A difficulty of the standard Galerkin finite element method has been the ability to accurately resolve oscillating wave solutions at higher frequencies. Many alternative methods have been developed including high-order methods, stabilized Galerkin methods, multi-scale variational methods, and other wave-based discretization methods. In this work, consistent residuals, both in the form of least-squares and gradient least-squares are linearly combined and added to the Galerkin variational Helmholtz equation to form a new generalized Galerkin least-squares method (GGLS). By allowing the stabilization parameters to vary spatially within each element, we are able to select optimal parameters which reduce dispersion error for all wave directions from second-order to fourth-order in nondimensional wavenumber; a substantial improvement over standard Galerkin elements. Furthermore, the stabilization parameters are frequency independent, and thus can be used for both time-harmonic solutions to the Helmholtz equation as well as direct time-integration of the wave equation, and eigenfrequency/eigenmodes analysis. Since the variational framework preserves consistency, high-order accuracy is maintained in the presence of source terms. In the case of homogeneous source terms, we show that our consistent variational framework is equivalent to integrating the underlying stiffness and mass matrices with optimally selected numerical quadrature rules. Optimal GGLS stabilization parameters and equivalent quadrature rules are determined for several element types including: bilinear quadrilateral, linear triangle, and linear tetrahedral elements. Numerical examples on unstructured meshes validate the expected high-order accuracy.

1.
Harari
I.
, and
Hughes
T.
,
1992
. “
Galerkin/least-squares finite element methods for the reduced wave equation with non-reflecting boundary conditions in unbounded domains
.”
Comput. Methods Appl., Mech. Engrg.
,
98
[], pp.
411
454
.
2.
Harari
I.
, and
Nogueira
C.
,
2002
. “
Reducing dispersion of linear triangular elements for the Helmholtz equation
.”
J. of Engineering Mechanics
,
128
(
3
) [], pp.
351
358
.
3.
Thompson
L.
, and
Pinsky
P.
,
1995
. “
A Galerkin least squares finite element method for the two-dimensional Helmholtz equation
.”
Int. J. Num. Meth. Eng.
,
38
[], pp.
371
397
.
4.
Harari
I.
,
Grosh
K.
,
Hughes
T.
,
Malhotra
M.
,
Pinsky
P.
,
Stewart
J.
, and
Thompson
L.
,
1996
. “
Recent developments in finite element methods for structural acoustics
.”
Archives of Computational Methods in Engineering
,
3
[], pp.
132
311
.
5.
Oberai
A.
, and
Pinsky
P.
,
2000
. “
A residual-based finite element method for the Helmholtz equation
.”
Int. J. Num. Meth. Eng.
,
49
[], pp.
399
419
.
6.
Thompson
L.
, and
Sankar
S.
,
2001
. “
Dispersion analysis of stabilized finite element methods for acoustic fluid interaction with Reissner-Mindlin plates
.”
Int. J. Num. Meth. Eng.
,
50
(
11
) [], pp.
2521
2545
.
7.
Thompson
L.
, and
Thangavelu
S.
,
2002
. “
A stabilized MITC element for accurate wave response in Reissner-Mindlin plates
.”
Comp. & Struct.
,
80
(
9–10
) [], pp.
769
789
.
8.
Thompson
L.
,
2003
. “
On optimal stabilized MITC4 plate bending elements for accurate frequency response analysis
.”
Computers & Structures
,
81
[], pp.
995
1008
.
9.
Hughes
T.
,
Feijoo
G.
,
Mazzei
L.
, and
Quincy
J.-B.
,
1998
. “
The variational multiscale method - a paradigm for computational mechanics
.”
Comput. Methods Appl. Mech. Engrg.
,
166
[], pp.
3
24
.
10.
Franca
L.
,
Farhat
C.
,
Macedo
A.
, and
Lesoinne
M.
,
1997
. “
Residual-free bubbles for the Helmholtz equation
.”
Int. J. Num. Meth. Eng.
,
40
(
21
) [], pp.
4003
4009
.
11.
Barbone
P.
, and
Harari
I.
,
2001
. “
Nearly H1-optimal finite element methods
.”
Comput. Methods Appl. Mech. Engrg.
,
190
[], pp.
5679
5690
.
12.
Harari, I., 2004. “Acoustics.” In Finite Element Methods: 1970’s and Beyond, L. Franca, T. Tezduyar, and A. Masud, Eds. CIMNE, Barcelona.
13.
Thompson, L., and Pinsky, P., 2004. “Acoustics.” In Encyclopedia of Computational Mechanics, E. Stein, R. D. Borst, and T. J. Hughes, Eds., vol. 2. Wiley InterScience, ch. 22.
14.
Franca
L.
, and
Dutra do Carmo
E.
,
1989
. “
The Galerkin gradient least-squares method
.”
Computer Methods in Applied Mechanics and Engineering
,
74
[], pp.
41
54
.
15.
Harari
I.
, and
Hughes
T.
,
1994
. “
Stabilized finite element methods for steady advection-diffusion with production
.”
Comput. Methods Appl. Mech. Engrg.
,
114
[], pp.
165
191
.
16.
Challa, S., 1998. High-order accurate spectral elements for wave propagation problems. Master’s thesis, Clemson University, Mechanical Engineering, August.
17.
Guddati
M.
, and
Yue
B.
,
2004
. “
Modified integration rules for reducing dispersion error in finite element methods
.”
Comput. Methods Appl. Mech. Engrg.
,
193
(
3–5
) [], pp.
275
287
.
18.
Thompson
L.
, and
Pinsky
P.
,
1994
. “
Complex wavenumber Fourier analysis of the p-version finite element method
.”
Computational Mechanics
,
13
(
4
) [], pp.
255
275
.
19.
Ihlenburg
F.
, and
Babuska
I.
,
1995
. “
Dispersion analysis, and error estimation of Galerkin finite element methods for the Helmholtz equation
.”
Int. J. Num. Meth. Eng.
,
38
[], pp.
3745
3774
.
20.
Ihlenburg, E., 1998. Finite Element Analysis of Acoustic Scattering. Springer-Verlag.
21.
Bayliss
A.
,
Goldstein
C.
, and
Turkel
E.
,
1985
. “
On accuracy conditions for the numerical computation of waves
.”
J. Comput. Phys.
,
59
[], pp.
396
404
.
22.
Ihlenburg
F.
,
2003
. “
The medium-frequency range in computational acoustics: practical and numerical aspects
.”
J. Comput. Acoustics
,
11
(
2
) [], pp.
175
194
.
23.
Harari
I.
, and
Magoules
F.
,
2004
. “
Numerical investigations of stabilized finite element computations for acoustics
.”
Wave Motion
,
39
(
4
) [], pp.
339
349
.
This content is only available via PDF.
You do not currently have access to this content.