Adjoint Method Research Articles

Compact SOI Dual-Mode (De)multiplexer Based on the Level Set Method

Mode (de)multiplexer is an essential device in integrated multimode photonic systems. Here, we present a dual-mode (de)multiplexer that separates two input modes, TE0 and TE1, into two output ports while converting TE1 to TE0 mode. Based on the adjoint and level set method, the device features a small footprint of 9.4 μm × 2.9 μm, and a minimum feature size over 200 nm is achieved, affirming stable and reliable fabrication. Through simulations, we observed insertion losses of less than 0.28 dB for TE0 mode and 0.35 dB for TE1 mode within the wavelength range of 1500–1600 nm, accompanied by crosstalk levels lower than −30 dB. In our experimental tests, we achieved insertion losses of less than 0.89 dB for TE0 mode and 0.44 dB for TE1 modes within the 1530 nm to 1570 nm range, with crosstalk maintained below −25 dB. Furthermore, we conducted an experimental verification of the differences between the standard device and the boundary dilation/erosion device, observing an insertion loss degradation by 0.61 dB within a deviation range of ±40 nm, which demonstrates the device’s robustness to the fabrication. The proposed devices exhibit exceptional performance and feature a compact structure, thus holding significant potential for the development of future multimode integrated photonic circuits.

An Adjoint Method for High-Resolution EPMA Based on the Spherical Harmonics (PN) Model of Electron Transport

Electron Probe Microanalysis (EPMA) is a nondestructive technique to determine the chemical composition of material samples in the micro- to nanometer range. Based on intensity measurements of x-radiation, the reconstruction of the material composition and structure poses an inverse problem. The reconstruction methods currently applied are based on models that assume a homogeneous or a layered structure of the studied material. To increase the spatial resolution of reconstruction in EPMA the combination of a more sophisticated reconstruction method, which is based on a model that allows complex material structure, together with multiple measurements with varying beam configurations is required. We present a deterministic k-ratio model that is based on the PN model, an approximation of the radiative transfer equation for electron transport. Our goal is to approximate a maximum likelihood solution of the inverse problem using gradient-based optimization. We detail the application of the model in the context of algorithmic differentiation, in particular by deriving its continuous adjoint formulation. Algorithmic differentiation provides the flexibility to adapt the reconstruction method to various material parametrizations and thus to regularize and take into account prior knowledge. Through examples, we verify our implementation and demonstrate the flexibility of the reconstruction/differentiation method.

Design method for large-scale wide field-of-view monochromatic metalenses

We propose a novel design method for wide field-of-view monochromatic metalenses. The proposed technique partitions the outer region of a metalens into supercells, which are generated by dividing the outer region into intervals along the radial direction, where the target phase changes by 2π, and along the angular direction with a constant angular periodicity. Therefore, the shape of each supercell can be approximated as a rectangle with its size comparable to a wavelength. The arrangement of pillars within this supercell is determined by metagrating optimization via the adjoint method. The optimization process considers both inter-pillar couplings and the range of incidence angles. This makes the design of large-scale wide field-of-view high-efficiency metalenses more tractable than the conventional unit-cell-based method, which is prone to efficiency decrease especially near the lens periphery. Furthermore, it has a potential advantage in terms of computational cost over other recently proposed optimization-based methods.

Concurrent topology optimization of multiscale piezoelectric actuators

As an important component of smart structural devices, piezoelectric composites have strong macroscopic electromechanical coupling properties which often depend on their microstructural geometries and material parameters. Therefore, optimizing the microstructural design is crucial for achieving the desired macroscale effective properties. To this end, this paper conducts a concurrent multiscale topology optimization (TO) to optimize the material distribution of piezoelectric actuators, aiming to maximize the electrical and mechanical energy transmission to meet specific engineering requirements. Firstly, an energy method is used to homogenize the microstructures and obtain the effective parameters of the piezoelectric material. Then, the piezoelectric material with penalization and polarization (PEMAP-P) approach in conjunction with the adjoint method is employed to calculate the sensitivity of objective and constraint functions. The sensitivities at both the macroscale and microscale are obtained by considering the interscale coupling effects. Finally, the concurrent bi-level iteration based on the optimality criteria (OC) or the method of moving asymptotes (MMA) is conducted. Through this research, we have illuminated the relations between the optimized performance of the actuator and the material volume fractions at each scale. We have also compared in detail the differences between two-scale and single-scale designs, as well as the differences between the simplified half-symmetric geometric model and the full geometric model. We find that a larger volume fraction of material either at the macroscale or microscale generates a larger transmitted displacement magnitude under the same applied electric field, and for a given total material, the transmitted maximum increases with increasing microstructural volume fraction.

Organ registration from partial surface data in augmented surgery from an optimal control perspective

We address the problem of organ registration in augmented surgery, where the deformation of the patient’s organ is reconstructed in real-time from a partial observation of its surface. Physics-based registration methods rely on adding artificial forces to drive the registration, which may result in implausible displacement fields. In this paper, we look at this inverse problem through the lens of optimal control, in an attempt to reconstruct a physically consistent surface load. The resulting optimization problem features an elastic model, a least-squares data attachment term based on orthogonal projections, and an admissible set of surface loads defined prior to reconstruction in the mechanical model. After a discussion about the existence of solutions, we analyse the necessary optimality conditions and use them to derive a suitable optimization algorithm. We implement an adjoint method and we test our approach on multiple examples, including the so-called Sparse Data Challenge . We obtain very promising results, that illustrate the feasibility of our approach with linear and nonlinear models.

ECCD studies for EU-DEMO plasmas

The possibility of driving a large fraction of the plasma current in a tokamak reactor employing electron cyclotron waves is investigated for scenarios as envisaged in the European DEMO through beam tracing calculations performed with the TORBEAM code, which assumes a linear regime for power absorption and employs an adjoint method for the determination of the driven current. Comparatively high ECCD efficiency in the inner half of the plasma column can be achieved by injecting the wave from an elevated position. On the other hand, the efficiency deteriorates in the colder, outer part of the plasma, so that a prohibitive amount of power would be needed to sustain the plasma current non-inductively by ECCD only in the considered scenarios. As an alternative scheme, the injection of slow extraordinary wave below the fundamental resonance is considered. The basic physics features of this scenario are discussed.

Seasonality of Water Exchange in the Northern South China Sea from Hydrodynamic Perspective

In this study, we utilized exposure time (θ¯) as a key metric to investigate water exchange and its spatiotemporal variations in the Northern South China Sea (NSCS). The Eulerian adjoint method and Lagrangian tracking were adopted to capture a comprehensive view of water exchange in coastal regions. Our findings reveal distinct spatial and seasonal variations in θ¯. Spatially, a long θ¯ (exceeding 150 days) appears in the coastal region, and the largest values occur in the Beibu Gulf (300 days). Temporally, θ¯ exhibits clear seasonal patterns across the extensive shelf area, influenced by the seasonal monsoon which induced seasonally reversing shelf current and results in symmetrical distribution patterns of θ¯ across the board shelf during both winter and summer months. θ¯ is longer in winter than in summer. The study also revealed pronounced vertical contrasts in cross-isobath transport over the NSCS shelf, though significant vertical variations in net exchange time were noted only in specific locations, including the northeast side of Hainan Island, the Beibu Gulf mouth, and along the west side of Taiwan Island. The Beibu Gulf emerged as a critical factor in the NSCS’s water exchange dynamics in both seasons. In summer, it impacts more than 20% of the water exchange over adjacent areas, particularly through its westward transport against typical northeastward shelf currents. This highlights the combined effect of the westward spread of the Pearl River freshwater and the stable slope current on regional hydrodynamics. In winter, the Gulf’s retention characteristics profoundly affected even distant areas, contributing to up to 50% of water exchange, showing its broad impact on the NSCS’s water dynamics throughout the year.

Eikonal-equation-based characteristic reflection traveltime tomography

Automatically picking reflection traveltimes from the prestack shot gathers and positioning and updating the reflector depths during inversion are challenging in conventional eikonal-equation-based adjoint-state reflection traveltime tomography methods. To solve these problems, we develop an eikonal-equation-based adjoint-state characteristic reflection traveltime tomography (ASCRT) method. This method automatically extracts the reflection traveltimes by sequentially applying characteristic reflector picking in the depth migration section, followed by eikonal-equation-based traveltime calculation and an event-tracking algorithm. We also develop an efficient eikonal-equation-based kinematic imaging method to rapidly position and update the reflector depths. With our ASCRT method, the characteristic reflector positions and the velocity above the reflector are alternately updated. Our ASCRT method is performed with a layer-stripping strategy that sequentially uses selected characteristic reflectors to update the velocity model from shallow to deep. Compared with the wave equation-based reflection traveltime inversion methods, the efficiency of the ASCRT method is improved by several orders of magnitude, and the memory consumption is remarkably reduced. Synthetic and field data examples are presented to demonstrate the effectiveness and efficiency of our method.

Compact simply-connected SOI spot size converters for TE and TM polarizations

Spot size converters (SSCs) are pivotal building blocks of silicon on insulator (SOI) on-chip optical systems, which enable efficient coupling of modes between waveguides of different widths. In this paper, we propose two SSCs for TE and TM polarizations based on level-set and adjoint methods, which can convert modes between 0.5 µm and 10 µm wide waveguides. The devices’ lengths are 22.4 µm and 23.7 µm, respectively, with simply-connected structures and the 200 nm minimum feature size. The simulation results demonstrate minimal insertion losses of 0.04 dB and 0.12 dB for the two SSCs, with respective 1 dB bandwidths exceeding 170 nm and 250 nm. Within the wavelength range of 1520–1590 nm, the insertion losses are below 0.42 dB and 0.24 dB. Experimental results, normalized with a referenced 250 µm length adiabatic linear taper, indicates insertion losses of less than dB and dB within the 1520–1590 nm wavelength range for the proposed SSCs, demonstrating a ten-fold reduction in length while maintaining comparable performance to the reference taper. Our devices hold the potential to facilitate the utilization of grating couplers in high-efficiency and high integration applications.

Three-dimensional shape optimization of fins for application in compact supercritical CO2 solar receivers

The compact solar receiver is a promising option for S-CO2 solar receivers, designed to operate safely and efficiently under high-temperature, high-pressure, and high solar flux conditions. Adding fins with outstanding thermal-hydraulic performance in the mini-channel contributes to maximizing heat transfer performance and minimizing flow resistance for compact solar receivers. However, most of the research on high-performance fins is limited to two-dimensional optimization, ignoring the effect of shape changes in the height direction of the fins. The present study employs a combination of the finite volume method and the adjoint method to optimize single-row cylindrical fin shapes in three-dimensional for compact solar receivers, and the strengthening mechanism of high efficiency and low resistance is further revealed in the view of entropy generation. The results indicate: (1) three-dimensional fins exhibit significant variation in cross-section along the height direction, and each fin is unique; (2) three-dimensional fins exhibit more excellent thermal-hydraulic performance than cylindrical fins, showing an increase of 13 % and 20 % in the performance evaluation criterion (PEC) in case 1 (qw = 160 kW·m−2, vin = 0.487 m·s−1) and in case 2 (qw = 584 kW·m−2, vin = 1.778 m·s−1), respectively; (3) the entropy generation due to heat transfer is reduced by 4.9% and 3.9%; as well as the entropy generation due to pressure drop is decreased by 8.5 % and 16.5 %, in case 1 and case 2, respectively; (4) robustness results show that three-dimensional fins can maintain higher thermal-hydraulic performance than the initial cylindrical fin even under wide working conditions.

Converted-wave reflection traveltime inversion with free-surface multiples for ocean-bottom-node data

Advancements in ocean-bottom-node (OBN) technologies have enabled the recording of high-quality multicomponent seismic data, which can provide crucial information about subsurface properties through elastic seismic imaging. However, imaging multicomponent data is challenging due to the requirement for P- and S-wave velocity models. Conventional processing relies on estimating the S-wave velocity model using a primary converted wave (C wave) because the pure S-wave primaries are not available if using standard air-gun sources. When a free surface is present, the C waves recorded with geophones can be contaminated by source-side P-wave multiples, especially water-layer reverberations, which are difficult to remove even using state-of-the-art wavefield decomposition methods. These C-wave multiples can cause significant mismatches between synthetic and observed data during reflection traveltime or waveform inversion if only primary P-to-S (PS) conversions are simulated. To address these issues and reliably build an S-wave velocity model for PS imaging, we develop a C-wave reflection traveltime inversion method that takes source-side free-surface multiples into account. The traveltime misfit of C-wave data is extracted using dynamic image warping to ensure a better match between the synthetic and observed data. The functional gradient of the objective function is derived using the adjoint state method. Based on sensitivity kernel analyses, a convenient gradient preconditioning strategy is developed to robustly separate the contribution of primary and multiple C waves during backpropagation. This strategy can effectively mitigate the high-wavenumber crosstalk associated with nonphysical wavepaths in the gradient and thus allow more accurate updating for the S-wave velocity model. The synthetic data and shallow-water OBN data from the East China Sea indicate that this approach can reliably recover the S-wave macrovelocity structures for PS imaging.

Optimization of Non-Newtonian Flow through a Coat-Hanger Die Using the Adjoint Method

The use of coat-hanger dies is prevalent in the plastic film and sheet extrusion industry. The product quality and the power of the extrusion machine depend on the uniformities of the fluid velocity at the exit and the pressure drop. Die manufacturers face the challenge of producing coat-hanger dies that can extrude materials uniformly and with a minimal pressure drop. Previous studies have analyzed the die outlet’s flow homogeneity and pressure drop using various numerical simulations. However, the combination of the scheme programming language together with the Adjoint Method of Optimization has yet to be attempted. The adjoint optimization method has been demonstrated to be beneficial in addressing issues related to shape optimization problems and it may also be beneficial in optimizing the design of dies used in polymer melt extrusion. In this study, the proposed innovations involve incorporating both the Scheme programming language and Adjoint solver to examine and optimize the coat hanger’s flow homogeneity and pressure drop. Before optimization, the outlet velocity was almost 10 times higher at the die center than at the edges but after optimization, it became more uniform. The proposed optimized coat-hanger die geometry results in more uniform melt flow as demonstrated by the velocity contour plot and the outlet velocity graph in the die slit area, reducing the deviation value from 0.097 to 0.015. Additionally, the mass flux variance across the die outlet decreased by 71.6% from 0.015069 kg m−2 s−1 to 0.004281 kg m−2 s−1. Therefore, using this method reduces the amount of time wasted on trial and error or other optimization techniques that may be limited by design constraints.

Enhancing formula student car performance: Nose shape optimization via adjoint method

For a formula student vehicle, the nose is the first point of flow contact and decides the flow direction to help other aero components work efficiently. Therefore, an optimized nose shape is crucial for the optimum running condition of the vehicle. In recent years, Gradient-based optimization in the automotive industry has emerged with an adjoint approach. The adjoint solver is used to modify the geometry such as the nose of a car to the best optimal shape with respect to other design variables. Hence, the present study encompasses an examination conducted under six distinct Reynolds number conditions, specifically, 8.15 × 105, 9.05 × 105, 9.96 × 105, 10.87 × 105, 11.77 × 105, and 12.68 × 105. The simulation setup is validated using the Ahmed Body's 30° slant angle. The setup uses the k-omega SST turbulence model to capture the airflow phenomenon over the racing car. The simulation setup shows only a 0.07 % variation from the existing experimental findings. The results are represented in terms of different 2D plots, contours and streamline plots. The optimized nose has 27 % less drag and consumes 27.09 % less fuel than the base model at a Reynolds number of 10.8 × 105. To the automotive industry and race community, this optimization technique will be expected to enhance the vehicle's overall performance.

Numerical and experimental investigations on shape optimization of submerged floating tunnels with a discrete adjoint method

The shape of the tube in submerged floating tunnels (SFTs) plays a critical role in determining their performance and safety in marine environments. In this study, a gradient optimization procedure based on the discrete adjoint method is performed to minimize the drag force under uniform currents action. The free form deformation approach is employed to parameterize the design variables. The physical mechanisms of the optimization process are revealed via unsteady numerical simulations and experimental tests. Moreover, the hydrodynamic performance of the tube with the optimal shapes subjected to a wave–current combination is also evaluated. The results show that the drag coefficient is significantly reduced through optimization by reducing the pressure amplitude in the negative-pressure area. Additionally, the lift oscillation is also suppressed to delay structural fatigue, owing to the strength reduction and increased distance of the wake vortex. The experimental analysis indicates the advantage of the optimal shape in reducing the loads under wave–current actions, accompanied by changes in the frequency distribution of the force and vortex structure. The elliptical shape of the SFT's tube provides significant advantages in drag reduction at high Reynold number. Suggestions on the shape design of the section of SFT are given according to different types of constraints.

Adjoint Inversion of Near‐Field Pressure Gauge Recordings for Rapid and Accurate Tsunami Source Characterization

AbstractWe explore the potential of the adjoint‐state tsunami inversion method for rapid and accurate near‐field tsunami source characterization using S‐net, an array of ocean bottom pressure gauges. Compared to earthquake‐based methods, this method can obtain more accurate predictions for the initial water elevation of the tsunami source, including potential secondary sources, leading to accurate water height and wave run‐up predictions. Unlike finite‐fault tsunami source inversions, the adjoint method achieves high‐resolution results without requiring densely gridded Green's functions, reducing computation time. However, optimal results require a dense instrument network with sufficient azimuthal coverage. S‐net meets these requirements and reduces data collection time, facilitating the inversion and timely issuance of tsunami warnings. Since the method has not yet been applied to dense, near‐field data, we test it on synthetic waveforms of the 2011 Mw 9.0 Tohoku earthquake and tsunami, including triggered secondary sources. The results indicate that with a static source model without noise, using the first 5 min of the waveforms yields a favorable performance with an average accuracy score of 93%, and the largest error of predicted wave amplitudes ranges between −5.6 and 1.9 m. Using the first 20 min, secondary sources were clearly resolved. We also demonstrate the method's applicability using S‐net recordings of the 2016 Mw 6.9 Fukushima earthquake. The findings suggest that lower‐magnitude events require a longer waveform duration for accurate adjoint inversion. Moreover, the estimated stress drop obtained from inverting our obtained tsunami source, assuming uniform slip, aligns with estimations from recent studies.

Damage identification based on XFEM: inverse analysis and experimental validation

AbstractModel‐based inverse analyses emerge as an alternative means to reconstruct buried objects in structures. In this paper, we introduced an inverse analysis using XFEM and adjoint method to identify the cracks, which is based on laboratory experimental data. XFEM model allows precisely reproducing the crack forms in the structures, whilst adjoint state with level set for inverse problem allows calculating the gradient of complex geometrical structures and avoiding the local minima traps. Besides proposing a modification XFEM to model reinforced concrete behaviors, we also establish the adjoint formulas for inverse analysis based on the measured displacement and strain in the experiments. Finally, the preliminary results are verified with experimental data.

Adjoint Data Assimilation of the Flow on the Southern Flank of Georges Bank: March–June 1999

An adjoint data assimilation method was incorporated into the ECOM-si coastal ocean circulation model and applied to assimilate the flow field on the southern flank of Georges Bank from March to June 1999. The model was driven by tidal forcing consisting of ten tidal constituents at the open boundary, observed winds, and surface heat fluxes. Numerical experiments were conducted following a strategy to understand the critical issues affecting the efficiency and accuracy of the assimilated flow field. The adjoint data assimilation method significantly improved the computational accuracy of subtidal currents, especially for the along-isobath velocity. The integration window length and iteration number were two parameters affecting the assimilation convergence rate toward the observations. In such a nonlinear dynamical system, using a window length close to the M2 tidal period could make the adjoint model difficult to converge, no matter how many iterations were made. Reducing the time interval for the window length speeded up the convergence rate, but it was paid out with the sacrifice of statistical confidence. This assimilation experiment used a 6-h window length, which led to a faster convergence rate in the first ten iterations. The best-assimilated fields that satisfied the error criteria were obtained as the iteration number increased.

GIA imaging of 3-D mantle viscosity based on palaeo sea level observations - Part I: Sensitivity kernels for an Earth with laterally varying viscosity.

A key initial step in geophysical imaging is to devise an effective means of mapping the sensitivity of an observation to the model parameters, that is to compute its Fréchet derivatives or sensitivity kernel. In the absence of any simplifying assumptions and when faced with a large number of free parameters, the adjoint method can be an effective and efficient approach to calculating Fréchet derivatives and requires just two numerical simulations. In the Glacial Isostatic Adjustment problem, these consist of a forward simulation driven by changes in ice mass and an adjoint simulation driven by fictitious loads that are applied at the observation sites. The theoretical basis for this approach has seen considerable development over the last decade. Here, we present the final elements needed to image 3-D mantle viscosity using a dataset of palaeo sea-level observations. Developments include the calculation of viscosity Fréchet derivatives (i.e. sensitivity kernels) for relative sea-level observations, a modification to the numerical implementation of the forward and adjoint problem that permits application to 3-D viscosity structure, and a recalibration of initial sea level that ensures the forward simulation honours present-day topography. In the process of addressing these items, we build intuition concerning how absolute sea-level and relative sea-level observations sense Earth's viscosity structure and the physical processes involved. We discuss examples for potential observations located in the near field (Andenes, Norway), far field (Seychelles), and edge of the forebulge of the Laurentide ice sheet (Barbados). Examination of these kernels: (1) reveals why 1-D estimates of mantle viscosity from far-field relative sea-level observations can be biased; (2) hints at why an appropriate differential relative sea-level observation can provide a better constraint on local mantle viscosity and (3) demonstrates that sea-level observations have non-negligible 3-D sensitivity to deep mantle viscosity structure, which is counter to the intuition gained from 1-D radial viscosity Fréchet derivatives. Finally, we explore the influence of lateral variations in viscosity on relative sea-level observations in the Amundsen Sea Embayment and at Barbados. These predictions are based on a new global 3-D viscosity inference derived from the shear-wave speeds of GLAD-M25 and an inverse calibration scheme that ensures compatibility with certain fundamental geophysical observations. Use of the 3-D viscosity inference leads to: (1) generally greater complexity within the kernel; (2) an increase in sensitivity and presence of shorter length-scale features within lower viscosity regions; (3) a zeroing out of the sensitivity kernel within high-viscosity regions where elastic deformation dominates and (4) shifting of sensitivity at a given depth towards distal regions of weaker viscosity. The tools and intuition built here provide the necessary framework to explore inversions for 3-D mantle viscosity based on palaeo sea-level data.

Integrated optimization of components’ layout and structural topology with considering the interface stress constraint

Structural designs in the engineering field commonly require the prior embedding of several fixed-shape components to satisfy desired performance or functional requirements. In the past few decades, topology optimization has been regarded as an effective approach to deal with the optimization problems of continuous structures embedded with multiple components. However, the interfaces between the host supporting structure and embedded components are assumed perfect-bonded, which may result that the optimal design occurs interface failure and structural damage under extreme or long-time service conditions. In this study, a new topology optimization method of structures with embedded components considering the tension/compression-asymmetric interface stress constraint is proposed under the solid isotropic material with penalization (SIMP) method framework. This proposed method optimizes simultaneously the topology of the host structure and the layout of embedded components. A new interpolation model is developed to determine interface layers between the host structure and embedded components. Then, interface stress constraints are derived and imposed on each element of interface layers. For relieving the computational burden caused by enormous stress constraints, the global p-norm stress aggregation method is adopted to quantify the peak stresses. Subsequently, the interface stress constraint is introduced into the integrated optimization of multi-component layout and structural topology, the objective function is structural compliance and constraint functions are composed of volume, non-overlapping and tension/compression-asymmetric interface stress constraints. The corresponding sensitivity analyses are computed by using the adjoint method. Several two-dimensional and three-dimensional numerical examples are provided to verify the effectiveness of the proposed method.

Study on continuous adjoint-based optimization method for internal cooling structures in turbine blades with conjugate heat transfer

A continuous adjoint-based shape optimization system is presented in this paper for conjugate heat transfer (CHT) problems and is applied to the optimization for internal cooling structures of gas turbine blades. With the consideration for the variation of heat conduction equation in the solid regions, the adjoint equations and adjoint boundary conditions in different regions are derived. The adjoint energy correction equation is also derived successfully based on the SIMPLE algorithm, and the adjoint CHT solver is implemented by imposing the values of adjoint energy at the interfaces. Two cases of full 3D optimization for the C3X hollow blade and cooling channel with pin fins are conducted by using the optimization system. In both cases, their flow and heat transfer characteristics are optimized and analyzed in detail. This study demonstrates the potential advantages and basic capabilities of the continuous adjoint method in dealing with CHT optimization problems.