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.

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.

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.

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.

A Geometric Space-Time Multigrid Algorithm for the Heat Equation

We study the time-dependent heat equation on its space-time domain that is discretised by a k-spacetree.k-spacetrees are a generalisation of the octree concept and are a discretisation paradigm yielding a multiscale representation of dynamically adaptive Cartesian grids with low memory footprint.The paper presents a full approximation storage geometric multigrid implementation for this setting that combines the smoothing properties of multigrid for the equation’s elliptic operator with a multiscale solution propagation in time.While the runtime and memory overhead for tackling the all-in-one space-time problem is bounded,the holistic approach promises to exhibit a better parallel scalability than classical time stepping,adaptive dynamic refinement in space and time fall naturally into place,as well as the treatment of periodic boundary conditions of steady cycle systems,on-time computational steering is eased as the algorithm delivers guesses for the solution’s long-term behaviour immediately,and,finally,backward problems arising from the adjoint equation benefit from the the solution being available for any point in space and time.