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This paper presents the development of the continuous adjoint method for incompressible fluid flows, solved for the absolute velocity in the relative reference frame, allowing the optimization of rotating machines. The development is conducted using an extended version of the OpenFOAM-based continuous adjoint solver adjointOptimisationFoam. This implements and solves the adjoint to the Navier–Stokes system of equations, coupled with the differentiation of the Spalart–Allmaras turbulence model. Its application to the aerodynamic shape optimization of the MEXICO and NREL Phase VI wind turbines blades follows, targeting the maximization of the axial moment. The flow solution for the two cases is compared with the outcome of other CFD solvers and experimental data, where available. Blades and the displacements of the surrounding grid nodes are parameterized using a volumetric B-Splines morphing box. A number of optimizations are conducted using different operating conditions and geometric constraints. [ABSTRACT FROM AUTHOR]

The continuous adjoint method for transitional flows of compressible fluids is developed and assessed, for the first time in the literature. The gradient of aerodynamic objective functions (aerodynamic forces) with respect to design variables, in problems governed by the compressible Navier–Stokes equations coupled with the Spalart–Allmaras turbulence model and the γ−R˜eθt$$ \gamma -\tilde{R}{e}_{\theta t} $$ transition model (in three, non‐smooth and smooth, variants of it), is computed based on the continuous adjoint method. The development of the adjoint to the smooth transition model variant proved to be beneficial. The accuracy of the computed sensitivity derivatives is verified against finite differences. Programming is performed in an in‐house, vertex‐centered finite‐volume code, efficiently running on GPUs. The proposed continuous adjoint method is used in 2D and 3D aerodynamic shape optimization problems, namely the constrained optimization of the NLF(1)–0416 isolated airfoil and that of the ONERA M6 wing. The impact of "frozen transition" (assumption according to which the adjoint to the transition model equations are not solved) or "frozen turbulence" (by additionally ignoring the adjoint to the turbulence model) are evaluated; it is shown that both lead to inaccurate sensitivities. [ABSTRACT FROM AUTHOR]

This research employs a continuous adjoint data assimilation (DA) algorithm to enhance the prediction of three-dimensional flow behavior in the inclined round jet in crossflow (JICF). To rectify model-form errors arising from the Boussinesq approximation, a force correction is implemented, and the linear part of the eddy viscosity is incorporated for numerical stability. The DA model is theoretically derived to minimize discrepancies between particle image velocimetry (PIV) measurement and the numerical predictions of the primary-adjoint system, thus enabling determination of the optimal contribution of the force correction. Observational data are acquired through planar PIV measurement in the centerplane of JICF with a fixed velocity ratio of 1.2 and a bulk Reynolds number of 150,000. Delayed detached eddy simulation is validated using PIV results and serves as supplementary data for evaluating the reconstruction capabilities of the DA method. The research explores various regularization parameters in the data assimilation procedure, emphasizing their impact on eddy viscosity, correction force, and overall flow prediction accuracy. The findings underscore the effectiveness of regularization in promoting smoothness in the optimized field, thereby mitigating overfitting and irregular solutions. In-depth analyses of critical flow features, including counter-rotating vortex pairs and Reynolds stress forcing, provide insights into the accuracy achieved by the data assimilation procedure. Despite limited measurement data, the study demonstrates the capability of the presented method to successfully recover global flow fields. [ABSTRACT FROM AUTHOR]

This paper presents a topology optimization (TopO) method for conjugate heat transfer (CHT), with turbulent flows. Topological changes are controlled by an artificial material distribution field (design variables), defined at the cells of a background grid and used to distinguish a fluid from a solid material. To effectively solve the CHT problem, it is crucial to impose exact boundary conditions at the computed fluid–solid interface (FSI); this is the purpose of introducing the cut-cell method. On the grid, including also cut cells, the incompressible Navier–Stokes equations, coupled with the Spalart–Allmaras turbulence model with wall functions, and the temperature equation are solved. The continuous adjoint method computes the derivatives of the objective function(s) and constraints with respect to the material distribution field, starting from the computation of derivatives with respect to the positions of nodes on the FSI and then applying the chain rule of differentiation. In this work, the continuous adjoint PDEs are discretized using schemes that are consistent with the primal discretization, and this will be referred to as the "Think Discrete–Do Continuous" (TDDC) adjoint. The accuracy of the gradient computed by the TDDC adjoint is verified and the proposed method is assessed in the optimization of two 2D cases, both in turbulent flow conditions. The performance of the TopO designs is investigated in terms of the number of required refinement steps per optimization cycle, the Reynolds number of the flow, and the maximum allowed power dissipation. To illustrate the benefits of the proposed method, the first case is also optimized using a density-based TopO that imposes Brinkman penalization terms in solid areas, and comparisons are made. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper presents a topology optimization (TopO) method for conjugate heat transfer (CHT), with turbulent flows. Topological changes are controlled by an artificial material distribution field (design variables), defined at the cells of a background grid and used to distinguish a fluid from a solid material. To effectively solve the CHT problem, it is crucial to impose exact boundary conditions at the computed fluid–solid interface (FSI); this is the purpose of introducing the cut-cell method. On the grid, including also cut cells, the incompressible Navier–Stokes equations, coupled with the Spalart–Allmaras turbulence model with wall functions, and the temperature equation are solved. The continuous adjoint method computes the derivatives of the objective function(s) and constraints with respect to the material distribution field, starting from the computation of derivatives with respect to the positions of nodes on the FSI and then applying the chain rule of differentiation. In this work, the continuous adjoint PDEs are discretized using schemes that are consistent with the primal discretization, and this will be referred to as the “Think Discrete–Do Continuous” (TDDC) adjoint. The accuracy of the gradient computed by the TDDC adjoint is verified and the proposed method is assessed in the optimization of two 2D cases, both in turbulent flow conditions. The performance of the TopO designs is investigated in terms of the number of required refinement steps per optimization cycle, the Reynolds number of the flow, and the maximum allowed power dissipation. To illustrate the benefits of the proposed method, the first case is also optimized using a density-based TopO that imposes Brinkman penalization terms in solid areas, and comparisons are made.

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.

The method of characteristics is a classical method for gaining understanding in the solution of a partial differential equation. It has recently been applied to the adjoint equations of the 2D steady-state Euler equations and the first goal of this paper is to present a linear algebra analysis that greatly simplifies the discussion of the number of independent characteristic equations satisfied along a family of characteristic curves. This method may be applied for both the direct and the adjoint problem. Our second goal is to directly derive in conservative variables the characteristic equations of 2D compressible inviscid flows. Finally, the theoretical results are assessed for a nozzle flow with a classical scheme and its dual consistent discrete adjoint. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

• Optimize incompressible turbulent flow at bidimensional and tridimensional domains employing the continuous adjoint approach • Analyze the influence of integer design variables in topology optimization of incompressible turbulent fluids • Develop the adjoint near wall distance calculation to treat the adjoint spalart allmaras turbulence model Turbulent fluid flow is a challenging physical problem to be addressed with topology optimization due to the vortex generation and the near wall distance calculation that difficult the calibration of topology optimization parameters when continuous design variables based-optimizers are used. The intrinsic nature of topology optimization inverse permeability interpolation clashes with the accuracy requirements from turbulent fluid flow dynamics. To solve the mentioned challenge, the present paper investigates the influence of an integer continuous variable-based-optimizer, i.e. a [0, 1] binary behaviour, that avoids grey regions during the optimization process. Its influence on the near wall distance calculation is analyzed as it affects the permeability of the fluid domain. The turbulence phenomenon is treated via the Spalart-Allmaras model and its near wall distance calculation is approached by the Hamilton-Jacobi formulation. Permeability independence on the topology optimization formulation constraints is proposed firstly at the continuous adjoint approach to avoid miscalculations in the fluid cells. The finite volume method is employed to solve the Reynolds Averaged Navier-Stokes governing equations and 2D and 3D tests are developed, where discussions on the solid-fluid boundaries definition, the material penalization dependence, objective function comparisons and accuracy of the CFD analysis and the wall distance calculation are presented. [ABSTRACT FROM AUTHOR]

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. • An coupled fully implicit method for solving the adjoint equations of steady incompressible flows is developed. • The pressure-weighted interpolation (PWI) method is used for coupling of adjoint velocity and adjoint pressure. • The adjoint transpose convection (ATC) term is discretized implicitly, so that no masking or damping is needed. • A parallel sparse linear solver which effectively combines direct and iterative methods is applied. [ABSTRACT FROM AUTHOR]

The method of characteristics is a classical method for gaining understanding in the solution of a partial differential equation. It has recently been applied to the adjoint equations of the 2D steady-state Euler equations and the first goal of this paper is to present a linear algebra analysis that greatly simplifies the discussion of the number of independent characteristic equations satisfied along a family of characteristic curves. This method may be applied for both the direct and the adjoint problem. Our second goal is to directly derive in conservative variables the characteristic equations of 2D compressible inviscid flows. Finally, the theoretical results are assessed for a nozzle flow with a classical scheme and its dual consistent discrete adjoint.

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.

The present work focuses on the application of density-based topology optimization to the design of a surface cooler. This kind of device is used to cool down the oil circuit in aircraft engines thanks to the cold air in the bypass stream, and is subject to severe heat duty and pressure drop requirements. The optimization is carried out with an in-house implementation of the density method in OpenFOAM. A continuous adjoint strategy is employed to compute the sensitivities with respect to the design variables. Avoiding the so-called “frozen turbulence” assumption, the variations of the turbulent viscosity are taken into account in the computation of the sensitivity. The proposed model also considers the influence of the design variables on the wall distance function occurring in the formulation of the Spalart–Allmaras turbulence model. A simplified two-dimensional model is first employed to tune the optimization and the density model parameters. Then, the methodology is applied to a large-scale three-dimensional case, and the results are compared to a reference straight-fin geometry. The performance is finally evaluated with a reference solver, showing that the density model overestimates both the heat exchange and the total pressure loss, but that the methodology is still able to provide efficient designs in turbulent flow, starting from a very remote initialization. [ABSTRACT FROM AUTHOR]

Decades of research have progressed geometrically exact beam theory to the point where it is now an invaluable resource for analyzing and modeling highly flexible slender structures. Large-scale optimization using geometrically exact beam theory remains nontrivial, however, due to the inability of gradient-free optimizers to handle large numbers of design variables in a computationally efficient manner and the difficulties associated with obtaining smooth, accurate, and efficiently calculated design sensitivities for gradient-based optimization. To overcome these challenges, this paper presents a finite-element implementation of geometrically exact beam theory which has been developed specifically for gradient-based optimization. A key feature of this implementation of geometrically exact beam theory is its compatibility with forward and reverse-mode automatic differentiation. Another key feature is its support for both continuous and discrete adjoint sensitivity analysis. Other features are also presented which build upon previous implementations of geometrically exact beam theory, including a singularity-free rotation parameterization based on Wiener-Milenković parameters, an implementation of stiffness-proportional structural damping using a discretized form of the compatibility equations, and a reformulation of the equations of motion for geometrically exact beam theory as a semi-explicit system. Several examples are presented which verify the utility and validity of each of these features. • Geometrically exact beam theory sensitivities may be computed automatically. • Automatic differentiation is augmented with analytic derivatives to minimize costs. • Continuous adjoint sensitivities are facilitated using a semi-explicit formulation. • Discrete adjoint sensitivity analysis is implemented using automatic differentiation. [ABSTRACT FROM AUTHOR]

Fluid topology optimisation has become a popular approach for optimisation of geometries in aero-thermal applications. However, one of the main limitations of current approaches considering turbulent flow is the fidelity of the Reynolds Averaged Navier–Stokes models employed. In response, this paper shows the development of the first data-driven fluid topology optimisation technique based on the continuous adjoint method. The technique first extracts data from a high fidelity simulation of a standard topology-optimised geometry. These data are fed through a symbolic regression-based machine learning algorithm called gene expression programming, to learn an explicit model for the anisotropy tensor. The novel aspect of the work is the derivation of the adjoint form of the generalised explicit algebraic stress model such that the developed turbulence model can be inserted directly into the primal and adjoint system of equations. This allows a second, data-driven optimisation to be performed. Finally, a high fidelity simulation of the resulting geometry is also conducted to allow comparison against the standard geometry. The method is first applied to the back-facing step to verify the approach and then to a 2D u-bend configuration. The data-driven optimisation was able to find geometries exhibiting significant reductions in pressure loss when compared with results from the standard optimisation. [ABSTRACT FROM AUTHOR]

This article contributes to the development of methods for shape optimization under uncertainties, associated with the flow conditions, based on intrusive Polynomial Chaos Expansion (iPCE) and continuous adjoint. The iPCE to the Navier–Stokes equations for laminar flows of incompressible fluids is developed to compute statistical moments of the Quantity of Interest which are, then, compared with those obtained through the Monte Carlo method. The optimization is carried out using a continuous adjoint‐enabled, gradient‐based loop. Two different formulations for the continuous adjoint to the iPCE PDEs are derived, programmed, and verified. Intrusive PCE methods for the computation of the statistical moments require mathematical development, derivation of a new system of governing equations and their numerical solution. The development is presented for a chaos order of two and two uncertain variables and can be used as a guide to those willing to extend this development to a different set of uncertain variables or chaos order. The developed method and software, programmed in OpenFOAM, is applied to two optimization problems pertaining to the flow around isolated airfoils with uncertain farfield conditions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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.

This article presents a continuous adjoint-enabled, gradient-based optimization tool for multi-point, multi-objective industrial optimization problems and its application to the shape optimization of a concept car. Apart from the adjoint to the incompressible Reynolds-averaged Navier-Stokes equations, the adjoint to the Spalart-Allmaras turbulence model equation is also solved, in order to support the optimization with accurate gradients. Part of the mathematical development related to the sensitivity derivative terms resulting from the differentiation of the Reynolds-averaged Navier-Stokes (RANS) variant of the Spalart-Allmaras model when using an adjoint formulation consisting of field integrals is presented for the first time in the literature. In the industrial application, two operating points are considered, corresponding to two flow velocity angles with respect to the car symmetry plane, with a different objective (drag and yaw moment coefficients) for each of them. With the aforesaid targets, the Pareto front of optimal solutions is computed and discussed. Each point on this front is computed by minimizing a single objective function, resulting from the linear combination of the objective functions defined on the different operating points, using appropriate weights. Finally, some of the Pareto front members are re-evaluated using delayed detached eddy simulation (DDEs). The overall optimization tool is developed in the open-source CFD toolbox OpenFOAM. [ABSTRACT FROM AUTHOR]

Efficient Robust Design Optimization (RDO) strategies coupling a parsimonious uncertainty quantification (UQ) method with a surrogate-based multi-objective genetic algorithm (SMOGA) are investigated for a test problem in computational fluid dynamics (CFD), namely the inverse robust design of an expansion nozzle. The low-order statistics (mean and variance) of the stochastic cost function are computed through either a gradient-enhanced kriging (GEK) surrogate or through the less expensive, lower fidelity, first-order method of moments (MoM). Both the continuous (non-intrusive) and discrete (intrusive) adjoint methods are evaluated for computing the gradients required for GEK and MoM. In all cases, the results are assessed against a reference kriging UQ surrogate not using gradient information. Subsequently, the GEK and MoM UQ solvers are fused together to build a multi-fidelity surrogate with adaptive infill enrichment for the SMOGA optimizer. The resulting hybrid multi-fidelity SMOGA RDO strategy ensures a good tradeoff between cost and accuracy, thus representing an efficient approach for complex RDO problems.

Summary: A novel acceleration technique using a reduced‐order model is presented to speed up convergence of continuous adjoint solvers. The acceleration is achieved by projecting to an improved solution within an iterative process solely using early solution results. This is achieved by forming basis vectors from early iteration adjoint solutions to perform model order reduction of the adjoint equations. The reduced‐order model of the adjoint equations is then substituted into the full‐order discretized governing equations to determine weighting coefficients for each basis vector. With these coefficients, a linear combination of the basis vectors is used to project to an improved solution. The method is applied to 3 inviscid quasi‐1D nozzle flow cases including fully subsonic flow, subsonic inlet to supersonic outlet flow, and transonic flow with a shock. Significant cost reductions are achieved for a single application as well as repeated applications of the convergence acceleration technique. [ABSTRACT FROM AUTHOR]

We present a method for estimating spatial and temporal numerical errors in scalar outputs of unsteady fluid dynamics simulations using continuous-in-time adjoint solutions and general time-integration methods. A continuous formulation decouples the primal and adjoint temporal discretizations and allows for the use of standard time-integration schemes for the adjoint. For non-variational methods, a scheme-agnostic temporal reconstruction of the primal and adjoint solutions replaces the functional representation in between time nodes. The output error is still estimated through an adjoint-weighted residual, which takes the form of a space–time integral. Separate temporal and spatial error estimates arise from projection of the adjoint to semi-refined spaces. These estimates drive adaptive refinement, which first requires a calculation of the appropriate cost distribution between the spatial and temporal discretizations. The adaptive mechanics consist of uniform temporal refinement/coarsening, and several localized spatial refinements: order adaptation, hanging-node refinement, and unstructured mesh optimization. Results for scalar advection–diffusion and for the compressible Navier–Stokes equations demonstrate the effectivity of the error estimates, the efficiency of the adaptive refinement, independence of optimized meshes to the starting mesh, and the importance of high-order spatial and temporal approximations. [ABSTRACT FROM AUTHOR]

A range of optimization cases of two-dimensional Stefan problems, solved using a tracking-type cost-functional, is presented. A level-set method is used to capture the interface between the liquid and solid phases and an immersed boundary (cut cell) method coupled with an implicit time-advancement scheme is employed to solve the heat equation. A conservative implicit-explicit scheme is then used for solving the level-set transport equation. The resulting numerical framework is validated with respect to existing analytical solutions of the forward Stefan problem. An adjoint-based algorithm is then employed to efficiently compute the gradient used in the optimization algorithm (L-BFGS). The algorithm follows a continuous adjoint framework, where adjoint equations are formally derived using shape calculus and transport theorems. A wide range of control objectives are presented, and the results show that using parameterized boundary actuation leads to effective control strategies in order to suppress interfacial instabilities or to maintain a desired crystal shape. • Simulations of two-phase Stefan problem with Gibbs-Thomson effects. • Novel cut cell method for the solution of the heat equation. • Implicit-explicit scheme for the level set advection equation. • Validation of the method on melting and solidification problems. • Continuous adjoint-based optimization with a tracking-type cost functional. [ABSTRACT FROM AUTHOR]

This paper presents a continuous adjoint approach for topology optimization of a coupled heat transfer and laminar fluid flow system under tangential thermal gradient (TTG) constraints. In this system, the thermal gradient along the boundary of multiple heat sources needs to be controlled. The design goals are to minimize the temperature of the domain, the fluid power dissipation and the TTG along the boundary of the heat sources. The first two goals are combined into a single cost function with weight variables. The TTG is constrained in one of two forms, an integral form where the integral of TTG squares along the boundaries of heat sources is constrained, or a point-wise form where TTG is constrained point-wise. A gradient-based approach is developed to obtain the optimized designs. A salient feature of our approach is the use of the continuous adjoint approach to derive gradients of both the cost function and two forms of TTG constraints. Numerical examples demonstrate that the continuous adjoint approach leads to successful topological optimization of the constrained thermal-fluid system. The use of TTG constraint is effective in lowering the TTG along the heat source boundaries. The resulting designs exhibit clear black/white contrast. [ABSTRACT FROM AUTHOR]

In this article, the gradient of aerodynamic objective functions with respect to design variables, in problems governed by the incompressible Navier–Stokes equations coupled with thek–ωSSTturbulence model, is computed using the continuous adjoint method, for the first time. Shape optimization problems for minimizing drag, in external aerodynamics (flows around isolated airfoils), or viscous losses in internal aerodynamics (duct flows) are considered. Sensitivity derivatives computed with the proposed adjoint method are compared to those computed with finite differences or a continuous adjoint variant based on the frequently used assumption of frozen turbulence; the latter proves the need for differentiating the turbulence model. Geometries produced by optimization runs performed with sensitivities computed by the proposed method and the ‘frozen turbulence’ assumption are also compared to quantify the gain from formulating and solving the adjoint to the turbulence model equations. [ABSTRACT FROM AUTHOR]

The development of adjoint procedures for general compositional flow problems is much more challenging than for oil-water problems, due to significantly higher complexity of the underlying physics. The treatment of nondifferentiable constraints, an example of which is a maximum gas rate specification in injection or production wells, when the control variables are well-bottom-hole pressures, poses an additional major challenge. A new formal approach for handling these constraints is presented and compared against a formal treatment within the optimizer employing constraint lumping and a simpler heuristic treatment in the forward model. The three constraint-handling methods are benchmarked for three example cases of increasing complexity. Moreover, the new approach allows the optimizer to converge to optimal solutions exhibiting higher objective values, since unlike the formal lumping-based methods, where a pressure reduction suggested by the optimizer propagates through the smoothing function to all well rates, it handles constraints individually on a per well and per time step basis. The numerical examples show that the new formal constraint-handling approach allows the optimizer to converge significantly faster than formal lumping-based techniques independently of the initial guess used for the optimization. [ABSTRACT FROM AUTHOR]

The continuous adjoint to the incompressible Reynolds-averaged Navier–Stokes equations coupled with the low Reynolds number Launder–Sharmak–ε turbulence model is presented. Both shape and active flow control optimization problems in fluid mechanics are considered, aiming at minimum viscous losses. In contrast to the frequently used assumption of frozen turbulence, the adjoint to the turbulence model equations together with appropriate boundary conditions are derived, discretized and solved. This is the first time that the adjoint equations to the Launder–Sharmak–ε model have been derived. Compared to the formulation that neglects turbulence variations, the impact of additional terms and equations is evaluated. Sensitivities computed using direct differentiation and/or finite differences are used for comparative purposes. To demonstrate the need for formulating and solving the adjoint to the turbulence model equations, instead of merely relying upon the ‘frozen turbulence assumption’, the gain in the optimization turnaround time offered by the proposed method is quantified. [ABSTRACT FROM AUTHOR]

An adjoint formulation for the gradient-based optimization of oil-gas compositional reservoir simulation problems is presented. The method is implemented within an automatic differentiation-based compositional flow simulator (Stanford's Automatic Differentiation-based General Purpose Research Simulator, AD-GPRS). The development of adjoint procedures for general compositional problems is much more challenging than for oil-water problems due to the increased complexity of the code and the underlying physics. The treatment of nonlinear constraints, an example of which is a maximum gas rate specification in injection or production wells, when the control variables are well bottom-hole pressures, poses a particular challenge. Two approaches for handling these constraints are presented-a formal treatment within the optimizer and a simpler heuristic treatment in the forward model. The relationship between discrete and continuous adjoint formulations is also elucidated. Results for four example cases of increasing complexity are presented. Improvements in the objective function (cumulative oil produced) relative to reference solutions range from 4.2 to 11.6 %. The heuristic treatment of nonlinear constraints is shown to offer a cost-effective means for obtaining feasible solutions, which are, in some cases, better than those obtained using the formal constraint handling procedure. [ABSTRACT FROM AUTHOR]

In fluid mechanics, topology optimization is used for designing flow passages, connecting predefined inlets and outlets, with optimal performance based on selected criteria. In this article, the continuous adjoint approach to topology optimization in incompressible ducted flows with heat transfer is presented. A variable porosity field, to be determined during the optimization, is the means to define the optimal topology. The objective functions take into account viscous losses and the amount of heat transfer. Turbulent flows are handled using the Spalart–Allmaras model and the proposed adjoint is exact,i.e.the adjoint to the turbulence model equation is formulated and solved, too. This is an important novelty in this article which extends the porosity-based method to account for heat transfer flow problems in turbulent flows. In problems such as the design of manifolds, constraints on the outlet flow direction, rates and mean outlet temperatures are imposed. [ABSTRACT FROM AUTHOR]

Abstract: Data assimilation obtains improved estimates of the state of a physical system by combining imperfect model results with sparse and noisy observations of reality. In the four dimensional variational (4D-Var) framework data assimilation is formulated as an optimization problem, which is solved using gradient based optimization methods. The 4D-Var gradient is obtained by forcing the adjoint model with observation increments. The construction of the adjoint model requires considerable development effort. In the continuous approach the adjoint differential equations are discretized. In the discrete approach the numerical solution of the forward equations is differentiated. The two routes lead to different gradients. In this paper we investigate numerically the effect of using discrete and continuous adjoints of the advection equation in chemical transport modeling. Continuous advection adjoints are easily implemented by calling the same advection subroutines as the forward model, with a sign change for the winds, and with rescaling the solution. Discrete advection adjoints involve the differentiation of a nonlinear, monotonic advection scheme like the piecewise parabolic method. The numerical experiments are carried out with CMAQ-ADJ. The results show that, while discrete advection adjoints are more accurate in point-to-point comparisons against finite differences, the continuous adjoints of advection perform better as gradients for optimization in 4D-Var data assimilation. [Copyright &y& Elsevier]

Abstract: A continuous adjoint formulation is used to determine optimal airfoil shapes in unsteady viscous flows at Re =1×104. The Reynolds number is based on the free-stream speed and the chord length of the airfoil. A finite element method based on streamline-upwind Petrov/Galerkin (SUPG) and pressure-stabilized Petrov/Galerkin (PSPG) stabilizations is used to solve both the flow and adjoint equations. The airfoil is parametrized via a Non-Uniform Rational B-Splines (NURBS) curve. Three different objective functions are used to obtain optimal shapes: maximize lift, minimize drag and minimize ratio of drag to lift. The objective functions are formulated on the basis of time-averaged aerodynamic coefficients. The three objective functions result in diverse airfoil geometries. The resulting airfoils are thin, with the largest thickness to chord ratio being only 5.4%. The shapes obtained are further investigated for their aerodynamic performance. Maximization of time-averaged lift leads to an airfoil that produces more than six times more lift compared to the NACA 0012 airfoil. The excess lift is a consequence of the large peak and extended region of high suction on the upper surface and high pressure on the lower surface. Minimization of drag results in an airfoil with a sharp leading edge. The flow remains attached for close to 70% of the chord length. Minimization of the ratio of drag to lift results in an airfoil with a shallow dimple on the upper surface. It leads to a fairly large value of the time-averaged ratio of lift to drag (∼17.8). The high value is mostly achieved by a 447% increase in lift and 16% reduction in drag, compared to a NACA 0012 airfoil. Imposition of volume constraint, for the cases studied, is found to result in airfoils that have lower aerodynamic performance. [Copyright &y& Elsevier]

A new model reduction approach is presented for time-dependent topology optimization of elastodynamic problems. Two salient features of the model reduction approach are on-the-fly and dual reduction. That is, the snapshots and reduced basis are obtained during the optimization rather than through off-line training; primal and adjoint equations are reduced independently. For the on-the-fly reduction, we use two complementary dynamic sampling strategies for snapshots: nearest n f full-order solutions and rolling transient ensemble of n t solutions. For the dual reduction, we apply the proper orthogonal decomposition and Galerkin procedure independently to the primal equations and to the adjoint equations for reduction. Central to our dual reduction strategy is the use of continuous adjoint to obtain the adjoint equations so that the adjoint equations can be reduced independently and along side the primal equation reduction. The residuals of the governing primal and adjoint equations are used to determine whether full-order model (FOM) based solutions should be used. Two-dimensional and three-dimensional numerical examples on topology optimization of elastodynamics problems are presented. Our numerical results demonstrate that the on-the-fly dual-reduction approach leads to two orders of magnitude improvement in FOM reduction (in terms of the number of FOM solution procedures being invoked) and data storage (in terms of storage of primal solutions required to solve the backward time-dependent adjoint equations and compute the sensitivity). • On-the-fly reduction: snapshots and reduced basis are obtained on-the-fly rather than off-line training. • Dual reduction: reduction is applied to primal equilibrium equations and derived adjoint equations independently. • Two snapshot strategies are used: nearest full-order solutions and a moving transient ensemble of solutions. • Continuous adjoint is used to derive the adjoint equations and sensitivity expressions. • Over 99% full-order solutions can be skipped. This corresponds to over two orders of magnitude saving in storage cost. [ABSTRACT FROM AUTHOR]