3.2 Reduced basis methods in vibroacoustics
3.3.2 Geometry variations of the violin bridge
We apply the approximation introduced above to geometry variations of the violin bridge, see Figure 3.3 for the multi-patch isogeometric discretization.
The bridge of a violin is a very convenient example for geometry variations, as there exists a large variety of shapes that are used in practice. In addition to the previously considered three-dimensional model, we also consider a two-dimensional model. Four boundary modifications are defined manually and
d1 d2
d3 d4
Figure 3.10: Four two-dimensional displacement modes.
then extended into the interior as described above. First, we compute the surrogate model based on the affine-decomposition using adjDF paired with EIM for the (detDF)−1-term, on each patch individually. Then a POD is used to construct the reduced basis for the approximation of the firstK eigenvalues.
We start with a two-dimensional model of the violin bridge and consider the four predefined deformation modes presented in Figure 3.10. Sample eigen-modes are shown in Figure 3.11 for the reference domain and a domain with maximal geometry change, Ω(µ) with µ = (1,1,1,1). We see a large influ-ence of the geometry parameter on the eigenvalues of interest. Using empirical interpolation with a stopping tolerance of 10−4 resulted in a total of 1 155 parameter-independent parts of the stiffness matrix. Based on this decom-position, we perform a POD-based reduced basis model order reduction. For different numbers of eigenvalues, the convergence is shown in Figure 3.12, which is as promising as in the case of only a few affine pars. Again a large growth in the error for a given basis size is observed, when we increaseK, the number of eigenvalues of interest.
For the three-dimensional setting, we extend the deformation modes in the
Figure 3.11: Comparison of two-dimensional eigenfunctions. From left to right: Second, third and sixth eigenvalue. Top: Undeformed. Bottom: Maximal deformation.
50 150 250 350
Figure 3.12: Model order reduction for the two-dimensional case. Top: Eigenvalue conver-gence. Bottom: Eigenfunction converconver-gence. Left to right: K= 5, 10, 15, 50.
Figure 3.13: Some possible geometries. From left to right: Reference domain, maximal deformation, random deformation.
thickness direction, taking care of the non-constant thickness. See Figure 3.13 for an illustration of the geometry variations. Empirical interpolation leads to 10 010 affine matrices. This large amount is due to the curse of dimen-sions as the affine parts for different components multiply. In Figure 3.14 the convergence of a POD-based reduced basis method is shown. As in the two-dimensional setting, we observe a fast convergence of the eigenvalues and eigenfunctions even for such a challenging problem.
50 150 300
10-10 10-8 10-6 10-4 10-2 100
EV1 EV2 EV3 EV4 EV5
50 150 300
10-6 10-4 10-2 100
EF1 EF2 EF3 EF4 EF5
Figure 3.14: Model order reduction for the three-dimensional case withK= 5. Left: Eigen-value convergence. Right: Eigenfunction convergence.
In summary, we have seen the possibility to apply reduced basis meth-ods to geometry variations combined with the empirical interpolation method even for linear elasticity. The amount of affine matrices limits the success-ful use of this model reduction to situations where the geometry handling, integration or the eigenvalue solver are the bottleneck of the detailed sys-tem. The two-dimensional example was significantly more efficient than the three-dimensional problem as it is less affected by the curse of dimensional-ity. More sophisticated methods to generate a decomposable approximation of the stiffness and mass matrix, e.g., the multi-component empirical interpo-lation method introduced in [213, Chapter 4.3.2], could significantly improve the efficiency especially for three-dimensional problems.
for Signorini-type problems
In this chapter, we consider the Poisson equation with unilateral Signorini boundary conditions and provide optimal order convergence rates in norms associated with the Signorini boundary ΓS. More precisely, we consider a pri-ori error estimates for the trace in the H001/2(ΓS) norm and for the Lagrange multiplier, i.e., the flux, in the H−1/2(ΓS) norm. As a corollary we show im-proved a priori estimates in theL2 norm for the primal variable on Ω and for the dual variable on ΓS. While convergence rates for traces can often be es-tablished using estimates in the domain, these rates are typically not optimal.
The order of the finite element approximation of variational inequalities in the L2(Ω) norm is firstly addressed in the early paper [163]. However, the theo-retical results are limited to very special situations. A generalization can be found in [54, 205], but for a straightforward application to Signorini problems, the required dual regularity is lacking, so we do not follow these ideas. Re-cently introduced techniques allow optimal estimates on interfaces and bound-aries for linear problems under moderately stronger regularity assumptions, see [5, 6, 157, 158, 222]. These techniques can also be used to compensate a lack of regularity in the dual problem, see [110]. A reformulation of the pri-mal variational inequality on the boundary, as applied in [73, 200, 203], and a Strang lemma for variational inequalities allow us to use these techniques for the non-linear Signorini problem.
Results of this section have been published by the author in collaboration with O. Steinbach and B. Wohlmuth in the article “Trace and flux a priori error estimates in finite-element approximations of Signorini-type problems”
in the year 2016, [204].
4.1 Optimal a priori estimates
In this section the problem setting and main result are stated. We state the discretization of the Signorini-type problem as a primal formulation. Two re-formulations which play an important part in the analysis are briefly recalled:
a saddle point problem and a variational formulation of the Schur complement.