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Natural Science Forum / Physics / Acoustics / February 2008treatment of harmonic point source in FEM
hi i am solving Helmholtz equation for a rectangular waveguide using Finite Element Method with a Fortran code. Governing equation is del ^ 2 (p) + k^2 (p) = 0 Left (vertical ) boundary of the domain carries the source. Right ( vertical ) boundary is the absorbing boundary with Sommerfield Radiation Condition. Top boundary is the pressure release. Bottom is rigid I have verified the computer code by imposing a sine function at the left boundary. This becomes an oscillating cylinder problem. I could recover the mode shape in the near as well as far field. The numerical error associated with this exercise was < 2 % (quadratic element). Now I want to use a harmonic point source. How do I incorporate a point source ( Hankel function H (k,0 ) becomes singular at r = 0 ) in the finite element mesh. Scooping out a small area (fraction of wavelength )around the source, and then imposing unbounded pressure values at nodes surrounding the source would lead to a problem same as stress concentration. What can I do to avoid such a thing? The pressure thus obtained differ from the theory badly. Has anybody come across such a problem ? > Now I want to use a harmonic point source. How do I incorporate a > point source ( Hankel function H (k,0 ) becomes singular at r = 0 ) [quoted text clipped - 4 lines] > pressure thus obtained differ from the theory badly. Has anybody come > across such a problem ? The "stress" may be real. Cetainly we know that for tiny sources, compressibility and nonlinearity abound. So you need to scale the boundary surface (size and shape) and pressure amplitudes, and frequency range accurately to achieve a meaningful result. Angelo Campanella. > So you need to scale the boundary surface (size and shape) and pressure > amplitudes, and frequency range accurately to achieve a meaningful result. hi what did you mean by 'scale the boundary surface' ? could you please elaborate ? did you mean refining mesh near the boundary and the source ? thank you. -ganesh diwan. >> So you need to scale the boundary surface (size and shape) and pressure >>amplitudes, and frequency range accurately to achieve a meaningful result. > what did you mean by 'scale the boundary surface' ? > could you please elaborate ? > did you mean refining mesh near the boundary and the source ? In the vicinity of the source, as the size of the source gets smaller, there is a tendency for the mesh to not represent the true shape of the boundary unless it is refined as you say. But also, the actual pressure, including the acoustic pressure becomes very high, leading to compression/rarefaction nonlinearities. In the least, these lead to harmonic generation, not represented in your original DE. The extreme example of this nonnearity is gas flow through an orifice. for very high pressure at one side of the orifice, the speed of the gas through that orifice canot exceed the speed of sound, regardless of that pressure. So, along with a refined mesh, if you want to see the harmonic generation, you must add another tern to the DE. I am not familiar with the detals of such; I only know that this is a problem in small sources. In addition some wavelength effects are to be considered. If the source surface is comparable to or larger than a wavelength of the sound you are considering, then the shape of the surface is important. If the source surface is much smaller than the wavelength of sound you are considereing AND the wavelength of all the harmonics you may consider, then the surface shape is unimportant. BUT whatever surfaces you use must represent the physics of the gas flow that really occurs. I.e., you almost need to know the answer before you can solve the problem! The ultimate result will be the product of modeling iteration. Build a model. Make a real-world test. Compare predicted and actual results. Rebuild the model. Repeat test and comparison. Continue until ou are either satisfied, or run out of time, or run out of funds, or the problem becomes no longer important... Sincerely, Angelo Campanella |
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