Exponentially fitted difference scheme for singularly perturbed mixed integro-differential equations

In this study, singularly perturbed mixed integro-differential equations (SPMIDEs) are taken into account. First, the asymptotic behavior of the solution is investigated. Then, by using interpolating quadrature rules and an exponential basis function, the finite difference scheme is constructed on a uniform mesh. The stability and convergence of the proposed scheme are analyzed in the discrete maximum norm. Some numerical examples are solved, and numerical outcomes are obtained.

Application of model order reduction with Cauer ladder networks to industrial inductors

Purpose The Cauer ladder network (CLN) model order reduction (MOR) method is applied to an industrial inductor. This paper aims to to anaylse the influence of different meshes on the CLN method and their parameters. Design/methodology/approach The industrial inductor is simulated with the CLN method for different meshes. Meshes considering skin effect are compared with equidistant meshes. The inductor is also simulated with the eddy current finite element method (ECFEM) for frequencies 1 kHz to 1 MHz. The solution of the CLN method is compared with the ECFEM solutions for the current density in the conductor and the total impedance. Findings The increase of resistance resulting from the skin effect can be modelled with the CLN method, using a uniform mesh with elements much larger than the skin depth. Meshes taking account of the skin depth are only needed if the electromagnetic fields have to be reconstructed. Additionally, the convergence of the impedance is used to define a stopping criterion without the need for a benchmark solution. Originality/value The work shows that the CLN method can generate a network, which is capable of mimicking the increase of resistance usually accompanied by the skin effect without using a mesh that takes the skin depth into account. In addition, the proposed stopping criterion makes it possible to use the CLN method as an a priori MOR technique.

MPAS-Seaice (v1.0.0): Sea-ice dynamics on unstructured Voronoi meshes

We present MPAS-Seaice, a sea-ice model which uses the Model for Prediction Across Scales (MPAS) framework and Spherical Centroidal Voronoi Tessellation (SCVT) unstructured meshes. As well as SCVT meshes, MPAS-Seaice can run on the traditional quadrilateral grids used by sea-ice models such as CICE. The MPAS-Seaice velocity solver uses the Elastic-Viscous-Plastic (EVP) rheology, and the variational discretization of the internal stress divergence operator used by CICE, but adapted for the polygonal cells of MPAS meshes, or alternatively an integral (“weak”) formulation of the stress divergence operator. An incremental remapping advection scheme is used for mass and tracer transport. We validate these formulations with idealized test cases, both planar and on the sphere. The variational scheme displays lower errors than the weak formulation for the strain rate operator but higher errors for the stress divergence operator. The variational stress divergence operator displays increased errors around the pentagonal cells of a quasi-uniform mesh, which is ameliorated with an alternate formulation for the operator. MPAS-Seaice shares the sophisticated column physics and biogeochemistry of CICE, and when used with quadrilateral meshes can reproduce the results of CICE. We have used global simulations with realistic forcing to validate MPAS-Seaice against similar simulations with CICE and against observations. We find very similar results compared to CICE with differences explained by minor differences in implementation such as with interpolation between the primary and dual meshes at coastlines. We have assessed the computational performance of the model, which, because it is unstructured, runs 70 % as fast as CICE for a comparison quadrilateral simulation. The SCVT meshes used by MPAS-Seaice allow culling of equatorial model cells and flexibility in domain decomposition, improving model performance. MPAS-Seaice is the current sea-ice component of the Energy Exascale Earth System Model (E3SM).

Effect of uniform and non-uniform mesh on the development of turbulent boundary layer

For incompressible flow simulation, it is commonly accepted to use uniform meshes to solve the governing equation of turbulent boundary layer. It follows the laws of conservation stabilizing the flow field in the domain and preventing odd-even decoupling in the pressure field. In this study, Large Eddy Simulation (LES) has been conducted in a long channel. In order to calculate the turbulent boundary layer in the channel, the unsteady Navier-Stokes equations has been adopted at a Reynolds number =180, which is based on mean centerline velocity and the half-width of the channel. The mesh used in this study was based on both stretch and uniform mesh having grid points, which is corresponding to . Turbulence statistics were also calculated to compare to the existing results. In the results, the turbu lent boundary layer was fully developed at around . In addition, fully developed channel flow was achieved at the non-dimensional time of .

A Local Adaptive Mesh Refinement for JFO Cavitation Model on Cartesian Meshes

Nonuniform mesh is beneficial to reduce computational cost and improve the resolution of the interest area. In the paper, a cell-based adaptive mesh refinement (AMR) method was developed for bearing cavitation simulation. The bearing mesh can be optimized by local refinement and coarsening, allowing for a reasonable solution with special purpose. The AMR algorithm was constructed based on a quadtree data structure with a Z-order filling curve managing cells. The hybrids of interpolation schemes on hanging nodes were applied. A cell matching method was used to handle periodic boundary conditions. The difference schemes at the nonuniform mesh for the universal Reynolds equation were derived. Ausas’ cavitation algorithm was integrated into the AMR algorithm. The Richardson extrapolation method was employed as an a posteriori error estimation to guide the areas where they need to be refined. The cases of a journal bearing and a thrust bearing were studied. The results showed that the AMR method provided nearly the same accuracy results compared with the uniform mesh, while the number of mesh was reduced to 50–60% of the number of the uniform mesh. The computational efficiency was effectively improved. The AMR method is suggested to be a potential tool for bearing cavitation simulation.

A parameter-uniform finite difference scheme for singularly perturbed parabolic problem with two small parameters

A parameter-uniform finite difference scheme is constructed and analyzed for solving singularly perturbed parabolic problems with two parameters. The solution involves boundary layers at both the left and right ends of the solution domain. A numerical algorithm is formulated based on uniform mesh finite difference approximation for time variable and appropriate piecewise uniform mesh for the spatial variable. Parameter-uniform error bounds are established for both theoretical and experimental results and observed that the scheme is second-order convergent. Furthermore, the present method produces a more accurate solution than some methods existing in the literature.

Stable and Convergent Finite Difference Schemes on NonuniformTime Meshes for Distributed-Order Diffusion Equations

In this work, stable and convergent numerical schemes on nonuniform time meshes are proposed, for the solution of distributed-order diffusion equations. The stability and convergence of the numerical methods are proven, and a set of numerical results illustrate that the use of particular nonuniform time meshes provides more accurate results than the use of a uniform mesh, in the case of nonsmooth solutions.

A Higher-Order Finite Difference Scheme for Singularly Perturbed Parabolic Problem

In this paper, we deal with a singularly perturbed parabolic convection-diffusion problem. Shishkin mesh and a hybrid third-order finite difference scheme are adopted for the spatial discretization. Uniform mesh and the backward Euler scheme are used for the temporal discretization. Furthermore, a preconditioning approach is also used to ensure uniform convergence. Numerical experiments show that the method is first-order accuracy in time and almost third-order accuracy in space.

Correlation of choriocapillaris hemodynamic data from dynamic indocyanine green and optical coherence tomography angiography

To investigate the correlation between posterior pole choroidal blood flow evaluated with digital subtraction indocyanine green angiography and enface optical coherence tomography angiography (OCTA). Imaging in animal study. The anatomy of 2 cynomogulus monkeys was studied. Each monkey was given a 0.75 mg/kg injection of indocyanine green in the saphenous vein. The dynamic angiographic filling sequence was recorded at 15 frames per second using the Heidelberg Spectralis. After image registration, sequential frame subtraction was used to image the dye front moving through the choroid. The OCTA was obtained by frame averaging nine separate choriocapillaris slab flow images obtained from the Zeiss Plex Elite 9000. Posterior pole choriocapillaris filling pattern in relation to the choriocapillaris anatomy as imaged by OCTA. In the posterior pole, the choriocapillaris fills in the pattern of discrete units with variable sizes and shapes. The cycle of dye filling begins in the peripapillary area and progresses toward the periphery in a wavelike manner. This filling pattern repeats in a cyclical manner, consistent with the cardiac cycle. OCTA shows a uniform mesh of vessels. While OCTA shows a uniform meshwork appearance of the choriocapillaris, the dynamic dye angiography suggests an irregular configuration of functional units partitioned by pressure gradients as opposed to structural boundaries. Disturbance of local perfusion pressure within choroidal vasculature may result in abnormal flow patterns, which could be evaluated in the clinic using commercially available equipment.

STUDY AND VALIDATION OF MESHES IN TURBULENT ISOTHERMAL PROBLEMS OF NATURAL CONVECTION IN FLAT PLATES

The study of natural convection on flat plates is of great interest in the areas of Engineering, both for the simplicity of geometry and the wide variety of applications. In the study and definition of a numerical model, an ideal mesh configuration is the one that best represents physically, with minimal numerical influence and with the lowest computational cost, the problem addressed. The influence of two mesh configurations (non-uniform staggered and entirely uniform), at different refinement levels, was studied to evaluate natural convection heat transfer rates in flat plates of AR = 5; in isothermal conditions, in turbulent regime with the κ−ω SST RANS model and using free and open-source software OpenFOAM®. The physical-numerical methodology applied, and the numerical results obtained were validated from experimental results in the literature. The non-uniform staggered mesh configuration proved to be more adequate in precision, and computational cost to the problem situation studied. The entirely uniform mesh proved to be infeasible due to the high number of elements and computational cost demanded.