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3.5.3 Spatial Discretization
The velocity and Jacobian are spatially interpolated over the domain using the same interpolation functions , with to where is the number of nodes in an element), Here, and are nodal values of and that evolve with time. In contrast to classical mixed formulations for incompressible flow , which solve for the pressure using instead of eq.(2.11.1-6), equal order interpolation is acceptable in this formulation since the governing equations for and involve spatial derivatives of both variables ( and ). The expressions of eq.(3.5.3-1) may be used to evaluate , , , , , etc. Similar interpolations are used for virtual increments and , as well as real increments and .
When substituted into eq.(184.108.40.206-1), we find that the discretized form of may be written as where Similarly, the discretized form of in eq.(3.5.2-5) becomes where whereas that of in eq.(3.5.2-7) becomes where
For the external work in eq.(220.127.116.11-3), its discretized form is where The discretized form of in eq.(3.5.2-10) is where