Search Results (36)

This work constructs an orthogonal function system on bounded intervals $[0,R]$ associated with the Atangana–Baleanu–Caputo (ABC) fractional derivative for $\alpha \in (1/2,1)$. Starting from a fractional Laguerre-type equation involving the ABC operator, solutions are obtained via a generalized Frobenius method, yielding series representations with characteristic exponent $\alpha -1$. Rather than postulating a weight function by analogy with classical or Caputo settings, the weight is derived directly from the requirement that the underlying fractional operator be formally self-adjoint on a suitable admissible domain. This operator-theoretic approach leads to the explicit Mittag–Leffler weight $w_{\alpha }\left(x\right)=x^{-(2\alpha -1)}{E}_{\alpha }(-x^{\alpha })$, which intrinsically reflects the nonlocal memory structure of the ABC kernel. A similarity transformation removes the universal singular factor and produces regularized eigenfunctions that are continuous on $[0,R]$ and orthogonal in the weighted $L^2$ space. The weight identity and formal self-adjointness are rigorously verified through a right-Volterra uniqueness argument. Numerical experiments confirm orthogonality up to machine precision, demonstrate spectral convergence for a model ABC differential equation, and illustrate consistency with classical Laguerre polynomials in the limit $\alpha \to 1^{-}$. The resulting framework provides a self-consistent orthogonal system suitable for spectral approximations of problems governed by the ABC operator on bounded domains. Full article

Frequency stability becomes more and more important with the increase in inverter-based resources in power systems. To enhance the frequency stability, this paper proposes a novel data-driven method for frequency stability assessment and control for energy storage systems (ESSs). Leveraging the neural ordinary differential Equation (ODE) framework, the method learns a continuous-time model of the system frequency dynamics. A neural-network-based controller is then designed according to the learned neural ODE model to enhance frequency stability. The training process involves an alternating optimization algorithm that updates both the neural ODE model and the controller parameters via the adjoint sensitivity method. Simulation results demonstrate that the proposed method accurately predicts frequency trajectories. Moreover, the designed controller reduces frequency deviation by 53%. Full article

Topology optimization is increasingly employed to design fluid flow systems capable of achieving optimal performance under specific constraints. This study presents a density-based topology optimization approach specifically tailored for second-order reactive flows. The fluid-solid distribution within the domain is represented by continuous design variables expressed as an inverse permeability field. An adjoint method is used to efficiently compute gradients of the objective function, enabling the application of gradient-based algorithms to solve the optimization problem. The methodology is validated on a benchmark bend-pipe case, reproducing known optimal geometry. Subsequently, the method is applied to optimize a system involving second-order chemical reactions, aiming to maximize a desired reaction while limiting undesirable side reactions. Results demonstrate significant performance improvements, achieving gains in reaction efficiency ranging from 90.4% to 98.7% for the porous geometries and from 94.6% to 105.2% for real geometries. The optimization strategy successfully generates flow configurations analogous to those observed in modern gas turbines, highlighting the practical relevance and potential impact of the developed methodology. Full article

This study presents the development of a discrete algorithm for interpreting ground-penetrating radar (GPR) data in vertically inhomogeneous media for the diagnostics of road structures. Experimental data were obtained using an OKO-2 GPR system, followed by primary radargram processing using the CartScan software. This included noise and interference filtering, as well as the initial estimation of the dielectric permittivity of detected layers. The resulting dataset was used to validate numerical algorithms for solving the forward and inverse problems of geolectrics. The proposed approach is based on minimizing a quadratic misfit functional between the calculated and observed values of the horizontal component of the electromagnetic field. The gradient of the functional required for optimization is obtained via the numerical solution of an adjoint problem. A discrete version of this problem was developed, which satisfies the properties of conservativeness and uniformity according to finite difference theory. The inverse problem reconstruction of dielectric permittivity is considered a non-destructive method for radargram interpretation. Assuming a piecewise-continuous medium structure eliminates the need for computing gradients at material interfaces. The proposed methodology enhances the accuracy and reliability of pavement condition assessment and holds practical value for road infrastructure monitoring. Full article

This study proposes a new computational framework for the multi-objective topology optimization of conjugate heat transfer systems using a continuous adjoint approach. It relies on a monolithic solver for the coupled steady-state Navier–Stokes and heat equations, which combines finite elements stabilized by the variational multi-scale method, level set representations of the fluid–solid interfaces and immersed modeling of heterogeneous materials (fluid–solid) to ensure that the proper amount of heat is exchanged to the ambient fluid by solid objects in arbitrary geometry. At each optimization iteration, anisotropic mesh adaptation is applied in near-wall regions automatically captured by the level set. This considerably cuts the computational effort associated with calling the finite element solver, in comparison to traditional topology optimization algorithms operating on isotropic grids with a comparable refinement level. Given that we operate within the constraint of a specified number of nodes in the mesh, this allows not only to improve the accuracy of interface representation and motion but also to retain the high fidelity of the numerical solutions at the grid points just adjacent to the interface. Finally, the remeshing and resolution steps both run within a highly parallel environment, which makes it possible for the proposed algorithm to tackle large-scale problems in three dimensions with several tens of millions of state degrees of freedom. The developed solver is validated first by minimizing dissipation in a flow splitter device, for which the method delivers relevant optimal designs over a wide range of volume constraints and flow rate distributions over the multiple outlet orifices but yields better accuracy compared to reference data from literature obtained using uniform meshes (in the sense that the layouts are more smooth, and the solutions are better resolved). The scheme is then applied to a two-dimensional heat transfer problem, using bi-objective cost functionals combining flow resistance and thermal recoverable power. A comprehensive parametric study reveals a complex arrangement of optimal solutions on the Pareto front, with multiple branches of symmetric and asymmetric designs, some of them previously unreported. Finally, the algorithmic developments are substantiated with several three-dimensional numerical examples tackled under fixed weights for heat transfer and flow resistance, for which we show that the optimal layouts computed at low Reynolds number, that are intrinsically relevant to a broad range of microfluidic application, can also serve as smooth solutions to high-Reynolds-number engineering problems of practical interest. Full article

This paper presents a topology optimization (TopO) method for conjugate heat transfer (CHT), with turbulent flows. Topological changes are controlled by an artificial material distribution field (design variables), defined at the cells of a background grid and used to distinguish a fluid from a solid material. To effectively solve the CHT problem, it is crucial to impose exact boundary conditions at the computed fluid–solid interface (FSI); this is the purpose of introducing the cut-cell method. On the grid, including also cut cells, the incompressible Navier–Stokes equations, coupled with the Spalart–Allmaras turbulence model with wall functions, and the temperature equation are solved. The continuous adjoint method computes the derivatives of the objective function(s) and constraints with respect to the material distribution field, starting from the computation of derivatives with respect to the positions of nodes on the FSI and then applying the chain rule of differentiation. In this work, the continuous adjoint PDEs are discretized using schemes that are consistent with the primal discretization, and this will be referred to as the “Think Discrete–Do Continuous” (TDDC) adjoint. The accuracy of the gradient computed by the TDDC adjoint is verified and the proposed method is assessed in the optimization of two 2D cases, both in turbulent flow conditions. The performance of the TopO designs is investigated in terms of the number of required refinement steps per optimization cycle, the Reynolds number of the flow, and the maximum allowed power dissipation. To illustrate the benefits of the proposed method, the first case is also optimized using a density-based TopO that imposes Brinkman penalization terms in solid areas, and comparisons are made. Full article

This article presents single- and multi-disciplinary shape optimizations of a generic business jet wing at two transonic cruise flow conditions. The studies performed are based on two high-fidelity gradient-based optimization tools, assisted by the adjoint method (following both discrete and continuous approaches). Single discipline and coupled multi-disciplinary sensitivity derivatives computed from the two tools are compared and verified against finite differences. The importance of not making the frozen turbulence assumption in adjoint-based optimization is demonstrated. Then, a number of optimization runs, ranging from a pure aerodynamic with a rigid structure to an aerostructural one exploring the trade-offs between the involved disciplines, are presented and discussed. The middle-ground scenario of optimizing the wing with aerodynamic criteria and, then, performing an aerostructural trimming is also investigated. Full article

The scale of cities is increasing with continuous urban development. Effective methods, such as the source term estimation (STE) method, must be established for identifying the sources of air pollution in cities to prevent economic losses and casualties caused by pollutant leakage. Herein, methods for optimizing sensor configuration and identifying pollution sources are discussed, and an STE method based on the regularized minimum residual method is proposed. Urban wind environments were simulated using a computational fluid dynamics (CFD) model, and the results were compared with experimental data pertaining to the wind tunnel of an architectural ensemble to verify the model’s accuracy. The sensor layout was optimized using the simulated annealing (SA) algorithm and adjoint entropy, and the relationship between sensor responses and potential pollution sources was established using the CFD model. Pollutant concentrations measured using sensors were combined with the regularization method to extrapolate the pollution source strength, and the regularized minimum residual method was used to obtain the locations of the real pollution sources. The results show that compared with the Bayesian methods, the proposed method can more accurately identify pollution sources (100%), with a smaller source strength error of 2.01% for constant sources and one of 2.62% for attenuation sources. Full article

The Monte Carlo neutron transport method is used to accurately estimate various quantities, such as k-eigenvalue and integral neutron flux. However, in the case of estimating a distribution of a desired quantity, the Monte Carlo method does not typically provide continuous distribution. Recently, the functional expansion tally (FET) and kernel density estimation (KDE) methods have been developed to provide a continuous distribution of a Monte Carlo tally. In this paper, we propose a method to estimate a continuous distribution of a quantity in all phase-space variables using a fully connected feedforward artificial neural network (ANN) model with Monte Carlo-based training data. As a proof of concept, a continuous distribution of iterated fission probability (IFP) was estimated by ANN models in two distinct fissile systems. The ANN models were trained on the training data created using the Monte Carlo IFP method. The estimated IFP distributions by the ANN models were compared with the Monte Carlo-based data that include the training data. Additionally, the IFP distributions by the ANN models were also compared with the adjoint angular neutron flux distributions obtained with the deterministic neutron transport code PARTISN. The comparisons showed varying degrees of agreement or discrepancy; however, it was observed that the ANN models learned the general trend of the IFP distributions from the Monte Carlo-based training data. Full article

With the increasing number of deep rock engineering projects, many different types of tunnels have emerged, such as cross-tunnels. These tunnels intersect with each other in rock, which causes potential safety hazards. We must analyze the stability of the surrounding rock, to ensure worker safety. This article presents a method for dynamically assessing the stability of the surrounding rock in deep-buried cross-tunnels. The method consists of two main analysis steps: (1) P-wave velocity field inversion; and (2) Stability analysis of the surrounding rock. The P-wave velocity field inversion involves inverting the S-wave velocity field by Rayleigh wave and inverting the P-wave velocity field by adjoint state traveltime tomography. Then, a method of stability analysis is proposed which is used to update the mechanical properties of the rock (based on the continuously updated wave velocity field). The elastic modulus of the surrounding rock is approximated throughout the excavation process. CASRock V1.0 (Cellular Automation Software for engineering Rockmass fracturing processes) is used to assess rock damage via the equivalent plastic shear strain and local energy release rate. The new method is used to analyze the stability of a new tunnel excavated in Jinping (in China). The results reveal the severity and spatial distribution of the damage caused. The yield depth is concentrated near the sidewalls, while the top and bottom of the tunnel exhibit a smaller depth. The yield depths present a particular pattern of change (high–low–high–low) with increasing distance from tunnel #2. Finally, this research enriches our understanding of excavating deep cross-tunnels and makes an important contribution to improving worker safety in deep cross-tunnels. Full article

The method of characteristics is a classical method for gaining understanding in the solution of a partial differential equation. It has recently been applied to the adjoint equations of the 2D steady-state Euler equations and the first goal of this paper is to present a linear algebra analysis that greatly simplifies the discussion of the number of independent characteristic equations satisfied along a family of characteristic curves. This method may be applied for both the direct and the adjoint problem. Our second goal is to directly derive in conservative variables the characteristic equations of 2D compressible inviscid flows. Finally, the theoretical results are assessed for a nozzle flow with a classical scheme and its dual consistent discrete adjoint. Full article

We obtain an analytical solution for the time-optimal control problem in the induction phase of anesthesia. Our solution is shown to align numerically with the results obtained from the conventional shooting method. The induction phase of anesthesia relies on a pharmacokinetic/pharmacodynamic (PK/PD) model proposed by Bailey and Haddad in 2005 to regulate the infusion of propofol. In order to evaluate our approach and compare it with existing results in the literature, we examine a minimum-time problem for anesthetizing a patient. By applying the Pontryagin minimum principle, we introduce the shooting method as a means to solve the problem at hand. Additionally, we conducted numerical simulations using the MATLAB computing environment. We solve the time-optimal control problem using our newly proposed analytical method and discover that the optimal continuous infusion rate of the anesthetic and the minimum required time for transition from the awake state to an anesthetized state exhibit similarity between the two methods. However, the advantage of our new analytic method lies in its independence from unknown initial conditions for the adjoint variables. Full article

In the last decade, parameter-free approaches to shape optimization problems have matured to a state where they provide a versatile tool for complex engineering applications. However, sensitivity distributions obtained from shape derivatives in this context cannot be directly used as a shape update in gradient-based optimization strategies. Instead, an auxiliary problem has to be solved to obtain a gradient from the sensitivity. While several choices for these auxiliary problems were investigated mathematically, the complexity of the concepts behind their derivation has often prevented their application in engineering. This work aims to explain several approaches to compute shape updates from an engineering perspective. We introduce the corresponding auxiliary problems in a formal way and compare the choices by means of numerical examples. To this end, a test case and exemplary applications from computational fluid dynamics are considered. Full article

A continuous adjoint-based aeroacoustic optimization, based on a hybrid model including the Ffowcs Williams–Hawkings (FW–H) acoustic analogy, to account for the multidisciplinary design of aero-engine intakes with an axisymmetric geometry, is presented. To optimize such an intake, the generatrix of its lips is parameterized using B-Splines, and the energy contained in the sound pressure spectrum, at the blade passing frequency at receivers located axisymmetrically around the axis of the engine, is minimized. The engine is not included in the optimization and manifests its presence through an independently computed time-series of static pressure over the annular boundary of the simulation domain that corresponds to the inlet to the fan. Taking advantage of the case axisymmetry, the steady 3D RANS equations are solved in the rotating frame of reference and post-processed to compute the flow quantities’ time-series required by the FW–H analogy. The numerical solution of the unsteady flow equations and the otherwise excessive overall cost of the optimization are, thus, avoided. The objective function gradient is computed using the continuous adjoint method, coupled with the analytical differentiation of the FW–H analogy. The adjoint equations are also solved in the rotating frame via steady solver. Full article

In this paper, we study a kind of Stackelberg game where the controlled systems are described by backward stochastic differential delayed equations (BSDDEs). By introducing a new kind of adjoint equation, we establish the sufficient verification theorem for the optimal strategies of the leader and the follower in a general case. Then, we focus on the linear–quadratic (LQ) backward Stackelberg game with delay. The backward Stackelberg equilibrium is presented by the generalized fully coupled anticipated forward–backward stochastic differential delayed Equation (AFBSDDE), which is composed of anticipated stochastic differential equations (ASDEs) and BSDDEs. Moreover, we obtain the unique solvability of the AFBSDDE using the continuation method. As an application of the theoretical results, the pension fund problem with delay effect is considered. Full article