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The paper presents a study on four adaptive sampling methods of a multi-fidelity (MF) metamodel, based on stochastic radial basis functions (RBF), for global design optimisation based on expensive ...

This paper presents a comparison of two methods for the forward uncertainty quantification (UQ) of complex industrial problems. Specifically, the performance of Multi-Index Stochastic Collocation (MISC) and adaptive multi-fidelity Stochastic Radial Basis Functions (SRBF) surrogates is assessed for the UQ of a roll-on/roll-off passengers ferry advancing in calm water and subject to two operational uncertainties, namely the ship speed and draught. The estimation of expected value, standard deviation, and probability density function of the (model-scale) resistance is presented and discussed; the required simulations are obtained by the in-house unsteady multi-grid Reynolds Averaged Navier-Stokes (RANS) solver $\chi$navis. Both MISC and SRBF use as multi-fidelity levels the evaluations on the different grid levels intrinsically employed by the RANS solver for multi-grid acceleration; four grid levels are used here, obtained as isotropic coarsening of the initial finest mesh. The results suggest that MISC could be preferred when only limited data sets are available. For larger data sets both MISC and SRBF represent a valid option, with a slight preference for SRBF, due to its robustness to noise.

International audience; An adaptive-fidelity approach to metamodeling from noisy data is presented for design-space exploration and design optimization. Computational fluid dynamics (CFD) simulations with different numerical accuracy (spatial discretization) provides metamodel training sets affected by unavoidable numerical noise. The-fidelity approximation is built by an additive correction of a low-fidelity metamodel with metamodels of differences (errors) between higher-fidelity levels whose hierarchy needs to be provided. The approach encompasses two core metamodeling techniques, namely: i) stochastic radial-basis functions (SRBF) and ii) Gaussian process (GP). The adaptivity stems from the sequential training procedure and the auto-tuning capabilities of the metamodels. The method is demonstrated for an analytical test problem and a CFD-based optimization of a NACA airfoil, where the fidelity levels are defined by an adaptive grid refinement technique of a Reynolds-averaged Navier-Stokes (RANS) solver. The paper discusses: i) the effect of using more than two fidelity levels; ii) the use of least squares regression as opposed to exact interpolation; iii) the comparison between SRBF and GP; and iv) the use of two sampling approaches for GP. Results show that in presence of noise, the use of more than two fidelity levels improves the model accuracy with a significant reduction of the number of high-fidelity evaluations. Both least squares SRBF and GP provide promising results in dealing with noisy data.

The paper presents partitioned tightly coupled fluid-structure interaction (FSI) simulations for composite panel slamming of a high-speed planing hull, including comparison with full-scale experiments. Panels with different layout/stiffness are investigated. Computational fluid dynamics (CFD) is performed using the URANS code CFDShip-Iowa. Computational structural dynamics (CSD) uses modal expansion by ANSYS finite elements. One- and two-way tightly coupled FSI is performed. The complexity of sea-trial conditions is reduced by statistical/frequency analysis, allowing for a simplified representation by one regular wave. Simulations provide details of slamming, including correlation of re-entering pressure peaks with motions and strain peaks. Numerical/modeling issues are discussed. Expected value and associated uncertainty of experimental pressure/strain peak and duration are used for validation. The difference of panels’ dynamics is well predicted. Validation errors and uncertainties (average 25% and 14%) are quite large. Nevertheless, errors always fall within one standard deviation of experimental-data individual readings. Results are promising especially if compared to earlier slamming studies for regular/irregular waves in controlled towing tank tests, which show average error and validation uncertainty of 25% and 10%. The current study lays the groundwork for research on high-fidelity CFD/CSD FSI of real-world geometry slamming and ultimately multidisciplinary design optimization of structural and hull-form parameters.

Hull-form stochastic optimization methods are presented and evaluated for resistance reduction and operational efficiency (operability), addressing stochastic sea state and operations. The cost/benefit analysis of the optimization procedure is presented by comparison of four hierarchical problems, from stochastic most general to deterministic least general. The parent hull is a high-speed catamaran, with geometrical constraints for maximum variation of length, beam, draft, and displacement. Problem 1 is used as a benchmark for the evaluation of the other problem formulations and is defined as a multi-objective stochastic optimization for resistance and operability, considering stochastic sea state and speed, but limited to head waves. Problem 2 is a multi-objective stochastic optimization for resistance and motions at fixed sea state and speed. Problem 3 is a multi-objective deterministic optimization for resistance and motions using a single regular wave at fixed speed. Problem 4 is a single-objective deterministic optimization for calm-water resistance at fixed speed. The design optimization is based on hull-form modifications by the Karhunen-LoSve expansion of a free-form deformation, URANS-based CFD simulations, regular wave approximations for irregular waves, metamodels and multi-objective particle swarm. The design optimization achieves an 8.7, 23, 53, and 10% average improvements for problems 1, 2, 3, and 4, respectively. Comparing to problem 1, problem 2, 3, 4 optimized designs have average performances 1, 2.1 and 1.7% worse, respectively. The most efficient problem, from the computational cost/benefit viewpoint, is problem 3. Nevertheless, problem 1 is needed to evaluate and compare the stochastic performance of the designs and finally assess the optimization cost/benefit.

The curse of dimensionality represents a relevant issue in simulation-based shape optimization, especially when complex physics and high-fidelity computationally-expensive solvers are involved in the process and a global optimum is sought after. In order to have a deeper insight into this problem and indicate possible remedies, the present paper studies the effects of both design-space dimensionality reduction (DR) and optimization methods on the shape optimization efficiency. Linear and non-linear DR methods are used for the design-space DR, based on principal component analysis and deep autoencoders. Global and hybrid global/local deterministic derivative-free optimization algorithms (Deterministic Particle Swarm Optimization, DIviding RECTangles, Dolphin Pod Optimization, LSDFPSO, and DIRMIN-2) are applied to the original and the reduced-dimensionality design-spaces, investigating their efficiency and effectiveness. Example application is shown for the shape optimization of a destroyer-type vessel sailing in calm water at fixed speed.

The objective of this work is to provide and use both experimental fluid dynamics (EFD) data and computational fluid dynamics (CFD) results to validate a regular-wave uncertainty quantification (UQ) model of ship response in irregular waves, based on a set of stochastic regular waves with variable frequency. As a secondary objective, preliminary statistical studies are required to assess EFD and CFD irregular wave errors and uncertainties versus theoretical values and evaluate EFD and CFD resistance and motions uncertainties and, in the latter case, errors versus EFD values. UQ methods include analysis of the autocovariance matrix and block-bootstrap of time series values (primary variable). Additionally, the height (secondary variable) associated with the mean-crossing period is assessed by the bootstrap method. Errors and confidence intervals of statistical estimators are used to define validation criteria. The application is a two-degrees-of-freedom (heave and pitch) towed Delft catamaran with a length between perpendiculars equal to 3 m (scale factor equal to 33), sailing at Froude number equal to 0.425 in head waves at scaled sea state 5. Validation variables are x-force, heave and pitch motions, vertical acceleration of bridge, and vertical velocity of flight deck. Autocovariance and block-bootstrap methods for primary variables provide consistent and complementary results; the autocovariance is used to assess the uncertainty associated with expected values and standard deviations and is able to identify undesired self-repetition in the irregular wave signal; block-bootstrap methods are used to assess additional statistical estimators such as mode and quantiles. Secondary variables are used for an additional assessment of the quality of experimental and simulation data as they are generally more difficult to model and predict than primary variables. Finally, the regular wave UQ model provides a good approximation of the desired irregular wave statistics, with average errors smaller than 5% and validation uncertainties close to 10%.

An adaptive multi-fidelity metamodel is presented for efficient simulation-based design optimization processes. A multi-fidelity approximation is built by correcting a low-fidelity-trained metamodel with the metamodel of the difference (error) between high- and low-fidelity simulations. The metamodel prediction is based on the expected value of stochastic radial basis functions, which also provide the uncertainty associated to the prediction. New training points are placed where the prediction uncertainty is maximum. The prediction uncertainty of both the low-fidelity and the metamodel of the error is considered for an adaptive refinement of the low- and high-fidelity training sets, respectively. The method is demonstrated for the hull-form optimization of a SWATH (small water-plane area twin hull). The hydrodynamic performance is evaluated with RANSE (high-fidelity) and potential flow (low-fidelity) solvers and the optimization aims at the minimization of the hydrodynamic resistance and maximization of the payload.

An accurate multidisciplinary analysis (MDA) and design optimization (MDO) process coupling hydro-and structural dynamics is required to assess and design flexible marine structures. When performing MDO, the large computational cost associated to high-fidelity simulations within MDA can be mitigated by reducing the number of design variables and using surrogate models. An architecture for performing efficient MDO is here presented. Sequential and variable accuracy surrogate-based optimization is performed driven by a multi-criterion adaptive sampling method. The latter is guided by the objective function value and the uncertainty associated to both multidisciplinary coupling (in the MDA) and surrogate model prediction. Multidisciplinary consistency and accurate surrogate-model training are achieved at the end of the MDO process. The Karhunen-Loève expansion is used to assess the design space and reduce its dimensionality preserving the original geometric variability. The method is suitable for derivative-free optimization allowing for the use of expensive black box simulation tools. Outer shape and inner fiber layout of a flexible 3D hydrofoil in carbon fiber-reinforced plastic are optimized aiming at minimizing the drag.

The paper presents a methodology to reduce the design-space dimensionality in reliability-based robust optimization problems for shape design, with the aim of improving the efficiency of the optimization process. The method is based on a generalized Karhunen-Loève expansion (KLE) and the definition of geometric and physics-based variance associated with the design space. The geometric variance originates from the shape modification method, whereas the physics-based variance stems from multidisciplinary multi-physics computations (performed with low-fidelity solvers) at different design conditions. Combining geometric and multidisciplinary multiple-conditions physics-based variability with KLE provides the foundation for a design-space augmented dimensionality reduction (ADR) method, which extends authors' previous work. Herein, the ADR effectiveness and efficiency is demonstrated for the optimization of a destroyer-type vessel to improve calm water and seakeeping performance in a stochastic environment. The method presented goes beyond the current application and is suitable in all areas where shape design is of primary importance (such as fluid dynamics, structural, and heat transfer applications), involving complex and expensive simulations.

A statistical approach for computational fluid dynamics (CFD) state-of-the-art (SoA) assessment is presented for specified benchmark test cases and validation variables, based on the combination of solution and N-version verification and validation (V&V). Solution V&V estimates the systematic numerical and modeling errors/uncertainties. N-version verification estimates the random simulation uncertainty. N-version validation estimates the random absolute error uncertainty, which is combined with the experimental and systematic numerical uncertainties into the SoA uncertainties and then used to determine whether or not the individual codes/simulations and the mean code are N-version validated at the USoAi and USoA intervals, respectively. The scatter is due to differences in models and numerical methods, grid types, domains, boundary conditions, and other setup parameters. Differences between codes/simulations and implementations are due to myriad possibilities for modeling and numerical methods and their implementation as CFD codes and simulation applications. Industrial CFD codes are complex software with many user options such that even in solving the same application, different results may be obtained by different users, not necessarily due to user error but rather the variability arising from the selection of various models, numerical methods, and setup options. Examples are shown for ship hydrodynamics applications using results from the Seventh CFD Ship Hydrodynamics and Second Ship Maneuvering Prediction Workshops. The role and relationship of individual code solution V&V versus N-version V&V and SoA assessment are discussed.

A method based on the Karhunen–Loeve expansion (KLE) is formulated for the assessment of arbitrary design spaces in shape optimization, assessing the shape modification variability and providing the definition of a reduced-dimensionality global model of the shape modification vector. The method is based on the concept of geometric variance and does not require design-performance analyses. Specifically, the KLE is applied to the continuous shape modification vector, requiring the solution of a Fredholm integral equation of the second kind. Once the equation is discretized, the problem reduces to the principal component analysis (PCA) of discrete geometrical data. The objective of the present work is to demonstrate how this method can be used to (a) assess different design spaces and shape parameterization methods before optimization is performed and without the need of running simulations for the performance prediction, and (b) reduce the dimensionality of the design space, providing a shape reparameterization using KLE/PCA eigenvalues and eigenmodes. A demonstration for the hull-form optimization of the DTMB 5415 model in calm water is shown, where three design spaces are investigated, namely provided by free-form deformation, radial basis functions, and global modification functions.

An overview is provided of stochastic uncertainty quantification (UQ) and validation methods for application to realistic problems in ship hydrodynamics. The paper summarizes the research activities conducted by the hydrodynamics team within the NATO Task Group AVT-191 "Application of Sensitivity Analysis and Uncertainty Quantification to Military Vehicle Design." UQ methods assess the expected value (EV), standard deviation (SD) and cumulative distribution function or probability density function (CDF/PDF) of the hydrodynamic performance of interest. The hydrodynamic solvers include unsteady Reynolds-averaged Navier-Stokes and potential flow. Three UQ problems are presented. The first is a unit study, addressing the effects of stochastic Reynolds number on the lift and drag of a NACA 0012 2D hydrofoil at constant angle of attack. The second and the third are industrial problems of a high-speed catamaran, advancing in calm water and waves respectively, with stochastic operating conditions and geometry. Numerical validation benchmarks include the deterministic V&V for selected conditions, along with converged (quasi) Monte Carlo (MC) samples. Non-intrusive UQ methods are applied and discussed. MC simulations with direct CFD simulations and with metamodels are shown. Metamodels include linear and Hermite-cubic interpolation, inverse distance weighting, quadratic and cubic response surface, Hermite and Legendre polynomial expansions, least-square support vector machine, thin plate spline, radial basis functions network with multiquadric kernels, polyharmonic splines, Kriging using linear and exponential covariance functions, dynamic radial basis functions and dynamic Kriging. Polynomial chaos method and quadrature formulas (trapezoidal, Simpson's rule, Gaussian quadrature) are also assessed and compared to metamodel-based methods. Convergence criteria for UQ methods include deterministic and stochastic convergence criteria for MC simulation, along with convergence criteria for MC with metamodels, and quadrature formulas. UQ methods have shown their maturity for application to realistic stochastic design optimization problems. MC with dynamic metamodels is found the most promising method overall, due to its function-adaptation capability and high computational efficiency, which make the method also recommended for stochastic optimization.

Simulation-based design optimization methods integrate computer simulations, design modification tools, and optimization algorithms. In hydrodynamic applications, often objective functions are computationally expensive and noisy, their derivatives are not directly provided, and the existence of local minima cannot be excluded a priori, which motivates the use of deterministic derivative-free global optimization algorithms. The enhancement of two algorithms of this type, DIRECT (DIviding RECTangles) and DPSO (Deterministic Particle Swarm Optimization), is presented based on global/local hybridization with derivative-free line search methods. The hull-form optimization of the DTMB 5415 model is solved for the reduction of the calm-water resistance at Fr = 0.25, using potential flow and RANS solvers. Six and eleven design variables are used respectively, modifying both the hull and the sonar dome. Hybrid algorithms show a faster convergence towards the global minimum than the original global methods and are a viable option for ship hydrodynamic optimization. A significant resistance reduction is achieved both by potential flow and RANS-based optimizations, showing the effectiveness of the optimization procedure.

The paper presents a multi-fidelity global metamodel for expensive computer simulations, developed as an essential part of efficient simulation-based design optimization under uncertainty. High- and low-fidelity solvers are managed through a multi-fidelity adaptive sampling procedure. The multi-fidelity approximation is built as the sum of a low-fidelity-trained metamodel and the metamodel of the difference (error) between high- and low-fidelity simulations. The metamodels are based on dynamic stochastic radial basis functions, which provide the prediction along with the associated uncertainty. New training points are placed where the prediction uncertainty is maximum. The prediction uncertainty of both the low-fidelity and the error metamodel is considered for the adaptive refinement of the low- and high-fidelity training set, respectively. The method is demonstrated through three analytical test problems and one simple industrial application in ship hydrodynamics. The fitting error of the multi-fidelity metamodel is used as evaluation metric. The comparison with a high-fidelity-trained metamodel shows the effectiveness of the present method.

International audience; The paper presents recent research conducted within the NATO RTO Task Group AVT-204 "Assess the Ability to Optimize Hull Forms of Sea Vehicles for Best Performance in a Sea Environment." The objective is the improvement of the hydrodynamic performances (resistance/powering requirements, seakeeping, etc.) of naval vessels, by integration of computational methods used to generate, evaluate, and optimize hull-form variants. Several optimization approaches are brought together and compared. A multi-objective optimization of the DTMB 5415 (specifically the MARIN variant 5415M) is used as a test case and results obtained so far using low-fidelity solvers show an average improvement for resistance and seakeeping performances of nearly 10 and 9%, respectively.

International audience; The paper presents a novel approach to include community noise considerations based on sound quality in Multidisciplinary Conceptual Design Optimization (MCDO) of civil transportation aircraft. The novelty stems from the use of an unconventional objective function, defined as a measure of the distance between the noise emission of the aircraft under analysis and a reference "weakly annoying" noise, the target sound. The minimization of such a merit factor yields an aircraft concept with a noise signature as close as possible to the given target. The reference sound is one of the outcomes of the European Research Project SEFA (Sound Engineering For Aircraft, VI Framework Programme, 2004-2007), and used here as an external input. The aim of the present work is to address the definition ad the inclusion of the sound-matching-based objective function in the MCDO of aircraft.

The present paper presents an offline dimensionality reduction technique for single- and multi-disciplinary shape optimization problems, based on a generalized Karhunen-Loève expansion (KLE) of the shape modification vector. The formulation is based on the geometric variability and the method does not require objective function or gradient evaluations. The mathematical derivation of the design-space dimensionality reduction is presented for a global representation of the shape modification. The associated structure and breakdown of the geometric variance is investigated through eigenvalues and eigenmodes provided by the KLE. The dimensionality reduction for the shape optimization problem is based on the eigenvalues, which represent the geometric variance associated to the corresponding eigenmodes. The reduced-dimensionality design space is defined using the eigenmodes as new basis functions. Two example applications are presented. The first example is the hydrodynamic hull-form optimization for resistance reduction of an USS Arleigh Burke-class destroyer ship. The results show the KLE capability of reducing the design-space dimensionality, while retaining a prescribed level of design variability and achieving the same optimization results as the original design space. The second example is the design-space dimensionality reduction of a NACA 0009 3D hydrofoil, used as multi-disciplinary design optimization test case in ongoing research by the authors. The formulation presented goes beyond the current applications and is suitable in all areas where shape design is of primary importance (such as aerodynamics, aeroelasticity, structural, and heat transfer applications), involving complex single and multi-disciplinary simulations.