Continuous Adjoint Research Articles

Efficient Aerodynamic Design Method Using a Tightly Coupled Algorithm

An efe cient aerodynamic design optimization method for the compressible Euler equations is presented. A gradient-based optimization method is used to e nd the optimal design, in conjunction with a continuous adjoint sensitivity analysis method. To improve the efe ciency of the design method, a tightly coupled algorithm based on a step-size estimation isproposed. The efe ciency of thepresentmethod is demonstrated through drag minimizations of a wing and an airfoil. The result shows that the cost of the present method is signie cantly lower than that of the loosely coupled algorithm. Nomenclature Ci = e ux Jacobian matrix in the computational domain d = design variables vector Fi = e ux vector in the computational domain fi = e ux vector in the physical domain G = gradient vector of the object function I = object function q = conservative e ow variables vector R = residual vector in the computational domain S = searching direction ≤ = step size along the searching direction S A = Lagrangian multiplier vector Subscript face = value at cell face Superscript T = transpose of vector or matrix

Optimal Control of Aeroacoustic Noise Generated by Cylinder Vortex Interaction

This paper presents an optimal control formulation and solution for an idealized Blade Vortex Interaction (BVI) problem. This problem consists of the interaction of an inviscid vortex pair with a circular cylinder in a steady Mach 0.3 uniform flow with wall-normal velocity used as control on the cylinder surface. This model problem captures the fundamental noise generation process of the BVI phenomena while mitigating many of the complexities of the full rotorcraft problem. The optimal control problem is solved using a gradient based method where gradient information is computed from a continuous adjoint analysis of the governing unsteady Euler equations. The BVI wave packet is targeted by defining an objective function that measures the square amplitude of pressure fluctuations in an observation region over a time interval of interest. When the observation region encloses the entire flow a 6 dB reduction in overall BVI noise is obtained. When the observation region is limited to the region of greatest BVI noise, a 13dB reduction is obtained. The optimal control, unlike most common mitigation methods, does not target the interaction directly — instead, the computed boundary control modulates the potential flow about the cylinder, producing a wave packet of the correct amplitude and phase to approximately cancel the BVI noise in the observation region.

비정렬 격자계에서 Continuous Adjoint 방정식을 이용한 공력 형상 최적 설계

비정렬 격자계에서 continuous adjoint 방정식을 사용하여 비점성 압축성 유동장에서의 이차원 날개꼴에 대한 공력 형상 최적설계를 수행하였다. 정확한 민감도를 구하기 위한 재구성 기법으로는 Laplacian averaging을 사용하였으며, extended stencil을 사용한 최소자승법으로 유동변수의 미분을 구하였다. 또한 유동장과 adjoint 방정식의 적절한 수렴 조건에 대해서 알아보았으며, 이를 이용하여 천음속 비점성 유동장에서 RAE2822 및 NACA0012 날개꼴에 대해서 조파항력 최소화 설계를 수행하였다. 설계된 날개꼴에서 다시 충격파를 복원하는 역설계문제를 적용하여 본 연구에서 사용된 최적설계 기법을 검증하였다. Aerodynamic shape optimization of two-dimensional airfoils in inviscid compressible flows is performed using a continuous adjoint formulation on unstructured meshes. Accurate evaluation of the gradient is achieved by using a reconstruction scheme based on the Laplacian averaging. A least-square method with extended stencil is used for flow gradient calculations. Proper convergence criterion is studied on Euler and adjoint equations for efficient design. The present method has been applied to RAE2822 and NACA0012 airfoils such that wave drag can be minimized by removing the shock wave. An inverse design is also performed to recover the shock wave on the designed RAE2822 airfoil.

Singular Characteristics of the HJBI Equation in State Constraint Optimal Control Problems

For a state constraint optimal control problem the equations of motion along the constraint surface are derived in the form of singular characteristics which are given by an ODE system in closed form (no uncertain multipliers). A sufficient condition is found for the Bellman function’s gradient to be a continuous adjoint vector in maximum principle. It is shown that the solution of the original problem can be composed from the solutions of two dynamic programming problems which provides an interpretation and new proof of the jump condition for the adjoint vector. An illustrative example is considered.

Continuous Adjoint Sensitivities for Optimization with General Cost Functionals on Unstructured Meshes

A continuous adjoint approach is developed to obtain the sensitivity derivatives for the Euler equations. The completederivationofthecostateequations andtheirtransversality (boundary)conditionsarepresented. Both the state and the costate equations are second-order e nite volume discretized for unstructured meshes, and they are coupled with a constrained optimization algorithm. Also integrated into the overall methodology are a geometry parameterization method for the shape optimization, and a dynamic unstructured mesh method for the shape evolution and the consequent volume mesh adaptations. For the proof of concept, three transonic airfoil optimization problems are presented. This method accepts general cost functionals, which are not necessarily functions of pressure only. It is also shown that a switch to the natural coordinate system in conjunction with the reduction of the governing state equation to the control surface results in sensitivity integrals that are only a function of the tangential derivativesof the state variables. Thisapproach eliminates the need fornormal derivativecomputations that can be erroneous.

Continuous adjoint sensitivities for optimization with general cost functionals on unstructured meshes

Continuous adjoint sensitivities for optimization with general cost functionals on unstructured meshes

Accurate Aerodynamic Sensitivity Analysis Using Adjoint Equations

An accurate aerodynamic sensitivity analysis technique is described. The current method uses the continuous adjoint approach to avoid excessive memory requirement and the DADI relaxation with multigrid acceleration for the fast convergence. A derivation of the continuous adjoint operator is described for the 2nd-order TVD scheme and a new anti-diffusive flux is proposed for the adjoint equation to enhance the accuracy of sensitivity. The accuracy of current method is demonstrated by solving sensitivity problem for 2 dimensional airfoils. Comparisons are also presented with finite-difference results.

Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation

A continuous adjoint approach for obtaining sensitivity derivatives on unstructured grids is developed and analyzed. The derivation of the costate equations is presented, and a second-order accurate discretization method is described. The relationship between the continuous formulation and a discrete formulation is explored for inviscid, as well as for viscous flow. Several limitations in a strict adherence to the continuous approach are uncovered, and an approach that circumvents these difficulties is presented. The issue of grid sensitivities, which do not arise naturally in the continuous formulation, is investigated and is observed to be of importance when dealing with geometric singularities. A method is described for modifying inviscid and viscous meshes during the design cycle to accommodate changes in the surface shape. The accuracy of the sensitivity derivatives is established by comparing with finite-difference gradients and several design examples are presented.

An adjoint method for the inverse design of solidification processes with natural convection

This paper presents a finite element algorithm based on the adjoint method for the design of a certain class of solidification processes. In particular, the paper addresses the design of directional solidification processes for pure materials such that a desired freezing front heat flux and growth velocity are achieved. This is the first time that an infinite-dimensional continuum adjoint formulation is obtained and implemented for the solution of such inverse/design problems with moving boundaries and Boussinesq incompressible flow. The present design problem belongs to a category of inverse problems in which one is looking for the unknown conditions in part of the boundary, while overspecified boundary conditions are supplied in another part of the boundary (here the freezing interface). The solidification design problem is mathematically posed as a whole time-domain optimization problem. The gradient of the cost functional is calculated using the solution of an appropriately defined continuous adjoint problem. The minimization process is realized by the conjugate gradient method via the solutions of the direct, adjoint and sensitivity sub-problems. The proposed methodology is demonstrated with the solidification of an initially superheated liquid aluminum confined in a square mold. The non-uniformity in the casting product in the direction of gravity due to the existence of natural convection in the melt is emphasized. The inverse design problem is then posed as finding the appropriate spatial-temporal variations of the boundary heat flux on the vertical mold walls that can eliminate or reduce the effects of convection on the freezing interface heat fluxes and growth velocity. The numerical example demonstrates the accuracy and convergence of the adjoint formulation. Finally, open related research design problems are discussed. © 1998 John Wiley & Sons, Ltd.

Finite Difference of Adjoint or Adjoint of Finite Difference?

Abstract Adjoint models are used for atmospheric and oceanic sensitivity studies in order to efficiently evaluate the sensitivity of a cost function (e.g., the temperature or pressure at some target time tf, averaged over some region of interest) with respect to the three-dimensional model initial conditions. The time-dependent sensitivity, that is the sensitivity to initial conditions as function of the initial time ti, may be obtained directly and most efficiently from the adjoint model solution. There are two approaches to formulating an adjoint of a given model. In the first (“finite difference of adjoint”), one derives the continuous adjoint equations from the linearized continuous forward model equations and then formulates the finite-difference implementation of the continuous adjoint equations. In the second (“adjoint of finite difference”), one derives the finite-difference adjoint equations directly from the finite difference of the forward model. It is shown here that the time-dependent sensiti...

Application of the continuous orthonormalization and adjoint methods to the computation of solar eigenfrequencies and eigenfrequency sensitivities

view Abstract Citations (3) References (21) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Application of the Continuous Orthonormalization and Adjoint Methods to the Computation of Solar Eigenfrequencies and Eigenfrequency Sensitivities Rosenwald, R. D. ; Rabaey, G. F. Abstract Existing numerical techniques in helioseismology are applied in new ways to compute two sets of basic quantities: solar eigenfrequencies, computed via the continuous orthonormalization (CON) method, and their sensitivities with respect to changes in the solar equilibrium model, computed via the adjoint method. The CON method integrates an eighth-order nonlinear system of ordinary differential equations (ODEs) which defines the linear adiabatic nonradial oscillation modes of the sun. The adjoint method integrates a related eighth-order linear inhomogeneous system of ODEs. From the resultant solution, an eigenfrequency's partial derivatives with repsect to an extensive set of solar model parameters may be computed simultaneously. Numerical tests confirm the validity of the two methods. Eigenfrequencies obtained via the CON method agree with the corresponding eigenfrequencies obtained via finite-difference or mesh approaches. Eigenfrequency sensitivities for the three different types of modes (pressure, gravity, and fundamental) obtained via the adjoint method are presented. Publication: The Astrophysical Journal Supplement Series Pub Date: September 1991 DOI: 10.1086/191600 Bibcode: 1991ApJS...77...97R Keywords: Computational Astrophysics; Helioseismology; Solar Oscillations; Eigenvalues; Orthonormal Functions; Solar Interior; Solar Physics; NUMERICAL METHODS; SUN: INTERIOR; SUN: OSCILLATIONS full text sources ADS |

Sufficient conditions for optimality in problems with state constraints

In this paper the sufficient conditions for optimality are obtained for problems with state constraints. These constraints may be active. It means that the adjoint function may have points of discontinuity or jumps. Similar results in the case of absolutely continuous adjoint function were given by the author in [1] and [2].

Continuous posets and adjoint sequences

Continuous posets and adjoint sequences

An approximation to linear bounded phase coordinate control problems

Approximation to linear bounded phase coordinate control problem and continuous adjoint solution for optimum controller