UPWIND SCHEME FOR IDEAL 2-D MHD FLOWS BASED ON UNSTRUCTURED MESH

An upwind scheme based on the unstructured mesh is developed to solve ideal 2-D magnetohydrodynamics （MHD） equations. The inviscid fluxes are approximated by using the modified advection upstream splitting method （AUSM） scheme, and a 5-stage explicit Runge-Kutta scheme is adopted in the time integration. To avoid the influence of the magnetic field divergence created during the simulation, the hyperbolic divergence cleaning method is introduced. The shock-capturing properties of the method are verified by solving the MHD shock-tube problem. Then the 2-D nozzle flow with the magnetic field is numerically simulated on the unstructured mesh. Computational results demonstrate the effects of the magnetic field and agree well with those from references.

Three-dimensional magnetotellurics modeling using edgebased finite-element unstructured meshes

Three-dimensional forward modeling magnetotellurics （MT） problems. We present a is a challenge for geometrically complex new edge-based finite-element algorithm using an unstructured mesh for accurately and efficiently simulating 3D MT responses. The electric field curl-curl equation in the frequency domain was used to deduce the H （curl） variation weak form of the MT forward problem, the Galerkin rule was used to derive a linear finite-element equation on the linear-edge tetrahedroid space, and, finally, a BI-CGSTAB solver was used to estimate the unknown electric fields. A local mesh refinement technique in the neighbor of the measuring MT stations was used to greatly improve the accuracies of the numerical solutions. Four synthetic models validated the powerful performance of our algorithms. We believe that our method will effectively contribute to processing more complex MT studies.

High Accuracy Finite Volume Method for Solving Nonlinear Aeroacoustics Problems on Unstructured Meshes

The paper presents a finite volume numerical method universally applicable for solving both linear and nonlinear aeroacoustics problems on arbitrary unstructured meshes. It is based on the vertexcentered multi-parameter scheme offering up to the 6th accuracy order achieved on the Cartesian meshes. An adaptive dissipation is added for the numerical treatment of possible discontinuities. The scheme properties are studied on a series of test cases, its efficiency is demonstrated at simulating the noise suppression in resonance-type liners.

High-order Lagrangian cell-centered conservative scheme on unstructured meshes

A high-order Lagrangian cell-centered conservative gas dynamics scheme is presented on unstructured meshes. A high-order piecewise pressure of the cell is intro- duced. With the high-order piecewise pressure of the cell, the high-order spatial discretiza- tion fluxes are constructed. The time discretization of the spatial fluxes is performed by means of the Taylor expansions of the spatial discretization fluxes. The vertex velocities are evaluated in a consistent manner due to an original solver located at the nodes by means of momentum conservation. Many numerical tests are presented to demonstrate the robustness and the accuracy of the scheme.

Matrix-Free Higher-Order Finite Element Method for Parallel Simulation of Compressible and Nearly-Incompressible Linear Dedicated to Professor Karl Stark Pister for his 95th birthday Elasticity on Unstructured Meshes

Higher-order displacement-based finite element methods are useful for simulating bending problems and potentially addressing mesh-locking associated with nearly-incompressible elasticity,yet are computationally expensive.To address the computational expense,the paper presents a matrix-free,displacement-based,higher-order,hexahedral finite element implementation of compressible and nearly-compressible(ν→0.5)linear isotropic elasticity at small strain with p-multigrid preconditioning.The cost,solve time,and scalability of the implementation with respect to strain energy error are investigated for polynomial order p=1,2,3,4 for compressible elasticity,and p=2,3,4 for nearly-incompressible elasticity,on different number of CPU cores for a tube bending problem.In the context of this matrix-free implementation,higher-order polynomials(p=3,4)generally are faster in achieving better accuracy in the solution than lower-order polynomials(p=1,2).However,for a beam bending simulation with stress concentration(singularity),it is demonstrated that higher-order finite elements do not improve the spatial order of convergence,even though accuracy is improved.

Reverse time migration imaging of tunnels via the finite element method using an unstructured mesh

Wavefield extrapolation is critical in reverse time migration(RTM).The finite diff erence method is primarily used to achieve wavefi eld extrapolation in case of the RTM imaging of tunnels.However,complex tunnel models,including those for karsts and fault fracture zones,are constructed using regular grids with straight curves,which can cause numerical dispersion and reduce the imaging accuracy.In this study,wavefi eld extrapolation was conducted for tunnel RTM using the finite element method,wherein an unstructured mesh was considered to be the body-fi tted partition in a complex model.Further,a Poynting vector calculation equation suitable for the unstructured mesh considered in the fi nite element method was established to suppress the interference owing to low-frequency noise.The tunnel space was considered during wavefi eld extrapolation to suppress the mirror artifacts based on the fl exibility of mesh generation.Finally,the infl uence of the survey layouts(one and two sidewalls)on the tunnel imaging results was investigated.The RTM results obtained for a simple tunnel model with an inclined interface demonstrate that the method based on unstructured meshes can effectively suppress the low-frequency noise and mirror artifacts,obtaining clear imaging results.Furthermore,the two-sidewall tunnel survey layout can be used to accurately obtain the real position of the inclined interface ahead of the tunnel face.The complex tunnel numerical modeling and actual data migration results denote the eff ectiveness of the fi nite element method in which an unstructured mesh is used.

DISCONTINUITY-CAPTURING FINITE ELEMENT COMPUTATION OF UNSTEADY FLOW WITH ADAPTIVE UNSTRUCTURED MESH

A discontinuity-capturing scheme of finite element method(FEM)is proposed.The unstructured-grid technique combined with a new type of adaptive mesh approach is developed for both compressible and incompressible unsteady flows,which exhibits the capability of capturing the shock waves and/or thin shear layers accurately in an unsteady viscous flow at high Reynolds number. In particular,a new testing variable,i.e.,the disturbed kinetic energy E,is suggested and used in the adaptive mesh computation,which is universally applicable to the capturing of both shock waves and shear layers in the inviscid flow and viscous flow at high Reynolds number.Based on several calculated examples,this approach has been proved to be effective and efficient for the calculations of compressible and incompressible flows.

NUMERICAL SIMULATION FOR 2D SHALLOW WATER EQUATIONS BY USING GODUNOV-TYPE SCHEME WITH UNSTRUCTURED MESH

In order to establish a well-balanced scheme, 2D shallow water equations were transformed and solved by using the Finite Volume Method （FVM） with unstructured mesh. The numerical flux from the interface between cells was computed with an exact Riemann solver, and the improved dry Riemann solver was applied to deal with the wet/dry problems. The model was verified through computing some typical examples and the tidal bore on the Qiantang River. The results show that the scheme is robust and accurate, and could be applied extensively to engineering problems.

Development and validation of a three-dimensional,wave-current coupled model on unstructured meshes

Using unstructured meshes provides great flexibility for modeling the flow in complex geomorphology of tidal creeks,barriers and islands,with refined grid resolution in regions of interest and not elsewhere.In this paper,an unstructured three-dimensional fully coupled wave-current model is developed.Firstly,a parallel,unstructured wave module is developed.Variations in wave properties are governed by a wave energy equation that includes wave-current interactions and dissipation representative of wave breaking.Then,the existing Finite-Volume Coastal Ocean Model(FVCOM) is modified to couple with the wave module.The couple procedure includes depth dependent wave radiation stress terms,Stokes drift,vertical transfer of wave-generated pressure transfer to the mean momentum equation,wave dissipation as a source term in the turbulence kinetic energy equation,and mean current advection and refraction of wave energy.Several applications are presented to evaluate the developed model.In particular the wind and wave-induced storm surge generated by Hurricane Katrina is investigated.The obtained results have been compared to the in situ measurements with respect to the wave heights and water level elevations revealing good accuracy of the model in reproduction of the investigated events.In a comparison to water level measurements at Dauphin Island,inclusion of the wave induced water level setup reduced the normalized root mean square error from 0.301 to 0.257 m and increased the correlation coefficient from 0.860 to 0.929.Several runs were carried out to analyze the effects of waves.The experiments show that among the processes that represent wave effects,radiation stress and wave-induced surface stress are more important than wave-induced bottom stress in affecting the water level.The Hurricane Katrina simulations showed the importance of the inclusion of the wave effects for the hindcast of the water levels during the storm surge.

An Improved LU-SGS Implicit Scheme for High Reynolds Number Flow Computations on Hybrid Unstructured Mesh

The lower-upper symmetric Gauss-Seidel （LU-SGS） implicit relaxation has been widely used because it has the merits of less dependency on grid topology, low numerical complexity and modest memory requirements. In original LU-SGS scheme, the implicit system matrix is constructed based on the splitting of convective flux Jacobian according to its spectral radius. Although this treatment has the merit of reducing computational complexity and helps to ensure the diagonally dominant property of the implicit system matrix, it can also cause serious distortions on the implicit system matrix because too many approximations are introduced by this splitting method if the contravariant velocity is small or close to sonic speed. To overcome this shortcoming, an improved LU-SGS scheme with a hybrid construction method for the implicit system matrix is developed in this paper. The hybrid way is that： on the cell faces having small contravariant velocity or transonic contravariant velocity, the accurate derivative of the convective flux term is used to construct more accurate implicit system matrix, while the original Jacobian splitting method is adopted on the other cell faces to reduce computational complexity and ensure the diagonally dominant property of the implicit system matrix. To investigate the convergence performance of the improved LU-SGS scheme, 2D and 3D turbulent flows around the NACA0012 airfoil, RAE2822 airfoil and LANN wing are simulated on hybrid unstructured meshes. The nu- merical results show that the improved LU-SGS scheme is significantly more efficient than the original LU-SGS scheme.

An unstructured mesh method for numerical simulation of violent sloshing flows

An unstructured mesh Reynolds-averaged Navier-Stokes(RANS)solver has been developed for numerical simulation of violent sloshing flows inside a tank with complicated inner structures.The numerical solver employs the unstructured multi-dimensional tangent hyperbolic interface capturing method(UMTHINC)for free-surface capturing combined with various turbulence models.The sloshing motion is numerically modeled using the body-force method which introduces a source term into the momentum equation corresponding to the tank motion profile.Numerical simulations of the tank sloshing problems are performed for different test cases with various oscillation frequencies.The performance of the interface capturing method has been discussed and the effect of turbulence model choice on loading predictions is highlighted by studying several RANS models and analyzing its effect on fluid motion and impact pressure.Numerical simulations of the sloshing inside the tank with a vertical baffle has also been conducted and a discussion is provided on different numerical treatment of the baffle.

Automatic differentiation based discrete adjoint method for aerodynamic design optimization on unstructured meshes

A systematic methodology for formulating,implementing,solving and verifying discrete adjoint of the compressible Reynolds-averaged Navier-Stokes（RANS） equations for aerodynamic design optimization on unstructured meshes is proposed.First,a general adjoint formulation is constructed for the entire optimization problem,including parameterization,mesh deformation,flow solution and computation of the objective function,which is followed by detailed formulations of matrix-vector products arising in the adjoint model.According to this formulation,procedural components of implementing the required matrix-vector products are generated by means of automatic differentiation（AD） in a structured and modular manner.Furthermore,a duality-preserving iterative algorithm is employed to solve flow adjoint equations arising in the adjoint model,ensuring identical convergence rates for the tangent and the adjoint models.A three-step strategy is adopted to verify the adjoint computation.The proposed method has several remarkable features：the use of AD techniques avoids tedious and error-prone manual derivation and programming;duality is strictly preserved so that consistent and highly accurate discrete sensitivities can be obtained;and comparable efficiency to hand-coded implementation can be achieved.Upon the current discrete adjoint method,a gradient-based optimization framework has been developed and applied to a drag reduction problem.

A large-scale wave-current coupled module with wave diffraction effect on unstructured meshes

Based on the extended mild-slope equation,a large-scale wave module is developed.By combining the eikonal equation and the modified wave action equation,the wave model can account for diffraction in most situations such as in the lee of islands and breakwaters,and using unstructured meshes provides great flexibility for modelling the wave in the complex geomorphology of barriers and islands,also allowing for refinement of the grid resolution within computationally important domains.The numerical implementation of the module is based on the explicit second-order upwind finite-volume schemes in geographic space,the Flux-Corrected Transport(FCT)algorithm in frequency space and the implicit Crank-Nicolson method in directional space.The three-dimensional hydrodynamic module is then modified to couple with the wave model,where the wave readily provides the depth-dependent radiation stress and the wave-induced turbulence coefficient for the current fields,and the wave propagation takes into account the current-induced advection,refraction and diffraction of wave energy and the effect of water level.The applicability of the proposed model to calculate Snell’s Law,wave transformation over the breakwaters and the elliptic shoal,wave propagation over the rip current field and the undertow on a sloping beach is evaluated.Numerical results show that the present model makes better predictions of the near-shore wave propagation and complex three-dimensional(3D)near-shore circulation driven by the waves,considering analytical solutions and experimental values.

A Priori Subcell Limiting Approach for the FR/CPR Method on Unstructured Meshes

A priori subcell limiting approach is developed for high-order flux reconstruction/correction procedure via reconstruction(FR/CPR)methods on twodimensional unstructured quadrilateralmeshes.Firstly,a modified indicator based on modal energy coefficients is proposed to detect troubled cells,where discontinuities exist.Then,troubled cells are decomposed into nonuniform subcells and each subcell has one solution point.A second-order finite difference shock-capturing scheme based on nonuniform nonlinear weighted(NNW)interpolation is constructed to perform the calculation on troubled cells while smooth cells are calculated by the CPR method.Numerical investigations show that the proposed subcell limiting strategy on unstructured quadrilateral meshes is robust in shock-capturing.

Mesh Motion Approach Based on Spring Analogy Method for Unstructured Meshes

Mesh motion strategy is one of the key points in many fluid-structure interaction problems. One popular technique used to solve this problem is known as the spring analogy method. In this paper a new mesh update approach based on the spring analogy method is presented for the effective treatment of mesh moving boundary problems. The proposed mesh update technique is developed to avoid the generation of squashed invalid elements and maintain mesh quality by considering each element shape and grid scale to the definition of the spring stiffness. The method is applied to several 2D and 3D boundary correction problems for fully unstructured meshes and evaluated by a mesh quality indicator. With these applications,it is demonstrated that the present method preserves mesh quality even under large motions of bodies. We highlight the advantages of this method with respect to robustness and mesh quality.

Runge-Kutta Discontinuous Galerkin Method with Front Tracking Method for Solving the Compressible Two-Medium Flow on Unstructured Meshes

In this paper,we extend using the Runge-Kutta discontinuous Galerkin method together with the front tracking method to simulate the compressible twomedium flow on unstructured meshes.A Riemann problem is constructed in the normal direction in the material interfacial region,with the goal of obtaining a compact,robust and efficient procedure to track the explicit sharp interface precisely.Extensive numerical tests including the gas-gas and gas-liquid flows are provided to show the proposed methodologies possess the capability of enhancing the resolutions nearby the discontinuities inside of the single medium flow and the interfacial vicinities of the two-medium flow in many occasions.

A Novel VOF-Type Volume-Tracking Method for Free-Surface Flows Based on Unstructured Triangular Mesh

A novel VOF-type volume-tracking method for two-dimensional free-surface flows based on the unstructured triangular mesh is presented. Owing to the inherent merit of the unstructured triangular mesh in fitting curved boundaries, this method can handle the free-surface problems with complex geometries accurately and directly, without introducing any complicated boundary treatment or artificial diffusion. The method solves the volume transport equation geometrically through the Modified Lagrangian-Eulerian Re-map （MLER） method, which is applied to advective fluid volumes. Moreover, the PLIC method is adopted to give a second-order reconstructed interface approximation. To validate this method, two advection tests were performed for the establishment of the accuracy and convergence rate of the solutions. Numerical results for these complex tests provide convincing evidence for the excellent solution quality and fidelity of the method.

Application of the VOF method based on unstructured quadrilateral mesh

To simulate two-dimensional free-surface flows with complex boundaries directly and accurately, a novel VOF (Volume-of-fluid) method based on unstructured quadrilateral mesh is presented. Without introducing any complicated boundary treatment or artificial diffusion, this method treated curved boundaries directly by utilizing the inherent merit of unstructured mesh in fitting curves. The PLIC (Piecewise Linear Interface Calculation) method was adopted to obtain a second-order accurate linearized reconstruction approximation and the MLER (Modified Lagrangian-Eulerian Re-map) method was introduced to advect fluid volumes on unstructured mesh. Moreover, an analytical relation for the interface’s line constant vs. the volume clipped by the interface was developed so as to improve the method’s efficiency. To validate this method, a comprehensive series of large straining advection tests were performed. Numerical results provide convincing evidences for the method’s high volume conservative accuracy and second-order shape error convergence rate. Also, a dramatic improvement on computational accuracy over its unstructured triangular mesh counterpart is checked.

An improved two-dimensional unstructured CE/SE scheme for capturing shock waves

In this paper, the accuracy of Chang＇s unstructured space-time conservation element and solution element （CE/SE） scheme is analysed for the first time. Based on a redefinition of conservation elements and solution elements, an improved two-dimensional （2D） unstructured CE/SE scheme with an adjustable parameter β is proposed to accurately capture shock waves. The new scheme can be applied to any type of grid without special treatnmnt. Compared with Chang＇s original parameter a, larger/5 dose not cost extra computational resources. Numerical tests reveal that the new scheme is not only clear in physical concept, compact and highly accurate but also more capable of capturing shock waves than the popular fifth-order accurate weighted essentially non-oscillatory scheme.

High-order discontinuous Galerkin solver on hybrid anisotropic meshes for laminar and turbulent simulations

Efficient and robust solution strategies are developed for discontinuous Galerkin （DG） discretization of the Navier-Stokes （NS） and Reynolds-averaged NS （RANS） equations on structured/unstructured hybrid meshes. A novel line-implicit scheme is devised and implemented to reduce the memory gain and improve the computational eificiency for highly anisotropic meshes. A simple and effective technique to use the mod- ified Baldwin-Lomax （BL） model on the unstructured meshes for the DC methods is proposed. The compact Hermite weighted essentially non-oscillatory （HWENO） limiters are also investigated for the hybrid meshes to treat solution discontinuities. A variety of compressible viscous flows are performed to examine the capability of the present high- order DG solver. Numerical results indicate that the designed line-implicit algorithms exhibit weak dependence on the cell aspect-ratio as well as the discretization order. The accuracy and robustness of the proposed approaches are demonstrated by capturing com- plex flow structures and giving reliable predictions of benchmark turbulent problems.