23rd Nordic Seminar on Computational Mechanics NSCM-23 A. Eriksson and G. Tibert (Eds)
c
KTH, Stockholm, 2010
Kjetil A. Johannessen and Trond Kvamsdal
3 Problems containing singularities
We will here present how the T-splines along with a posteriori error estimators perform on prob-lems containing singularities. We will present a model problem which we will solve using the presented method. Obviously, T-splines will outperform the homogeneous ref nement strategies, but as we shall see, even more sophisticated NURBS ref nement strategies are not comparable with T-splines due to the fact that they are limited to tensor product ref nement. For our model problem containing a singularity, we are going to solve the stationary heat equation, or Laplace equation
∇2u= f on an L-shape geometry with appropriate boundary conditions.
3.1 A posteriori error estimator
For the adaptivity we will need an estimate on what parts of the grid is contributing the most to the global error. For this purpose, we will be using a resiudal-based a posteriori error estimator. From Ainsworth3, we have
|||e|||2 ≤ C
X
K∈P
h2Kkrk2L2(K)+X
γ∈Γ
hKkRk2L2(γ)
, (1)
where R is a compact notation for the edge residual
R=
g− ∂u∂n ,h on ∂ΩN
h∂uh
∂n i
K−h
∂uh
∂n i
L, on ∂K ∩ ∂L ∀ K, L ∈ P
0, else
(2)
and r is the internal residual r = f − ∇2u. This allows us to quantify an error estimate for each element to be
ηK2 = h2Kkrk2L
2(K)+ hKkRk2L
2(∂K) (3)
For comparison purposes, we provide three adaptivity strategies.
a) Uniform ref nement using NURBS.
b) Rule of thumb ref nement, where we recursively ref ne the element closest to the singularity using NURBS.
c) Adaptive T-splines where we use the error estimator ηK to ref ne the α percent elements with the highest contribution to the error.
3.2 Results
Due to the singularity the convergence rates of the uniform ref nement is completely ruined. How-ever the rule-of-thumb ref nement keeps an optimal slope up until the point where the error from other parts of the grid than the singularity becomes dominant. The T-splines however gains an optimal convergence.
4 Smooth problems
In this section we take a look at a smooth problem, namely the static elasticity problem on an inf nite plate with a circular hole. For full details on the problem see4.
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Kjetil A. Johannessen and Trond Kvamsdal
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
T−spline spatial domain
(a) T-spline grid with adaptive solver
101 102 103 104
10−3 10−2 10−1 100
Adaptive T−splines Rule of thumb Uniform NURBS
(b) Convergence rates for the three proposed ref nement strategies
Figure 1: Results obtained from the singularity problem
0 1 2 3 4
0 0.5 1 1.5 2 2.5 3 3.5 4
(a) The parametrization control mesh
0 1 2 3 4
0 0.5 1 1.5 2 2.5 3 3.5 4
(b) The physical mapping
Figure 2: Parametrization of the hole problem
4.1 Parametrization
What is interesting here is the choice of parametrization. We will chose a particular parametriza-tion which is given in f gure2. The traditional way of creating sharp corners using NURBS is by making the knot vector be interpolating by creating a knot with multiplicity p where p is the degree of the NURBS. However we have created the upper right corner by stacking p control points on top of each other. Multiple control points is making the derivatives vanish, much in the same manner as multiple knots and is thus allowing us to create the sharp corner. This parametrization has some propterties which is rendering the particular error estimator presented earlier, useless. We will use the exact error for adaptation purposes and argue that given an appropriate error estimator, then T-splines will contain some remarkable properties, even for smooth problems without singulari-ties. The problem with our choice of parametrization for the hole-problem is that it is not uniform.
That is a uniform ref nement in the parametric domain, will result in a biased ref nement in the physical space. This is illustrated in f gure2bwhere we have drawn one such uniform ref nement.
We clearly see that the ref nements are completely biased towards the upper right corner, leaving the lower right and upper left corners with unnaturally large elements. It is however possible to bypass this by weighting the ref nement in the parametric space, but this requires hand-tailoring by the implementer, which we want to avoid as much as possible.
4.2 Refinement using T-splines
When using an adaptive T-spline ref nement strategy, this will detect the large elements that occur in the upper left and lower right corners and ref ne those as appropriate. This is a true local ref ne-ment and will not spread out to the rest of the domain, resulting in a better ref nene-ment. T-splines is
Kjetil A. Johannessen and Trond Kvamsdal
0 1 2 3 4
0 0.5 1 1.5 2 2.5 3 3.5 4
(a) Ref ned T-spline grid in the physical space
0 0.5 1 1.5 2
0 0.2 0.4 0.6 0.8 1
(b) Ref ned T-spline grid in the parametric space
101 102 103 104
10−4 10−3 10−2 10−1
T−spline refinement Uniform refinement
(c) Convergence rates of T-splines versus uniform NURBS
Figure 3: Results from solving a smooth problem
as such, negating the effect of the bad parametrization by enforcing a true uniform ref nement in the physical space.
This is shown in f gure3a where the physical T-mesh is illustrated. The parametric mesh corre-sponding to this is depicted in f gure3bwhere it is plotted in the parametric space.
4.3 Results
Since this problem has a smooth solution, we expect the uniform NURBS ref nement to be close to optimal. As was seen, however, the (parametric) uniform ref nement scheme was not uniform at all when viewed in the physical space. This is then resulting in a non-optimal ref nement scheme.
The adaptive T-splines countered this, and is thus providing a better convergence rate. These are shown in f gure3c.
5 Concluding remarks
T-splines shows great promise as a basis for adaptive FEM. Not only have they superior properties when it comes to true local ref nement around singularities, but they also have the remarkable property of negating the effect of badly parameterized models.
REFERENCES
[1] Hughes, T. J. R., Cottrell, J. A. & Bazilevs, Y. Isogeometric analysis: Cad, f -nite elements, nurbs, exact geometry and mesh ref nement. Computer Meth-ods in Applied Mechanics and Engineering 194, 4135–4195 (2005). URL http://dx.doi.org/10.1016/j.cma.2004.10.008.
[2] Sederberg, T. W., Zheng, J., Bakenov, A. & Nasri, A. T-splines and t-nurccs. In SIGGRAPH
’03: ACM SIGGRAPH 2003 Papers, 477–484 (ACM, New York, NY, USA, 2003).
[3] Ainsworth, M. & Oden, J. T. A posteriori error estimation in f nite element analysis. Computer Methods in Applied Mechanics and Engineering 142, 1 – 88 (1997).
[4] Kvamsdal, T. & Okstad, K. M. Error estimation based on superconvergent patch recovery using statically admissable stress f elds. Int. J. Numer. Meth. Engng. 42, 443–472 (1998).
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23rd Nordic Seminar on Computational Mechanics NSCM-23 A. Eriksson and G. Tibert (Eds)
⃝KTH, Stockholm, 2010c