p*-version</caption><thead><tr><th>DOF</th><th>energy</th><th>relative error</th><th>β</th></tr></thead><tbody><tr><td>22</td><td>65.602828</td><td>1.9370%</td><td></td></tr><tr><td>64</td><td>66.720910</td><td>0.2657%</td><td>1.86</td></tr><tr><td>127</td><td>66.878978</td><td>0.0294%</td><td>3.21</td></tr><tr><td>211</td><td>66.897338</td><td>0.0020%</td><td>5.30</td></tr><tr><td>316</td><td>66.898308</td><td>0.0005%</td><td>3.22</td></tr><tr><td>442</td><td>66.898586</td><td>0.0001%</td><td>4.30</td></tr><tr><td>589</td><td>66.898646</td><td>0.0000%</td><td>4.17</td></tr></tbody></table>
# 4 Conclusion

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

# References

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