Adjoint-based Shape Optimization Research Articles

Geometric design of electric motors using adjoint-based shape optimization

Geometric design of electric motors using adjoint-based shape optimization

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.

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.

Aeroacoustic and Aerodynamic Adjoint-Based Shape Optimization of an Axisymmetric Aero-Engine Intake

A continuous adjoint-based aeroacoustic optimization, based on a hybrid model including the Ffowcs Williams–Hawkings (FW–H) acoustic analogy, to account for the multidisciplinary design of aero-engine intakes with an axisymmetric geometry, is presented. To optimize such an intake, the generatrix of its lips is parameterized using B-Splines, and the energy contained in the sound pressure spectrum, at the blade passing frequency at receivers located axisymmetrically around the axis of the engine, is minimized. The engine is not included in the optimization and manifests its presence through an independently computed time-series of static pressure over the annular boundary of the simulation domain that corresponds to the inlet to the fan. Taking advantage of the case axisymmetry, the steady 3D RANS equations are solved in the rotating frame of reference and post-processed to compute the flow quantities’ time-series required by the FW–H analogy. The numerical solution of the unsteady flow equations and the otherwise excessive overall cost of the optimization are, thus, avoided. The objective function gradient is computed using the continuous adjoint method, coupled with the analytical differentiation of the FW–H analogy. The adjoint equations are also solved in the rotating frame via steady solver.

Adjoint-based shape optimization for radiative transfer using level-set function and volume penalization method

A inverse design of a radiative heat transfer surface is established based on an adjoint analysis combined with a level-set function and a volume penalization method. The level-set function is employed to represent an arbitrary complex geometry embedded in a Cartesian grid system, whereas the volume penalization method is applied to simulate a forward radiative transfer problem. Then, the sensitivity of a current shape with respect to a prescribed cost functional is obtained from adjoint analysis, and it is used to update the shape. The entire inverse design procedures including the forward and adjoint analyses as well as the shape update based on the adjoint analysis are implemented to an open-source CFD solver, OpenFOAM. Two pure radiation problems from the existing literature are considered for verifying the proposed method. The resultant optimal geometries in both problems show good agreement with the reference data. Finally, the present method is applied to a more complex three-dimensional problem to validate the strength of its applicability to complex geometries.

Cathode Shape Design for Steady-State Electrochemical Machining

The inverse or cathode shape design problem of electrochemical machining (ECM) deals with the computation of the shape of the tool cathode required for producing a workpiece anode of a desired shape. This work applied the complex variable method and the continuous adjoint-based shape optimization method to solve the steady-state cathode shape design problem with anode shapes of different smoothnesses. An exact solution to the cathode shape design problem is proven to exist only in cases when the function describing the anode shape is analytic. The solution’s physical realizability is shown to depend on the aspect ratio of features on the anode surface and the width of the standard equilibrium front gap. In cases where an exact and physically realizable cathode shape exists, the continuous adjoint-based shape optimization method is shown to produce accurate numerical solutions; otherwise, the method produces cathode shapes with singularities. For the latter cases, the work demonstrates how perimeter regularization can be applied to compute smooth approximate cathode shapes suitable for producing workpieces within the range of manufacturing tolerance.

Mixing enhancement of passive type T-mixer through shape optimization

This study looked at optimizing the geometrical shape of a simple T-mixer using Bernstein polynomials-based shape optimization technique to improve the mixing of the T-mixer. Passive micromixers of planar geometry are preferred in a wide range of applications such as lab-on-chips and chemical processing applications, due to their ease of fabrication and low processing costs. Studies conducted on T-mixers have revealed that the performance of T-mixers at low Re (<30) is dismal. At low Reynolds number flows, the mixing is completely dominated by diffusion because of laminar flow conditions. In the present work, an attempt to improve the mixing performance of the T-mixer was made and a nearly three-fold improvement in performance was reported. The adjoint-based shape optimization technique was employed to optimize the wall profile without losing the advantage of the ease of fabrication. The T-mixer boundaries were represented parametrically using Bernstein polynomials that could take any shape within a constrained plane. Different shapes can be generated for different polynomial orders. A limit on the minimum channel thickness (60 microns) was imposed, while the inlet and outlet boundary lengths were fixed. For this particular geometry, the 12th-order polynomial exhibits an optimized shape for maximum mixing performance. The optimized shape of the T-mixer also shows significant improvement in mixing compared to a conventional T-mixer with a reduced channel thickness of 60 microns.

Continuous Adjoint-Based Optimization of an Internally Cooled Turbine Blade—Mathematical Development and Application

This paper presents an adjoint-based shape optimization framework and its demonstration in a conjugate heat transfer problem in a turbine blading. The gradient of the objective function is computed based on the continuous adjoint method, which also includes the adjoint to the turbulence model. Differences in the gradient resulting from making the frozen turbulence assumption are discussed. The developed software was used to optimize both the blade shape of the internally cooled linear C3X turbine blade and the position of cooling channels aiming at (a) minimum total pressure drop of the hot gas flow and (b) minimum highest temperature within the blade. A two-step optimization procedure was used. A free-form parameterization tool, based on volumetric NURBS, controls the blade airfoil contour, while the cooling channels are free to move following changes in the coordinates of their centers. Geometric and flow constraints are included in the performed optimizations, keeping the cooling channels away from the airfoil sides and retaining the turbine inlet capacity and flow turning.

Shape Optimization of Thermoacoustic Systems Using a Two-Dimensional Adjoint Helmholtz Solver

Abstract Thermoacoustic instabilities, which arise due to the interaction between flames and acoustics, are sensitive to small changes to the system parameters. In this paper, we apply adjoint-based shape optimization to a 2D finite element Helmholtz solver to find accurately and inexpensively the shape changes that most stabilize a 2D thermoacoustic system in the linear regime. We examine two cases: a Rijke tube and a turbulent swirl combustor. Both systems exhibit an unstable longitudinal mode and we suppress the instability by slightly modifying the geometry. In the case of the turbulent swirl combustor, the sensitivities are higher in the plenum and in the burner than in the combustion chamber, mainly due to the effect of the mean temperature. In the cooler regions, the local wavelength is shorter, which means that geometry changes of a given distance have more influence than they do where the local wavelength is longer. This is the first time adjoint-based shape optimization is applied to 2D Helmholtz solvers in thermoacoustics, after being previously applied to low-order thermoacoustic networks. But Helmholtz solvers have an intrinsic advantage: they can handle complex geometries. The easy scalability of this method to complex 3D geometries makes this tool a strong candidate for the iterative design of thermoacoustically stable combustors.

Adjoint-based shape optimization for the minimization of flow-induced hemolysis in biomedical applications

This paper reports on the derivation and implementation of a shape optimization procedure for the minimization of hemolysis induction in blood flows through biomedical devices.Despite the significant progress in relevant experimental studies, the ever-growing advances in computational science have made computational fluid dynamics an indispensable tool for the design of biomedical devices. However, even the latter can lead to a restrictive cost when the model requires an extensive number of computational elements or when the simulation needs to be overly repeated. This work aims at the formulation of a continuous adjoint complement to a power-law hemolysis prediction model dedicated to efficiently identifying the shape sensitivity to hemolysis. The proposed approach can accompany any gradient-based optimization method at the cost of approximately one additional flow solution per shape update. The approach is verified against analytical solutions of a benchmark problem and computed sensitivity derivatives are validated by a finite differences study on a generic 2D stenosed geometry. The included application addresses a 3D ducted geometry which features typical characteristics of blood-carrying devices. An optimized shape, leading to a potential improvement up to 22%, is identified. It is shown that the improvement persists for different hemolysis-evaluation parameters.

Efficient Airframe Noise Reduction Framework via Adjoint-Based Shape Optimization

This paper presents an efficient adjoint-based shape optimization framework for airframe noise reduction in which algorithmic differentiation (AD) is applied to the open-source multiphysics solver SU2 to obtain design sensitivities. An AD-based consistent discrete adjoint solver is developed that directly inherits the convergence properties of the primal flow solver due to the differentiation of the entire nonlinear fixed-point iterator. In addition, a 2D and 3D far-field noise prediction framework coupling the Computational Fluid Dynamics (CFD) and Computational Aeroacoustics (CAA) using a permeable surface Ffowcs Williams-Hawkings (FWH) approach in the time domain is also developed. The resultant AD-based discrete adjoint solver is applied to the reduction of noise radiation from 2D and 3D rod–airfoil configurations. The results suggest that the unsteady adjoint information provided by this AD-based discrete adjoint framework is accurate and robust, due to the AD of the entire design chain including the dynamic mesh movement routine and various turbulence models, as well as the hybrid CFD-CAA model. This framework is shown to be highly effective in reducing the tonal noise component due to the turbulent wake impingement on the airfoil leading edge. This study also compares the optimization results obtained on the basis of 3D unsteady Reynolds-averaged Navier–Stokes equation simulations and analyzes them using scale-resolving simulations of higher fidelity.

Simulation and Adjoint-Based Design for Variable Density Incompressible Flows with Heat Transfer

This paper details the development and implementation of a density-based solution approach for simulation and design in variable density incompressible flows with heat transfer. In the low-Mach approximation of the Navier–Stokes equations, density can vary as a function of transported scalars, and in this case, density varies with temperature from a coupled energy equation. These governing equations are spatially discretized using a finite volume method on unstructured grids and solved in a coupled manner with a generalized artificial compressibility method. Complete details of the numerical implementation are provided, and it has been algorithmically differentiated to construct a discrete adjoint for efficient sensitivity analysis. Results demonstrating the primal solver on a set of standard verification and validation cases and adjoint-based shape optimization are presented.

Adjoint-based shape optimization of the microchannels in an inkjet printhead

In drop-on-demand inkjet printheads, ink is pumped steadily through small channels, each of which contains an actuator and a nozzle. When an actuator pulses, a droplet is forced through the nozzle, after which acoustic oscillations reverberate within the channel. Manufacturers would like to damp the residual reverberations, without increasing the pressure drop required to drive the steady flow. In this paper we use gradient-based optimization to show that this can be achieved by constricting the channel where the acoustic velocity is largest and enlarging the channel where the acoustic velocity is smallest. This increases the viscothermal dissipation of the acoustics without changing the viscous dissipation of the steady flow. We separate the compressible Navier–Stokes equations into equations for a steady flow with no oscillations and equations for oscillations with no steady flow. We define two objective functions: the viscous dissipation of the steady flow and the dissipation of the oscillations. We then derive the adjoints for both sets of equations, and obtain expressions for the gradient of each objective function with respect to boundary deformations in Hadamard form. We combine these with a gradient-based optimization algorithm, incorporating constraints such as the shapes of the actuator and nozzle. This algorithm quickly converges to a design that has the same viscous dissipation for the steady flow but a 50 % larger decay rate for the oscillating flow. We show that this design is nearly optimal. It is a shape that inkjet manufacturers, using physical insight and trial and error, have probably not yet considered. We also show how the adjoint fields provide physical insight into the mechanisms affecting each objective function. The main requirements of this method are that the steady flow Mach number and oscillating flow Mach number are small, and that dissipation is dominated by thermoviscous mechanisms. These requirements are often satisfied in microfluidics, so the method in this paper could be applied to many other applications.

Multipoint high-fidelity CFD-based aerodynamic shape optimization of a 10 MW wind turbine

Abstract. The wind energy industry relies heavily on computational fluid dynamics (CFD) to analyze new turbine designs. To utilize CFD earlier in the design process, where lower-fidelity methods such as blade element momentum (BEM) are more common, requires the development of new tools. Tools that utilize numerical optimization are particularly valuable because they reduce the reliance on design by trial and error. We present the first comprehensive 3-D CFD adjoint-based shape optimization of a modern 10 MW offshore wind turbine. The optimization problem is aligned with a case study from International Energy Agency (IEA) Wind Task 37, making it possible to compare our findings with the BEM results from this case study and therefore allowing us to determine the value of design optimization based on high-fidelity models. The comparison shows that the overall design trends suggested by the two models do agree, and that it is particularly valuable to consult the high-fidelity model in areas such as root and tip where BEM is inaccurate. In addition, we compare two different CFD solvers to quantify the effect of modeling compressibility and to estimate the accuracy of the chosen grid resolution and order of convergence of the solver. Meshes up to 14×106 cells are used in the optimization whereby flow details are resolved. The present work shows that it is now possible to successfully optimize modern wind turbines aerodynamically under normal operating conditions using Reynolds-averaged Navier–Stokes (RANS) models. The key benefit of a 3-D RANS approach is that it is possible to optimize the blade planform and cross-sectional shape simultaneously, thus tailoring the shape to the actual 3-D flow over the rotor. This work does not address evaluation of extreme loads used for structural sizing, where BEM-based methods have proven very accurate, and therefore will likely remain the method of choice.

Adjoint-based fluid dynamic design optimization in quasi-periodic unsteady flow problems using a harmonic balance method

Shape optimization in unsteady flow problems enables the consideration of dynamic effects on design. The ability to treat unsteady effects is attractive, as it can provide performance gains when compared to steady-state design methods for a variety of applications in which time-varying flows are of paramount importance. This is the case, for example, in turbomachinery or rotorcraft design. Given the high computational cost involved in time-accurate design problems, adjoint-based shape optimization is a promising option. However, efficient sensitivity analysis should also be accompanied by a significant decrease in computational cost for the primal flow solution, as well. Reduced-order models, like those based on the harmonic balance concept, in combination with the calculation of gradients via adjoint methods, are proposed for the efficient solution of a certain class of aerodynamics optimization problems. The harmonic balance method is applicable if the flow is characterized by discrete finite dominant flow frequencies that do not need to be integer multiples of a fundamental harmonic. A fully-turbulent harmonic balance discrete adjoint formulation based on a duality-preserving approach is proposed. The method is implemented by leveraging algorithmic differentiation and is applied to two test cases: the constrained shape optimization of both a pitching airfoil and a turbine cascade. A key advantage of the current approach is the accurate computation of gradients as compared to second order finite differences without any approximation in the linearization of the turbulent viscosity. The shape optimization results show significant improvements for the selected time-dependent objective functions, demonstrating that design problems involving almost-periodic unsteady flows can be tackled with manageable computational effort.

Adjoint-based shape optimization of complex heat-transfer surfaces in turbulent flows

Adjoint-based shape optimization of complex heat-transfer surfaces in turbulent flows

A continuous adjoint-based aeroacoustic shape optimization for multi-mode duct acoustics

A multi-mode adjoint-based optimization method is proposed for the noise reduction optimization in multi-mode duct acoustics problems. The objective is to minimize the amplitude of sound from an inlet duct on the wall and integral line while maintaining the aerodynamic performance. The complete detailed derivation of the adjoint equations and their corresponding adjoint boundary conditions are presented firstly based on the multi-mode linear Euler equations. With the solved adjoint variables, the final expression of the cost function gradient with respect to the design variables is formulated. The sensitivity derivative computed by the continuous adjoint method is validated by comparing with that obtained using finite difference method. Up to 50 design variables are involved in the adjoint optimization to ensurely provide an adequate design space. And a quasi-Newton Broyden–Fletcher–Goldfarb–Shanno algorithm is utilized to determine an improved intake duct geometry based on the objective function gradient provided by the adjoint solution. Finally, two multi-mode optimization of a typical inlet duct confirms the flexibility of the multi-mode adjoint-based framework and the efficiency of the multi-mode adjoint-based technique.

CAD Integrated Multipoint Adjoint-Based Optimization of a Turbocharger Radial Turbine

The adjoint method is considered as the most efficient approach to compute gradients with respect to an arbitrary number of design parameters. However, one major challenge of adjoint-based shape optimization methods is the integration into a computer-aided design (CAD) workflow for practical industrial cases. This paper presents an adjoint-based framework that uses a tailored shape parameterization to satisfy geometric constraints due to mechanical and manufacturing requirements while maintaining the shape in a CAD representation. The system employs a sequential quadratic programming (SQP) algorithm and in-house developed libraries for the CAD and grid generation as well as a 3D Navier–Stokes flow and adjoint solver. The developed method is applied to a multipoint optimization of a turbocharger radial turbine aiming at maximizing the total-to-static efficiency at multiple operating points while constraining the output power and the choking mass flow of the machine. The optimization converged in a few design cycles in which the total-to-static efficiency could be significantly improved over a wide operating range. Additionally, the imposed aerodynamic constraints with strict convergence tolerances are satisfied and several geometric constraints are inherently respected due to the parameterization of the turbine. In particular, radial fibered blades are used to avoid bending stresses in the turbine blades due to centrifugal forces. The methodology is a step forward towards robustness and consistency of gradient-based optimization for practical industrial cases, as it maintains the optimal shape in CAD representation. As shown in this paper, this avoids shape approximations and allows manufacturing constraints to be included.

Adjoint-based shape optimization of fin geometry for heat transfer enhancement in solidification problem

Adjoint-based shape optimization of fin geometry for heat transfer enhancement in solidification problem

G111 随伴解析に基づく凝固プロセス促進用フィンの形状最適化(OS-14:凝固・融解伝熱および結晶成長の新展開(1))

This paper presents an adjoint-based shape optimization method for heat transfer processes involving liquid-solid phase-change phenomena. We have formulated an adjoint-based shape optimization scheme using the enthalpy method. With the present method, the fin shape can be optimized for enhanced solidification process, in which the liquid is solidified through heat conduction. Based on a meshless local Petrov-Galerkin (MLPG) method, the characteristics of the present shape-optimization method have been examined through analyses with different terminal time in the cost functional and the initial fin geometries.