On the theory of semi‐implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 2: Implementation

Abstract

SUMMARYEver since the expansion of the finite element method (FEM) into unsteady fluid mechanics, the 'consistent mass matrix' has been a relevant issue. Applied to the time-dependent incompressible Navier-Stokes equations, it virtually demands the use of implicit time integration methods in which full 'velocity-pressure coupling' is also inherent. The high cost of such (high-quality) FEM calculations led to the development of simpler but ad hoc methods in which the 'lumped' mass matrix is employed and the velocity and pressure are uncoupled to the maximum extent possible. Resulting computer codes were less expensive to use but suffered a significant loss of accuracy, caused by lumping the mass when the flow was advection-dominated and accurate transport of 'information' was important. In the second part of this paper we re-introduce the consistent mass matrix into some semi-implicit projection methods in such a way that the cost advantage of lumped mass and the accuracy advantage of consistent mass are simultaneously realized. KEY WORDS Incompressible flows Navier-Stokes equations Projection methods Consistent mass DISCRETE PROJECTION METHODS*In this second part of the paper we go from semi-discrete to fully discrete in the form of several finite element methods that we have designed and tested. In addition to implementing some of the projection methods discussed in Part 1, we also introduce a variation on the GFEM that utilizes the consistent mass matrix in a cost-effective way. That is, the object of the second part of this paper, besides implementing some of the techniques derived above, is to put the 'mass' back where it belongsdistributed according to the Galerkin principle-not concentrated at node points as it is in the lumped mass approximation used in our (and many others') most recent FEM schemes for solving the time-dependent incompressible Navier-Stokes equations-see Gresho et al. ' for an explicit time integration method and Gresho and Chan2 for a semi-implicit one.

Distributed control of linearized Navier‐Stokes equations via discretized input/output maps

“…Recalling that the projection algorithms produce a time integration error of first order, cf. [18], this error may be neglected.…”

“…(10), and the function B 1 u : [0, T ] → R nv has to be sufficiently smooth. Since B 1 as a bounded operator maintains regularity, one can infer from (18) and Proposition 4.2 the regularity requirements…”

“…The behavior of the fluid is modeled by the LNSE, spatially discretized by stabilized Q 1 − P 0 finite elements, which are piecewise linear for the velocity and piecewise constant for the pressure, on a uniform 128 × 128 grid. The resulting, still time dependent linear DAE system is integrated numerically using a projection method as introduced in [18].…”

“…Recall that the actual output of the system is recovered by adding the affine component y 0 to Gu, cf. (18).…”

IFP-C3D: an Unstructured Parallel Solver for Reactive Compressible Gas Flow with Spray

“…To assess the validity of our development, the inviscid Gresho case (Gresho, 1990) can be solved with IFP-C3D. A stationary vortex is set in a 3D cubic box of length 0.02 m. The velocity profile along the radius is triangular as shown in Figure 4.…”