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Acoustic simulations with higher order finite and infinite elements
Otto von Estorff*,1
, Steffen Petersen**,2
, and Jan Biermann3
1,3Institute of Modelling and Computation, Hamburg University of Technology, Denickestraße 17, 21073 Hamburrg,
Germany. 2 Department of Mechanical Engineering, Stanford University, Mail Code 3035, Stanford, CA 94305, USA.
The efficiency of finite element based simulations of Helmholtz problems is primarily affected by two facts. First, the
numerical solution suffers from the so-called pollution effect, which leads to very high element resolutions at higher
frequencies. Furthermore, the spectral properties of the resulting system matrices, and hence the convergence of iterative
solvers, deteriorate with increasing wave numbers. In this contribution the influence of different types of polynomial basis
functions on the efficiency and stability of interior as well as exterior acoustic simulations is analyzed. The current
investigations show that a proper choice for the polynomial shape approximation may significantly increase the performance
of Krylov subspace methods. In particular, the efficiency of higher order finite and infinite elements based on Bernstein
polynomial shape approximation and the corresponding iterative solution strategies is assessed for practically relevant numerical examples including the sound radiation from rolling vehicle tires.
1 Introduction
The need for the simulation of acoustical phenomena is constantly increasing in order to enable cost efficient acoustical
optimization of systems. Using the automobile as an example, this includes the solution of interior problems, like a car
compartement (for the sake of increasing passenger comfort), as well as of exterior problems, like tire simulations (in order
to fullfill legal requirements). The Finite Element Method is an appropriate instrument for the solution of this kind of
problems but it suffers from two contrary effects that will be outlined briefly as follows. The application of high order
elements is preferable in order to reduce the so-called dispersion error that can be quantified in the H1-norm by the wave
number k, the polynomial oder p, the characteristic element length h and two cconstants C1 and C2 [1]:
pp
p
khkC
p
khCe
2
211
22+ . (1)
It can be seen that at high wave numbers the second term dominates the error. This term can be reduced more effecticely by
increasing the polynomial order p rather than decreasing the element size h. But those high order elements leed to
undesirable spectral properties of the system matrix A which is a result of the discretization of of the weak formulation of the
Helmholtz equation (with p being the pressure vector, f the excitation vector with normal surface velocities, K, C, M being
the stiffness, damping and mass matrix respectively, is the fluid density and the angular velocity):
[ ] fpMCKpA ikikk =+= 2:)( . (2)
These spectral properties negatively influence the convergence properties of Krylov subspace based iterative solvers.
However, iterative solvers must be used rather than direct solvers because of the considerable size of the matrices to be
solved, which is depending on the size of the computational domain and the large frequency ranges to be considered.
Consequently it is desirable to choose a trial/test function space that allows for high polynomial orders without deteriorating
the spectral properties of the system matrix.
For the acoustical p-FEM hierarchic elements based on integrated Legendre polynomials however [2] are widely used. They
aim for the othogonalization of the operator K in equation (2) but with increasing wave number k the mass matrix M gains of
influence (neglecting acoustical damping, C= 0) and the positive properties of the elements degrade. That is why elements
based on Bernstein polynomials are proposed here. They preserve the spectral properties that are required for the Krylov
subspace based iterative solvers over a wider range in frequency and polynomial order [3].
2 Numerical Investigations
2.1 Interior Acoustical Simulations
In order to demonstrate the superior properties of the Bernstein Finite Elements the results of a small academic 2D-example
will be shown. A L-shape domain filled with air and discretized with a regular grid of triangular elements (Fig.1 ) is excited
at the bottom with a unit velocity of varying frequency f. A TFQMR solver with a prescribed residual for convergence of ____________________
* Corresponding author: e-mail [email protected], Phone: +0049 4042878 3032, Fax: +0049 4042878 4353 ** e-mail [email protected]
© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
PAMM · Proc. Appl. Math. Mech. 7, 4120013–4120014 (2007) / DOI 10.1002/pamm.200700947
6101r has been used. The solution was considered to diverge if the convergence criteria have not been met after 10,000
iterations. Finally, the simulations were conducted for 34 frequencies between 25Hz and 850Hz using polynomial orders
between two and six and leaving the number of degrees of freedom constant at 1825.
Fig. 1 Computational domain with mesh and pressure amplitudes for 750Hz (left), number of iterations vs. frequency for
varying polynomial orders for elements based on integrated Legendre polynomials (middle) and Bernstein polynomials (right)
It can be seen that the elements based on integrated Legendre polynomials yield efficient solutions for low frequencies but
the stability limit is lower compared to the Bernstein elements. This difference becomes larger for increasing polynomial
orders (Fig. 1). The elements have also been tested for realistic 3D applications like a car compartment (Fig. 2 left) that has
been computed with second order elements and about 3 Mio degrees of freedom. Here, savings of computational time of a
factor of three were achieved compared to conventional Lagrange elements.
2.2 Exterior Acoustical Simulations
In addition to the improvements for the interior acoustical simulations it was tried to transfer the good properties of the
Bernstein polynomials to exterior acoustics as well. This was done by using the Bernstein polynomials for the basis
approximation of infinite elements, that have previousely been optimized by Dreyer [4] for the radial approximation. It was
observed that this modification of the element basis lead to further reduction of computation time. An application is the
simulation of rolling tires (Fig. 2) that leads to models with about 0.5 Mio. degrees of freedom for second order
approximation. For the solution of this problem with a BICGSTAB solver together with an iLU preconditioner, savings in
computation time of a factor between 1.5 and 2 were achieved (Fig. 2). The total benefit will be emphasised by considering
that for realistic simulations the system of equations has to be solved several hundred times.
Fig. 2 Computational domain and mesh for a car compartment (left), tire geometry with parts of the adjacent
acoustical elements (middle), comparison of the computation time of the tire model with Lagrange and Bernstein
FE/IFE
3 Conclusion
It can be concluded, that the use of Bernstein polynomials for finite/infinite higher order elements significantly enhances the
performance of interior/exterior acoustical simulations in conjunction with the application of Krylov subspace based iterative
solvers. Beside this speedup, an improved robustness of the solution procedure can be observed.
References
[1] F. Ihlenburg, B. Babuska, Finite Element Solution of the Helmholtz equation with high wave number part II: The h-p version of the
FEM, SIAM Journal onNumerical Analysis, 34(1), 315-358 (1997).
[2] B.A. Szabo, B. Babuska, Finite Element Analysis (Wiley, Chichester, 1991).
[3] S. Petersen, Adaptive finite und infinite Elementemethoden in der Akustik (PhD Thesis, TU Hamburg-Harburg, Shaker Verl. 2007).
[4] D. Dreyer, Efficient infinite elements for exterior acoustics (PhD Thesis, TU Hamburg-Harburg, Shaker Verlag 2004).
© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
GAMM Sections 4120014
