Discrete Adjoint Sensitivity Research Articles

Numerical simulation of differential-algebraic equations with embedded global optimization criteria

We are considering differential-algebraic equations with embedded optimization criteria (DAEOs), in which the embedded optimization problem is solved by global optimization. This leads to differential inclusions for cases in which there are multiple global optimizers at the same time. Jump events from one global optimum to another result in nonsmooth DAEs and reduce the order of convergence of the numerical integrator to first-order. Implementation of event detection and location as introduced in this work preserves the higher-order convergence behaviour of the integrator. This allows the computation of discrete tangent and adjoint sensitivities for optimal control problems.

Geometrically exact beam theory for gradient-based optimization

Decades of research have progressed geometrically exact beam theory to the point where it is now an invaluable resource for analyzing and modeling highly flexible slender structures. Large-scale optimization using geometrically exact beam theory remains nontrivial, however, due to the inability of gradient-free optimizers to handle large numbers of design variables in a computationally efficient manner and the difficulties associated with obtaining smooth, accurate, and efficiently calculated design sensitivities for gradient-based optimization. To overcome these challenges, this paper presents a finite-element implementation of geometrically exact beam theory which has been developed specifically for gradient-based optimization. A key feature of this implementation of geometrically exact beam theory is its compatibility with forward and reverse-mode automatic differentiation. Another key feature is its support for both continuous and discrete adjoint sensitivity analysis. Other features are also presented which build upon previous implementations of geometrically exact beam theory, including a singularity-free rotation parameterization based on Wiener-Milenković parameters, an implementation of stiffness-proportional structural damping using a discretized form of the compatibility equations, and a reformulation of the equations of motion for geometrically exact beam theory as a semi-explicit system. Several examples are presented which verify the utility and validity of each of these features.

A discrete-adjoint framework for optimizing high-fidelity simulations of turbulent reacting flows

An adjoint-based framework is presented that measures exact sensitivity from high-fidelity simulations of turbulent reacting flows. The framework leverages and extends state-of-the-art numerical methods in a manner that is compatible with a discrete adjoint solver. To ensure energy stability and accuracy, high-order narrow-stencil finite-difference operators satisfying the summation-by-parts (SBP) property are combined with simultaneous-approximation-term boundary treatment. An adaptive SBP dissipation operator and its corresponding adjoint are formulated in a way to dampen unresolved modes and preserve scalar boundedness. A flamelet/progress variable approach is employed to handle chemical reactions using tabulated chemistry. The utility of using pre-computed lookup tables is that discrete adjoint sensitivity can be computed efficiently for arbitrary chemical mechanisms. Gradient-based optimization utilizing the sensitivity obtained from the adjoint solution is demonstrated on two configurations. First, momentum actuation is applied upstream in a spatial mixing layer to control the evolution of a passive scalar in a target region downstream. Then, the framework is used to optimize an acoustic actuator in a three-dimensional turbulent jet with the aim of anchoring an H2 lifted flame at a target location. Both cases involve manipulating discrete space-time fields with O(108)−O(109) degrees of freedom, which would not be possible via a brute-force trial-and-error approach.

Sensitivity analysis of the 1-D SFR thermal stratification model via discrete adjoint sensitivity method

This paper presents a parameter sensitivity analysis on a thermal stratification (TS) model by using the discrete sensitivity method. The TS model was recently developed in our research group to efficiently predict the TS phenomenon in pool-type Sodium-cooled Fast Reactors. The fluid temperature gradient was considered as the figure of merit in the sensitivity analysis because it best characterizes the thermal stratification phenomenon. The sensitivities of the fluid temperature gradient with respect to four different parameters were investigated, including jet volumetric flow rate Qjet, jet temperature Tjet, heat capacity of the ambient fluid Cp,amb, and static thermal conductivity of the ambient fluid kc,amb. The sensitivity analysis was conducted through both the conventional forward sensitivity method and the advanced adjoint sensitivity method, which is more effective in cases where the number of outputs is small and the number of input parameters is large.The sensitivities obtained in this study suggested that perturbations in Qjet, Cp,amb, and kc,amb could introduce either positive or negative changes to the temperature gradient, depending on the axial location and the elapsed time of the experiment. However, an increase in Tjet always decreased the temperature gradient. Moreover, the impact of Tjet on the maximum temperature gradient was several times higher than that of the other three parameters, which indicated that additional attention may need to be paid to the occurrence of thermal stratification in the sodium pool when the impinging jet has a large temperature change. This study also provides a step-by-step example for the application of the discrete adjoint sensitivity method to the time-dependent nonlinear systems.

Reduced-order model-based convergence acceleration of reverse mode discrete adjoint solvers

This work presents a new technique to reduce the computational cost of sensitivities calculated using a discrete adjoint solver developed via reverse mode automatic differentiation. A fixed-point iterative method is built for the discrete adjoint sensitivity equations by employing the primal time-stepping adjoint approach. The fixed-point sensitivity solution is then accelerated by building a reduced-order model (ROM) that maps the relationship between the sensitivity solution and its corresponding residual. This model is then used to approximate the converged solution, corresponding to a zero residual. While the approximation might not produce the fully converged solution, it typically provides an improved solution that the fixed-point solver can be re-initialized with, which is one of the novel aspects of the present work. After re-initializing the solution, the ROM acceleration technique can be reapplied until the desired convergence criterion is reached. A key feature of the proposed ROM is that it is formed on the fly during a single sensitivity solution. Additionally, its implementation requires only minor modifications to an existing fixed-point iterative solver. The ROM projection technique is evaluated by considering design optimization of horizontal wind turbine blade profiles and cost reductions of 57 to 80% were achieved.

FDOT: A Fast, memory-efficient and automated approach for Discrete adjoint sensitivity analysis using the Operator overloading Technique

A new toolbox based on operator overloading is introduced for automatic differentiation of scientific computing codes – and in particular legacy computational fluid dynamics solvers that are developed using Fortran. The method can be readily implemented into existing iterative solvers with minimal changes to the primal code. The integrated toolbox can efficiently calculate the sensitivities of any objective function with respect to all variables (design or intermediate) that can later be used for gradient-based design optimization, uncertainty quantification, error estimation, and mesh adaptation. The underlying definition of the current automatic differentiation is directly related to the discrete adjoint sensitivity analysis. Unlike most traditional operator overloading-based adjoint approaches reported in the literature, the current technique offers huge reductions in the memory footprint. To demonstrate the advantages of the current approach, various solvers/problems are considered. It is shown that the proposed technique can be used as an efficient toolbox for automatic differentiation of scientific solvers requiring only a handful additional lines of coding.

Parallel Finite Element Framework for Rotorcraft Multibody Dynamics and Discrete Adjoint Sensitivities

A parallel finite element framework for high-fidelity structural dynamic analysis and gradient evaluation using the discrete adjoint method is presented. The framework is intended to be used for gradient-based design optimization of flexible multibody dynamic systems such as rotorcraft. The formulation of governing equations, the treatment of kinematic constraints, and the evaluation of functionals of interest and their derivatives are addressed. A minimal set of routines needed to implement the discrete adjoint method is proposed. The governing equations are integrated in time using a diagonally implicit Runge–Kutta method for second-order systems of equations. The formulation of the corresponding time-dependent discrete adjoint equations are presented and are numerically verified using the complex-step method. A verification of the dynamics, an assessment of parallel scalability of the analysis and derivative evaluation techniques, and a demonstration of the design capability are presented.

Application of discrete adjoint method to sensitivity and uncertainty analysis in steady-state two-phase flow simulations

Adjoint method is efficient for computing sensitivities of a response to a large number of input parameters; however, a successful application of adjoint method to sensitivity analysis in two-phase flow simulations is rare. In this article, a discrete adjoint sensitivity analysis framework is developed for steady-state two-phase flow problems. The new framework is based on a new implicit forward solver. The residual function of the discretized forward governing equation is used to formulate the adjoint problem. The framework is verified with the faucet flow problem and the Boiling Water Reactor Full-size Fine-mesh Bundle Test (BFBT) benchmark. For the faucet flow problem, adjoint sensitivities are shown to match analytical sensitivities very well. For the BFBT benchmark, the adjoint sensitivities are shown to match the sensitivities calculated with a perturbation equation. The adjoint sensitivities are also used to propagate uncertainties in input parameters to the uncertainty in the response. The uncertainty propagation with the adjoint method is verified with the Monte Carlo method and is shown to be accurate and efficient.

Eliminating mesh sensitivities in microstructure design with an adjoint algorithm

A microstructure design approach utilizing a discrete adjoint sensitivity analysis scheme is addressed. The microstructure is modeled with a one-point probability descriptor, known as Orientation Distribution Function (ODF). The ODF measures the volume densities of the unique orientations in a polycrystalline material. It can be discretized within an orientation space to describe the microstructure-sensitive features and calculate volume-averaged material properties. In this work, a finite element scheme is applied to discretize the ODF values in the Rodrigues orientation space to obtain the macro-scale properties. An adjoint sensitivity analysis is integrated into the solution framework to perform a design optimization for the microstructure to minimize/maximize a macro-scale property. Next, a gradient-based design optimization which utilizes the outputs of the sensitivity analysis is performed to enhance the material properties. The adjoint technique is shown to save a significant computational time compared to the existing design approaches. This is because the adjoint method is independent of the number of design variables. Thus, it eliminates the solution dependence to the finite element mesh that is used to model the problem.

Centralized control of district heating networks during failure events using discrete adjoint sensitivities

Real-time control of district heating networks in the case of failures requires for accurate and fast strategies able to guarantee thermal comfort to all connected users. In this paper, we demonstrate a control framework that responds to these essential requirements. We minimize a global measure of discomfort based on a smooth maximum approximation. The optimization problem is solved through a gradient-based algorithm that can be naturally integrated with distributed meter readings leading to high accuracy of both forward and sensitivity analysis. Objective function gradients are computed by a discrete adjoint method, which is fast and nearly insensitive to the dimensionality of the optimization problem. The proposed framework is tested with numerical experiments on a reference medium-size distribution network in Turin. Results show that the thermal comfort of most critical users increases quickly, yielding to a nearly homogeneous discomfort distribution at the end of the optimization process. Studying the effect of the inlet pressure head on the optimized system performance reveals that a centralized operation results in increased robustness of the network and allows reducing backup pumping equipment. Furthermore, applying the proposed framework at the distribution network level yields remarkable benefits also in case of failures in the main transportation network.

Discrete Adjoint Sensitivity Analysis of Hybrid Dynamical Systems With Switching

Sensitivity analysis is an important tool for describing power system dynamic behavior in response to parameter variations. It is a central component in preventive and corrective control applications. The existing approaches for sensitivity calculations, namely, finite-difference and forward sensitivity analysis, require a computational effort that increases linearly with the number of sensitivity parameters. In this paper, we investigate, implement, and test a discrete adjoint sensitivity approach whose computational effort is effectively independent of the number of sensitivity parameters. The proposed approach is highly efficient for calculating sensitivities of larger systems and is consistent, within machine precision, with the function whose sensitivity we are seeking. This is an essential feature for use in optimization applications. Moreover, our approach includes a consistent treatment of systems with switching, such as dc exciters, by deriving and implementing the adjoint jump conditions that arise from state-dependent and time-dependent switchings. The accuracy and the computational efficiency of the proposed approach are demonstrated in comparison with the forward sensitivity analysis approach. This paper focuses primarily on the power system dynamics, but the approach is general and can be applied to hybrid dynamical systems in a broader range of fields.

Topology optimization for effective energy propagation in rate-independent elastoplastic material systems

A topology optimization framework for effective energy management in dynamically loaded structures with rate-independent elastoplastic material behavior is presented. The elastoplastic constitutive equations are integrated using an implicit numerical scheme based on a multi-level Newton procedure. This primal analysis procedure requires the computation of the incremental algorithmic consistent tangent operator derived from the mappings that enforce the evolving constitutive relations. For our gradient based optimization, we develop discrete adjoint sensitivity expressions. Finite element implementation is discussed and the optimization framework is exemplified via the design of three-dimensional structures subjected to dynamic loads.

Discrete adjoint sensitivity analysis for fluid flow topology optimization based on the generalized lattice Boltzmann method

A discrete adjoint sensitivity analysis for fluid flow topology optimization based on the lattice Boltzmann method (LBM) with multiple-relaxation-times (MRT) is developed. The lattice Boltzmann fluid solver is supplemented by a porosity model using a Darcy force. The continuous transition from fluid to solid facilitates a gradient based optimization process of the design topology of fluidic channels. The adjoint LBM equation, which is used to compute the gradient of the optimization objective with respect to the design variables, is derived in moment space and found to be as simple as the original LBM. The moment based spatial momentum derivatives used to express the discrete objective functional (cost function) have the advantage that the local stress tensor is a local quantity avoiding the numerical computation of gradients of the discrete velocity field. This is particularly useful if dissipation is a design criterion as demonstrated in this paper. The method is validated by a detailed comparison with results obtained by Borrvall et al. for Stokes flow. While their approach is only valid for Stokes flow (i.e. very low Reynolds numbers) our approach in its present form can in principle be applied for flows of different Reynolds numbers just like the Navier–Stokes equation based approaches. This point is demonstrated with a bending pipe example for various Reynolds numbers.

Topology optimisation for natural convection problems

SummaryThis paper demonstrates the application of the density‐based topology optimisation approach for the design of heat sinks and micropumps based on natural convection effects. The problems are modelled under the assumptions of steady‐state laminar flow using the incompressible Navier–Stokes equations coupled to the convection‐diffusion equation through the Boussinesq approximation. In order to facilitate topology optimisation, the Brinkman approach is taken to penalise velocities inside the solid domain and the effective thermal conductivity is interpolated in order to accommodate differences in thermal conductivity of the solid and fluid phases. The governing equations are discretised using stabilised finite elements and topology optimisation is performed for two different problems using discrete adjoint sensitivity analysis. The study shows that topology optimisation is a viable approach for designing heat sink geometries cooled by natural convection and micropumps powered by natural convection. Copyright © 2014 John Wiley & Sons, Ltd.

Space–time adaptive solution of inverse problems with the discrete adjoint method

This paper develops a framework for the construction and analysis of discrete adjoint sensitivities in the context of time dependent, adaptive grid, adaptive step models. Discrete adjoints are attractive in practice since they can be generated with low effort using automatic differentiation. However, this approach brings several important challenges. The space–time adjoint of the forward numerical scheme may be inconsistent with the continuous adjoint equations. A reduction in accuracy of the discrete adjoint sensitivities may appear due to the inter-grid transfer operators. Moreover, the optimization algorithm may need to accommodate state and gradient vectors whose dimensions change between iterations. This work shows that several of these potential issues can be avoided through a multi-level optimization strategy using discontinuous Galerkin (DG) hp-adaptive discretizations paired with Runge–Kutta (RK) time integration. We extend the concept of dual (adjoint) consistency to space–time RK-DG discretizations, which are then shown to be well suited for the adaptive solution of time-dependent inverse problems. Furthermore, we prove that DG mesh transfer operators on general meshes are also dual consistent. This allows the simultaneous derivation of the discrete adjoint for both the numerical solver and the mesh transfer logic with an automatic code generation mechanism such as algorithmic differentiation (AD), potentially speeding up development of large-scale simulation codes. The theoretical analysis is supported by numerical results reported for a two-dimensional non-stationary inverse problem.

On the adaptive solution of space–time inverse problems with the adjoint method

Abstract Adaptivity in space and time is ubiquitous in modern numerical simulations. The large number of unknowns associated with today's typical inverse problem may run in the millions, or more. To capture small scale phenomena in regions of interest, adaptive mesh and temporal step refinements are required, since uniform refinements quickly make the problem computationally intractable. To date, there is still a considerable gap between the state–of–the–art techniques used in direct (forward) simulations, and those employed in the solution of inverse problems, which have traditionally relied on fixed meshes and time steps. This paper describes a framework for building a space-time consistent adjoint discretization for a general discrete forward problem, in the context of adaptive mesh, adaptive time step models. The discretize–then–di_erentiate approach to optimization is a very attractive approach in practice, because the adjoint model code may be generated using automatic di_erentiation (AD). However, several challenges are introduced when using an adaptive forward solver. First, one may have consistency problems with the adjoint of the forward numerical scheme. Similarly, intergrid transfer operators may reduce the accuracy of the discrete adjoint sensitivities. The optimization algorithm may need to be specifically tailored to handle variations in the state and gradient vector sizes. This work shows that several of these potential issues can be avoided when using the Runge–Kutta discontinuous Galerkin (DG) method, an excellent candidate method for h=p-adaptive parallel simulations. Selective application of automatic di_erentiation on individual numerical algorithms may simplify considerably the adjoint code development. A numerical data assimilation example illustrates the e_ectiveness of the primal/dual RK–DG methods when used in inverse simulations.

Forward, tangent linear, and adjoint Runge–Kutta methods for stiff chemical kinetic simulations

This paper investigates numerical methods for direct decoupled sensitivity and discrete adjoint sensitivity analysis of stiff systems based on implicit Runge–Kutta schemes. Efficient implementations of tangent linear and adjoint schemes are discussed for two families of methods: fully implicit three-stage Runge–Kutta and singly diagonally-implicit Runge–Kutta. High computational efficiency is attained by exploiting the sparsity patterns of the Jacobian and Hessian. Numerical experiments with a large chemical system used in atmospheric chemistry illustrate the power of the stiff Runge–Kutta integrators and their tangent linear and discrete adjoint models. Through the integration with the Kinetic PreProcessor KPP–2.2 these numerical techniques become readily available to a wide community interested in the simulation of chemical kinetic systems.

Discrete direct and adjoint sensitivity analysis for arbitrary Mach number flows

Parallel discrete direct and adjoint sensitivity analysis capabilities are developed for arbitrary Mach flows on mixed-element unstructured grids. The discrete direct and adjoint methods need a consistent and complete linearization of the flow-solver to obtain accurate derivatives. The discontinuous nature of the commonly used unstructured flux-limiters, like Barth–Jespersen and Venkatakrishnan, make them unsuitable for sensitivity analysis. A modification is proposed to make these limiters piecewise continuous and numerically differentiable, without compromising the monotonicity conditions. An improved version of Symmetric Gauss–Seidel that significantly reduces the computational cost is implemented. A distributed-memory message passing model is employed for the parallelization of sensitivity analysis solver. These algorithms are implemented within a three-dimensional unstructured grid framework and results are presented for inviscid, laminar and turbulent flows. Copyright © 2005 John Wiley & Sons, Ltd.

NON-LINEAR PROGRAMMING IN GROUNDWATER DECONTAMINATION PROBLEMS

The contaminant distribution into an aquifer is simulated through steady state groundwater flow and transient convective-dispersive transport. A minimal cost pumpage strategy for groundwater decontamination is found as the solution of a non-linearly constrained optimization problem. The exact gradient of the constrains is computed at a minimal cost through the introduction of the discrete sensitivity equations or the discrete adjoint sensitivity equations. We propose a new initialization procedure of the Lagrangian function Hessian for the Sequential Quadratic Programming method (SQP). For this peculiar application, it appears to be very efficient when combined with the SQP implementation of Schittkowski and the discrete gradient compulation. Numerical experiments with problems with up to thirty-five extraction wells have been solved in a computer time equivalent to less than one hundred state equations simulations.