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Fluid Control Using the Adjoint Method
 ACM TRANS. GRAPH. (SIGGRAPH PROC
, 2004
"... We describe a novel method for controlling physicsbased fluid simulations through gradientbased nonlinear optimization. Using a technique known as the adjoint method, derivatives can be computed efficiently, even for large 3D simulations with millions of control parameters. In addition, we introdu ..."
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Cited by 70 (1 self)
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We describe a novel method for controlling physicsbased fluid simulations through gradientbased nonlinear optimization. Using a technique known as the adjoint method, derivatives can be computed efficiently, even for large 3D simulations with millions of control parameters. In addition, we introduce the first method for the full control of freesurface liquids. We show how to compute adjoint derivatives through each step of the simulation, including the fast marching algorithm, and describe a new set of control parameters specifically designed for liquids.
Multipoint and Multiobjective Aerodynamic
 Shape Optimization,” AIAA Journal
"... A gradientbased Newton–Krylov algorithm is presented for the aerodynamic shape optimization of single and multielement airfoil configurations. The flow is governed by the compressible Navier–Stokes equations in conjunction with a oneequation transport turbulence model. The preconditioned general ..."
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Cited by 22 (15 self)
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A gradientbased Newton–Krylov algorithm is presented for the aerodynamic shape optimization of single and multielement airfoil configurations. The flow is governed by the compressible Navier–Stokes equations in conjunction with a oneequation transport turbulence model. The preconditioned generalized minimal residual method is applied to solve the discreteadjoint equation, which leads to a fast computation of accurate objective function gradients. Optimization constraints are enforced through a penalty formulation, and the resulting unconstrained problem is solved via a quasiNewton method. The new algorithm is evaluated for several design examples, including the lift enhancement of a takeoff configuration and a liftconstrained drag minimization at multiple transonic operating points. Furthermore, the new algorithm is used to compute a Pareto front based on competing objectives, and the results are validated using a genetic algorithm. Overall, the new algorithm provides an efficient approach for addressing the issues of complex aerodynamic design.
Optimal Control of Flow With Discontinuities
 Journal of Computational Physics
, 2003
"... Optimal control of the 1D Riemann problem of Euler equations whose solution is characterized by discontinuities is carried out by both nonsmooth and smooth op timization methods. By matching a desired flow to the numerical model for a given time window we effectively change the location of discont ..."
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Cited by 15 (1 self)
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Optimal control of the 1D Riemann problem of Euler equations whose solution is characterized by discontinuities is carried out by both nonsmooth and smooth op timization methods. By matching a desired flow to the numerical model for a given time window we effectively change the location of discontinuities. The control pa rameters are chosen to be the initial values for pressure and density fields. Existence of solutions for the optimal control problem is proven. A high resolution model and a model with artificial viscosity, implementing two different numerical methods, are used to solve the forward model. The cost functional is the weighted difference be tween the numerical values and the observations for pressure, density and velocity. The observations are constructed from the analytical solution. We consider either distributed observations in time or observations calculated at the end of the assimi lation window. We consider two different time horizons and two sets of observations. The gradient (respectively a subgradient) of the cost functional, obtained from the adjoint of the discrete forward model, are employed for the smooth minimization (respectively for the nonsmooth minimization) algorithm. Discontinuity detection improves the performance of the minimizer for the model with artificial viscosity by selecting the points where the shock occurs (and these points are then removed from Preprint submitted to Elsevier Science 26 March 2002 the cost functional and its gradient). The numerical flow obtained with the optimal initial conditions obtained from the nonsmooth minimization matches very well the observations. The algorithm for smooth minimization converges for the shorter time horizon but fails to perform satisfactorily for the longer time horizon.
Analytic adjoint solutions for the quasionedimensional Euler equations
 J. Fluid Mechanics
, 2001
"... The analytic properties of adjoint solutions are examined for the quasionedimensional Euler equations. For shocked flow, the derivation of the adjoint problem reveals that the adjoint variables are continuous with zero gradient at the shock, and that an internal adjoint boundary condition is requir ..."
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Cited by 15 (6 self)
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The analytic properties of adjoint solutions are examined for the quasionedimensional Euler equations. For shocked flow, the derivation of the adjoint problem reveals that the adjoint variables are continuous with zero gradient at the shock, and that an internal adjoint boundary condition is required at the shock. A Green’s function approach is used to derive the analytic adjoint solutions corresponding to supersonic, subsonic, isentropic and shocked transonic flows in a converging–diverging duct of arbitrary shape. This analysis reveals a logarithmic singularity at the sonic throat and confirms the expected properties at the shock. 1.
On the use of RungeKutta timemarching and multigrid for the solution of steady adjoint equations
, 2000
"... This paper considers the solution of steady adjoint equations using a class of iterative methods which includes preconditioned RungeKutta timemarching with multigrid. It is shown that, if formulated correctly, equal numbers of iterations of the direct and adjoint iterative solvers will result in t ..."
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Cited by 14 (5 self)
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This paper considers the solution of steady adjoint equations using a class of iterative methods which includes preconditioned RungeKutta timemarching with multigrid. It is shown that, if formulated correctly, equal numbers of iterations of the direct and adjoint iterative solvers will result in the same value for the linear functional being sought. The precise details of the adjoint iteration are formulated for the case of RungeKutta timemarching with partial updates, which is commonly used in CFD computations. The theory is supported by numerical results from a MATLAB program for two model problems, and from programs for the solution of the linear and adjoint 3D NavierStokes equations.
Adjoint Error Correction for Integral Outputs
"... Introduction 1.1 Output functionals Why do engineers perform CFD calculations? In the case of a transport aircraft at cruise conditions, a calculation might be performed to investigate whether there is an adverse pressure gradient near the leading edge of the wing, causing boundary layer separatio ..."
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Cited by 14 (2 self)
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Introduction 1.1 Output functionals Why do engineers perform CFD calculations? In the case of a transport aircraft at cruise conditions, a calculation might be performed to investigate whether there is an adverse pressure gradient near the leading edge of the wing, causing boundary layer separation and premature transition. Alternatively, one might be concerned about wing/pylon/nacelle integration, in which case one might be looking to see if there are any shocks on the pylon, leading to unacceptable integration losses. In both of these examples, qualitative information is being obtained from the computed ow eld to understand and interpret the impact of the phenomena on the quantitative outputs of most concern to the aeronautical engineer, the lift and drag on the aircraft. The quality of the CFD calculation is judged, rst and foremost, by the accuracy of the lift and drag predictions. The details of the ow eld are much less important, and are used in a more qualitative manner t
Modeling and rendering of heterogeneous translucent materials using the diffusion equation
, 2007
"... In this article, we propose techniques for modeling and rendering of heterogeneous translucent materials that enable acquisition from measured samples, interactive editing of material attributes, and realtime rendering. The materials are assumed to be optically dense such that multiple scattering c ..."
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Cited by 13 (3 self)
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In this article, we propose techniques for modeling and rendering of heterogeneous translucent materials that enable acquisition from measured samples, interactive editing of material attributes, and realtime rendering. The materials are assumed to be optically dense such that multiple scattering can be approximated by a diffusion process described by the diffusion equation. For modeling heterogeneous materials, we present the inverse diffusion algorithm for acquiring material properties from appearance measurements. This modeling algorithm incorporates a regularizer to handle the illconditioning of the inverse problem, an adjoint method to dramatically reduce the computational cost, and a hierarchical GPU implementation for further speedup. To render an object with known material properties, we present the polygrid diffusion algorithm, which solves the diffusion equation with a boundary condition defined by the given illumination environment. This rendering technique is based on representation of an object by a polygrid, a grid with regular connectivity and an irregular shape, which facilitates solution of the diffusion equation in arbitrary volumes. Because of the regular connectivity, our rendering algorithm can be implemented on the GPU for realtime performance. We demonstrate our techniques by capturing materials from physical samples and performing realtime rendering and editing with these materials.
Discrete adjoint approximations with shocks
 CONFERENCE ON HYPERBOLIC PROBLEMS
, 2002
"... In recent years there has been considerable research into the use of adjoint flow equations for design optimisation (e.g. [Jam95]) and error analysis (e.g. [PG00, BR01]). In almost every case, the adjoint equations have been formulated under the assumption that the original nonlinear flow solution i ..."
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Cited by 9 (3 self)
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In recent years there has been considerable research into the use of adjoint flow equations for design optimisation (e.g. [Jam95]) and error analysis (e.g. [PG00, BR01]). In almost every case, the adjoint equations have been formulated under the assumption that the original nonlinear flow solution is smooth. Since most applications have been for incompressible or subsonic flow, this has been valid, however there is now increasing use of such techniques in transonic design applications for which there are shocks. It is therefore of interest to investigate the formulation and discretisation of adjoint equations when in the presence of shocks.
The reason that shocks present a problem is that the adjoint equations are defined to be adjoint to the equations obtained by linearising the original nonlinear flow equations. Therefore, this raises the whole issue of linearised perturbations to the shock. The validity of linearised shock capturing for harmonically oscillating shocks in flutter analysis was investigated by Lindquist and Giles [LG94] who showed that the shock capturing produces the correct prediction of integral quantities such as unsteady lift and moment provided the shock is smeared over a number of grid points. As a result, linearised shock capturing is now the standard method of turbomachinery aeroelastic analysis [HCL94], benefitting from the computational advantages of the linearised approach, without the many drawbacks of shock fitting.
There has been very little prior research into adjoint equations for flows with shocks. Giles and Pierce [GP01] have shown that the analytic derivation of the adjoint equations for the steady quasionedimensional Euler equations requires the specification of an internal adjoint boundary condition at the shock. However, the numerical evidence [GP98] is that the correct adjoint solution is obtained using either the "fully discrete" approach (in which one linearises the discrete equations and uses the transpose) or the "continuous" approach (in which one discretises the analytic adjoint equations). It is not
clear though that this will remain true in two dimensions, for which there is a similar adjoint boundary condition along a shock.
In this paper, we consider unsteady onedimensional hyperbolic equations with a convex scalar flux, and in particular obtain numerical results for Burgers equation. Tadmor [Tad91] developed a Lip' topology for the formulation of adjoint equations for this problem, with application to linear postprocessing functionals. Building on this and the work of Bouchut and James [BJ98], Ulbrich has very recently introduced the concept of shiftdifferentiability [Ulb02a, Ulb02b] to handle nonlinear functionals of the type considered in this paper. This supplies the analytic adjoint solution against which the numerical solutions in this paper will be compared. An alternative derivation of this analytic solution is presented in an expanded version of this paper [Gil02].
Analytic Adjoint Solutions for the Quasi1D Euler Equations
"... this paper we have undertaken a detailed investigation of adjoint solutions for the quasi1D Euler equations, focusing in particular on the solution behaviour at a shock or a sonic point where there is a change in sign of one of the hyperbolic characteristics. Formulating the adjoint equations using ..."
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Cited by 6 (1 self)
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this paper we have undertaken a detailed investigation of adjoint solutions for the quasi1D Euler equations, focusing in particular on the solution behaviour at a shock or a sonic point where there is a change in sign of one of the hyperbolic characteristics. Formulating the adjoint equations using Lagrange multipliers to enforce the RankineHugoniot shock jump conditions proves that, contrary to previous literature, the adjoint variables are continuous at the shock. This result is supported by the derivation of a closed form solution to the adjoint equations using a Green's function approach. In addition to proving the existence of a log(x) singularity at the sonic point, this closed form solution should be very helpful as a test case for others developing numerical methods for the adjoint equations. Future research will attempt to extend this analysis to two dimensions. Preliminary analysis, supported by the results of numerical computations (Giles & Pierce 1997), shows that the adjoint variables are again continuous at a shock, and that an adjoint boundary condition is required along the length of the shock. However, since adjoint computations currently employed for transonic aerofoil optimisation do not enforce this internal boundary condition, it remains an open question as to whether there is a consistency error in the limit of increasing grid resolution. In two dimensions, numerical evidence suggests that there is no longer a singularity at a sonic line if (as is usually the case) it is not orthogonal to the ow. This can be explained qualitatively by considering the region of inuence of points in the neighbourhood of the sonic line (Giles & Pierce 1997). An important new feature that must be considered for twodimensional ows is the behavior of the adjoint sol...
The Harmonic Adjoint Approach to Unsteady Turbomachinery Design
 ICFD Conference
, 2001
"... This paper demonstrates how the worksum output produced by the linear harmonic ow analysis can be obtained by an adjoint harmonic analysis which, under certain conditions, is a more ecient alternative to the usual linear approach. The adjoint approach has been developed for aeronautical optimal desi ..."
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Cited by 6 (2 self)
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This paper demonstrates how the worksum output produced by the linear harmonic ow analysis can be obtained by an adjoint harmonic analysis which, under certain conditions, is a more ecient alternative to the usual linear approach. The adjoint approach has been developed for aeronautical optimal design by Jameson [10, 11]. At each optimisation step, a single adjoint ow calculation determines the sensitivity of a steadystate functional (e.g. lift or drag) to a large number of geometric design parameters. The same idea is applied in this paper in the context of linear unsteady ow analysis, to compute the worksum values corresponding to any input unsteady ow perturbations, whereas the usual approach would require a separate linear unsteady ow calculation for each set of inputs