Discrete Adjoint Formulation for Continuum Sensitivity Analysis

Sensitivity analysis is essential for understanding how changes in design variables affect system performance. Numerical methods for calculating sensitivities, such as finite difference methods, are often easy to implement but can suffer from high computational costs and limited accuracy. Continuum Sensitivity Analysis (CSA) is an alternative approach for calculating analytical derivatives with respect to shape or value parameters. It is easy to implement and can be as accurate as conventional analytical sensitivity methods. By employing Spatial Gradient Reconstruction (SGR), continuous sensitivity equations can be solved in a nonintrusive manner. This method has been applied and validated on a wide range of problems. This work aims to extend the range of applicability of CSA to composite laminates. Composite materials may be characterized by several cross-coupling between loads and deformations, which leads to complex stress fields. A general formulation is introduced that applies to any plate-discretized Finite Element (FE) problem. The developed methodology utilizes the plate elements’ resultant forces and moments instead of the stresses to reconstruct spatial gradients and apply boundary conditions. Because of that, the method does not depend on the material type or lamination sequence and is also computationally inexpensive. It can be used indifferently for isotropic or composite materials, with any lamination sequence. This feature makes CSA attractive for classical FE models used in design optimization problems. This novelty extends the range applicability of CSA to any possible plate-based structural problem.

The sensitivity analysis of static aeroelastic loads can be analytically performed by virtue of a modified stiffness matrix. However, introducing a modified stiffness matrix into the calculation will incur extra computational cost. Additionally, the intrinsic drawback of the direct method is disadvantageous for the sensitivity calculation with a large number of design variables. This paper therefore presents a novel iteration-based adjoint method for the sensitivity analysis of static aeroelastic loads acting on flexible wing, the basis of which is the static aeroelastic calculation via loosely coupled iteration between the potential-flow panel model and the structural linear finite element model. By using an iterative approach for evaluating the adjoint variable, modification to the original stiffness matrix can be obviated. Moreover, this method is more competent for the structural sensitivity analysis with a very large number of design variables, since the adjoint variable is unrelated to the designs. A rectangular wing and a swept wing are employed to demonstrate the verification of the algorithms. The design sensitivities of applied nodal forces on the structure, lift per unit span, total lift and root bending moment are calculated and analyzed. The computational cost is also discussed to further demonstrate the efficiency of the proposed method.

This paper presents a zeroth-order (non-gradient based) methodology for topology optimization of continuum structures. The solid and void elements in the design domain are represented by binary design variables. As in the SIMP method, design variables are assumed to represent the non-dimensional density of the individual finite elements and their contribution to structural stiffness in the model. The optimization procedure begins by specifying a desired volume fraction over a discretized domain and associated boundary conditions. Based on a randomized distribution of solid and void elements, a finite element analysis is performed to identify the strain energy of individual elements. While keeping the volume fraction fixed, the solid elements with low strain energy are converted into void elements whereas the void elements with high strain energy are converted into solid elements. The strain energy threshold for element exchange is specified at the beginning and can be adjusted in the course of optimization. Through subsequent repetition of analysis and exchange steps, the optimal topology corresponding to minimum compliance emerges with no gray elements to filter out. The general characteristics of the Element Exchange Method are demonstrated through solutions of several benchmark problems.

The FengYun-2F (FY-2F) geostationary satellite land surface temperature (LST) and its diurnal variation are important when evaluating climate change, the land-atmosphere energy budget, and the hydrological cycle. However, the presence of clouds generates numerous meaningless pixels that constrain the potential application of the available satellite LST products. These pixels covered by cloud are assigned –2, and otherwise are the LST values, based on the result of a double-channel threshold cloud detection algorithm. This paper proposes a combined temporal and spatial information reconstruction method for the missing FY-2F LST data reconstruction with a good spatial continuity, where cloud detection has already been undertaken. Compared with the methods used in the past, the main characteristics of the proposed method are: 1) the consideration of a free parameter $\delta T$ when modeling the diurnal temperature cycle curve; 2) the introduction of the genetic algorithm for solving the parameters; 3) the adoption of the spectral multimanifold clustering algorithm for clustering the multitemporal geostationary satellite LST data; and 4) the accurate and efficient combined temporal and spatial reconstruction method. The proposed combined temporal and spatial reconstruction method was tested and quantitatively assessed with both simulated and real data experiments, using the FY-2F LST products. The results indicate that the combined reconstruction method is accurate to within about 2 °C, which can significantly improve the practical value of FY-2F LST datasets.

Computational fluid dynamics (CFD) is increasingly used to analyze wind turbines, and the next logical step is to develop CFD-based optimization to enable further gains in performance and reduce model uncertainties. We present an aerodynamic shape optimization framework consisting of a Reynolds-averaged Navier Stokes solver coupled to a numerical optimization algorithm, a geometry modeler, and a mesh perturbation algorithm. To efficiently handle the large number of design variables, we use a gradient-based optimization technique together with an adjoint method for computing the gradients of the torque coefficient with respect to the design variables. To demonstrate the effectiveness of the proposed approach, we maximize the torque of the NREL VI wind turbine blade with respect to pitch, twist, and airfoil shape design variables while constraining the blade thickness. We present a series of optimization cases with increasing number of variables, both for a single wind speed and for multiple wind speeds. For the optimization at a single wind speed performed with respect to all the design variables (1 pitch, 11 twist, and 240 airfoil shape variables), the torque coefficient increased by 22.4% relative to the NREL VI design. For the multiple-speed optimization, the torque increased by an average of 22.1%. Depending on the CFD mesh size and number of design variables, the optimization time ranges from 2 to 24h when using 256 cores, which means that wind turbine designers can use this process routinely. Copyright © 2016 John Wiley & Sons, Ltd.

The paper presents an optimization method based on optimality criterion for minimum weight design of structures with geometric nonlinear behavior. The nonlinear critical load is determined by finding the load level at which the Hessian of the potential energy ceases to be positive definite. A recurrence relation based on the criterion that at optimum the nonlinear strain energy density be equal in all the members is used to develop an algorithm. Sample problems are given to illustrate the application of the method to truss type structures with a large number of design variables. NUMBER of algorithms based on the optimality criterion approach have been developed to design a minimum weight structure with specified constraints on nodal displacements, element stresses, system stability, etc.1'2 The optimization algorithm consisted of analyzing the structure by the finite element method and using a recurrence relation derived from the appropriate optimality criterion to modify the design variables. The optimality criterion was derived by, differentiating the Lagrangian with respect to the design variables. The displacements were assumed to be small and, in the finite element analysis, linear equilibrium equations were solved to determine the response of the structure to the ap- plied loads. In the case of system stability the constraints were defined (see Refs. 3-6) by the associated linear eigenvalue problem. This definition of system stability for some structures may not be valid because of the nonlinear behavior of the structure which may be due to the geometry of the structure or the presence of geometric imperfections. The correct procedure for these structures would be to analyze the structure* by using nonlinear equilibrium equations. This will be particularly true for large space structures (LSS). In this paper an optimization method based on the optimality criterion approach is presented for structures with geometric nonlinear behavior. In the case of a structure optimized with constraints on linear stability the optimum structure can have more than one critical buckling mode. This design tends to become im- perfection sensitive and a small deviation in the geometry of the structure can reduce its load carrying capacity sub- stantially. The imperfection sensitivity of the optimized structure can be reduced by designing the structure so that buckling loads associated with the critical buckling modes are not equal (see Ref. 7). A real structure has geometric im- perfections and these structures tend to have nonlinear

Various optimization problems have been studied for the purpose of structural morphogenesis by many researchers. Structural optimization problems can be classified into three different sub-problems, namely size, shape and topology optimization problem. Size optimization problem is consisted of small sets of design variables. The shape and topology of a structure are defined by a set of design variables, and these design variables are adjusted to achieve given objectives, such as minimum volume. Such optimization problems can be solved iteratively, using gradient-based techniques. Introducing more design variables increases the complexity of the optimization problem. Therefore, it becomes difficult to solve the optimization problem by using MP techniques with large sets of design variables. Heuristics can be improved the difficulty. There are many studies using heuristics like genetic algorithm and simulated annealing. An effective method for structural morphogenesis inspired by self-organization phenomena is presented in this paper. Self-organization is a phenomenon that an entire structure gradually emerges by interaction between elements of the structure, and the elements are affected by the entire structure. Since the self-organization algorithm consists of simple calculation iteration, it can be applied to problems with large number of design variables. Proposed method is applied to problems of uniform member length, minimization of strain energy and cross-section design for frame structures, and the effectiveness is demonstrated through some examples. On the other hand, the number of projects incorporating computational design has increased in recent year, and designs with complicated forms composed of free surfaces are also increasing. In order to design such a shape, the designer needs to consider the rationality of the structure from the initial stage. So, it seems necessary to develop simple software for the structural morphogenesis. Therefore, we developed components of Grasshopper that works within Rhinoceros, so that it will be possible to reassemble algorithms in an intuitive way for designers who have never experienced programming. Grasshopper is one of Graphical Algorithm Editor (GAE), and can be visually constructed an algorithm by connecting components that is a function of modeling. Analysis by gradient-based MP techniques often involves jumping of solutions and analytical instability peculiar to non-linear problems, and it is difficult to apply to the computer aided systems as described above. Analysis algorithms based on self-organizing algorithms are considered to be suitable for the above system due to their high robustness.

<p>Multidisciplinary design optimization (MDO) refers to the process of designing systems characterized by the interaction of multiple interconnected disciplines. High-fidelity MDO usually requires large computational resources due to the computational cost of achieving multidisciplinary consistent solutions by coupling high-fidelity physics-based solvers. Gradient-based minimization algorithms are generally applied to find local minima, due to their efficiency in solving problems with a large number of design variables. This represents a limitation to performing global MDO and integrating black-box type analysis tools, usually not providing gradient information. The latter issues generally inhibit a wide use of MDO in complex industrial applications.</p><p>An architecture named multi-criterion adaptive sampling MDO (MCAS-MDO) is presented in the current research for complex simulation-based applications. This research aims at building a global derivative-free optimization tool able to employ high-fidelity/expensive black-box solvers for the analysis of the disciplines. MCAS-MDO is a surrogate-based architecture featuring a variable level of coupling among the disciplines and is driven by a multi-criterion adaptive sampling (MCAS) assessing coupling and sampling uncertainties. MCAS uses the dynamic radial basis function surrogate model to identify the optimal solution and explore the design space through parallel infill of new solutions.</p><p>The MCAS-MDO is tested versus a global derivative-free multidisciplinary feasible (MDF) approach, which solves fully-coupled multidisciplinary analyses, for two analytical test problems. Evaluation metrics include number of function evaluations required to achieve the optimal solution and sample distribution. The MCAS-MDO outperforms the MDF showing a faster convergence by clustering refined function evaluations in the optimum region.</p><p>The architecture is applied to a steady fluid-structure interaction (FSI) problem, namely the design of a tapered three-dimensional carbon fiber-reinforced plastic hydrofoil for minimum drag. The objective is the design of shape and composite material layout subject to hydrodynamic, structural, and geometrical constraints. Experimental data are available for the original configuration of the hydrofoil and allow validating the FSI analysis, which is performed coupling computational fluid dynamics, solving the Reynolds averaged Navier-Stokes equations, and finite elements, solving the structural equation of elastic motion. Hydrofoil forces, tip displacement, and tip twist are evaluated for several materials providing qualitative agreement with the experiments and confirming the need for the two-way versus one-way coupling approach in case of significantly compliant structures.</p><p>The free-form deformation method is applied to generate shape modifications of the hydrofoil geometry. To reduce the global computational expense of the optimization, a design space assessment and dimensionality reduction based on the Karhunen–Loève expansion (KLE) is performed off-line, i.e. without the need for high-fidelity simulations. It provides with a selection of design variables for the problem at hand through basis rotation and re-parametrization. By using the KLE, an efficient design space is identified for the current problem and the number of design variables is reduced by 92%.</p><p>A sensitivity analysis is performed prior to the optimization to assess the variability associated with the shape design variables and the composite material design variable, i.e. the fiber orientation. These simulations are used to initialize the surrogate model for the optimization, which is carried out for two models: one in aluminum and one in composite material. The optimized designs are assessed by comparison with the original models through evaluation of the flow field, pressure distribution on the body, and deformation under the hydrodynamic load. The drag of the aluminum and composite material hydrofoils is reduced by 4 and 11%, respectively, increasing the hydrodynamic efficiency by 4 and 7%. The optimized designs are obtained by evaluating approximately 100 designs. The quality of the results indicates that global derivative-free MDO of complex engineering applications using expensive black-box solvers can be achieved at a feasible computational cost by minimizing the design space dimensionality and performing an intelligent sampling to train the surrogate-based optimization.</p>

Design space exploration (DSE) is the process whereby a designer seeks to understand some results across a set of design variations. Structural DSE of turbomachinery compressor blades is often challenging because the large number of design variables make it difficult to learn the effect that each variable has upon the stress contours. Principal component analysis (PCA) of the stress contours is used as a way to understand how the stress contours change over the design space. Two methods are introduced to address the challenge of understanding how the stress changes over a large number of variables. First, a two-point correlation is applied to relate the design variables to the scores of each principal component. Second, a coupling of the stress and coordinate location of each node in PCA is developed which also indicates how the stress variations relate to geometric variations. These provide insight to how design variables influence the stress. It is shown how these methods use PCA as DSE tools to better explore the structural design space of compressor blades. Better DSE can improve compressor blades and the computational cost needed for their design.

Traditionally, computational design optimization of commercial aircraft is performed by considering a small number of representative operating conditions. These conditions are based on the design Mach number, altitude, payload, and range for which the aircraft will be flown. However, the design also influences which routes and mission parameters are optimal, and so there is coupling that is ignored when using the traditional approach. Here, the aircraft design, mission profiles, and the allocation of aircraft to routes in an airline network are simultaneously optimized. This is a mixed-integer nonlinear programming problem that is reformulated as a nonlinear programming problem because of the large number of design variables. The reformulated problem is solved using a gradient-based optimization approach with a parallel computational framework that facilitates the multidisciplinary analysis and the derivative computation. A surrogate model is used for the computational fluid dynamics analysis that is retrained in each optimization iteration given the new set of shape design variables. The resulting optimization problem contains over 4,000 design variables and close to 14,000 constraints. The optimization results show a 2% increase in airline profit compared with the traditional multipoint optimization approach. The wing area increases to the upper bound, enabling a higher cruise altitude that improves propulsive efficiency. This study finds that simultaneously optimizing the allocation, mission, and design to maximize airline profit results in a different optimized wing design from that resulting from the multipoint optimization approach.

In order to decrease the fluid drag on an underwater robot manipulator, an optimal trajectory method based on the variational method is presented. By introducing the adjoint variables, which are Lagrange multipliers, we formulate a Lagrange function under certain constraints related to the target angle, target angular velocity, and dynamic equation of the robot manipulator. The state equation (the partial differentiation of the Lagrange function with respect to the state variables), adjoint equation (the partial differentiation of the Lagrange function with respect to the adjoint variables), and sensitivity equation (the partial differentiation of the Lagrange function with respect to torques) can be derived from the stationary conditions of the Lagrange function. Using the state equation, we can calculate the state variables (angles, angular velocities, and angular acceleration) at every time step in the forward time direction. These state variables are stored as data at every time step. Next, by using the adjoint equation, we can calculate the adjoint variables by using these state variables at every time step in the backward time direction. These adjoint variables are stored as data at every time step. Third, the sensitivity equation is calculated by using both the state variables and the adjoint variables. Finally, the optimal trajectory of the manipulator is obtained using the sensitivities. The proposed method is applied to the problem of two-link manipulators. It can obtain the optimal drag reduction trajectory of the manipulator under the constraints mentioned above.

This work develops an analytical framework to study the effects of enforcing simulator-based constraints while performing gradient-based production optimization. In particular, this work studies how the enforcement of this type of constraints affects the consistency of the adjoint-based gradient and the performance of a gradient-based algorithm. In reservoir management, production optimization is commonly performed using gradient-based algorithms that rely on the efficient computation of control gradients through an adjoint formulation. Often, production optimization is implicitly coupled with economic constraints, which typically are implemented through well performance limits enforced within the reservoir simulator. We show that enforcing simulator-based economic constraints triggers non-differentiable unscheduled changes in both the well model equations and in the functions defining the economic criteria. These discontinuities lead to inconsistencies within the adjoint gradient formulation that eventually translate into decreased algorithmic performance. The analytical framework developed in this study allows us to devise an efficient implementation of the simulator-based constraints that provides gradients that are consistent with the adjoint formulation. These analytical results are described using the theoretical framework developed in this paper, and implemented for a production optimization test case where they are shown to outperform common modes of economic constraint enforcement.

A method to apply the discrete adjoint for computing sensitivity derivatives in two-dimensional unsteady flow problems is presented. The approach is to first develop a forward or tangent linearization of the nonlinear flow problem in which each individual component building up the complete flow solution is differentiated against the design variables using the chain rule. The reverse or adjoint linearization is then constructed by transposing and reversing the order of multiplication of the forward problem. The developed algorithm is very general in that it applies directly to the arbitrary Lagrangian-Eulerian form of the governing equations and includes the effect of deforming meshes in unsteady flows. It is shown that an unsteady adjoint formulation is essentially a single backward integration in time and that the cost of constructing the final sensitivity vector is close to that of solving the unsteady flow problem integrated forward in time. It is also shown that the unsteady adjoint formulation can be applied to time-integration schemes of different orders of accuracy with minimal changes to the base formulation. The developed technique is then applied to three optimization examples, the first in which the shape of a pitching airfoil is morphed to match a target time-dependent load profile, the second in which the shape is optimized to match a target time-dependent pressure profile, and the last in which the time-dependent drag profile is minimized without any loss in lift.

Eigenvalue problem derivatives computation for a complex matrix using the adjoint method

This paper deals with an adjoint-based design optimization of vortex generators for the performance improvement of an S-shaped subsonic inlet, the Royal Aircraft Establishment intake model 2129. To enhance the flow quality entering the engine face maximally, the vortex generators are independently optimized with five design parameters per each vortex generator (a total of 55 design variables). To increase the design efficiency, the source term model of the vortex generator is employed. The original source term model, which does not reflect a small change in position and thus has difficulties in differentiation for sensitivity analysis, is modified into a differentiable source term model. To deal with a large number of design variables, the gradient-based design optimization method using the discrete adjoint approach is employed to minimize the distortion coefficient while maintaining the baseline total pressure recovery ratio. A total of five design cases are conducted to validate the proposed design approach, to obtain the optimized vortex generators, and to confirm their enhanced performance. Through the proposed design process,theperformanceofthetargetinletisremarkablyimproved,showingthatthedistortioncoefficientdecreases well over 70% while maintaining the total pressure recovery ratio.