Discretization Research Articles

On nonlinear dynamic analysis of hyperbolic tangent FG-GPL-reinforced shallow spherical cap under supersonic flow

Shallow spherical caps are commonly encountered in the design of spacecraft components, such as fuel tanks or structural elements. Understanding the dynamic behavior can contribute to the identification of potential failure modes. The modulus of elasticity of the augmented composite upper layer is obtained using the model Halpin–Tsai, and this layer is augmented with graphene platelets (GPLs). Also, GPLs’ volume fraction from the top layer is considered for different cases of GPLs. The bottom layer is assumed functionally graded (FG) with two kinds, type A: FG layers and homogeneous core, and type B: FG core and homogeneous layers. To approximate the displacement field, the new hyperbolic tangent shear deformation theory (HTSDT) has been used which is no need for shear modification coefficients. By adopting Hamilton’s principle, the equations of motion are obtained and are solved by the discrete differential quadrature method (DQM) as well as Newmark's time marching scheme by implementing the famous Newton–Raphson iterative technique. When the FG index is infinite, it is observed that the point of maximum displacement decreases by 12.5%, and critical load increases by about 23% compared to the case where it is zero.

Numerical analysis of coupled thermal–hydraulic processes based on the embedded discrete fracture modeling method

The enhanced geothermal system (EGS) technology is considered an effective way to extract energy from deep hot dry rock (HDR) geothermal resources, which involves the coupling of complex thermal–hydraulic processes in fractured geological media. In this work, we propose a numerical approach based on the concept of embedded discrete fracture modeling method (EDFM), to estimate heat production performance from the EGS. We validated the method by comparing the outputs of our numerical models with experimental data and analytical solutions. A simplified, idealized two-dimensional model with discrete fractures was applied to gain insights on mass and heat transfer mechanisms expected under field conditions. Simulation results indicated that isolated and dead-end fractures have little influence on the overall performance of EGS, under the initial and boundary conditions of our numerical experiments (e.g., fracture density and permeability differences between fractures and matrix). It emerges that geometry and hydraulic transmissivity of fractures are key factors in controlling the effectiveness of the heat exchange between host rocks and circulating fluids. This work emphasizes the importance of implementing increasingly reliable descriptions of the fluid-dynamic and thermodynamic behavior of fractures, to increase the confidence in the numerical assessment of the performance of EGS installations.

Second-order Sobolev gradient flows for computing ground state of ultracold Fermi gases

Based on density functional theory, the ground state for ultracold Fermi gases with dipole–dipole interaction is a functional minimization problem with orthonormality constraints. Many methods such as direct minimization, self-consistent iteration method and first-order gradient flow had been proposed to solve such problem. In this paper, we introduce the second-order Sobolev gradient flows, solve them to the steady state and get the ground state of ultracold Fermi gases numerically. Two contributions have been made: firstly we extend the usual first-order gradient flows to the second-order gradient flows; secondly, we extend the usual L2 gradient to the Sobolev gradient and get the Sobolev gradient flows. We prove that the second-order Sobolev gradient flows have the properties of orthonormality preserving and ‘modified’ energy diminishing, which are desirable for the computation of the ground state solution. Several numerical techniques have been used to discretize the time-dependent second-order Sobolev gradient flows. These include explicit leap-frog method and stabilized semi-implicit method for temporal discretization and Fourier pseudo-spectral method for spatial variables. Extensive numerical examples for the ground state are reported to demonstrate the power of the numerical methods.

A novel discretized physics-informed neural network model applied to the Navier–Stokes equations

The advancement of scientific machine learning (ML) techniques has led to the development of methods for approximating solutions to nonlinear partial differential equations (PDE) with increased efficiency and accuracy. Automatic differentiation has played a pivotal role in this progress, enabling the creation of physics-informed neural networks (PINN) that integrate relevant physics into machine learning models. PINN have shown promise in approximating the solutions to the Navier–Stokes equations, overcoming the limitations of traditional numerical discretization methods. However, challenges such as local minima and long training times persist, motivating the exploration of domain decomposition techniques to improve it. Previous domain decomposition models have introduced spatial and temporal domain decompositions but have yet to fully address issues of smoothness and regularity of global solutions. In this study, we present a novel domain decomposition approach for PINN, termed domain-discretized PINN (DD-PINN), which incorporates complementary loss functions, subdomain-specific transformer networks (TRF), and independent optimization within each subdomain. By enforcing continuity and differentiability through interface constraints and leveraging the Sobolev (H 1) norm of the mean squared error (MSE), rather than the Euclidean norm (L 2), DD-PINN enhances solution regularity and accuracy. The inclusion of TRF in each subdomain facilitates feature extraction and improves convergence rates, as demonstrated through simulations of threetest problems: steady-state flow in a two-dimensional lid-driven cavity, the time-dependent cylinder wake, and the viscous Burgers equation. Numerical comparisons highlight the effectiveness of DD-PINN in preserving global solution regularity and accurately approximating complex phenomena, marking a significant advancement over previous domain decomposition methods within the PINN framework.

Relationship between TIGRE solar S-index and USET Ca ii K full disk images

Full disk observations of the solar chromosphere in the Ca ii K line represent a valuable dataset for studies of solar magnetic activity. The well known S-index is widely used to investigate the magnetic activity of stars, however, its connection to the coverage of stellar magnetic structure is still poorly understood. We use the archives of full disk Ca ii K images taken by the Royal Observatory of Belgium with the Uccle Solar Equatorial Table (USET) to derive the area fraction of the brightest chromospheric structures over the last decade. These data have allowed us to study the end of the solar cycle 24 and the beginning of the solar cycle 25. Our aim is to compare this dataset with the solar S-index from the Telescopio Internacional de Guanajuato Robotico Espectroscopico (TIGRE) lunar spectroscopy to analyze the relationship between a disk coverage index and an integrated spectrum. We also searched for periodic modulations in our two datasets to detect the solar rotation period. We used more than 2700 days of observations since the beginning of the Ca ii K observations with USET in July 2012. We performed a calibration of the images (re-centering and center-to-limb variation correction). The brightest regions of the solar surface (plages and enhanced network) were then segmented using an algorithm based on an intensity threshold. We computed the area fraction over the solar disk and compared it with the S-index from TIGRE. For the detection of periodic modulations, we applied a discrete Fourier power spectrum method to both datasets. A tight linear relationship was found between the USET area fraction and the TIGRE S-index, with an improved correlation obtained in the low-activity regime by considering the enhanced network. In both time series, we detected the modulation caused by the rotation of bright structures on the solar disk. However, this detection is constrained in the case of TIGRE due to its observation strategy. We studied the correlation between the disk coverage with chromospheric structures and the variability of the S-index on an overlapping period of ten years. We concluded that the disk coverage index is a good proxy for the S-index and will be useful in future studies of the magnetic activity of solar-type stars. The USET area fraction dataset is most appropriate for evaluating the solar rotation period and will be used in future works to analyze the impact of the inclination of the stellar rotation axis on the detectability of such periodic modulations in solar-type stars.

A discrete spectral method for time fractional fourth-order 2D diffusion-wave equation involving [formula omitted]-Caputo fractional derivative

In this work, the ψ-Caputo fractional derivative, as a generalization of the classical Caputo fractional derivative in which the fractional derivative of a sufficiently differentiable function is defined with respect to another strictly increasing function, ψ, is utilized to define the time fractional fourth-order 2D diffusion-wave equation. A Chebyshev–Gauss–Lobatto scheme is developed to solve this equation. In this way, a new operational matrix for the ψ-Riemann–Liouville fractional integration of the Chebyshev polynomials is derived. The solution of the equation is obtained by determining the solution of the algebraic system extracted from approximation the ψ-Caputo fractional derivative term via a finite discrete Chebyshev series and employing the expressed operational matrix. The validity of the established approach is examined by solving two examples.

A DEIM-CUR factorization with iterative SVDs

A CUR factorization is often utilized as a substitute for the singular value decomposition (SVD), especially when a concrete interpretation of the singular vectors is challenging. Moreover, if the original data matrix possesses properties like nonnegativity and sparsity, a CUR decomposition can better preserve them compared to the SVD. An essential aspect of this approach is the methodology used for selecting a subset of columns and rows from the original matrix. This study investigates the effectiveness of one-round sampling and iterative subselection techniques and introduces new iterative subselection strategies based on iterative SVDs. One provably appropriate technique for index selection in constructing a CUR factorization is the discrete empirical interpolation method (DEIM). Our contribution aims to improve the approximation quality of the DEIM scheme by iteratively invoking it in several rounds, in the sense that we select subsequent columns and rows based on the previously selected ones. Thus, we modify A after each iteration by removing the information that has been captured by the previously selected columns and rows. We also discuss how iterative procedures for computing a few singular vectors of large data matrices can be integrated with the new iterative subselection strategies. We present the results of numerical experiments, providing a comparison of one-round sampling and iterative subselection techniques, and demonstrating the improved approximation quality associated with using the latter.

Interpolation-based immersogeometric analysis methods for multi-material and multi-physics problems

AbstractImmersed boundary methods are high-order accurate computational tools used to model geometrically complex problems in computational mechanics. While traditional finite element methods require the construction of high-quality boundary-fitted meshes, immersed boundary methods instead embed the computational domain in a structured background grid. Interpolation-based immersed boundary methods augment existing finite element software to non-invasively implement immersed boundary capabilities through extraction. Extraction interpolates the structured background basis as a linear combination of Lagrange polynomials defined on a foreground mesh, creating an interpolated basis that can be easily integrated by existing methods. This work extends the interpolation-based immersed isogeometric method to multi-material and multi-physics problems. Beginning from level-set descriptions of domain geometries, Heaviside enrichment is implemented to accommodate discontinuities in state variable fields across material interfaces. Adaptive refinement with truncated hierarchically refined B-splines (THB-splines) is used to both improve interface geometry representations and to resolve large solution gradients near interfaces. Multi-physics problems typically involve coupled fields where each field has unique discretization requirements. This work presents a novel discretization method for coupled problems through the application of extraction, using a single foreground mesh for all fields. Numerical examples illustrate optimal convergence rates for this method in both 2D and 3D, for partial differential equations representing heat conduction, linear elasticity, and a coupled thermo-mechanical problem. The utility of this method is demonstrated through image-based analysis of a composite sample, where in addition to circumventing typical meshing difficulties, this method reduces the required degrees of freedom when compared to classical boundary-fitted finite element methods.

A hybrid analysis method for calculating the cogging torque of consequent pole hybrid excitation synchronous machine

PurposeWhen solving the cogging torque of complex electromagnetic structures, such as consequent pole hybrid excitation synchronous (CPHES) machine, traditional methods have a huge computational complexity. The notable feature of CPHES machine is the symmetric range of field-strengthening and field-weakening, but this type of machine is destined to be equipped with a complex electromagnetic structure. The purpose of this paper is to propose a hybrid analysis method to quickly and accurately solve the cogging torque of complex 3D electromagnetic structure, which is applicable to CPHES machine with different magnetic pole shapings.Design/methodology/approachIn this paper, a hybrid method for calculating the cogging torque of CPHES machine is proposed, which considers three commonly used pole shapings. Firstly, through magnetic field analysis, the complex 3D finite element analysis (FEA) is simplified to 2D field computing. Secondly, the discretization method is used to obtain the distribution of permeance and permeance differential along the circumference of the air-gap, taking into account the effect of slots. Finally, the cogging torque of the whole motor is obtained by using the idea of modular calculation and the symmetry of the rotor structure.FindingsThis method is applicable to different pole shapings. The experimental results show that the proposed method is consistent with 3D FEA and experimental measured results, and the average calculation time is reduced from 8 h to 4 min.Originality/valueThis paper proposes a new concept for calculating cogging torque, which is a hybrid calculation of dimension reduction and discretization modules. Based on magnetic field analysis, the 3D problem is simplified into a 2D issue, reducing computational complexity. Based on the symmetry of the machine structure, a modeling method for discretized analytical models is proposed to calculate the cogging torque of the machine.

Front row or backstage? Evidence on concert ticket preferences from a discrete choice experiment

Sophisticated ticketing practices have become widespread in the concert industry in recent years, with a wider range of musicians now experimenting with different ticket pricing schemes. The aim of these practices is to help musicians manage ticket capacity and maximize their concert income. However, there is limited evidence on how musicians can optimally allocate and price tickets with respect to how consumers value different ticket attributes. This study uses a stated preference discrete choice experiment and choice modeling methods to analyze consumer preferences for different attributes of concert tickets. The results of the modeling exercise highlight patterns in consumer preferences across different seating areas within a hypothetical venue, as well as average preferences for other common attributes of concert tickets. Finally, this study provides evidence of the significant welfare consumers derive from the availability of new ticketing innovations in the form of VIP packages.

Wearable Sensors for Athletic Performance: A Comparison of Discrete and Continuous Feature-Extraction Methods for Prediction Models

Wearable Sensors for Athletic Performance: A Comparison of Discrete and Continuous Feature-Extraction Methods for Prediction Models

Numerical solution of static and spatial kinetics self-adjoint angular flux neutron transport equation

This paper elucidates a comprehensive derivation of the variational formulation pertaining to the static and spatial kinetics self-adjoint angular flux (SAAF) neutron transport equation. The methodology employed for discretization of the spatial variable is the finite element method, while the energy group discretization is executed via the group method, and the directional discretization is conducted using the discrete ordinates method. Analytic expressions for the discretization to variable separation under both vacuum and reflective boundary conditions are furnished. Constructed upon the MOOSE framework, a code designated as SAAFCGSN has been developed for the resolution of the SAAF neutron transport equation. The zero-order scattering matrix within this computational framework is managed through an innovative "decoupling" method, thereby enhancing the computational efficiency significantly. The functionality and robustness of the SAAFCGSN code are corroborated through meticulous evaluation involving seven distinct steady-state scenarios as well as two transient states. Empirical outcomes verify the compatibility of the SAAFCGSN code with both structured and unstructured mesh, inclusive of their amalgamations, thus facilitating maintenance and ensuring elevated computational accuracy. In addition, a performance analysis benchmarked against IAEA standards reveals that the adoption of the scattering matrix "decoupling" method propels a computational speed increase exceeding 30 %, signifying a notable advancement in calculation efficiency.

Level-set topology optimization of heat sinks with phase-change material

Phase-change materials (PCMs) excel in storing significant thermal energy through the latent heat of fusion during phase changes. However, they often suffer from low thermal conductivity, requiring the incorporation of materials with higher thermal conductivity to overcome this limitation. In this study, we employ the level-set method to topologically optimize the distribution of both PCM and high thermal conductivity materials within heat sinks.The heat transfer process is modeled as an unsteady diffusion problem, with PCM integrated using the apparent heat capacity method. The finite element method, implemented using FEniCS, is utilized to compute the temperature distribution at each time step. The optimization problem is solved using ParaLeSTO, a modular topology optimization software written in C++, seamlessly interfacing with FEniCS through its Python interface.To compute boundary point sensitivities for level-set topology optimization, the discrete adjoint method is employed, combining a perturbation scheme with automatic differentiation. The proposed methodology is first applied to a two-dimensional example of temperature oscillation minimization with a heat flux boundary condition, drawn from existing literature. Subsequently, a two-dimensional example with multiple volumetric heat sources is studied, followed by an extension to a three-dimensional design domain.Comparisons with optimized designs without PCM, serving as a reference, highlight the superior thermal performance of heat sink designs with PCM. The results showcase a maximum temperature reduction of up to 42%, a mean temperature reduction of up to 42%, and a temperature variance reduction of up to 94%. This research contributes to heat sink design by offering discrete solutions that significantly improve thermal performance as compared to designs without PCM. Its impact extends to addressing crucial challenges in thermal management across various applications, including automotive cooling systems, aerospace components, and consumer electronics.

Numerical and experimental studies of airflows at exhaust hoods with inlet extensions

Local exhaust ventilation is used to capture contaminants and maintain a comfortable atmosphere in premises. Exhaust hoods with extension units (canopy hoods) are commonly used in open-type local exhaust devices. The goal of this study was to use computational and experimental methods to determine the effect of adding an extension to a local exhaust hood on the separated-flow streamlines, vortex zones, exhaust velocity distribution, and local drag coefficient. We used the discrete vortex method together with computational fluid dynamics to investigate air flows near round and slotted extended exhaust hoods. We then compared the computed airflow velocity field, vortex zone outlines at the inlet of a local exhaust hood, and local drag coefficient factor with data from a field experiment. Outlines of vortex zones resulting from flow separation at the inlet of the hood were considered for a hood length of 5 times gauge (the radius of a round hood or half-width of a slotted exhaust hood), hood tilt angles of 90°, 75°, 60°, 45°, and 30°, and extension lengths of 0; 0.5; 1; and 2 times gauge. We found that longer extensions caused the first vortex zone to enlarge, and the contaminant capture velocity of the local exhaust hood to decrease. We identified the behavior of the local drag coefficient at various hood tilt angles, extension lengths (as listed above), and hood lengths (including 2, 3, and 5 times gauge), and translated them into equations for computing the local drag coefficient.

A simple displacement perturbation method for phase-field modeling of ferroelectric thin film

A displacement perturbation method is proposed for phase-field simulations of the ferroelectric domain structures. A temperature-misfit strain phase diagram is computed to validate this method's accuracy, encompassing rhombohedral, orthorhombic, tetragonal, cubic, and mixed phases by comparing with previous phase-field simulations. The change in free energy density surface with temperature and strain distribution is computed to clarify the mechanism of the mixed phase. The phase diagram, ferroelectric hysteresis, and domain structure all demonstrate that the numerical method of displacement discretization is reliable and practical in solving the time-dependent Ginzburg–Landau equation, consistent with previous simulated and experimental results. This new method offers the advantages of simple programming and easy parallelism, paving the way for advancements in phase-field modeling.

The impact of wedge and wavy-wall combustor side walls on shock-induced mixing and combustion in a hydrogen-fuelled model scramjet combustor

The mixing and combustion processes are linked and challenging for a supersonic combustion engine. Supersonic air doesn't spend much time in the combustor, making it challenging to blend fuel with supersonic air. The expansion of shear-mixing layer with shock waves and their interactions are the primary characteristics of mixing augmentation. This paper investigates the same methodology for a model scramjet combustion chamber with wedge and wavy sides. The novel scramjet combustion models include wedge and way-wall sidewalls based on the basic scramjet model. By combining the finite volume approach with a second-order upwind discretization method, the Reynolds-averaged Navier-Stokes equations are numerically analyzed. The interplay between wedge and wavy-wall effects and free stream flows is taken into consideration by using the shear stress transport (SST) k−ω turbulence model. The effect of the wedge and wavy wall combustor has been evaluated by analysing the shock production and their interactions with the mixing layer and internal flow structure. The turbulence characteristics are also examined to analyse the degree of flow disturbance for wedge and wavy-wall combustion. The wavy wall combustor model results in a 14.5% increase in turbulence intensity, which raises air-fuel mixing and there by improves mixing and combustion efficiency by 12.59% and 11.23%, compared to the base wedge type combustor.

Computational assessment of soot models in ethylene/air laminar diffusion flames

To improve the accuracy of soot formation and evolution predictions, several physical and chemical models have been developed over the last decades. These models include (i) detailed chemical kinetic mechanisms describing both gas-phase chemistry related to combustion processes and reaction pathways leading to large-sized aromatic molecules, which are needed for modelling soot formation, and (ii) soot models providing a comprehensive description of soot particle dynamics and interactions with gas-phase chemical species. Accordingly, in this work, two detailed soot models, the method of moments (MOM) and the discrete sectional method (DSM), are evaluated in ethylene/air laminar diffusion flames, and their corresponding results are compared with experimental measurements. Furthermore, the NBP and KM2 chemical kinetic mechanisms are assessed and compared with each other by examining key chemical species related to soot formation and evolution. To compute gas mixture’s radiative properties, the weighted sum of grey gases model considering a grey medium is also utilised. Finally, the contributions of the soot precursors known as PAH (polycyclic aromatic hydrocarbon) to soot formation are also analysed. The main results show that the discrepancies in PAH concentrations obtained with different chemical kinetic mechanisms can be significant. In addition, compared to MOM ones, DSM results obtained here show a better agreement with experimental data. Finally, the analysis of PAH shows that those with two (A2) to four (A4) aromatic rings impact the most on soot modelling. Specifically, contributions of A4 were found to be more significant at lower heights above the burner, whereas A2 was found to be more impactful downstream as the flame develops. Maximum contributions of A2 and A4 to the soot inception rate were 66% and 85%, respectively, whereas the maximum summed contribution of PAH with five (A4R5) to seven (A7) aromatic rings accounted for only 13% of the inception rate.

A dynamic discrete model of ventilated concrete floor integrated with solar air collector under the effect of uneven direct solar radiation

Abstract A system combining the ventilated concrete floor (VCF) with solar air collector (SAC) in buildings has been applied in the western Sichuan Plateau for nighttime heating. However, the dynamic model coupled with uneven solar radiation of SAC-VCF is absent, thus the thermal behavior of the system is difficult to predict in practice. In this research, a coupled model is built to predict the thermal characteristics of the room with the SAC-VCF system under uneven solar radiation. The calculation model of VCF considers the uneven distribution of solar radiation on the floor surface, established by combining the resistance-capacitance (RC) network model and the number of transfer unit model (NTU) using the discrete method and validated by experimental data. A 3R2C model is utilized to model envelopes, validated by simulation results. The calculation error of surface temperature and room air temperature is within 5%. Then a case study is conducted with the validated model to predict the thermal performance of the SAC-VCF system with even and uneven solar radiation. Results indicate that under uneven solar radiation, the local surface temperature significantly increases to 35.1 °C, 9.1 °C higher than even solar radiation. Meanwhile, under uneven solar radiation, the heat transfer of supply air and surface of VCF is increased by 14% and 6%, respectively. Besides, the room air temperature is almost equal of two cases, while the operative temperature is 0.4 °C lower under uneven solar radiation. The model is beneficial to further study the influencing factors of this system.

A second-order Strang splitting scheme for the generalized Allen–Cahn type phase-field crystal model with FCC ordering structure

In this paper, we consider the generalized Allen–Cahn-type phase-field crystal model with face-centered-cubic ordering structure (PFC-FCC). Due to the combined complexity of the eighth-order spatial derivative and inherent nonlinearity, it poses a significant challenge to design a numerical scheme of high accuracy, stability, and efficiency to solve the PFC-FCC model. Endeavoring towards this objective, we adopt operator splitting method to address the PFC-FCC model. Based on Fourier spectral method for spatial discretization and SSP-RK method for temporal discretization, we propose an effective and easy-to-implement second-order scheme. We are also able to proffer an analytical proof for discrete mass conservation, and engage a discussion on an optimal error estimate for the second-order scheme. In addition, the energy stability of the numerical scheme is derived. Ultimately, a series of numerical experiments specifically designed to substantiate its accuracy, efficiency, and capability for phase transition are performed.

Establishment and Application of a Radiation Dose Rate Model for Nuclear Medicine Examinees.

Traditional methods for determining radiation dose in nuclear medicine include the Monte Carlo method, the discrete ordinate method, and the point kernel integration method. This study presents a new mathematical model for predicting the radiation dose rate in the vicinity of nuclear medicine patients. A new algorithm was created by combining the physical model of "cylinder superposition" of the human body with integral analysis to assess the radiation dose rate in the vicinity of nuclear medicine patients. The model accurately predicted radiation dose rates within distances of 0.1-3.0 m, with a deviation of less than 11% compared to observed rates. The model demonstrated greater accuracy at shorter distances from the radiation source, with a deviation of only 1.55% from observed values at 0.1 m. The model proposed in this study effectively represents the spatial and temporal distribution of the radiation field around nuclear medicine patients and demonstrates good agreement with actual measurements. This model has the potential to serve as a radiation dose rate alert system in hospital environments.