Anthony T. Patera, D. V. Rovas and Luc Machiels

Dept. of Mechanical Engineering, Massachusetts Institute of Technology

September, 2000

RTA 705-01-03-02

Research Objective. The objective of this work is to develop rigorous methods for quantifying the numerical error arising from the use of reduced-basis approximations to high-fidelity codes for solving elliptic partial differential equations (PDEs). The approximations are more efficient for use in a repetitive design environment than the high-fidelity codes, but the associated errors need to be quantified.

Approach. A specific class of reduced-basis approximations to elliptic PDEs is developed. They are suited for use with high-fidelity codes that solve a weak formulation of the PDE. The first step in construction of the approximation is to solve the high-fidelity problem for a representative sample set of, say, N, combinations of the design variables. These N solutions form the basis functions. They are computed in a pre-processing, or off-line, stage. Then for any arbitrary point in the design space, one constructs the reduced-basis approximation as that linear combination of the basis functions that solves the weighted residual problem constructed from the bilinear form appropriate for the PDE. The cost of constructing a single reduced-basis approximation in the on-line design stage is typically orders of magnitude cheaper than the cost of solving a single high-fidelity problem. Such reduced-basis approximations are amenable to rigorous error analysis. Therefore, they can be used in a design process with assured bounds on their associated numerical error.

Accomplishment Description. The theoretical error analysis for elliptic, coercive PDEs has been completed. The method has been demonstrated on the thermal fin problem illustrated in the figure. The fin consists of a central "post" and four "subfins." The fin conducts heat from a prescribed uniform flux "source" at the root through the subfins to the surrounding flowing air. There are 7 design variables: the k ⁱ are the thermal conductivities of the subfins; Bi is the Biot number (nondimensional heat transfer coefficient): L and t are the length and thickness of the subfins. The output quantity is the average temperature at the root of the central fin. The right half of the figure illustrates the achievable set, in terms of the volume of the central fin and the average temperature at the root over the domain of two of the design variables L and t. The area was computed from the reduced-basis approximation, and the upper bound on the root temperature was plotted (using the error analysis results).

Significance. Variable-fidelity approximations are quite popular, but rarely come with rigorous quantification of the approximation error. This work provides an efficient scheme for not only constructing such approximations, but also for rigorously quantifying the numerical errors (uncertainties) associated to the use of them as a low-fidelity approximation to the high-fidelity code.

Future Plans. Extend the reduced-basis methodology to a broader class of PDEs, including non-coercive problems, non-symmetric problems, hyperbolic problems and to such challenging applications as compressible, Navier-Stokes equations.

Figure: Reduced-Basis Model of Thermal Analysis

Related Publications:

Rovas, D. V.: "An overview of blackbox reduced-basis output bound methods for elliptic partial differential equations," Proceedings 16th IMACS World Congress, Lausanne, Switzerland, August 2000

Patera, A. T.: "Output bounds for elliptic partial differential equations: a general formulation," Proceedings 16th IMACS World Congress, Lausanne, Switzerland, August 2000.

Patera, A. T. and Rönquist, E. M.: A general output bound result: application to discretization and iteration error estimation and control," Mathematical Models and Methods in Applied Science, 2000, in press.

Machiels, L.: "A Posteriori Finite Elements Bounds for Output Functionals of Discontinuous Galerkin Discretizations of Parabolic Problems," Comp. Meth. Appl. Engrg., submitted, 1999.

Machiels, L., Patera, A. T. and Peraire, J.: "A Posteriori Finite Element Bounds of the Incompressible Navier-Stokes Equations; Application to a Natural Convection Problem,: J. Comput. Phys., submitted, 1999.

Machiels, L., Maday, Y. and Patera, A. T.: "Output Bounds for Reduced-Order Approximations for Elliptic Partial Differential Equations," Technical Report FML 99-5-1, M.I.T., 1999.

Machiels, L., Peraire, J. and Patera, A. T.: "Output Bound Approximations for Partial Differential Equations; Application to the Incompressible Navier-Stokes Equations," Proceedings of the Istanbul Workshop on Industrial and Environmental Applications of Direct and Large Eddy Numerical Simulation, S. Biringen (ed.), Springer-Verlag, 1998.