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

Mohammad Rahimi, Kuan Zhang, Marino Arroyo

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)519-525
Number of pages7
JournalComputational Mechanics
Issue number3
StatePublished - Mar 2015

Bibliographical note

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

Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.


  • Follower load
  • Periodic surface
  • Pressure
  • Volume


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