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Colloquium - Paraschivoiu

Bounds for the Output of the Steady Incompressible Navier-Stokes Equations in Three Space Dimensions
Concordia University, Canada
5/12/2003
2:00pm-3:00pm

A design engineer is often interested in optimizing a system based on some specific characteristic values of that system. For fluid mechanics or heat/mass transfer problems, these characteristic values, namely `outputs', may be the temperature at some location, or the flow rate of the underlying system, and can be expressed as linear functionals of the system's field variables, which are temperature, density, and velocities. For simulation based engineering design, the relevant system may be modeled by partial differential equations (PDEs) which yield field variable solutions. The accuracy of numerical solution of the PDEs depends of the discretization size and may required large computational resources.

In this work, an implicit a posteriori error estimation method is used to provide bounds to the linear functional output of the numerical solutions to the three dimensional Navier-Stokes equations at a fraction of the cost of calculating the output directly. The finite element bound method is based on a domain decomposition technique for efficiently generating lower and upper bounds. The novelty of this research is to utilize the posteriori error estimation method with Lagrange multipliers and the finite element tearing and interconnecting (FETI) procedure to extend the bound method to three space dimensions for Navier-Stokes (N-S) problems. The approach herein exploits a coarse mesh linearized N-S problem and demonstrates that the omitted nonlinear term is very small.

The computational advantage of the bound procedure is that a single coupled nonsymmetric/symmetric large problem can be decomposed into several uncoupled symmetric small problems. A simple benchmark problem, which is selected here to illustrate the procedure, is to find the lower and upper bounds of the average velocity of a pressure driven, incompressible, steady Newtonian fluid flow moving at low Reynolds numbers through an endless square channel which has an array of rectangular obstacles. Numerical results from the three dimensional Navier--Stokes equations show that the bounds are sharp and that the required computational resources decrease significantly. Parallel implementation on a Beowulf cluster is also reported. The bound method has the potential to be an inexpensive and reliable approach to incorporate numerical simulation of large-scale engineering problems in engineering design.

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University of Colorado Boulder
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