Gregory R. Shubin

Problem Formulation for Multidisciplinary Optimization

This paper is about multidisciplinary (design) optimization, or MDO, the coupling of two or more analysis disciplines with numerical optimization.The paper has three goals. First, it is an expository introduction to MDO aimed at those who do research on optimization algorithms, since the optimization community has much to contribute to this important class of computational engineering problems. Second, this paper presents to the MDO research community a new abstraction for multidisciplinary analysis and design problems as well as new decomposition formulations for these problems. Third, the “individual discipline feasible” (IDF) approaches introduced here make use of existing specialized analysis codes, and they introduce significant opportunities for coarse-grained computational parallelism particularly well suited to heterogeneous computing environments.The key distinguishing characteristic of the three fundamental approaches to MDO formulation discussed here is the kind of disciplinary feasibility that...

An unsplit, higher order Godunov method for scalar conservation laws in multiple dimensions

Abstract In this paper we develop an unsplit, higher order Godunov method for scalar conservation laws in two dimensions. The method represents an extension, for the special case being considered, of methods developed by Colella and by Van Leer. In our method we begin with a piecewise bilinear representation of the solution on each grid cell. A new piecewise bilinear representation at the next time level is then obtained from a two-step procedure. In the first step, a conservative predictor-corrector scheme derived from the integral form of the differential equation and characteristic considerations is used to obtain average values of the solution over grid cells at the new time level. Next, these new average values are then used to construct a limited, piecewise bilinear profile for each cell at the new time level. The resulting method is shown to satisfy a maximum principle for constant coefficient linear advection. Computational results are presented comparing the new method to Colellas method for linear advection. The method is also applied to two model problems from porous media flow, miscible and immiscible displacement. The new scheme provides accurate resolution of sharp fronts without any significant distortion.

A comparison of optimization-based approaches for a model computational aerodynamics design problem

Abstract The objective of this paper is to compare three optimization-based methods for solving aerodynamic design problems. We use the Euler equations for one-dimensional duct flow as a model problem. The optimization methods are (i) the black-box method with finite difference gradients, (ii) a modification where gradients are found by an algorithm based on the implicit function theorem, and (iii) an all-atonce method where the flow and design variables are simultaneously altered. The three methods are applied to the model problem and compared for efficiency, robustness, and implementation difficulty. We also show that the black-box (implicit gradient) method is equivalent to applying the “variational” or “optimal control” approach to design optimization directly to the discretized analysis problem, rather than to the continuous problem as is usually done. The black-box method with implicit gradients seems to provide a good compromise of features, and can be retrofitted to most existing analysis codes to turn them into design codes. Although the all-at-once method was found to be less robust than the black-box methods, when it succeeded it was considerably more efficient.

Conservation laws of mixed type describing three-phase flow in porous media

In this paper we examine the mathematical structure of a model for three-phase, incompressible flow in a porous medium. We show that, in the absence of diffusive forces, the system of conservation laws describing the flow is not necessarily hyperbolic. We present an example in which there is an elliptic region in saturation space for reasonable relative permeability data. A linearized analysis shows that in nonhyperbolic regions solutions grow exponentially. However, the nonhyperbolic region, if present, will be of limited extent which inherently limits the exponential growth. To examine these nonlinear effects we resort to fine grid numerical experiments with a suitably dissipative numerical method. These experiments indicate that the solutions of Riemann problems remain well behaved in spite of the presence of a linearly unstable elliptic region in saturation space. In particular, when initial states are outside the elliptic region the Riemann problem solution appears to stay outside the region. Further...

A modified equation approach to constructing fourth order methods for acoustic wave propagation

In this paper we use a modified equation analysis to develop formally fourth order accurate finite difference and pseudospectral methods for the one-dimensional wave equation. The difference scheme is constructed by performing a modified equation analysis of a centered, second-order conservative scheme to determine its dominant error term. Subtracting a centered discretization of this term from the scheme cancels the second order truncation errors. This technique yields a formally fourth order accurate explicit difference scheme that employs only three time levels. Similarly, the modified equation technique can be used to achieve fourth order time accuracy for the pseudospectral method with no increase in storage. The difference and pseudospectral schemes are fourth order convergent for constant coefficients even when a spatially singular forcing term is used for a source. Numerical results are given comparing the accuracy and efficiency of these methods for some model problems. Finally, we present a gene...

A method for reducing numerical dispersion in two-phase black-oil reservoir simulation

Abstract In this paper a new sequential method for 1-dimensional two-phase black-oil reservior simulation is presented. The model includes compressibility and mass transfer between phases. The method decouples the system of equations into a parabolic equation and a hyperbolic system. One-dimensional numerical examples are presented using both first- and second-order Godunov discretizations for the hyperbolic part of the system. The results are compared to a fully implicit method using upwind differences. The second-order Godunov method shows dramatic improvement over those obtained with the more dispersive first-order methods.