Engineering topics

Arian, Eyal

Publications and source records attributed to Arian, Eyal.

7 records

NASA NTRS1998

A Preconditioning Method for Shape Optimization Governed by the Euler Equations

We consider a classical aerodynamic shape optimization problem subject to the compressible Euler flow equations. The gradient of the cost functional with respect to the shape variables is derived with the adjoint method at the continuous level. The Hessian (second order derivative of the cost functional with respect to the shape variables) is approximated also at the continuous level, as first introduced by Arian and Ta'asan (1996). The approximation of the Hessian is used to approximate the Newton step which is essential to accelerate the numerical solution of the optimization problem. The design space is discretized in the maximum dimension, i.e., the location of each point on the intersection of the computational mesh with the airfoil is taken to be an independent design variable. We give numerical examples for 86 design variables in two different flow speeds and achieve an order of magnitude reduction in the cost functional at a computational effort of a full solution of the analysis partial differential equation (PDE).

Arian, Eyal↗

NASA NTRS1997

Admitting the Inadmissible: Adjoint Formulation for Incomplete Cost Functionals in Aerodynamic Optimization

We derive the adjoint equations for problems in aerodynamic optimization which are improperly considered as "inadmissible." For example, a cost functional which depends on the density, rather than on the pressure, is considered "inadmissible" for an optimization problem governed by the Euler equations. We show that for such problems additional terms should be included in the Lagrangian functional when deriving the adjoint equations. These terms are obtained from the restriction of the interior PDE to the control surface. Demonstrations of the explicit derivation of the adjoint equations for "inadmissible" cost functionals are given for the potential, Euler, and Navier-Stokes equations.

Arian, Eyal↗

NASA NTRS1997

Convergence Estimates for Multidisciplinary Analysis and Optimization

A quantitative analysis of coupling between systems of equations is introduced. This analysis is then applied to problems in multidisciplinary analysis, sensitivity, and optimization. For the sensitivity and optimization problems both multidisciplinary and single discipline feasibility schemes are considered. In all these cases a "convergence factor" is estimated in terms of the Jacobians and Hessians of the system, thus it can also be approximated by existing disciplinary analysis and optimization codes. The convergence factor is identified with the measure for the "coupling" between the disciplines in the system. Applications to algorithm development are discussed. Demonstration of the convergence estimates and numerical results are given for a system composed of two non-linear algebraic equations, and for a system composed of two PDEs modeling aeroelasticity.

Arian, Eyal↗

NASA NTRS1996

Smoothers for Optimization Problems

We present a multigrid one-shot algorithm, and a smoothing analysis, for the numerical solution of optimal control problems which are governed by an elliptic PDE. The analysis provides a simple tool to determine a smoothing minimization process which is essential for multigrid application. Numerical results include optimal control of boundary data using different discretization schemes and an optimal shape design problem in 2D with Dirichlet boundary conditions.

Arian, Eyal↗

NASA NTRS1996

Analysis of the Hessian for Aerodynamic Optimization: Inviscid Flow

In this paper we analyze inviscid aerodynamic shape optimization problems governed by the full potential and the Euler equations in two and three dimensions. The analysis indicates that minimization of pressure dependent cost functions results in Hessians whose eigenvalue distributions are identical for the full potential and the Euler equations. However the optimization problems in two and three dimensions are inherently different. While the two dimensional optimization problems are well-posed the three dimensional ones are ill-posed. Oscillations in the shape up to the smallest scale allowed by the design space can develop in the direction perpendicular to the flow, implying that a regularization is required. A natural choice of such a regularization is derived. The analysis also gives an estimate of the Hessian's condition number which implies that the problems at hand are ill-conditioned. Infinite dimensional approximations for the Hessians are constructed and preconditioners for gradient based methods are derived from these approximate Hessians.

Arian, Eyal↗

NASA NTRS1995

Analysis of the Hessian for aeroelastic optimization

The symbol of the Hessian for an aeroelastic optimization model problem is analyzed. The flow is modeled by the small-disturbance full potential equation and the structure is modeled by an isotropic (von Karman) plate equation. The cost function consists of both aerodynamic and structural terms. In the new analysis the symbol of the cost function Hessian near the minimum is computed. The result indicates that under some conditions, which are likely fulfilled in most applications, the system is decoupled for the non-smooth components. The result also shows that the structure part in the Hessian is well-conditioned while the aerodynamic part is ill-conditioned. Applications of the result to optimization strategies are discussed.

Arian, Eyal↗

NASA NTRS1994

Multigrid one shot methods for optimal control problems: Infinite dimensional control

The multigrid one shot method for optimal control problems, governed by elliptic systems, is introduced for the infinite dimensional control space. ln this case, the control variable is a function whose discrete representation involves_an increasing number of variables with grid refinement. The minimization algorithm uses Lagrange multipliers to calculate sensitivity gradients. A preconditioned gradient descent algorithm is accelerated by a set of coarse grids. It optimizes for different scales in the representation of the control variable on different discretization levels. An analysis which reduces the problem to the boundary is introduced. It is used to approximate the two level asymptotic convergence rate, to determine the amplitude of the minimization steps, and the choice of a high pass filter to be used when necessary. The effectiveness of the method is demonstrated on a series of test problems. The new method enables the solutions of optimal control problems at the same cost of solving the corresponding analysis problems just a few times.

Arian, Eyal↗