1. Computational Solutions of Helmholtz Equation Yau Shu Wong Department of Mathematical & Statistical Sciences University of Alberta Edmonton, Alberta, Canada

2. MD-11

4. Noise radiation from NACA 0012 airfoil at zero incidence

5. Sound radiation from a jet pipe Near field pressure at 1623 Hz at 0.45 Mach number

6. Wave Equation: where For a time - periodic solution, Helmholtz Equation: orin homogeneous media

7. K denotes the wave number, it is related to the frequency of the wave propagation The wavelength The number of points per wavelength

8. Discretization Boundary condition Solution of the resulting system of equations Computational methods for Helmholtz Equation Problem: high frequency, large wave number and short wave

9. Finite-difference method: Exact solution Hence, Discretization

10. Sixth-order Compact Finite Difference: Recall

11. New Finite Differene – Lambe, Luczak & Nehrbass Taylor’s expansion: Recall:

12. 2D Helmholtz equation: Finite Difference: New Finite Difference:

13. Boundary condition: Radiation condition Absorption condition Damping Layer Perfectly Matched Layer

14. Noise radiation in turbofan

15. Radiation boundary condition: Finite Difference: New Finite Difference: Uniqueness of numerical solutions using new finite difference schemes for the 1D and 2D problems.

16. Solution of system of linear equations: The matrix A is usually large, non-diagonally dominant, indefinite and with complex coefficients The system is very ill-conditioned Solution methods: Direct: Gaussian elimination Iterative: Conjugate Gradient, bi-CG, CGS, preconditioning, multi-grid, GMRES Direct: Iterative:

17. Generalized Minimum Residual (GMRES) is a Krylov subspace method, compute an approximate solution from the space GMRES algorithm ensures that the residual has a minimal 2-norm.

18. GMRES algorithm Matrix-vector Full GMRES GMRES(m)

19. m=5 m=1 m=50 m=100

20. K=20,kh=0.625 ResidualError

21. Numerical Simulations: Elliptic BVP:

22. Numerical wave number: 1D Helmholtz equation

23. 1D Problem, h = 0.01

24. 1D Problem, k = 50

27. 2D Helmholtz equation Exact solution Let

32. Concluding Remarks: Solving Helmholtz equation numerically is a difficult task for large wave numbers The effects on discretization, boundary conditions and linear solver are reported For 1D problem, exact solution can be computed For 2D and 3D problems, more works and new ideas are needed

33. Thank You!