Adjoint Variable Method Research Articles

Entropy generation analysis during adjoint variable-based topology optimization of porous reaction-diffusion systems under various design dimensionalities

Several studies attempted to enhance the performance of various reaction-diffusion systems, such as electrodes of electrochemical devices, by modifying the spatial distribution of porosity throughout the porous reactor. However, research is proceeding without knowing the theoretical limitation of the improvement. To connect optimization techniques with a well-established theoretical approach, this paper presents topology optimization for the design of diffusion fields in reaction-diffusion systems together with entropy generation analysis using nonequilibrium thermodynamics. Topology optimization of porosity distribution in reaction-diffusion systems was carried out using the adjoint variable methods. The optimization problem was formulated to maximize the reaction in the design domain. The results obtained from the investigation under various dimensionalities were compared and discussed. A formula for the local entropy generation of reaction-diffusion systems was derived and used to assess the topology optimization results. While the findings showed an insignificant difference between the overall reaction rate of 0D and 1D cases, optimization of higher dimensionalities (2D and 3D) considerably enhanced the overall reaction rate. The results revealed that the optimum porosity distribution corresponds to the most uniform and minimum entropy generation.

Modal laminar–turbulent transition delay by means of topology optimization of superhydrophobic surfaces

When submerged under a liquid, the microstructure of a SuperHydrophobic Surface (SHS) traps a lubricating layer of gas pockets, which has been seen to reduce the skin friction of the overlying liquid flow in both laminar and turbulent regimes. More recently, spatially homogeneous SHS have also been shown to delay laminar–turbulent transition in channel flows, where transition is triggered by modal mechanisms. In this study, we investigate, by means of topology optimization, whether a spatially inhomogeneous SHS can be designed to further delay transition in channel flows. Unsteady direct numerical simulations are conducted using the spectral element method in a 3D periodic wall-bounded channel. The effect of the SHS is modelled using a partial slip length on the walls, forming a 2D periodic optimization domain. Following a density-based approach, the optimization procedure uses the adjoint-variable method to compute gradients and a checkpointing strategy to reduce storage requirements. This methodology is adapted to optimizing over an ensemble of initial perturbations.This study presents the first application of topology optimization to laminar–turbulent transition. We show that this method can design surfaces that delay transition significantly compared to a homogeneous counterpart, by inhibiting the growth of secondary instability modes. By optimizing over an ensemble of streamwise phase-shifted perturbations, designs have been found with comparable mean transition time and lower variance.

Adjoint variable method for sensitivity analysis of performance metrics involving mode shapes

An adjoint variable method is proposed for sensitivity analysis of performance metrics involving mode shapes. First, a functional is defined to render these performance metrics in scalar form so that their sensitivity analysis can be performed in a unified form. The adjoint variable method is then performed with the eigenproblem and normalization conditions as constraints on these performance metrics to form Lagrangian functional. After obtaining the Lagrangian multipliers, the sensitivity of these performance metrics can be directly calculated. In the framework of the adjoint variable method, three different methods are proposed to obtain these Lagrangian multipliers. Afterward, the performance of the direct differentiation method and the adjoint variable method are compared. Furthermore, modal assurance criteria (MAC) and elemental modal strain energy are used as examples of performance metrics, and two numerical examples are applied to verify the performance of the adjoint variable method and the direct differentiation method. The results show that the proposed method is more effective for sensitivity analysis of performance metrics with multiple design variables without loss of accuracy.

Self-adjusting inverse design method for nanophotonic devices.

Nanophotonic devices, which consist of multiple cell structures of the same size, are easy to manufacture. To avoid the optical proximity effect in the ultraviolet lithography process, the cell structures must be maintained at a distance from one another. In the inverse design process, the distance is maintained by limiting the optimized range of the location. However, this implementation can weaken the performance of the devices designed during transmission. To solve this problem, a self-adjusting inverse design method based on the adjoint variable method is developed. By introducing artificial potential field method, the location of one cell structure is modified only when the distances between this cell structure and other cell structures are smaller than a threshold. In this case, the range of the location can be expanded, and thus the performance of the designed devices can be improved. A wavelength demultiplexer with a channel spacing of 1.6 nm is designed to verify the performance of the proposed method. The experiment reveals that the transmission of the designed devices can be improved by 20%, and the self-adjusting inverse design process is 100 times faster than the inverse-design process based on the genetic algorithm.

DeepAdjoint: An All-in-One Photonic Inverse Design Framework Integrating Data-Driven Machine Learning with Optimization Algorithms

In recent years, hybrid design strategies combining machine learning (ML) with electromagnetic optimization algorithms have emerged as a new paradigm for the inverse design of photonic structures and devices. While a trained, data-driven neural network can rapidly identify solutions near the global optimum with a given data set’s design space, an iterative optimization algorithm can further refine the solution and overcome data set limitations. Furthermore, such hybrid ML-optimization methodologies can reduce computational costs and expedite the discovery of novel electromagnetic components. However, existing hybrid ML-optimization methods have yet to optimize across both materials and geometries in a single integrated and user-friendly environment. In addition, due to the challenge of acquiring large data sets for ML, as well as the exponential growth of isolated models being trained for photonics design, there is a need to standardize the ML-optimization workflow while making the pretrained models easily accessible. Motivated by these challenges, here we introduce DeepAdjoint, a general-purpose, open-source, and multiobjective “all-in-one” global photonics inverse design application framework that integrates pretrained deep generative networks with state-of-the-art electromagnetic optimization algorithms such as the adjoint variables method. DeepAdjoint allows a designer to specify an arbitrary optical design target, then obtain a photonic structure that is robust to fabrication tolerances and possesses the desired optical properties, all within a single user-guided application interface. We demonstrate DeepAdjoint for the design of infrared-controlled metasurfaces and show that a wide range of structures and absorption spectra can be achieved and optimized, including single- and multiresonance behavior through single- and supercell-class structures, respectively. Our framework, thus, paves a path toward the systematic unification of ML and optimization algorithms for photonic inverse design.

FFT-based Inverse Homogenization for Cellular Material Design

• An FFT-based inverse homogenization approach is proposed for efficiently designing cellular materials. • Rotating architected materials with negative Poisson's ratio are generated. • Architected cellular materials with maximum stiffness and isotropy are sought out. • The inverse homogenization method for scaffold design with specified mechanical performance is validated. Besides constituting components, the properties of composites are highly relevant to their microstructures. The work proposed a fast Fourier transform (FFT)-based inverse homogenization method implemented by the bi-directional evolutionary structural optimization (BESO) technique to explore the vast potential of cellular materials. The periodic boundary condition of self-repeated representative volume elements can be naturally satisfied in the FFT-based homogenization scheme. The objective function of the optimization problem is the specific moduli or the quadratic difference between the effective value and the target, which are obtained in terms of mutual strain energies. Its sensitivity to the design variable, namely the elemental density, is derived from the adjoint variable method and used as the criterion to remove or add material in local elements. Numerical examples show that the proposed method generates a series of architected cellular materials with maximum modulus, negative Poisson's ratio, and specific elasticity tensor. FFT-based homogenization in the method demands less memory usage but has high efficiency. Thus, it can achieve topology optimization of unit cell with one million hexahedral elements. This approach contributes to the extended application of FFT-based homogenization and can guide the microstructure design of mechanical metamaterials. :

An improved Material Mask Overlay Strategy for the desired discreteness of pressure-loaded optimized topologies

This paper presents a Material Mask Overlay topology optimization approach with the improved material assignment at the element level for achieving the desired discreteness of the optimized designs for pressure-loaded problems. Hexagonal elements are employed to parametrize the design domain. Such elements provide nonsingular local connectivity; thus, checkerboard patterns and point connections inherently get subdued. Elliptical negative masks are used to find the optimized material layout. Each mask is represented via seven parameters that describe the location, shape, orientation, material dilation, and erosion variables of the mask. The latter two variables are systematically varied in conjunction with a grayscale measure constraint to achieve the solutions' sought 0-1 nature. Darcy's law with a drainage term is used to model the pressure load. The obtained pressure field is converted into the consistent nodal forces using Wachspress shape functions. Sensitivities of the objective and pressure load are evaluated using the adjoint-variable method. The efficacy and robustness of the approach are demonstrated by solving various pressure-loaded structures and pressure-driven compliant mechanisms. Compliance is minimized for loadbearing structures, whereas a multicriteria objective is minimized for mechanism designs. The boundary smoothing scheme is implemented within each optimization iteration to subdue the designs' undulated boundaries.

Large‐Scale Worst‐Case Topology Optimization

AbstractWe propose a novel topology optimization method to efficiently minimize the maximum compliance for a high‐resolution model bearing uncertain external loads. Central to this approach is a modified power method that can quickly compute the maximum eigenvalue to evaluate the worst‐case compliance, enabling our method to be suitable for large‐scale topology optimization. After obtaining the worst‐case compliance, we use the adjoint variable method to perform the sensitivity analysis for updating the density variables. By iteratively computing the worst‐case compliance, performing the sensitivity analysis, and updating the density variables, our algorithm achieves the optimized models with high efficiency. The capability and feasibility of our approach are demonstrated over various large‐scale models. Typically, for a model of size 512×170×170 and 69934 loading nodes, our method took about 50 minutes on a desktop computer with an NVIDIA GTX 1080Ti graphics card with 11 GB memory.

A three‐dimensional shape optimization for transient acoustic scattering problems using the time‐domain boundary element method

AbstractWe develop a three‐dimensional shape optimization (SO) framework for the wave equation with taking the unsteadiness into account. Resorting to the adjoint variable method, we derive the shape derivative (SD) with respect to a deformation (perturbation) of an arbitrary point on the target surface of acoustic scatterers. Successively, we represent the target surface with non‐uniform rational B‐spline patches and then discretize the SD in term of the associated control points (CPs), which are useful for manipulating a surface. To solve both the primary and adjoint problems, we apply the time‐domain boundary element method (TDBEM) because it is the most appropriate when the analysis domain is the ambient air and thus infinitely large. The issues of the severe computational cost and instability of the TDBEM are resolved by exploiting the fast and stable TDBEM proposed by the present authors. Instead, since the TDBEM is mesh‐based and employs the piecewise‐constant element for space, we introduce some approximations in evaluating the discretized SD from the two solutions of TDBEM. By regarding the evaluation scheme as the computation of the gradient of the objective functional, given as the summation of the absolute value of the sound pressure over the predefined observation points, we can solve SO problems with a gradient‐based non‐linear optimization solver. To assess the developed SO system, we performed several numerical experiments from the perspective of verification and application with satisfactory results.

A BEM-based topology optimization for acoustic problems considering tangential derivative of sound pressure

This paper employs a new tangential derivative of boundary integral equation for the optimization problems in the acoustic field with a objective function involving tangential derivatives of the sound pressure on the boundary. The level set method is adopted to generate the topological structure by updating the level set function which defines the boundary of the material domain with its zero contour line. The hyper singular integral is directly derived and singular terms are canceled due to the form of the tangential derivative at the boundary. The topological derivative is derived through the adjoint variable method(AVM) and the most of the unknowns in the variation of objective function can be canceled by evaluating the adjoint field. However, one of the terms which includes the variation of the tangential derivative of the sound pressure is evaluated using integration by parts. The remaining part having unknown variation of sound pressure is neglected by extending the objective function defined boundary by one elements at its start and end points. Numerical implementations demonstrate the effectiveness and correctness of the proposed method for topology optimization problems with the objective function involving tangential derivative quantities.

Computational morphology design of duplex structure considering interface debonding

A finite volume interface is an interface region with interface strength, and is most often generated during the fabrication (3D printing) of a duplex structure. However, it is often neglected in morphology design due to numerical complexity and computational difficulties. In addition, a sharp and perfect-bonding interface is usually assumed in literatures. However, such assumptions bring failure risks, thus limiting the industrial applicability of the morphology designs of duplex structures. This study aims to identify the optimal morphology design though a computational design method, considering the finite volume interface and debonding of a duplex structure. This method is based on topology optimization, which utilizes a level-set function for optimizing material distribution in the design space. To introduce finite volume interfaces in morphology design, a simple interface debonding model is integrated into implicit finite element analysis, based on the finite strain theory. Moreover, a distance function is employed to describe the interface region in addition to a level-set function for topology optimization. Further, a topological derivative based on an adjoint variable method is formulated for a debonding interface state in a nonlinear finite element analysis, which is incorporated in topology optimization to obtain the optimal duplex structures. The numerical demonstrations verified the applicability of the proposed approach. The zigzag interface was proven to be one of the features of the optimal duplex structures, considering interface debonding. The results also indicated the optimal duplex structures considering interface debonding to be non-symmetric, in which the interfaces are primarily enriched in compression areas to ensure structural integrity.

Level set-based topology optimization for thermal-fluid system based on the radial basis functions

This work proposes a novel topology optimization methodology derived from a parametric level set model to the weakly-coupled heat flow system. In proposed approach, traditional level set function is explicitly through compactly supported radial basis functions and adjoint variable method (AVM) is utilized to derive the sensitivity analysis. The optimal design goal is treated as maximizing heat exchange and minimizing total power dissipation, simultaneously. To achieve comparable results within fewer iterations and more stable convergence process than solving Ordinary Differential Equations, the method of moving asymptotes is employed to drive design variables update. Earth Mover’s Distance which can weigh similarity between two structures, has been chosen as the index instead of visual observation to estimate the dependence of the developed method and density-based method on the initial guesses. Three dimensional results and distribution characteristics of each physical fields are presented.

Simultaneous optimization of materials and fiber angles on laminated composite shells for reducing transient sound radiation

Researchers generally study optimization problems against noise using frequency-domain analysis. Nonetheless, ubiquitous transient behaviors indicate that time-domain optimization approaches cannot be ignored. Thus, this paper investigates a simultaneous optimization performed on fiber, material, and layer scales to reduce the transient noise from laminated shells with variable stiffness design. The optimization goal is to minimize the integral of the squared sound pressure from structures over a time interval of interest; the Heaviside Penalization of Discrete Material Optimization (HPDMO) model is employed to choose an adequate fiber angle or soft material for each element and determine its stacking sequence. The transient sound pressure is predicted by the finite element/time-domain boundary element method (FEM/TDBEM). Considering that vibration response on each boundary node needs to be known beforehand in TDBEM, an efficient combination scheme is proposed to derive the sensitivity formulation. Namely, the direct method is applied to obtain the dynamic response sensitivities of all nodes at once; and the results are input into the sensitivity calculation of transient sound radiation addressed by using the adjoint variable method. Numerical examples discuss the influences of the number of candidate fiber angles, usage of soft material, size of patches, selection of objective function, eccentric excitation, asymmetric layers, and geometric feature on optimal design.

Sensitivity-Based Topology Optimization of Squirrel-Cage Induction Motor in Time Domain Using Multi-Material Level-Set Method

Although the output power of an induction motor (IM) is generally lower than that of the interior permanent magnet synchronous motor, IMs are widely used in industrial world due to their solidity, low cost, self-starting, etc. For designing high-performance IMs, a computer-aided engineering approach using the finite-element method is essential. However, because a finite-element analysis of IMs takes a long time due to the long transient state in the time domain, it is difficult to perform the optimization using evolutionary algorithms. To overcome this difficulty, a sensitivity-based topology optimization (TO) method for IMs in the frequency domain was proposed. However, this method has a drawback: because the magnetic nonlinearity in the rotor and stator cores cannot be considered, the physical quantities also cannot be accurately estimated. To consider the magnetic nonlinearity in the TO of the IM, the sensitivity analysis with the rotor rotation of the IM using the time-domain adjoint variable method is proposed here. The proposed sensitivity analysis method for the steady state of the IM was evaluated, and it was applied to the TO of a squirrel-cage IM. The effective rotor structure, which is composed of the iron core and secondary conductor, was derived from the multi-material level-set method.

Parametrized level set method for a coupled thermal–fluid problem using radial basis functions

This paper develops a parametric level set-based topology optimization method for the micro-channel heat sink design. In this study, the optimal objective functional is formulated as a bi-objective functional consisting of heat transfer maximization and pressure drop minimization based on the two- and three-dimensional steady-state Navier–Stokes and energy balance equations. The normal velocity of the interface propagation, which is derived from the steepest descent method and adjoint variable method, is used to drive the evolution of the level set function. Influence of the governing equations on the optimized heat sink structure and the distribution characteristics of adjoint variables are analysed through numerical experiments, which proved that governing equations can improve the heat dissipation capacity of the optimal structure. Based on Earth Mover’s Distance, the developed method is proved to have less dependence on initial structures compared with the density-based topology optimization method. Finally, through the experimental tests and numerical simulations, the effectiveness of the developed model is confirmed.

Antenna Beamforming With Multiple-Input, Multiple-Output Metastructures: Controlling the Amplitude and Phase of Antenna Aperture Fields

The experimental realization of a multiple-input, multiple-output (MIMO) metastructure for antenna beamforming is reported. The metastructured beamformer is designed using a recently reported computational inverse design procedure that significantly reduces the required time and computational resources needed to design MIMO metastructures. This reduction in time and resources is achieved by circumventing full-wave simulations to evaluate device responses and using the adjoint variable method to evaluate gradients. To experimentally verify the MIMO metastructure’s performance, the beamformer is patterned on a microwave substrate and interfaced with a 3D-printed aperture antenna to form a multibeam antenna. Measurement results for the multibeam antenna’s performance are provided.

Topology optimization of DC-biased pot-type reactor using design sensitivity in steady state of electromagnetic field with magnetic nonlinearity

Because reactors applied to electric vehicles are typically driven under a DC-biased current, a constant inductance property is required. When a high-performance DC-biased reactor is designed, the topology optimization (TO) is an effective design method owing to the high degree of structural change in the magnetic circuit. However, there exists no report regarding the TO method in the steady-state time domain with magnetic nonlinearity. In this paper, based on the time-domain adjoint variable method, a novel sensitivity analysis approach for the steady-state time domain with thorough consideration of the magnetic saturation is proposed. Through the proposed method, the TO of the iron core and winding structures in a pot-type reactor was carried out to enhance the DC-biased characteristics. Furthermore, the eddy current loss occurring on the aluminum plate installed outside the reactor was suppressed by the multi-objective TO. The interesting structure of a pot-type reactor that enhanced the DC-biased characteristics and reduced the eddy current loss on the outer conductor plate was also illustrated.

Design of Planar and Conformal, Passive, Lossless Metasurfaces That Beamform

A general technique for synthesizing both planar and conformal beamforming metasurfaces is presented that utilizes full-wave modeling techniques and rapid optimization methods. The synthesized metasurfaces consist of a patterned metallic cladding supported by a finite-size grounded dielectric substrate. The metasurfaces are modeled using integral equations which accurately account for mutual coupling and the metasurface's finite dimensions. The synthesis technique consists of three phases: a direct solve phase to obtain an initial metasurface design with complex-valued impedances satisfying the desired far-field beam specifications, a subsequent optimization phase that converts the complex-valued impedances to purely reactive ones, and a final patterning phase to realize the purely reactive impedances as a patterned metallic cladding. The optimization phase introduces surface waves which facilitate passivity. The metasurface is optimized using gradient descent with a semi-analytic gradient obtained using the adjoint variable method. Three examples are presented: a low-profile directly-fed metasurface antenna with near perfect aperture efficiency, a scanned-beam reflectarray design with controlled sidelobes, and a conformal metasurface reflectarray. The far-field and near-field performance of the metasurfaces are verified and the bandwidth and loss tolerance of the metasurfaces are investigated.

A topology optimisation of acoustic devices based on the frequency response estimation with the Padé approximation

We propose a topology optimisation of acoustic devices that work in a certain bandwidth. To achieve this, we define the objective function as the frequency-averaged sound intensity at given observation points, which is represented by a frequency integral over a given frequency band. It is, however, prohibitively expensive to evaluate such an integral naively by a quadrature. We thus estimate the frequency response by the Padé approximation and integrate the approximated function to obtain the objective function. The corresponding topological derivative is derived with the help of the adjoint variable method and chain rule. It is shown that the objective as well as its sensitivity can be evaluated semi-analytically. We present efficient numerical procedures to compute them and incorporate them into a topology optimisation based on the level-set method. We confirm the validity and effectiveness of the present method through some numerical examples.

Topology optimization of the support structure for heat dissipation in additive manufacturing

A support structure is required to successfully create structural parts in the powder bed fusion process for additive manufacturing. In this study, we present the topology optimization of a support structure that improves the heat dissipation in the building process. First, we construct a numerical method that obtains the temperature field in the building process, represented by the transient heat conduction phenomenon with the volume heat flux. Next, we formulate an optimization problem for maximizing heat dissipation and develop an optimization algorithm that incorporates a level-set-based topology optimization. A sensitivity of the objective function is derived using the adjoint variable method. Finally, several numerical examples are provided to demonstrate the effectiveness and validity of the proposed method.