Unstructured Grid Method Research Articles

MachLine: Development of a Dirichlet-Based, Subsonic/Supersonic Panel Method for Unstructured Grids

To meet the growing need for fast supersonic flow solutions, the theory and implementation of a modern subsonic/supersonic, unstructured panel method are here presented. This work presents several improvements over legacy supersonic panel codes and available modern codes. Among these improvements are automatic methods for generating unstructured wakes, enforcing the Kutta condition at trailing edges, and placing interior control points. An efficient method for calculating domains of dependence on an unstructured mesh is also presented. For the first time in the literature regarding supersonic panel methods, a complete method for calculating influence coefficients is also described. The numerical sensitivities of the method are examined. Compared to analytic and experimental data, the method is accurate for small-perturbation flows. For example, the relative error in the root pressure coefficient on a straight wing in supersonic flow (M∞=1.5) was calculated to be within approximately 7% of the analytic solution for a range of angles of attack. For a delta wing in supersonic flow (M∞=1.62), the method is found to calculate lift slope and zero-lift drag to within 5% of experimental data. The method is also fast, having run times on the order of 1 minute for meshes with around 26,000 panels.

A robust phase-field method for two-phase flows on unstructured grids

A phase-field method for unstructured grids that is accurate, conservative, and robust is proposed in this work. The proposed method also results in bounded transport of volume fraction, and the interface thickness adapts automatically to local grid size. In addition to this, we present a novel formulation for two-phase flows on collocated grids that is provably energy stable, which is a critical feature for robust simulations of two-phase turbulent flows. The proposed method is first evaluated on canonical test cases, including scenarios like drop advection and drop in a shear flow. The accuracy of the proposed method in the presence of grid transitions, which is important for simulations in complex geometries using unstructured grids, is also evaluated. We assess the robustness of our method by performing simulations of a drop in homogeneous isotropic turbulence at infinite Reynolds number with varying density ratios. Furthermore, as a step toward verification and validation, simulations of test cases in complex geometries and conditions, such as, a damped surface wave, infinitesimal interface distortion on a liquid jet, and the atomization of a liquid jet from the engine combustion network's Spray A nozzle are presented. These simulations illustrate the accuracy, robustness, and applicability of the proposed method in various kinds of complex two-phase flow applications of engineering interest.

A numerical model for simulating separated gas-liquid two-phase flow with low GVF in a V-cone flowmeter

One of the relatively new types of differential pressure flowmeters is the V-Cone (conical or cone) meter. In addition to having many advantages over other types, this flowmeter can also be used in multiphase flows. In the last few decades, many numerical works have been presented for single-phase flows. But in the discussion of two-phase flow, most of the available works are related to experimental research. Therefore, in this paper, the separated two-phase flow with a low gas volume fraction (GVF) has been numerically investigated. For this purpose, an unstructured grid and finite volume numerical method were used. In order to model the two-phase flow and turbulence, the existing approaches were compared. According to the results obtained, the volume of fluid (VOF) method for simulating two-phase flow and the Realizable k-ε model for turbulence modeling lead to better results than other investigated models. Also, by stimulating the flow with the aforementioned methods, it was found that the accuracy of the pressure calculation decreases with the reduction of the superficial velocity and volume fraction of the gas. Furthermore, for a more detailed analysis, the superiority of the Realizable k-ε turbulence model compared to other investigated models was proved quantitatively.

Aerodynamic characteristics of a super high-speed elevator with different toe guards

To reduce the drag and noise of a super high-speed elevator, a certain type of elevator in operation with different toe guards was taken as the subject to conduct research on its aerodynamic characteristics. In this study, a numerical simulation model was established and used to carry out experiments, and by comparing the theoretical and experimental results, the validity of the method was verified. First, a 3-D unsteady numerical model was established based on improved unstructured dynamic grid method, and the Reynolds average N–S equation was solved by SIMPLEC algorithm. Then, the whole running process of the elevator was simulated, including the velocity distribution, vorticity strength, and pressure gradient in the flow field were analyzed. Finally, the aerodynamic characteristics of the elevator with different vertical heights and four structure layouts of the toe guard were analyzed. The results show that the drag reduction effect is quite apparent when toe guards are installed at the car's top and bottom. In addition, it was found that decreasing the vertical height of the toe guard is conducive to reducing resistance and saving space in the elevator car. The inclusive analysis not only explores the aerodynamic characteristics of super high-speed elevators but also recommends structural optimization design.

Enhancement on parallel unstructured overset grid method for complex aerospace engineering applications

In the present study, an efficient overset grid method by means of parallel implicit hole-cutting is proposed for the sake of simulating unsteady flows in aerospace engineering involving multiple bodies in relative movement. In view of the degraded computational efficiency and robustness for conventional overset grid assembly, several innovative techniques are developed within the overset grid assembly process, viz., a bookkeeping alternative digital tree method to speed up the donor-cell searching, a fast parallel advancing front algorithm to accelerate the wall-distance calculation and a message-passing strategy with efficient information communication and lower storage expenditure within distributed computational architecture. The contribution of the developed techniques is evidenced by comparison with the existing alternative ways in terms of computing efficiency. Subsequently, the overset grid method is embedded into an in-house programed URANS solver to examine its capability in predicting the flow field of complex applications such as helicopter, store separation and component deploying. Results show that the developed overset grid methodology is, in practice, able to resolve the aerodynamic characteristics of complex aerospace engineering with a high-fidelity flow topology and accuracy.

Research on Internal Structure and Mechanism of Landslide Based on Hydrogeophysical Investigation (Quan’an Landslide, Southwest China)

The survey area is located in the planning area of Longquan national urban park, with frequent human activities, numerous water conservancy, transportation projects, and frequent geological disasters such as landslides. The Red Mudstone has low strength and easy to soften in the case of water, and it is very prone to landslide in rainy season. At present, there are many studies on Red Mudstone Landslides, but most of them are analyzed by geological means, and the monitoring process is also dominated by surface deformation and settlement; there is less analysis with hydrogeological that is caused by rainfall and human activities. For Red Mudstone Landslides, to effectively reveal and analyze the process of the Slip Mechanism, we need to carry out the characteristics of hydrogeological structure and underground physical structure, and compare with surface monitoring and point geological structure. In this study, a one-year field survey and hydrogeophysical survey were carried out. Combined with the characteristics of environmental geology, the geophysical forward modeling problem based on finite element method was used to verify the influence of rainfall on the electrical characteristics of underground media. The smoothness constrained least square method and three-dimensional unstructured grid method are used to interpret the ERT data, analyze the abnormal patterns of electrical structure of landslide in different seasons, and correspond well with the GPR data. According to the characteristics of surface vegetation and weathering, as well as the displacement and settlement monitoring after landslide, the spatial characteristics of red bed landslide are revealed, and the relationship between landslide, rainfall, and human activities is clarified.

Uniform block-diagonal preconditioners for divergence-conforming HDG Methods for the generalized Stokes equations and the linear elasticity equations

Abstract We propose a uniform block-diagonal preconditioner for condensed $H$(div)-conforming hybridizable discontinuous Galerkin schemes for parameter-dependent saddle point problems, including the generalized Stokes equations and the linear elasticity equations. An optimal preconditioner is obtained for the stiffness matrix on the global velocity/displacement space via the auxiliary space preconditioning technique (Xu (1994) The Auxiliary Space Method and Optimal Multigrid Preconditioning Techniques for Unstructured Grids, vol. 56. International GAMM-Workshop on Multi-level Methods (Meisdorf), pp. 215–235). A spectrally equivalent approximation to the Schur complement on the element-wise constant pressure space is also constructed, and an explicit computable exact inverse is obtained via the Woodbury matrix identity. Finally, the numerical results verify the robustness of our proposed preconditioner with respect to model parameters and mesh size.

Comparative Assessment of Methods for Coupling Regional and Local Groundwater Flow Models: A Case Study in the Beijing Plain, China

A coupled regional and local model is required when groundwater flow and solute transport are to be simulated in local areas of interest with a finer grid while regional aquifer boundary and major stresses should be retained with a coarser grid. The coupled model should also maintain interactions between the regional and local flow systems. In the Beijing Plain (China), assessment of managed aquifer recharge (MAR), groundwater pollution caused by rivers, capture zone of well fields, and land subsidence at the cone of depression requires a coupled regional and local model. This study evaluates three methods for coupling regional and local flow models for simulating MAR in the Chaobai River catchment in the Beijing Plain. These methods are the conventional grid refinement (CGR) method, the local grid refinement (LGR) method and the unstructured grid (USG) method. The assessment included the comparison of the complexity of the coupled model construction, the goodness of fit of the computed and observed groundwater heads, the consistency of regional and local groundwater budgets, and the capture zone of a well filed influenced by the MAR site. The results indicated that the CGR method based on MODFLOW-2005 is the easiest to implement the coupled model, capable of reproducing regional and local groundwater heads and budget, and already coupled with density and viscosity dependent model codes for transport simulation. However, the CGR method inherits shortcomings of finite difference grids to create multiple local models with inefficient computing efforts. The USG method based on MODFLOW-USG has the advantage of creating multi-scale models and is flexible to simulate rivers, wells, irregular boundaries, heterogeneities and the MAR site. However, it is more difficult to construct the coupled models with the unstructured grids, therefore, a good graphic user interface is necessary for efficient model construction. The LGR method based on MODFLOW-LGR can be used to create multiple local models in uniform aquifer systems. So far, little effort has been devoted to upgrade the LGR method for complex aquifer structures and develop the coupled transport models.

Mesh-free radial-basis function generated finite differences and its application in full-waveform inversion

As the key point of full-waveform inversion (FWI), numerical solution of wave equation is required to simulate the forward propagated and adjoint wavefield. When traditional regular grids are used for forward modelling, scattering artifacts may occur due to the stepped approximation of layer interfaces and rugged topography. Unstructured mesh or irregular grids method can achieve certain geometric flexibility, however, its algorithm is quite complex. Mesh-free FWI can effectively reduce the scattered artifacts under regular grids and avoid the extra computation in the process of irregular grids generation. For the implementation of mesh-free FWI method, an algorithm with fast generation of node distributions is used to discretize the velocity model, radial-basis function generated finite difference is used to realise seismic wave propagation numerical simulation, and Limited-memory BFGS (L-BFGS) algorithm is used for iteration. The mesh-free FWI method we proposed achieves flexibility of simulation region and abundant wavefield information. It reduces the storage required for FWI and illustrates the applicability of high-precision reconstruction of underground velocity in the case of rugged topography, which can provide more accurate velocity information for oil and gas exploration under complex geological conditions.

An efficient reconstruction algorithm for diffusion on triangular grids using the nodal discontinuous Galerkin method

High-energy-density (HED) hydrodynamics studies such as those relevant to inertial confinement fusion and astrophysics require highly disparate densities, temperatures, viscosities, and other diffusion parameters over relatively short spatial scales. This presents a challenge for high-order accurate methods to effectively resolve the hydrodynamics at these scales, particularly in the presence of highly disparate diffusion. A significant volume of engineering and physics applications use an unstructured discontinuous Galerkin (DG) method developed based on the finite element mesh generation and algorithmic framework. This work discusses the application of an affine reconstructed nodal DG method for unstructured grids of triangles. Solving the diffusion terms in the DG method is non-trivial due to the solution representations being piecewise continuous. Hence, the diffusive flux is not defined on the interface of elements. The proposed numerical approach reconstructs a smooth solution in a parallelogram that is enclosed by the quadrilateral formed by two adjacent triangle elements. The interface between these two triangles is the diagonal of the enclosed parallelogram. Similar to triangles, the mapping of parallelograms from a physical domain to a reference domain is an affine mapping, which is necessary for an accurate and efficient implementation of the numerical algorithm. Thus, all computations can still be performed on the reference domain, which promotes efficiency in computation and storage. This reconstruction does not make assumptions on choice of polynomial basis. Reconstructed DG algorithms have previously been developed for modal implementations of the convection–diffusion equations. However, to the best of the authors’ knowledge, this is the first practical guideline that has been proposed for applying the reconstructed algorithm on a nodal discontinuous Galerkin method with a focus on accuracy and efficiency. The algorithm is demonstrated on a number of benchmark cases as well as a challenging substantive problem in HED hydrodynamics with highly disparate diffusion parameters.

Numerical Simulation of Gravity Anomaly Based on the Unstructured Element Grid and Finite Element Method

Finite element method is an important method to solve mathematical problems in engineering. Many mathematical equations are difficult to solve, but it becomes very simple after using the finite element method. In this paper, the finite element method is applied to the calculation of gravity anomaly. First, the variational equation of gravity anomaly calculation is established, and then the gravity anomaly value ten times the distance away from the anomaly body is used as the boundary condition. By comparing the gravity anomaly obtained by solving the stiffness matrix with the analytical solution, it can be found that the method in this paper has high accuracy. Finally, the model of Jinchuan copper nickel deposit is used for calculation, and the calculated gravity anomaly field is inverted with Growth3D. It can be found that the inversion result is very close to the model, which verifies the effectiveness of the algorithm in this paper.

Scalable parallel finite volume lattice Boltzmann method for thermal incompressible flows on unstructured grids

A parallel cell-centered finite volume thermal lattice Boltzmann method (FV-TLBM) for two-dimensional unstructured grids is proposed to simulate thermal incompressible flows. The governing equations for the velocity and temperature fields are solved using the D2Q9 lattice model in the lattice Boltzmann method and the cell-centered finite volume method. In this paper, the advective fluxes in the governing equations are evaluated by a low-diffusion Roe scheme, and the gradients of the particle distribution functions are computed with the least-square approach. To simulate large-scale complex flows within a reasonable time, a parallel algorithm for the FV-TLBM on unstructured grids is devised. The present method is validated by numerical simulations of the natural convection in a square cavity and natural convection in a concentric annulus at different Rayleigh numbers. The results obtained by the proposed method agree well with previous studies. The parallel performance results show that the proposed algorithm has considerable scalability and the parallel efficiency is as high as 96.50% for a case with over 5 billion grid cells using 6000 processor cores.

Extension of B-spline Material Point Method for unstructured triangular grids using Powell\u2013Sabin splines

The Material Point Method (MPM) is a numerical technique that combines a fixed Eulerian background grid and Lagrangian point masses to simulate materials which undergo large deformations. Within the original MPM, discontinuous gradients of the piecewise-linear basis functions lead to the so-called grid-crossing errors when particles cross element boundaries. Previous research has shown that B-spline MPM (BSMPM) is a viable alternative not only to MPM, but also to more advanced versions of the method that are designed to reduce the grid-crossing errors. In contrast to many other MPM-related methods, BSMPM has been used exclusively on structured rectangular domains, considerably limiting its range of applicability. In this paper, we present an extension of BSMPM to unstructured triangulations. The proposed approach combines MPM with C^1-continuous high-order Powell–Sabin spline basis functions. Numerical results demonstrate the potential of these basis functions within MPM in terms of grid-crossing-error elimination and higher-order convergence.

Impact of Number Representation for High-Order Implicit Large-Eddy Simulations

High-order numerical methods for unstructured grids combine the superior accuracy of high-order spectral or finite difference methods with the geometric flexibility of low-order finite volume or finite element schemes. Over the past few years, they have shown promise in enabling implicit large-eddy simulations within the vicinity of complex geometrical configurations. However, the cost of such simulations remains prohibitive. In this paper, the impact of number representation on the efficiency and efficacy of these methods is considered. Theoretical performance models are derived and assessed. Specifically, four test cases are considered: 1) the viscous Taylor–Green vortex, 2) flow over a circular cylinder, 3) flow through a T106c low-pressure turbine cascade, and 4) flow around a NACA 0021 in deep stall. It is found that the use of single (in lieu of double) precision arithmetic does not have a significant impact on accuracy. The performance of the resulting simulations was found to improve between 1.4 and 2.9 times.

A volume-conserving balanced-force level set method on unstructured meshes using a control volume finite element formulation

The level set method for modeling multi-fluid interfaces gives a simple approach for the simulation of multiphase flow problems. However, many of the components of this method, such as the redistancing of the level set function and discretization of the singular surface tension force, become more difficult on unstructured meshes in a control volume framework; furthermore, these operations are shown to lead to errors in conservation of mass of the individual phases, as well as unphysical currents near the interface. This paper presents a novel formulation for the level set method combined with a balanced-force algorithm using a control volume finite element method (CVFEM) for unstructured grids. A diffuse interface description is used, allowing a straightforward inclusion of surface tension forces; we show that by consistently evaluating the pressure gradient and surface tension, we are able to discretely balance these two forces and suppress any parasitic currents given a prescribed curvature. By converting the redistancing equation to a form that is suitable for CVFEM and enforcing a constraint directly linked with the phase fraction in each cell, we are able conserve mass and preserve the shape of the interface to a high degree of accuracy in multiphase flow problems with high density and viscosity ratios. The accuracy of the of our proposed method is assessed by comparing with benchmark results from the literature.

On the use of polyhedral unstructured grids with a moving immersed boundary method

A moving immersed boundary method for unstructured grids is presented which offers several advantages over the standard ones with Cartesian grids. The flexibility provided by the unstructured grids allows for smaller grid sizes and meshing of complex flow configurations. The method features a conservative fluid domain characterization together with a flexible least-squares flow reconstruction to emulate the immersed body. The method is validated with several test cases and different grid types: polyhedral, triangular and Cartesian.A static problem is first simulated in order to demonstrate the expected second order accuracy. Then, a benchmark moving body problem is studied, where the method is shown to compute the correct velocity and pressure fields independently of the grid type. The effects of the cell topology in the spurious force oscillations (SFO) are also studied and the polyhedral grids are proven to be superior to their Cartesian and triangular counterparts.Finally, some examples of moving bodies in a computational domain with complex static boundaries are provided. The new method allows the use of unstructured grids for the outer fixed boundary, which allows good geometry conformance and therefore a better flow resolution. These grids can include a region with a reduced grid size and smooth transition, located at the body’s path.

A parallelized hybrid N-S/DSMC-IP approach based on adaptive structured/unstructured overlapping grids for hypersonic transitional flows

A new hybrid N-S/DSMC-IP approach based on adaptive structured/unstructured overlapping grids is proposed and implemented to simulate hypersonic transitional flows. This approach uses the N–S method based on structured grids and the DSMC-IP method based on unstructured grids in the simulation of continuum and rarefied flow fields. The adaptive overlapping grid method is utilized and the interface between the continuum and rarefied regions is partitioned using KnGL number. The information in the different regions is strongly coupled to the continuum/rarefied interface, and the values in the coupling cells are interpolated with the weight coefficients. Finally, an adaptive parallel strategy is proposed to further improve computational efficiency. This hybrid approach combines the advantages of the structured grid and unstructured grid methods, meaning it can investigate the various characteristics of transitional flow fields with high efficiency and accuracy, while also enhancing the approach's fitness. This hybrid approach is implemented to simulate the continuum/rarefied hypersonic transitional flows around a cylinder and a nozzle; with the results indicating that the computational speed for this hybrid approach is much higher than for the DSMC-IP method and that it can be used to investigate variation in the characteristics of continuum/rarefied regions in transitional flows. Therefore, it is an efficient algorithm for the investigation of hypersonic transitional flows.

A coupled volume of fluid and level set method based on analytic PLIC for unstructured quadrilateral grids

We develop a coupled volume of fluid and level set (VOSET) method for unstructured quadrilateral grids to simulate incompressible two-phase interfacial flows in irregular domains. In the method, an analytic piecewise linear interface calculation (PLIC) is first proposed and its solution speed is much faster (about five times) than that of Brent's iterative method; then an iterative geometric operation by which the level set function near interfaces can be calculated, is extended to unstructured quadrilateral grids; moreover, the volume fraction advection is solved by a Lagrangian-Eulerian advection scheme. Finally, the present method is validated by the single vortex flow problem, bubble rising problem and falling droplet problem. Our simulation results show good agreement with those in previous studies.

Numerical analysis of a high-order unstructured overset grid method for compressible LES of turbomachinery

Large-Eddy Simulation (LES) is recognized as a promising method for high-fidelity flow predictions in turbomachinery applications. The presented approach consists of the coupling of several instances of the same LES unstructured solver through an overset grid method. A high-order interpolation, implemented within this coupling method, is introduced and evaluated on several test cases. It is shown to be third order accurate, to preserve the accuracy of various second and third order convective schemes and to ensure the continuity of diffusive fluxes and subgrid scale tensors even in detrimental interface configurations. In this analysis, three types of spurious waves generated at the interface are identified. They are significantly reduced by the high-order interpolation at the interface. The latter having the same cost as the original lower order method, the high-order overset grid method appears as a promising alternative to be used in all the applications.

Analysis of non-conservative interpolation techniques in overset grid finite-volume methods

Abstract This paper reports on the challenges of using an overset grid approach when simulating incompressible flows with a cell-centred finite-volume solver. The focal point is on the grid coupling approach. Usually, overset grid methods for unstructured three-dimensional grids apply a local interpolation of field values onto the partner grid as a coupling procedure. Such local interpolation is obviously insufficient to guarantee that the global sum of the mass fluxes across the overlapping interfaces vanishes. Hence, the inherent mass conservation of finite volume methods is violated by the non-conservative property of the overset grid coupling. Since incompressible finite-volume solvers directly use the mass defect when solving for the pressure, severe pressure fluctuations may be provoked. A sensitivity study of the residual mass defect on overlapping grids and the related pressure fluctuations is performed for a variety of coupling strategies. Emphasis is given to five different aspects, i.e. relative grid motion, the order of accuracy of the employed baseline interpolation, the formulation of global mass conservation enforcing practices, the resolution quality of the overlapping grids and multiphase-flow issues. Examples included refer to a simple 2D cylinder flow, a 2D/3D lid-driven cavity flow, a 2D body force disturbed channel flow and 2D oscillating hydrofoil flows in fully wetted single-phase and immiscible two-phase conditions. Results reveal that transient flows and simulations with relative grid motion are subject to significant mass and pressure fluctuations. High-order interpolation and grid refinement prove beneficial, but can still be afflicted with severe disturbances, particularly when the resolution properties of the overlapping grids disagree. Two-phase flows with a high density ratio show a distinct compensating behaviour, while the introduction of flux or wetted volume correction practices leads to notable improvements for single-phase flows in both homogeneous and heterogeneous resolution conditions.