Improvement of PUFEM for the numerical solution of high-frequency elastic wave scattering on unstructured triangular mesh grids

International Journal for Numerical Methods in Engineering

Volumn 84, Issue 3, 2010, Pages 330-350

  Kacimi, A El ^a   Laghrouche, O ^a  

^a HERIOT WATT UNIVERSITY   (United Kingdom)

Author keywords

Elastic waves; Exact integration; Plane waves; PUFEM; Unstructured triangular mesh

Indexed keywords

CLOSED FORM SOLUTIONS; COARSE MESHES; COMPUTATIONAL TIME; DISCRETIZATIONS; DISPLACEMENT FIELD; ELASTIC PLANE WAVES; ELASTIC WAVE SCATTERING; ELEMENT MASS MATRIX; ELEMENT MATRIX; EXACT INTEGRATION; FINITE ELEMENT; GLOBAL MATRIX; HIGH FREQUENCY; HIGH FREQUENCY HF; IMPROVE-A; INTEGRATION SCHEME; LOCAL COORDINATE; MATRIX; NODAL POINTS; NUMBER OF DEGREES OF FREEDOM; NUMERICAL SCHEME; NUMERICAL SOLUTION; NUMERICAL TESTS; OSCILLATORY INTEGRAL; PARTITION OF UNITY FINITE ELEMENT METHODS; PHASE VARIATION; PLANE WAVE; PUFEM; RIGID BODY; STANDARD FINITE ELEMENT; STRESS TENSORS; TIME HARMONIC; TWO-DIMENSIONAL WAVES; UNSTRUCTURED TRIANGULAR MESHES; WAVE PROBLEMS;

ASTATINE; AUTOMOBILE RADIATORS; FINITE ELEMENT METHOD; FUNCTION EVALUATION; INFORMATION DISSEMINATION; MATRIX ALGEBRA; NUMERICAL METHODS; SOIL STRUCTURE INTERACTIONS; TWO DIMENSIONAL; WAVE EQUATIONS; WAVES;

ELASTIC WAVES;

EID: 78650824050     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2901     Document Type: Article

References (43)

1. Bettess P, Laghrouche O, Perrey-Debain E (eds). Theme issue on short wave scattering. Philosophical Transactions of the Royal Society of London A 2004; 362:415-696.
2. Melenk JM, Babuška I. The partition of unity finite element method: basic theory and applications. Computer Methods in Applied Mechanics and Engineering 1996; 139:289-314.
3. Babuška I, Melenk JM. The partition of unity method. International Journal for Numerical Methods in Engineering 1997; 40:727-758.
4. Laghrouche O, Bettess P. Short wave modelling using special finite elements. Journal of Computational Acoustics 2000; 8(1):189-210.
5. Laghrouche O, Bettess P. Short wave modelling using special finite elements: towards an adaptive approach. In Mathematics of Finite Elements and Applications, Chapter 10, Whiteman JR (ed.). Elsevier: Amsterdam, 2000; 181-194.
6. Ortiz P, Sanchez E. An improved partition of unity finite element model for diffraction problems. International Journal for Numerical Methods in Engineering 2001; 50:2727-2740.
7. Laghrouche O, Bettess P, Astley RJ. Modelling of short wave diffraction problems using approximating systems of plane waves. International Journal for Numerical Methods in Engineering 2002; 54:1501-1533.
8. Laghrouche O, Bettess P, Perrey-Debain E, Trevelyan J. Plane wave basis for wave scattering in three dimensions. Communications in Numerical Methods in Engineering 2003; 19:715-723.
9. Perrey-Debain E, Laghrouche O, Bettess P, Trevelyan J. Plane wave basis finite lements and boundary elements for three dimensional wave scattering. Philosophical Transactions of the Royal Society of London A 2004; 362:561-577.
10. Astley RJ, Gamallo P. Special short wave elements for flow acoustics. Computer Methods in Applied Mechanics and Engineering 2005; 194:341-353.
11. Ortiz P. Finite elements using plane wave basis for scattering of surface water waves. Philosophical Transactions of the Royal Society of London A 2004; 362:525-540.
12. Laghrouche O, Bettess P, Perrey-Debain E, Trevelyan J. Wave interpolation finite elements for Helmholtz problems with jumps in the wave speed. Computer Methods in Applied Mechanics and Engineering 2005; 194:367-381.
13. Strouboulis T, Babuška I, Hidajat R. The generalized finite element method for Helmholtz equation: Theory, computation, and open problems. Computer Methods in Applied Mechanics and Engineering 2006; 195: 4711-4731.
14. Strouboulis T, Babuška I, Copps K. The design and analysis of the generalized finite element method. Computer Methods in Applied Mechanics and Engineering 2000; 181:43-69.
15. Strouboulis T, Zhang L, Babuška I. Assessment of the cost and accuracy of the generalized FEM. International Journal for Numerical Methods in Engineering 2007; 69:250-283.
16. Farhat C, Harari I, Franca L. The discontinuous enrichment method. Computer Methods in Applied Mechanics and Engineering 2001; 190:6455-6479.
17. Farhat C, Harari I, Hetmanuk U. A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime. Computer Methods in Applied Mechanics and Engineering 2003; 192:1389-1419.
18. Tezaur R, Zhang L, Farhat C. A discontinous enrichmment method for capturing envanescent waves in multiscale fluid and fluid/solid problems. Computer Methods in Applied Mechanics and Engineering 2008; 197:1680-1698.
19. Cessenat O, Després B. Application of an ultra weak variational formulation of elliptic PDEs to the twodimensional Helmholtz problem. SIAM Journal on Numerical Analysis 1998; 35(1):255-299.
20. Cessenat O, Després B. Using plane waves as base functions for solving time harmonic equations with the ultra weak variational formulation. Journal of Computational Acoustics 2003; 11(22):227-238.
21. Huttunen T, Monk P, Kaipio JP. Computation aspects of the ultra weak variational formulation. Journal of Computational Physics 2002; 182:27-46.
22. Monk P, Wang DQ. A least-squares method for the Helmholtz equation. Computer Methods in Applied Mechanics and Engineering 1999; 175:121-136.
23. Gillman A, Djellouli R, Amara M. A mixed hybrid formulation based on oscillated finite element polynomials for solving Helmholtz problems. Journal of Computational and Applied Mathematics 2007; 204:515-525.
24. Perrey-Debain E, Trevelyan J, Bettess P. P-wave and S-wave decomposition in boundary integral equation for plane elastodynamic problems. Communications in Numerical Methods in Engineering 2003; 19:945-958.
25. Huttunen T, Monk P, Collino F, Kaipio JP. The ultra weak variational formulation for elastic wave problems. SIAM Journal on Scientific Computing 2004; 25:1717-1742.
26. Huttunen T, Kaipio JP, Monk P. An ultra-weak method for acoustic fluid-solid interaction. Journal of Computational and Applied Mathematics 2008; 213:166-185.
27. Zhang L, Tezaur R, Farhat C. The discontinuous enrichment method for elastic wave propagation in the medium-frequency regime. International Journal for Numerical Methods in Engineering 2006; 66:2086-2114.
28. EL Kacimi A, Laghrouche O. Numerical modelling of elastic wave scattering in frequency domain by the Partition of Unity Finite Element Method. International Journal for Numerical Methods in Engineering 2009; 77:1646-1669.
29. Sugimoto R, Bettess P, Trevelyan J. A Numerical integration scheme for special quadrilateral finite elements for Helmholtz equation. Communications in Numerical Methods in Engineering 2002; 19:233-245.
30. Bettess P, Shirron J, Laghrouche O, Peseux B, Sugimoto R, Trevelyan J. A numerical integration scheme for special finite elements for Helmholtz equation. International Journal for Numerical Methods in Engineering 2003; 56:531-552.
31. Honnor ME, Trevelyan J, Bettess P, El Hachemi M, Hassan O, Morgan K, Shirron J. An integration scheme for electromagnetic scattering using plane wave edge elements. Advances in Engineering Software 2009; 40:58-65.
32. Trevelyan J, Honnor ME. A numercial coordinate transformation for efficient evaluation of oscilallory integrals over boundary element. Journal of Integral Equations and Applications 2009; 21:447-468.
33. Honnor ME, Trevelyan J, Huybrechs D. Numerical evaluation of 2D partition of unity boundary integrals for Helmholtz problems. Journal of Computational and Applied Mathematics 2007; DOI: 10.1016/j.cam.2009.08.012.
34. Huybrechs D, Vandewalle S. On the evaluation of highly oscillatory integrals by analytic continuation. SIAM Journal on Numerical Analysis 2006; 44:1026-1048.
35. Huybrechs D, Vandewalle S. A sparse discretisation for integral equation formulations of high frequency scattering problems. SIAM Journal on Scientific Computing 2007; 6:2305-2328.
36. Gordon WB. Far-field approximation to the Kirchhoff-Helmholtz representations of scattered fields. IEEE Transactions on Antennas and Propagation 1975; 23:590-592.
37. Gabard G. Exact integration of polynomial-exponential products with application to wave-based numerical methods. Communications in Numerical Methods in Engineering 2009; 25:237-246.
38. Graff F. Wave Motion in Elastic Solids. Ohio State University: Ohio, 1975.
39. Adams RA. Sobolev Spaces. Academic Press: New York, 1975.
40. Gilbarg D, Trudinger NS. Elliptic Partial Differential Equations of Second Order. Springer: Berlin, 1983.
41. Leis R. Boundary Value Problems in Mathematical Physics. Teubner, Wiley: Stuttgart, New York, 1989.
42. HSL, a collection of Fortran codes for large-scale scientific computation, 2007. See http://www.hsl.rl.ac.uk.
43. Perrey-Debain E. Plane wave decomposition in the unit disc: convergence estimates and computational aspects. Journal of Computational and Applied Mathematics 2006; 193:140-156.