Analytical Test Cases Research Articles

Bayesian optimization using deep Gaussian processes with applications to aerospace system design

Bayesian Optimization using Gaussian Processes is a popular approach to deal with optimization involving expensive black-box functions. However, because of the assumption on the stationarity of the covariance function defined in classic Gaussian Processes, this method may not be adapted for non-stationary functions involved in the optimization problem. To overcome this issue, Deep Gaussian Processes can be used as surrogate models instead of classic Gaussian Processes. This modeling technique increases the power of representation to capture the non-stationarity by considering a functional composition of stationary Gaussian Processes, providing a multiple layer structure. This paper investigates the application of Deep Gaussian Processes within Bayesian Optimization context. The specificities of this optimization method are discussed and highlighted with academic test cases. The performance of Bayesian Optimization with Deep Gaussian Processes is assessed on analytical test cases and aerospace design optimization problems and compared to the state-of-the-art stationary and non-stationary Bayesian Optimization approaches.

Marching On-In-Time Unstructured PEEC Method for Electrically Large Structures with Conductive, Dielectric, and Magnetic Media

The Marching On-In-Time (MOT) unstructured Partial Element Equivalent Circuit (PEEC) method for time domain electromagnetic problems is presented. The method allows the transient analysis of electrically large electromagnetic devices consisting of conductive, dielectric, and magnetic media coupled with external lumped circuits. By re-formulating PEEC following the Coulombian interpretation of magnetization phenomena and by using electric and magnetic vector potentials, the proposed approach allows for a completely equivalent treatment of electric and magnetic media and inhomogeneous and anisotropic materials are accounted for as well. With respect to the recently proposed Marching On-In-Time PEEC approach, based on the standard (structured) discretization of PEEC, the method presented in this paper uses a different space and time MOT discretization, which allows for a reduction in the number of the unknowns. Analytical and industrial test cases consisting in electrically large devices are considered (e.g., the model of a Neutral Beam Injector adopted in thermonuclear fusion applications). Results obtained from the simulations show that the proposed method is accurate and yields good performances. Moreover, when rich harmonic content transient phenomena are considered, the unstructured MOT–PEEC method allows for a significant reduction of the memory and computation time when compared to techniques based on Inverse Discrete Fourier Transform applied to the frequency domain unstructured PEEC approach.

An implicit wetting–drying algorithm for the discontinuous Galerkin method: application to the Tonle Sap, Mekong River Basin

The accurate simulation of wetting–drying processes in floodplains and coastal zones is a challenge for hydrodynamic modelling, especially for long time simulations. Indeed, dedicated numerical procedures are generally time-consuming, instabilities can occur at the wet/dry front, rapid transition of wet/dry interface and mass conservation are not always ensured. We present the extension of an existing wetting–drying algorithm in two space dimensions and its application to a real case. The wetting–drying algorithm is implemented in Second-generation Louvain-la-Neuve Ice-ocean Model ( www.slim-ocean.be ), a discontinuous Galerkin finite element model solving the shallow water equations in a fully implicit way. This algorithm consists in applying a threshold value of fluid depth for a thin layer and a blending parameter in order to guarantee positive values of the water depth, while preserving local mass conservation and the well balanced property at wet/dry interfaces. The technique is first validated against standard analytical test cases (Balzano 1, Balzano 3 and Thacker test cases) and is subsquently applied in a realistic domain, the Tonle Sap Lake in the Mekong River Basin, where the water level can vary by about 10 m between the dry and the wet season.

Aerodynamic/aeroacoustic variable-fidelity optimization of helicopter rotor based on hierarchical Kriging model

Rotor noise is one of the most important reasons for restricting helicopter development; hence, the optimization design of rotor blade considering aeroacoustic and aerodynamic performance at the same time has always been the focus of research attention. For complex rotor design problems with a large number of design variables, the efficiency of the traditional Kriging model needs to be improved. Thus, Hierarchical Kriging (HK) model is employed in this study for rotor optimization design. By using the validated RANS solver and acoustic method based on the FW–Hpds equation, an efficient aerodynamic/aeroacoustic optimization method for high-dimensional problem of rotors in hover based on HK model is developed. By using present HK model and new infill-sampling criteria, the number of design variables is increased from less than 20–53. Results of two analytical function test cases show that the HK model is efficient and accurate in calculation. Subsequently, the helicopter rotor blade is optimally designed for aerodynamic/aeroacoustic performance in hover based on the HK model with high dimensional design variables. The objective function is adopted to improve the rotational noise characteristics by reducing the absolute peak of the acoustic pressure. In addition, the constraints of thrust, hover efficiency, solidity, and airfoils thickness are strictly satisfied. Optimization results show that the Kriging model finds the objective of reducing the noise by 2.87 dB after 248 iterations while the HK model does it only after 164 iterations. The optimization efficiency of the HK model is significantly higher than that of the traditional Kriging model. In the case analyzed, the HK model saves 35% of the time used by the Kriging model.

Applications of surrogate-assisted and multi-fidelity multi-objective optimization algorithms to simulation-based aerodynamic design

Purpose The purpose of this work is to apply and compare surrogate-assisted and multi-fidelity, multi-objective optimization (MOO) algorithms to simulation-based aerodynamic design exploration. Design/methodology/approach The three algorithms for multi-objective aerodynamic optimization compared in this work are the combination of evolutionary algorithms, design space reduction and surrogate models, the multi-fidelity point-by-point Pareto set identification and the multi-fidelity sequential domain patching (SDP) Pareto set identification. The algorithms are applied to three cases, namely, an analytical test case, the design of transonic airfoil shapes and the design of subsonic wing shapes, and are evaluated based on the resulting best possible trade-offs and the computational overhead. Findings The results show that all three algorithms yield comparable best possible trade-offs for all the test cases. For the aerodynamic test cases, the multi-fidelity Pareto set identification algorithms outperform the surrogate-assisted evolutionary algorithm by up to 50 per cent in terms of cost. Furthermore, the point-by-point algorithm is around 27 per cent more efficient than the SDP algorithm. Originality/value The novelty of this work includes the first applications of the SDP algorithm to multi-fidelity aerodynamic design exploration, the first comparison of these multi-fidelity MOO algorithms and new results of a complex simulation-based multi-objective aerodynamic design of subsonic wing shapes involving two conflicting criteria, several nonlinear constraints and over ten design variables.

On the assessment, implementation, validation, and verification of drag and lift forces in gas–liquid applications for the CFD codes FLUENT and CFX

The understanding of two-phase gas–liquid flows is of utmost importance in a large range of industrial applications, including the petrochemical, pharmaceutical, biochemical, nuclear, and metallurgical industries. At ANSYS, significant effort is being made in assessing the physical force models present in both FLUENT and CFX. This includes the investigation of the interfacial closures (drag, lift, wall lubrication, turbulent dispersion, and virtual mass), heat and mass transfer, cavitation, wall boiling, population balance approaches, bubble breakup and coalescence, and turbulence modeling. This assessment is being done with the objective to conduct an audit/validation of the current models, features, and capabilities, as well as the identification and closing of gaps and differences between CFX and FLUENT. The work presented here is mostly focused on the assessment, implementation, and validation of the drag and lift interfacial closures. The numerical assessment and validation are performed using both analytical and industrial-like test cases for complex bubbly flows (both with wall and bulk void fraction maximums), as well as transitional flow from bubbly to slug regime using the Helmholtz-Zentrum Dresden-Rossendorf (HZDR) experimental facility known as MT-Loop.

Implementation of an immersed boundary method in the Meso-NH v5.2 model: applications to an idealized urban environment

Abstract. This study describes the numerical implementation, verification and validation of an immersed boundary method (IBM) in the atmospheric solver Meso-NH for applications to urban flow modeling. The IBM represents the fluid–solid interface by means of a level-set function and models the obstacles as part of the resolved scales. The IBM is implemented by means of a three-step procedure: first, an explicit-in-time forcing is developed based on a novel ghost-cell technique that uses multiple image points instead of the classical single mirror point. The second step consists of an implicit step projection whereby the right-hand side of the Poisson equation is modified by means of a cut-cell technique to satisfy the incompressibility constraint. The condition of non-permeability is achieved at the embedded fluid–solid interface by an iterative procedure applied on the modified Poisson equation. In the final step, the turbulent fluxes and the wall model used for large-eddy simulations (LESs) are corrected, and a wall model is proposed to ensure consistency of the subgrid scales with the IBM treatment. In the second of part of the paper, the IBM is verified and validated for several analytical and benchmark test cases of flows around single bluff bodies with an increasing level of complexity. The analysis showed that the Meso-NH model (MNH) with IBM reproduces the expected physical features of the flow, which are also found in the atmosphere at much larger scales. Finally, the IBM is validated in the LES mode against the Mock Urban Setting Test (MUST) field experiment, which is characterized by strong roughness caused by the presence of a set of obstacles placed in the atmospheric boundary layer in nearly neutral stability conditions. The Meso-NH IBM–LES reproduces with reasonable accuracy both the mean flow and turbulent fluctuations observed in this idealized urban environment.

Modeling discontinuous potential distributions using the finite volume method, and application to liquid metal batteries

The electrical potential in a battery jumps at each electrode-electrolyte interface. We present a model for computing three-dimensional current and potential distributions, which accounts for such internal voltage jumps. Within the framework of the finite volume method we discretize the Laplace and gradient operators such that they account for internal jump boundary conditions. After implementing a simple battery model in OpenFOAM we validate it using an analytical test case, and show its capabilities by simulating the current distribution and discharge curve of a Li‖Bi liquid metal battery.

A Novel and Systematic Approach to Inhomogeneous Dielectric Lens Design Based on Curved Ray Geometrical Optics and Particle Swarm Optimization

A numerical synthesis algorithm is presented to design inhomogeneous 3-D-printed lenses for any aperture field and, hence, beam pattern. The algorithm is based on geometrical optics (GO), a numerical method for wave propagation in inhomogeneous media, particle swarm optimization, and a global stochastic optimization technique. The approach allows the synthesis of either optimal geometry with known constant permittivity (shape only), optimal nonconstant permittivity with known geometry (material only), or optimal geometry and nonconstant permittivity (shape and material). The presented approach is validated against the analytic test case of the Maxwell fish eye and is shown to provide excellent agreement. A spherical lens is optimized for nonconstant permittivity function and is shown to recover the permittivity of the Luneburg lens. Instructions on how to fabricate the resultant inhomogeneous lenses via 3-D-printing techniques is given. Examples of optimum inhomogeneous lenses synthesized with our approach are provided along with the full-wave simulation verification. A 30 cm inhomogeneous lens is fabricated via 3-D-printing and measured in the UCLA’s plane bipolar near-field measurement range and compared with the results of the GO algorithm.

Efficient aerodynamic shape optimization using variable-fidelity surrogate models and multilevel computational grids

A variable-fidelity method can remarkably improve the efficiency of a design optimization based on a high-fidelity and expensive numerical simulation, with assistance of lower-fidelity and cheaper simulation(s). However, most existing works only incorporate “two” levels of fidelity, and thus efficiency improvement is very limited. In order to reduce the number of high-fidelity simulations as many as possible, there is a strong need to extend it to three or more fidelities. This article proposes a novel variable-fidelity optimization approach with application to aerodynamic design. Its key ingredient is the theory and algorithm of a Multi-level Hierarchical Kriging (MHK), which is referred to as a surrogate model that can incorporate simulation data with arbitrary levels of fidelity. The high-fidelity model is defined as a CFD simulation using a fine grid and the lower-fidelity models are defined as the same CFD model but with coarser grids, which are determined through a grid convergence study. First, sampling shapes are selected for each level of fidelity via technique of Design of Experiments (DoE). Then, CFD simulations are conducted and the output data of varying fidelity is used to build initial MHK models for objective (e.g. CD) and constraint (e.g. CL, Cm) functions. Next, new samples are selected through infill-sampling criteria and the surrogate models are repetitively updated until a global optimum is found. The proposed method is validated by analytical test cases and applied to aerodynamic shape optimization of a NACA0012 airfoil and an ONERA M6 wing in transonic flows. The results confirm that the proposed method can significantly improve the optimization efficiency and apparently outperforms the existing single-fidelity or two-level-fidelity method.

Implicit Solvation Using a Generalized Finite-Difference Approach in CRYSTAL: Implementation and Results for Molecules, Polymers, and Surfaces.

We present the implementation of an implicit solvation model in the CRYSTAL code. The solvation energy is separated into two components: the electrostatic contribution arising from a self-consistent reaction field treatment obtained within a generalized finite-difference Poisson model, augmented by a nonelectrostatic contribution proportional to the solvent-accessible surface area of the solute. A discontinuous dielectric boundary is used, along with a solvent-excluded surface built from interlocking atom-centered spheres on which apparent surface point charges are mapped. The procedure is general and can be performed at both the Hartree-Fock and density functional theory levels, with pure or hybrid functionals, for systems periodic in 0, 1, and 2 directions, that is, for isolated molecules and extended polymers and surfaces. The Poisson equation resolution and apparent surface charge formalism is first validated on model analytical test cases. The good agreement obtained on solvation free energies is further confirmed by calculations performed on a large test set of 501 neutral molecules, for which a mean unsigned error of 1.3 kcal/mol is obtained when compared to the available experimental data. Importantly, the self-consistent reaction field procedure converges well for all molecules tested. This is further verified for all polymers and surfaces considered. In particular, for periodic systems, results obtained on an infinite glycine chain and on the wettability parameters of SiO2 surfaces are in good agreement with previously published data. The size extensivity of the energetic terms involved in the electrostatic contribution to the solvation energy is also well verified. These encouraging results constitute a first step to take into account complex environments in the CRYSTAL code, potentially allowing for a more accurate modeling of complex processes for both periodic and nonperiodic systems.

Hybrid Computational-Fluid-Dynamics Platformto Investigate Aircraft Trailing Vortices

This paper outlines the development of a parallel three-dimensional hybrid finite volume finite difference capability. The specific application area under consideration is modeling the trailing vortices shed from the wings of aircraft under transonic flight conditions. For this purpose, the Elemental finite volume code is employed in the vicinity of the aircraft, whereas the ESSENSE finite difference software is employed to accurately resolve the trailing vortices. The former method is spatially formally second-order, and the latter is set to sixth-order accuracy. The coupling of the two methods is achieved in a stable manner through the use of summation-by-parts operators and weak imposition of boundary conditions using simultaneous approximation terms. The developed hybrid solver is successfully validated against an analytical test case. This is followed by demonstrating the ability to model the flowfield, including trailing vortex structures, around the NASA Common Research Model under transonic flow conditions. The interface treatment is shown to describe the intersecting vortices in a smooth manner. In addition, insights gained in resolving the vortices include violation of underlying assumptions of analytical vortex modeling methods.

A discrete forcing method dedicated to moving bodies in two‐phase flow

SummaryThe numerical simulation of interaction between structures and two‐phase flows is a major concern for many industrial applications. To address this challenge, the motion of structures has to be tracked accurately. In this work, a discrete forcing method based on a porous medium approach is proposed to follow a nondeformable rigid body with an imposed velocity by using a finite‐volume Navier‐Stokes solver code dedicated to multiphase flows and based on a two‐fluid approach. To deal with the action reaction principle at the solid wall interfaces in a conservative way, a porosity is introduced allowing to locate the solid and insuring no diffusion of the fluid‐structure interface. The volumetric fraction equilibrium is adapted to this novelty. Mass and momentum balance equations are formulated on a fixed Cartesian grid. Interface tracking is addressed in detail going from the definition of the porosity to the changes in the discretization of the momentum balance equation. This so‐called time‐ and space‐dependent porosity method is then validated by using analytical and elementary test cases.

CHAOS: An octree-based PIC-DSMC code for modeling of electron kinetic properties in a plasma plume using MPI-CUDA parallelization

A new computational framework for a coupled PIC-DSMC tool using multiple GPUs to model the kinetic behavior of electrons in plasma plumes is presented in this work. The disparate length scales of the Debye length and the collisional mean free path are resolved by using separate, independent linearized Morton Z-ordered forest of trees. A 2:1 restraint is imposed for the PIC module to solve partial differential equations in the context of an AMR/Octree framework. The MPI-CUDA parallelization strategies used to implement a preconditioned conjugate gradient method for solving the electrostatic Poisson's equation on the 2:1 octree are discussed and the scaling of the code to near ideal speedup as a function of the number of GPUs is demonstrated. The PIC method is validated using analytical test cases, and the octree-based PIC simulations are found to be ten times more efficient compared to the uniform grid method especially for plume simulations which have large density variations. The computational strategies are then demonstrated with the simulation of collisionless, mesothermal plasma plumes using a kinetic approach for both ions and electrons. The effect of ion mass and electron source location are analyzed by comparing plume dynamics and electron velocity distribution functions. It is shown that a more confined mesothermal plume is observed for the heavier xenon ions, present in electric propulsion devices, compared to protons. The confinement of the xenon plume traps the electrons resulting in higher electron temperatures compared to the proton plasma case. In both the simulations, however, the electron temperature is found to be anisotropic. Finally, when a shifted electron source location case is considered the electron velocity distributions in all three directions are found to be unequal and non-Maxwellian, contrary to the co-located case. The ion beam is observed to attract the electrons, which initially oscillate between the radial edges of the ion beam as confirmed by the bi-modal velocity distribution at early times. As the plume evolves, it electrostatically traps these electrons within the ion beam, allowing them to thermalize, as observed from the single-peak electron velocity distribution functions at later times. These results demonstrate that GPUs can efficiently accelerate computations of a fully kinetic approach to enable the study of electron kinetics and neutralization with the highest physical fidelity.

Investigating the Effectiveness of the Proposed Algorithms to Reduce the Simulation Time of the EBAC FEM Gun Code

This paper presents the results of a newly developed 2-D electron beam analyzing code (EBAC) and its capabilities. This finite element cylindrical axis-symmetric code tracks particles by the fourth-order Runge–Kutta method in the local coordinate of elements and calculates steady-state results for both particle trajectories and the emitted current under space-charge limited condition. Two analytical test cases, along with an experimental case are presented to validate the EBAC results. Some techniques have also been developed to augment the basic algorithm of gun codes, dealing with computation of both the emitted current and particle trajectories. These proposed techniques in EBAC have proven to be efficient for reducing convergence time of the basic steady-state algorithm in Pierce-type structures for producing rectilinear electron beam.

Bayesian-Enhanced Low-Fidelity Correction Approach to Multifidelity Aerospace Design

This work discusses a novel approach, Bayesian-inspired multifidelity optimization, for solving complex optimization requiring computationally expensive simulations. Bayesian-inspired multifidelity optimization aims to minimize computational cost in achieving accurate high-fidelity results through the use of low-fidelity (computationally cheaper) models in combination with a surrogate correction model. A novel Bayesian hybrid bridge function was developed to serve as the low-fidelity correction technique. This Bayesian hybrid bridge function is a Bayesian weighted average of two standard bridge functions, additive and multiplicative. The correction technique is implemented in parallel with a modified trust region model management optimization scheme. It is shown that optimization on the corrected low-fidelity model converges to the same local optimum as optimization on the high-fidelity model in fewer high-fidelity model. This technique is developed to reach that optimum in fewer high-fidelity function evaluations than traditional optimization, ultimately reducing computational cost. This work also extends the low-fidelity correction optimization beyond the traditional bifidelity optimization to that of optimization with multiple-fidelity objective and constraint functions. The proposed solution technique allows for use of commercial optimizers and is demonstrated on three separate problems. The first demonstration is on a one-dimensional analytical test case in which the Bayesian correction is shown. Then, the ability to handle multiple fidelities in both the objective and constraint functions is presented and discussed using a two-dimensional analytical test case. An airfoil shape optimization problem is then used to illustrate the effectiveness of this approach in an engineering design scenario in which the various fidelities arise as a difference in computational physics. It is shown via these demonstrations that implementation of this Bayesian low-fidelity correction optimization approach results in convergence to a high-fidelity optimum at a reduced computational cost in comparison to traditional optimization techniques.

Variable-fidelity expected improvement method for efficient global optimization of expensive functions

The efficient global optimization method (EGO) based on kriging surrogate model and expected improvement (EI) has received much attention for optimization of high-fidelity, expensive functions. However, when the standard EI method is directly applied to a variable-fidelity optimization (VFO) introducing assistance from cheap, low-fidelity functions via hierarchical kriging (HK) or cokriging, only high-fidelity samples can be chosen to update the variable-fidelity surrogate model. The theory of infilling low-fidelity samples towards the improvement of high-fidelity function is still a blank area. This article proposes a variable-fidelity EI (VF-EI) method that can adaptively select new samples of both low and high fidelity. Based on the theory of HK model, the EI of the high-fidelity function associated with adding low- and high-fidelity sample points are analytically derived, and the resulting VF-EI is a function of both the design variables x and the fidelity level l. Through maximizing the VF-EI, both the sample location and fidelity level of next numerical evaluation are determined, which in turn drives the optimization converging to the global optimum of high-fidelity function. The proposed VF-EI is verified by six analytical test cases and demonstrated by two engineering problems, including aerodynamic shape optimizations of RAE 2822 airfoil and ONERA M6 wing. The results show that it can remarkably improve the optimization efficiency and compares favorably to the existing methods.

Uncertainty quantification for the aeroacoustics of rotating blades in the time domain

Aeroacoustics has received great attention in the past decade, owing to the ever stricter noise regulations. Despite the stochastic nature of most aeroacoustic systems, non-deterministic investigations in regards to computational aeroacoustics (CAA) are limited. In this paper, uncertainty quantification has been achieved for the noise propagation stage of hybrid CAA, and also on the noise prediction of a non-lifting helicopter rotor in hover. Analytical and computational fluid dynamics test cases have been analyzed, with uncertainties propagated through these systems using non-intrusive polynomial chaos methods. It is shown here that the source of the uncertainty in the noise is dominated by the major characteristic properties of the simulations, such as the mean flow Mach number and blade tip Mach number. Only at a low tip Mach number uncertainties in the blade thickness may contribute significantly to the noise uncertainty. Apart from this, it is seen to be unlikely that small uncertainties in the geometry, ambient conditions and observer properties will contribute significantly to the noise uncertainty. A peak pressure uncertainty of up to 20% is seen in the hovering helicopter test case, from small, realistic uncertainties. This highlights the importance of considering uncertainties in CAA investigations.

Unified Convergence Analysis of Numerical Schemes for a Miscible Displacement Problem

This article performs a unified convergence analysis of a variety of numerical methods for a model of the miscible displacement of one incompressible fluid by another through a porous medium. The unified analysis is enabled through the framework of the gradient discretisation method for diffusion operators on generic grids. We use it to establish a novel convergence result in $$L^\infty (0,T; L^2(\Omega ))$$ of the approximate concentration using minimal regularity assumptions on the solution to the continuous problem. The convection term in the concentration equation is discretised using a centred scheme. We present a variety of numerical tests from the literature, as well as a novel analytical test case. The performance of two schemes is compared on these tests; both are poor in the case of variable viscosity, small diffusion and medium to small time steps. We show that upstreaming is not a good option to recover stable and accurate solutions, and we propose a correction to recover stable and accurate schemes for all time steps and all ranges of diffusion.

Assessment of the pseudo-tracking approach for the calculation of material acceleration and pressure fields from time-resolved PIV: part II. Spatio-temporal filtering

The present study characterises the spatio-temporal filtering associated with pseudo-tracking. A combined theoretical and numerical assessment is performed that uses the relatively simple flow case of a two-dimensional Taylor vortex as analytical test case. An additional experimental assessment considers the more complex flow of a low-speed axisymmetric base flow, for which time-resolved tomographic PIV measurements and microphone measurements were obtained. The results of these assessments show how filtering along Lagrangian tracks leads to amplitude modulation of flow structures. A cut-off track length and spatial resolution are specified to support future applications of the pseudo-tracking approach. The experimental results show a fair agreement between PIV and microphone pressure data in terms of fluctuation levels and pressure frequency spectra. The coherence and correlation between microphone and PIV pressure measurements were found to be substantial and almost independent of the track length, indicating that the low-frequency behaviour of the flow could be reproduced regardless of the track length. It is suggested that a spectral analysis can be used inform the selection of a suitable track length and to estimate the local error margin of reconstructed pressure values.