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Computing the volume enclosed by a periodic surface and its variation to model a follower pressure


In modeling and numerically implementing a follower pressure in a geometrically nonlinear setting, one needs to compute the volume enclosed by a surface and its variation. For closed surfaces, the volume can be expressed as a surface integral invoking the divergence theorem. For periodic systems, widely used in computational physics and materials science, the enclosed volume calculation and its variation is more delicate and has not been examined before. Here, we develop simple expressions involving integrals on the surface, on its boundary lines, and point contributions. We consider two specific situations, a periodic tubular surface and a doubly periodic surface enclosing a volume with a nearby planar substrate, which are useful to model systems such as pressurized carbon nanotubes, supported lipid bilayers or graphene. We provide a set of numerical examples, which show that the familiar surface integral term alone leads to an incorrect volume evaluation and spurious forces at the periodic boundaries.

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European Research Council (FP7/20072013)/ ERC Grant Agreement no. 240487.

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Corresponding author

Correspondence to Marino Arroyo.

Additional information

Mohammad Rahimi and Kuan Zhang have contributed equally to this work.

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Rahimi, M., Zhang, K. & Arroyo, M. Computing the volume enclosed by a periodic surface and its variation to model a follower pressure. Comput Mech 55, 519–525 (2015).

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  • Periodic surface
  • Volume
  • Pressure
  • Follower load