Search Results (17)

A composite-coated inclusion is embedded in a matrix, where the conductivity (permittivity) of the phases is assumed to be complex-valued. The purpose of this paper is to demonstrate that a non-zero flux can arise under specific conditions related to the conductivities of the components in the absence of external sources. These conditions are unattainable with conventional positive conductivities but can be satisfied when the conductivities are negative or complex—a scenario achievable in the context of metamaterials. The problem is formulated as a spectral boundary value problem for the Laplace equation, featuring a linear conjugation condition defined on a smooth curve L. This curve divides the plane ${\mathbb{R}}^2$ into two regions, $D^{+}$ and $D^{-}\ni \infty$. The spectral parameter appears in the boundary condition, drawing parallels with the Steklov eigenvalue problem. The case of a circular annulus is analyzed using the method of functional equations. The complete set of eigenvalues is derived by applying the classical theory of self-adjoint operators in Hilbert space. Full article

Mesh adaptation techniques can significantly impact Computational Fluid Dynamics by improving solution accuracy and reducing computational costs. In this review, we begin by defining the concept of mesh adaptation, its core components and the terminology commonly used in the community. We then categorise and evaluate the main adaptation strategies, focusing both on error estimation and mesh modification techniques. In particular, we analyse the two most prominent families of error estimation: feature-based techniques, which target regions of high physical gradients and goal-oriented adjoint methods, which aim to reduce the error in a specific integral quantity of interest. Feature-based methods are advantageous due to their reduced computational cost: they do not require adjoint solvers, and they have a natural ability to introduce anisotropy. A substantial portion of the literature relies on second-order derivatives of scalar flow quantities to construct sensors that can be equidistributed to minimise discretisation error. However, when used carelessly, these methods can lead to over-refinement, and they are generally outperformed by adjoint-based techniques when improving specific target quantities. Goal-oriented methods typically achieve higher accuracy in fewer adaptation steps with coarser meshes. It will be seen that various approaches have been developed to incorporate anisotropy into adjoint-based adaptation, including hybrid error sensors that combine feature-based and goal-oriented indicators, sequential strategies and adjoint weighting of fluxes. After years of limited progress, recent work has demonstrated promising results, including certifiable solutions and applications to increasingly complex cases such as transonic compressor blades and film-cooled turbines. Despite these advances, several critical challenges remain: efficient parallelisation, robust geometry integration, application to unsteady flows and deployment in high-order discretisation frameworks. Finally, examples of the potential role of artificial intelligence in guiding or accelerating mesh adaptation are also discussed. Full article

An important aspect of pressurized water reactor (PWR) lifetime monitoring is supporting radiation shielding analyses which can quantify various in-core and out-core effects induced in reactor materials by varying neutron–gamma fields. A good understanding of such a radiation environment during normal and accidental operating conditions is required by plant regulators to ensure proper shielding of equipment and working personnel. The complex design of a typical PWR is posing a deep penetration shielding problem for which an elaborate simulation model is needed, not only in geometrical aspects but also in efficient computational algorithms for solving particle transport. This paper presents such a hybrid shielding approach of FW-CADIS for characterization of the reactor pressure vessel (RPV) irradiation using SCALE6.2.4 code package. A fairly detailed Monte Carlo model (MC) of typical reactor internals was developed to capture all important streaming paths of fast neutrons which will backscatter the biological shield and thus enhance RPV irradiation through the cavity region. Several spatial differencing and angular segmentation options of the discrete ordinates S_N flux solution were compared in connection to a S_N mesh size and were inspected by VisIt code. To optimize MC neutron transport toward the upper RPV head, which is a particularly problematic region for particle transport, a deterministic solution of discrete ordinates in forward/adjoint mode was convoluted in a so-called contributon flux, which proved to be useful for subsequent S_N mesh refinement and variance reduction (VR) parameters preparation. The pseudo-particle flux of contributons comes from spatial channel theory which can locate spatial regions important for contributing to a shielding response. Full article

This work presents an illustrative application of the newly developed “Second-Order Features Adjoint Sensitivity Analysis Methodology for Neural Ordinary Differential Equations (2nd-FASAM-NODE)” methodology to determine most efficiently the exact expressions of the first- and second-order sensitivities of NODE decoder responses to the neural net’s underlying parameters (weights and initial conditions). The application of the 2nd-FASAM-NODE methodology will be illustrated using the Nordheim–Fuchs phenomenological model for reactor safety, which describes a short-time self-limiting power transient in a nuclear reactor system having a negative temperature coefficient in which a large amount of reactivity is suddenly inserted. The representative model responses that will be analyzed in this work include the model’s time-dependent total energy released, neutron flux, temperature and thermal conductivity. The 2nd-FASAM-NODE methodology yields the exact expressions of the first-order sensitivities of these decoder responses with respect to the underlying uncertain model parameters and initial conditions, requiring just a single large-scale computation per response. Furthermore, the 2nd-FASAM-NODE methodology yields the exact expressions of the second-order sensitivities of a model response requiring as few large-scale computations as there are features/functions of model parameters, thereby demonstrating its unsurpassed efficiency for performing sensitivity analysis of NODE nets. Full article

We are interested in the determination of the local nonlinear magnetic material behaviour in electrical steel sheets due to cutting and punching effects. For this purpose, the inverse problem has to be solved, where the objective function, which penalises the difference between the measured and the simulated magnetic flux density, has to be minimised under a constraint defined according to the corresponding partial differential equation model. We use the adjoint method to efficiently obtain the gradients of the objective function with respect to the material parameters. The optimisation algorithm is low-memory Broyden–Fletcher–Goldfarb–Shanno (BFGS), the forward and adjoint formulations are solved using the finite element (FE) method and the ill-posedness is handled via Tikhonov regularisation, in combination with the discrepancy principle. Realistic numerical case studies show promising results. Full article

The licensing of new reactors implies the use of verified and validated neutronic codes. Numerical validation can rely on sensitivity and uncertainty studies, but they require repeated execution of time-consuming neutron flux and depletion calculations. The computational costs can be reduced by using perturbation theories. However, the uncoupled Depletion Perturbation Theory is restricted to single integral values such as nuclide density. Relying on reduced-basis approaches, which reconstruct all nuclide densities at once, is one way to get around this restriction. Furthermore, the adjoint-based reduced-order model uses the direct and adjoint equations for projection. For diffusion or transport calculations, the Exact-to-Precision Generalized Perturbation Theory was developed. Still, no models for depletion calculations are readily available. Therefore, this paper describes a novel adjoint-based reduced-order model for the Bateman Equation. It uses a range-finding algorithm to create the basis and the uncoupled Depletion Perturbation Theory for the reconstruction of the first order replaced by with a first order formulation. Our paper shows that for several perturbed cases, the depletion reduced-order model successfully reconstructs the nuclide densities. As a result, this serves as a proof of concept for our adjoint-based reduced-order model, which can perform sensitivity and uncertainty burn-up analysis in a shorter time. Full article

Even though inhalation dosimetry is determined by three factors (i.e., breathing, aerosols, and the respiratory tract), the first two categories have been more widely studied than the last. Both breathing and aerosols are quantitative variables that can be easily changed, while respiratory airway morphologies are difficult to reconstruct, modify, and quantify. Although several methods are available for model reconstruction and modification, developing an anatomically accurate airway model and morphing it to various physiological conditions remains labor-intensive and technically challenging. The objective of this study is to explore the feasibility of using an adjoint–CFD model to understand airway shape effects on vapor deposition and control vapor flux into the lung. A mouth–throat model was used, with the shape of the mouth and tongue being automatically varied via adjoint morphing and the vapor transport being simulated using ANSYS Fluent coupled with a wall absorption model. Two chemicals with varying adsorption rates, Acetaldehyde and Benzene, were considered, which exhibited large differences in dosimetry sensitivity to airway shapes. For both chemicals, the maximal possible morphing was first identified and then morphology parametric studies were conducted. Results show that changing the mouth–tongue shape can alter the oral filtration by 3.2% for Acetaldehyde and 0.27% for Benzene under a given inhalation condition. The front tongue exerts a significant impact on all cases considered, while the impact of other regions varies among cases. This study demonstrates that the hybrid adjoint–CFD approach can be a practical and efficient method to investigate morphology-associated variability in the dosimetry of vapors and nanomedicines under steady inhalation. Full article

The use of coat-hanger dies is prevalent in the plastic film and sheet extrusion industry. The product quality and the power of the extrusion machine depend on the uniformities of the fluid velocity at the exit and the pressure drop. Die manufacturers face the challenge of producing coat-hanger dies that can extrude materials uniformly and with a minimal pressure drop. Previous studies have analyzed the die outlet’s flow homogeneity and pressure drop using various numerical simulations. However, the combination of the scheme programming language together with the Adjoint Method of Optimization has yet to be attempted. The adjoint optimization method has been demonstrated to be beneficial in addressing issues related to shape optimization problems and it may also be beneficial in optimizing the design of dies used in polymer melt extrusion. In this study, the proposed innovations involve incorporating both the Scheme programming language and Adjoint solver to examine and optimize the coat hanger’s flow homogeneity and pressure drop. Before optimization, the outlet velocity was almost 10 times higher at the die center than at the edges but after optimization, it became more uniform. The proposed optimized coat-hanger die geometry results in more uniform melt flow as demonstrated by the velocity contour plot and the outlet velocity graph in the die slit area, reducing the deviation value from 0.097 to 0.015. Additionally, the mass flux variance across the die outlet decreased by 71.6% from 0.015069 kg m⁻² s⁻¹ to 0.004281 kg m⁻² s⁻¹. Therefore, using this method reduces the amount of time wasted on trial and error or other optimization techniques that may be limited by design constraints. Full article

An adjoint data assimilation method was incorporated into the ECOM-si coastal ocean circulation model and applied to assimilate the flow field on the southern flank of Georges Bank from March to June 1999. The model was driven by tidal forcing consisting of ten tidal constituents at the open boundary, observed winds, and surface heat fluxes. Numerical experiments were conducted following a strategy to understand the critical issues affecting the efficiency and accuracy of the assimilated flow field. The adjoint data assimilation method significantly improved the computational accuracy of subtidal currents, especially for the along-isobath velocity. The integration window length and iteration number were two parameters affecting the assimilation convergence rate toward the observations. In such a nonlinear dynamical system, using a window length close to the M₂ tidal period could make the adjoint model difficult to converge, no matter how many iterations were made. Reducing the time interval for the window length speeded up the convergence rate, but it was paid out with the sacrifice of statistical confidence. This assimilation experiment used a 6-h window length, which led to a faster convergence rate in the first ten iterations. The best-assimilated fields that satisfied the error criteria were obtained as the iteration number increased. 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

The shielding calculation of neutron streaming problems with ducts is characterized by the strong anisotropy of angular flux, which poses a challenge for the analysis of nuclear installations. The discrete ordinate method is one of the most commonly deterministic techniques to solve the neutron transport equation, in which the accuracy and efficiency neutron are crucial to ensure the reliability of the streaming shielding simulation. We implemented the goal-oriented multi-collision source algorithm in the 3D transport code ARES. This algorithm can determine the importance factor based on the adjoint transport calculation, obtain the response function to enable problem-dependent, goal-oriented spatial decomposition, and provide the error estimation as a driving force behind the dynamic quadrature to optimize the source iteration. This study focuses on verifying the goal-oriented multi-collision source algorithm under the neutron streaming problems, and the capabilities of the algorithm have been tested on IRI-TUB benchmark of SINBAD database. The numerical results show that the algorithm can effectively control the angular discretization error for the neutron streaming problems, which is more economical than the traditional discrete ordinate calculation. Full article

This work illustrates the application of the n^th-order comprehensive adjoint sensitivity analysis methodology for response-coupled forward/adjoint linear systems (abbreviated as “n^th-CASAM-L”) to a paradigm model that describes the transmission of particles (neutrons and/or photons) through homogenized materials, as encountered in radiation protection and shielding. The first-, second-, and third-order sensitivities of responses that depend on both the forward and adjoint particle fluxes are obtained exactly, in closed-form, underscoring the principles and methodology underlying the n^th-CASAM-L. The results presented in this work underscore the fundamentally important role of the n^th-CASAM-L in the quest to overcome the “curse of dimensionality” in sensitivity analysis, uncertainty quantification and predictive modeling. Full article

In this paper, we study the effects of rotation in the spin-1/2 non-relativistic Aharonov-Bohm problem for bound states. We use a technique based on the self-adjoint extension method and determine an expression for the energies of the bound states. The inclusion of the spin element in the Hamiltonian guarantees the existence of bound state solutions. We perform a numerical analysis of the energies and verify that both rotation and the spin degree of freedom affect the energies of the particle. The main effect we observe in this analysis is a cutoff value manifested in the Aharonov-Bohm flux parameter that delimits the values for the positive and negative energies. Full article

In this work, the existence and nonexistence of stationary radial solutions to the elliptic partial differential equation arising in the molecular beam epitaxy are studied. Since we are interested in radial solutions, we focus on the fourth-order singular ordinary differential equation. It is non-self adjoint, it does not have exact solutions, and it admits multiple solutions. Here, $\lambda \in \mathbb{R}$ measures the intensity of the flux and G is stationary flux. The solution depends on the size of the parameter $\lambda$. We use a monotone iterative technique and integral equations along with upper and lower solutions to prove that solutions exist. We establish the qualitative properties of the solutions and provide bounds for the values of the parameter $\lambda$, which help us to separate existence from nonexistence. These results complement some existing results in the literature. To verify the analytical results, we also propose a new computational iterative technique and use it to verify the bounds on $\lambda$ and the dependence of solutions for these computed bounds on $\lambda$. Full article

The article is devoted to the development and creation of a multiphysics simulator that can, on the one hand, simulate the most significant physical processes in the IPMC actuator, and on the other hand, unlike commercial products such as COMSOL, can use computing resources economically. The developed mathematical model is an adjoint differential equation describing the transport of charged particles and water molecules in the ion-exchange membrane, the electrostatic field inside, and the mechanical deformation of the actuator. The distribution of the electrostatic potential in the interelectrode space is located by means of the solution of the Poisson equation with the Dirichlet boundary conditions, where the charge density is a function of the concentration of cations inside the membrane. The cation distribution was obtained by means of the solution of the equation system, in which the fluxes of ions and water molecules are described by the modified Nernst-Planck equations with boundary conditions of the third kind (the Robin problem). The cantilever beam forced oscillation equation in the presence of resistance (allowing for dissipative processes) with assumptions of elasticity theory was used to describe the actuator motion. A combination of the following computational methods was used as a numerical algorithm for the solution: the Poisson equation was solved by a direct method, the modified Nernst-Planck equations were solved by the Newton-Raphson method, and the mechanical oscillation equation was solved using an explicit scheme. For this model, a difference scheme has been created and an algorithm has been described, which can be implemented in any programming language and allows for fast computational experiments. On the basis of the created algorithm and with the help of the obtained experimental data, a program has been created and the verification of the difference scheme and the algorithm has been performed. Model parameters have been determined, and recommendations on the ranges of applicability of the algorithm and the program have been given. Full article