Adjoint-based Optimization Research Articles

Adjoint-Based Design Optimization of Stability Constrained Systems

A partial differential equation (PDE) constrained design optimization problem usually optimizes a characteristic of a dynamical system around an equilibrium point. However, a commonly omitted constraint is the linear stability constraint at the equilibrium point, which undermines the optimized solution’s applicability. To enforce the linear stability constraint in practical gradient-based optimization, the derivatives must be computed accurately, and their computational cost must scale favorably with the number of design variables. In this paper, we propose an algorithm based on the coupled adjoint method and the algorithmic differentiation method that can compute the derivative of such constraint accurately and efficiently. We verify the proposed method using several simple low-dimensional dynamical systems. The relative difference between the adjoint method and the finite differences is between 10−6 to 10−8. The proposed method is demonstrated through several optimizations, including a nonlinear aeroelastic optimization. The proposed algorithm has the potential to be applied to more complex problems involving large-scale nonlinear PDEs, such as aircraft flutter and buffet suppression.

Adjoint-based shape optimization for compressible flow based on volume penalization method

AbstractReducing the resistance of compressible flow around a blunt body is of great interest in engineering applications, while an efficient shape optimization method for compressible flows remains far from well established, especially for high Mach numbers. To this end, a volume penalization method for simulating compressible flows past a no-slip and isothermal solid is established by introducing an artificial body force and a heat sink into the governing equations. The level-set functions are introduced as design variables, and the cost functional is defined as the total drag acting on the solid. Then, a continuous adjoint-based shape optimization algorithm for drag reduction is developed by deriving the adjoint equations, the adjoint boundary conditions, and the shape update formula. Both the forward and adjoint simulations are verified by existing studies. The results show that the relative deviations of the drag coefficients obtained in the present study from those reported in the reference studies are around 5% at most, and also a comparable drag reduction rate and also optimal shapes can be reproduced by the present optimization scheme for benchmark problems at relatively low Mach numbers considered in previous studies. Finally, the present method is applied to shape optimization of an initially two-dimensional cylinder and also a three-dimensional sphere in the transonic regime of Ma∞ = 1.2. The drag reduction of over 20% is achieved for both two-dimensional and three-dimensional cases.

Adjoint-based inversion for stress and frictional parameters in earthquake modeling

We present an adjoint-based optimization method to invert for stress and frictional parameters used in earthquake modeling. The forward problem is linear elastodynamics with nonlinear rate-and-state frictional faults. The misfit functional quantifies the difference between simulated and measured particle displacements or velocities at receiver locations. The misfit may include windowing or filtering operators. We derive the corresponding adjoint problem, which is linear elasticity with linearized rate-and-state friction and, for forward problems involving fault normal stress changes, nonzero fault opening, with time-dependent coefficients derived from the forward solution. The gradient of the misfit is efficiently computed by convolving forward and adjoint variables on the fault. The method thus extends the framework of full-waveform inversion to include frictional faults with rate-and-state friction. In addition, we present a space-time dual-consistent discretization of a dynamic rupture problem with a rough fault in antiplane shear, using high-order accurate summation-by-parts finite differences in combination with explicit Runge–Kutta time integration. The dual consistency of the discretization ensures that the discrete adjoint-based gradient is the exact gradient of the discrete misfit functional as well as a consistent approximation of the continuous gradient. Our theoretical results are corroborated by inversions with synthetic data. We anticipate that adjoint-based inversion of seismic and/or geodetic data will be a powerful tool for studying earthquake source processes; it can also be used to interpret laboratory friction experiments.

Efficient Forced Response Minimization Using a Full-Viscosity Discrete Adjoint Harmonic Balance Method

A forced response induced by inlet distortion and blade row interactions can lead to high-cycle fatigue failure, especially when the unsteady excitation frequency is close to the natural vibration frequency of a blade. This paper presents the study of forced response sensitivity analysis and minimization using a full-viscosity unsteady discrete adjoint method. The goal is to improve the aeroelastic performances of turbomachinery blades and simultaneously constrain/improve aerodynamic performances. The decoupled modal reduction method is used to compute the forced response, which is considered the objective function in adjoint-based design optimization. To analyze aeroforcing and aerodamping flow and adjoint fields efficiently, the harmonic balance method and its adjoint counterpart developed by an automatic differentiation tool are applied. Two cases—the NASA Rotor 67 with inlet distortion and a three-row configuration with multiple fundamental frequencies in the second row—are used to demonstrate the effectiveness of the aerodynamic and aeroelastic coupled design optimization system developed in this work. For the latter case, the Fourier-transform-based method that first decomposes and then matches the time and space modes at two sides of an interface is used for interface coupling. The almost-periodic Fourier transform method is used to determine the time instances for cases with multiple fundamental frequencies.

Hybrid deep learning for design of nanophotonic quantum emitter lenses

Inverse design of nanophotonic structures has allowed unprecedented control over light. These design processes however are accompanied with challenges, such as their high sensitivity to initial conditions, computational expense, and complexity in integrating multiple design constraints. Machine learning approaches, however, show complementary strengths, allowing huge sample sets to be generated nearly instantaneously, and with transfer learning, allowing modifications in design parameters to be integrated with limited retraining. Herein we investigate a hybrid deep learning approach, leveraging the accuracy and performance of adjoint-based topology optimization to produce a high-quality training set for a convolutional generative network. We specifically explore this in the context of 3D nanophotonic lenses, used for focusing light between plane-waves and single-point, single-wavelength sources such as quantum emitters. We demonstrate that this combined approach allows higher performance than adjoint optimization alone when additional design constraints are applied; can generate large datasets (which further allows faster iterative training to be performed); and can utilize transfer learning to be retrained on new design parameters with very few new training samples. This process can be used for general nanophotonic design, and is particularly beneficial when a range of design parameters and constraints would need to be applied.

Adjoint-Based Optimization for the Venturi Mixer of A Burner

Adjoint-Based Optimization for the Venturi Mixer of A Burner

Adjoint-based optimisation of time- and span-periodic flow fields with Space–Time Spectral Method: Application to non-linear instabilities in compressible boundary layer flows

We aim at computing time- and span-periodic flow fields in span-invariant configurations. The streamwise and cross-stream derivatives are discretised with finite volumes while time and the span-direction are handled with pseudo-spectral Fourier-collocation methods. Doing so, we extend the classical Time Spectral Method (TSM) to a Space–Time Spectral Method (S-TSM), by considering non-linear interactions of a finite number of time and span harmonics. For optimisation, we introduce an adjoint-based framework that allows efficient computation of the gradient of any cost functional with respect to a large-dimensional control parameter. Both theoretical and numerical aspects of the methodology are described: evaluation of matrix–vector products with S-TSM Jacobian (or its transpose) by algorithmic differentiation, solution of fixed-points with quasi-Newton method and de-aliasing in time and space, solution of direct and adjoint linear systems by iterative algorithms with a block-circulant preconditioner, performance assessment in CPU time and memory. We illustrate the methodology on the case of 3D instabilities (first Mack mode) triggered within a developing adiabatic boundary layer at M = 4.5. A gradient-ascent method allows to identify a finite-amplitude 3D forcing that triggers a non-linear response exhibiting the strongest time- and span-averaged drag on the flat-plate. In view of flow control, a gradient-descent method finally determines a finite amplitude 2D wall-heat flux that minimises the averaged drag of the plate in presence of the previously determined non-linear optimal forcing.

Topology Optimization Enables High-Q Metasurface for Color Selectivity.

Nonlocal metasurfaces, exemplified by resonant waveguide gratings (RWGs), spatially and angularly configure optical wavefronts through narrow-band resonant modes, unlike the broad-band and broad-angle responses of local metasurfaces. However, forward design techniques for RWGs remain constrained at lower efficiency. Here, we present a topology-optimized metasurface resonant waveguide grating (MRWG) composed of titanium dioxide on a glass substrate capable of operating simultaneously at red, yellow, green, and blue wavelengths. Through adjoint-based topology optimization, while considering nonlocal effects, we significantly enhance its diffraction efficiency, achieving numerical efficiencies up to 78% and Q-factors as high as 1362. Experimentally, we demonstrated efficiencies of up to 59% with a Q-factor of 93. Additionally, we applied our topology-optimized metasurface to color selectivity, producing vivid colors at 4 narrow-band wavelengths. Our investigation represents a significant advancement in metasurface technology, with potential applications in see-through optical combiners and augmented reality platforms.

A coupled fully implicit method for solving drag-adjoint equations of steady incompressible flows

Numerical instability is one of the main concerns when applying adjoint-based gradient computation in aerodynamic design optimization for a variety of engineering problems. An adjoint pressure-adjoint velocity-coupled fully implicit method for solving the adjoint equations of steady incompressible flows is developed in the present work, taking the aerodynamic drag as the optimization target. The finite volume method for cell-centered, collocated variables on unstructured grids is used for spatial discretization. A stabilization term based on pressure-weighted interpolation (PWI) of the adjoint velocity is added to the adjoint continuity equation to avoid spurious oscillations of adjoint pressure. All the terms of the continuous adjoint equations are discretized implicitly, including the adjoint transpose convection (ATC) term, which is regarded as one of the major sources of numerical instability. The resulting discretized equations are integrated into one linear system, and the adjoint fields can be obtained simultaneously. To prevent the diagonal element of the coefficient matrix of the linear system from getting too small, a singular value decomposition is applied to the diagonal block of the coefficient matrix, and a limiter is applied to the diagonal elements. The condition number of the coefficient matrix may be quite large, so a parallel sparse linear solver that effectively combines direct and iterative methods through a Schur complement approach is applied to solve the linear system. Compared to the staggered iterative solver, no damping or masking is needed for the ATC term in the proposed fully implicit method. There is no need to sacrifice numerical accuracy for numerical stability. This method has been tested in several benchmark cases, and the numerical results show good numerical stability and accuracy. The present method provides a sound basis for industrial applications of adjoint-based aerodynamic optimization methods.

Adjoint-Optimized Large Dielectric Metasurface for Enhanced Purcell Factor and Directional Photon Emission.

Extracting photons efficiently from quantum sources, such as atoms, molecules, and quantum dots, is crucial for various nanophotonic systems used in quantum communication, sensing, and computation. To improve the performance of these systems, it is not only necessary to provide an environment that maximizes the number of optical modes, but it is also desirable to guide the extracted light toward specific directions. One way to achieve this goal is to use a large area metasurface that can steer the beam. Previous work has used small aperture devices that are fundamentally limited in their ability to achieve high directivity. This work proposes an adjoint-based topology optimization approach to design a large light extractor that can enhance the spontaneous decay rate of the embedded quantum transition and collimate the extracted photons. With the help of this approach, we present all-dielectric metasurfaces for a quantum transition emitting at λ = 600 nm. These metasurfaces achieve a broadband improvement of spontaneous emission compared to that in the vacuum, reaching a 10× enhancement at the design frequency. Furthermore, they can beam the extracted light into a narrow cone (±10°) along a desired direction that is predefined through their respective design process.

Geometric design of electric motors using adjoint-based shape optimization

Geometric design of electric motors using adjoint-based shape optimization

An adjoint-based approach for the surgical correction of nasal septal deviations

Deviations of the septal wall are widespread anatomic anomalies of the human nose; they vary significantly in shape and location, and often cause the obstruction of the nasal airways. When severe, septal deviations need to be surgically corrected by ear–nose–throat (ENT) specialists. Septoplasty, however, has a low success rate, owing to the lack of suitable standardized clinical tools for assessing type and severity of obstructions, and for surgery planning. Moreover, the restoration of a perfectly straight septal wall is often impossible and possibly unnecessary. This paper introduces a procedure, based on advanced patient-specific Computational Fluid Dynamics (CFD) simulations, to support ENT surgeons in septoplasty planning. The method hinges upon the theory of adjoint-based optimization, and minimizes a cost function that indirectly accounts for viscous losses. A sensitivity map is computed on the mucosal wall to provide the surgeon with a simple quantification of how much tissue removal at each location would contribute to easing the obstruction. The optimization procedure is applied to three representative nasal anatomies, reconstructed from CT scans of patients affected by complex septal deviations. The computed sensitivity consistently identifies all the anomalies correctly. Virtual surgery, i.e. morphing of the anatomies according to the computed sensitivity, confirms that the characteristics of the nasal airflow improve significantly after small anatomy changes derived from adjoint-based optimization.

Application of Multi-objective Adjoint-based Aerodynamic Optimisation on Generic Road Vehicle with Rear Spoiler

Finding possible solutions where there are multiple conflicting objectives to be simultaneously satisfied is a challenging situation. Multi-objective optimisation of a rear spoiler on a generic road vehicle model is carried out by using adjoint-based optimisation coupled with Computational Fluid Dynamics. The study aims to reduce the vehicle drag and increase vehicle downforce simultaneously by optimising the shape of the spoiler, by allowing the deformation to achieve the most optimised shape assuming no manufacturing constraint. The OpenFOAM software was used for the solver. A strategy for multi-objective optimisation was proposed by assigning appropriate objective function weight, leading to some possible solutions and Pareto front of the proposed design family. Five optimisation solutions of the non-dominated solution Pareto front resulting from the spoiler shape optimisation are presented, explaining the trade-off between conflicting drag and downforce objectives on the vehicle model. The baseline geometry of the simulation is in good agreement with the experimental measurement. The analysis of the shape changes in the proposed optimisation is deeply investigated in terms of the optimised geometry deformation, velocity contour comparison, recirculating region on the base, pressure coefficient comparison and stream-wise velocity component at the slant region of the model. The adjoint-based optimisation method in the presence study can handle multiple objective optimisations and generate possible optimised spoiler shapes to reduce drag and increase downforce. Free deformation of the shape yields in the unique shapes of the spoiler, enabling to manipulate of the base flow at the rear of the vehicle model.

Discrete and Continuous Adjoint-Based Aerostructural Wing Shape Optimization of a Business Jet

This article presents single- and multi-disciplinary shape optimizations of a generic business jet wing at two transonic cruise flow conditions. The studies performed are based on two high-fidelity gradient-based optimization tools, assisted by the adjoint method (following both discrete and continuous approaches). Single discipline and coupled multi-disciplinary sensitivity derivatives computed from the two tools are compared and verified against finite differences. The importance of not making the frozen turbulence assumption in adjoint-based optimization is demonstrated. Then, a number of optimization runs, ranging from a pure aerodynamic with a rigid structure to an aerostructural one exploring the trade-offs between the involved disciplines, are presented and discussed. The middle-ground scenario of optimizing the wing with aerodynamic criteria and, then, performing an aerostructural trimming is also investigated.

Adjoint-based shape optimization for radiative transfer in porous structure for volumetric solar receiver

Due to their large surface areas, porous structures are often used for volumetric solar receivers for efficient utilization of the solar energy. However, the precise simulation and optimization of radiative heat transfer inside porous structures are still challenging due to their complex geometries. In the present study, an adjoint-based shape optimization framework for radiative transfer in porous structures is first proposed. Specifically, an arbitrary complex geometry of a porous structure is represented by a level-set function embedded in a Cartesian grid system. The volume penalization method combined with the discrete ordinate method is employed to simulate a forward radiative transfer problem in a porous structure. Then, a cost functional for an adjoint analysis is defined by the volume average of the divergence of the radiative heat flux, which corresponds to the total heat absorbed by the porous structure with a negative sign. The adjoint equations for minimizing the cost functional, and the corresponding shape update formula are derived and implemented in an open-source software, OpenFOAM. The developed adjoint-based shape optimization framework is applied to three types of initial porous structures, i.e., SC Kelvin, BCC Kelvin, and square honeycomb models. For all the initial porous models, the front surface commonly extends towards the direction of the incident radiation, and consequently notable improvements of the absorption efficiency can be achieved through the optimization. In addition, a systematic parametric survey for the initial shape of the SC Kelvin model is conducted to reveal the impacts of the number of unit cells, the solid emissivity, and the solid temperature on the optimal shapes and the resulting absorption efficiencies.

A novel accelerated convergence method for solving adjoint equations based on modal reduction

The efficiency of adjoint-based aerodynamic shape optimization depends critically on the solution efficiency of adjoint equations. In this letter, we employ the Proper Orthogonal Decomposition (POD) method to analyze the adjoint field samples and project them from the physical space into a low-order modal space. Subsequently, the full-order adjoint equations are reduced to low-order equations using the POD modes. Thus, we can efficiently predict the initial values for pseudo-time marching, thereby accelerating the solution of adjoint equations. Results indicate that the high-order POD modes are crucial for constructing the low-dimensional system. Moreover, this method can be seamlessly integrated with our previously established Dynamic Mode Decomposition (DMD) acceleration method to form a POD+DMD acceleration approach. Application of this approach to the flow past a National Advisory Committee for Aeronautics 0012 airfoil demonstrates a noteworthy 80.9% reduction in iteration numbers when solving the adjoint equations. Even for the airfoil located on the upper boundary of sampling space, the number of iterations is still reduced by 72.6%. Therefore, we believe that the proposed method holds significant promise for improving the efficiency of adjoint-based aerodynamic shape optimization in future research.

Exploring the fundamental limits of integrated beam splitters with arbitrary phase via topology optimization.

Optical beam splitters are essential for classical and quantum photonic on-chip systems. In integrated optical technology, a beam splitter can be implemented as a beam coupler with two input and two output ports. The output phases are constrained by the conservation of energy. In lossless beam splitters, the phase shift between the output fields is π and zero for excitation from the first and second input ports, respectively. Therefore, for excitation from both inputs, the phase between the output fields, defined as beam splitter phase (BSP), is π. The BSP leads to several phenomena, such as the quantum interference between two photons, known as the Hong-Ou-Mandel effect. By introducing losses, BSP values different than π become theoretically possible, but the design of 2 × 2 beam couplers with an arbitrary phase is elusive in integrated optics. Inspired by the growing interest on fundamental limits in electromagnetics and inverse design, here we explore the theoretical limits of symmetrical integrated beam splitters with an arbitrary BSP via adjoint-based topology optimization. Optimized 2D designs accounting for fabrication constraints are obtained for several combinations of loss and phase within the theoretical design space. Interestingly, the algorithm does not converge for objectives outside of the theoretical limits. Designs of beam splitters with arbitrary phase may find use in integrated optics for quantum information processing.

Adjoint-based design optimization of a Kenics static mixer

Mixing processes are important in many applications like food processing or in chemical reactors. Achieving homogeneous mixtures over short distances with a small pressure drop is a challenge. Kenics static mixers usually provide a good balance between pressure drop and mixing quality. Nevertheless, little attention has been paid to the use of this mixing device in mixing gases, and consequently relatively large pressure drops are obtained. In this paper, the discrete adjoint method is used to obtain optimized Kenics static mixer designs with a negligible increase in pressure drop. First, a framework of a composition-dependent model based on the ideal gas mixing laws for thermochemical properties is implemented and coupled to the existing discrete adjoint solver within SU2, which is an open-source software suite for multiphysics simulations and design optimization. Subsequently, in order to identify the key geometrical parameters that affect the pressure drop in the mixer, numerical simulations are performed for different aspect ratios, blade thicknesses, and Reynolds numbers. These results are compared with available simulations and correlations reported in the literature, showing good agreement. These simulations indicate that reducing the aspect ratio and increasing the blade thickness enhance the mixing process in the mixing units. However, a substantial increase in pressure drop is observed. The results of the parameter study are used as a starting point to optimize the blade designs inside the Kenics static mixer using a discrete adjoint approach. The objective is to minimize the variance of the mass fraction at the outlet, which is a measure of mixture homogeneity, without substantially increasing the pressure drop along the mixing device. The results indicate that an optimized design can be obtained with a negligible increase in pressure drop, highlighting the capabilities of the discrete adjoint design approach as an optimization tool for mixing devices.

Shape optimization to enhance energy harvesting from vortex-induced vibration of a circular cylinder

In previous research on energy harvesting from vortex-induced vibrations (VIVs), the cross section of the structure commonly utilizes basic geometric shapes like circular, ellipse, square, and semicircle. Nevertheless, exploring optimized shapes for energy harvesting from VIV remains an understudied area. To address this gap, this paper employs adjoint-based unsteady shape optimization to increase the efficiency of energy harvesting from VIV of a circular cylinder at low Reynolds numbers. The goal of the optimization is to maximize the plunge-damping derivative of a single-degree-of-freedom transversely vibrating cylinder, which represents the rate of energy injected into the structure by the flow. To facilitate this process, an efficient method to evaluate the gradient of the objective function with respect to shape parameters is provided via the proposed unsteady discrete adjoint method. Results show that, through optimization, the low-pressure region behind the cylinder is significantly enlarged and the separation points move forward, resulting in the faster development of separation vortex and reduced stability of the fluid–structure coupling system. As a consequence, the intensity of VIV as well as the corresponding power generation efficiency is remarkably enhanced, accompanied by a notable expansion in the energy harvesting region.

Aeroacoustic simulation of bluff bodies with protrusions at moderate Reynolds number

This paper presents an evaluation of passive control methods that employ surface protrusions to mitigate the aerodynamic sound generated from a cylinder wake flow. Building on previous designs optimized for low Reynolds numbers (Re = 150) through adjoint-based aeroacoustic shape optimization, this study investigated the performance under a moderate Reynolds number (Re = 67 000) condition typical of mechanical engineering applications using aeroacoustic simulations based on the lattice Boltzmann method. Three configurations of surface protrusions were tested, all of which were found to significantly reduce the mean drag by at least 45% compared with that of an unmodified circular cylinder. Designs featuring rear protrusions outperformed the conventional splitter plate in terms of the sound reduction performance, with symmetrical protrusions on both the front and rear surfaces achieving a tonal sound reduction of 13 dB. However, a specific protrusion design increased the low-frequency sound owing to the intensified large-scale flow separation. These findings highlight the effectiveness of rear protrusions in suppressing wake oscillations and dipole sound generation in the subcritical Reynolds number range. Moreover, the study revealed the need to tailor the front protrusion shape to the Reynolds number for performance optimization.