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Posted:
4 years ago
9 juil. 2020, 08:38 UTC−4
Updated:
4 years ago
9 juil. 2020, 09:19 UTC−4
Hello Harsha,
This is an interesting question.
I do not have a full answer that would work out of the box in COMSOL, but I see two work-arounds. Hopefully someone with more experience can provide a better solution
- Export the volume fraction and pressure fields, then work out the problem in python
- Compute an approximated average using a Gaussian as a substitute for the Dirac distribution
I will focus on solution n°2:
The point is that you are interested in the quantity
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAYsAAAAyCAQAAAAHWz8JAAAAAmJLR0QA/4ePzL8AAAAJcEhZcwAAAHgAAAB4AJ31WmAAAAAHdElNRQfoBwwGHigrTLtwAAAKc0lEQVR42u2d732rvA7Hv7qfu4CfEXhG8BmBMwIdgTMCGYGOkIyQjJCMUEZoRigj+L6AJOaPwThQ0nP58aZUYCzZsiVZdsSwYcOGJv6zdgU2jEO05GvXwapNJHtRa9diWWxq8fIQTc772rV4wFwpOP/diiGbETUFkgMlF1PMQfH6oubIW9i7M/IdmWvjPiMyf9at06Iw2+V9kXIm4ovjHBSvLyo+SFbmOiLn3PnvkXzt9lju+u/aavmjkLGj5EIxC8XvixdzWo9h0SSUxJQd0h8+5bT2LLYY1tbLn3ORYFBzUby+GPMV/vaMnJ+7s4WBlI+1a7bUtbnc/ogpTDkbxQd7Dk+8vTDMASXZ2rVYBpta+CPhMiNlFJKiXikC1YN3sr8zIrWphSckRvX7ByEUL2SvPFcAmAMl6dq1WAKbWvgiBkcXD6GMQhIiDmszPYrTphb/39CUzdj9U5RxJBTB734fTkQSr12J+bGphS9i57gfQhmBKBJWDMz6whRcSdauxfzY1MIL4jSHQigeiCHcWf9WnH6KWoiWVGLJRY8/u6mFH77ZsyCm/CFLZQXKp6O9APZczIWC4/ijL6QWkvqIVyIZcfJkiaBhBFxno4wjDpkrXBJcNIh6oRoAFscQd15tXg0zJc4nH2U0kj8kJWoW9H2JBxKj7PGxnZx2R9ocgUVRLSkpLhQk5p0DOXOnscVgitkot7onaCIS3s2uxVM07llIahqRqrYEbZnJNbQlJUehQfaUHLptYkq5MsNsMdb3hrkbanPJ0WB+16mNKTtcuMupoRbmIJoP/lTilohcMn5/R+xcFKl5u99FpGh+9z4a211INDl/qsaSjIx3MKWcJDc7ZoNoVP/oHUKxoM2u9kFa/2fU/BLdnIlaEtQti38vgXEts4ORQeba6tBBGO57Y9zxC2ebm53k1Xwmuu2zOeXUyXMxRPc7hfmePEn2xPVfmpyMj74snIrauP941NbAPf+Is/3/p2uXYtjPRbGe+XT8P8Pc5OF8N2tmSz0kWN/H1vVBtmDb5ZjZpOzoez7cuducs1VSK8+sX07d7tloKkx/95xZsKrdQfqT0wzkaOsuaT716IYkY11yUv32OIaHEIr1zLG/u/p0tNbwoFwqZkCTLtp6OcZulSek7Oh7fty52xxDQmR1+Wy4pG6qYMPVk4TvCRP6x+h1w77ULffpbu4ZR9iwvflT0jEHvv6OSw4hlAcOZL1u5HSjZECCpjDLrpaX4HJjJ8nb3fe8uHO2eWU4xdV6vCgcBqotp6bLHRE1XkkoviUBIfZTC0laDBVkYmUONWzLQmLT7ZZX2T92lUlK6vBgmtC4YkohFPuZkiP/dv4/GFURTYxCSW65wA0J1mGIwpwkQ3FdWC2GvCBveQ/2PV/uGm0uESlQoClMKSciSShJeL89M1BSYxqxrDsUe85P7BXQDavtcfVMt3y0LWlHhv++bT1yxGA4k3Uoe8dUmd6mWlI+fPgjxvA1F8WSz56IGNM1cVwmpAFFTmYgM6A43j0yS4Kou629R5HMY+KMyMdpLvrKe6jv+XJntzm6KoGYr1vt0MQN78VZUnN3nqYkldvd4akFJffaZ7dUbfxMtU7Q1rxJgiYmJ5NfDaojPm0Oksre/KlGLq8omzsuFEIBQBRH3syVqxTEE2bkI6e70VDKgX0919gSTO/llaaUK7s1Fwa95T3U93y5u7e5KM68mRLMRe4RwZYcBkpqqkXMZa7A5pwB0prVtM9WNydOIJozmd9qhTlIKmeUd+g5wtXFQygV8vuRBxf/qL/k0DCIyl4v5FLzpTmAKdr1EOX44kLr6p7y9u17I9zVyLlWqiQxOAbdgZIstZDoxVOZ42a3F8v9NoUcJi0qRXehjGOB2YLE4sV/XSiptiWJrsvunw+L+hnXuolyrEp3FajnWBgjhGBE3v59b5S7m6Qefsllekn2bDFreppkLscxbB4RBbZgRdF1wP1KSkn5RWI7g4OYXS0kRt3XcPUEmUe1Ex+batfeUKjCuV3WXPFsgUAVaHPrI++pfW9wM3Cjq+uRftFbkq0Wun8alZRqXDqZq6REHMxV8mpxXjQJBRFFZ6K64J+Jc+2NGjXR3v7ZHvF0q7FdO+ZuNu5BUjnyZ2zOkBj6d02EUGrcvykKPSFR5faeApCI5B7X6UqwVjdRpGbFra+e8taDJpwvd8qaRW+lxbx3E2VGS7JiAZ99ZxmRk9drjoqUqI6EpOQGdBVbcEV9vGMZx/ZZSN1YTDsqRs6nFatI27EQzr0xr0Y0xCcWRYZjSTOEYsk6qp/sieFwdsS38urpukUsDh8SJMagUbf4S3s1fKFIlGNN3lfe/X1vKncPifB1l69BobtnbQ2XVM8WkhMRQSMWXo1IGf8AURX7NaWk/Mstyz7jZEp42icp0A9zoC85TSKurXFG80YuBVcgoeyYZrp39Iltx88cxLZC++Fc/gmi3PBGLgciVK9J6Yqi7SSXnGttov6xWsqWYIEi4ze5xJN8qFAMLT56yNvV9yz4cvdo8x2ZnIg5cSGucs96SnWXNDpWnu1Rjbi+T1EGDLqaSZ4cb6KxE4fIOvOJruuTV3VpUWdL/uCM6T/XL4TSGq9cOTwDyR+odn5QW4KoWjZqLK/KUwLKXdNbbRdeGfHirtnmt+cMzplsQE5jFaoXYziTkFrTeLUQ8lU/83TeVF9TN+gTD+rqLvwF18zgOhotgOL5zcFUQWcO1ixK0FOXnIw9H66uP1eq4DP9Y942Hz9s80QsMfE9efjAXuIq0gvsJKcM37Fs4Z3cHYmQsWhC9/nwowWaJUXQMd+CKd4IkeegBIP5z27rK5Jzbi2Y3qCDN1vNyN18bQ7McdhmN3UjqJTUnefZzJv1GltmcjOJcSWOB1C8vzqQ0N/nPo5LMLgmZyshw5Ur/Bl27PR8/WPeNjc8ddim5JKBaOcq4jT1PFA6N61GU9ZfJWM3m5v53Xu4K1mU7q09xrljclCC4fzv73Wib8FU1H01ZVEMczdrm8NTJ5YXaMkpvXJQfRh3Jw+/TSpnzij9MnGocVxC9kcvsMX4rbEG39f9Y/ieo3uGuJt9ZWb56e8nX3y48mBDKJNMBjPn/sIZJJHQGwYgf57b17te6OSPl4Q7NSOE4o8TvNj5S9ljn0IDP+KYt6nY1GIA4sxsCqFMgSk5fc8xM56S2FP0LYlJRPRDjnmbhE0thrDUZlUfnIhlhhM15oCk4EjzSwk+aueVsanFENwJbCGUSTAnrq9xGrikt5+P7FHTv9KE2tRiGNrZ6CGUqXh/BbUQbW0FaJl1P+CHaUK53n6A2AWJ+OStz0QIoQTV4JPDmknh9fF01tzXOv3wY77dnC+GtUNhr3uROkOwAZSgGqz+k5J8NW6bR5il7tObfvq1GVEdSFafbtRNgQ6gPANz4cKqP9po/jFiXfY5TmqBk35fB2vr5etdfPJlILa3OYVTnqyL4nOZzNina/ZX/1z95lt0ICkRV5LuQTIhlKdrozny+9V+Lkwy9LSUnJ+FTS1eHqLJqzOPXgUTztj6odh8i5eHKdit62E0IVFzK+rfiG222LChg/8B5MPhsnF3DbYAAAAtdEVYdGljYzpjb3B5cmlnaHQAQ29weXJpZ2h0IEFydGlmZXggU29mdHdhcmUgMjAxMQi6xbQAAAAxdEVYdGljYzpkZXNjcmlwdGlvbgBBcnRpZmV4IFNvZnR3YXJlIHNSR0IgSUNDIFByb2ZpbGUTDAGGAAAAInRFWHRwczpIaVJlc0JvdW5kaW5nQm94ADIzN3gzMCsxODcrNjI0pTgL+gAAAB50RVh0cHM6TGV2ZWwAUFMtQWRvYmUtMi4wIEVQU0YtMi4wQfkzEwAAAABJRU5ErkJggg==)
where
is the volume fraction field,
is the pressure field and
is the meniscus area
![](data:image/png;base64,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)
The idea is to approximate
with a narrow normal distribution
![](data:image/png;base64,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)
So for
small enough, you can take
. Where
has the dimension of u and is to be interpreted as the width of a small averaging window around the meniscus value
.
So, I would define this expression in COMSOL using the usual integration operators, then trace this quantity has a function of
. As it get smaller,
should converge, until
becomes too small compared to the mesh spacing I presume.
I wish I knew a more efficient answer to your question.
Hello Harsha,
This is an interesting question.
I do not have a full answer that would work out of the box in COMSOL, but I see two work-arounds. Hopefully someone with more experience can provide a better solution
1. Export the volume fraction and pressure fields, then work out the problem in python
2. Compute an approximated average using a Gaussian as a substitute for the Dirac distribution
I will focus on solution n°2:
The point is that you are interested in the quantity
P_{avg} = (1/S) \times \iiint \delta\left(\phi(\vec{x} - \frac{1}{2}\right) \times P(\vec{x}) d^3\vec{x}
where \phi(\vec x) is the volume fraction field, P(\vec x) is the pressure field and S is the meniscus area
S = \iiint \delta\left(\phi(\vec{x}) - \frac{1}{2}\right) d^3\vec{x}
The idea is to approximate \delta with a narrow normal distribution
\delta(u) = \lim_{\sigma \to 0} \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{u}{\sigma}\right)^2}
So for \sigma small enough, you can take
\delta(u) \simeq \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{u}{\sigma}\right)^2}. Where \sigma has the dimension of u and is to be interpreted as the width of a small averaging window around the meniscus value \phi_m = 1/2.
So, I would define this expression in COMSOL using the usual integration operators, then trace this quantity has a function of \sigma. As it get smaller, P_{avg} should converge, until \sigma becomes too small compared to the mesh spacing I presume.
I wish I knew a more efficient answer to your question.