Adjoint Sensitivity Analysis Method Research Articles

Adjoint methods of sensitivity analysis for Lyapunov equation

The existing direct sensitivity analysis of optimal structural vibration control based on Lyapunov's second method is computationally expensive when applied to finite element models with a large number of degree-of-freedom and design variables. A new adjoint sensitivity analysis method is proposed in this paper. Using the new method the sensitivity of the performance index, a time integral of a quadratic function of state variables, with respect to all design variables is calculated by solving two Lyapunov matrix equations. In consideration of computational cost reduction, the new adjoint method is further extended to the reduced order model by Guyan method. This makes the method applicable to large finite element models. Two numerical examples demonstrate the accuracy and efficiency of the proposed method.

Tool for preliminary structural sizing, weight estimation, and aeroelastic optimization of lifting surfaces*

This paper presents the development and implementation of a tool for wing structural sizing and aeroelastic optimization in early design steps, where the amount of available data about the wing structure is not enough to allow high-fidelity finite element analysis and optimization. The proposed tool consists of two levels. The first level is a quasi-analytical method for wing structural weight estimation and initially sizing of the wing box structure. The second level is an aeroelastic tool that uses a vortex lattice method and a finite beam element to compute the stress distribution in the wing box structure. The Newton method is used to solve the coupled system. The coupled adjoint sensitivity analysis method is used to compute the sensitivity of any function of interest with respect to the design variables. The tool was used for a series of wing aeroelastic optimizations to minimize the wing weight with a series of constraints on the wing structural failure modes and aileron effectiveness. Another series of optimizations is also used to find the wing jig shape for a predefined cruise shape. The outputs of the optimizations showed that the wing box weight varies quadratically with the required value for the aileron effectiveness.

Air pollution and early deaths in the United States. Part II: Attribution of PM2.5 exposure to emissions species, time, location and sector

Combustion emissions constitute the largest source of anthropogenic emissions in the US, and lead to the degradation of air quality and human health. In Part I we computed the population fine particulate matter (PM2.5) exposure and number of early deaths caused by emissions from six major sectors: electric power generation, industry, commercial and residential activities, road transportation, marine transportation and rail transportation. In Part II we attribute exposure and early deaths to sectors, emissions species, time of emission, and location of emission. We apply a long-term adjoint sensitivity analysis and calculate the four dimensional sensitivities (time and space) of PM2.5 exposure with respect to each emissions species. Epidemiological evidence is used to relate increased population exposure to premature mortalities. This is the first regional application of the adjoint sensitivity analysis method to characterize long-term air pollution exposure. (A global scale application has been undertaken related to intercontinental pollution.) We find that for the electric power generation sector 75% of the attributable PM2.5 exposure is due to SO2 emissions, and 80% of the annual impacts are attributed to emissions from April to September. In the road transportation sector, 29% of PM2.5 exposure is due to NOx emissions and 33% is from ammonia (NH3), which is a result of emissions after-treatment technologies. We estimate that the benefit of reducing NH3 emissions from road transportation is ∼20 times that of NOx per unit mass. 75% of the road transportation ammonia impacts occur during the months October to March. We publicly release the sensitivity matrices computed, noting their potential use as a rapid air quality policy assessment tool.

Adjoint design sensitivity analysis of dynamic crack propagation using peridynamic theory

Based on the peridynamics of the reformulated continuum theory, an adjoint design sensitivity analysis (DSA) method is developed for the solution of dynamic crack propagation problems using the explicit scheme of time integration. Non-shape DSA problems are considered for the dynamic crack propagation including the successive branching of cracks. The adjoint variable method is generally suitable for path-independent problems but employed in this bond-based peridynamics since its path is readily available. Since both original and adjoint systems possess time-reversal symmetry, the trajectories of systems are symmetric about the u-axis. We take advantage of the time-reversal symmetry for the efficient and concurrent computation of original and adjoint systems. Also, to improve the numerical efficiency of large scale problems, a parallel computation scheme is employed using a binary space decomposition method. The accuracy of analytical design sensitivity is verified by comparing it with the finite difference one. The finite difference method is susceptible to the amount of design perturbations and could result in inaccurate design sensitivity for highly nonlinear peridynamics problems with respect to the design. It is demonstrated that the peridynamic adjoint sensitivity involving history-dependent variables can be accurate only if the path of the adjoint response analysis is identical to that of the original response.

Shape Design Sensitivity Analysis of Dynamic Crack Propagation Problems using Peridynamics and Parallel Computation

Using the bond-based peridynamics and the parallel computation with binary decomposition, an adjoint shape design sensitivity analysis(DSA) method is developed for the dynamic crack propagation problems. The peridynamics includes the successive branching of cracks and employs the explicit scheme of time integration. The adjoint variable method is generally not suitable for path-dependent problems but employed since the path of response analysis is readily available. The accuracy of analytical design sensitivity is verified by comparing it with the finite difference one. The finite difference method is susceptible to the amount of design perturbations and could result in inaccurate design sensitivity for highly nonlinear peridynamics problems with respect to the design. It turns out that C1-continuous volume fraction is necessary for the accurate evaluation of shape design sensitivity in peridynamic discretization.

Adjoint design sensitivity analysis of molecular dynamics in parallel computing environment

An adjoint design sensitivity analysis method is developed for molecular dynamics using a parallel computing scheme of spatial decomposition in both response and design sensitivity analyses to enhance the computational efficiency. Molecular dynamics is a path-dependent transient dynamic problem with many design variables of high nonlinearity. Adjoint variable method is not appropriate for path-dependent problems but employed in this paper since the path is readily available from response analysis. The required adjoint system is derived as a terminal value problem. To compute the interaction forces between atoms in different spatial boxes, only atomic positions in the neighboring boxes are required to minimize the amount of data communications. Through some numerical examples, the high nonlinearity of the selected design variables is discussed. Also, the accuracy of the derived adjoint design sensitivity is verified by comparing with finite difference sensitivity and the efficiency of parallel adjoint variable method is demonstrated.

Topological layout design of electro-fluid-thermal-compliant actuator

While the sequential solution procedure (the sequence of solutions) for a steady-state electro-thermal-compliant (ETC) actuator device has been considered in previous studies, very few researches have concurrently considered the mutual couplings between fluid and thermal domains and between fluid and structural domains. Such an analysis allows a straightforward and accurate finite element (FE) simulation of various flow boundary conditions and different flow types, but makes the involved differential equations highly nonlinear. Thus, the goal of this research is to develop a new and rigorous monolithic analysis and optimization framework for an electro-fluid-thermal-compliant (EFTC) microactuator. To accommodate the coupling effect of the fluid and thermal domains, a modified incompressible Navier–Stokes equation with Darcy’s force and a heat transfer equation coupled with fluid motion are additionally formulated in the present research. In addition, a new nonlinear monolithic FE modeling method that involves the use of the deformation tensor is employed in order to consider the mutual coupling between fluid and structural domains. Using the solid isotropic material with penalization (SIMP) approach to interpolate seven material properties (Young’s modulus, electrical conductivity, heat conductivity, heat capacity, mass density, heat capacity, and Darcy’s force coefficient) with respect to a density design variable, it is possible to achieve topology optimization (TO) upon consideration of the strong nonlinear couplings of the EFTC actuator. To efficiently solve four sets of nonlinearly coupled equations, a sequence of iterative solutions to the equations is proposed and a nonlinear sensitivity analysis without consideration of the iterative solution sequence is derived based on the adjoint sensitivity analysis method. Several three-dimensional examples are also examined in order to demonstrate the validity and potential of the present formulations for analysis and TO.

A shape optimization method for nonlinear axisymmetric magnetostatics using a coupling of finite and boundary elements

AbstractIn this paper we propose a method for constrained shape optimization governed with a nonlinear axisymmetric magnetostatic state problem and we apply it to an optimal shape design of an electromagnet. The state problem is solved via Hiptmair’s symmetric coupling of finite elements employed in the interior ferromagnetic domain and boundary elements modelling the exterior air domain as well as current excitations. As a novelty we derive Duffy regularization transforms of the boundary element integrals for the axisymmetric case, which are then evaluated using a tensor-product Gaussian quadrature. Nonlinear ferromagnetic behaviour is resolved by Newton iterations. The optimization method under both linear and nonlinear constraints relies on the active-set steepest-descent search, projections onto the set of linearized constraints, and an adjoint method of shape sensitivity analysis. Shape perturbations influence grid deformation via a solution to an auxiliary torsion-free linear elasticity problem. Finally, numerical results are presented.

An adaptive approach to constraint aggregation using adjoint sensitivity analysis

Constraint aggregation is the key for efficient structural optimization when using a gradient-based optimizer and an adjoint method for sensitivity analysis. We explore different methods of constraint aggregation for numerical optimization. We analyze existing approaches, such as considering all constraints individually, taking the maximum of the constraints and using the Kreisselmeier–Steinhauser (KS) function. A new adaptive approach based on the KS function is proposed that updates the aggregation parameter by taking into account the constraint sensitivity. This adaptive approach is shown to significantly increase the accuracy of the results without additional computational cost especially when a large number of constraints are active at the optimum. The characteristics of each aggregation method and the performance of the proposed adaptive approach are shown by solving a wing structure weight minimization problem.

A sequential coupling of optimal topology and multilevel shape design applied to two-dimensional nonlinear magnetostatics

In this paper, a sequential coupling of two-dimensional (2D) optimal topology and shape design is proposed so that a coarsely discretized and optimized topology is the initial guess for the following shape optimization. In between, we approximate the optimized topology by piecewise Bezier shapes via least square fitting. For the topology optimization, we use the steepest descent method. The state problem is a nonlinear Poisson equation discretized by the finite element method and eliminated within Newton iterations, while the particular linear systems are solved using a multigrid preconditioned conjugate gradients method. The shape optimization is also solved in a multilevel fashion, where at each level the sequential quadratic programming is employed. We further propose an adjoint sensitivity analysis method for the nested nonlinear state system. At the end, the machinery is applied to optimal design of a direct electric current electromagnet. The results correspond to physical experiments.

Topology design optimization of structures at high frequencies using power flow analysis

A density–based topology optimization method is developed for power flow problems in steady state using an adjoint design sensitivity analysis (DSA) method. Design variables such as Young's modulus, density, and damping factor are parameterized using a bulk material density function. A generalized compliance of structures, defined as the inner product of prescribed loading and response function, is selected as an objective function. The objective is to minimize the generalized compliance within allowable material volume. Since already factorized system matrix obtained from the power flow analysis is utilized to evaluate the adjoint response and sensitivity, computational cost for the DSA is trivial. Through several numerical examples, the developed DSA method is verified to yield very accurate sensitivity, requiring only 0.15% of CPU cost for finite difference sensitivity. Also, it turns out that the developed method can yield an optimal structural layout to transmit the vibrational power effectively.

THEORETICAL AND ALGORITHMIC ON TOPOLOGY OPTIMIZATION DESIGN OF DISTRIBUTED COMPLIANT MECHANISMS

Topological optimization technology of continuum structures is extended to compliant mechanism designs. A kind of explicit updating scheme of design variables is worked out based on variable density method and optimality criteria. A new multi-criteria formulation is proposed for optimal topology design of compliant mechanisms using artificial spring model and dummy load method, in which the mechanical efficiency of mechanism problems and stiffness of structure problems are both involved. Sensitivity analysis of objective function is performed by applying adjoint sensitivity analysis method. Numerical examples have been presented for demonstration of the effectiveness of the proposed approaches. The influences of some optimal parameters upon the resulting mechanisms are discussed and the reasonable value scopes have been suggested. Prototype manufacturing methodology is also employed to manufacture the typical compliant mechanism to prove the validation of the conceptual design.

Efficient topology optimization of thermo-elasticity problems using coupled field adjoint sensitivity analysis method

We develop a unified and efficient adjoint design sensitivity analysis (DSA) method for weakly coupled thermo-elasticity problems. Design sensitivity expressions with respect to thermal conductivity and Young's modulus are derived. Besides the temperature and displacement adjoint equations, a coupled field adjoint equation is defined regarding the obtained adjoint displacement field as the adjoint load in the temperature field. Thus, the computing cost is significantly reduced compared to other sensitivity analysis methods. The developed DSA method is further extended to a topology design optimization method. For the topology design optimization, the design variables are parameterized using a bulk material density function. Numerical examples show that the DSA method developed is extremely efficient and the optimal topology varies significantly depending on the ratio of mechanical and thermal loadings.

Topological shape optimization of power flow problems at high frequencies using level set approach

Using a level set method we develop a topological shape optimization method applied to power flow problems in steady state. Necessary design gradients are computed using an efficient adjoint sensitivity analysis method. The boundaries are implicitly represented by the level set function obtainable from the “Hamilton–Jacobi type” equation with the “Up-wind scheme.” The implicit function is embedded into a fixed initial domain to obtain the finite element response and sensitivity. The developed method defines a Lagrangian function for the constrained optimization. It minimizes a generalized compliance, satisfying the constraint of allowable volume through the variations of implicit boundary. During the optimization, the boundary velocity to integrate the Hamilton–Jacobi equation is obtained from the optimality condition for the Lagrangian function. Compared with the established topology optimization method, the developed one has no numerical instability such as checkerboard problems and easy representation of topological shape variations.

A Coupled-Adjoint Sensitivity Analysis Method for High-Fidelity Aero-Structural Design

This paper presents an adjoint method for sensitivity analysis that is used in an aero-structural aircraft design framework. The aero-structural analysis uses high-fidelity models of both the aerodynamics and the structures. Aero-structural sensitivities are computed using a coupled-adjoint approach that is based on previously developed single discipline sensitivity analysis. Alternative strategies for coupled sensitivity analysis are also discussed. The aircraft geometry and a structure of fixed topology are parameterized using a large number of design variables. The aero-structural sensitivities of aerodynamic and structural functions with respect to these design variables are computed and compared with results given by the complex-step derivative approximation. The coupled-adjoint procedure is shown to yield very accurate sensitivities and to be computationally efficient, making high-fidelity aero-structural design feasible for problems with thousands of design variables.

Error Estimation of Numerical Solutions of One-dimensional Burgers Equation by Adjoint Sensitivity Analysis Method

In this study, we present a new numerical method to analyze quantitatively the error of numerical solution by using the adjoint sensitivity analysis method and the modified equation approaches. If a reference case of typical parameters is once calculated with the method, no additional calculation is required to estimate the results of the other numerical parameters such as those of more detailed resolution. Furthermore, we can estimate the strict solution from the sensitivity analysis results and can quantitatively evaluate the error of numerical solutions based on only the numerical procedures.

Adjoint state method for shape sensitivity analysis in non‐steady forming applications

AbstractThis paper introduces the adjoint state method in non‐steady‐state forming processes, such as forging, whereas this method is more frequently used for steady‐state problems. Some practical details for an efficient implementation are given. The emphasis is put on the accuracy of the sensitivity analysis for problems with remeshings and transfers of variables. The precision of the calculated derivatives is verified by comparing its results with the direct differentiation method. The adjoint state method is then applied to shape optimization problems. Copyright © 2003 John Wiley & Sons, Ltd.

Efficient Aerodynamic Design Method Using a Tightly Coupled Algorithm

An efe cient aerodynamic design optimization method for the compressible Euler equations is presented. A gradient-based optimization method is used to e nd the optimal design, in conjunction with a continuous adjoint sensitivity analysis method. To improve the efe ciency of the design method, a tightly coupled algorithm based on a step-size estimation isproposed. The efe ciency of thepresentmethod is demonstrated through drag minimizations of a wing and an airfoil. The result shows that the cost of the present method is signie cantly lower than that of the loosely coupled algorithm. Nomenclature Ci = e ux Jacobian matrix in the computational domain d = design variables vector Fi = e ux vector in the computational domain fi = e ux vector in the physical domain G = gradient vector of the object function I = object function q = conservative e ow variables vector R = residual vector in the computational domain S = searching direction ≤ = step size along the searching direction S A = Lagrangian multiplier vector Subscript face = value at cell face Superscript T = transpose of vector or matrix

An adjoint variable method for design sensitivity analysis of elastoplastic structures

Design sensitivity analysis of structural problems obeying an elastoplastic material behavior is developed using adjoint variable method. An elastoplastic constitutive equation with yield surface and kinematic hardening is considered to describe the material behavior. The traditional incremental procedure and its design variation need special treatments in order to predict the discontinuity of the structural response sensitivity because the contribution from the design sensitivity at the material transition point is lost during the calculation. In this study, discontinuities of the design variations at the material transition points are alleviated in the adjoint variable method. Analytical and numerical examples are used not only to demonstrate the developed sensitivity procedure but also to gain insights of numerical implementation for the design sensitivity analysis of the elastoplastic structure based on the adjoint variable method. The comparisons between adjoint variable and direct variation methods are also discussed.

Statistical Sensitivity Analysis of Lattice Space Structures

This paper presents a statistical sensitivity analysis for shape distortions of space structures. The approach is based on a statistical shape‐distortion analysis on the structural errors and an adjoint method of sensitivity analysis. The statistical shape‐distortion analysis allows the stochastic errors to be represented by member‐length tolerances. The sensitivity analysis is performed to predict the effects of member‐length errors for lattice space antennas on the surface accuracy. The formulas presented in this paper give an effective approach to predict the effects of member‐length errors on the shape distortions, to obtain effective structural elements to correct the shape distortions, and to design tolerance errors of the structural elements. Numerical examples for statically determinate and indeterminate two‐dimensional truss beams have been demonstrated to identify the members contributing most to the errors. These results show that the errors of the longitudinal elements of the structure are imp...