Mixed Grids Research Articles

Cache blocking for flux reconstruction: Extension to Navier-Stokes equations and anti-aliasing

In this article, cache blocking is implemented for the Navier Stokes equations with anti-aliasing support on mixed grids in PyFR for CPUs. In particular, cache blocking is used as an alternative to kernel fusion to eliminate unnecessary data movements between kernels at the main memory level. Specifically, kernels that exchange data are grouped together, and these groups are then executed on small sub-regions of the domain that fit in per-core private data cache. Additionally, cache blocking is also used to efficiently implement a tensor product factorisation of the interpolation operators associated with anti-aliasing. By using cache blocking, the intermediate results between application of the sparse factors are stored in per-core private data cache, and a significant amount of data movement from main memory is avoided. In order to assess the performance gains a theoretical model is developed, and the implementation is benchmarked using a compressible 3D Taylor-Green vortex test case on both hexahedral and prismatic grids, with third-, fourth-, and fifth-order solution polynomials. The expected performance gains based on the theoretical model range from 1.99 to 2.83, and the speedups obtained in practice range from 1.51 to 3.91 compared to PyFR v1.11.0.

Projection-Grid Schemes on Irregular Grids for a Parabolic Equation

A family of projection-grid schemes has been constructed for approximating parabolic equations with a variable diffusion coefficient in tensor form. The schemes are conservative and retain the self-adjointness of the original differential operator and are destined for calculations on 3D irregular difference grids, including tetrahedral, mixed (grids of arbitrary polyhedra), and locally adaptive (octal-tree type).

A Comparison of Seven Medium Resolution Impervious Surface Products on the Qinghai–Tibet Plateau, China from a User’s Perspective

As a vital land cover type, impervious surface directly reflects human activities and urbanization, significantly impacting the environment, climate, and biodiversity, especially in ecologically fragile areas such as the Qinghai–Tibet Plateau (QTP) in China. Thus, precise knowledge of impervious surface information on the QTP is essential for its ecological protection and social development. In order to improve the application of products and inform further studies, we assessed the accuracy of seven medium resolution (10–30 m) impervious surface products in the QTP, including GAIA, CISC, GlobalLand30 (GL30), GLC-FCS30 (FCS30), GHS-BUILT-S2 (GHSB), ESA WorldCover10 (WC10), and Dynamic World NRT products (DW). The validation set labeled according to domestic GF-1 images was used to calculate the precision, recall, and F1-Score of these products, and two impervious surface vote maps were generated to analyze their spatial consistency. The results showed that CISC and DW had the highest overall quality among the 30 m and 10 m products, with F1-Scores of 0.5701 and 0.5670, respectively. We also validated the accuracy of different data combinations and their intersection and union sets to provide guidance based on the results for data selection in impervious surface studies on the QTP. For results calculated by the strict validation set, which was exclusive of mixed grids, precision decreased slightly while recall increased significantly for all products, indicating that the omissions were mostly mixed pixels with a smaller percentage of impervious surface. In terms of spatial consistency, the maximum impervious surface range voted by the seven products jointly only accounts for 0.82% of the QTP, which is 2,786,800 km2 in total. Additionally, the high consistency area (votes > 4), with a distribution concentrated in large cities and dense buildings, only accounts for 15.18% of this maximum range. In summary, each product’s regional accuracy in the QTP was lower than their published accuracy, and they omitted many impervious surfaces, especially those with a background of bare land.

The impact of the thermal distribution network operating temperature and system design on different communities' energy profiles

The use of heat pumps to increase the flexibility of heating networks is critical for overcoming the intermittent nature of renewable energy. This article introduces a novel thermal network operation strategy, referred to as “Smart Network with Peak Control,” which entails adjusting the network temperature to control the peak electricity demand created by the electrification of heating. A comparison between four different operating scenarios—namely, low-temperature (fourth-generation), ultra-low temperature (fifth-generation), smart network (hybrid of low and ultra-low temperatures), and smart network with peak control—is conducted for different sites of different energy profiles. For heating-dominant sites, the application of the smart network and smart network with peak control resulted in 10% lower emissions than the other scenarios. While the peak electricity demand in the smart network scenario was double that of the conventional system, the smart network with peak control scenario was able to counter this effect by adjusting the network temperature. The effects of integrating combined heat and power with different sizes and operating schedules based on the hours of natural-gas peaking generators in mixed electrical grids were also investigated. The developed model was applied to 1139 sites in Canada, revealing significant carbon emissions reductions of up to 73%.

An information extraction method based on improved mixed text density web pages

AbstractTo improve the effect of web page information extraction, this paper proposes an improved information extraction method of mixed text density grids. Under various power constraints of the relay node itself, this paper proposes a design scheme of joint beamforming and artificial noise based on safety and rate maximisation. Furthermore, with the help of semidefinite relaxation techniques and first‐order approximations, it can be approximated as a semidefinite programming problem that is easy to solve. In addition, this paper uses an iterative algorithm based on a continuous convex approximation to process data to improve the accuracy of web page data extraction. The experimental results show that the information extraction method based on improved mixed text density webpages proposed in this paper has a good information extraction effect.

Efficient buckling analysis and optimization method for rotationally periodic stiffened shells accelerated by Bloch wave method

Rotationally periodic stiffened shells have been widely used as load-bearing parts in engineering structures, however, the buckling analysis and optimization of stiffened shells suffer from large computational time owing to the large-scale and complex development tendency of engineering structures. Considering that most of the traditional equivalent buckling analysis methods do not retain the features of stiffeners and cannot capture local buckling, their prediction accuracies in buckling load and mode are not high enough although they can reduce the analysis cost. In this paper, an efficient buckling analysis and optimization method is proposed for rotationally periodic stiffened shells accelerated by Bloch wave method. Firstly, the derivation of Bloch wave boundary conditions for buckling analysis of the substructure is presented, along with the corresponding numerical implementation procedure. Based on the buckling analysis method accelerated by Bloch wave method, the Kriging-based Efficient Global Optimization method is used to improve the optimization efficiency. Then, typical examples are carried out for the buckling analyses of stiffened cylindrical shells with various grid patterns including orthogonal grids, triangle grids, rotated triangle grids, mixed triangle grids and hierarchical orthogonal grids, as well as the integrally stiffened shell with non-straight generatrix. Compared with buckling results from the direct finite element analysis, the proposed method achieves high prediction accuracy in buckling load and remarkable improvement in modelling and analysis efficiency simultaneously. Particularly, the proposed method succeeds in capturing both local buckling mode and global buckling mode. Finally, the optimization design of the integrally stiffened shell with non-straight generatrix is carried out based on the proposed method aiming at maximizing the buckling load. Results indicate that the proposed method can reduce the total optimization time significantly and improve the buckling load by 94.5%, contributing to providing meaningful references for the design large-scale engineering structures.

High-order compact finite volume schemes for solving the Reynolds averaged Navier-Stokes equations on the unstructured mixed grids with a large aspect ratio

In this paper, high-order compact finite volume schemes on the unstructured grids based on the variational reconstruction are developed to solve the Reynolds averaged Navier-Stokes equations closed by the Spalart-Allmaras one-equation turbulence model. Encouraging progress is made in addressing the following two challenging problems: reducing the numerical errors on the large aspect ratio grids and avoiding the negative turbulent viscosity associated with the high-order methods. On grids with large aspect ratios, a three-step procedure is designed to optimize the functional parameters of variational reconstruction. In addition, an exponential decay procedure is proposed to cure the negative turbulent viscosity problem of the Spalart-Allmaras model. The exponential decay procedure has the advantage of being able to be used with any spatial discretization method and with the implicit temporal discretization. Numerical tests show significant benefits of the high-order schemes in predicting the skin frictions, capturing some important flow structures, and achieving grid-independent solutions. The numerical tests also show that the proposed schemes are sufficiently robust for practical applications.

Comparative Assessment of Multi-Port MMCs for High-Power Applications

Multi-port converters can interconnect different dc and ac systems while reducing semiconductor requirements and losses by eliminating redundant power conversion stages. Modular multilevel converter (MMC)-based multiport systems are well suited for application in mixed ac-dc grids containing high-voltage dc (HVDC) and medium-voltage dc (MVDC) systems. This paper conducts a detailed comparative assessment of multi-port dc-dc-ac MMCs for high power applications. Four representative topologies are chosen for study due to their contrasting internal power processing characteristics. Three different network scenarios are investigated that include HVDC and MVDC applications, covering several different power flow cases. The multi-port MMCs are compared in terms of losses, semiconductor effort, internal energy storage and magnetics requirements. The results are extensively discussed and general conclusions are summarized.

Simulation of flow induced by single-dielectric-barrier-discharge plasma actuator using a high-order flux-reconstruction scheme

The starting vortex flow generated by a single-dielectric-barrier-discharge (SDBD) plasma actuator in still air is investigated numerically by direct numerical simulation (DNS) using a 4th order Navier–Stokes solver and Suzen's model for the plasma actuator. The Navier–Stokes equations and the equations for the electric field and the charge density distribution in the plasma body force model are solved by the same high-order method based on the flux reconstruction algorithm on arbitrary mixed grids, allowing easy extension to configurations of complex geometry. The computational method and its high orders of accuracy are verified for the problem of the translation of an isentropic vortex and validated for the vortex shedding problem behind a circular cylinder. The plasma body force model is then implemented in the Navier–Stokes code to perform highly resolved DNS of the starting vortex generated by an SDBD plasma actuator. The computed vortex pattern and its trajectory with time are compared with available experimental data. The computational results confirm similarity laws on the motion of the vortex.

A three-dimensional non-hydrostatic mathematical model with mixed triangle and quadrilateral grids

The three-dimensional numerical models based on the hydrostatic pressure hypothesis can’t accurately and effectively simulate the cases of estuaries or natural rivers with the presence of short wave flow, stratified gravity flow, sudden change of local topography or flow near underwater buildings. Therefore it is useful to resort to a more accurate model in which the hydrostatic assumption is removed. Based on the mixed grids of triangle and quadrilateral, a three-dimensional non hydrostatic mathematical model is presented. The control equations are discretized by the semi-implicit fractional step method. The pressure is divided into the hydrostatic pressure and non-hydrostatic pressure terms. At each step, the water level is calculated by solving the sparse equations, and then the non-hydrostatic pressure is obtained by the pressure Poisson equation. Three typical examples were adopted to simulate strong three-dimensional flow. As a result, the results show that the model can accurately and effectively simulate strong three-dimensional flow with a few layers.

Research on Influence of Angle of Deflector on Air Flow in Spray Area Based on Numerical Simulation

The movement and distribution of the airflow field of the sprayer are very important for the distribution and penetration of the droplets. It is the focus and difficulty of the research to find the airflow field distribution of the fan outlet that matches the parameters of the canopy of the fruit tree. In order to study the influence of the angle change of the deflector of the orchard air-driven sprayer on the three-dimensional spatial distribution of the external air velocity field, this paper numerically simulated the external flow field based on computational fluid dynamics. The spray requirements of different fruit tree crown shapes were analyzed. The ICEM CFD software was used to model the external flow field. In order to improve the calculation efficiency, the model was divided into mixed grids. The k-ε turbulence model and Fluent solver are used for numerical solution. The flow fields of different deflectors are analyzed, and the influence of different positions and different sizes of deflector angles on the external flow field is determined.

A large‐scale parallel hybrid grid generation technique for realistic complex geometry

SummaryHigh‐Performance Computing (HPC) systems and Computational Fluid Dynamics (CFD) have made significant progress in recent years; however, as the basis of the large‐scale parallel computing, the massive grid generation of billions of cells has become a bottleneck problem. In this study, a parallel grid generation technique is proposed to generate large‐scale mixed grids with arbitrary cell types and scales. The basic idea of our method is analogous to the global mesh refinement technique. An initial coarse grid with arbitrary cell types is regarded as a background mesh which is partitioned into subzones, and subzones are assigned onto different CPU cores. After the cells and faces in each subzone are split, the inserted new points of the solid wall are projected onto the original CAD entities to preserve the geometry accurately. Finally, the tangled cells caused by the projection in the boundary layer are untangled by a local Radial Basis Function mesh deformation technique. Furthermore, a parallel partition approach and an efficient wall distance computing technique for massive grids are developed also to shorten the preprocessing time. The tests show that the preprocessing efficiency has been increased by two or three orders compared with traditional methods. Billions of grids are generated for the AIAA JSM high‐lift model and the Chinese CHN‐T1 transport model to test the ability of the parallel grid generation technique. The maximum scale up to 19 billion mixed elements is generated using 16 384 CPU cores in parallel, and the mesh quality is acceptable for CFD simulations.

Non-linear aspects of the tidal dynamics in the Sylt-Rømø Bight, south-eastern North Sea

Abstract. This study is dedicated to the tidal dynamics in the Sylt-Rømø Bight with a focus on the non-linear processes. The FESOM-C model was used as the numerical tool, which works with triangular, rectangular or mixed grids and is equipped with a wetting/drying option. As the model's success at resolving currents largely depends on the quality of the bathymetric data, we have created a new bathymetric map for an area based on recent studies of Lister Deep, Lister Ley, Højer Deep and Rømø Deep. This new bathymetric product made it feasible to work with high-resolution grids (up to 2 m in the wetting/drying zone). As a result, we were able to study the tidal energy transformation and the role of higher harmonics in the domain in detail. For the first time, the tidal ellipses, maximum tidally induced velocities, energy fluxes and residual circulation maps were constructed and analysed for the entire bight. Additionally, tidal asymmetry maps were introduced and constructed. The full analysis was performed on two grids with different structures and showed a convergence of the results as well as fulfilment of the energy balance. A great deal of attention has been paid to the selection of open-boundary conditions, model validation against tide gauges and recent in situ current data. The tidal residual circulation and asymmetric tidal cycles largely define the circulation pattern, transport and accumulation of sediment, and the distribution of bedforms in the bight; therefore, the results presented in the article are necessary and useful benchmarks for further studies in the area, including baroclinic and sediment dynamics investigations.

A coupled volume-of-fluid and level-set method (VOSET) for capturing interface of two-phase flows in arbitrary polygon grid

This article presents an extension of coupled volume-of-fluid and level-set method (VOSET) for simulating free surfaces flows in arbitrary 2D polygon meshes. A series of techniques are introduced for geometric calculations in convex polygons. Newton iteration is adopted for the interface reconstruction in polygons, and incremental remapping approach is employed for the propagation of the volume fractions. The interface tracking test results suggested a second order accuracy in mixed and hexagonal grids. For the validation purpose, a Rayleigh-Taylor instability problem, a liquid column collapse problem and a single bubble rising problem were numerically studied, and the obtained results show excellent agreements with experimental data and benchmark solutions in literatures. Finally the proposed VOSET method was applied in simulating the working process of a flow-focusing microfluidic droplet generator.

On Calibration and Simulation of Local Volatility Model with Stochastic Interest Rate

Local volatility model is a relatively simple way to capture volatility skew/smile. In spite of its drawbacks, it remains popular among practitioners for derivative pricing and hedging. For long-dated options or interest rate/equity hybrid products, in order to take into account the effect of stochastic interest rate on equity price volatility stochastic interest rate is often modelled together with stochastic equity price. Similar to local volatility model with deterministic interest rate, a forward Dupire PDE can be derived using Arrow-Debreu price method, which can then be shown to be equivalent to adding an additional correction term on top of Dupire forward PDE with deterministic interest rate. Calibrating a local volatility model by the forward Dupire PDE approach with adaptively mixed grids ensures both calibration accuracy and efficiency. Based on Malliavin calculus an accurate analytic approximation is also derived for the correction term incorporating impacts from both interest rate volatility and correlation, which integrates along a more likely straight line path for better accuracy. Eventually, the hybrid local volatility model can be calibrated in a two-step process, namely, calibrate local volatility model with deterministic interest rate and add adjustment for stochastic interest rate. Due to the lack of analytic solution and path-dependency nature of some products, Monte Carlo is a simple but flexible pricing method. In order to improve its convergence, we develop a scheme to combine merits of different simulation schemes and show its effectiveness.

A high-order control volume finite element method for thermoelastic analysis of functionally graded solids with mixed grids

In this article, a new two-dimensional control volume finite element method has been developed for thermoelastic analysis in functionally graded materials. A nine-node quadrilateral element and a six-node triangular element are employed to deal with the mixed-grid problem. The unknown variables and material properties are defined at the node. The high-order shape functions of six-node triangular and nine-node quadrilateral element are employed to obtain the unknown variables and their derivatives. In addition, the material properties in functionally graded structure are also modeled by applying the high-order shape functions. The capabilities of the presented method to heat conduction problem, elastic problem, and thermoelastic problem have been validated. First, the defined location of material properties is found to be important for the accuracy of the numerical results. Second, the presented method is proven to be efficient and reliable for the elastic analysis in multi-phase materials. Third, the presented method is capable of high-order mixed grids. The memory and computational costs of the presented method are also compared with other numerical methods.

High-order multi-dimensional limiting strategy with subcell resolution I. Two-dimensional mixed meshes

The present paper deals with a new improvement of hierarchical multi-dimensional limiting process for resolving the subcell distribution of high-order methods on two-dimensional mixed meshes. From previous studies, the multi-dimensional limiting process (MLP) was hierarchically extended to the discontinuous Galerkin (DG) method and the flux reconstruction/correction procedure via reconstruction (FR/CPR) method on simplex meshes. It was reported that the hierarchical MLP (hMLP) shows several remarkable characteristics such as the preservation of the formal order-of-accuracy in smooth region and a sharp capturing of discontinuities in an efficient and accurate manner. At the same time, it was also surfaced that such characteristics are valid only on simplex meshes, and numerical Gibbs–Wilbraham oscillations are concealed in subcell distribution in the form of high-order polynomial modes. Subcell Gibbs–Wilbraham oscillations become potentially unstable near discontinuities and adversely affect numerical solutions in the sense of cell-averaged solutions as well as subcell distributions. In order to overcome the two issues, the behavior of the hMLP on mixed meshes is mathematically examined, and the simplex-decomposed P1-projected MLP condition and smooth extrema detector are derived. Secondly, a troubled-boundary detector is designed by analyzing the behavior of computed solutions across boundary-edges. Finally, hMLP_BD is proposed by combining the simplex-decomposed P1-projected MLP condition and smooth extrema detector with the troubled-boundary detector. Through extensive numerical tests, it is confirmed that the hMLP_BD scheme successfully eliminates subcell oscillations and provides reliable subcell distributions on two-dimensional triangular grids as well as mixed grids, while preserving the expected order-of-accuracy in smooth region.

The discontinuous Galerkin spectral element methods for compressible flows on two-dimensional mixed grids

This paper presents a practical algorithm for constructing high order discontinuous Galerkin spectral element methods (DGSEM) on mixed triangular and quadrilateral grids. The traditional DGSEM belongs to the collocation-type nodal discontinuous Galerkin method which is computationally efficient on one-dimensional and tensor-product grids. This work generalizes DGSEM to triangular grids using symmetric quadrature points in sphere close packed arrangement. The dispersion and dissipation analysis shows that the scheme is stable for linear problems on triangles. A problem independent limiting procedure is proposed for capturing the shock waves. The main components are a modified KXRCF shock detector, a smoothness indicator and a high-resolution WENO limiter based on candidate polynomials constructed by newly developed projection approach. The implementation of the DGSEM on the mixed grids is also presented. Numerical examples on triangular and mixed grids verify the scheme's high order accuracy and robustness in computing inviscid and viscous compressible flows.

$hp$-Adaptation Driven by Polynomial-Degree-Robust A Posteriori Error Estimates for Elliptic Problems

We devise and study experimentally adaptive strategies driven by a posteriori error estimates to select automatically both the space mesh and the polynomial degree in the numerical approximation of diffusion equations in two space dimensions. The adaptation is based on equilibrated flux estimates. These estimates are presented here for inhomogeneous Dirichlet and Neumann boundary conditions, for spatially-varying polynomial degree, and for mixed rectangular-triangular grids possibly containing hanging nodes. They deliver a global error upper bound with constant one and, up to data oscillation, error lower bounds on element patches with a generic constant only dependent on the mesh regularity and with a computable bound. We numerically asses the estimates and several hp-adaptive strategies using the interior penalty discontinuous Galerkin method. Asymptotic exactness is observed for all the symmetric, nonsymmetric (odd degrees), and incomplete variants on non-nested unstructured triangular grids for a smooth solution and uniform refinement. Exponential convergence rates are reported on nonmatching triangular grids for the incomplete version on several benchmarks with a singular solution and adaptive refinement.

Heterogeneous computing on mixed unstructured grids with PyFR

PyFR is an open-source high-order accurate computational fluid dynamics solver for unstructured grids. In this paper we detail how PyFR has been extended to run on mixed element meshes, and a range of hardware platforms, including heterogeneous multi-node systems. Performance of our implementation is benchmarked using pure hexahedral and mixed prismatic-tetrahedral meshes of the void space around a circular cylinder. Specifically, for each mesh performance is assessed at various orders of accuracy on three different hardware platforms; an NVIDIA Tesla K40c GPU, an Intel Xeon E5-2697 v2 CPU, and an AMD FirePro W9100 GPU. Performance is then assessed on a heterogeneous multi-node system constructed from a mix of the aforementioned hardware. Results demonstrate that PyFR achieves performance portability across various hardware platforms. In particular, the ability of PyFR to target individual platforms with their ‘native’ language leads to significantly enhanced performance cf. targeting each platform with OpenCL alone. PyFR is also found to be performant on the heterogeneous multi-node system, achieving a significant fraction of the available FLOP/s. Finally, each mesh is used to undertake nominally fifth-order accurate long-time simulations of unsteady flow over a circular cylinder at a Reynolds number of 3900 using a cluster of NVIDIA K20c GPUs. Long-time dynamics of the wake are studied in detail, and results are found to be in excellent agreement with previous experimental/numerical data. All results were obtained with PyFR v0.2.2, which is freely available under a 3-Clause New Style BSD license (see www.pyfr.org).