Non-myopic multipoint multifidelity Bayesian framework for multidisciplinary design

The adoption of high-fidelity models in multidisciplinary design optimization (MDO) permits to enhance the identification of superior design configurations, but would prohibitively rise the demand for computational resources and time. Multifidelity Bayesian Optimization (MFBO) efficiently combines information from multiple models at different levels of fidelity to accelerate the MDO procedure. State-of-the-art MFBO methods currently meet two major limitations: (i) the sequential adaptive sampling precludes parallel computations of high-fidelity models, and (ii) the search scheme measures the utility of new design evaluations only at the immediate next iteration. This paper proposes a Non-Myopic Multipoint Multifidelity Bayesian Optimization (NM3-BO) algorithm to sensitively accelerate MDO overcoming the limitations of standard methods. NM3-BO selects a batch of promising design configurations to be evaluated in parallel, and quantifies the expected long-term improvement of these designs at future steps of the optimization. Our learning scheme leverages an original acquisition function based on the combination of a two-step lookahead policy and a local penalization strategy to measure the future utility achieved evaluating multiple design configurations simultaneously. We observe that the proposed framework permits to sensitively accelerate the MDO of a space vehicle and outperforms popular algorithms.

where the thrust vector F = [F V , F N ] is given by the components tangential F V and normal F N to the descend orbit, c is the effective exhaust velocity, t on is the initial time of the maneuver, t o f f corresponds to the time when the maneuver is completed, and ∆ = t o f f − t on .

Trajectory Model
The trajectory model computes the descend orbit determined by the re-entry maneuver F. This disciplinary analysis requires in input the aerodynamic coefficients that characterize the re-entry vehicle computed through either the high-fidelity or the low-fidelity aerothermodynamic model, and the trajectory design parameters reported in Table 2.
The re-entry profile in terms of descend velocity V , flight path angle γ, altitude during the re-entry h, and longitude angle β is obtained considering a planar trajectory assuming the Earth with negligible rotation and constant flight path azimuth angle: where t is the re-entry time, M is the overall mass of the vehicle, D and L are respectively the aerodynamic drag and lift, g is the acceleration of gravity, and R E is the Earth radius.
The system of non-linear ODEs (Equation ( 2)) is numerically solved adopting the Runge-Kutta method integrating the equations over the re-entry time.The entry maneuver is considered as impulsive given the contained burning time that characterize the chemical thrusters, allowing to consider the effect of the thrust components F V and F N exclusively during the first integration step.

High-Fidelity Aerothermodynamic Model
The high-fidelity aerothermodynamic model computes the heat flux q at the stagnation point of the thermal protection system and the aerodynamic coefficients of lift C L and drag C D .This model requires in input the geometric features of the re-entry vehicle (Table 3) together with the re-entry profile in output from the trajectory model and the temperature of the heat shield T T PS computed with the thermo-structural model.The flow-field experienced by the vehicle during the re-entry is modelled through the full set of Reynolds-Averaged Navier-Stokes (RANS) equations to account for the effects of turbulence and unsteadiness.We use the finite volume method to discretize the RANS equations in space, adopting a standard edge-based data structure where the convective and viscous fluxes are computed at the midpoint of the edges.The fluid domain is geometrically defined as a semicircle of radius 6.3R N to avoid shock reflections, and is discretized with a total of 9.2 • 10 4 quads elements.The computational grid is characterizes by a refined density of the mesh in proximity of the thermal protection system of the vehicle, to accurately capture the severe aerothermodynamic phenomena critical for the structural frame.The RANS equations are numerically solved through SU2 (2) version 7.0.3computational fluid dynamic solver, and the computational grid is generated adopting the gmsh software (3).The governing equations are integrated through the Euler implicit scheme, where the convergence criteria is set for computational residuals 10 −6 .

Low-Fidelity Aerothermodynamic Model
The low-fidelity aerothermodynamic model provides a fast approximation of the aerodynamic coefficients C L and C D of the vehicle and the stagnation heat flux q affecting the TPS frame, given the re-entry profile and the geometry of the capsule (Table 3).This representation uses two physics-based surrogate models that allow to evaluate the aerothermodynamic phenomena with a fraction of the computational cost required to compute the high-fidelity model.Specifically, the aerodynamic coefficients are approximated through the Oswatitsch Mach number independence principle (4), and the stagnation heat flux is computed as the sum of convective and radiative effects adopting the Sutton-Grave (5) and Tauber-Sutton (6) formulations.The Mach number independence principle defines the aerodynamic coefficients constant with altitude assuming that the flow-field is governed by inviscid Euler equations; this implies that the flow-field tends to a limit condition at high values of the Mach number that characterizes the re-entry trajectory, for which C L and C D assume constant limit values.
The total heat flux at the stagnation point is computed as follows: The convective heat flux is evaluated following the Sutton-Grave formulation ( 5): where k s = 5.1564 • 10 −5 is a constant for the Earth atmosphere, R N is the radius of the nose of the capsule and V is the re-entry flight velocity.
The radiative heat flux is estimated according to the Tauber-Sutton formulation (6): where C = 4.736 • 10 4 and b = 1.22 are constants for the Earth atmosphere, a = 1.072 • 10 6 V −1.88 ρ −0.325 ∞ is given in function of the descend velocity V and the density of the atmosphere ρ ∞ (h), and f (V ) is a tabulated function of velocity.

Thermo-Structural Model
The thermo-structural representation models the interactions between the flow field heat fluxes affecting the structure and the frame of the thermal protection system.This model evaluates the temperature of the TPS structure T T PS and the mass of the TPS frame m T PS , given the total heat load q provided by the high-fidelity or low-fidelity aerothermodynamic analysis, the thickness of the TPS structure s T PS , the geometry of the capsule (Table 3), and the material property of the TPS (Table 4).
The thermo-structural interaction is evaluated adopting the finite element method to numerically approximate the governing heat equation.In particular, the TPS frame is represented as an arc of circumference discretized with n e = 1000 linear elements; this permits to formulate the heat equation for the generic e-th finite element considering a linear formulation where the thermal conductivity κ T PS and the thickness of the thermal protection system s T PS are assumed uniform: where ρ T PS , c P and ε T PS are respectively the density, the specific heat at constant pressure and the emissivity coefficient of the TPS material, σ is the Stephan-Boltzmann constant, qs is the heat source term and T ∞ (h) is the temperature of the atmosphere.The numerical solution of Equation ( 6) is computed through the Galerkin method over the discretized domain that models the structure of the TPS.
In addition, the thermo-structural model estimates the total mass of the structural frame of the TPS m T PS as a function of the thickness s T PS : m T PS = ρ T PS S T PS s T PS (7) where S T PS is the frontal surface of the structure of the TPS which corresponds to the circular are with radius equal to the radius of the nose of the capsule R N .

Table 1 .
Propulsion System Design Parameters.
A re f = 78.54m 2 Altitude for parachute deployment h = 5000 m

Table 3 .
Geometry Design Paramaters of the Re-Entry Vehicle.