| Author | Message | |||
     Mehdi Naderi (mnader4) Member Username: mnader4 Post Number: 43 Registered: 12-2006 |
Dear Mr. Nelson, I will apprecite if you help me in below question. We working together on fluctuating beam (interface problem discussion). Accroding to Grasho: p=-penalty div(u) I check that formula and as you said in the other topics, it is very harsh. You are right. Also, in Smoothing Operators in PDE's there is a method to smooth a pde solution. so, the pressure equation: dt(p)-eps*Div(grad(p))=-penalty*div(U), in my problem (time dependent), AS you corrected is Div(grad(p))=penalty*div(U) We eliminate dt(p). Can it be true in time dependent problem? (I checked timedependent sample in fpde, you did the same way) | |||
| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 1043 Registered: 06-2003 |
A term dt(p) in the pressure equation means that a change in the source terms will not immediately appear in the pressure, but that there will be a response delay. Ignoring the Div(grad(p)) term and dividing by penalty shows that (1/penalty) is related to the delay time. [dP/d(t*penalty) = div(u) ] Ignoring the dt(p) term merely means that changes in div(u) are felt immediately in the pressure term, and therefore in the velocities. This is a desirable effect, since you don't want the velocity errors to continue compressing a cell for some delay time after the error is detected. | |||
| Mehdi Naderi (mnader4) Member Username: mnader4 Post Number: 44 Registered: 12-2006 |
Dear Mr. Nelson, I really apparecite your help. I am still confused about what's happening in "Smoothing Operators in PDE" in a phsical sense (I read fpde help and ...). I was wondering if you let me know a refference which helps me understand it better. | |||
| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 1046 Registered: 06-2003 |
I notice that the fonts in our Tech Note "Smoothing Operators in PDEs" are garbled on some platforms. For completeness, I have herewith attached a PDF of this note. I had hoped that the derivation given in the note would be sufficient to establish the validity of the process. I do not at this time know of an independent source of this information. Physically speaking, the del2 operator acts as a low-pass filter on the spatial frequencies of the solution. Rapid oscillations are damped in such a way that oscillations with spatial wavelength D are damped by a factor of 2. Higher spatial frequencies are damped by increasingly larger factors. The rigorous application of the principles stated in McGillem and Cooper (quoted in the Note) require uniform sample spacing, and this is of course not true in a FlexPDE model. Nevertheless, the general effect of the del2 operator is (in the instance of the pressure equation) to smear a sharp pressure spike over a distance of order D. A forest of pressure spikes caused by numerical uncertainties in the velocity solution, and feeding back as oscillations of the velocity field, are converted to a more general pressure field that damps local oscillations but imposes larger scale motions consistent with mass conservation.
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