Aerodynamic design is a computationally expensive process because high-fidelity compu-tational fluid dynamics (CFD) must repeatedly solve nonlinear governing equations for a large number of degrees of freedom. This section presents an adjoint-based formulation for an aerodynamic design problem described by a CFD model and then presents results of a multifidelity supersonic airfoil design problem.
To minimize the drag of a supersonic airfoil we formulate an optimization problem in the form given in Eq. 4.1. Jhigh(x, u) is the surface integral of pressure acting in the flow direction, Rhigh(x, u) = 0 represents the discretized Euler equations, and g(x) ≤ 0 comprises two constraints on the airfoil geometry. The state variables u are the primal flow variables for all of the control volumes in a finite volume discretization of the governing equations.
In order to use the multifidelity optimization technique presented, we need to compute the
gradient of Jhigh(x, u) with respect to the design variables x given that Rhigh(x, u) = 0. We start by writing the gradient of Jhigh(x, u),
dJhigh
where M(x) represents the dependence of each nodal vertex in the volume mesh on the design vector. For this analysis, a meshing tool was developed that created a volume mesh around the airfoil with an analytical derivative. Accordingly, for any airfoil that could be generated with the parameterization used, the volume mesh and mesh derivative were known a priori.
Since the flow residual must always be zero for a converged solution we know that, dRhigh
Therefore, combing Eqs. 4.31 and 4.32 with the adjoint equation, Eq. 4.25, we may write the gradient of an objective function with the state equations satisfied as
dfhigh
This reformulation shows how to convert the constrained objective function into an objective function from which the state variables have been eliminated through the solution of the residual equations. Accordingly, the gradient dfdxhigh represents the gradient of drag with respect to the design variables given that the discretized Euler equations are satisfied. For further discussion see Jameson [49] or Nemec et al. [82]. The computational effort required to compute this gradient requires one flow solution, one adjoint solution, one flow iteration per design variable (about 1/500 the effort of a flow solution), and the matrix multiplications shown above. Therefore, the gradient of the objective with respect to all of the design variables requires the computational effort of about 2 flow solutions plus n/500 flow solutions, where n is the number of design variables. For comparison, a finite difference gradient estimate requires at least n + 1 flow solutions, so this is a considerable savings and shows the
cost of the gradient estimate using an adjoint solution is almost independent of the number of design variables.
In addition to the Euler equations as the high-fidelity method, we use a panel method as a low-fidelity analysis. A supersonic panel method can be derived from supersonic small-disturbance theory, and is only a function of the airfoil geometry, the freestream Mach number, M∞, and the gas specific heat ratio. In small disturbance theory, the change in the pressure coefficient, δCp is proportional, to the flow turning angle δϑ,
δCp = −2δϑ
pM∞2 − 1, (4.34)
and by integrating the pressure coefficient around the airfoil surface the wave drag can be easily estimated [66]. Since the drag coefficient is only a function of the freestream Mach number and airfoil geometry, the analytical derivative of drag coefficient with respect to the airfoil shape design parameters is easy to compute.
The airfoil optimization problem is to minimize the drag of an airfoil at M∞ = 1.5, by changing the angle of attack, five upper surface spline points, and five lower surface spline points. The airfoil is required to have positive thickness everywhere, and to have a maximum thickness to chord ratio that is at least five percent. Figure 4-4 shows the optimal airfoil and spline control points for the panel method. Figure 4-5 shows the optimal airfoil pressure contours and adjoint solution for the streamwise momentum from the Euler method solutions.
Table 4.3 presents the number of high-fidelity function calls to find the minimum drag airfoil with respect to the Euler code. The results show that the conventional first-order consistent trust-region and the Bayesian model calibration approach reduce the number of high-fidelity function calls by nearly the same amount, about 80%. In addition, because the low-fidelity model is computationally very inexpensive, and the dimension of the parameter space is small, this is nearly a 70% reduction in wall-clock time. However, it should be noted that the conventional trust-region approach does use on average fewer high-fidelity function calls than the Bayesian calibration method for the ten random initial airfoils.
Figure 4-4: Minimum drag airfoil computed with the supersonic panel method showing the spline control points.
SQP First-Order TR Calibration Mean 81.5 (-) 12.9 (-84%) 14.7 (-82%)
Std. Dev. 14.0 2.86 4.19
Table 4.3: The average number of high-fidelity function evaluations to minimize the drag of a supersonic airfoil with respect to an Euler solution using a panel method as a low-fidelity estimate. The numbers in parentheses indicate the percentage reduction in high-fidelity function evaluations relative to SQP.
The same optimization problem is solved using the same methods, but without any low-fidelity information. The results of this optimization using flow(x) = 0 as the low-fidelity function are presented in Table 4.4. In this case the Bayesian model calibration approach performed noticeably better than the conventional trust-region method. The results suggest that when using a “good” low-fidelity model, the Bayesian calibration approach is not neces-sary and the computational effort of constructing the Cokriging surrogates is not worthwhile.
However, when the low-fidelity function is poor, or when the error between the high- and low-fidelity function behaves in a highly non-linear fashion, then the Bayesian calibration approach may provide a computational savings for low-dimensional optimization problems.
A possible performance issue that affects Bayesian calibration approach significantly more than the conventional trust-region approach is when high-fidelity gradients are computed
in-SQP First-Order TR Calibration Mean 81.5 (-) 91.1 (+12%) 64.2 (-21%)
Std. Dev. 14.0 61.2 21.8
Table 4.4: The average number of high-fidelity function evaluations to minimize the drag of a supersonic airfoil with respect to an Euler solution. No low-fidelity information is used. The numbers in parentheses indicate the percentage reduction in high-fidelity function evaluations relative to SQP.
accurately. The calibration approach reuses high-fidelity gradient information, so incorrect gradient information will propagate into the surrogate models generated at future trust-region iterations. In contrast, the conventional approach only uses the high-fidelity gradient to create a single surrogate model so the approach is likely less affected by inaccurate gradient information.
(a) Optimal airfoil pressure solution. (b) Optimal airfoil adjoint solution.
Figure 4-5: Minimum drag supersonic airfoil parameterized by 5 upper surface spline points, 5 lower surface spline points, and angle of attack.
4.4 Summary
This chapter has presented a multifidelity optimization algorithm using Cokriging-based Bayesian model calibration that is provably convergent to an optimum of the original high-fidelity optimization problem. Inexpensive derivative information was obtained through
adjoint solutions for both structural design and aerodynamic shape optimization problems.
The derivative estimates and function values from previously visited design sites are used to calibrate the low-fidelity model to the high-fidelity function. Using this strategy, the op-timization algorithm is likely to quickly find an optimum of the high-fidelity function. The sample results show that for poor low-fidelity models the Bayesian calibration exceeds perfor-mance of conventional trust-region algorithms; however, for cases with good low-fidelity mod-els the performance of the two algorithms is similar. However, the multifidelity approaches still significantly outperform single-fidelity sequential quadratic programming methods. The calibration technique developed is recommended for low-dimensional optimization problems where the quality of the low-fidelity model is unknown or known to be poor in certain portions of the design space.
This algorithm complements the work in Chapters 2 and 3, in that although a first-order consistent calibration strategy satisfies the requirement for a fully linear calibration, an optimization algorithm based on a fully linear calibration does not take advantage of gradient information. The algorithm in this chapter attempts to truly exploit gradient information to find the optimal design as quickly as possible by efficiently using available gradient information. This strategy still supports the use of multiple lower-fidelity models through the filtering technique presented in Section 2.5, and both fully linear and first-order consistent surrogates can be combined with that technique.
Chapter 5
Multidisciplinary and Multifidelity Optimization
Chapters 2, 3, and 4 presented techniques for multifidelity optimization and demonstrated how to accelerate finding optimal designs using computationally expensive simulations. This chapter considers the case of designing a system with multiple interacting disciplines or sub-systems, and focuses on the case where each is modeled with an expensive high-fidelity simulation. The challenge of optimizing multidisciplinary systems is two-fold. First, it is necessary to resolve the coupling between all disciplines in order to find a feasible design, one where the inputs and outputs from communicating disciplines match, and second, it is nec-essary to analyze the trade-offs among all disciplines in order to find an optimal design. For systems where each discipline analysis is a high-fidelity simulation, multifidelity optimization alone may be insufficient to make designing an optimal system tractable. Therefore, this chapter develops multidisciplinary optimization methods that enable multifidelity optimiza-tion at the discipline level and that enable the design and analysis of each subsystem to be conducted in parallel.
This chapter will first formulate a multidisciplinary optimization problem in Section 5.1.
It then presents two multidisciplinary and multifidelity optimization methods. The first method uses an Individual Discipline Feasible (IDF) approach, which decomposes the system optimization problem into a collection of smaller discipline-level optimization problems that
may be conducted in parallel. The IDF approach is presented in Section 5.2. The second MDO optimization method is based on an All-At-Once (AAO) formulation, though it is in actuality just a gradient-free method for solving equality constrained optimization problems.
The AAO approach is presented in Section 5.3. The two approaches are compared with some baseline methods on two analytical test problems in Section 5.4. This chapter concludes with Section 5.5 which contains a discussion about these methods, their applications, and possible extensions.
5.1 Problem Formulation
We consider a system design problem comprising m disciplines or subystems. Each discipline produces a response that is a function of both the system design and the output of the other disciplines. The response of a discipline may be thought of as the coupling between the disciplines. For example, the ith discipline has a response, ri = ri(x, r1,...,m\i, ui(x, r1,...,m\i)), that is a function of the n-dimensional system design vector, x, the responses of all other disciplines, r1,...,m\i, and the discipline state variables, ui(x, r1,...,m\i). The vector r without superscript is the vector of all ri, r = [(r1)>, (r2)>, . . . , (rm)>]>, and has length v. To find a multidisciplinary feasible design, an iterative solution scheme is typically required to find the set of state variables and discipline responses that enable a feasible coupling among all disciplines. This iterative solve is typically called a multidisciplinary analysis (MDA).
When optimizing a multidisciplinary system using a multidisciplinary feasible (MDF) formulation, the first step in evaluating the performance of a given design vector, x, is the MDA, to calculate the state and coupling variables that satisfy the multidisciplinary feasibility [34, 111]. In this case, we may write an MDF optimization problem as
min
x∈Rn F (x, r(x)) (5.1a)
s.t. G(x, r(x)) ≤ 0 (5.1b)
H(x, r(x)) = 0, (5.1c)
(a) Block diagram for an MDF framework.
(b) Block diagram for a decoupled framework like IDF.
Figure 5-1: Block diagram for a system with, (a), and without, (b), interdisciplinary com-munication.
where we are minimizing a scalar objective function, F (x, r(x)), subject to inequality con-straints, G(x, r(x)) ≤ 0, and equality concon-straints, H(x, r(x)) = 0. In the MDF formula-tion we have omitted the dependence of the discipline responses on the state variables and other discipline responses since they are satisfied by construction (i.e., each discipline per-forms an internal iteration to satisfy its state constraints, Ri(x, r1,...,m\i, ui(x, r1,...,m\i)) = 0).
This formulation is presented schematically as Figure 5-1(a). Each time the system per-formance, F (x, r(x)), is evaluated an iterative solution process amongst the disciplines has been undertaken to satisfy feasibility. Some challenges with a MDF framework are that if finite-differences are necessary to generate sensitivity information, then an iterative solve is necessary for each perturbed component of x. In addition, an MDF formulation only supports multifidelity optimization and parallelization at the system level, evaluating per-formance of multiple complete system designs at the same time. To parallelize the system optimization we develop two alternative formulations of (5.1). The first decomposes the large system optimization problem into a collection of smaller subsystem optimization problems,
as shown in Figure 5-2(a). This framework is presented in Section 5.2. The other evalu-ates all computationally expensive functions in parallel, as shown in Figure 5-2(b). This framework is presented in Section 5.3.
(a) Parallel discipline optimizations.
(b) Parallel discipline evaluations.
Figure 5-2: Block diagram for optimizing a system with parallel discipline optimizations, (a), and with parallel discipline evaluations, (b). It may also be possible in both frame-works to evaluate multiple design sites of the same discipline in parallel during the discipline optimizations or during the discipline surrogate creation.