A linear solver based on algebraic multigrid and defect correction for the solution of adjoint RANS equations

A new solution approach for discrete adjoint Reynolds‐averaged Navier–Stokes equations is suggested. The approach is based on a combination of algebraic multigrid and defect correction. The efficient interplay of these two methods results in a fast and robust solution technique. Algebraic multigrid deals very well with unstructured grids, and defect correction completes it in such a way that a required second‐order accuracy of the solution is achieved. The performance of the suggested solution approach is demonstrated on a number of representative benchmarks.Copyright © 2014 John Wiley & Sons, Ltd.     

Use of structural components of specific work of internal forces for estimating the strength of viscoelastic structures in local stress concentration regions

Material fracture experiments on specimens and structures testify that materials can resist greater stresses in local stress concentration regions than in regions with a nearly homogeneous stress state. Taking this fact into account in design stress analysis permits one to reveal additional structure loading and/or service life margins. One approach aimed at taking into account the increased strength in local stress concentration regions is to use averaged limit characteristics parametrically depending on the characteristic size L of the averaging region. One version of this approach is the concept of “elementary block” of a material [1, 2]. The averaged limit characteristics are determined by an experiment-calculation method involving the analysis of the stress-strain state of a material specimen with a stress concentrator at the time when the specimen attains the limit state preceding macrofracture.In [3], the dependence of the averaged limit separation stresses on the size of the averaging region was determined on the basis of numerical analysis of the singular stress state of the specimen used to determine the standard characteristics of the adhesion strength of a filled polymer material. In the present paper, we generalize the above approach to the case of a viscoelastic material. For the limit characteristics of the material in the local stress concentration region we take the volume-averaged components of the specific work of internal forces [4, 5] (the averaged specific absorbed energy and the averaged specific instantaneously reversible energy). The introduction of two limit energies originates from the fact that, to initiate the process of macrofracture, it is necessary to satisfy the following two conditions simultaneously: the material must be “damaged” sufficiently strongly by the preceding loading, and the “damaged” material must be loaded sufficiently strongly. As an example of determining the material averaged limit energy characteristics in a local stress concentration region, we consider the problem about the strain of a viscoelastic specimen used to determine the standard adhesion strength characteristics. The problem is solved numerically under the following assumptions: the specimen material is assumed to be linearly viscoelastic, and the specific absorbed energy in the stress concentration region is assumed to coincide in magnitude with the specific scattered energy. To estimate the accuracy of the numerical method, we use the solution of the model problem about the action of a plane circular die on a half-space consisting of a linearly viscoelastic incompressible material.     

Stochastic Stabilization of Quasi-Partially Integrable Hamiltonian Systems by Using Lyapunov Exponent

A procedure for designing a feedback control to asymptoticallystabilize, with probability one, a quasi-partially integrableHamiltonian system is proposed. First, the averaged stochasticdifferential equations for controlled r first integrals are derived fromthe equations of motion of a given system by using the stochasticaveraging method for quasi-partially integrable Hamiltonian systems.Second, a dynamical programming equation for the ergodic control problemof the averaged system with undetermined cost function is establishedbased on the dynamical programming principle. The optimal control law isderived from minimizing the dynamical programming equation with respectto control. Third, the asymptotic stability with probability one of theoptimally controlled system is analyzed by evaluating the maximalLyapunov exponent of the completely averaged Itô equations for the rfirst integrals. Finally, the cost function and optimal control forces aredetermined by the requirements of stabilizing the system. An example isworked out in detail to illustrate the application of the proposedprocedure and the effect of optimal control on the stability of thesystem.     

Optimal bounded control for minimizing the response of quasi-integrable Hamiltonian systems

A procedure for designing optimal bounded control to minimize the response of quasi-integrable Hamiltonian systems is proposed based on the stochastic averaging method for quasi-integrable Hamiltonian systems and the stochastic dynamical programming principle. The equations of motion of a controlled quasi-integrable Hamiltonian system are first reduced to a set of partially completed averaged Itô stochastic differential equations by using the stochastic averaging method for quasi-integrable Hamiltonian systems. Then, the dynamical programming equation for the control problems of minimizing the response of the averaged system is formulated based on the dynamical programming principle. The optimal control law is derived from the dynamical programming equation and control constraints without solving the dynamical programming equation. The response of optimally controlled systems is predicted through solving the Fokker-Planck-Kolmogrov equation associated with fully completed averaged Itô equations. Finally, two examples are worked out in detail to illustrate the application and effectiveness of the proposed control strategy.     

Optimal Bounded Control for Minimizing the Response of Quasi Non-Integrable Hamiltonian Systems

A strategy for designing optimal bounded control to minimize theresponse of quasi non-integrable Hamiltonian systems is proposed basedon the stochastic averaging method for quasi non-integrable Hamiltoniansystems and the stochastic dynamical programming principle. Theequations of motion of a controlled quasi non-integrable Hamiltoniansystem are first reduced to an one-dimensional averaged Itô stochasticdifferential equation for the Hamiltonian by using the stochasticaveraging method for quasi non-integrable Hamiltonian systems. Then, thedynamical programming equation for the control problem of minimizing theresponse of the averaged system is formulated based on the dynamicalprogramming principle. The optimal control law is derived from thedynamical programming equation and control constraints without solvingthe equation. The response of optimally controlled systems is predictedthrough solving the Fokker–Planck–Kolmogrov (FPK) equation associatedwith completely averaged Itô equation. Finally, two examples are workedout in detail to illustrate the application and effectiveness of theproposed control strategy.     

Stochastic fractional optimal control of quasi-integrable Hamiltonian system with fractional derivative damping

A stochastic fractional optimal control strategy for quasi-integrable Hamiltonian systems with fractional derivative damping is proposed. First, equations of the controlled system are reduced to a set of partially averaged It $\hat{o}$ stochastic differential equations for the energy processes by applying the stochastic averaging method for quasi-integrable Hamiltonian systems and a stochastic fractional optimal control problem (FOCP) of the partially averaged system for quasi-integrable Hamiltonian system with fractional derivative damping is formulated. Then the dynamical programming equation for the ergodic control of the partially averaged system is established by using the stochastic dynamical programming principle and solved to yield the fractional optimal control law. Finally, an example is given to illustrate the application and effectiveness of the proposed control design procedure.     

Averaging of stochastic evolution equations of transport in porous media

A study is made of the problem of averaging the simplest one-dimensional evolution equations of stochastic transport in a porous medium. A number of exact functional equations corresponding to distributions of the random parameters of a special form is obtained. In some cases, the functional equations can be localized and reduced to differential equations of fairly high order. The first part of the paper (Secs. 1–6) considers the process of transport of a neutral admixture in porous media. The functional approach and technique for decoupling the correlations explained by Klyatskin [4] is used. The second part of the paper studies the process of transport in porous media of two immiscible incompressible fluids in the framework of the Buckley—Leverett model. A linear equation is obtained for the joint probability density of the solution of the stochastic quasilinear transport equation and its derivative. An infinite chain of equations for the moments of the solution is obtained. A scheme of approximate closure is proposed, and the solution of the approximate equations for the mean concentration is compared with the exactly averaged concentration.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 127–136, September–October, 1985.We are grateful to A. I. Shnirel'man for pointing out the possibility of obtaining an averaged equation in the case of a velocity distribution in accordance with a Cauchy law.     

Excursion probabilities of non-linear systems

This paper suggests a procedure for estimating excursion probabilities for linear and non-linear systems subjected to Gaussian excitation processes. In this paper, the focus is on non-linear systems which might also have stochastic properties. The approach is based on the so-called “averaged excursion probability flow” which allows for a simple solution for the interaction in excursion problems. Simplifying, the dynamic reliability problem can be reduced to a simpler “static” problem by considering the probability flow at fixed time instances. The proposed approach is very general and can be applied to both linear and non-linear systems of which the response can be determined by deterministic methods. Hence, the procedure applies to arbitrary structures and any suitable mathematical model including large FE-models solved by deterministic FE-codes.     

Time-delayed stochastic optimal control of strongly non-linear systems with actuator saturation by using stochastic maximum principle

A time-delayed stochastic optimal bounded control strategy for strongly non-linear systems under wide-band random excitations with actuator saturation is proposed based on the stochastic averaging method and the stochastic maximum principle. First, the partially averaged Itô equation for the system amplitude is derived by using the stochastic averaging method for strongly non-linear systems. The time-delayed feedback control force is approximated by a control force without time delay based on the periodically random behavior of the displacement and velocity of the system. The partially averaged Itô equation for the system energy is derived from that for the system amplitude by using Itô formula and the relation between system amplitude and system energy. Then, the adjoint equation and maximum condition of the partially averaged control problem are derived based on the stochastic maximum principle. The saturated optimal control force is determined from maximum condition and solving the forward–backward stochastic differential equations (FBSDEs). For infinite time-interval ergodic control, the adjoint variable is stationary process and the FBSDE is reduced to a ordinary differential equation. Finally, the stationary probability density of the Hamiltonian and other response statistics of optimally controlled system are obtained from solving the Fokker–Plank–Kolmogorov (FPK) equation associated with the fully averaged Itô equation of the controlled system. For comparison, the optimal control forces obtained from the time-delayed bang–bang control and the control without considering time delay are also presented. An example is worked out to illustrate the proposed procedure and its advantages.     

Modeling the acidizing of a carbonate formation

Two approaches to the determination of the optimum conditions of the well bottom zone treatment in a carbonate formation are considered and compared. The first, bench approach is based on the investigation of the process of wormhole formation in rock cores, while the second approach is the mathematical modeling of the process within the framework of mechanics of multi-component seepage flows using averaged values of the reservoir porosity and permeability. A mathematical model of the process, that takes account for fluid flow and the chemical reaction between the acid and the rock matrix proceeding in the kinetic regime, is presented. Both approaches indicate the existence of an optimum rate of acid solution injection into the formation at a certain given value of the dimensionless Damköller number.     

The analysis of the square array of van der pol oscillators using the averaged potential

This paper describes an approach to finding stable oscillations in van der Pol oscillators with many degrees of freedom. We summarize a new concept of the “averaged potential” which is derived from “mixed potential” defined by Brayton and Moser. The averaged potential is the time average of the losses (dissipation function) in the system. It is shown that the averaged equations of a system take the form of the gradient system of the averaged potential. Hence, the stable oscillations of this system correspond to the minimal points of the averaged potential. Therefore, finding the stable oscillations is reduced to constructing the averaged potential and finding its minimal points. This method is successfully applied to the analysis of a square array of van der Pol oscillators coupled by inductors. It is shown that the triple and quadruple mode oscillations can be stably excited as well as simple and double mode oscillations.     

Stochastic optimal bounded control of MDOF quasi nonintegrable-Hamiltonian systems with actuator saturation

A new bounded optimal control strategy for multi-degree-of-freedom (MDOF) quasi nonintegrable-Hamiltonian systems with actuator saturation is proposed. First, an n-degree-of-freedom (n-DOF) controlled quasi nonintegrable-Hamiltonian system is reduced to a partially averaged Itô stochastic differential equation by using the stochastic averaging method for quasi nonintegrable-Hamiltonian systems. Then, a dynamical programming equation is established by using the stochastic dynamical programming principle, from which the optimal control law consisting of optimal unbounded control and bang–bang control is derived. Finally, the response of the optimally controlled system is predicted by solving the Fokker–Planck–Kolmogorov (FPK) equation associated with the fully averaged Itô equation. An example of two controlled nonlinearly coupled Duffing oscillators is worked out in detail. Numerical results show that the proposed control strategy has high control effectiveness and efficiency and that chattering is reduced significantly compared with the bang–bang control strategy.     

Effective equations for two-phase flow in porous media: the effect of trapping on the microscale

In this article, we consider a two-phase flow model in a heterogeneous porous column. The medium consists of many homogeneous layers that are perpendicular to the flow direction and have a periodic structure resulting in a one-dimensional flow. Trapping may occur at the interface between a coarse and a fine layer. Assuming that capillary effects caused by the surface tension are in balance with the viscous effects, we apply the homogenization approach to derive an effective (upscaled) model. Numerical experiments show a good agreement between the effective solution and the averaged solution taking into account the detailed microstructure.     

VERTICAL DISCRETIZATION IN TIDAL FLOW SIMULATIONS

We propose an empirical law for vertical nodal placement in tidal simulations that depends on a single parameter p. The influence of dimensionless numbers on the optimal value of pis analysed through a series of numerical experiments for an individual vertical and a single value of pis found to be adequate for all cases. The proposed law can lead to gains in accuracy of over two orders of magnitude relative to a uniform grid and compares favourably with non-uniform grids previously used in the literature. In practical applications the most effective use of this law may require each vertical to have a different number of nodes. Criteria for the distribution of the total number of nodes among different verticals are also proposed, based on the concept of equalizing errors across the domain. The usefulness of the overall approach is demonstrated through a two-dimensional laterally averaged application to a synthetic estuary.     

Optimal control of film casting processes

We present an optimal control approach for the isothermal film casting process with free surfaces described by averaged Navier–Stokes equations. We control the thickness of the film at the take‐up point using the shape of the nozzle and the initial thickness. The control goal consists in finding an even thickness profile. To achieve this goal, we minimize an appropriate cost functional. The resulting minimization problem is solved numerically by a steepest descent method. The gradient of the cost functional is approximated using the adjoint variables of the problem with fixed film width. Numerical simulations show the applicability of the proposed method. Copyright © 2008 John Wiley & Sons, Ltd.     

NUMERICAL SOLUTIONS OF OPTIMAL CONTROL FOR THERMALLY CONVECTIVE FLOWS

We study the numerical solution of optimal control problems associated with two-dimensional viscous incompressible thermally convective flows. Although the techniques apply to more general settings, the presentation is confined to the objectives of minimizing the vorticity in the steady state case and tracking the velocity field in the non-stationary case with boundary temperature controls. In the steady state case we develop a systematic way to use the Lagrange multiplier rules to derive an optimality system of equations from which an optimal solution can be computed; finite element methods are used to find approximate solutions for the optimality system of equations. In the time-dependent case a piecewise-in-time optimal control approach is proposed and the fully discrete approximation algorithm for solving the piecewise optimal control problem is defined. Numerical results are presented for both the steady state and time-dependent optimal control problems. © 1997 John Wiley & Sons, Ltd.     

Nonlinear robust and optimal control of robot manipulators

In this paper, we propose a new optimal control method for robust control of nonlinear robot manipulators. Many industrial robot systems are required to perform relatively large angular movement with sufficient accuracy. In real circumstances, highly nonlinear manipulator dynamics and uncertainties such as unknown load placed on the manipulator, external disturbance, and joint friction make the precise control of manipulators a very challenging task. The main contribution of this work is to develop a new robust control strategy to accomplish the precise control of robot manipulators under load uncertainty using a nonlinear optimal control formulation and solution. This methodology is based on the underlying relation between the robust stability and performance optimality. A class of robust control problems can be transformed to an equivalent optimal control problem by incorporating the uncertainty bounds into the cost functional. The θ-D optimal control approach is utilized to find an approximate closed-form feedback solution to the resultant nonlinear optimal control problem via a perturbation process. Numerical simulations show that the proposed robust controller is able to control the robot manipulator precisely under large load variations.     

A semi‐implicit finite element model for non‐hydrostatic (dispersive) surface waves

The objective of this research is to develop a model that will adequately simulate the dynamics of tsunami propagating across the continental shelf. In practical terms, a large spatial domain with high resolution is required so that source areas and runup areas are adequately resolved. Hence efficiency of the model is a major issue. The three‐dimensional Reynolds averaged Navier–Stokes equations are depth‐averaged to yield a set of equations that are similar to the shallow water equations but retain the non‐hydrostatic pressure terms. This approach differs from the development of the Boussinesq equations where pressure is eliminated in favour of high‐order velocity and geometry terms. The model gives good results for several test problems including an oscillating basin, propagation of a solitary wave, and a wave transformation over a bar. The hydrostatic and non‐hydrostatic versions of the model are compared for a large‐scale problem where a fault rupture generates a tsunami on the New Zealand continental shelf. The model efficiency is also very good and execution times are about a factor of 1.8 to 5 slower than the standard shallow water model, depending on problem size. Moreover, there are at least two methods to increase model accuracy when warranted: choosing a more optimal vertical interpolation function, and dividing the problem into layers. Copyright © 2005 John Wiley & Sons, Ltd.     

航天器最优受控绕飞轨迹推力幅值延拓设计方法

针对航天器燃料最优可变周期绕飞轨迹的求解问题, 提出了一种以推力幅值为延拓参数的延拓方法. 问题求解从最为简单的双脉冲绕飞轨迹出发, 首先利用有限推力替代脉冲推力, 设定推力序列为“开— 关— 开”,然后逐步减小推力幅值, 最终得到最小推力绕飞轨迹; 此后, 再逐步增加推力幅值, 结合主矢量曲线判断最优推力开关序列, 将最小推力解延拓至有限推力以及脉冲推力燃料最优解. 该方法通过对推力幅值的延拓, 实现了有限推力bang-bang 控制与脉冲推力燃料最优绕飞轨迹优化问题的一并求解, 同时避免了最优控制问题中协态变量的随机猜测. 慢速与快速绕飞算例的优化结果验证了所提出方法的有效性.      

Solving the fully nonlinear weakly dispersive Serre equations for flows over dry beds

We describe a numerical method for solving the Serre equations that can simulate flows over dry bathymetry. The method solves the Serre equations in conservation law form with a finite volume method. A finite element method is used to solve the auxiliary elliptic equation for the depth‐averaged horizontal velocity. The numerical method is validated against the lake at rest analytic solution, demonstrating that it is well‐balanced. Since there are currently no known nonstationary analytical solutions to the Serre equation that involve bathymetry, a nonstationary forced solution, involving bathymetry was developed. The method was further validated and its convergence rate established using the developed nonstationary forced solution containing the wetting and drying of bathymetry. Finally, the method is also validated against experimental results for the run‐up of a solitary wave on a sloped beach. The finite‐volume finite‐element approach to solving the Serre equation was found to be accurate and robust.