## Dispersion error reduction for acoustic problems using the edge-based smoothed finite element method (ES-FEM).(English)Zbl 1235.76065

Summary: The paper reports a detailed analysis on the numerical dispersion error in solving 2D acoustic problems governed by the Helmholtz equation using the edge-based smoothed finite element method (ES-FEM), in comparison with the standard FEM. It is found that the dispersion error of the standard FEM for solving acoustic problems is essentially caused by the ‘overly stiff’ feature of the discrete model. In such an ‘overly stiff’ FEM model, the wave propagates with an artificially higher ‘numerical’ speed, and hence the numerical wave-number becomes significantly smaller than the actual exact one. Owing to the proper softening effects provided naturally by the edge-based gradient smoothing operations, the ES-FEM model, however, behaves much softer than the standard FEM model, leading to the so-called very ‘close-to-exact’ stiffness. Therefore the ES-FEM can naturally and effectively reduce the dispersion error in the numerical solution in solving acoustic problems. Results of both theoretical and numerical studies will support these important findings. It is shown clearly that the ES-FEM suits ideally well for solving acoustic problems governed by the Helmholtz equations, because of the crucial effectiveness in reducing the dispersion error in the discrete numerical model.

### MSC:

 76M10 Finite element methods applied to problems in fluid mechanics 76Q05 Hydro- and aero-acoustics
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### References:

 [1] Deraemaeker, Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimension, International Journal for Numerical Methods in Engineering 46 pp 471– (1999) · Zbl 0957.65098 · doi:10.1002/(SICI)1097-0207(19991010)46:4<471::AID-NME684>3.0.CO;2-6 [2] Harari, Galerkin/least-squares finite element methods for the reduced wave equation with nonreflecting boundary conditions in unbounded domains, Computer Methods in Applied Mechanics and Engineering 98 (3) pp 411– (1992) · Zbl 0762.76053 · doi:10.1016/0045-7825(92)90006-6 [3] Thompson, A Galerkin least-squares finite element method for the two-dimensional Helmholtz equation, International Journal for Numerical Methods in Engineering 38 pp 371– (1995) · Zbl 0844.76060 · doi:10.1002/nme.1620380303 [4] Harari, Reducing dispersion of linear triangular elements for the Helmholtz equation, Journal of Engineering Mechanics 128 pp 351– (2002) · doi:10.1061/(ASCE)0733-9399(2002)128:3(351) [5] Babuška, A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution, Computer Methods in Applied Mechanics and Engineering 128 (3-4) pp 325– (1995) · Zbl 0863.73055 · doi:10.1016/0045-7825(95)00890-X [6] Franca, Residual-free bubbles for the Helmholtz equation, International Journal for Numerical Methods in Engineering 40 pp 4003– (1997) · Zbl 0897.73062 · doi:10.1002/(SICI)1097-0207(19971115)40:21<4003::AID-NME199>3.0.CO;2-Z [7] Melenk, The partition of unity finite element method: basic theory and applications, Computer Methods in Applied Mechanics and Engineering 139 (1-4) pp 289– (1996) · Zbl 0881.65099 · doi:10.1016/S0045-7825(96)01087-0 [8] Cessenat, Application of an ultra weak variational formulation of elliptic PDES to the two-dimensional Helmholtz problem, SIAM Journal on Numerical Analysis 35 (1) pp 255– (1998) · Zbl 0955.65081 · doi:10.1137/S0036142995285873 [9] Farhat, The discontinuous enrichment method, Computer Methods in Applied Mechanics and Engineering 190 pp 6455– (2001) · Zbl 1002.76065 · doi:10.1016/S0045-7825(01)00232-8 [10] Belytschko, Element-free Galerkin methods, International Journal for Numerical Methods in Engineering 37 pp 229– (1994) · Zbl 0796.73077 · doi:10.1002/nme.1620370205 [11] Wang, On the optimal shape parameters of radial basis functions used for 2-D meshless methods, Computer Methods in Applied Mechanics and Engineering 191 (23-24) pp 2611– (2002) · Zbl 1065.74074 · doi:10.1016/S0045-7825(01)00419-4 [12] Wang, A point interpolation meshless method based on radial basis functions, International Journal for Numerical Methods in Engineering 54 (11) pp 1623– (2002) · Zbl 1098.74741 · doi:10.1002/nme.489 [13] Liu, Meshfree Methods: Moving Beyond the Finite Element Method (2009) · Zbl 1205.74003 · doi:10.1201/9781420082104 [14] Bouillard, Element-free Galerkin solutions for Helmholtz problems: formulation and numerical assessment of the pollution effect, Computer Methods in Applied Mechanics and Engineering 162 (1) pp 317– (1998) · Zbl 0944.76033 · doi:10.1016/S0045-7825(97)00350-2 [15] Suleau, Dispersion and pollution of meshless solution for the Helmholtz equation, Computer Methods in Applied Mechanics and Engineering 190 pp 639– (2000) · Zbl 1006.76072 · doi:10.1016/S0045-7825(99)00430-2 [16] Wenterodt, Dispersion analysis of the meshfree radial point interpolation method for the Helmholtz equation, International Journal for Numerical Methods in Engineering 77 pp 1670– (2009) · Zbl 1158.76415 · doi:10.1002/nme.2463 [17] Petersen, Assessment of finite and spectral element shape functions or efficient iterative simulations of interior acoustics, Computer Methods in Applied Mechanics and Engineering 195 pp 6463– (2006) · Zbl 1119.76043 · doi:10.1016/j.cma.2006.01.008 [18] Biermann, Higher order finite and infinite elements for the solution of Helmholtz problems, Computer Methods in Applied Mechanics and Engineering 198 pp 1171– (2009) · Zbl 1157.65482 · doi:10.1016/j.cma.2008.11.009 [19] Liu, A generalized gradient smoothing technique and the smoothed bilinear form for Galerkin formulation of wide class of computational methods, International Journal of Computational Methods 5 pp 199– (2008) · Zbl 1222.74044 · doi:10.1142/S0219876208001510 [20] Liu, On a G space theory, International Journal of Computational Methods 6 (2) pp 257– (2009) · Zbl 1264.74266 · doi:10.1142/S0219876209001863 [21] Liu, A G space theory and a weakened weak (W 2) form for a unified formulation of compatible and incompatible methods: part I: theory. Applications to solid mechanics problems, International Journal for Numerical Methods in Engineering 81 pp 1093– (2010) [22] Liu, A G space theory and a weakened weak (W 2) form for a unified formulation of compatible and incompatible methods: part II. Applications to solid mechanics problems, International Journal for Numerical Methods in Engineering 81 pp 1127– (2010) [23] Liu, A linearly conforming point interpolation method (LC-PIM) for 2D solid mechanics problems, International Journal of Computational Methods 2 (4) pp 645– (2005) · Zbl 1137.74303 · doi:10.1142/S0219876205000661 [24] Zhang, A linearly conforming point interpolation method (LC-PIM) for three-dimensional elasticity problems, International Journal for Numerical Methods in Engineering 72 (113) pp 1524– (2007) · Zbl 1194.74543 · doi:10.1002/nme.2050 [25] Liu, A node-based smoothed finite element method for upper bound solution to solid problems, Computers and Structures 87 pp 14– (2009) · doi:10.1016/j.compstruc.2008.09.003 [26] Liu, Upper bound solution to elasticity problems: a unique property of the linearly conforming point interpolation method (LC-PIM), International Journal for Numerical Methods in Engineering 74 pp 1128– (2008) · Zbl 1158.74532 · doi:10.1002/nme.2204 [27] Liu, An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analysis, Journal of Sound and Vibration 320 pp 1100– (2009) · doi:10.1016/j.jsv.2008.08.027 [28] Liu, Smoothed Finite Element Methods (2010) · doi:10.1201/EBK1439820278 [29] He, An edge-based smoothed finite element method (ES-FEM) for analyzing three-dimensional acoustic problems, Computer Methods in Applied Mechanics and Engineering 199 pp 20– (2009) · Zbl 1231.76147 · doi:10.1016/j.cma.2009.09.014 [30] Chen, Assessment of smoothed point interpolation methods for elastic mechanics, Communications in Numerical Methods in Engineering (2009) [31] Irimie, A residual a posteriori error estimator for the finite element solution of the Helmholtz equation, Computer Methods in Applied Mechanics and Engineering 190 pp 2027– (2001) · Zbl 0987.76051 · doi:10.1016/S0045-7825(00)00314-5 [32] Ihlenburg, Finite element solution of the Helmholtz equation with high wave-number, part I: the h-version of the FEM, Computers and Mathematics with Applications 30 (9) pp 9– (1995) · Zbl 0838.65108 · doi:10.1016/0898-1221(95)00144-N [33] Ihlenburg, Reliability of finite element methods for the numerical computation of waves, Advances in Engineering Software 28 pp 417– (1997) · doi:10.1016/S0965-9978(97)00007-0 [34] Ihlenburg, Finite element solution of the Helmholtz equation with high wave number, part II: the h-p version of the FEM, SIAM Journal on Numerical Analysis 34 (1) pp 315– (1997) · Zbl 0884.65104 · doi:10.1137/S0036142994272337
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