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Hide internal edge from the mesher

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Dear all,
I have a 2D geometry similar to the one shown in the attached file. I need the domains D1 and D2 to assign different fluids in each region. My problem has to do with inner edge E. Specifically, is there any way to instruct the mesher to ignore that edge? I need this because latter I want to use an ALE interface and the presence of edge E causes problems. I've tried some virtual operations but without luck, since removing edge E, the regions D1 and D2 are unified and, as a consequence, I have no way to initialise different fluids over these areas. I'd appreciate any help.

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