# Multi-Fidelity Gradient-Based Strategy for Robust Optimization in Computational Fluid Dynamics
## 2 Problem Definition
### Test Problem: Quasi-1D Supersonic Nozzle  
Various RDO strategies are assessed against an inexpensive test problem (also studied in [10; 17]), namely the inverse design of a supersonic quasi-1D diverging nozzle. This allows validating UQ methods against MC sampling. The nozzle geometry is assigned through the area distribution \(S(x)\) along the longitudinal axis \(x\). This is chosen to be of the form:  
\[S(x)=a+b\ tanh(cx-d) \tag{2}\]  
with \(a\), \(b\), \(c\), and \(d\) coefficients defining the geometry. The nozzle length is set to \(L=10\). A typical nozzle geometry is depicted in Figure 1.  
For this test case, the QoI \(J\) (Equation (3)) is a scalar, namely the mean quadratic error of the actual pressure distribution in the nozzle \(P(x)\) with respect to the target design pressure distribution \(P_{des}(x)\). The latter corresponds to the pressure distribution for a nozzle geometry with \(a=1.75\), \(b=0.699\)  
Figure 1: Baseline quasi-1D nozzle. \(c=1.00\), and \(d=3.80\), a reservoir pressure \(P_{T,in}=1\) bar, an outlet static pressure \(P_{out}=0.6\) bar, and a gas specific heat capacity ratio \(\gamma=1.4\). \[J=\frac{1}{2}\int_{0}^{L}{(P-P_{des})^{2}dx} \tag{3}\]  
The optimization goal is to determine the design parameters \(a,\ b,\ c,\ d\) in Equation (2) providing the best fit to the target pressure distribution under multiple uncertainties, in the sense of Equation (1). The flow is assumed to be governed by the steady Euler equations for quasi-1D flows (Equation (4)):  
\[R(w,x)=\frac{\partial(fS)}{\partial x}+K\frac{dS}{dx}=0 \tag{4}\]  
where \(w=[\rho,\rho v,\rho e_{t}]^{T}\) and \(f(w)=[\rho v,\rho v^{2}+P,\rho vh_{t}]^{T}\) are, respectively, the conservative variable and the physical flux vectors [59]. In the preceding equations, \(\rho\) is the fluid density, \(v\) is the velocity along the nozzle axis, \(e_{t}\) and \(h_{t}\) are the total specific energy and enthalpy, and \(K=[0,-P,0]^{T}\). The system of equations is supplemented by the equation of state for thermally and calorically perfect gases, \(P=(\gamma-1)\rho(e_{T}-v^{2}/2)\). The governing equations are discretized by a cell-centered finite volume formulation, using [PERSON]'s first-order upwind scheme for space integration. The steady state solution is computed iteratively by solving a false transient with four-stage explicit Runge-Kutta time-stepping [59]. Characteristic boundary conditions based on Riemann invariants are imposed at the nozzle inlet and outlet. Sonic flow conditions are prescribed at the inlet, so that all Riemann invariants enter the domain. The range of variation of the total pressure is such that a shock is always created in the divergent nozzle. As a consequence, outlet flow conditions are always subsonic. In this case, we impose the outlet static pressure, which is treated as deterministic, and fixed to \(P_{out}=0.6\) bar. Based on a preliminary mesh study, a computational grid of 300 uniformly spaced cells is used in all of the following calculations. The system is assumed to be subject to uncertainties of various nature, specifically:  
* geometric tolerances on the nozzle shape, modeled by treating the shape parameters \(a,\ b,\ c,\ d\) as normally distributed random variables, with mean \(\mu\) and coefficient of variation \(CoV=\sigma/\mu\), with \(\sigma\) the standard deviation;
* uncertainties in inlet total pressure \(P_{T,in}\) described as a uniformly-distributed random variable with imposed lower and upper bounds;
* uncertainties in the gas properties, here represented by the specific heat ratio \(\gamma\), which is also assumed as uniformly distributed. The characteristics of the random parameters are listed in Table 1. In the inverse design process, the uncertain geometric parameters are also (uncertain) design variables: for this reason, their mean is not fixed, but varies within the ranges corresponding to the bounds of the design space.