Adjoint Variable Method Research Articles

Direct sensitivity analysis of multibody systems with holonomic and nonholonomic constraints via an index-3 augmented Lagrangian formulation with projections

Optimizing the dynamic response of mechanical systems is often a necessary step during the early stages of product development cycle. This is a complex problem that requires to carry out the sensitivity analysis of the system dynamics equations if gradient-based optimization tools are used. These dynamics equations are often expressed as a highly nonlinear system of ordinary differential equations or differential-algebraic equations, if a dependent set of generalized coordinates with its corresponding kinematic constraints is used to describe the motion. Two main techniques are currently available to perform the sensitivity analysis of a multibody system, namely the direct differentiation and the adjoint variable methods. In this paper, we derive the equations that correspond to the direct sensitivity analysis of the index-3 augmented Lagrangian formulation with velocity and acceleration projections. Mechanical systems with both holonomic and nonholonomic constraints are considered. The evaluation of the system sensitivities requires the solution of a tangent linear model that corresponds to the Newton–Raphson iterative solution of the dynamics at configuration level, plus two additional nonlinear systems of equations for the velocity and acceleration projections. The method was validated in the sensitivity analysis of a set of examples, including a five-bar linkage with spring elements, which had been used in the literature as benchmark problem for similar multibody dynamics formulations, a point-mass system subjected to nonholonomic constraints, and a full-scale vehicle model.

Optimal mass distribution in carbon nanotubes for extreme thermal conductivity: Analytical manipulation of isotope effects

Using a gradient-based optimization method, an optimal mass distribution for a single-walled carbon nanotube (CNT) is determined for extremal thermal conductivity. The variation of atomic mass distribution or the isotope impurity in the CNT could lead to either extremely high or low thermal conductivity. The adjoint variable method is employed for the efficient and accurate design sensitivity analysis of thermal conductivity, which can be obtained from non-equilibrium molecular dynamics (NEMD) approach. The obtained analytical sensitivity can be extensively utilized to determine the optimal distribution of atomic masses, using a density-based topology optimization method. The resulting thermal conductivity is reduced by more than 90% or increased by 70% of the one of the pristine CNT. The density of states (DOS) after the optimization indicates that the optimization process yields the effective and physically meaningful result as we interpret the resulting distribution of mass.

A framework of shape optimisation based on the isogeometric boundary element method toward designing thin-silicon photovoltaic devices

We propose a gradient-based optimisation framework to design the shape of the interfaces in periodic layered structures, which can model photovoltaic devices, optical gratings, etc., according to a desired electromagnetic property. To this end, we first develop an isogeometric boundary element method (IGBEM) to analyse singly periodic boundary value problems for 2D Helmholtz equation in layered media. By nature, the IGBEM is not only numerically accurate but also suitable for the shape optimisation in comparison with the conventional boundary- and domain-type solvers. Next, we derive the shape derivative (or shape sensitivity) of the objective function in terms of the magnetic field strength or energy absorption rate on the basis of the adjoint variable method. The shape derivative can be computed by solving the primal and adjoint problems with the IGBEM. Subsequently, we map our shape optimisation problems to nonlinear programming problems in order to exploit a general-purpose software for the latter problems. After verifying our optimisation framework rigorously by comparing with the exact solutions, we demonstrate a shape optimisation of the silicon-metal interface in a model of thin-silicon photovoltaic devices: it is crucial to determine the interface so that the energy of the incident light can be confined in the silicon layer as much as possible when the thickness of the layer is unconventionally small; one micrometre in our model. The optimal shape achieved 8.6 times higher absorption rate than the reference (flat) shape. This simulation shows that the proposed framework is capable of making a fundamental contribution to analysing and designing photovoltaic devices as well as photonic and plasmonic devices that consist of singly periodic layers.

Layout design of piezoelectric patches in structural linear quadratic regulator optimal control using topology optimization

This article investigates topology optimization for piezoelectric thin-shell structures under the linear quadratic regulator optimal control. In the optimization model, the structural dynamic compliance is taken as the measure of control performance, and the relative densities describing the distribution of the piezoelectric material are considered as design variables. An artificial material model with penalization on both mechanical and piezoelectric properties is employed. For the purpose of improving computational efficiency of the sensitivity and response analysis, modal superposition method is adopted. The derivative of the Riccati equation governing the linear quadratic regulator control with respect to the design variables is shown to be a Lyapunov equation. In conjunction with the adjoint variable method, the design sensitivities of the dynamic compliance are obtained using the solution of the Lyapunov equation. Numerical examples demonstrate the validity of the proposed method and show the significance of layout design of piezoelectric sensors/actuators. The influences of some key factors on the optimization solutions are discussed. It is shown that the optimized layout of the piezoelectric patches may be significantly influenced by the excitation frequency, but only slightly affected by the choice of the weighting matrix in the linear quadratic regulator control. This work aims to provide an efficient gradient-based mathematical programming method for guiding the layout design of actuators and sensors in smart structures under optimal vibration control. However, the considered model is a purely mathematical one without consideration of engineering realization, thus the optimization result may only serve as an upper bound for practically realizable control performance.

Simultaneous identification of damage in bridge under moving mass by Adjoint variable method

In this paper, a theoretical and numerical study on bridge simultaneous damage detection procedure for identifying both the system parameters and input excitation mass, are presented. This method is called \'Adjoint Variable Method\' which is an iterative gradient-based model updating method based on the dynamic response sensitivity. The main advantage of proposed method is inclusion of an analytical method to augment the accuracy and speed of the solution. Moving mass is a model which takes into account the inertia effects of the vehicle. This interaction model is a time varying system and proposed method is capable of detecting damage in this variable system. Robustness of proposed method is illustrated by correctly detection of the location and extension of predetermined single, multiple and random damages in all ranges of speed and mass ratio of moving vehicle. A comparison study of common sensitivity and proposed method confirms its efficiency and performance improvement in sensitivity-based damage detection methods. Various sources of errors including the effects of measurement noise and initial assumption error in stability of method are also discussed.

A comparative study of design sensitivity analysis based on adjoint variable method for transient response of non-viscously damped systems

Abstract In this paper, design sensitivity analysis methods for the transient response of non-viscously damped systems are considered. The damping forces of the non-viscously damped systems depend on the past history of motion via convolution integrals over some suitable kernel functions. The adjoint variable method (AVM) is adopted to develop the design sensitivity analysis. Two numerical solution schemes, namely the discretize-then-differentiate method and the differentiate-then-discretize method, are introduced to complete the AVM for the sensitivity analysis of non-viscously damped systems. The discretize-then-differentiate AVM discretizes the equations of motion based on the Newmark- β implicit integration method first and then differentiates the discrete equations. On the contrary, the differentiate-then-discretize AVM firstly differentiates the equations of motion after transforming it into a state-space form, and then discretizes the equations based on a modified precise integration method (PIM). The numerical accuracy, efficiency, consistency and implementation effort are discussed and compared. Two numerical examples are presented to show the effectiveness of both methods. The results indicate that, by considering both computational accuracy and efficiency, the PIM based differentiate-then-discretize method is more suitable than the Newmark- β based discretize-then-differentiate method for the sensitivity analysis of transient responses for non-viscously damped systems.

Multiple Level-Set Methods for Optimal Design of Nonlinear Magnetostatic System

This paper proposes an optimal design method using continuum sensitivity analysis and the multiple level-set methods for multi-material shape optimization in nonlinear magnetostatic systems. The general shape sensitivity formula for the nonlinear magnetostatic system is derived based on the material derivative concept of continuum mechanics and the adjoint variable method. In the multiple level-set methods, the sign combination of level-set functions is used to identify the different material regions and their interfaces. The multi-material shapes are simultaneously deformed by solving the multiple level-set equations coupled with the velocity fields in the sensitivity formula. Two numerical examples are employed to show the usefulness of the proposed design method.

Adjoint-based optimization of active nanophotonic devices.

We show that the adjoint variable method can be combined with the multi-frequency finite-difference frequency-domain method for efficient sensitivity calculations, enabling the systematic optimization of active nanophotonic devices. As a proof of principle demonstration, we have optimized a dynamic isolator structure in two-dimensions, resulting in the reduction of the length of the modulated regions by a factor of two, while retaining good performance in the isolation ratio and insertion loss.

A method for topology optimization of structures under harmonic excitations

This paper deals with topology optimization of large-scale structures with proportional damping subjected to harmonic excitations. A combined method (CM) of modal superposition with model order reduction (MOR) for harmonic response analysis is introduced. In the method, only the modes corresponding to a frequency range which is a little bigger than that of interest are used for modal superposition, the contribution of unknown higher modes is complemented by a MOR technique. Objective functions are the integral of dynamic compliance of a structure, and that of displacement amplitude of a certain user-defined degree of freedom in the structure, over a range of interested frequencies. The adjoint variable method is applied to analyze sensitivities of objective functions and the accuracy of the sensitivity analyses can also be ensured by CM. Topology optimization procedure is illustrated by three examples. It is shown that the topology optimization based on CM not only remarkably reduce CPU time, but also ensure accuracy of results.

Efficient gradient‐based optimisation of pixel antenna with large‐scale connections

An efficient gradient-based optimisation method is proposed for designing pixel antennas with hundreds of connections, where the connections between adjacent square pixel patches are modelled by lossy materials with continuous density variables. The gradients of antenna objective functions with respect to these density variables are derived by the adjoint variable method, at the cost of less than two times of simulation. With a gradient-based optimisation solver, four antennas with 220 connections are optimised. The optimised antennas achieve different expected polarisation and/or radiation patterns. All the optimisations in this work converge in only 80 iterations which is considerably faster than the heuristic algorithms handling the same number of variables. Two of the optimised antennas are fabricated to validate the optimisations. The simulated and measured results verify the accuracy and effectiveness of this method.

Topology optimization of viscoelastic damping layers for attenuating transient response of shell structures

This paper presents topology optimization of viscoelastic damping layers attached to shell structures for attenuating the amplitude of transient response under dynamic loads. The transient response is evaluated using an implicit time integration scheme. Dynamic performance indices are defined to measure the transient response. In the optimization formulation, three different types of the performance indices are considered as the objective function. The density-based topology optimization scheme is applied to find the optimal distribution of the viscoelastic material. The artificial densities of the shell elements of the viscoelastic layers are taken as the design variables. The constraint is the maximum volume fraction of the viscoelastic material. A sensitivity analysis method of the transient response is developed based on the adjoint variable method. Several numerical examples are presented to demonstrate the validity of the proposed method. The transient responses of the optimized structures are compared to those of the uniformly distributed structures to show the effectiveness of the proposed method. Also, the influences of the performance indices are discussed.

Shape optimization of unsteady state viscous flow fields for drag minimization and lift maximization

This paper describes a numerical solution to the shape optimization problems of viscous flow fields. The drag minimization problem and lift maximization problem for an isolated body in uniform flow were formulated in the domain of steady-state viscous flow fields. The shape gradient of the shape optimization problems were derived theoretically using the adjoint variable method, the Lagrange multiplier method and the formulae of the material derivative. Reshaping was accomplished using a traction method that was proposed as a solution to domain optimization problems. The validity of the proposed method was confirmed by results of 2D numerical analyses using finite element method.

Application of the adjoint variable method for high accurate numerical evaluation in hammering test

Application of the adjoint variable method for high accurate numerical evaluation in hammering test

Sensitivity analysis for elastic problems by the adjoint variable method based on the LBM

Sensitivity analysis for elastic problems by the adjoint variable method based on the LBM

Topology optimization of piezoelectric smart structures for minimum energy consumption under active control

This paper investigates topology optimization of the electrode coverage over piezoelectric patches attached to a thin-shell structure to reduce the energy consumption of active vibration control under harmonic excitations. The constant gain velocity feedback control method is employed, and the structural frequency response under control is analyzed with the finite element method. In the mathematical formulation of the proposed topology optimization model, the total energy consumption of the control system is taken as the objective function, and a constraint of the maximum allowable dynamic compliance is considered. The pseudo-densities indicating the distribution of surface electrode coverage over the piezoelectric layers are chosen as the design variables, and a penalized model is employed to relate the active damping effect and these design variables. The sensitivity analysis scheme of the control energy consumption with respect to the design variables is derived with the adjoint-variable method. Numerical examples demonstrate that the proposed optimization model is able to generate optimal topologies of electrode coverage over the piezoelectric layers, which can effectively reduce the energy consumption of the control system. Also, numerical comparisons with a minimum-volume optimization model show the advantage of the proposed method with respect to energy consumption. The proposed method may provide useful guidance to the layout optimization of piezoelectric smart structures where the energy supply is limited, such as miniature vibration control systems.

A RBRDO approach based on structural robustness and imposed reliability level

A new approach for the multi-objective optimization of composite structures under the effects of uncertainty in mechanical properties, structural parameters and external loads is proposed, to guarantee higher levels of accuracy exclusively with Evolutionary Algorithms (EA). The concepts of Reliability-Based Robust Design Optimization (RBRDO) are applied. Optimality is defined as the minimization of the structural weight and robustness as the minimization of the determinant of the variance-covariance matrix of the structural responses. Reliability assessment is performed through a mathematical reformulation of the Performance Measure Approach, suitable for EA, where the standard normal uncertainty space was defined in directional coordinates and reduced to the surface of the hypersphere of radius β^a. A binary reliability constraint, that allowed avoiding unnecessary runs of the reliability inner-cycle is defined. The Robust Design Optimization cycle is solved by a multi-objective EA, based on constrained-dominance. Sensitivities of the structural responses, necessary for uncertainty analysis only, are calculated analytically by the Adjoint Variable Method. A numerical example considering a balanced angle-ply laminate shell is presented. Results show an effective convergence of the Pareto-optimal Front (POF). Uncertainty analysis shows that the variability of the critical displacements increases along the POF. For the stresses, variability is stable but of higher values. The incorporation of the reliability constraint prevents the natural decrease of the reliability index, along the POF, to reach levels too close, or inside, of the failure domain. The distribution of the reliability measures along the POF is similar and demonstrates the effects of reliability in the RBRDO procedure.

Sensitivity Analysis of Ferrites With TLM

We propose a wideband adjoint variable method for sensitivity analysis of structures comprised of dispersive and anisotropic magnetized ferrite. Our algorithm efficiently estimates the gradients of the desired response with respect to all the designable parameters using only one extra simulation regardless of the number of parameters.

Non-probabilistic reliability-based topology optimization with multidimensional parallelepiped convex model

In this paper, a new non-probabilistic reliability-based topology optimization (NRBTO) method is proposed to account for interval uncertainties considering parametric correlations. Firstly, a reliability index is defined based on a newly developed multidimensional parallelepiped (MP) convex model, and the reliability-based topology optimization problem is formulated to optimize the topology of the structure, to minimize material volume under displacement constraints. Secondly, an efficient decoupling scheme is applied to transform the double-loop NRBTO into a sequential optimization process, using the sequential optimization & reliability assessment (SORA) method associated with the performance measurement approach (PMA). Thirdly, the adjoint variable method is used to obtain the sensitivity information for both uncertain and design variables, and a gradient-based algorithm is employed to solve the optimization problem. Finally, typical numerical examples are used to demonstrate the effectiveness of the proposed topology optimization method.

Adjoint sensitivity analysis and optimization of hysteretic dynamic systems with nonlinear viscous dampers

In this paper we discuss the adjoint sensitivity analysis and optimization of hysteretic systems equipped with nonlinear viscous dampers and subjected to transient excitation. The viscous dampers are modeled via the Maxwell model, considering at the same time the stiffening and the damping contribution of the dampers. The time-history analysis adopted for the evaluation of the response of the systems relies on the Newmark-β time integration scheme. In particular, the dynamic equilibrium in each time-step is achieved by means of the Newton-Raphson and the Runge-Kutta methods. The sensitivity of the system response is calculated with the adjoint variable method. In particular, the discretize-then-differentiate approach is adopted for calculating consistently the sensitivity of the system. The importance and the generality of the sensitivity analysis discussed herein is demonstrated in two numerical applications: the retrofitting of a structure subject to seismic excitation, and the design of a quarter-car suspension system. The MATLAB code for the sensitivity analysis considered in the first application is provided as “Supplementary Material”.

環境工学分野における随伴変数法の適用に関する研究 （第1報）気流抑制・排熱促進を目的とした駅舎内間仕切りの最適配置に関する設計手法の提案

環境工学分野における随伴変数法の適用に関する研究 （第1報）気流抑制・排熱促進を目的とした駅舎内間仕切りの最適配置に関する設計手法の提案