Euler solution using adaptive Cartesian grid with a gridless boundary treatment

A nested multi-grid solution algorithm has been developed for an adaptive Cartesian/Quad grid viscous flow solver. Body-fitted adaptive Quad (quadrilateral) grids are generated around solid bodies through ‘surface extrusion’. The Quad grids are then overlapped with an adaptive Cartesian grid. Quadtree data structures are employed to record both the Quad and Cartesian grids. The Cartesian grid is generated through recursive sub-division of a single root, whereas the Quad grids start from multiple roots—a forest of Quadtrees, representing the coarsest possible Quad grids. Cell-cutting is performed at the Cartesian/Quad grid interface to merge the Cartesian and Quad grids into a single unstructured grid with arbitrary cell topologies (i.e., arbitrary polygons). Because of the hierarchical nature of the data structure, many levels of coarse grids have already been built in. The coarsening of the unstructured grid is based on the Quadtree data structure through reverse tree traversal. Issues arising from grid coarsening are discussed and solutions are developed. The flow solver is based on a cell-centered finite volume discretization, Roe's flux splitting, a least-squares linear reconstruction, and a differentiable limiter developed by Venkatakrishnan in a modified form. A local time stepping scheme is used to handle very small cut cells produced in cell-cutting. Several cycling strategies, such as the saw-tooth, W- and V-cycles, have been studies. The V-cycle has been found to be the most efficient. In general, the multi-grid solution algorithm has been shown to greatly speed up convergence to steady state—by one to two orders. Copyright © 2000 John Wiley & Sons, Ltd.

A FV-TD electromagnetic solver using adaptive Cartesian grids

A hybrid Cartesian grid and gridless method for compressible flows

A block LU-SGS implicit dual time-stepping algorithm for hybrid dynamic meshes

An adaptive Cartesian grid (ACG) concept is presented for the fast and robust numerical solution of the 3D Poisson-Boltzmann Equation (PBE) governing the electrostatic interactions of large-scale biomolecules and highly charged multi-biomolecular assemblies such as ribosomes and viruses. The ACG offers numerous advantages over competing grid topologies such as regular 3D lattices and unstructured grids. For very large biological molecules and multi-biomolecule assemblies, the total number of grid-points is several orders of magnitude less than that required in a conventional lattice grid used in the current PBE solvers thus allowing the end user to obtain accurate and stable nonlinear PBE solutions on a desktop computer. Compared to tetrahedral-based unstructured grids, ACG offers a simpler hierarchical grid structure, which is naturally suited to multigrid, relieves indirect addressing requirements and uses fewer neighboring nodes in the finite difference stencils. Construction of the ACG and determination of the dielectric/ionic maps are straightforward, fast and require minimal user intervention. Charge singularities are eliminated by reformulating the problem to produce the reaction field potential in the molecular interior and the total electrostatic potential in the exterior ionic solvent region. This approach minimizes grid-dependency and alleviates the need for fine grid spacing near atomic charge sites. The technical portion of this paper contains three parts. First, the ACG and its construction for general biomolecular geometries are described. Next, a discrete approximation to the PBE upon this mesh is derived. Finally, the overall solution procedure and multigrid implementation are summarized. Results obtained with the ACG-based PBE solver are presented for: (i) a low dielectric spherical cavity, containing interior point charges, embedded in a high dielectric ionic solvent - analytical solutions are available for this case, thus allowing rigorous assessment of the solution accuracy; (ii) a pair of low dielectric charged spheres embedded in a ionic solvent to compute electrostatic interaction free energies as a function of the distance between sphere centers; (iii) surface potentials of proteins, nucleic acids and their larger-scale assemblies such as ribosomes; and (iv) electrostatic solvation free energies and their salt sensitivities - obtained with both linear and nonlinear Poisson-Boltzmann equation - for a large set of proteins. These latter results along with timings can serve as benchmarks for comparing the performance of different PBE solvers.

Compared to traditional body-fitted moving grids and overset grids, adaptive Cartesian grids offer significant advantages, including no need for grid deformation, highly automated generation process, natural adaptability to complex geometries, and ease of implementing solution-based adaptive mesh refinement. However, in high Reynolds number simulations involving large motions, these grids face the issue of “grid catastrophe” due to the extreme grid resolution required for turbulent boundary layer simulation. To address this, we employ the immersed boundary-dynamic wall function model combined with an implicit dual-time-stepping method, effectively resolving near-wall flow structures without significantly increasing grid count. To handle the assignment of newly emerged Eulerian cells during motion, a three-point interpolation method is used for flow field initialization. This computational framework has been validated through a series of steady-state benchmark cases. Through simulations of typical dive processes and typical pitching airfoil benchmarks, we demonstrate the high efficiency and fidelity of our method in solving turbulent flow fields under dynamic boundary conditions at subsonic and critical sonic speeds. This method provides efficient and reliable numerical simulation support for the aerodynamic design and optimization of various new aircraft.

Efficient dimension-by-dimension higher order finite-volume methods for a Cartesian grid with cell-based refinement

Iterative schemes and algorithms for adaptive grid generation in one dimension

An automatic adaptive hybrid Cartesian grid generation and solution system is presented together with applications for turbulent flows around complicated three-dimensional geometries. The primary computational grid is an octree Cartesian grid with a cut-cell approach. A quasi-prismatic grid may be added for resolving the boundary layer region of viscous flow around the solid body. For external flow simulations the flow solver TAU from the Deutsches Zentrum fiir Luftund Raumfahrt is integrated in the simulation system. Coarse grids are generated automatically, which are required for multilevel smoothing. The initial hybrid Cartesian grid is improved with help of several adaptation features, which are optimized in order to avoid any user interaction. Therefore, the final grid depends only on the particular geometry and flow structure (selforganized grid).

An automatic adaptive hybrid Cartesian grid generator is presented together with solutions of flows around complicated two- and three-dimensional geometries. The primary computational grid is a Cartesian grid, which is generated based on an octree-data structure. A secondary grid may be added for resolving the boundary layer region of viscous flow around the solid body. It consists of quasi triangular-prismatic cells generated by marching the body surface triangulation outward. A modified unstructured flow solver is applied for the compressible Euler and Navier Stokes equations. For turbulent flows, the eddy viscosity is calculated using the Spalart-Allmaras turbulence model. The hybrid Cartesian grid is improved due to manifold adaptation features, which are optimized in order to avoid any user interaction. Therefore, the final grid depends only on the particular geometry and flow structure (self-organized grid).

: This paper describes a new domain-connectivity module developed to support Chimera-based interfacing of different CFD solvers for performing time-dependent adaptive moving-body calculations of external aerodynamic flows. The domain-connectivity module coordinates the data transfer between different solvers applied in different parts of the computational domain -- body fitted structured or unstructured to capture viscous near-wall effects, and Cartesian adaptive mesh refinement to capture effects away from the wall. The CFD solvers and the domain-connectivity module are executed within a Python-based computational infrastructure. The domain-connectivity module is fully parallel and performs all its operations (identification of holes and fringe points, donor cell searches and data interpolation) on the partitioned grid data. In addition, the connectivity procedures are completely automated using the implicit hole-cutting methodology such that no user intervention or explicit hole-map specification is necessary. The capabilities and performance of the package are presented for several test problems, including flow over a NACA 0015 wing, AGARD A2 slotted airfoil, hover simulation of scaled V-22 rotor, and a dynamic simulation of UH-60A rotor in forward flight.

A block LU-SGS implicit unsteady incompressible flow solver on hybrid dynamic grids for 2D external bio-fluid simulations

A geometry-based adaptive unstructured grid generation algorithm for complex geological media

A scheme to generate dynamic unstructured grids and an implicit upwind Euler solver are presented to compute solutions on dynamic grids in threedimensions. The grid generation scheme produces Delaunay unstructured triangular or tetrahedral meshes for static and dynamic grid cases. To obtain a solution for multiple bodies in relative motion, dynamic grid restructuring and smoothing are implemented to provide a high quality unstructured grid for the flow solver. The scheme uses the boundary node location and a user specified clustering parameter to determine where nodes on the interior should be added and deleted to maintain a smooth variation in cell volume. In this analysis, numerical results are compared to experimental data for a complex wing-pylonlfinned-store configuration.

A new adaptive grid generation procedure is proposed, which combines a previously developed method that used Anderson's adaptive grid scheme along with a method for controlling grid spacing and orthogonality on all of the boundaries. The proposed method assigns the desired grid stretching over the smooth region during initial grid system generation and before grid adaptation is performed. After properly interpreting the smoothing term of the weighting function, the desired grid stretching is added to the adaptive grid scheme. Several test cases illustrate the method's feasibility.