A finite volume method for global electromagnetic induction forward modeling on collocated unstructured grids

Global electromagnetic induction provides an efficient way to probe the electrical conductivity in the Earth’s deep interior.Owing to the increasing geomagnetic data especially from high-accuracy geomagnetic satellites,inverting the Earth’s three-dimensional conductivity distribution on a global scale becomes attainable.A key requirement in the global conductivity inversion is to have a forward solver with high-accuracy and efficiency.In this study,a finite volume method for global electromagnetic induction forward modeling is developed based on unstructured grids.Arbitrary polyhedral grids are supported in our algorithms to obtain high geometric adaptability.We employ a cell-centered collocated variable arrangement which allows convenient discretization for complex geometries and straightforward implementation of multigrid technique.To validate the method,we test our code with two synthetic models and compare our finite volume results with an analytical solution and a finite element numerical solution.Good agreements are observed between our solution and other results,indicating acceptable accuracy of the proposed method.

Hermite WENO-based limiters for high order discontinuous Galerkin method on unstructured grids

A novel class of weighted essentially nonoscillatory （WENO） schemes based on Hermite polynomi- als, termed as HWENO schemes, is developed and applied as limiters for high order discontinuous Galerkin （DG） method on triangular grids. The developed HWENO methodology utilizes high-order derivative information to keep WENO re- construction stencils in the von Neumann neighborhood. A simple and efficient technique is also proposed to enhance the smoothness of the existing stencils, making higher-order scheme stable and simplifying the reconstruction process at the same time. The resulting HWENO-based limiters are as compact as the underlying DG schemes and therefore easy to implement. Numerical results for a wide range of flow conditions demonstrate that for DG schemes of up to fourth order of accuracy, the designed HWENO limiters can simul- taneously obtain uniform high order accuracy and sharp, es- sentially non-oscillatory shock transition.

A fast and reliable overset unstructured grids approach

A cell-centred overset unstructured grids approach is developed.In this approach,the intergrid boundary is initially established based on the wall distance from the cell centre,and is then optimized.To accelerate the intergrid-boundary definition much more,a neighbor-toneighbor donor search algorithm based on advancing-front method is modified with the help of minimum cuboid boxes.To simplify the communications between different grid cell types and to obtain second-order spatial accuracy,a new interpolation method is constructed based on linear reconstruction,which employs only one layer of fringe cells along the intergrid boundary.For unsteady flows with relative motion,the intergrid boundary can be redefined fast and automatically.Several numerical results show that the present dynamic overset unstructured grids approach is accurate and reliable.

Development of A Depth-Integrated Longshore Current Model with Unstructured Grids

A depth-integrated model for simulating wave-induced longshore current was developed with unstructured grids. Effects of surface roller and horizontal mixing under combined waves and currents were incorporated in the numerical model. Recommended values of model coefficients were also proposed based on sensitivity analysis. Field observations and three series of laboratory measurements including two cases conducted on the plane beach and one implemented on the ideal inlet were employed to examine the predictive capability of this model. For the field case and laboratory cases conducted on the plane beach, numerical results were compared favorably with the measured data. For the case with an ideal inlet, simulated circulation pattern is supposed to be reasonable although some deviations between numerical results and measured data still can be detected.

Two-dimensional shallow water equations with porosity and their numerical scheme on unstructured grids

In this study, porosity was introduced into two-dimensional shallow water equations to reflect the effects of obstructions, leading to the modification of the expressions for the flux and source terms. An extra porosity source term appears in the momentum equation. The numerical model of the shallow water equations with porosity is presented with the finite volume method on unstructured grids and the modified Roe-type approximate Riemann solver. The source terms of the bed slope and porosity are both decomposed in the characteristic direction so that the numerical scheme can exactly satisfy the conservative property. The present model was tested with a dam break with discontinuous porosity and a flash flood in the Toce River Valley. The results show that the model can simulate the influence of obstructions, and the numerical scheme can maintain the flux balance at the interface with high efficiency and resolution.

A typhoon-induced storm surge numerical model with GPU acceleration based on an unstructured spherical centroidal Voronoi tessellation grid

Storm surge is often the marine disaster that poses the greatest threat to life and property in coastal areas.Accurate and timely issuance of storm surge warnings to take appropriate countermeasures is an important means to reduce storm surge-related losses.Storm surge numerical models are important for storm surge forecasting.To further improve the performance of the storm surge forecast models,we developed a numerical storm surge forecast model based on an unstructured spherical centroidal Voronoi tessellation(SCVT)grid.The model is based on shallow water equations in vector-invariant form,and is discretized by Arakawa C grid.The SCVT grid can not only better describe the coastline information but also avoid rigid transitions,and it has a better global consistency by generating high-resolution grids in the key areas through transition refinement.In addition,the simulation speed of the model is accelerated by using the openACC-based GPU acceleration technology to meet the timeliness requirements of operational ensemble forecast.It only takes 37 s to simulate a day in the coastal waters of China.The newly developed storm surge model was applied to simulate typhoon-induced storm surges in the coastal waters of China.The hindcast experiments on the selected representative typhoon-induced storm surge processes indicate that the model can reasonably simulate the distribution characteristics of storm surges.The simulated maximum storm surges and their occurrence times are consistent with the observed data at the representative tide gauge stations,and the mean absolute errors are 3.5 cm and 0.6 h respectively,showing high accuracy and application prospects.

PRESSURE CORRECTION METHOD ON UNSTRUCTURED GRIDS

In this paper, an unstructured, collocated finite volume method for solvingthe Navier-Stokes equations was developed by virtue of auxiliary points. The derivatives weredetermined by the Gauss theorem. The proposed method could provide control volumes with arbitrarygeometry and preserve the second-order accuracy even if highly distorted grids are used. Althougharbitrary number of cell faces can be used, the hybrid quadrilateral/triangular grids are moredesirable for the simplicity of implementation and applications to engineering problems. Thepressure-velocity coupling was treated using a SIMPLE-like algorithm. The Generalized MinimumResidual (GMRES) method with the Incomplete LU (ILU) preconditioner was used to solve linearequations. Four test cases were studied for validating the proposed method. In using this method,grid quality is not important. Thus, engineers can pay mostly attention to physical mechanism ofproblems. Turbulence models can be simply integrated and the method can be straightforwardlyextended to treat three-dimensional problems.

High-order compact finite volume methods on unstructured grids with adaptive mesh refinement for solving inviscid and viscous flows

In the present paper, high-order finite volume schemes on unstructured grids developed in our previous papers are extended to solve three-dimensional inviscid and viscous flows. The highorder variational reconstruction technique in terms of compact stencil is improved to reduce local condition numbers. To further improve the efficiency of computation, the adaptive mesh refinement technique is implemented in the framework of high-order finite volume methods. Mesh refinement and coarsening criteria are chosen to be the indicators for certain flow structures. One important challenge of the adaptive mesh refinement technique on unstructured grids is the dynamic load balancing in parallel computation. To solve this problem, the open-source library p4 est based on the forest of octrees is adopted. Several two-and three-dimensional test cases are computed to verify the accuracy and robustness of the proposed numerical schemes.

A Parameter–Free Generalized Moment Limiter for High-Order Methods on Unstructured Grids

A parameter-free limiting technique is developed for high-order unstructured-grid methods to capture discontinuities when solving hyperbolic conservation laws.The technique is based on a“troubled-cell”approach,in which cells requiring limiting are first marked,and then a limiter is applied to these marked cells.A parameter-free accuracy-preserving TVD marker based on the cell-averaged solutions and solution derivatives in a local stencil is compared to several other markers in the literature in identifying“troubled cells”.This marker is shown to be reliable and efficient to consistently mark the discontinuities.Then a compact highorder hierarchical moment limiter is developed for arbitrary unstructured grids.The limiter preserves a degree p polynomial on an arbitrary mesh.As a result,the solution accuracy near smooth local extrema is preserved.Numerical results for the high-order spectral difference methods are provided to illustrate the accuracy,effectiveness,and robustness of the present limiting technique.

Modified QUICK Schemes for 3D Advection-Diffusion Equation of Pollutants on Unstructured Grids

In the framework of finite volume method(FVM),two modified schemes of quadratic upstream interpolation for convective kinematics(QUICK),namely quasi-QUICK(Q-QUICK) and normal quasi-QUICK(NQ-QUICK),for improving the precision of convective flux approximation are verified in 3D unsteady advectiondiffusion equation of pollutants on unstructured grids.The constructed auxiliary nodes for Q-QUICK or NQQUICK are composed of two neighboring nodes plus the next upwind node;the later node is generated from intersection of the line of current neighboring nodes and their corresponding interfaces.The numerical results show that Q-QUICK and NQ-QUICK overwhelm central differencing scheme(CDS) in computational accuracy and behave similar numerical stability to upwind difference scheme(UDS),hybrid differencing scheme(HDS) and power difference scheme(PDS) after applying the deferred correction method.Their corresponding CPU time is approximately equivalent to that of traditional difference schemes.In addition,their abilities for adapting high grid deformation are robust.It is so promising to apply the suggested schemes to simulate pollutant transportation on arbitrary 3D natural boundary in the hydraulic or environmental engineering.

A Generalised Lattice Boltzmann Equation on Unstructured Grids

This paper presents a new finite-volume discretization of a generalised LatticeBoltzmann equation (LBE) on unstructured grids. This equation is the continuumLBE, with the addition of a second order time derivative term (memory), and is derivedfrom a second-order differential form of the semi-discrete Boltzmann equationin its implicit form. The new scheme, named unstructured lattice Boltzmann equationwith memory (ULBEM), can be advanced in time with a larger time-step than the previousunstructured LB formulations, and a theoretical demonstration of the improvedstability is provided. Taylor vortex simulations show that the viscosity is the same aswith standard ULBE and demonstrates that the new scheme improves both stabilityand accuracy. Model validation is also demonstrated by simulating backward-facingstep flow at low and moderate Reynolds numbers, as well as by comparing the reattachmentlength of the recirculating eddy behind the step against experimental andnumerical data available in literature.

A Nominally Second-Order Cell-Centered Finite Volume Scheme for Simulating Three-Dimensional Anisotropic Diffusion Equations on Unstructured Grids

We present a finite volume based cell-centered method for solving diffusion equations on three-dimensional unstructured grids with general tensor conduction.Our main motivation concerns the numerical simulation of the coupling between fluid flows and heat transfers.The corresponding numerical scheme is characterized by cell-centered unknowns and a local stencil.Namely,the scheme results in a global sparse diffusion matrix,which couples only the cell-centered unknowns.The space discretization relies on the partition of polyhedral cells into sub-cells and on the partition of cell faces into sub-faces.It is characterized by the introduction of sub-face normal fluxes and sub-face temperatures,which are auxiliary unknowns.A sub-cellbased variational formulation of the constitutive Fourier law allows to construct an explicit approximation of the sub-face normal heat fluxes in terms of the cell-centered temperature and the adjacent sub-face temperatures.The elimination of the sub-face temperatures with respect to the cell-centered temperatures is achieved locally at each node by solving a small and sparse linear system.This system is obtained by enforcing the continuity condition of the normal heat flux across each sub-cell interface impinging at the node under consideration.The parallel implementation of the numerical algorithm and its efficiency are described and analyzed.The accuracy and the robustness of the proposed finite volume method are assessed by means of various numerical test cases.

APPLICATION OF A MODIFIED QUICK SCHEME TO DEPTH-AVERAGED k-ε TURBULENCE MODEL BASED ON UNSTRUCTURED GRIDS

The modified QUICK scheme on unstructured grid was used to improve the advection flux approximation, and the depth-averaged κ-ε turbulence model with the scheme based on FVM by SIMPLE series algorithm was established and applied to spur-dike flow computation. In this model, the over-relaxed approach was adopted to estimate the diffusion flux in view of its advantages in reducing errors and sustaining numerical stability usually encountered in non-orthogonal meshes. Two spur-dike cases with different defection angles （90°and 135°） were analyzed to validate the model. Computed results show that the predicted velocities and recirculation lengths are in good agreement with the observed data. Moreover, the computations on structured and unstructured grids were compared in terms of the approximately equivalent grid numbers. It can be concluded that the precision with unstructured grids is higher than that with structured grids in spite that the CPU time required is slightly more with unstructured grids Thus, it is significant to apply the method to numerical simulation of practical hydraulic engineering.

NUMERICAL SIMULATION OF SLOSHING IN RECTANGULAR TANK WITH VOF BASED ON UNSTRUCTURED GRIDS

A new method for sloshing simulation in a sway tank is present, in which the two phase interface is treated as a physical discontinuity, which can be captured by a well-designed high order scheme. Based on Normalized Variable Diagram （NVD）, a high order discretization scheme with unstructured grids is realized, together with a numerical method for free surface flow with a fixed grid. This method is implemented in an in-house code General Transport Equation Analyzer （ GTEA ） which is an unstructured grids finite volume solver. The present method is first validated by available analytical solutions. A simulation for a 2-D rectangular tank at different excitation frequencies of the sway is carried out. A comparison with experimental data in literature and results obtained by commercial software CFX shows that the sloshing load on the monitor points agrees well with the experimental data, with the same grids, and the present method gives better results on the secondary peak. It is shown that the present method can simulate the free surface overturning and breakup phenomena.

A Finite Volume Method with Unstructured Triangular Grids for Numerical Modeling of Tidal Current

The finite volume method （FVM） has many advantages in 2-D shallow water numerical simulation. In this study, the finite volume method is used with unstructured triangular grids to simulate the tidal currents. The Roe scheme is applied in the calculation of the intercell numerical flux, and the MUSCL method is introduced to improve its accuracy. The time integral is a two-step scheme of forecast and revision. For the verification of the present method, the Stoker＇s problem is calculated and the result is compared with the mathematically analytic solutions. The comparison indicates that the method is feasible. A sea area of a port is used as an example to test the method established here. The result shows that the present computational method is satisfactory, and it could be applied to the engineering fields.

Applications of multi-dimensional schemes on unstructured grids for high-accuracy heat flux prediction

The simulation of hypersonic flows with fully unstructured(tetrahedral)grids has severe problems with respect to the prediction of stagnation region heating,due to the random face orientation without alignment to the bow shock.To improve the accuracy of aero-heating predictions,three multi-dimensional approaches on unstructured grids are coupled in our Reynolds-averaged Navier-Stokes(RANS)solver,including multi-dimensional upwind flux reconstruction(MUP),multi-dimensional limiter(MLP-u2)and multi-dimensional gradient reconstruction(MLR).The coupled multi-dimensional RANS solver is validated by several typical verification and validation(V&V)cases,including hypersonic flows over a cylinder,a blunt biconic,and a double-ellipsoid,with commonly used prism/tetrahedral hybrid grids.Finally,the coupled multi-dimensional solver is applied to simulating the heat flux distribution over a 3D engineering configuration,i.e.a Hermes-like space shuttle model.The obtained numerical results are compared with experimental data.The predicted results demonstrate that the coupled multi-dimensional approach has a good prediction capability on aerodynamic heating over a wide range of complex engineering configurations.

Optimizing Parallel S_n Sweeps on Unstructured Grids for Multi-Core Clusters

In particle transport simulations, radiation effects are olden described by the discrete ordinates （Sn） form of Boltzmann equation. In each ordinate direction, the solution is computed by sweeping the radiation flux across the grid. Parallel Sn sweep on an unstructured grid can be explicitly modeled as topological traversal through an equivalent directed acyclic graph （DAG）, which is a data-driven algorithm. Its traditional design using MPI model results in irregular communication of massive short messages which cannot be efficiently handled by MPI runtime. Meanwhile, in high-end HPC cluster systems, multicore has become the standard processor configuration of a single node. The traditional data-driven algorithm of Sn sweeps has not exploited potential advantages of multi-threading of multicore on shared memory. These advantages, however, as we shall demonstrate, could provide an elegant solution resolving problems in the previous MPI-only design. In this paper, we give a new design of data-driven parallel Sn sweeps using hybrid MPI and Pthread programming, namely Sweep-H, to exploit hierarchical parallelism of processes and threads. With special multi-threading techniques and vertex schedule policy, Sweep-H gets more efficient communication and better load balance. We further present an analytical performance model for Sweep-H to reveal why and when it is advantageous over former MPI counterpart. On a 64-node multicore cluster system with 12 cores per node, 768 cores in total, Sweep-H achieves nearly linear scalability for moderate problem sizes, and better absolute performance than the previous times speedup on 64 nodes）. MPI algorithm on more than 16 nodes （by up to two

ADAPTIVE LAYERED CARTESIAN CUT CELL METHOD FOR THE UNSTRUCTURED HEXAHEDRAL GRIDS GENERATION

Adaptive layered Cartesian cut cell method is presented to solve the difficulty of the tmstructured hexahedral anisotropic Cartesian grids generation from the complex CAD model. ＂Vertex merging algorithm based on relaxed AVL tree is investigated to construct topological structure for stereo lithography （STL） files, and a topology-based self-adaptive layered slicing algorithm with special features control strategy is brought forward. With the help of convex hull, a new points-in-polygon method is employed to improve the Cartesian cut cell method. By integrating the self-adaptive layered slicing algorithm and the improved Cartesian cut cell method, the adaptive layered Cartesian cut cell method gains the volume data of the complex CAD model in STL file and generates the unstructured hexahedral anisotropic Cartesian grids.

Three-dimensional inversion of controlled-source audio-frequency magnetotelluric data based on unstructured fi nite-element method

We propose a new 3D inversion scheme to invert the near-and transition-zone data of CSAMT with topography accurately.In this new method,the earth was discretized into unstructured tetrahedra to fit the ragged topography and the vector fi nite-element method was adopted to obtain precise responses and good sensitivity.To simulate the attitude and shape of the transmitter,we divided a long-grounded transmitter into dipoles and integrated these dipoles to obtain good responses in the near-and transition-fi eld zones.Next,we designed an L2 norm-based objective functional and applied a standard quasi-Newton method as the optimization method to solve the inverse problem and guarantee steady convergence.We tested our 3D inversion method first on synthetic data and then on a field dataset acquired from select sites near Changbai Mountain,China.In both tests,the new inversion algorithm achieved excellent fitting between the predicted and observed data,even in near-and transition-fi eld zones,and the inversion results agreed well with the true model.These fi ndings reveal that the proposed algorithm is eff ective for 3D inversion of CSAMT data.

An efficient unstructured WENO method for supersonic reactive flows

An efficient high-order numerical method for supersonic reactive flows is proposed in this article.The reactive source term and convection term are solved separately by splitting scheme.In the reaction step,an adaptive time-step method is presented,which can improve the efficiency greatly.In the convection step,a third-order accurate weighted essentially non-oscillatory（WENO）method is adopted to reconstruct the solution in the unstructured grids.Numerical results show that our new method can capture the correct propagation speed of the detonation wave exactly even in coarse grids,while high order accuracy can be achieved in the smooth region.In addition,the proposed adaptive splitting method can reduce the computational cost greatly compared with the traditional splitting method.