Diffusion Approximation Method Research Articles

Phase-field formulated meshless simulation of axisymmetric Rayleigh-Taylor instability problem

A formulation of the immiscible Newtonian two-liquid system with different densities and influenced by gravity is based on the Phase-Field Method (PFM) approach. The solution of the related governing coupled Navier-Stokes (NS) and Cahn-Hillard (CH) equations is structured by the meshless Diffuse Approximate Method (DAM) and Pressure Implicit with Splitting of Operators (PISO). The variable density is involved in all the terms. The related moving boundary problem is handled through single-domain, irregular, fixed node arrangement in Cartesian and axisymmetric coordinates. The meshless DAM uses weighted least squares approximation on overlapping subdomains, polynomial shape functions of second-order and Gaussian weights. This solution procedure has improved stability compared to Chorin's pressure-velocity coupling, previously used in meshless solutions of related problems. The Rayleigh-Taylor instability problem simulations are performed for an Atwood number of 0.76. The DAM parameters (shape parameter of the Gaussian weight function and number of nodes in a local subdomain) are the same as in the authors’ previous studies on single-phase flows. The simulations did not need any upwinding in the range of the simulations. The results compare well with the mesh-based finite volume method studies performed with the open-source code Gerris, Open-source Field Operation and Manipulation (OpenFOAM®) code and previously existing results.

Phase-field formulated meshless simulation of Rayleigh-Taylor instability problem

The interface between two immiscible Newtonian liquids with different densities and the same viscosity, influenced by gravity, is based on the Phase-Field Method (PFM) formulation. The solution of the related governing coupled Navier-Stokes (NS) and Cahn-Hillard (CH) equations is structured by the meshless Diffuse Approximate Method (DAM) and Pressure Implicit with Splitting of Operators (PISO). The variable density is involved in the inertial and buoyancy terms (non-Boussinesq formulation). The related moving boundary problem is handled through single-domain, irregular, fixed node arrangement in two-dimensional Cartesian coordinates. The meshless DAM uses weighted least squares approximation on overlapping subdomains, polynomial shape functions of second-order and Gaussian weights. Implicit time discretisation is performed for the NS and CH equations in the momentum predictor and Phase-Field (PF) variable corrector steps of PISO, while the momentum corrector steps solve the NS equation explicitly. This solution procedure has improved stability compared to Chorin’s pressure-velocity coupling, previously used in meshless solutions of related problems. The Rayleigh-Taylor instability problem simulations are performed for an Atwood number of 0.76. The DAM parameters (shape parameter of the Gaussian weight function and number of nodes in a local subdomain) are the same as in the author’s previous studies on single-phase flows. The simulations did not need any upwinding in the range of the simulations. The results compare well with the mesh-based finite volume method studies performed with the open-source code Gerris.

Radiation transport methods in star formation simulations

Radiation transport plays a crucial role in star formation models, as certain questions within this field cannot be accurately addressed without taking it into account. Given the high complexity of the interstellar medium from which stars form, numerical simulations are frequently employed to model the star formation process. This study reviews recent methods for incorporating radiation transport into star formation simulations, discussing them in terms of the used algorithms, treatment of radiation frequency dependence, the interaction of radiation with the gas, and the parallelization of methods for deployment on supercomputers. Broadly, the algorithms fall into two categories: i) moment-based methods, encompassing the flux-limited diffusion approximation, M1 closure, and variable Eddington tensor methods, and ii) methods directly solving the radiation transport equation, including forward and reverse ray tracing, characteristics-based methods, and Monte Carlo techniques. Beyond discussing advantages and disadvantages of these methods, the review also lists recent radiation hydrodynamic codes implemented the described methods.

Kernel-based diffusion approximated Markov decision processes for autonomous navigation and control on unstructured terrains

We propose a diffusion approximation method to the continuous-state Markov decision processes that can be utilized to address autonomous navigation and control in unstructured off-road environments. In contrast to most decision-theoretic planning frameworks that assume fully known state transition models, we design a method that eliminates such a strong assumption that is often extremely difficult to engineer in reality. We first take the second-order Taylor expansion of the value function. The Bellman optimality equation is then approximated by a partial differential equation, which only relies on the first and second moments of the transition model. By combining the kernel representation of the value function, we design an efficient policy iteration algorithm whose policy evaluation step can be represented as a linear system of equations characterized by a finite set of supporting states. We first validate the proposed method through extensive simulations in 2D obstacle avoidance and 2.5D terrain navigation problems. The results show that the proposed approach leads to a much superior performance over several baselines. We then develop a system that integrates our decision-making framework with onboard perception and conduct real-world experiments in both cluttered indoor and unstructured outdoor environments. The results from the physical systems further demonstrate the applicability of our method in challenging real-world environments.

Physics-guided neural network for tissue optical properties estimation.

Finding the optical properties of tissue is essential for various biomedical diagnostic/therapeutic applications such as monitoring of blood oxygenation, tissue metabolism, skin imaging, photodynamic therapy, low-level laser therapy, and photo-thermal therapy. Hence, the research for more accurate and versatile optical properties estimation techniques has always been a primary interest of researchers, especially in the field of bioimaging and bio-optics. In the past, most of the prediction methods were based on physics-based models such as the pronounced diffusion approximation method. In more recent years, with the advancement and growing popularity of machine learning techniques, most of the prediction methods are data-driven. While both methods have been proven to be useful, each of them suffers from several shortcomings that could be complemented by their counterparts. Thus, there is a need to bring the two domains together to obtain superior prediction accuracy and generalizability. In this work, we proposed a physics-guided neural network (PGNN) for tissue optical properties regression which integrates physics prior and constraint into the artificial neural network (ANN) model. With this method, we have demonstrated superior generalizability of PGNN compared to its pure ANN counterpart. The prediction accuracy and generalizability of the network were evaluated on single-layered tissue samples simulated with Monte Carlo simulation. Two different test datasets, the in-domain test dataset and out-domain dataset were used to evaluate in-domain generalizability and out-domain generalizability, respectively. The physics-guided neural network (PGNN) showed superior generalizability for both in-domain and out-domain prediction compared to pure ANN.

Meshless simulation of a macrosegregation benchmark considering the solid motion

We have extended the existing two-dimensional rigid solid phase benchmark for binary substance with the solid phase motion in the present paper. Incompressible laminar Newtonian flow is assumed, and a standard mixture formulation is used for the mass, momentum, energy, and solute transport. A coherency solid motion model accounts for the free-floating grains, assuming that the solid velocity is proportional to the mixture velocity and the liquid fraction. The lever rule is used to describe the mass fractions of the phases. A two-dimensional benchmark is solved using the semi-implicit meshless diffuse approximate method with an adaptive subdomain upwinding strategy. The results of the meshless method are compared to the finite volume method results with a reasonable agreement. The new benchmark results show that the solid motion has an essential effect on the macrosegregation pattern.

Simulations of dendritic solidification via the diffuse approximate method

The current work presents simulations of two-dimensional dendritic solidification via the meshless Diffuse Approximate Method (DAM). The presumed Stefan problem is studied through the phase-field model. Isotropic and anisotropic materials are considered for comparisons with the benchmark tests. Investigations on the change of some constants are carried out to discover their effects on the obtained patterns. The nodal DAM (with the Kronecker delta property of its shape functions) for the spatial discretization and the forward Euler temporal discretization for the coupled phase-field and heat equations provide results that comply with the patterns of previous works. The significant feature of the proposed numerical method manifests through the explicit time marching and the local property of the chosen meshless method that ensues solving small-size linear systems for each subdomain.

A meshless solution of the compressible viscous flow in axisymmetric tubes with varying cross-sections

The meshless Diffuse Approximate Method (DAM) is for the first time formulated and applied to simulate the compressible Newtonian flow of an ideal gas in an axisymmetric tube with varying cross-sections. The problem is structured by coupled partial differential equations for conservation of mass, momentum and energy, and the equation of state closure. These equations are solved in primitive variables and strong form. DAM is formulated on irregular node arrangement by using the second and third-order polynomial shape functions and Gaussian weights, leading to weighted least squares approximation on overlapping local subdomains. Pressure-velocity coupling is performed by the Pressure Implicit with Splitting of Operators (PISO) scheme. The solution of the represented novel application of DAM is verified by matching the meshless results with the fine-mesh finite-volume method results. The characteristics of meshless DAM for this kind of problem are systematically assessed by a detailed investigation of the varying node density, shape function order, Gaussian weight's shape, and the number of nodes in a local subdomain. The sensitivity study of DAM parameters shows that for the tested problems, the most suitable values of the Gaussian weight function and the number of nodes in a local subdomain are 5.0 and 13, respectively. Third-order convergence rate with better results is observed while using third-order polynomial shape functions.

Development and verification of meshless diffuse approximate method for simulation of compressible flow between parallel plates

A meshless numerical model is developed to simulate single-phase, Newtonian, compressible flow in the Cartesian coordinate system. The coupled set of partial differential equations, i.e., mass conservation, momentum conservation, energy conservation, and equation of state is solved by using Diffuse Approximate Method (DAM) and Pressure Implicit with Splitting of Operators (PISO) pressure correction algorithm on an irregular node arrangement. DAM is structured by using the second-order polynomial basis functions and the Gaussian weight function, leading to the weighted least squares approximation on overlapping sub-domains. Implicit time discretization is performed for the predictor step of PISO, while in the corrector steps the equations are discretized explicitly. The numerical model is validated for flow between parallel plates with helium obeying ideal gas law. The solver’s accuracy is assessed by investigating the shape of the Gaussian weight and the number of the nodes in the local subdomains. The calculated velocity, temperature and pressure fields are compared with the Finite Volume Method (FVM) results obtained by OpenFOAM software and show a reasonably good agreement.

A Data-Driven Space-Time-Parameter Reduced-Order Model with Manifold Learning for Coupled Problems: Application to Deformable Capsules Flowing in Microchannels.

An innovative data-driven model-order reduction technique is proposed to model dilute micrometric or nanometric suspensions of microcapsules, i.e., microdrops protected in a thin hyperelastic membrane, which are used in Healthcare as innovative drug vehicles. We consider a microcapsule flowing in a similar-size microfluidic channel and vary systematically the governing parameter, namely the capillary number, ratio of the viscous to elastic forces, and the confinement ratio, ratio of the capsule to tube size. The resulting space-time-parameter problem is solved using two global POD reduced bases, determined in the offline stage for the space and parameter variables, respectively. A suitable low-order spatial reduced basis is then computed in the online stage for any new parameter instance. The time evolution of the capsule dynamics is achieved by identifying the nonlinear low-order manifold of the reduced variables; for that, a point cloud of reduced data is computed and a diffuse approximation method is used. Numerical comparisons between the full-order fluid-structure interaction model and the reduced-order one confirm both accuracy and stability of the reduction technique over the whole admissible parameter domain. We believe that such an approach can be applied to a broad range of coupled problems especially involving quasistatic models of structural mechanics.

Modeling Bus Capacity for Bus Stops Using Queuing Theory and Diffusion Approximation

For bus service quality and line capacity, one critical influencing factor is bus stop capacity. This paper proposes a bus capacity estimation method incorporating diffusion approximation and queuing theory for individual bus stops. A concurrent queuing system between public transportation vehicles and passengers can be used to describe the scenario of a bus stop. For most of the queuing systems, the explicit distributions of basic characteristics (e.g., waiting time, queue length, and busy period) are difficult to obtain. Therefore, the diffusion approximation method was introduced to deal with this theoretical gap in this study. In this method, a continuous diffusion process was applied to estimate the discrete queuing process. The proposed model was validated using relevant data from seven bus stops. As a comparison, two common methods— Highway Capacity Manual (HCM) formula and M/M/S queuing model (i.e., Poisson arrivals, exponential distribution for bus service time, and S number of berths)—were used to estimate the capacity of the bus stop. The mean absolute percentage error (MAPE) of the diffusion approximation method is 7.12%, while the MAPEs of the HCM method and M/M/S queuing model are 16.53% and 10.23%, respectively. Therefore, the proposed model is more accurate and reliable than the others. In addition, the influences of traffic intensity, bus arrival rate, coefficient of variation of bus arrival headway, service time, coefficient of variation of service time, and the number of bus berths on the capacity of bus stops are explored by sensitivity analyses.

Diffusion Model of Preemptive-Resume Priority Systems and Its Application to Performance Evaluation of SDN Switches.

The increasing use of Software-Defined Networks brings the need for their performance analysis and detailed analytical and numerical models of them. The primary element of such research is a model of a SDN switch. This model should take into account non-Poisson traffic and general distributions of service times. Because of frequent changes in SDN flows, it should also analyze transient states of the queues. The method of diffusion approximation can meet these requirements. We present here a diffusion approximation of priority queues and apply it to build a more detailed model of SDN switch where packets returned by the central controller have higher priority than other packets.

Meshless simulation of a lid-driven cavity problem with a non-Newtonian fluid

The purpose of the present paper is to solve the lid-driven cavity problem for a non-Newtonian power-law shear thinning and shear thickening fluid by a meshless method. Results are presented for Re=100 and Re=1000, where different levels of shear thinning and thickening are considered. Furthermore, the lid-driven cavity case is made geometrically more complex by adding several circular-shaped obstacles in the geometry. The Navier-Stokes equations are solved with the local meshless diffuse approximate method. The weighted least squares approximation is structured by using the second-order polynomial basis vector and the Gaussian weight function. The explicit Euler scheme is used to perform the artificial time stepping. The non-incremental pressure correction scheme is used to couple the pressure and the velocity fields. Results are presented in terms of viscosity contours, stream function, velocity vectors, mid-plane velocity and viscosity profiles for the steady state. The numerical method is verified by a comparison with the results obtained from the literature, which are computed with the least squares finite element formulation. Furthermore, an investigation of the node density and node distribution type is performed. A near perfect match with the reference solution and between the structured and unstructured node distribution is found.

Global connections with some genomic differentiation occur between Indo‐Pacific and Atlantic Ocean wahoo, a large circumtropical pelagic fish

AbstractAimGlobally distributed pelagic fishes are typified by very low to negligible genetic differentiation at oceanic scales arising from high gene flow and (or) large population sizes. However, genomic approaches employing thousands of loci to characterise genetic variation can illuminate subtle patterns of genetic structure and facilitate demographic inference, such that effects arising from gene flow and population size can be partially decoupled. We used a population genomics approach to identify putative stocks in a circumtropical pelagic fish, wahoo, and to assess global connectivity in this species.LocationIndo‐Pacific and Atlantic Oceans.TaxonWahoo, Acanthocybium solandri (Cuvier, 1832).MethodsGlobally distributed wahoo samples from 11 locations (representing a total of 296 individuals) were sequenced using a pool‐seq ezRAD approach to obtain 1289–9825 genome‐wide SNP loci per population pair for analyses of genetic structure at a minor allele frequency (MAF) >0.05. Demographic inference using a diffusion approximation method (∂a∂i) was performed using 11,495–12,812 SNPs per population pair at a MAF >0.02.ResultsGenetic structure, measured as FST, was overall low, suggesting very little heterogeneity amongst sample pairs (pairwise FST ≤ 0.021). However, there was a clear signal of regional genetic structuring between ocean basins. A principal coordinate analysis separated samples from the Indo‐Pacific with those from the Atlantic and an analysis of molecular variance suggested that ~77% of variation in genetic structure was amongst regions. Our demographic analyses found greater support for models including migration over simple models of isolation.Main conclusionsOur study provides the most thorough genetic investigation of wahoo to date. We provide evidence for global connectivity of wahoo populations over their evolutionary history, but we also provide the first indication of subtle regional structure between the Indo‐Pacific and Atlantic Oceans, which occurs against a background of high gene flow. The identification of regional stocks will inform new management strategies and guide future investigations in wahoo, an increasingly important species in global fisheries.

Lambda modes comparison for different approximations of the neutron transport equation: Diffusion, SN and SP3

The methods presented in this paper solve the Simplified Spherical Harmonics approximation to the multidimensional neutron transport equation. 1D, 2D and 3D systems were modeled with Cartesian geometry using the finite difference method to discretize the spatial variables. The method is able to simulate any energy group discretization, including up-scattering terms. The Krylov Shur method was used to calculate the solution of the steady-state equation by solving a generalized eigenvalue problem. This methodology has the capability to calculate any number of eigenfunctions. A formulation review of the Simplified Spherical Harmonics is explained in this work, as well as, a study of the boundary conditions for different approaches of the finite difference method. The results calculated by this methodology are compared with the discrete ordinates and diffusion approximation methods, all of them, using the same spatial discretization in order to show the different accuracy of each method without influence of the method used for discretizing the spatial variable. The results show the validity of each method for different benchmark problems.

Base Station Wake-Up Strategy in Cellular Networks With Hybrid Energy Supplies for 6G Networks in an IoT Environment

To reduce carbon footprint, a hybrid energy powered cellular network (HybE-Net) in the Internet-of-Things (IoT) environment is widely sought after. Different from cellular network powered on-grid energy, the base station (BS) wakeup in HybE-Net needs to consider the solar energy of the BS and the traffic load in the network. Thus, in this article, a fuzzy logic-based wakeup strategy is proposed, which comprehensively considers the energy wakeup level and the available network resource ratio. Then, the solar energy states are analyzed mathematically by using the diffusion approximation method. Finally, to prevent BSs from switching frequently between the sleeping and nonsleeping model, the awakening threshold is optimized by the penalty function method. The simulation results demonstrate that in the proposed wakeup strategy, the wakeup threshold based on the energy state is used to avoid the phenomenon of frequent handoff of the BSs, resulting in fewer handoffs. The proposed BS wakeup strategy can be further applied to both the current and sixth-generation (6G) mobile communication networks, which will be powered by other forms of renewable energy and on-grid energy in the future.

Optimal Stopping of Adaptive Dose-Finding Trials

The primary objective of this paper is to develop computationally efficient methods for optimal stopping of an adaptive Phase II dose-finding clinical trial, where the decision maker may terminate the trial for efficacy or abandon it as a result of futility. We develop two solution methods and compare them in terms of computational time and several performance metrics such as the probability of correct stopping decision. One proposed method is an application of the one-step look-ahead policy to this problem. The second proposal builds a diffusion approximation to the state variable in the continuous regime and approximates the trial’s stopping time by optimal stopping of a diffusion process. The secondary objective of the paper is to compare these methods on different dose-response curves, particularly when the true dose-response curve has no significant advantage over a placebo. Our results, which include a real clinical trial case study, show that look-ahead policies perform poorly in terms of the probability of correct decision in this setting, whereas our diffusion approximation method provides robust solutions.

A meshless solution of a of lid-driven cavity containing a heterogeneous porous medium

The main objective of the present paper is to define and evaluate a benchmark test for solving the Navier-Stokes equations in heterogeneous porous medium. The benchmark is of high relevance for proper numerical simulation of flow pattern in the liquid-solid phase-change related phenomena. The test case represents a lid-driven square cavity which consists of a solid, a liquid, and a mushy part. Three different variations of the problem are solved, in which the ratio between the liquid and the mushy part is modified. The Reynolds and the Darcy number of this case are equal to 400 and 2.5×10−10, respectively. The Navier-Stokes equation, with an additional source term to take into account the Darcy effect, is solved with the local meshless diffuse approximate method. The method is structured by using the second-order polynomial basis vector and the Gaussian function to evaluate the weighted least squares approximation. The explicit Euler scheme is used to perform the time discretisation. The non-incremental pressure correction scheme is used to couple the pressure and velocity fields. Results are presented in terms of stream function, velocity magnitude, and mid-plane velocity profiles in the steady state. The accuracy of the numerical method is evaluated by a comparison with the finite volume solution obtained with a commercial solver and an investigation of the node arrangement effect.

Equivalent-PDE based stabilization of strong-form meshless methods applied to advection-dominated problems

In this paper a new criterion for stability of strong-form meshless discretizations of advection-dominated problems is developed. The criterion follows from the analysis of the equivalent PDE. It incorporates the influence of the positions of the nodes in the local subdomain, the influence of free parameters of local approximation, such as the weight function and the order of local approximation, and the time stepping formula used. We use the developed criterion in an adaptive upwinding scheme and test the developed scheme on a simple two-dimensional advection problem for regular and irregular node arrangement by using second-, third- and fourth-order accurate explicit diffuse approximate method. The proposed upwinding algorithm significantly improves the accuracy of the resulting solution in comparison with the existing approaches which do not consider the parameters of the local approximation and the irregular positioning of nodes, which is the usual setting for meshless methods.

Simulation of macrosegregation in direct-chill casting—A model based on meshless diffuse approximate method

A multiphysics numerical model has been developed for prediction of macrosegregation in direct-chill casting. The model uses volume averaged mass, momentum, energy, and species conservation equations to simulate the solidification of axisymmetric aluminium-alloy billets. The boundary conditions for the heat transfer include the effects of hot top, mould chill, and direct chill. The volume averaged conservation equations are solved with the explicit local meshless diffuse-approximate method. The approximation coefficients are evaluated on subdomains containing 14 nodes with the second order polynomial basis and the Gaussian weight function. Problems with instabilities due to strong convection are successfully eliminated with an adaptive shift of the weight function and evaluation position in the upstream direction. The automatic adaptive generation of computational node arrangement decreases the calculation time and in a straightforward way enables investigation of the complex flow for geometrically different inflow structures, including sharp and curved edges. The model is used for casting simulation of an Al-5.25wt%Cu alloy billet with a 120 mm radius for various inlet designs. The effect of inlet geometry on the temperature, solid fraction, melt flow, and macrosegregation is investigated. The described developments represent a first meshless numerical approach for simulation of macrosegregation of an important industrial process.