Adjoint Equation Method Research Articles

Parameter Identification of Elastic Modulus of Rock Based on Blasting Waves in Tunnel Excavation

The objective of this research is to present an identification method for elastic moduli of ground rock, through the first-order adjoint equation method using the measurements of the blasting vibration in tunnel excavation. Parameter identification is a minimization problem of the square sum of discrepancy between the computed and observed velocities. For the identification of these parameters, the magnitudes of the blasting force should be identified beforehand. In this study, propagation of an elastic wave is assumed because the amplitude of such a wave is infinitesimal. After the identification of the blasting force, the elastic moduli of three layers are identified simultaneously. We assume that the damping of vibration is linear. By applying the identification technique at the Ohyorogi tunnel site, we verify that the method is useful for tunnel excavation. Using measured data from actual tunnel excavation sites, the numerical identification method presented in this paper is shown to be useful for practical tunnel excavation.

Optimal Control of Shallow Water Flows Using Adjoint Equation Method

This paper presents a method to control a flow behavior using the first-order adjoint equation method. A flood causes large-scale and extensive damage to the human property. It is expected that the damage can be suppressed to minimum if the water level or velocity of the river can be controlled. Therefore, the optimal control of a water flow is carried out in this study. In the control theory, the performance function which is defined by the square sum of the discrepancy between the computed and the objective water elevation at the target points is used. The extended performance function is given by the performance function and the state equation. The first-order adjoint equation can be derived by the condition that the first variations of the extended performance function and constraint condition are zero. The gradient of the extended performance function is obtained by solving the first-order adjoint equation. As the minimization technique, the weighted gradient method is applied. The shallow water equation based on the water velocity and elevation is used as a state equation. As the spatial discretization, the stabilized bubble function is employed. As the temporal discretization, the Crank-Nicolson method is applied. In numerical studies, the optimal control of water elevation in the Ikari dam lake is carried out. In this research, the optimal water velocity sent by the pump to minimize the water rise in the Ikari dam lake is computed. The inflow boundary conditions are presupposed as sinusoidal wave water elevation. The results of optimal control is performed effectively.

Variational data assimilation: mathematical foundations and one application

Variational Data Assimilation and Adjoint Equation Method are presented here as a general methodology designed to improve the quality of computational simulation when are given the dynamics and a set of observation of the system under study. The mathematical foundations the procedures to obtain the adjoint of a given computational program, a fundamental task in order to apply the methodology, are carefully examined.

Shape optimization of a body located in incompressible flow using adjoint method and partial control algorithm

The purpose of this study is to obtain an optimal shape of a body located in an incompressible viscous flow. The optimal shape of the body is defined so as to minimize the fluid forces acting on it by determining the surface coordinates based on the finite element method and the optimal control theory. The performance function, which is used to judge the optimality of a shape, is defined as the square sum of the drag and lift forces. The minimization problem is solved using an adjoint equation method. The gradient in the adjoint equation is affected by the finite element configuration. The use of a finite element mesh whose shape is appropriate for the procedure is important in shape optimization. If the finite element mesh used is not suitable for computations, the exact gradient is not calculated. Therefore, a structured mesh is used for the adjacent area of the body and all finite element meshes are refined using the Delaunay triangulation at each iteration computation. The weighted gradient method is applied as the minimization technique. Using an algorithm in which all nodal coordinates on the surface of the body are employed and starting from a circle as an initial shape, a front-edged and rear-round shape is obtained because of the vortices at the back of the body. To overcome this difficulty, we introduced the partial control algorithm, in which some of the nodal coordinates on the surface of the body are updated. From four cases of computational studies, we reveal that the optimal shape has both sharp front and sharp rear edges. All computations are conducted at Reynolds number Re=250. The minimum value of the performance function is obtained. Copyright © 2010 John Wiley & Sons, Ltd.

Examination for adjoint boundary conditions in initial water elevation estimation problems

AbstractI present here a method of generating a distribution of initial water elevation by employing the adjoint equation and finite element methods. A shallow‐water equation is employed to simulate flow behavior. The adjoint equation method is utilized to obtain a distribution of initial water elevation for the observed water elevation. The finite element method, using the stabilized bubble function element, is used for spatial discretization, and the Crank–Nicolson method is used for temporal discretizations. In addition to a method for optimally assimilating water elevation, a method is presented for determining adjoint boundary conditions. An examination using the observation data including noise data is also carried out. Copyright © 2009 John Wiley & Sons, Ltd.

Parameter identification method for determination of elastic modulus of rock based on adjoint equation and blasting wave measurements

AbstractThe purpose of this paper is to present a parameter identification method to determine the force of a blast and the elastic modulus of the ground using the measurements of a dynamic elastic wave, the adjoint equation method of optimal control theory, and the finite element method. Before the excavation of rocky ground, it is important to estimate the ground properties. In this paper, the elastic modulus is determined as the performance function is minimized using a technique based on the first‐order adjoint method. The performance function is a square sum of the discrepancies between the computed and the observed values of the velocities. After the determination of the magnitude of the blasting force, we can determine the elastic modulus of the rock. As the basic equation to calculate the velocities of dynamic elastic body, elastic equilibrium equations with linear viscosity are employed. The adjoint equation method has been utilized in order to calculate the gradient of the performance function with respect to the parameters. The gradient of the performance function is calculated using the first‐order adjoint equation. The weighted gradient method is applied for minimization. In order to solve the state equations in space and time, the finite element method and the Newmark \documentclass{article}\usepackage{amssymb}\footskip=0pc\pagestyle{empty}\begin{document}$\frac{1}{4}$\end{document} method are used. In this paper, we tested the practical application of our proposed method for determination of the elastic modulus of rock at the Ikawa tunnel located in the Tokushima prefecture, Japan. Copyright © 2009 John Wiley & Sons, Ltd.

An analysis of 4D variational data assimilation and its application

In this paper, we give an analysis and a general procedure for 4D variational data assimilation (4D-Var). In functional partial differential equation setting, the adjoint equation method, sensitivity analysis, and multicomponent operator splitting are discussed. Nonlinear optimization methods and convergence analysis are also investigated for 4D-Var.

Examinations for terminal condition of Lagrange multiplier for heat transfer control problems

AbstractThis paper presents a method to express the terminal condition of Lagrange multiplier by the solution of the steady adjoint equation. In the control problem based on the adjoint equation method, the terminal condition of Lagrange multiplier has been set zero to satisfy the stationary condition of the extended performance function. Therefore, there is a possibility that the control value near the terminal time cannot be appropriately expressed. To solve this problem, a method to express the terminal condition of Lagrange multiplier by the solution of the steady adjoint equation is proposed and examined. Finally, to confirm the propriety of the proposed method, this method is applied to the sequential control problem, and some remarks are made. Copyright © 2007 John Wiley & Sons, Ltd.

Parameter estimation in convection dominated nonlinear convection–diffusion problems by the relaxation method and the adjoint equation

The development of numerical methods for strongly nonlinear convection–diffusion problems with dominant convection is an ongoing topic in numerical analysis. For inverse problems in this setting, there is a need of fast and accurate solvers. Here, we present operator splitting with a Riemann solver for the convective part and a relaxation method for the diffusive part, as a means to achieve this goal. Combined with the adjoint equation method this allows us to solve inverse problems within reasonable time frames and with modest computing power. As an example, the dual-well experiment is considered and the adjoint method is compared with a conjugate gradient algorithm and a Levenberg–Marquardt type of iteration method.

On the adjoint solution of the quasi‐1D Euler equations: the effect of boundary conditions and the numerical flux function

AbstractThis work compares a numerical and analytical adjoint equation method with respect to boundary condition treatments applied to the quasi‐1D Euler equations. The effect of strong and weak boundary conditions and the effect of flux evaluators on the numerical adjoint solution near the boundaries are discussed. Copyright © 2005 John Wiley & Sons, Ltd.

Approach for optimal power flow with transient stability constraints

Optimal power flow with transient stability (OTS) is a nonlinear optimisation problem in functional space that can be transformed to a static optimisation problem via constraint transcription techniques. However, the solution of the transformed OTS by ordinary numerical methods involves a massive calculation due to the stability constraints. Hence, the authors propose a penalty-based approach for OTS, in which the adjoint equation method is applied to evaluate the gradient of the penalty term associated with the stability constraints. It is demonstrated that the implementation of the adjoint equation method significantly reduces the computational burden in solving the OTS. A numerical test on a power system using prototype code is presented.

Optimal control of unsteady compressible viscous flows

AbstractThe control of complex, unsteady flows is a pacing technology for advances in fluid mechanics. Recently, optimal control theory has become popular as a means of predicting best case controls that can guide the design of practical flow control systems. However, most of the prior work in this area has focused on incompressible flow which precludes many of the important physical flow phenomena that must be controlled in practice including the coupling of fluid dynamics, acoustics, and heat transfer. This paper presents the formulation and numerical solution of a class of optimal boundary control problems governed by the unsteady two‐dimensional compressible Navier–Stokes equations. Fundamental issues including the choice of the control space and the associated regularization term in the objective function, as well as issues in the gradient computation via the adjoint equation method are discussed. Numerical results are presented for a model problem consisting of two counter‐rotating viscous vortices above an infinite wall which, due to the self‐induced velocity field, propagate downward and interact with the wall. The wall boundary control is the temporal and spatial distribution of wall‐normal velocity. Optimal controls for objective functions that target kinetic energy, heat transfer, and wall shear stress are presented along with the influence of control regularization for each case. Copyright © 2002 John Wiley & Sons, Ltd.

Splitting-up method and adjoint equation method in the ocean dynamics problem

Splitting-up method and adjoint equation method in the ocean dynamics problem

Approximate sensitivities for the electromagnetic inverse problem

SUMMARY We present an approximate method for generating the Jacobian matrix of sensitivities required by linearized, iterative procedures for inverting electromagnetic measurements. The approximation is based on the adjoint-equation method in which the sensitivities are obtained by integrating, over each cell, the scalar product of an adjoint electric field (the adjoint Green's function) with the electric field produced by the forward modelling at the end of the preceding iteration. Instead of computing the adjoint field in the multidimensional conductivity model, we compute an approximate adjoint field, either in a homogeneous or layered half-space. Such an approximate adjoint field is significantly faster to compute than the true adjoint field. This leads to a considerable reduction in computation time over the exact sensitivities. The speed-up can be one or two orders of magnitude, with the relative difference increasing with the size of the problem. Sensitivities calculated using the approximate adjoint field appear to be good approximations to the exact sensitivities. This is verified by comparing true and approximate sensitivities for 2- and 3-D conductivity models, and for sources that are both finite and infinite in extent. The approximation is sufficiently accurate to allow an iterative inversion algorithm to converge to the desired result, and we illustrate this by inverting magnetotelluric data to recover a 2-D conductivity structure. Our approximate sensitivities should enable larger inverse problems to be solved than is currently feasible using exact sensitivities and present-day computing power.

Design sensitivity analysis for transient eddy current problems using finite element discretization and adjoint variable method

For shape design problems subjected to the transient eddy current equation, a shape design sensitivity expressed explicitly in terms of design variables is derived using a discrete system equation of finite elements and an adjoint equation method. The original state equation is an initial-value problem which is to be solved by time-stepping finite element method. On the other hand, the adjoint equation is obtained as a terminal-value problem which is also to be solved by time-stepping finite element method. With the state and adjoint variables solved, the sensitivity is evaluated, which is employed in a gradient-based optimization algorithm. As a numerical example, an eddy current distribution control on a metal surface is treated using the proposed method in an induction heating system.

A proposed quasi-Newton method for parameter identification in a flow and transport system

A quasi-Newton algorithm allied with the adjoint state equation method is proposed to solve the inverse problem of parameter estimation. The proposed method utilizes the information from both groundwater flow and solute transport in parameter estimation. It is capable of identifying parameters which control both flow and transport simultaneously. It also avoids the calculation of sensitivity coefficients, which normally requires much computational effort. The BFGS (Broyden, 2 Fletcher, 10 Goldfarb, 13 and Shanno 21) formula is used to approximate the Hessian matrix required in the optimization procedure. The proposed inverse procedure is applied to a two-dimensional flow-transport problem. Two dimensionless numbers, the Peclet number and the Damkoehler number, are used to characterize the behavior of the transport processes. Numerical experiments which were conducted demonstrate the usefulness of the proposed methodology and show good results in practical application.

The Algorithms of Accuracy Research of Nonstationary Linear Systems with Continuous and Discrete Elements

This paper describes tlie algorithms for effective calculation of statistical characteristics of linear nonstationary composite systems. First the adjoint equation method for calculation of weight functions is developed for such systems with nonhomo-geneous means of description (vectors and matrices of different dimension, different types of equations etc.), in particular for the systems with continuous and discrete elements. Then the exact structure of the shaping filter method and the new forming equation method axe suggested for nonstationary regimes. By means of these methods a covariation matrix of the system can be determined by simultaneous integration of certain equations with weight matrix equations.The different procedures are formed to determine covariation matrix for fixed and running time.