Adjoint Sensitivity Analysis Method Research Articles

Locating leakage in pipelines based on the adjoint equation of inversion modeling

This paper presents an adjoint method for locating potential leakage in a single-phase fluid pipeline based on the analytic solution of inversion modeling. By studying the mechanism of pipeline leakage pressure, the adjoint equation based on the governing equation of transient flow is established in the single-liquid phase aspect using inverse adjoint theory and sensitivity analysis method. The inverse transient adjoint equation is primarily derived from the single linear fluid pipeline in the semi-infinite domain. The Laplace method is then used to obtain an analytical solution that determines the location of pipeline leakage. The experimental results indicate that the analytic solution can quickly and accurately judge the leakage location of the pipeline. Furthermore, it presents a new approach to engineering applications, such as gas-liquid two-phase flow complex pipe networks, etc.

Multihazard life-cycle cost optimization of active mass dampers in tall buildings

This article proposes an optimization-based-methodology for designing multiple active mass dampers (AMDs) in tall buildings in multihazard environments. A realistic cost of the AMD system is formulated as the objective-function, and the multihazard Life-Cycle-Cost (LCC) that represents the long-term hazard-related losses, is selected as the main performance-constraint. This allows joining winds and earthquakes in a single decision parameter. Additional constraints limit the dampers forces and strokes. To ensure stability of the actively controlled building, the LQR algorithm is nested within the formal optimization problem. In addition, for the first time in the context of AMDs, their masses are considered as design variables, emerging directly as an optimization output. As the LQR control law is not aimed at optimizing costs and LCC, the lower triangular matrices related to the LQR weighting matrices are used as additional design variables. Thus, their values converge throughout the optimization to optimize the formulated cost function while satisfying the constraints. As this strategy produces large-scale problems, an efficient gradient-based algorithm is developed, using the adjoint sensitivity analysis method. The framework is applied to several examples for a benchmark-building, providing consistent results. One example demonstrates that AMDs can offer similar performance to its passive TMDs counterpart, requiring a mass 40% lower.

Adjoint sensitivity analysis method based on lattice Boltzmann equation for flow-induced sound problems

This paper proposes an adjoint sensitivity analysis method that enables the evaluation of the sensitivities of far-field sounds with respect to object shapes. In this method, flow-induced sounds are directly simulated based on the lattice Boltzmann equation (LBE) with an athermal model under low-Mach-number conditions. In addition, wall boundaries of complex geometries are considered using the interpolated bounce-back (IBB) condition, which also allows the adoption of arbitrary parameters that define object geometries as design variables in the sensitivity analysis. The sensitivities of far-field sounds to numerous design variables can be evaluated by solving the adjoint equation, which is derived from the LBE with the IBB condition. This study demonstrates the validity of the proposed method using two test problems. In these tests, the sensitivities evaluated by the adjoint method are compared with those evaluated by the finite difference method. Furthermore, this paper presents new shapes optimized to suppress the generation of the Aeolian tone using the sensitivity analysis method.

Adjoint sensitivity analysis method for dynamic optimization of multibody systems considering collision and friction

Adjoint sensitivity analysis method for dynamic optimization of multibody systems considering collision and friction

Stress-based topology optimization of continuum structures for the elastic contact problems with friction.

Structural problems have various nonlinearities in the real world and these nonlinearities should be accommodated in structural topology optimization. This work proposes a topology optimization method for minimizing the maximum von Mises stress of elastic continuum structures with frictional contact under material usage constraint, using an extended Bi-directional Evolutionary Structural Optimization (BESO) method. Stresses are treated as global performance (objective) function, the global von Mises stress is measured by the p-norm stress aggregation approach, and the friction behavior is governed by the Coulomb friction law regularized in analogy with the perfect elasto-plastic theory. BESO method based on discrete variables which can avoid the well-known stress singularity and the numerical instability issue in frictional contact problems. The adjoint sensitivity analysis method is adopted to derive the sensitivity numbers. The effectiveness of the proposed method is validated through a series of comparison studies including elastic-rigid and elastic-elastic contact problems. The influence of varying friction coefficient on the optimized results and the stress distributions are investigated in comparison with the maximum stiffness design. The effect of different parameters including p-norm, volume fraction and mesh density on the optimized results are discussed. The optimized results, for elastic-rigid contact, indicate that the maximum stress can be reduced compared with elastic-elastic contact. The optimized stress decreases as the friction coefficient increases because the friction behavior resists the tangential deformation at the contact interface. The results also show that the proposed approach can achieve a reasonable design that effectively controls the stress level and reduces the stress concentration effect at the critical stress areas.

A stochastic topology optimization algorithm for improved fluid dynamics systems

Abstract The use of topology optimization in the design of fluid dynamics systems is still in its infancy. With the decreasing cost of additive manufacture, the application of topology optimization in the design of structural components has begun to increase. This paper provides a method for using topology optimization to reduce the power dissipation of fluid dynamics systems, with the novelty of it being the first application of stochastic mechanisms in the design of 3D fluid–solid geometrical interfaces. The optimization algorithm uses the continuous adjoint method for sensitivity analysis and is optimized against an objective function for fluid power dissipation. The paper details the methodology behind a vanilla gradient descent approach before introducing stochastic behavior through a minibatch-based system. Both algorithms are then applied to a novel case study for an internal combustion engine's piston cooling gallery before the performance of each algorithm's resulting geometry is analyzed and compared. The vanilla gradient descent algorithm achieves an 8.9% improvement in pressure loss through the case study, and this is surpassed by the stochastic descent algorithm which achieved a 9.9% improvement, however this improvement came with a large time cost. Both approaches produced similarly unintuitive geometry solutions to successfully improve the performance of the cooling gallery.

Topology optimization of thermoelastic structures using MMV method

Thermoelastic problems refer to a coupled phenomenon existing in a wide range of practical applications where elastic responses are influenced by temperature variation. Meanwhile, Moving Morphable Void (MMV) approach receives extensive attentions owing to explicit topological geometric information. In MMV method, a closed B-spline curve defined by several design variables is adopted to describe void material in design domain. In present study, we intend to use the MMV method to topologically optimize the thermoelastic structures under uniform temperature increment and varying temperature field. The weak formulations of heat conduction and elasticity are combined to describe the thermoelastic problems. The optimization mathematical model with the aim of compliance minimization and volume constraint is constructed. The sensitivity of design variables is derived by material derivative method and adjoint sensitivity analysis method. The method of moving asymptote (MMA) algorithm is implemented to update the design variables for finding the best material distribution. Numerical examples are provided to demonstrate the effectiveness of the proposed method.

Stress-Based topology optimization under the stress relaxation effect

One of the effects of creep phenomena in structures is stress relaxation. Owing to this effect, the stiffness of each point is inversely related to the stress at that point, which causes a stress redistribution, a reduction of the maximum stress, and an increment in the displacement field. By contrast, for structural optimization, it is a practical and challenging problem to consider the maximum stress as a failure criterion or objective function. In the present study, the stress relaxation effect was investigated in stress-based topology optimization. The so-called solid isotropic material with penalization method and a nonlinear finite element analysis are utilized for this purpose. The maximum von Mises stress and the total volume of the structure are selected as the objective function and constraint, respectively. The main novelty of this research is to find the optimized topology with the least maximum stress by developing and implementing a time-dependent adjoint method for sensitivity analysis. Implementing the calculated sensitivity for stress-based problems in a gradient-based optimization method, such as the optimality criteria, is quite challenging, and some numerical instabilities in the optimization process are addressed. Several examples are solved with and without considering the stress relaxation effect. The results reveal that the influence of stress relaxation in the reduction of stress concentration makes the optimized layout more similar to the optimized layout obtained by minimizing the compliance for the linear structures.

Shape design optimization of dynamic crack propagation using peridynamics

Crack generation tends to be prevented in most of design problems but taking advantages of cracks with some flaw acceptability could provide better design margin in practical problems. To develop a crack control strategy, we propose a shape design optimization method using an adjoint shape sensitivity based on a peridynamics theory that does not require crack propagation path a priori and thus is advantageous for the problems in which the discontinuity initiates, propagates, and interacts dynamically. The adjoint design sensitivity analysis (DSA) method based on the peridynamics is developed for dynamic crack propagations in brittle materials, including the successive branching of cracks. For a performance measure of stretch between the designated points at final time, the design sensitivity with respect to the design parameters is derived to be used in the shape design optimization. Numerical examples are demonstrated to verify the accuracy of the developed design sensitivity to show excellent agreement with finite differencing for all the shape design variables. Also, the proposed DSA method works very well in the shape design optimization applications of creating or avoiding cracks in a specified region.

Adjoint sensitivity analysis method based on lattice Boltzmann equation for flow-induced sound problems

Adjoint sensitivity analysis method based on lattice Boltzmann equation for flow-induced sound problems

Evaluating oil and gas contributions to ambient nonmethane hydrocarbon mixing ratios and ozone-related metrics in the Colorado Front Range

Recently, oil and natural gas (O&NG) production activities in the Denver-Julesburg Basin have expanded rapidly. Associated nonmethane hydrocarbon (NMHC) emissions contribute to photochemical formation of ground-level ozone and include benzene as well as other hazardous air pollutants. Using positive matrix factorization (PMF) and chemical mass balance (CMB) methods, we estimate how much O&NG activities and other sources contribute to morning NMHC mixing ratios measured from 2013 to mid-2016 at a site in Platteville, CO, in the Denver-Julesburg Basin, and at a contrasting site in downtown Denver. A novel adjoint sensitivity analysis method is then used to estimate corresponding contributions to ozone and ozone-linked mortality in the Denver region. Average 6–9 am NMHC mixing ratios in Platteville were seven times higher than those in Denver in 2013 but four times higher in 2016. CMB estimates that O&NG activities contributed to the Platteville (Denver) site an average of 96% (56%) of NMHC on a carbon basis while PMF indicated 92% (33%). Average vehicle-related contributions of NMHC are estimated as 41% by CMB and 53% by PMF in Denver. Estimates of the fractional contribution to potential ozone and ozone-linked mortality from O&NG activities are smaller while those from vehicles are larger than the NMHC contributions. CMB (PMF) indicate that greater than 78% (40%) of annual average benzene in Denver is attributable to vehicle emissions while greater than 75% (67%) of benzene in Platteville is attributable to O&NG activities.

Development and assessment of adjoint sensitivity analysis method for transient two-phase flow simulations

The adjoint method is efficient for calculating sensitivities of a few responses to a large number of parameters. The cost of solving an adjoint equation is comparable to the cost of solving the forward governing equation. Once the adjoint solution is available, the sensitivities to any number of parameters can be calculated with little effort. There are two methods to develop the adjoint equations: continuous method and discrete method. In the continuous method, the control theory is applied to the forward governing equation and produces an analytical partial differential equation for solving the adjoint variable; in the discrete method, the control theory is applied to the discrete form of the forward governing equation and produces a linear system of equations for solving the adjoint variable. In this article, an adjoint sensitivity analysis framework is developed using both the continuous and discrete methods. These two methods are assessed with one steady-state and one transient test cases. Adjoint sensitivities from both methods are verified by sensitivities calculated with a perturbation method. Adjoint sensitivities from both methods are physically reasonable and match each other. Though continuous method is computationally more efficient than the discrete method, difficulties are observed in solving the continuous adjoint equation for cases where the adjoint equation contains sharp discontinuities in the source terms. In such cases, the continuous method is not as robust as the discrete adjoint method.

Adding CO2 channel 16 to AHI data assimilation over land further improves short-range rainfall forecasts

The new generation of geostationary environmental operational satellite imager, the Himawari-8 Advanced Himawari Imager (AHI), adds two more water vapour channels and four more other channels than its predecessor, MTSAT-2. But except for the three water vapour channels, AHI channels are often not assimilated over land due to large uncertainty in surface parameters. Using the relative adjoint sensitivity analysis method, we show that the brightness temperature of AHI channel 16 is much more sensitive to the low-tropospheric atmosphere than to the surface emissivity, similar to those high-level water vapour channels. We thus assimilated AHI channel 16 brightness temperature observations together with AHI water vapour channels over land, and assessed the added benefits on short-range quantitative precipitation forecasts for several convection-induced rainfall cases. Results show that adding channel 16 over land to AHI data assimilation further improves short-range rainfall forecasts. Assimilation of AHI channel 16 improves the upstream near surface atmospheric temperature analysis and influences the development of downstream precipitating weather systems.

Towards Overcoming the Curse of Dimensionality: The Third-Order Adjoint Method for Sensitivity Analysis of Response-Coupled Linear Forward/Adjoint Systems, with Applications to Uncertainty Quantification and Predictive Modeling

This work presents the Third-Order Adjoint Sensitivity Analysis Methodology (3rd-ASAM) for response-coupled forward and adjoint linear systems. The 3rd-ASAM enables the efficient computation of the exact expressions of the 3rd-order functional derivatives (“sensitivities”) of a general system response, which depends on both the forward and adjoint state functions, with respect to all of the parameters underlying the respective forward and adjoint systems. Such responses are often encountered when representing mathematically detector responses and reaction rates in reactor physics problems. The 3rd-ASAM extends the 2nd-ASAM in the quest to overcome the “curse of dimensionality” in sensitivity analysis, uncertainty quantification and predictive modeling. This work also presents new formulas that incorporate the contributions of the 3rd-order sensitivities into the expressions of the first four cumulants of the response distribution in the phase-space of model parameters. Using these newly developed formulas, this work also presents a new mathematical formalism, called the 2nd/3rd-BERRU-PM “Second/Third-Order Best-Estimated Results with Reduced Uncertainties Predictive Modeling”) formalism, which combines experimental and computational information in the joint phase-space of responses and model parameters, including not only the 1st-order response sensitivities, but also the complete hessian matrix of 2nd-order second-sensitivities and also the 3rd-order sensitivities, all computed using the 3rd-ASAM. The 2nd/3rd-BERRU-PM uses the maximum entropy principle to eliminate the need for introducing and “minimizing” a user-chosen “cost functional quantifying the discrepancies between measurements and computations,” thus yielding results that are free of subjective user-interferences while generalizing and significantly extending the 4D-VAR data assimilation procedures. Incorporating correlations, including those between the imprecisely known model parameters and computed model responses, the 2nd/3rd-BERRU-PM also provides a quantitative metric, constructed from sensitivity and covariance matrices, for determining the degree of agreement among the various computational and experimental data while eliminating discrepant information. The mathematical framework of the 2nd/3rd-BERRU-PM formalism requires the inversion of a single matrix of size Nr Nr, where Nr denotes the number of considered responses. In the overwhelming majority of practical situations, the number of responses is much less than the number of model parameters. Thus, the 2nd-BERRU-PM methodology overcomes the curse of dimensionality which affects the inversion of hessian matrices in the parameter space.

Optimal determination of force field parameters for reduced molecular dynamics model

Using a gradient-based optimization method, the time-consuming atomistic model of substrate is replaced by computationally efficient Lennard-Jones (L-J) potential walls whose parameters are determined to appropriately represent the interactions between the nanoparticles and the substrate. To obtain the required design sensitivity with respect to design variables for the constant temperature molecular dynamics (MD) simulations that use the Nosé–Hoover thermostat, the finite difference method is impractical due to the huge amount of computational costs. Thus, we developed an adjoint design sensitivity analysis (DSA) method that is efficient for the system of many design variables. In numerical examples, we replace the complicated and time-consuming silicate structure to a multiple layer model of L-J potential wall, through the design optimization that includes the design variables of ϵ, σ, and the positions of each layer. The objective is to minimize the squared difference of time averaged performance between the full and the reduced models during the whole time span. The proposed method could lead to a significant reduction of computational costs, together with comparable outcomes from MD simulations.

Application of adjoint sensitivity analysis method to supercritical CO2 power cycle optimization

Adjoint sensitivity based optimization methodology for a supercritical CO2 (S-CO2) cycle is developed in this paper. The adjoint sensitivity analysis method (adjoint method) is a way to analyze sensitivity very quickly. By using the developed methodology, the optimal state variation due to the change of cycle parameters is analyzed. The S-CO2 recompression Brayton cycle is used for an example case for the demonstration of the proposed method. The independence of time consumption for the developed adjoint sensitivity analysis method to the number of optimized variables is demonstrated. For the test case, the developed algorithm shows the ability to make the design parameters converge with a precision of 10−6 by more than ten times faster for calculating the sensitivity than conventional optimization methods. Validation of the obtained optimal point is also included in the paper. For the validation, a response surface analysis is performed to visualize the pathway during the iteration for optimization. It is very challenging to carry out the optimization having the same precision using a brute force algorithm or a probability-based optimization algorithm since the number of variables to be optimized is substantial for this type of a problem. The smoothness of an optimal cycle efficiency variation is observed in every case.

An adjoint method of sensitivity analysis for residual vibrations of structures subject to impacts

For structures subject to impact loads, the residual vibration reduction is more and more important as the machines become faster and lighter. An efficient sensitivity analysis of residual vibration with respect to structural or operational parameters is indispensable for using a gradient based optimization algorithm, which reduces the residual vibration in either active or passive way. In this paper, an integrated quadratic performance index is used as the measure of the residual vibration, since it globally measures the residual vibration response and its calculation can be simplified greatly with Lyapunov equation. Several sensitivity analysis approaches for performance index were developed based on the assumption that the initial excitations of residual vibration were given and independent of structural design. Since the resulting excitations by the impact load often depend on structural design, this paper aims to propose a new efficient sensitivity analysis method for residual vibration of structures subject to impacts to consider the dependence. The new method is developed by combining two existing methods and using adjoint variable approach. Three numerical examples are carried out and demonstrate the accuracy of the proposed method. The numerical results show that the dependence of initial excitations on structural design variables may strongly affects the accuracy of sensitivities.

Predictive Modeling Applied to a Spent Fuel Dissolver Model—II: Uncertainty Quantification and Reduction

This work presents the application of the comprehensive predictive modeling methodology for single multiphysics time-dependent systems, developed by Cacuci and Ionescu-Bujor (2010), to a paradigm spent fuel dissolver model of interest to nonproliferation objectives, in order to quantify uncertainties due to dissolver model parameters and subsequently to combine all of the available experimental and computational information to obtain best-estimate values for model responses and parameters, along with reduced predicted uncertainties. This predictive modeling methodology uses the maximum entropy principle to construct an optimal approximation of the unknown a priori distribution for the a priori known mean values and uncertainties characterizing the model parameters and the computed and experimentally measured model responses. This approximate a priori distribution is subsequently combined using Bayes’ theorem with the likelihood provided by the multiphysics computational models. Finally, the posterior distribution is evaluated using the saddle-point method to obtain analytical expressions for the optimally predicted values for the parameters and responses of both multiphysics models, along with corresponding reduced uncertainties.The weighting functions used within this predictive modeling methodology are provided by the first-order sensitivities (i.e., functional derivatives) of the model’s response with respect to the model’s parameters. The dissolver model comprises 619 model parameters related to the model’s equation of state and inflow conditions. The sensitivities to all model parameters of the acid concentrations at each of these instances in time were computed exactly and efficiently in an accompanying work by Peltz and Cacuci (2015), using the adjoint sensitivity analysis method. These sensitivities are used in this work to quantify the uncertainties in the acid concentration (system responses) in various dissolver compartments, arising from uncertainties in the model parameters. Subsequently, the sensitivities are used within the predictive modeling methodology to combine the computational results with the available experiments, which were performed solely in the compartment farthest from the inlet. The results of applying the predictive modeling methodology yield optimally calibrated values for all 619 model parameters, with reduced predicted uncertainties, as well as optimal (best-estimate) predicted values for the acid concentrations, also with reduced predicted uncertainties. Notably, even though the experimental data pertain solely to the compartment farthest from the inlet (where the data were measured), the predictive modeling methodology actually improves the predictions and reduces the predicted uncertainties not only in the compartment in which the data were actually measured but also throughout the entire dissolver, including the compartment farthest from the measurements (i.e., at the inlet). This is because the predictive modeling methodology combines and transmits information simultaneously over the entire phase-space, comprising all time steps and spatial locations.The results obtained in this work establish confidence in the dissolver model’s accuracy for simulating the acid concentrations required to dissolve used nuclear fuel. In turn, these results will be used to generate source terms for key reprocessing facility components downstream and to support material accountability for nuclear safeguards.

Predictive Modeling Applied to a Spent Fuel Dissolver Model—I: Adjoint Sensitivity Analysis

This work presents a comprehensive sensitivity analysis of a paradigm dissolver model that has been selected because of its applicability to material separations and its potential role in diversion activities associated with proliferation and international safeguards. This dissolver model consists of eight active compartments in which the time-dependent nonlinear differential equations modeling the physical and chemical processes comprise 16 time-dependent spatially dependent state functions and 635 model parameters related to the model’s equation of state and inflow conditions. The most important response for the dissolver model is the computed nitric acid in the compartment farthest away from the inlet, where measurements are available for comparisons. The sensitivities to all model parameters of the acid concentrations at each of these instances in time are computed exactly and efficiently using the adjoint sensitivity analysis method for nonlinear systems. The relative importance of the sensitivities in contributing to the uncertainties in the computed model responses is quantified numerically and analyzed in the dissolver’s physics context. The sensitivities computed in this work will be used in a companion paper for uncertainty analysis and predictive modeling, which aims at validating the paradigm dissolver model using the available experimental data and subsequently obtaining best-estimate predicted nominal values for the acid concentrations, with reduced predicted uncertainties, for the longer-term purpose of coupling this dissolver model to other nuclear facilities of interest to nonproliferation objectives.

Topology optimization of plate structures subject to initial excitations for minimum dynamic performance index

This paper studies optimal topology design of damped vibrating plate structures subject to initial excitations. The design objective is to minimize an integrated square performance measure. The artificial density of the plate element is the topology design variable and the material volume is given. The Lyapunov's second method is applied to reduce the calculation of performance measure to the solution of the Lyapunov equation. An adjoint sensitivity analysis method is used, which only needs to solve the Lyapunov equation twice. However, when the problem has a large number of degrees of freedom, the solution process of Lyapunov equation is computational costly. Thus, the full model is transformed to a reduced space by mode reduction method. To further reduce the scale of reduced model, we propose a mode screen method to decrease the number of eigenmodes. Numerical examples of optimum topology design of bending plates are presented for illustrating validity and efficiency of our new algorithm.