COMPARISON OF SENSITIVITY EQUATION AND ADJOINT EQUATION METHODS FOR PARAMETER IDENTIFICATION PROBLEMS

This paper deals with the inverse analysis of a thermal conduction problem, in which the thermal conductivity is identified as an unknown parameter, which is determined so as to minimize the cost function represented by the square of the difference between the computed and observed temperatures at pre-assigned observation points. To minimize the cost function, both sensitivity equation and adjoint equation methods can be adopted. The sensitivity equation can be introduced by differentiating the governing equation directly. The sensitivity coefficient is obtained by the sensitivity equation. The adjoint equation is introduced via a variational approach using a Lagrange multiplier. The Lagrange multiplier is solution to an adjoint equation. Both sensitivity coefficient and Lagrange multiplier are used to calculate the gradient of the cost function. The purpose of this paper is to compare the sensitivity equation and adjoint equation methods from the convergence and computational efficiency points of view. © 1997 by John Wiley & Sons, Ltd.     

Solution to the topology optimization problem using a time-evolution equation

This article describes a numerical solution to the topology optimization problem using a time-evolution equation. The design variables of the topology optimization problem are defined as a mathematical scalar function in a given design domain. The scalar function is projected to the normalized density function. The adjoint variable method is used to determine the gradient defined as the ratio of the variation of the objective function or constraint function to the variation of the design variable. The variation of design variables is obtained using the solution of the time-evolution equation in which the source term and Neumann boundary condition are given as a negative gradient. The distribution of design variables yielding an optimal solution is obtained by time integration of the solution of the time-evolution equation. By solving the topology optimization problem using the proposed method, it is shown that the objective function decreases when the constraints are satisfied. Furthermore, we apply the proposed method to the thermal resistance minimization problem under the total volume constraint and the mean compliance minimization problem under the total volume constraint.     

Robust topology optimization of optical cloaks under uncertainties in wave number and angle of incident wave

This article presents a robust topology optimization method for optical cloaks under uncertainties in the wave number and angle in the incident wave. We first discuss the governing equation derived from Maxwell's equation, and extend it to the entire domain including the dielectric material and air, based on the level set-based topology optimization method. Next, a robust optimization problem is formulated as a minimization problem of the weighted sum of the scattered wave norm and its standard deviation with respect to the wave number and angle of the incident wave. The standard deviation is mathematically expressed by the Taylor series approximation and the use of the adjoint variable method. The design sensitivity of the objective functional is also derived by the adjoint variable method. An optimization algorithm is then constructed, based on the proposed formulation for robust designs of optical cloaks. Several numerical examples are finally provided to demonstrate the validity and utility of the proposed method.     

The continuous adjoint approach to the k–ω SST turbulence model with applications in shape optimization

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 the k–ω SST turbulence 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.     

引水隧洞爆破开挖对邻近建筑物的振动影响分析

以发电引水隧洞开挖过程中的爆破振动实测数据为基础,对爆破振动衰减规律进行了二元回归分析,与萨道夫斯基公式相比,预测数据更接近实际工程.同时对隧洞爆破开挖过程中邻近建筑物的安全,根据实际监测成果作了进一步分析,给出了相应的爆破安全判据.     

FINITE ELEMENT SOLUTION OF A MODEL FREE SURFACE PROBLEM BY THE OPTIMAL SHAPE DESIGN APPROACH

Optimal shape design approach is applied to numerical computation of a model potential free boundary value problem. The problem is discretized using the finite element method. To test the approach the problem is formulated in both velocity potential and stream function formulation and four different finite element discretizations are used. Associated minimization problem is solved using the quasi-Newton method. Gradient of the cost function is computed by solving the algebraic adjoint equation. Gravity and surface tension forces are included in the model. Viability of the method is showed by solving problems with important effects of gravity and surface tension forces. © 1997 by John Wiley & Sons, Ltd.     

Adjoint Hamiltonian Monte Carlo algorithm for the estimation of elastic modulus through the inversion of elastic wave propagation data

Efficient inversion of noisy seismic waveform data produced due to elastic wave propagation for the estimation of a high-dimensional elastic modulus vector is achieved. Estimation is carried out in a Bayesian framework using Hamiltonian Monte Carlo (HMC) that enables efficient statistical estimation over high-dimensional parameters. The truncated Karhunen-Loève (K-L) expansion is introduced to reduce the dimensionality of the elastic modulus vector. Expensive computations of the gradient of the state vector with respect to the parameter vector at every step are also eliminated through the adjoint method, which is developed from a general one-step discretization of the governing second-order ordinary differential equations (ODEs). An Adjoint HMC algorithm that employs a truncated K-L expansion of the elastic modulus vector is presented. The efficacy of the algorithm is investigated with respect to two representative problems with varying geometric complexity. Adjoint HMC offers a significant speed up in gradient calculation time over the direct differentiation counterpart as the number of terms in the K-L expansion increases. The algorithm is able to estimate the true elastic modulus within the credible intervals for both cases.     

Identification of boundary displacements in plane elasticity by BEM

The purpose of this study is to present a possible application of BEM for numerical identification of the boundary conditions for Navier equations in plane elasticity with internal measurements, based on insufficient and noisy information for unique identification. The inverse problem is re-formulated as a minimization problem by the direct variational method. The minimization problem is then recast using the gradient method into successive primary and adjoint boundary value problems in the corresponding plane elasticity problem. For numerical solution of the elasticity problems, the conventional direct boundary element method is employed. From the simple numerical examples considered, it is concluded that our identification scheme is stable and the approximate solutions are convergent to the minimum.     

Optimization of a variable mouth acoustic horn

By using boundary shape optimization on the end part of a semi‐infinite waveguide for acoustic waves, we design transmission‐efficient interfacial devices without imposing an upper bound on the mouth diameter. The boundary element method solves the Helmholtz equation modeling the exterior wave propagation problem. A gradient‐based optimization algorithm solves the resulting least‐squares problem and the adjoint method provides the necessary gradients. The results demonstrate that there appears to be a natural limit on the optimal mouth diameter. Copyright © 2010 John Wiley & Sons, Ltd.     

An adjoint method for the inverse design of solidification processes with natural convection

This paper presents a finite element algorithm based on the adjoint method for the design of a certain class of solidification processes. In particular, the paper addresses the design of directional solidification processes for pure materials such that a desired freezing front heat flux and growth velocity are achieved. This is the first time that an infinite-dimensional continuum adjoint formulation is obtained and implemented for the solution of such inverse/design problems with moving boundaries and Boussinesq incompressible flow. The present design problem belongs to a category of inverse problems in which one is looking for the unknown conditions in part of the boundary, while overspecified boundary conditions are supplied in another part of the boundary (here the freezing interface). The solidification design problem is mathematically posed as a whole time-domain optimization problem. The gradient of the cost functional is calculated using the solution of an appropriately defined continuous adjoint problem. The minimization process is realized by the conjugate gradient method via the solutions of the direct, adjoint and sensitivity sub-problems. The proposed methodology is demonstrated with the solidification of an initially superheated liquid aluminum confined in a square mold. The non-uniformity in the casting product in the direction of gravity due to the existence of natural convection in the melt is emphasized. The inverse design problem is then posed as finding the appropriate spatial-temporal variations of the boundary heat flux on the vertical mold walls that can eliminate or reduce the effects of convection on the freezing interface heat fluxes and growth velocity. The numerical example demonstrates the accuracy and convergence of the adjoint formulation. Finally, open related research design problems are discussed. © 1998 John Wiley & Sons, Ltd.     

Control of the freezing interface motion in two-dimensional solidification processes using the adjoint method

The aim of this work is to calculate the optimum history of boundary cooling conditions that, in two-dimensional conduction driven solidification processes, results in a desired history of the freezing interface location/motion. The freezing front velocity and heat flux on the solid side of the front, define the obtained solidification microstructure that can be selected such that desired macroscopic mechanical properties and soundness of the final cast product are achieved. The so-called two-dimensional inverse Stefan design problem is formulated as an infinite-dimensional minimization problem. The adjoint method is developed in conjunction with the conjugate gradient method for the solution of this minimization problem. The sensitivity and adjoint equations are derived in a moving domain. The gradient of the cost functional is obtained by solving the adjoint equations backward in time. The sensitivity equations are solved forward in time to compute the optimal step size for the gradient method. Two-dimensional numerical examples are analysed to demonstrate the performance of the present method.     

Optimality criteria-based topology optimization of a bi-material model for acoustic–structural coupled systems

This article investigates topology optimization of a bi-material model for acoustic–structural coupled systems. The design variables are volume fractions of inclusion material in a bi-material model constructed by the microstructure-based design domain method (MDDM). The design objective is the minimization of sound pressure level (SPL) in an interior acoustic medium. Sensitivities of SPL with respect to topological design variables are derived concretely by the adjoint method. A relaxed form of optimality criteria (OC) is developed for solving the acoustic–structural coupled optimization problem to find the optimum bi-material distribution. Based on OC and the adjoint method, a topology optimization method to deal with large calculations in acoustic–structural coupled problems is proposed. Numerical examples are given to illustrate the applications of topology optimization for a bi-material plate under a low single-frequency excitation and an aerospace structure under a low frequency-band excitation, and to prove the efficiency of the adjoint method and the relaxed form of OC.     

Effect of stenosis on solitary waves in arteries

In the present work, a mathematical model of nonlinear blood flow through a stenosed artery is developed. Treating the artery as a stenosed, elastic, thin walled long tube and using the reductive perturbation method, the propagation of weakly nonlinear waves in such a fluid-filled elastic tube is studied. By considering the blood as an incompressible inviscid fluid, the evolution equation is obtained as the extended Stochastic Korteweg–de Vries equation. Progressive wave solution to this evolution equation is obtained and effect of stenosis on the wave profile and the wave speed is discussed.     

An inverse radiation problem of simultaneous estimation of heat transfer coefficient and absorption coefficient in participating media

An inverse radiation problem is investigated where the spatially varying heat transfer coefficient h(z) and the absorption coefficient κ in the radiant cooler are estimated simultaneously from temperature measurements. The inverse radiation problem is solved through the minimization of a performance function, which is expressed by the sum of square residuals between calculated and observed temperature, using the conjugate gradient method. The gradient of the performance function is evaluated by means of the improved adjoint variable method that can take care of both the function estimation and the parameter estimation efficiently. The present method is found to estimate h(z) and κ with reasonable accuracy even with noisy temperature measurements. Copyright © 2002 John Wiley & Sons, Ltd.     

Cost reduction techniques for the design of non‐linear flapping wing structures

This work details a computational framework for gradient‐based optimization of a non‐linear flapping wing structure with a large number of design variables, where analytical sensitivities of the unsteady finite element system are computed using the adjoint method. Two techniques are used to reduce the large computational cost of this structural design process. The first projects the finite element system onto a reduced basis of POD modes. The second uses a monolithic time formulation with spectral elements, and can be used to compute only the desired time‐periodic response. Results are given in terms of the trade‐off between accuracy and computational efficiency of these methods for both system response and adjoint computations, for a variety of mesh/time step refinements, degrees of non‐linearity (i.e. weakly or strongly non‐linear), and harmonic content. The work concludes with the structural design of a flapping wing: the elastic deformation at the wingtip is minimized through the flapping stroke by varying the thickness of each finite element. Significant improvements in computational cost are obtained at little expense to the accuracy of the results obtained via design optimization. Published in 2011 by John Wiley & Sons, Ltd.     

The continuous adjoint approach to the k–ε turbulence model for shape optimization and optimal active control of turbulent flows

The continuous adjoint to the incompressible Reynolds-averaged Navier–Stokes equations coupled with the low Reynolds number Launder–Sharma k–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–Sharma k–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.     

基坑上跨开挖诱发既有盾构隧道隆起变形研究EI北大核心CSCD

基坑上跨开挖将会引起既有盾构隧道隆起变形,危及既有盾构隧道的服役性能。目前基坑开挖引起的盾构隧道隆起变形的解析方法,通常将盾构隧道简化为埋置于弹性地基上的连续长梁,忽略了环间接头影响。针对前人研究的不足,提出带环间接头的盾构隧道计算模型,通过非线性Pasternak地基模型来考虑地基土变形的非线性特征,通过两阶段分析法,推导得到基坑上跨开挖作用下盾构隧道隆起位移和张开量简化解答。通过MINDLIN解计算基坑开挖引起的作用于盾构隧道上的附加荷载;建立基坑卸载下盾构隧道的隆起变形微分方程。利用有限差分法求解出基坑开挖引起的邻近盾构隧道隆起变形和内力分布。收集了三个工程实测数据,并将所提方法和实测数据、既有理论方法进行对比,验证所提方法的适用性。     

Optimization of an acoustic horn with respect to efficiency and directivity

We consider the problem of designing an acoustic horn in order to efficiently transmit the incoming wave energy and favorably distribute the energy in the far field. A finite element solution of the Helmholtz equation, in planar or cylindrical symmetry, models the wave propagation. The transmission efficiency is monitored by measuring the back reflections into the feeding waveguide, and the far‐field directivity pattern is computed using an integral expression known from scattering theory. The design problem is formulated as a non‐linear least‐squares problem, which is solved using a gradient‐based algorithm, where the gradients are provided by solutions of the associated adjoint equations. The results demonstrate that this approach can generate horns with almost perfect transmission in a wide frequency band. Due to the improved transmission properties at the lower‐frequency region, the optimization with respect to efficiency also generates improved far‐field directivity patterns, that is, patterns that vary less with frequency. It is possible to obtain even more uniform directivity patterns by explicitly including directivity requirements in the optimization. However, those improvements in directivity seem to be associated with a substantial loss of efficiency. Copyright © 2007 John Wiley & Sons, Ltd.     

Optimal control of a continuum model for single-product re-entrant manufacturing systems

A semiconductor manufacturing system that involves a large number of items and many steps can be modelled through conservation laws for a continuous density variable on a production process. In this paper, the basic hyperbolic partial differential equation (PDE) models for multiple re-entrant manufacturing systems are proposed. However, through numerical examples, the basic continuum models do not perform well for small-scale multiple re-entrant systems, so a new state equation taking into account the re-entrant degree of the product is introduced to improve the basic continuum models. The applicability of the modified continuum model is illustrated through a numerical example. Based on the modified continuous model, this paper studies the optimal control problems for multiple re-entrant manufacturing systems. The gradient of the cost function with respect to the influx is solved by the adjoint approach, and then the optimal influx is computed by the steepest descent method. Finally, numerical examples on optimal influx profiles for steps in demand rate, linear demand rate and periodically varying demand rate are given. The relationships among influx, outflux and demand are also discussed in detail.     

Output error estimation strategies for discontinuous Galerkin discretizations of unsteady convection‐dominated flows

We study practical strategies for estimating numerical errors in scalar outputs calculated from unsteady simulations of convection‐dominated flows, including those governed by the compressible Navier–Stokes equations. The discretization is a discontinuous Galerkin finite element method in space and time on static spatial meshes. Time‐integral quantities are considered for scalar outputs and these are shown to superconverge with temporal refinement. Output error estimates are calculated using the adjoint‐weighted residual method, where the unsteady adjoint solution is obtained using a discrete approach with an iterative solver. We investigate the accuracy versus computational cost trade‐off for various approximations of the fine‐space adjoint and find that exact adjoint solutions are accurate but expensive. To reduce the cost, we propose a local temporal reconstruction that takes advantage of superconvergence properties at Radau points, and a spatial reconstruction based on nearest‐neighbor elements. This inexact adjoint yields output error estimates at a computational cost of less than 2.5 times that of the forward problem for the cases tested. The calculated error estimates account for numerical error arising from both the spatial and temporal discretizations, and we present a method for identifying the percentage contributions of each discretization to the output error. Copyright © 2011 John Wiley & Sons, Ltd.