Feature study of body-fitted Cartesian grids used in casting numerical simulation

A type of mesh called a body-fi tted Cartesian mesh,very different from the traditional structured body-fi tted mesh,is established.At f irst,the right parallelepiped mesh is generated,then,a feature analysis is done on the cross sections.These cross sections are the intersections of the casting shape with the right parallelepiped grids(under the Cartesian coordinate system).On the basis of the feature analysis,two sorts of body-f itted boundary grids,shape-keeping grids and shape-distortion grids,are def ined.Shape-distortion grids can be removed or weaken by increasing the number of grids or moving the coordinates of the mesh generation region,so actually the body-fitted Cartesian mesh generation is to get shape-keeping grids.A shape-keeping grid mainly consists of two sorts of surfaces(I type face and II type face),and each of them is joined by two types of points(I type point and II type point).If only these two types of points were given,the shape-keeping mesh would be constructed.In this paper,the cases of the above two boundary grids being generated were discussed.An algorithm was put forward to get the shape-keeping grids.Several body-fi tted Cartesian meshes generated on castings show the validity of the algorithm.The mesh generation examples show that the body-fi tted Cartesian mesh is more excellent than the right parallelepiped mesh in aspects of decreasing grids number and being closer to the shape of the casting solid.

A Hybrid WENO Scheme for Steady Euler Equations in Curved Geometries on Cartesian Grids

For steady Euler equations in complex boundary domains,high-order shockcapturing schemes usually suffer not only from the difficulty of steady-state convergence but also from the problem of dealing with physical boundaries on Cartesian grids to achieve uniform high-order accuracy.In this paper,we utilize a fifth-order finite difference hybrid WENO scheme to simulate steady Euler equations,and the same fifth-order WENO extrapolation methods are developed to handle the curved boundary.The values of the ghost points outside the physical boundary can be obtained by applying WENO extrapolation near the boundary,involving normal derivatives acquired by the simplified inverse Lax-Wendroff procedure.Both equivalent expressions involving curvature and numerical differentiation are utilized to transform the tangential derivatives along the curved solid wall boundary.This hybrid WENO scheme is robust for steady-state convergence and maintains high-order accuracy in the smooth region even with the solid wall boundary condition.Besides,the essentially non-oscillation property is achieved.The numerical spectral analysis also shows that this hybrid WENO scheme has low dispersion and dissipation errors.Numerical examples are presented to validate the high-order accuracy and robust performance of the hybrid scheme for steady Euler equations in curved domains with Cartesian grids.

A fast and automatic full-potential finite volume solver on Cartesian grids for unconventional configurations

To meet the requirements of fast and automatic computation of subsonic and transonic aerodynamics in aircraft conceptual design,a novel finite volume solver for full potential flows on adaptive Cartesian grids is developed in this paper.Cartesian grids with geometric adaptation are firstly generated automatically with boundary cells processed by cell-cutting and cell-merging algorithms.The nonlinear full potential equation is discretized by a finite volume scheme on these Cartesian grids and iteratively solved in an implicit fashion with a generalized minimum residual（GMRES） algorithm.During computation,solution-based mesh adaptation is also applied so as to capture flow features more accurately.An improved ghost-cell method is proposed to implement the non-penetration wall boundary condition where the velocity-potential of a ghost cell is modified by an analytic method instead.According to the characteristics of the Cartesian grids,the Kutta condition is applied by specially computing the gradients on Kutta-faces without directly assigning the potential jump to cells adjacent wake faces,which can significantly improve the solution converging speed.The feasibility and accuracy of the proposed method are validated by several typical cases of sub/transonic flows around an ONERA M6 wing,a DLR-F4 wing-body,and an unconventional figuration of a blended wing body（BWB）.The validation cases demonstrate a fast convergence with fully automatic grid treatment and computation,and the results suggest its capacity in application for aircraft conceptual design.

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.

The Ghost Cell Method and its Applications for Inviscid Compressible Flow on Adaptive Tree Cartesian Grids

In this paper,an immersed boundary algorithm is developed by combining the ghost cell method with adaptive tree Cartesian grid method.Furthermore,the proposed method is successfully used to evaluate various inviscid compressible flow with immersed boundary.The extension to three dimensional cases is also achieved.Numerical examples demonstrate the proposed method is effective.

Cartesian Grid Method with Cut Cell in the Mold Filling Simulation

In this paper, a Cartesian grid method with cut cell has been developed to simulate mold filling of casting process. Cut cells at the cast-mold interface are generated on the Cartesian grid. With the boundary cut cells, a special treatment is necessary. That is Cartesian grid method with cut cell. A simple shape was tested and the cut cell method was compared with the traditional one on Cartesian grids. And, a developed method was applied to the real casting product simulation. Cartesian grid system causes momentum loss and unsound fluid flow patterns because of inaccurate generation of meshes. These problems have been improved by using cut cell method.

New Immersed Boundary Method on the Adaptive Cartesian Grid Applied to the Local Discontinuous Galerkin Method

Currently, many studies on the local discontinuous Galerkin method focus on the Cartesian grid with low computational e ciency and poor adaptability to complex shapes. A new immersed boundary method is presented, and this method employs the adaptive Cartesian grid to improve the adaptability to complex shapes and the immersed boundary to increase computational e ciency. The new immersed boundary method employs different boundary cells(the physical cell and ghost cell) to impose the boundary condition and the reconstruction algorithm of the ghost cell is the key for this method. The classical model elliptic equation is used to test the method. This method is tested and analyzed from the viewpoints of boundary cell type, error distribution and accuracy. The numerical result shows that the presented method has low error and a good rate of the convergence and works well in complex geometries. The method has good prospect for practical application research of the numerical calculation research.

Euler solution using adaptive Cartesian grid with a gridless boundary treatment

A quadtree-based adaptive Cartesian grid generator and flow solver were developed. The grid adaptation based on pressure or density gradient was performed and a gridless method based on the least-square fashion was used to treat the wall surface boundary condition, which is generally difficult to be handled for the common Cartesian grid. First, to validate the technique of grid adaptation, the benchmarks over a forward-facing step and double Mach reflection were computed. Second, the flows over the NACA 0012 airfoil and a two-element airfoil were calculated to validate the developed gridless method. The computational results indi- cate the developed method is reasonable for complex flows.

HYBRID CARTESIAN GRID/GRIDLESS METHOD FOR CALCULATING VISCOUS FLOWS OVER MULTI-ELEMENT AIRFOILS

A hybrid Cartesian grid/gridless method is developed for calculating viscous flows over multi-element airfoils.The method adopts an unstructured Cartesian grid to cover most areas of the computational domain and leaves only small region adjacent to the aerodynamic bodies to be filled with the cloud of points used in the gridless methods,which results in a better combination of the computational efficiency of the Cartesian grid and the flexibility of the gridless method in handling complex geometries.The clouds of points in the local gridless region are implemented in an anisotropic way according to the features of the thin boundary layer of the viscous flows over the airfoils,and the clouds of points at the vicinity of the interface between the grid and the gridless regions are also controlled by using an adaptive refinement technique during the generation of the unstructured Cartesian grid.An implementation of the resulting hybrid method is presented for solving two-dimensional compressible Navier-Stokes(NS)equations.The simulations of the viscous flows over a RAE2822airfoil or a two-element airfoil are successfully carried out,and the obtained results agree well with the available experimental data.

ADAPTIVE HYBRID CARTESIAN GRID METHOD FOR VORTEX-DOMINATED FLOWS

An efficient compressible Euler equation solver for vortex-dominated flows is presented based on the adaptive hybrid Cartesian mesh and vortex identifying method.For most traditional grid-based Euler solvers,the excessive numerical dissipation is the great obstruction for vortex capturing or tracking problems.A vortex identifying method based on the curl of velocity is used to identify the vortex in flow field.Moreover,a dynamic adaptive mesh refinement(DAMR)process for hybrid Cartesian gird system is employed to track and preserve vortex.To validate the proposed method,a single compressible vortex convection flow is involved to test the accuracy and efficiency of DAMR process.Additionally,the vortex-dominated flow is investigated by the method.The obtained results are shown as a good agreement with the previous published data.

Hybrid Cartesian Grid Method for Moving Boundary Problems

A hybrid Cartesian structured grid method is proposed for solving moving boundary unsteady problems. The near body region is discretized by using the body-fitted structured grids, while the remaining computational domain is tessellated with the generated Cartesian grids. As the body moves, the structured grids move with the body and the outer boundaries of inside grids are used to generate new holes in the outside adaptive Cartesian grid to facilitate data communication. By using the alternating digital tree （ADT） algorithm, the computational time of hole-cutting and identification of donor cells can be reduced significantly. A compressible solver for unsteady flow problems is developed. A cell-centered, second-order accurate finite volume method is employed in spatial discreti- zation and an implicit dual-time stepping low-upper symmetric Gauss-Seidei （LU-SGS） approach is employed in temporal discretization. Geometry-based adaptation is used during unsteady simulation time steps when boundary moves and the flow solution is interpolated from the old Cartesian grids to the new one with inverse distance weigh- ting interpolation formula. Both laminar and turbulent unsteady cases are tested to demonstrate the accuracy and efficiency of the proposed method. Then, a 2-D store separation problem is simulated. The result shows that the hybrid Cartesian grid method can handle the unsteady flow problems involving large-scale moving boundaries.

A Fast Cartesian Grid-Based Integral Equation Method for Unbounded Interface Problems with Non-Homogeneous Source Terms

This work presents a fast Cartesian grid-based integral equation method for unbounded interface problems with non-homogeneous source terms.The unbounded interface problem is solved with boundary integral equation methods such that infinite boundary conditions are satisfied naturally.This work overcomes two difficulties.The first difficulty is the evaluation of singular integrals.Boundary and volume integrals are transformed into equivalent but much simpler bounded interface problems on rectangular domains,which are solved with FFT-based finite difference solvers.The second one is the expensive computational cost for volume integrals.Despite the use of efficient interface problem solvers,the evaluation for volume integrals is still expensive due to the evaluation of boundary conditions for the simple interface problem.The problem is alleviated by introducing an auxiliary circle as a bridge to indirectly evaluate boundary conditions.Since solving boundary integral equations on a circular boundary is so accurate,one only needs to select a fixed number of points for the discretization of the circle to reduce the computational cost.Numerical examples are presented to demonstrate the efficiency and the second-order accuracy of the proposed numerical method.

A parallel methodology of adaptive Cartesian grid for compressible flow simulations

The combination of Cartesian grid and the adaptive mesh refinement(AMR)technology is an effective way to handle complex geometry and solve complex flow problems.Some high-efficiency Cartesian-based AMR libraries have been developed to handle dynamic changes of the grid in parallel but still can not meet the unique requirements of simulating flow around objects.In this paper,we propose an efficient Cartesian grid generation method and an information transmission approach for the wall boundary to parallelize the implementation of ghost-cell method(GCM).Also,the multi-valued ghost-cell method to handle multi-value points is improved to adapt to the parallel framework.Combining the mentioned methodologies with the open-source library p4est,an automatic and efficient simulation of compressible flow is achieved.The overall performance of the methodology is tested through a wide range of inviscid/viscous flow cases.The results indicate that the capability and parallel scalability of the present numerical methodology for solving multiple types of flows,involving shock and vortices,multi-body flow and unsteady flows are agreeable as compared with related reference data.

Development of Cartesian grid method for simulation of violent ship-wave interactions

We present a Cartesian grid method for numerical simulation of strongly nonlinear phenomena of ship-wave interactions. The Constraint Interpolation Profile （CIP） method is applied to the flow solver, which can efficiently increase the discretization accuracy on the moving boundaries for the Cartesian grid method. Tangent of Hyperbola for Interface Capturing （THINC） is imple- mented as an interface capturing scheme for free surface calculation. An improved immersed boundary method is developed to treat moving bodies with complex-shaped geometries. In this paper, the main features and some recent improvements of the Cartesian grid method are described and several numerical simulation results are presented to discuss its performance.

Positivity-Preserving Runge-Kutta Discontinuous Galerkin Method on Adaptive Cartesian Grid for Strong Moving Shock

In order to suppress the failure of preserving positivity of density or pres-sure,a positivity-preserving limiter technique coupled with h-adaptive Runge-Kutta discontinuous Galerkin(RKDG)method is developed in this paper.Such a method is implemented to simulate flows with the large Mach number,strong shock/obstacle interactions and shock diffractions.The Cartesian grid with ghost cell immersed boundary method for arbitrarily complex geometries is also presented.This ap-proach directly uses the cell solution polynomial of DG finite element space as the interpolation formula.The method is validated by the well documented test ex-amples involving unsteady compressible flows through complex bodies over a large Mach numbers.The numerical results demonstrate the robustness and the versatility of the proposed approach.

Numerical simulation of viscous flows with adaptively refined Cartesian grid

Quadtree-based Cartesian grid was automatically generated from specified geometry.Adaptive refinements were performed according to geometric parameters and solution of flow field.An altered CCST(curvature corrected symmetry technique)approach was proposed to apply solid wall boundary conditions. Driven flows in a square cavity and flows around NACA0012airfoil were simulated and compared with the result of published structured grid and stretched Cartesian grid.The results show that solid wall boundary condition are accurately applied by current altered CCST approach,while incompressible/compressible subsonic, transonic and supersonic viscous flows are adequately simulated with adaptively refined Cartesian grid.

Study on complicated solid modeling and Cartesian grid generation method

A Cartesian grid generation method is developed in this study.Two kinds of solid modeling methods,CSG and STL models,are used for complicated solid modeling.The staircase boundary approximation is implemented to handle irregular geometries and the computational domain is discretized using a regular Cartesian grid.Using the edge-based integral slice algorithm,the models are sliced with a set of parallel planes to generate 2D slices information.The scan line filling technique is used to achieve grid generation after slicing.Two grid generation examples with a CSG model and a STL model are given to test the capability of the grid generation method.For grid displaying,a method is proposed to remove the hidden surfaces fasten based on the topology of orthogonal hexahedral grids.The parallelization of grid displaying is achieved by employing multi-threaded parallel technique.Parallel test results show that the parallel algorithm has the absolute advantage on speed compared to the serial algorithm.

Marker-Based,3-D Adaptive Cartesian Grid Method for Multiphase Flow Around Irregular Geometries

Computational simulations of multiphase flow are challenging because many practical applications require adequate resolution of not only interfacial physics associated with moving boundaries with possible topological changes,but also around three-dimensional,irregular solid geometries.In this paper,we highlight recent efforts made in simulating multiphase fluid dynamics around complex geometries,based on an Eulerian-Lagrangian framework.The approach uses two independent but related grid layouts to track the interfacial and solid boundary conditions,and is capable of capturing interfacial as well as multiphase dynamics.In particular,the stationary Cartesian grid with time dependent,local adaptive refinement is utilized to handle the computation of the transport equations,while the interface shape and movement are treated by marker-based triangulated surface meshes which freely move and interact with the Cartesian grid.The markers are also used to identify the location of solid boundaries and enforce the no-slip condition there.Issues related to the contact line treatment,topological changes of multiphase fronts during merger or breakup of objects,and necessary data structures and solution techniques are also highlighted.Selected test cases including spacecraft fuel tank flow management and liquid plug flow dynamics are presented.

An Accurate Cartesian Method for Incompressible Flows with Moving Boundaries

An accurate cartesian method is devised to simulate incompressible viscous flows past an arbitrary moving body.The Navier-Stokes equations are spatially discretized onto a fixed Cartesian mesh.The body is taken into account via the ghost-cell method and the so-called penalty method,resulting in second-order accuracy in velocity.The accuracy and the efficiency of the solver are tested through two-dimensional reference simulations.To show the versatility of this scheme we simulate a threedimensional self propelled jellyfish prototype.

A Fourth-Order Kernel-Free Boundary Integral Method for Interface Problems

This paper presents a fourth-order Cartesian grid based boundary integral method(BIM)for heterogeneous interface problems in two and three dimensional space,where the problem interfaces are irregular and can be explicitly given by parametric curves or implicitly defined by level set functions.The method reformulates the governing equation with interface conditions into boundary integral equations(BIEs)and reinterprets the involved integrals as solutions to some simple interface problems in an extended regular region.Solution of the simple equivalent interface problems for integral evaluation relies on a fourth-order finite difference method with an FFT-based fast elliptic solver.The structure of the coefficient matrix is preserved even with the existence of the interface.In the whole calculation process,analytical expressions of Green’s functions are never determined,formulated or computed.This is the novelty of the proposed kernel-free boundary integral(KFBI)method.Numerical experiments in both two and three dimensions are shown to demonstrate the algorithm efficiency and solution accuracy even for problems with a large diffusion coefficient ratio.