Adjoint Variables Research Articles

Adjoint sensitivity analysis of hybrid multibody dynamical systems

Sensitivity analysis computes the derivatives of general cost functions that depend on the system solution with respect to parameters or initial conditions. This work develops adjoint sensitivity analysis for hybrid multibody dynamic systems. The adjoint sensitivity is commonly referred to as backward propagation. Hybrid systems are characterized by trajectories that are piecewise continuous in time, with finitely-many discontinuities being caused by events such as elastic/inelastic impacts or sudden changes in constraints. The corresponding direct and adjoint sensitivity variables are also discontinuous at the time of events. The framework discussed herein uses a jump sensitivity matrix to relate the jump conditions for the direct and adjoint sensitivities before and after the time event and provides analytical jump equations for the adjoint variables. The theoretical framework for sensitivities for hybrid systems is verified on a five-bar mechanism with non-smooth contacts.

Optimal Control Model and Cost Effectiveness Analysis of Maize Streak Virus Pathogen Interaction with Pest Invasion in Maize Plant

ABSTRACT In this paper, we proposed and analyzed an optimal control deterministic model for the dynamics of Maize Streak Virus pathogen interaction with a pest invasion on Maize plant to examine the best strategy for maize streak disease elimination. The Pontryagin’s maximum principle is applied to derive and study the necessary conditions for optimal control strategies. We obtained the Hamiltonian, the adjoint variables, the characterization of the controls and the optimality system. For minimizing the spread of maize streak disease, three controls namely prevention , quarantine and chemical control were incorporated and their impacts were pictorially shown using numerical simulations. We investigated cost-effectiveness using ICER to determine the best strategy. We found that the combination of prevention and quarantine is the best effective strategies in terms of cost as well as health benefits.

Dynamic programming and feedback analysis of the two dimensional tidal dynamics system

In this work, we consider the controlled two dimensional tidal dynamics equations in bounded domains. A distributed optimal control problem is formulated as the minimization of a suitable cost functional subject to the controlled 2D tidal dynamics equations. The existence of an optimal control is shown and the dynamic programming method for the optimal control of 2D tidal dynamics system is also described. We show that the feedback control can be obtained from the solution of an infinite dimensional Hamilton-Jacobi equation. The non-differentiability and lack of smoothness of the value function forced us to use the method of viscosity solutions to obtain a solution of the infinite dimensional Hamilton-Jacobi equation. The Bellman principle of optimality for the value function is also obtained. We show that a viscosity solution to the Hamilton-Jacobi equation can be used to derive the Pontryagin maximum principle, which give us the first order necessary conditions of optimality. Finally, we characterize the optimal control using the adjoint variable.

Convergence and quasi-optimality of <inline-formula><tex-math id="M1">$ L^2- $</tex-math></inline-formula>norms based an adaptive finite element method for nonlinear optimal control problems

<p style='text-indent:20px;'>This paper aims at investigating the convergence and quasi- optimality of an adaptive finite element method for control constrained nonlinear elliptic optimal control problems. We derive a posteriori error estimation for both the control, the state and adjoint state variables under controlling by <inline-formula><tex-math id="M2">$ L^2 $</tex-math></inline-formula>-norms where bubble function is a wonderful tool to deal with the global lower error bound. Then a contraction is proved before the convergence is proposed. Furthermore, we find that if keeping the grids sufficiently mildly graded, we can prove the optimal convergence and the quasi-optimality for the adaptive finite element method. In addition, some numerical results are presented to verify our theoretical analysis.</p>

Optimal control of HIV treatment and immunotherapy combination with state and control delays

SummaryIn this paper, we propose and analyze an optimal control problem where human immunodeficiency virus treatment and immunotherapy are described by two control functions that are subject to time delays representing pharmacological and absorption delays, respectively. The goal is to propose effective optimal control solutions for the combination of human immunodeficiency virus treatment and immunotherapy, ensuring a functional behavior of the immune system. The incubation period is mathematically represented by a time delay in the virus load, and the local asymptotic and Hopf bifurcation analysis of the CTL‐equilibrium point of the uncontrolled delayed system is studied. We obtain optimal controls of bang‐singular type both for the nondelayed and delayed optimal control problem with and without state constraints. We study boundary arcs of state constraints and junction properties of the control and adjoint variables at entry and exit points of boundary arcs. Moreover, we derive an explicit formula of the multiplier associated with the state constraint.

Another view of the maximum principle for infinite-horizon optimal control problems in economics

The authors present their recently developed complete version of the Pontryagin maximum principle for a class of infinite-horizon optimal control problems arising in economics. The main distinguishing feature of the result is that the adjoint variable is explicitly specified by a formula analogous to the Cauchy formula for solutions of linear differential systems. In certain situations this formula implies the ‘standard’ transversality conditions at infinity. Moreover, it can serve as an alternative to them. Examples demonstrate the advantages of the proposed version of the maximum principle. In particular, its applications are considered to Halkin’s example, to Ramsey’s optimal economic growth model, and to a basic model for optimal extraction of a non-renewable resource. Also presented is an economic interpretation of the characterization obtained for the adjoint variable. Bibliography: 62 titles.

Topology optimization using a continuous-time high-cycle fatigue model

We propose a topology optimization method that includes high-cycle fatigue as a constraint. The fatigue model is based on a continuous-time approach where the evolution of damage in each point of the design domain is governed by a system of ordinary differential equations, which employs the concept of a moving endurance surface being a function of the stress and back stress. Development of fatigue damage only occurs when the stress state lies outside the endurance surface. The fatigue damage is integrated for a general loading history that may include non-proportional loading. Thus, the model avoids the use of a cycle-counting algorithm. For the global high-cycle fatigue constraint, an aggregation function is implemented, which approximates the maximum damage. We employ gradient-based optimization, and the fatigue sensitivities are determined using adjoint sensitivity analysis. With the continuous-time fatigue model, the damage is load history dependent and thus the adjoint variables are obtained by solving a terminal value problem. The capabilities of the presented approach are tested on several numerical examples with both proportional and non-proportional loads. The optimization problems are to minimize mass subject to a high-cycle fatigue constraint and to maximize the structural stiffness subject to a high-cycle fatigue constraint and a limited mass.

Simulator-based training of generative neural networks for the inverse design of metasurfaces

Abstract Metasurfaces are subwavelength-structured artificial media that can shape and localize electromagnetic waves in unique ways. The inverse design of these devices is a non-convex optimization problem in a high dimensional space, making global optimization a major challenge. We present a new type of population-based global optimization algorithm for metasurfaces that is enabled by the training of a generative neural network. The loss function used for backpropagation depends on the generated pattern layouts, their efficiencies, and efficiency gradients, which are calculated by the adjoint variables method using forward and adjoint electromagnetic simulations. We observe that the distribution of devices generated by the network continuously shifts towards high performance design space regions over the course of optimization. Upon training completion, the best generated devices have efficiencies comparable to or exceeding the best devices designed using standard topology optimization. Our proposed global optimization algorithm can generally apply to other gradient-based optimization problems in optics, mechanics, and electronics.

Entropy and Adjoint Methods

Aerodynamic drag can be partially approximated by the entropy flux across fluid domain boundaries with a formula due to Oswatitsch. In this paper, we build the adjoint solution that corresponds to this representation of the drag and investigate its relation to the entropy variables, which are linked to the integrated residual of the entropy transport equation. For inviscid isentropic flows, the resulting adjoint variables are identical to the entropy variables, an observation originally due to Fidkowski and Roe, while for non-isentropic flows there is a significant difference that is explicitly demonstrated with analytic solutions in the shocked quasi-1D case. Both approaches are also investigated for viscous and inviscid flows in two and three dimensions, where the adjoint equations and boundary conditions are derived. The application of both approaches to mesh adaptation is investigated, with especial emphasis on inviscid flows with shocks.

Sensitivity of the Rayleigh criterion in thermoacoustics

Thermoacoustic instabilities are one of the most challenging problems faced by gas turbine and rocket motor manufacturers. The key instability mechanism is described by the Rayleigh criterion. The Rayleigh criterion does not directly show how to alter a system to make it more stable. This is the objective of sensitivity analysis. Because thermoacoustic systems have many design parameters, adjoint sensitivity analysis has been proposed to obtain all the sensitivities with one extra calculation. Although adjoint sensitivity analysis can be carried out in both the time and the frequency domain, the frequency domain is more natural for a linear analysis. Perhaps surprisingly, the Rayleigh criterion has not yet been rigorously derived and comprehensively interpreted in the frequency domain. The contribution of this theoretical paper is threefold. First, the Rayleigh criterion is interpreted in the frequency domain with integral formulae for the complex eigenvalue. Second, the first variation of the Rayleigh criterion is calculated both in the time and frequency domain, both with and without Lagrange multipliers (adjoint variables). The Lagrange multipliers are physically related to the system’s observables. Third, an adjoint Rayleigh criterion is proposed. The paper also points out that the conclusions of Juniper (Phys. Rev. Fluids, vol. 3, 2018, 110509) apply to the first variation of the Rayleigh criterion, not to the Rayleigh criterion itself. The mathematical relations of this paper can be used to compute sensitivities directly from measurable quantities to enable optimal design.

Boundary control problem and optimality conditions for the Cahn–Hilliard equation with dynamic boundary conditions

This paper is concerned with a boundary control problem for the Cahn–Hilliard equation coupled with dynamic boundary conditions. In order to handle the control problem, we restrict our analysis to the case of regular potentials defined on the whole real line, assuming the boundary potential to be dominant. The existence of optimal control, the Fréchet differentiability of the control-to-state operator between appropriate Banach spaces, and the first-order necessary conditions for optimality are addressed. In particular, the necessary condition for optimality is characterised by a variational inequality involving the adjoint variables.

Optimal Control of a Phase Field System Modelling Tumor Growth with Chemotaxis and Singular Potentials

A distributed optimal control problem for an extended model of phase field type for tumor growth is addressed. In this model, the chemotaxis effects are also taken into account. The control is realized by two control variables that design the dispensation of some drugs to the patient. The cost functional is of tracking type, whereas the potential setting has been kept quite general in order to allow regular and singular potentials to be considered. In this direction, some relaxation terms have been introduced in the system. We show the well-posedness of the state system, the Fr\'echet differentiability of the control-to-state operator in a suitable functional analytic framework, and, lastly, we characterize the first-order necessary conditions of optimality in terms of a variational inequality involving the adjoint variables.

Time fractional (2+1)-dimensional Wu–Zhang system: Dispersion analysis, similarity reductions, conservation laws, and exact solutions

This paper examines the time fractional (2+1)-dimensional Wu–Zhang system describing dispersive long waves. The dispersion relation of waves has been obtained from linear analysis and used to find the phase and group velocity. The dispersion relation shows an anomalous dispersion of waves. Lie symmetries and the corresponding similarity reductions are carried out using Riemann–Liouville fractional derivative and the order of the system is reduced to lower dimension. Furthermore, the system is analysed for nonlinear self-adjointness and the conservation laws are obtained associated with infinitesimal symmetries by applying new conservation theorem. The adjoint variables are used to convert conservation laws into original dependent variables. The exponential rational function method is employed to find exact solutions revealing kink and bell type profiles. The effect of fractional order α on the wave profile of solutions is discussed graphically.

Flow over natural or engineered surfaces: an adjoint homogenization perspective

Natural and engineered surfaces are never smooth, but irregular, rough at different scales, compliant, possibly porous, liquid impregnated or superhydrophobic. The correct numerical modelling of fluid flowing through and around them is important but poses problems. For media characterized by a periodic or quasi-periodic microstructure of characteristic dimensions smaller than the relevant scales of the flow, multiscale homogenization can be used to study the effect of the surface, avoiding the numerical resolution of small details. Here, we revisit the homogenization strategy using adjoint variables to model the interaction between a fluid in motion and regularly micro-textured, permeable or impermeable walls. The approach described allows for the easy derivation of auxiliary/adjoint systems of equations which, after averaging, yield macroscopic tensorial properties, such as permeability, elasticity, slip, transpiration, etc. When the fluid in the neighbourhood of the microstructure is in the Stokes regime, classical results are recovered. Adjoint homogenization, however, permits simple extension of the analysis to the case in which the flow displays nonlinear effects. Then, the properties extracted from the auxiliary systems take the name of effective properties and do not depend only on the geometrical details of the medium, but also on the microscopic characteristics of the fluid motion. Examples are shown to demonstrate the usefulness of adjoint homogenization to extract effective tensor properties without the need forad hocparameters. In particular, notable results reported herein include:(i)an original formulation to describe filtration in porous media in the presence of inertial effects;(ii)the microscopic and macroscopic equations needed to characterize flows through poroelastic media;(iii)an extended Navier’s condition to be employed at the boundary between a fluid and an impermeable rough wall, with roughness elements which can be either rigid or linearly elastic;(iv)the microscopic problems needed to define the relevant parameters for a Saffman-like condition at the interface between a fluid and a porous substrate; and(v)the macroscopic equations which hold at the dividing surface between a free-fluid region and a fluid-saturated poroelastic domain.

Rate optimization of fractional flow reservoir model based on the continuous adjoint method

In this paper, continuous adjoint method is applied to research the rate optimization of fractional flow reservoir model. The continuous adjoint equations are derived by the variational theory and the stable numerical schemes of adjoint equations are obtained through numerical methods. Firstly, the continuous adjoint equations are linear equations and evolve from the end simulation time. The convection velocity direction of the continuous adjoint saturation equation is opposite to the convection direction of the saturation equation. Secondly, the total simulation time has significant impact on the adjoint variables and the adjoint gradient. After water breakthrough in production well, the adjoint variables exhibit spatial variety that gradually changes from production well to injection well. With this observation, we find that this variety is related to the fractional flow function and its derivative value. The sharp change of fractional flow function and its derivative at the end of the simulation time can lead to large fluctuation of the adjoint gradient along the time step. Thirdly, the influence of time interval on optimization results is studied. Increasing the number of time steps will increase the optimal value, but the value of the difference between optimal and initial value will not increase significantly. When the variable number of time steps is adopted, this difference decreases as the number of time steps increases, which indicates that the coarse time step optimization can provide a better initial control value for the fine time step optimization. An adaptive time step from coarse step to fine step is thus recommended for optimization.

Discrete adjoint of fractional-step incompressible Navier-Stokes solver in curvilinear coordinates and application to data assimilation

The discrete adjoint of an incompressible Navier-Stokes algorithm in generalized coordinates is derived and applied to estimate the states of saturated and turbulent circular Couette flows. The forward Navier-Stokes model is based on the fractional-step algorithm in curvilinear coordinates on a structured grid [1], which has been widely adopted in direct numerical simulations of transitional and turbulent flows. The discrete adjoint equations adopt the same stencil and temporal scheme as the forward discretization, and expressions are derived that relate the discrete adjoint variables to their continuous counterpart. The key ingredients of the forward algorithm can be retained in the adjoint, including the computation of the cell geometry, the approximate factorization method, and the parallelization strategy. The accuracy, efficiency, and stability of the adjoint solver are also commensurate with the forward model. In addition, a novel symmetric projector is proposed to guarantee that the outcome of the adjoint algorithm is divergence free. The implementation of the algorithm in double precision satisfies the forward-adjoint relation up to eight significant figures, and further validation is performed using circular Couette flow. The forward and adjoint growth rates of instability modes from linear theory are accurately reproduced. In addition, an adjoint-variational data-assimilation algorithm (4DVar) is adopted to reconstruct the initial condition of circular Couette flows from limited measurements, obtained from an independent simulation. When the flow is comprised of saturated wavy vortices, wall measurements are sufficient to reconstruct an initial condition that latches onto the target state after a short time. For the more challenging turbulent case, coarse-grained velocity data are used to estimate the initial condition.

Effects of Inclusion of Adjoint Sea Ice Rheology on Backward Sensitivity Evolution Examined Using an Adjoint Ocean–Sea Ice Model

AbstractAs part of the ongoing development of an ocean data assimilation system for operational ocean monitoring and seasonal prediction, an adjoint sea ice model was developed that incorporates sea ice rheology, which was omitted from previously developed adjoint models to avoid model instability. The newly developed adjoint model was merged with the existing system to construct a global ocean–sea ice adjoint model. A series of sensitivity experiments, in which idealized initial values were given for the adjoint sea ice area fraction and thickness, were conducted, with particular attention to the differences between the cases with free-drift approximation in the adjoint sea ice model as in previous studies and with full sea ice dynamics including rheology. The internal stress effects represented in the adjoint rheology induced remarkable differences in the evolution of the initialized and generated adjoint variables, such as for the sea ice velocity by O(102) in magnitude, which highlighted the importance of the adjoint rheology in the central Arctic Ocean. In addition, sensitivities with respect to the nonprognostic variables associated with the sea ice dynamics were obtained only through the adjoint rheology. These results suggested a potential for providing an improved global atmosphere–ocean–sea ice state estimation through a four-dimensional variational approach with the adjoint sea ice model as developed in this study.

A Fast Computational Method for the Minimum Duration Transfer Trajectories of Space-to-Ground Vehicles

On the conditions that the spacecraft engine is in finite thrust mode and the maneuver time is given, it takes a long time to compute the minimum duration transfer trajectories of space-to-ground vehicles, which is mainly because the initial values of the adjoint variables involved in the optimization model have no definite physical meanings and the model is sensitive to them. In order to develop space-to-ground transfer trajectory programmes in real time in an uncertain environment for the decision makers, we propose a fast method for computing the minimum duration transfer trajectories of space-to-ground vehicles with the given position of the landing point and the arbitrary maneuver point. First, the optimization model based on the hybrid method is established to compute the minimum duration transfer trajectory. Then, the region composed of maneuverable points is gridded and the initial values of the adjoint variables and the values of partial state variables of the minimum duration transfer trajectories at all gridded points are computed and saved to a database. Finally, the predicted values of the initial values of the adjoint variables and the values of partial state variables at any maneuver point within the region composed of maneuverable points are computed by using a binary cubic interpolation method. Finally, the minimum duration transfer trajectory is obtained by the hybrid method which takes the neighborhood of the predicted values as the search ranges of the initial values of the adjoint variables and the values of partial state variables. Simulation results demonstrate that the proposed method, which requires only 2.93% of the computational time of the hybrid method, can improve substantially the computational time of the minimum duration transfer trajectory of a space-to-ground vehicle under the guarantee of ensuring accuracy. The methodology of converting the time domain into the space domain is well applied in this paper.

Distributed optimal control applied to Fluid Structure Interaction problems

In this paper we focus on an optimal control problem based on Lagrangian multipliers and adjoint variables applied to the steady state Fluid-structure interaction problem. FSI systems consist of one or more solid structures that deform by interacting with a surrounding fluid flow. We consider the Fluid-structure interaction problem written in variational monolithic form in order to balance automatically solid and liquid forces at the interface. The objective of the control problem is the minimization of a velocity matching functional and it is obtained with a distributed force control that deforms the shape of the solid domain. This distributed control for Fluid-structure interaction systems can be easily used to control the shape of the solid domain and the flow effects of the fluid-solid interactions. The optimality system is derived from the first order optimality conditions by taking the Fréchet derivatives of the Lagrangian with respect to all the variables involved. The optimal solution is then obtained through a steepest descent algorithm applied to the optimality system. To support this approach we report a few numerical tests where the distributed force deforms the solid structure in order to match the desired velocity profile. The results obtained show the feasibility and robustness of this approach and its possible use in many industrial relevant applications.

An adjoint based pressure boundary optimal control approach for fluid-structure interaction problems

In this work we investigate a new pressure boundary optimal control approach to the fluid-structure interaction problem based on Lagrangian multipliers and adjoint variables. We consider the steady FSI problem written in variational monolithic form in order to balance automatically solid and liquid forces at the interface and propose a pressure boundary optimal control method with the purpose to control the solid deformation in a well defined region by changing the fluid pressure on domain boundaries. The optimality system is obtained by imposing the first order necessary condition to the Lagrangian functional. In order to couple also the adjoint variables we must introduce a fictitious velocity field in the solid region that balances automatically interface adjoint forces as well. The system is solved in a segregated approach with different optimization schemes, such as the steepest descent and the quasi-Newton methods. We implemented the algorithms in a finite element code with mesh-moving capabilities for the study of large solid displacements. In order to support the proposed approach we perform numerical tests in two and three-dimensional spaces.