Tae Hee Lee, An adjoint variable method for design sensitivity analysis of elastoplastic structures, Ksme International Journal, 1999 9AAF40B3-A02C-4C31-8CFB-16B7204FD4CC, 13 (3) p. 246.
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, discontinu- ities 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.
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A McNamara et al., Fluid control using the adjoint method, Dl.Acm.Org, .
We describe a novel method for controlling physics-based fluid simulations through gradient-based nonlinear optimization. Using a technique known as the adjoint method, derivatives can be computed efficiently, even for large 3D simulations with millions of control parameters. In addition, we introduce the first method for the full control of free-surface liquids. We show how to compute adjoint derivatives through each step of the simulation, including the fast marching algorithm, and describe a new set of control parameters specifically designed for liquids.
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Seonho Cho and K K Choi, Design sensitivity analysis and optimization of non‐linear transient dynamics. Part I—sizing design, International Journal For Numerical Methods In Engineering, 48 (2000) 351-373.
A continuum-based sizing design sensitivity analysis (DSA) method is presented for the transient dynamic response of non-linear structural systems with elastic}plastic material and large deformation. The methodo- logy is aimed for applications in non-linear dynamic problems, such as crashworthiness design. The "rst-order variations of the energy forms, load form, and kinematic and structural responses with respect to sizing design variables are derived. To obtain design sensitivities, the direct di!erentiation method and updated Lagrangian formulation are used since they are more appropriate for the path-dependent problems than the adjoint variable method and the total Lagrangian formulation, respectively. The central di!erence method and the "nite element method are used to discretize the temporal and spatial domains, respectively. The Hughes}Liu truss/beam element, Jaumann rate of Cauchy stress, rate of deformation tensor, and Jaumann rate-based incrementally objective stress integration scheme are used to handle the "nite strain and rotation. An elastic}plastic material model with combined isotropic/kinematic hardening rule is employed. A key development is to use the radial return algorithm along with the secant iteration method to enforce the consistency condition that prevents the discontinuity of stress sensitivities at the yield point. Numerical results of sizing DSA using DYNA3D yield very good agreement with the "nite di!erence results. Design optimization is carried out using the design sensitivity information.
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P Dayan et al., The helmholtz machine, Neural Computation, 7 (1995) 889-904.
Discovering the structure inherent in a set of patterns is a fundamental aim of statistical inference or learning. One fruitful approach is to build a parameterized stochastic generative model, independent draws from which are likely to produce the patterns. For all but the simplest generative models, each pattern can be generated in exponentially many ways. It is thus intractable to adjust the parameters to maximize the probability of the observed patterns. We describe a way of finessing this combinatorial explosion by maximizing an easily computed lower bound on the probability of the observations. Our method can be viewed as a form of hierarchical self-supervised learning that may relate to the function of bottom-up and top-down cortical processing pathways.
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D Dyck et al., Design of electromagnetic devices using sensitivities computed withthe adjoint variable method, Computation In Electromagnetics, 1994.
Page 1. 231 USING SENSITIVITIES COMPUTED WITH THE DN Dyck, D A Lowther EM Freeman
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Jonathan B Freund, Adjoint-based optimization for understanding and suppressing jet noise, Procedia Engineering, 6 (2010) 54-63.
Advanced simulation tools, particularly large-eddy simulation techniques, are becoming capable of making qual- ity predictions of jet noise for realistic nozzle geometries and at engineering relevant flow conditions. Increasing computer resources will be a key factor in improving these predictions still further. Quality prediction, however, is only a necessary condition for the use of such simulations in design optimization. Predictions do not of themselves lead to quieter designs. They must be interpreted or harnessed in some way that leads to design improvements. As yet, such simulations have not yielded any simplifying principals that offer general design guidance. The turbulence mechanisms leading to jet noise remain poorly described in their complexity. In this light, we have implemented and demonstrated an aeroacoustic adjoint-based optimization technique that automatically calculates gradients that point the direction in which to adjust controls in order to improve designs. This is done with only a single flow solutions and a solution of an adjoint system, which is solved at computational cost comparable to that for the flow. Optimization requires iterations, but having the gradient information provided via the adjoint accelerates convergence in a manner that is insensitive to the number of parameters to be optimized.
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MB Giles and NA Pierce, Analytic adjoint solutions for the quasi-one-dimensional Euler equations, Journal Of Fluid Mechanics, 426 (2001) 327-345.
The analytic properties of adjoint solutions are examined for the quasi-one- dimensional Euler equations. For shocked flow, the derivation of the adjoint problem reveals that the adjoint variables are continuous with zero gradient at the shock, and that an internal adjoint boundary condition is required at the shock. A Green’s function approach is used to derive the analytic adjoint solutions corresponding to su- personic, subsonic, isentropic and shocked transonic flows in a converging–diverging duct of arbitrary shape. This analysis reveals a logarithmic singularity at the sonic throat and confirms the expected properties at the shock.
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MB Giles, On the iterative solution of adjoint equations, , 2000.
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R W Healy and T F Russell, A Finite-Volume Eulerian-Lagrangian Localized Adjoint Method for Solution of the Advection-Dispersion Equation, Water Resources Research, 29 (2007) 2399-2413.
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Hong-Lae Jang et al., Adjoint design sensitivity analysis of molecular dynamics in parallel computing environment, International Journal Of Mechanics And Materials In Design, 10 (2014) 379-394.
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Hong-Lae Jang and Seonho Cho, Adjoint design sensitivity analysis of constant temperature molecular dynamics, International Journal Of Mechanics And Materials In Design, 13 (2015) 243-252.
In this research, we proposed an efficient design sensitivity analysis (DSA) method for constant temperature molecular dynamics (MD). A Nose– Hoover thermostat is utilized to represent the possible state of a system that is in thermal equilibrium using a heat bath to maintain temperature constant. The design sensitivity of general performance measures is derived using an adjoint variable method. Since the adjoint system is path-dependent and derived in the form of a terminal value problem, the path of original MD analysis should be kept to be used with in the adjoint sensitivity computation. The time reversibility of the MD system with Nose–Hoover thermostat is investi- gated. The accuracy and efficiency of the developed adjoint DSA method are verified through demonstra- tive numerical examples.
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Gino I Montecinos et al., A numerical procedure and unified formulation for the adjoint approach in hyperbolic PDE-constrained optimal control problems, Arxiv.Org, 2017 1711.09297v2, math.OC.
The present paper aims at providing a numerical strategy to deal with PDE-constrained optimization problems solved with the adjoint method. It is done through out a unified formulation of the constraint PDE and the adjoint model. The resulting model is a non-conservative hyperbolic system and thus a finite volume scheme is proposed to solve it. In this form, the scheme sets in a single frame both constraint PDE and adjoint model. The forward and backward evolutions are controlled by a single parameter η and a stable time step is obtained only once at each optimization iteration. The methodology requires the complete eigenstructure of the system as well as the gradient of the cost functional. Numerical tests evidence the applicability of the present technique
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C Othmer et al., Implementation of a continuous adjoint for topology optimization of ducted flows, Arc.Aiaa.Org, 2007.
Topology optimization of fluid dynamical systems is still in its infancy, with its first academic realizations dating back to just four years ago. In this paper, we present an approach to fluid dynamic topology optimization that is based on a continuous adjoint. We briefly introduce the theory underlying the computation of topological sensitivity maps, discuss our implementation of this methodology into the professional CFD solver OpenFOAM and present results obtained for the optimization of an airduct manifold wrt. dissipated power.
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R E Plessix, A review of the adjoint-state method for computing the gradient of a functional with geophysical applications, Geophysical Journal International, 167 (2006) 495-503.
Estimating the model parameters from measured data generally consists of minimizing an error functional. A classic technique to solve a minimization problem is to successively determine the minimum of a series of linearized problems. This formulation requires the Fre ́chet derivatives (the Jacobian matrix), which can be expensive to compute. If the minimization is viewed as a non-linear optimization problem, only the gradient of the error functional is needed. This gradient can be computed without the Fre ́chet derivatives. In the 1970s, the adjoint-state method was developed to efficiently compute the gradient. It is now a well-known method in the numerical community for computing the gradient of a functional with respect to the model parameters when this functional depends on those model parameters through state variables, which are solutions of the forward problem. However, this method is less well understood in the geophysical community. The goal of this paper is to review the adjoint-state method. The idea is to define some adjoint-state variables that are solutions of a linear system. The adjoint- state variables are independent of the model parameter perturbations and in a way gather the perturbations with respect to the state variables. The adjoint-state method is efficient because only one extra linear system needs to be solved.
Several applications are presented. When applied to the computation of the derivatives of the ray trajectories, the link with the propagator of the perturbed ray equation is established.
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Radu Serban, Sensitivity Capabilities in SUNDIALS, , 2003 pp. 1-20.
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Radu Serban and AC Hindmarsh, CVODES: An ODE solver with sensitivity analysis capabilities, Proceedings Of The 5Th International Conference On Multibody Systems Nonlinear Dynamics, And, Control Long Beach, Ca, 2005.
CVODES, which is part of the SUNDIALS software suite, is a stiff and nonstiff ordinary differential equation initial value problem solver with sensitivity analysis capabilities. CVODES is written in a data-independent manner, with a highly modular structure to allow incorporation of different preconditioning and/or linear solver methods. It shares with the other SUNDIALS solvers several common modules, most notably the generic kernel of vector operations and a set of generic linear solvers and preconditioners.
CVODES solves the IVP by one of two methods – backward differentiation formula or Adams- Moulton – both implemented in a variable-step, variable-order form. The forward sensitivity module in CVODES implements the simultaneous corrector method, as well as two flavors of staggered corrector methods. Its adjoint sensitivity module provides a combination of checkpoint- ing and cubic Hermite interpolation for the efficient generation of the forward solution during the adjoint system integration.
We describe the current capabilities of CVODES, its design principles, and connection to the SUNDIALS suite, and the user interface. Finally, we mention current and future development efforts for CVODES, particularly in the direction of automatic generation of the sensitivity right- hand sides using automatic differentiation and/or complex-step techniques.
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Jelena Stefanović and Constantinos C Pantelides, Molecular Dynamics as a Mathematical Mapping. II. Partial Derivatives in the Microcanonical Ensemble, Molecular Simulation, 26 (2006) 167-192.
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Markus Towara, Discrete Adjoint Optimization with OpenFOAM, , .
Computer simulations and computer aided design in the past decades have evolved into a valuable instrument, penetrating just about every branch of engineering in industry and academia. More specifically, computational fluid dynamics (CFD) simulations allow to inspect flow phenomena in a variety of applications. As simulation methods evolve, mature, and are adopted by a rising number of users, the demand for methods which not only predict the result of a specific configuration, but can give indications on how to improve the design, increases.
This thesis is concerned with the efficient calculation of sensitivity information of CFD al- gorithms, and their application to numerical optimization. The sensitivities are obtained by applying Algorithmic Differentiation (AD).
A specific emphasis of this thesis is placed on the efficient application of adjoint methods, includ- ing parallelism, for commonly used CFD finite volume methods (FVM) and their implementation in the open source framework OpenFOAM
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JP Webb, Using adjoint solutions to estimate errors in global quantities, Ieee Transactions On Magnetics, 41 (2005) 1728-1731.
Adjoint solutions are widely used in computational electromagnetics to provide sensitivities of global quantities to variation in design parameters. They can also be used to provide estimates of the discretization error in these quantities. Electrostatic force and capacitance are considered. Results are obtained for two test problems, using different meshes and different polynomial orders of hierarchal, tetra- hedral finite elements. The estimates track the true errors well over a wide range. Furthermore, the estimates are good enough that, when added to the computed quantities, they reduce the error in those quantities, often substantially.
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Q Xu, Generalized adjoint for physical processes with parameterized discontinuities. Part IV: Problems in time discretization, Journal Of The Atmospheric Sciences, 54 (1997) 2722-2728.
It is shown analytically and graphically that when parameterized on/off switches are triggered at discrete time levels by a threshold condition in a numerical model, the model solution is not continuously dependent on the initial state. Consequently, the response function and costfunction contain small zigzag discontinuities; their gradients contain delta functions and thus are not good approximations of the original continuous gradients. The problem is caused by the traditional time discretization and cannot be solved by the conventional treatment of on/off switches. To solve the problem, the traditional time discretization is modified with the switch time determined by interpolation as a continuous function of the initial state. With this modification, the response function and costfunction become continuous in the space of the initial state and their gradients can be accurately computed by the generalized adjoint.
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