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.

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.

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 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.

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.

A multithreaded parallel upwind sweep algorithm for the S_(N) transport equations discretized with discontinuous finite elements

The complex structure and strong heterogeneity of advanced nuclear reactor systems pose challenges for high-fidelity neutron-shielding calculations. Unstructured meshes exhibit strong geometric adaptability and can overcome the deficiencies of conventionally structured meshes in complex geometry modeling. A multithreaded parallel upwind sweep algorithm for S_(N) transport was proposed to achieve a more accurate geometric description and improve the computational efficiency. The spatial variables were discretized using the standard discontinuous Galerkin finite-element method. The angular flux transmission between neighboring meshes was handled using an upwind scheme. In addition, a combination of a mesh transport sweep and angular iterations was realized using a multithreaded parallel technique. The algorithm was implemented in the 2D/3D S_(N) transport code ThorSNIPE, and numerical evaluations were conducted using three typical benchmark problems:IAEA, Kobayashi-3i, and VENUS-3. These numerical results indicate that the multithreaded parallel upwind sweep algorithm can achieve high computational efficiency. ThorSNIPE, with a multithreaded parallel upwind sweep algorithm, has good reliability, stability, and high efficiency, making it suitable for complex shielding calculations.

A Computational Fluid Dynamics (CFD) Analysis of an Undulatory Mechanical Fin Driven by Shape Memory Alloy

Many fishes use undulatory fin to propel themselves in the underwater environment. These locomotor mechanisms have a popular interest to many researchers. In the present study, we perform a three-dimensional unsteady computation of an undulatory mechanical fin that is driven by Shape Memory Alloy （SMA）. The objective of the computation is to investigate the fluid dynamics of force production associated with the undulatory mechanical fin. An unstructured, grid-based, unsteady Navier-Stokes solver with automatic adaptive remeshing is used to compute the unsteady flow around the fin through five complete cycles. The pressure distribution on fin surface is computed and integrated to provide fin forces which are decomposed into lift and thrust. The velocity field is also computed throughout the swimming cycle. Finally, a comparison is conducted to reveal the dynamics of force generation according to the kinematic parameters of the undulatory fin （amplitude, frequency and wavelength）.

Simulation of bluff body stabilized flows with hybrid RANS and PDF method

The motivation of this study is to investigate the turbulence-chemistry interactions by using probability density function （PDF） method. A consistent hybrid Reynolds Averaged Navier-Stokes （RANS）/PDF method is used to simulate the turbulent non-reacting and reacting flows. The joint fluctuating velocity-frequency-composition PDF equation coupled with the Reynolds averaged density, momentum and energy equations are solved on unstructured meshes by the Lagrangian Monte Carlo （MC） method combined with the finite volume （FV） method. The simulation of the axisymmetric bluff body stabilized non-reacting flow fields is presented in this paper. The calculated length of the recirculation zone is in good agreement with the experimental data. Moreover, the significant change of the flow pattern with the increase of the jet-to-coflow momentum flux ratio is well predicted. In addition, comparisons are made between the joint PDF model and two different Reynolds stress models.

Effects of mesh loop modes on performance of unstructured finite volume GPU simulations

In unstructured finite volume method,loop on different mesh components such as cells,faces,nodes,etc is used widely for the traversal of data.Mesh loop results in direct or indirect data access that affects data locality significantly.By loop on mesh,many threads accessing the same data lead to data dependence.Both data locality and data dependence play an important part in the performance of GPU simulations.For optimizing a GPU-accelerated unstructured finite volume Computational Fluid Dynamics(CFD)program,the performance of hot spots under different loops on cells,faces,and nodes is evaluated on Nvidia Tesla V100 and K80.Numerical tests under different mesh scales show that the effects of mesh loop modes are different on data locality and data dependence.Specifically,face loop makes the best data locality,so long as access to face data exists in kernels.Cell loop brings the smallest overheads due to non-coalescing data access,when both cell and node data are used in computing without face data.Cell loop owns the best performance in the condition that only indirect access of cell data exists in kernels.Atomic operations reduced the performance of kernels largely in K80,which is not obvious on V100.With the suitable mesh loop mode in all kernels,the overall performance of GPU simulations can be increased by 15%-20%.Finally,the program on a single GPU V100 can achieve maximum 21.7 and average 14.1 speed up compared with 28 MPI tasks on two Intel CPUs Xeon Gold 6132.

Numerical simulation of pedestrian flow past a circular obstruction

In this paper, a revisiting Hughes’ dynamic continuum model is used to investigate and predict the essential macroscopic characteristics of pedestrian flow, such as flow, density and average speed, in a two dimensional continuous walking facility scattered with a circular obstruction. It is assumed that pedestrians prefer to walk a path with the lowest instantaneous travel cost from origin to destination, under the consideration of the current traffic conditions and the tendency to avoid a high-density region and an obstruction. An algorithm for the pedestrian flow model is based on a cellcentered finite volume method for a scalar conservation law equation, a fast sweeping method for an Eikonal-type equation and a second-order TVD Runge-Kutta method for the time integration on unstructured meshes. Numerical results demonstrate the effectiveness of the algorithm. It is verified that density distribution of pedestrian flow is influenced by the position of the obstruction and the path-choice behavior of pedestrians.

3-D Marine CSEM Modeling in General Anisotropic Media by Using an Adaptive Finite Element Approach Based on the Vector-Scalar Potential

We present three-dimensional(3-D)modeling method of marine controlled-source electromagnetic(CSEM)fields in general anisotropic media using an adaptive finite element approach based on the vector-scalar potential.The modeling is based on the governing Helmholtz equations in the vector-scalar potential system.Unstructured tetrahedral grids are employed,which can exactly simulate the terrain relief and complex electrical structures.Moreover,based on the gradient recovery technology,the adaptive finite element approach is used to drive the mesh refinement,and make the finite element solutions converge gradually to the exact solutions.The primary-secondary field approach is used to improve the numerical accuracy of CSEM fields near the source point,where the primary field is calculated by using the quasi-analytical formula.The accuracy of this approach is verified by a one-dimensional model.Two 3-D models are used to demonstrate the effectiveness of the adaptive mesh refinement and the influences of dipping anisotropy layer on the marine CSEM responses for both inline and broadside geometries.The complex synthetic model is simulated to show the capability and flexibility of the approach for geometrically complex situations.

Lagrangian-Eulerian One-Step WENO Finite Volume Schemes on Unstructured Triangular Meshes

In this article we present a new class of high order accurate Arbitrary-Eulerian-Lagrangian(ALE)one-step WENO finite volume schemes for solving nonlinear hyperbolic systems of conservation laws on moving two dimensional unstructured triangular meshes.A WENO reconstruction algorithm is used to achieve high order accuracy in space and a high order one-step time discretization is achieved by using the local space-time Galerkin predictor proposed in[25].For that purpose,a new element-local weak formulation of the governing PDE is adopted on moving space-time elements.The space-time basis and test functions are obtained considering Lagrange interpolation polynomials passing through a predefined set of nodes.Moreover,a polynomial mapping defined by the same local space-time basis functions as the weak solution of the PDE is used to map the moving physical space-time element onto a space-time reference element.To maintain algorithmic simplicity,the final ALE one-step finite volume scheme uses moving triangular meshes with straight edges.This is possible in the ALE framework,which allows a local mesh velocity that is different from the local fluid velocity.We present numerical convergence rates for the schemes presented in this paper up to sixth order of accuracy in space and time and show some classical numerical test problems for the two-dimensional Euler equations of compressible gas dynamics.