Scalar Modeling Research Articles

Scalar far-field diffraction modelling using nonuniform fast Fourier transform for diffractive optical phase elements design.

Due to their advantages of compact geometries and lightweight, diffractive optical elements (DOEs) are attractive in various applications such as sensing, imaging and holographic display. When designing DOEs based on algorithms, a diffraction model is required to trace the diffracted light propagation and to predict the performance. To have more precise diffraction field tracing and optical performance simulation, different diffraction models have been proposed and developed. However, they are limited in diffraction angles or still suffer from serious aberrations within the nonparaxial region in the far-field, which are not desired for the aforementioned applications. In this work, we developed an optimized diffraction modelling method using a nonuniform fast Fourier transform (NUFFT) to minimize the aberrations in the nonparaxial diffraction area in the far field for DOE design. The simulation result shows that the imaging distortion of DOE designed using iterative Fourier transform algorithm (IFTA) with integration of our proposed diffraction modelling method was effectively optimized. Moreover, the designed DOE has a diffraction efficiency of 90.73% and a root mean square error (RMSE) of 0.4817. It exhibits 7.17% higher in diffraction efficiency and 8.59% smaller in RMSE (0.0453), respectively, compared to DOE designed with a diffraction modelling method by directly taking nonuniform diffraction sampling points that are mapped from the diffracted wavefronts surface on the output plane, which has a diffraction efficiency of 83.56% and a RMSE of 0.5270. Furthermore, a compensation matrix was introduced into the developed diffraction model to further improve the imaging quality of designed DOE. A further increase of diffraction efficiency by 0.18% and a decrease of RMSE by 12.43% (0.0599) were achieved. In addition, we also utilized the proposed approach for DOE design in the case of off-axis diffraction, and diffraction fields with an incident illumination angle up to 30° can be reconstructed and simulated.

An Efficient Immersed Free Surface Boundary Method for 3-D Scalar Seismic Waves Finite-Difference Modeling in Presence of Topography

An Efficient Immersed Free Surface Boundary Method for 3-D Scalar Seismic Waves Finite-Difference Modeling in Presence of Topography

A 17-point composite average-derivative optimal method for high-accuracy frequency-domain scalar wave modeling

The frequency-domain finite-difference (FD) technique, especially the average-derivative method (ADM), is widely utilized in wavefield modeling. However, conventional low-order FD techniques are insufficient for high-accuracy demands. Applying high-order FD operator and optimizing their coefficients are effective measures for improving the accuracy of seismic modeling. However, these approaches suffer from the limitations of constant density media and approximant perfectly matched layer (PML). To address this issue, we have developed a composite-ADM 17-point scheme for a scalar-wave equation. This novel scheme is based on the composition of ADM discretizations with different grid intervals, which successfully removes the above limitations and achieves high accuracy. To suppress the numerical dispersion, we take the L∞ norm to construct the objective function, which we minimize using the Grey Wolf Optimizer (GWO) and Particle Swarm Optimization (PSO) algorithms. Numerical analysis indicates that the composite-ADM greatly reduces phase velocity error compared with the traditional-ADM, requiring only 2.44 grid points per wavelength within a 1% phase velocity error. Using numerical examples in homogeneous and complex models, we demonstrate the accuracy and versatility of the composite-ADM 17-point scheme in variable density media with exact PML. Furthermore, we extend this optimal scheme to the case of 2D viscous and 3D scalar-wave equations, making it a more practical and versatile solution.

Joint subgrid velocity-scalar filtered mass density function method for compressible turbulent flows

In large eddy simulations (LESs) and scalar filtered mass density functions, a gradient diffusion model is generally used for subgrid-scale (SGS) scalar flux modeling. However, this hypothesis is known to generate evident errors under certain conditions, particularly when the counter-gradient scalar transport effect dominates. Herein, a joint subgrid velocity-scalar filtered mass density function (VSGSSFMDF) method is developed to address this problem. The exact FMDF transport equation is derived in detail. Based on the order of the magnitude analysis and simply Langevin model, the FMDF transport equation is modeled, and a system of stochastic differential equations is, thus, proposed. Theoretical derivation and analysis suggest that both the first- and second-order moments are consistent. A compressible mixing layer and hydrogen/air-reactive mixing layer are simulated to verify the proposed method. Based on the diffusion model, a direct numerical simulation and an LES are performed for comparative verification. A generally reasonable SGS velocity distribution is obtained using the proposed VSGSSFMDF method. Consequently, the counter-gradient scalar transport effect is effectively simulated using this method. Overall, the VSGSSFMDF produces better results than the SFMDF and LES in both cases, particularly in the reactive case.

Improved Delayed Detached Eddy Simulation with Reynolds-stress background modelling

A novel variant of Improved Delayed Detached-Eddy Simulation based on a differential Reynolds-stress background model is presented. The approach aims to combine the advantages of anisotropy-resolving Reynolds-stress closures in the modelled RANS regions with consistent LES and wall-modelled LES behaviour in the resolved flow regions. In computations of decaying isotropic turbulence with a low-dissipative flow solver it is shown that a straightforward hybridised Reynolds-stress model provides insufficient turbulent dissipation as sub-grid closure in the LES regions and is therefore locally replaced by scalar viscosity modelling. Simulations of periodic channel flows at different Reynolds numbers and grid resolutions are used to calibrate and validate the wall-modelled LES branch of the new model. A final application in embedded wall-modelled LES of a flat-plate boundary layer is widely consistent with results using the SST-RANS background model, but shows some deviations from the Coles–Fernholz skin-friction correlation. In this regard, initial sensitivity studies indicate possible adverse effects due to the synthetic-turbulence approach used in these simulations.

Efficient temporal high-order staggered-grid scheme with a dispersion-relation-preserving method for the scalar wave modeling

Staggered-grid finite-difference (FD) method is widely used to solve the wave equation for the numerical seismic modeling, and it is a key step of the advanced seismic imaging and inversion problem. However, the conventional FD method is prone to instability and dispersion error due to the insufficient approximation accuracy. In this work, we propose an efficient temporal high-order finite-difference (FD) scheme with the cross-rhombus stencil. Compared with the standard cross-rhombus method, the new method has less computational cost due to we simplify the FD scheme. Moreover, the dispersion relation of the new method is easy to obtain the dispersion-relation-preserving (DRP) FD coefficients, which can significantly alleviate the spatial and temporal dispersion errors. Dispersion and stability analyses indicate that the new scheme has better performance in seismic modeling than the conventional method, and numerical experiments also indicate that the new scheme can effectively mitigate dispersion error and improve the numerical accuracy.

2D scalar wave modelling of apodized RF BAW resonators for transverse mode analysis

This paper proposes the use of a two-dimensional scalar wave model for fast analysis of apodized AlN-based radio frequency bulk acoustic wave devices. First, the working principle and model setup is introduced. Only one particular Lamb mode (S1−) is considered, and its behavior is modeled by a simple scalar wave equation looking from the top surface. Multiple layers are added to the side boundaries so that mode conversion to other modes is treated as leakage to surroundings. The wavenumber of field distribution for each resonance mode is calculated, and then its excitation coefficient, quality factor, and resonance frequency are derived. The device input admittance is calculated by applying derived parameters to the equivalent circuit. A series of oval-shaped AlN-based film bulk acoustic resonator devices are simulated and compared with experimental results. The theoretical results agreed well with the experimental ones, and accuracy and effectiveness of the present analysis are verified.

Stochastic modelling of scalar time series of varicella incidence for a period of 92 years (1928-2019).

Varicella is an acute, highly contagious disease, characterised by generalised vesicular exanthema caused by the initial infection with varicella zoster virus (VZV) which usually affects children aged 2 to 8 years. To analyse the changes of varicella incidence in Bulgaria over the period of 1928-2019. The time series analysis is based on the official data for varicella incidence (per 100,000) in Bulgaria for ninety-two years (1928-2019), obtained from three major sources. We utilized the method to construct a time series model of overall incidence (1928-2019) using time series modeller in SPSS v. 25. We followed all three steps of the standard ARIMA methodology to establish the model - identification, parameter estimation, and diagnostic checking. Stochastic scalar time series modelling of the varicella incidence from 1928 to 2019 was performed. The stochastic ARIMA (0,1,1) was identified to be the most appropriate model. The decomposition of varicella incidence time series into a stochastic trend and a stationary component was reasoned based on the model defined. In addition, we assessed the importance of the long-term and immediate effect of one shock. The long-term forecast was also under discussion. The ARIMA model (0,1,1) in our study is an adequate tool for presenting the varicella incidence trend and is suitable to forecast near future disease dynamics with acceptable error tolerance.

A new linear optimized time–space domain spatial implicit and temporal high-order finite-difference scheme for scalar wave modeling

Suppressing the numerical dispersion is one of the key items for the finite-difference (FD) method. Usually, approximating the spatial derivatives by the implicit FD stencil is an effective approach to suppress the spatial dispersion for acoustic wave modeling. However, the temporal accuracy is still limited. To tackle this issue, we propose a new time–space domain (TS-D) implicit FD scheme, which could reach the arbitrary even-order temporal and spatial accuracy by adding a few additional grid points to the original implicit FD stencil. Compared with the existing spatial implicit and temporal high-order methods in the TS-D, our new scheme reduces 4N + 1 grid points, while reaches equivalent temporal and spatial high-order accuracy. To further enhance the simulation accuracy, a linear optimization strategy is developed by combining the Taylor series expansion (TE) and the least-squares methods. Our linear optimization strategy avoids the non-linear iterative optimization of the current spatial implicit and temporal high-order methods. Comparisons of the conventional TE- and optimization-based implicit methods demonstrate the accuracy and efficiency superiorities of our new linear optimized FD scheme.

Implications of inertial subrange scaling for stably stratified mixing

We investigate the effects of the turbulent dynamic range on active scalar mixing in stably stratified turbulence by adapting the theoretical passive scalar modelling arguments of Beguier, Dekeyser & Launder (1978) (Phys. Fluids, vol. 21 (3), pp. 307–310) and demonstrating their usefulness through consideration of the results of direct numerical simulations of statistically stationary homogeneous stratified and sheared turbulence. By analysis of inertial and inertial–convective subrange scalings, we show that the relationship between the active scalar and turbulence time scales is predicted by the ratio of the Kolmogorov and Obukhov–Corrsin constants, provided mean flow parameters permit the two subrange scalings to be appropriate approximations. We use the resulting relationship between time scales to parameterise an appropriate turbulent mixing coefficient $\varGamma \equiv \chi /\epsilon$ , defined here as the ratio of available potential energy ( $E_p$ ) and turbulent kinetic energy ( $E_k$ ) dissipation rates. With the analysis presented here, we show that $\varGamma$ can be estimated by $E_p,E_k$ and a universal constant provided an appropriate Reynolds number is sufficiently high. This large Reynolds number regime appears here to occur at $ {{Re_b}} \equiv \epsilon / \nu N^{2} \gtrapprox 300$ where $\nu$ is the kinematic viscosity and $N$ is the characteristic buoyancy frequency. We propose a model framework for irreversible diapycnal mixing with robust theoretical parametrisation and asymptotic behaviour in this high- $ {{Re_b}}$ limit.

Time–space domain scalar wave modeling by a novel hybrid staggered-grid finite-difference method with high temporal and spatial accuracies

Staggered-grid finite-difference (SFD) scheme is favored in wave equation simulation due to its superior accuracy and stability to center-grid finite-difference (CFD) scheme. However, for the scalar wave equation (SWE) modeling, the conventional SFD (CSFD) scheme only reaches second-order accuracy in space and time for the discrete SWE; the recently developed modified SFD (MSFD) scheme improves the accuracy to (2N)th-order (N<4) but is costly because the MSFD scheme is coined by adding many extra grid points to the CSFD scheme. To tackle these issues, we develop a cost-effective hybrid SFD (HSFD) scheme, which combines the features of the CSFD and MSFD schemes; we prove that the new HSFD scheme can simultaneously reach (2N)th-order accuracy in space and time for the discrete wave equation. In addition, to deal with the optimization difficulties due to the nonlinear dispersion relation of the SFD schemes, we propose a two-step linear optimization method to improve the accuracy of the new HSFD scheme. The analyses on dispersion, stability properties and numerical simulation examples demonstrate that the new HSFD scheme owns better accuracy and stability than the CSFD scheme. Computational cost analysis shows that the HSFD scheme can be more efficient than the MSFD scheme because it only requires half the additional grid points.

A novel hybrid method based on discontinuous Galerkin method and staggered‐grid method for scalar wavefield modelling with rough topography

ABSTRACTIn recent years, the discontinuous Galerkin method and the finite‐difference method have been widely applied to simulate seismic wave propagation. However, few studies have focused on the combination of the finite‐difference method with the discontinuous Galerkin method. We develop a new hybrid algorithm based on domain decomposition that combines the high efficiency of the finite‐difference method with the flexibility of the discontinuous Galerkin method. A computational domain is decomposed into two types of subregions where the finite‐difference method is applied in the main part of the model, whereas the discontinuous Galerkin method is employed to handle the complex structure with a free surface or caves. Total variation diminishing Runge–Kutta time discretization is used to enhance the stability of implementation. The approach inherits the advantages of the discontinuous Galerkin method and the finite‐difference method and does not need a transition zone, which reduces the computational cost caused by the additional conversion. In addition, some mesh patterns are provided to make the discontinuous Galerkin method domain discretized as needed without changing the grid size of the finite‐difference method region, which makes our hybrid algorithm more flexible. Numerical experiments prove that our hybrid scheme is capable of dealing with complex structures and maintains moderate computational efficiency.

On the generality of tensor basis neural networks for turbulent scalar flux modeling

This study focuses on Reynolds-averaged Navier Stokes (RANS) models for passive scalar transport. A recently developed machine learning turbulence closure, the tensor basis neural network for scalar flux modeling (TBNN-s), is applied on two different datasets, an inclined jet in crossflow and a wall mounted cube in crossflow. The TBNN-s consists of a deep neural network that predicts a turbulent diffusivity matrix after being trained on high fidelity datasets. The present work addresses the generality question (i.e. will a model trained in one turbulent flow perform well in a different turbulent flow?) by training and testing across different classes of flows. We show that when trained and tested in the same class of flow, the TBNN-s obtains significant improvements in the mean scalar field over a common baseline model. When trained and tested across different classes of flows, the TBNN-s has a similar performance to the baseline model, which is encouraging since training set extrapolation does not produce the nonphysical results some fear. Further analysis suggests that varying training sets for the same test case produces similar results in terms of alignment of the scalar flux vector, but different turbulent diffusivity matrix magnitudes.

Physical invariance in neural networks for subgrid-scale scalar flux modeling

In this paper we present a new strategy to model the subgrid-scale scalar flux in a three-dimensional turbulent incompressible flow using physics-informed neural networks (NNs). When trained from direct numerical simulation (DNS) data, state-of-the-art neural networks, such as convolutional neural networks, may not preserve well known physical priors, which may in turn question their application to real case-studies. To address this issue, we investigate hard and soft constraints into the model based on classical transformation invariances and symmetries derived from physical laws. From simulation-based experiments, we show that the proposed transformation-invariant NN model outperforms both purely data-driven ones as well as parametric state-of-the-art subgrid-scale models. The considered invariances are regarded as regularizers on physical metrics during the a priori evaluation and constrain the distribution tails of the predicted subgrid-scale term to be closer to the DNS. They also increase the stability and performance of the model when used as a surrogate during a large-eddy simulation. Moreover, the transformation-invariant NN is shown to generalize to regimes that have not been seen during the training phase.

Conditional scalar dissipation rate modeling for turbulent spray flames using artificial neural networks

Machine learning techniques have been used for the closure of the conditional scalar dissipation rate in turbulent spray flames. Statistical data are extracted from carrier-phase direct numerical simulation (CP-DNS) results for the generation of artificial neural networks that are trained to predict the conditional scalar dissipation rate such that they could serve as a sub-grid model for large eddy simulations of spray combustion. A quantitative error assessment suggests that predictions of the conditionally averaged dissipation rate are in excellent agreement with CP-DNS data. Further comparison with commonly used models for the unconditionally filtered dissipation rate promises significant improvements if ANNs are used for closure. The artificial neural networks also help to identify the important features that affect the local dissipation rates. The results suggest that few - mostly droplet related - parameters suffice as input features for accurate ANNs. This is in contradiction to standard modeling techniques that are solely based on gas phase properties and highlights the need to revisit scalar dissipation rate modeling for spray flames if analytical expressions are to be used.

Tensorial description of the plasticity of amorphous composites.

We use a continuous mesoscopic model to address the yielding properties of plastic composites, formed by a host material and inclusions with different elastic and/or plastic properties. We investigate the flow properties of the composed material under a uniform externally applied deviatoric stress. We show that due to the heterogeneities induced by the inclusions, a scalar modeling in terms of a single deviatoric strain of the same symmetry as the externally applied deformation gives inaccurate results. A realistic modeling must include all possible shear deformations. Implementing this model in a two-dimensional system, we show that the effect of harder inclusions is very weak to relatively high concentrations. For softer inclusions instead, the effect is much stronger; even a small concentration of inclusions affecting the form of the flow curve and the critical stress. We also present the details of a full three-dimensional simulation scheme and obtain the corresponding results for harder and softer inclusions.

Entropy solutions and flux identification of a scalar conservation law modelling centrifugal sedimentation

Centrifugal sedimentation of an ideal suspension in a rotating tube or basket can be modelled by an initial-boundary-value problem for a scalar conservation law with a nonconvex flux function. The sought unknown is the volume fraction of solids as function of radial distance and time for constant initial data. The method of characteristics is used to construct entropy solutions. The qualitatively different solutions, which depend on the initial value and the vessel radial coordinates, are presented in detail along with numerical simulations. Based on the entropy solutions, a new method of flux identification, which does not require any prescribed functional expression, is presented and illustrated with synthetic data.

Existence of periodic solutions for a scalar differential equation modelling optical conveyor belts

We study a one-dimensional ordinary differential equation modelling optical conveyor belts, showing in particular cases of physical interest that periodic solutions exist. Moreover, under rather general assumptions it is proved that the set of periodic solutions is bounded.

Double-grid quadrature with interpolation-projection (DoGIP) as a novel discretisation approach: An application to FEM on simplexes

This paper is focused on the double-grid integration with interpolation-projection (DoGIP), which is a novel matrix-free discretisation method of variational formulations introduced for Fourier–Galerkin approximation. Here, it is described as a more general approach with an application to the finite element method (FEM) on simplexes. The approach is based on treating the trial and a test function in variational formulation together, which leads to the decomposition of a linear system into interpolation and (block) diagonal matrices. It usually leads to reduced memory demands, especially for higher-order basis functions, but with higher computational requirements. The numerical examples are studied here for two variational formulations: weighted projection and scalar elliptic problem modelling, e.g. diffusion or stationary heat transfer. This paper also opens a room for further investigation, which is discussed in the conclusion.

A novel method for expressing anisotropy of subgrid-scale components for thermal and scalar fields

Some types of mixed subgrid-scale (SGS) models combining an isotropic eddy-viscosity model and a scale-similarity model can be used to effectively increase the accuracy of large eddy simulation (LES). For example, Abe (2013) recently proposed a stabilized mixed model, which can successfully express the anisotropy of the SGS stress, and remarkably improves the predictive performance for wall turbulence at coarse grid resolutions without sacrificing computational stability. In the present work, this approach is extended for thermal and scalar field modeling to express the anisotropy of SGS heat (scalar) flux. A priori tests using direct numerical simulation data demonstrated that the proposed model can accurately predict the anisotropic SGS heat flux in channel flow and its variation with Prandtl number in the range 0.1–2. Its effectiveness is also confirmed by a posteriori tests under flow conditions corresponding to those in the a priori tests.