Robust Acoustic Objective Functions and Sensitivities in Adjoint-Based Design Optimizations

The drive for improved flexibility and reusability in satellite launch systems has brought a resurgence in the popularity of scramjet-powered access-to-space launch vehicle concepts. A challenge in scramjet-powered accelerator vehicles is achieving positive thrust margins at the high-speed end of their flight envelopes. As a consequence, scramjet-powered vehicles typically rely on highly integrated airframe-engine configurations. However, due to the tight integration, the geometries and the flow physics are complex. An automated optimisation method seems an excellent candidate approach to explore the complex design space of hypersonic vehicles. This thesis focuses on the development and application of a particular optimisation method, adjoint-based optimisation, to aid in efficient aerodynamic design in hypersonic flows.Shape optimisation of scramjet-powered vehicles requires many design parameters to capture the geometric detail, and, since the flow physics is complex, the fidelity of Reynolds-Averaged Navier-Stokes (RANS) analyses is desirable. The combination of many design parameters and an expensive objective function evaluation drives the need for gradient evaluation methods that scale well with the number of design variables. An advantage of the adjoint method is that all shape sensitivities for an objective function are evaluated at the cost of only one flow solution and one adjoint solution. As a result of its efficiency, the adjoint method has become widely used for aircraft design, evolving to the design optimisation of full configurations. Despite the wide use of adjoint methods in aircraft optimisation, there has been very little application to hypersonic vehicle design.Several high-speed adjoint solvers have been reported in the literature, however, a majority of these works have only verified the adjoint sensitivities, few have followed on to demonstrate the method for use in design optimisation. The contribution of this work is the description and application of a discrete adjoint solver in high-speed compressible flow optimisation.What is unique in this work is a demonstration that complex-step differentiation works well to linearise a second-order spatially accurate unstructured RANS solver. In particular, the approach presented in this work utilises the k − ω turbulence model in high-speed ducted flow configurations.As part of this work, to provide flow analysis with a rapid turn-around, an unstructured steady-state RANS solver driven by a Jacobian-Free Newton-Krylov method was developed. Turbulence is modelled using the two-equation k − ω turbulence model. The Newton method is globalised by using the pseudo-transient approach. A restarted GMRES method is used to solve the system of linear equations arising when solving for the Newton steps. Evaluation of the matrix-vector products required in the GMRES algorithm is accomplished by Frechet derivatives using imaginary perturbations in the complex plane. This is necessary to achieve robust convergence of the types of turbulent hypersonic flows considered in this work. Equation scaling ensures that the linear solver provides an adequate solution of the linear system, especially for turbulent flows, where the flow and turbulence variables differ by several orders of magnitude. Incomplete lower-upper preconditioning with zero-fill is used to accelerate linear system convergence. To achieve adequate residual convergence for flows with embedded shocks, limiter freezing is required to prevent early stall. The developed flow solver is verified using several methods, including the Method of Manufactured Solutions.Several validation cases from the literature are presented to establish the appropriateness of the implemented physical models for design analysis in high-speed flow.An accompanying discrete adjoint solver was developed to provide efficient computation of the required shape sensitivities. The primary complication of linearising the flow solver routines is handled via a complex-step derivative approach. Targeted differentiation is employed to provide an efficient means of constructing the adjoint operator. The adjoint gradients are verified against a complex variable direct-differentiation method.The flow and adjoint solvers are coupled to the open-source optimisation library, DAKOTA, to perform design optimisation in high-speed flows. Design surfaces are parameterised using Bezier curves and mesh deformation is achieved by the inverse distance weighting method. Two optimisation applications are presented here as a demonstration of the development work: (a) wave drag minimisation of an axisymmetric body; and (b) hypersonic inlet design optimisation. For the case of axisymmetric bodies, the optimal shape determined by this work compares favourably to several minimum-drag power-law bodies published in the literature. The second application is the redesign of the P2 hypersonic inlet. The chosen objective function aimed at removing the reflected cowl shock whilst obtaining the desired compression ratio. The results presented show that the optimiser has removed the reflected shock while achieving the desired compression ratio, at no cost to the inlet performance metrics.The conclusions are that the developed discrete adjoint-based optimisation framework does work well in a hypersonic flow context and that the use of complex-step differentiation is a key enabler in the implementation. In particular, the inlet example presented in this work demonstrates the efficiency, accuracy, and applicability of discrete adjoint-based optimisation to design analysis in turbulent high-speed flow.

*† This study involves the application of fan flow deflectors for suppression of jet noise from low-bypass turbofan engines for next-generation supersonic aircraft. Experiments using canonical vane deflectors seek to establish correlations between aerodynamics, distortion of the mean velocity field in the jet plume, and noise reduction. The maximum radial velocity gradient, as a function of axial distance and azimuth angle, is used to quantify distortions of the plume. Vanes mounted at low azimuth angles (with respect to the downward vertical) reduce gradients in the downward direction, while vanes mounted at high azimuth angles reduce gradients in the sideline reduction. There is a significant correlation between gradient reduction and noise suppression in the azimuthal direction of the gradient. In addition, it is found that vanes with cambered airfoils perform better than vanes with symmetric airfoils. The trends obtained were used in the successful design of deflector configurations consisting of two pairs of vanes which resulted in cumulative peak overall sound pressure level (OASPL) and effective perceived noise level (EPNL) reductions of 8.8 dB and 6.7 dB respectively.

Frequency- and temperature-dependent properties of viscoelastic materials have a significant effect on vibration characteristics of constrained layer damping (CLD) structures. In this article, parametrized level set method (PLSM) is extended to solve the topology optimization problem of CLD plates, in which the viscoelastic material with frequency- and temperature-dependent characteristics is employed as damping core. The single/weighted modal loss factors, which are solved by an iterative method, are defined as objective functions, and topology optimization problem is formulated based on the finite element model for CLD plates which are described by combining parametric level set (PLS) functions. The sensitivities of objective functions with respect to expansion coefficients in PLS are calculated by the use of adjoint vector method. As the optimized results with complex constant shear modulus for viscoelastic core indicates, the proposed method is effective for topology optimization of CLD structures and insensitive to initial design. Numerical examples with the consideration of frequency- and temperature-dependent characteristics for “soft” and “hard” viscoelastic core are implemented. Through comparing the optimal results with different models for viscoelastic core, the influences of frequency- and temperature-dependent characteristics for viscoelastic core on natural frequencies, modal loss factors and optimized layouts are analyzed. Similarity indices are defined for distinguishing the difference among optimized layouts and initial design induced by modal strain energy contour. The optimal design dependence on layer thicknesses is further discussed with additional mass of CLD materials unchanged.

Isogeometric shape optimization of photonic crystals via Coons patches

Summary History matching generates detailed reservoir description that matches production data and can be used for forecasting and uncertainty estimation. Due to the ill-posedness of the history matching problem, the parametrization of high-dimensional fields in the model (such as permeability and porosity) is widely applied. The common approach of existing parametrization algorithms is to generate a dataset of possible fields realizations (prior models) and then convert this dataset to an orthogonal basis using PCA-based techniques. Model reduction is achieved by truncating the majority of basis components based on energy criteria. Due to high uncertainty and low quality of real data, the important pattern could be under-represented in the prior dataset and basis components with such structures could be truncated. We present a novel method where omitted components are defined not only by energy criteria but also by objective function sensitivity. In our Adaptive Strategies PCA (AS-PCA) technique we developed and advanced definition of the optimal basis and derived an efficient algorithm for basis recalculation using computational approaches from quantum mechanics. The algorithm requires gradient of an objective function w.r.t latent variables (only at the point of convergence). Then the new basis is obtained by a few linear transformations with negligible computational cost, and optimization continues. The method was tested on history matching of 2D reservoirs and have demonstrated improvements in terms of misfit value and field consistency in comparison with classic PCA parametrization. However, the applicability of gradient-based methods is constrained by local convergence and high implementation efforts (i.e adjoint technique). To overcome these constraints, we extend adaptive strategies for non-intrusive history matching approaches such as stochastic optimization and ensemble-based algorithms. Numerical gradient approximation is not well-suited for AS-PCA since it is inexact and takes additional simulation time. We developed the regression-based algorithm for gradient estimation using a set of field realizations, represented by an ensemble in ensemble-based methods or a population in evolution-based algorithms. In this study, we demonstrate the theory and examples of adaptive strategies application to history matching using PSO and enKF. The results of history matching with inconsistent prior datasets for 2D gaussian fields and applications to uncertainty quantification will be provided.

Simultaneous-source techniques have been proposed to reduce the acquisition cost and alleviate the computational overburden of data processing, and they have been applied to full waveform inversion (FWI) in recent years. These techniques are mainly based on the l2-norm objective function (least-squares criterion). However, if we consider that real field data contain noise such as outliers, it would be preferable to use the robust objective function in simultaneous-source FWI. In this study, we propose combining the simultaneous-source FWI with the l1-norm objective function (least-absolute criterion), which is known to be robust to data containing noise, specifically outliers. For the l1-norm-based simultaneous-source (SS-1) FWI, we first verify the crosstalk reduction and the robustness to data containing outliers. The expectations of the gradient direction directly give an evidence for the crosstalk reduction, and a signal-to-noise ratio analysis supports the convergence of our algorithm. The analysis shows that the crosstalk noise of the SS-1 FWI can be suppressed by random phase encoding. In addition, the spectrum of the weighted residuals indicates that the main property of the l1-norm objective function is preserved in the SS-1 FWI. Numerical examples show that the SS-1 FWI produces reliable results as the l2-norm simultaneous-source FWI does, similar to the case of the individual-source FWI. These results support the idea that combining the simultaneous-source FWI with the l1-norm objective function improves the computational efficiency and preserves the intrinsic characteristics of the l1-norm objective function.

This paper studies the problem of coordination and optimization of multi-machine PSS parameters under multiple operating modes. Firstly, there employs principal component analysis method based on trajectory sensitivity to select the optimal feedback input signals of PSS. Secondly, this paper proposes multi-objective optimization constraints including stability objective function and robust objective function. Then a special simplified method is proposed to filter partial oscillation curves into the objective function with larger correlation factors. Finally, employ the above optimization methods to design multi-machine PSS with EPRI-36 test system. Test and analysis results indicate that the proposed methods improve the positive damping of test system more effectively and implement better robust control effects for unknown operating modes and faults.

Two-dimensional Principal Component Analysis (2DP-CA) has been widely used for face image representation and recognition. However, 2DPCA, which is based on F-norm square, is sensitive to the presence of outliers. To enhance the robustness of 2DPCA model, we proposed a novel Robust 2DPCA objective function, called R-2DPCA. The criterion of R-2DPCA is maximizing the covariance of data in the projected subspace, while minimizing the reconstruction error of data. In addition, we use the efficient non-greedy optimization algorithms solving our objective function. Extensive experiments are done on the AR, CMU-PIE, Extended Yale B face image databases, and results illustrate that our method is more effective and robust than other robust 2DPCA algorithms, such as L1-2DPCA, L1-2DPCA-S, and N-2DPCA.

3D morphable model (3DMM) fitting on 2D data is traditionally done via unconstrained optimization with regularization terms to ensure that the result is a plausible face shape and is consistent with a set of 2D landmarks. This paper presents inequality-constrained 3DMM fitting as the first alternative to regularization in optimization-based 3DMM fitting. Inequality constraints on the 3DMM's shape coefficients ensure face-like shapes without modifying the objective function for smoothness, thus allowing for more flexibility to capture person-specific shape details. Moreover, inequality constraints on landmarks increase robustness in a way that does not require per-image tuning. We show that the proposed method stands out with its ability to estimate person-specific face shapes by jointly fitting a 3DMM to multiple frames of a person. Further, when used with a robust objective function, namely gradient correlation, the method can work "in-the-wild" even with a 3DMM constructed from controlled data. Lastly, we show how to use the log-barrier method to efficiently implement the method. To our knowledge, we present the first 3DMM fitting framework that requires no learning yet is accurate, robust, and efficient. The absence of learning enables a generic solution that allows flexibility in the input image size, interchangeable morphable models, and incorporation of camera matrix.

Multi-objective optimization of metal hydride hydrogen storage tank with phase change material

Often the parameters considered as constants in an optimization problem have some uncertainty and it is interesting to know how the optimum solution is modified when these values are changed. The only way to continue having the optimal solution is to perform a new optimization loop, but this may require a high computational effort if the optimization problem is large. However, there are several procedures to obtain the new optimal design, based on getting the sensitivities of design variables and objective function with respect to a fixed parameter. Most of these methods require obtaining second derivatives which has a significant computational cost. This paper uses the feasible direction-based technique updating the active constraints to obtain the approximate optimum design. This procedure only requires the first derivatives and it is noted that the updating set of active constraints improves the result, making possible a greater fixed parameter variation. This methodology is applied to an example of very common structural optimization problems in technical literature and to a real aircraft structure.

Modeling and thermo-economic optimization of heat recovery heat exchangers using a multimodal genetic algorithm

The authors show in this chapter, two different approaches for efficiently estimating the second-order derivatives (Hessian matrix) of a given objective function. The cost of evaluating the Hessian using classical finite difference approach is O(n2) where n is the number of parameters. The first adjoint approach reduces the cost of estimating all components of the Hessian matrix to only 2n extra simulations. This approach is simple, and it uses the algorithms developed in previous chapters. A second approach for estimating the complete Hessian is also presented. This approach is more complex than the first approach and requires extra memory storage. This approach requires only n + 1 extra simulations per Hessian evaluation. It follows that the computational cost is approximately one half of the first adjoint approach. This saving comes at the cost of a more complex algorithm and more extensive storage.

The non-uniform distributed pressure of impact wave is usually simplified into concentrated or uniform load equivalently in the optimization design of constrained layer damping structure. However, for the thin-walled structure, it becomes necessary to regard the load as a non-uniform distribution. In this paper, a topology optimization approach is proposed considering the blast load with non-uniform distribution, aiming to unveil its impact on optimization layouts and dynamic responses. Initially, the smoothed particle hydrodynamics (SPH) algorithm is used to obtain the blast pressure what are extracted and integrated into the optimization model. Subsequently, the relative density is regarded as design variable. The construction of material penalty model and the topology optimization model are based on polynomial interpolation scheme (PIS). The sensitivity of objective function is deduced employing an improved adjoint variable method (AVM) to fit the load forms, and the Newmark-[Formula: see text] method is used to calculate the dynamic response. The optimization criterion (OC) is adopted to update the design variables. Finally, two numerical examples are used to exhibit the validity and accuracy of the presented methodology. The findings indicate that the distributed form, spread velocity, excited position and excited amplitude of the blast load all exert a notable influence on optimization results and dynamic response. These results underscore the valuable engineering application of this research and introduce a fresh perspective to the challenge of topology optimization under the blast case.

Solving optimization problems with multiple uncertainties has always been a challenging task in different scopes of science. While different approaches have been developed to take advantage of the stochastic space of the problem, these methods are intensively dependent of the probabilistic information of various variables which are not always available. Relying on the severity of the failure, information-gap decision theory (IGDT) is a robust optimization approach which is entirely autonomous from probabilistic information. In this model, a forecasted amount is presumed for each uncertain variable, and the sensitivity of objective functions is analyzed according to the deviation of each of these uncertain parameters from their forecasted value. In this method, two main types of uncertainty set models including energy-bound model and envelope-bound model are handled. In this chapter, these principles and fundamentals of IGDT are described.