Linear shape function, mass matrix, and element volume
by Yi Zhang
In unsteady CFD problem, under certain time integration scheme, L2 inner product of shape functions gives what so called mass matrix, i.e. the matrix with its entries in the form , where is shape function. When shape function is linear, there is a better way to demonstrate is proportional to local element volume (area) than trivially plugging in and calculating or using Gaussian integral. Assume in 2D triangular element ABC whose area is , and stands for the shape function whose unit value is reached at A, B, C correspondingly.
If we interpret local mass entry over element ABC shown above as the mass of a piece of wood splinter in the shape of tetrahedron A1-A-B-C, where is the thickness function on the element, while is the density function, then it’s not difficult to see that , because they are both the mass of tetrahedron with the same volume and density distribution from bottom (A1-A-B or B1-A-B) to top (C). The next step is to show that the mass of tetrahedron C1-A-B-C () is the half of the total mass of the prism A1-B1-C1-A-B-C with the same density distribution. The argument is based on linearity, which allows us to assign a single parameter for above integrals. Set this single coordinate as the line parameter from C to line A-B, i.e. on AB, at C. Then, for mass of prism and tetrahedron, we both have
where gives the area for cross section parallel to A1-A-B-B1 at . is the thickness of ABC, while is the density distribution. We have
and for prism, for tetrahedron. Then we have
Since is nothing but the sum of the mass of three tetrahedrons, which is the mass of the prism. This sum is then . Using , finally we have
Note: for 3D tetrahedron element, linear shape function gives similar relation, in this case, above ratio over the volume of a tetrahedron would be 1/10 and 1/20: