Paraschivoiu, Marius

A Posteriori Finite Element Bounds for Linear-Functional Outputs of Coercive Partial Differential Equations and of the Stokes Problem

[PhD thesis]. Cambridge, MA: Massachusetts Institute of Technology

Patera, Anthony T. (supervisor)

October 1997

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- Abstract
Simulation-based engineering design and optimization are premised upon the accurate prediction of performance metrics such as flowrate, heat flux, and drag and lift forces. These metrics, which we will term "outputs," are functionals of the underlying field variables, such as velocity, temperature and pressure. We consider here the situation in which these functionals are linear, which includes many cases of practical interest.

There are two main problems that limit the use of simulations in engineering design: first, each appeal to the simulation is very expensive, which hinders interactivity in design; and second, evaluation of the reliability of the outputs is often unavailable, which precludes confident use of simulations in design. In this thesis we address these drawbacks in a new procedure focused on increasing the speed and reliability of simulation-based engineering design. The method exploits a fast "H-discretization" to compute bounds to the outputs that would have been obtained on a very fine mesh termed the "truth" mesh. These bounds inexpensively provide the desired assurance about the numerical error in the output.

The method is based upon the construction of an augmented Lagrangian, in which the objective is a quadratic "energy" reformulation of the desired output, and the constraints are the finite element equilibrium equations and the intersubdomain continuity requirements. Rigorous bounds are then obtained by application of quadratic-linear duality theory, in which the candidate Lagrange multipliers are obtained from the inexpensive H-discretization. The only computations required on the "truth" mesh are subdomaindecoupled symmetric local Neumann problems, which are very inexpensive to invert.

This technique is illustrated for the convection-diffusion equation both in one and two space dimensions. Outputs such as the flux, the pointwise value, and the average over a region are considered. Extension to the incompressible Stokes equations is then presented; for this problem, bounds for the lift force on an immersed body are calculated. The results indicate that this technique offers rigorous, quantitative, inexpensive, and relatively sharp bounds for "truth"-mesh engineering outputs, and thus provides for a fast and reliable design framework. Limitations and future work are briefly described.