COMSOL 6.3 - Volume Definition

Using the divergence theorem (or Gauss’s theorem), the undeformed volume, Vref, of a closed cavity can be computed from only a surface integral rather than a volume integral.

Here, S(V) is the surface enclosing the cavity, n is the boundary normal pointing from the solid boundary into the cavity volume; and R is a suitable vector for which the divergence is equal to 1 for any point on the boundary. For a 3D problem a suitable vector is

The coordinates of the reference point, Xref, are constants, which can be useful when dealing with symmetry planes or plane end caps in the model as described below.

The volume of a deformed cavity, V, is obtained in a similar way. By using Nanson’s relation the volume can be computed from an integration on the boundaries in the undeformed configuration (material frame).

Here, F is the deformation gradient, and J = det(F). The spatial divergence of r is again constant and equal to 1 for any point on the cavity wall. For 3D r is defined as

The integrals shown above are valid for closed surface integrals, that is, when the boundary selection over which to integrate forms a watertight volume. However, often closed volumes may have symmetry planes, or a rigid plane end cap, which is not explicitly modeled. In such cases the true volume is closed, while the model geometry is no longer watertight. The relations shown above do still apply in such cases, with some minor modifications.

When the model geometry does not form a closed surface, the vectors R and r must be chosen such that they are parallel along any symmetry plane or plane end cap. This can be achieved by shifting these vectors by selecting an appropriate reference point, Xref. In addition, the computed volume must be multiplied by a factor, fV, to get the true cavity volume. For instance, for half symmetry, the true volume will be twice as large as computed by the surface integral, and thus should be multiplied by a factor of two.