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

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

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

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.

: 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 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 indicate the developed method is reasonable for complex flows.

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.

Cell-centered and node-centered approaches have been compared for unstructured finite-volume discretization of inviscid fluxes. The grids range from regular grids to irregular grids, including mixed-element grids and grids with random perturbations of nodes. Accuracy, complexity, and convergence rates of defect-correction iterations are studied for eight nominally second-order accurate schemes: two node-centered schemes with weighted and unweighted least-squares (LSQ) methods for gradient reconstruction and six cell-centered schemes two node-averaging with and without clipping and four schemes that employ different stencils for LSQ gradient reconstruction. The cell-centered nearest-neighbor (CC-NN) scheme has the lowest complexity; a version of the scheme that involves smart augmentation of the LSQ stencil (CC-SA) has only marginal complexity increase. All other schemes have larger complexity; complexity of node-centered (NC) schemes are somewhat lower than complexity of cell-centered node-averaging (CC-NA) and full-augmentation (CC-FA) schemes. On highly anisotropic grids typical of those encountered in grid adaptation, discretization errors of five of the six cell-centered schemes converge with second order on all tested grids; the CC-NA scheme with clipping degrades solution accuracy to first order. The NC schemes converge with second order on regular and/or triangular grids and with first order on perturbed quadrilaterals and mixed-element grids. All schemes may produce large relative errors in gradient reconstruction on grids with perturbed nodes. Defect-correction iterations for schemes employing weighted least-square gradient reconstruction diverge on perturbed stretched grids. Overall, the CC-NN and CC-SA schemes offer the best options of the lowest complexity and secondorder discretization errors. On anisotropic grids over a curved body typical of turbulent flow simulations, the discretization errors converge with second order and are small for the CC-NN, CC-SA, and CC-FA schemes on all grids and for NC schemes on triangular grids; the discretization errors of the CC-NA scheme without clipping do not converge on irregular grids. Accurate gradient reconstruction can be achieved by introducing a local approximate mapping; without approximate mapping, only the NC scheme with weighted LSQ method provides accurate gradients. Defect correction iterations for the CC-NA scheme without clipping diverge; for the NC scheme with weighted LSQ method, the iterations either diverge or converge very slowly. The best option in curved geometries is the CC-SA scheme that offers low complexity, second-order discretization errors, and fast convergence.

A high-order accurate flow solution method for quadrilateral elements is developed and tested. The method employs the flux reconstruction (FR) approach proposed by Huynh. Ideal orders of accuracy are achieved for several problems on structured-based grids while some deterioration is observed on purely unstructured grids. Cares for curved wall boundary are also shown. The localized artificial diffusivity (LAD) method is incorporated to capture discontinuities that is crucial for compressible flow simulations. The FR+LAD works very well for 1D test cases but multi-dimensional shock problems requires further modifications to confine the artificial diffusivity to the shock location. Hybrid scheme of the FR and finite difference method taking advantage of 1D-feature alleviates the instability. I. Introduction Expanding demands for the computational fluid dynamics (CFD) requires more reliable numerical schemes. Specifically, turbulent flow simulations, acoustic simulation, combustion simulations and their combined problems. The key words in such simulations may be high accuracy and high resolution. In structured grid CFD, a compact finite difference scheme satisfies both requirements at higher level than existing other methods. If we put more emphasis on easier handling of complex geometries and flow-data dependent grid generation (grid adaptation), then unstructured grid method should also be explored and it should depart from 2nd-order scheme with limiter. Recent efforts in this area are development of the discontinuous Galerkin (DG) method, 1,2 the spectral finite volume (SV) method 3 and the staggered-grid multi-domain method spectral finite difference (SG) method 4,5 and its evolutional method called a spectral difference method. 6 The common and maybe the only way for those unstructured methods is to add degrees of freedom in each cell to achieve high order since the wide stencil of the original grid is not possible due to unstructured nature. The ENO-type method that uses original degrees of freedom in the grids and extends adaptive stencils to the flow solutions is also explored but the difficulty of extending stencils due to undirectional nature of unstructured grids and the unsuitability for parallel computing prohibits wide applications. A flux reconstruction (FR) scheme 7,8 studied in the paper is the new one among the former group. It uses differential form of conservation laws like the SD method but it is more general and includes equivalent counter part of the schemes in the group. In this paper, the performance of the FR is tested by a newly developed Navier-Stokes solver. Realistic flow applications of the FR in the Navier-Stokes equations are seldom seen since it is relatively new scheme. The Localized artificial diffusivity (LAD) 9–13 is incorporated for the shock capturing in the FR. The last item is motivated by the challenge and success of combining the SD and LAD in the recent study. 11,12 Wang 14 developed the lifting collocation penalty (LCP) method to extend the FR to triangular or tetrahedral grids and applied the LCP-FR combined method for hybrid unstructured grids. In the present study, only quadrilateral cells are treated to directly apply the LAD method that was developed on structured grids. The shock capturing by the FR+LAD high order scheme is tried.

Adaptive kinetic-fluid solvers for heterogeneous computing architectures

A simple diffuse interface approach for compressible flows around moving solids of arbitrary shape based on a reduced Baer–Nunziato model

Observations in polar regions show that sea ice deformations are often narrow linear features. These long bands of deformations are referred to as Linear Kinematic Features (LKFs). Viscous‐plastic sea ice models have the capability to simulate LKFs and more generally sea ice deformations. Moreover, viscous‐plastic models simulate a larger number and more refined LKFs as the spatial resolution is increased. Besides grid spacing, other aspects of a numerical implementation, such as the placement of velocities and the associated degrees of freedom, may impact the formation of simulated LKFs. To explore these effects this study compares numerical solutions of sea ice models with different velocity staggering in a benchmark problem. Discretizations based on A‐,B‐, and C‐grid systems on quadrilateral meshes have similar resolution properties as an approximation with an A‐grid staggering on triangular grids (with the same total number of vertices). CD‐grid approximations with a given grid spacing have properties, specifically the number and length of simulated LKFs, that are qualitatively similar to approximations on conventional Arakawa A‐grid, B‐grid, and C‐grid approaches with half the grid spacing or less, making the CD‐discretization more efficient with respect to grid resolution. One reason for this behavior is the fact that the CD‐grid approach has a higher number of degrees of freedom to discretize the velocity field. The higher effective resolution of the CD‐discretization makes it an attractive alternative to conventional discretizations.

A new two-dimensional hybrid grid generation method based on improved hole cutting

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).

The slip condition at the interface of a multiphase flow can occur in situations including micro-and nano-fluidic flow, flow over hydrophobic surfaces, rising bubbles in quiescent liquid, and polymer extrusion processes. The aim of this work is to implement the free-slip boundary condition with an adaptive Cartesian grid method. The Navier-Stokes (NS) equations are solved by a cell-centered collocated finite volume method with adaptive mesh refinement. The arbitrarily-shaped solids imbedded in the computational domain are treated by the cut-cell method where the geometric properties of cut-cells near the boundary are computed through robust geometric operations. In discretized NS equations, the second-order-accurate center difference method is used to estimate the surface fluxes of the regular Cartesian cells in the bulk region, whereas the least-squares method is used to estimate the fluxes of the cut-cells near the boundary. A local coordinate system aligned with the normal and tangential directions of the solid boundary is defined for each cut-cell in order to properly implement the free-slip condition. The tangential velocities at the curved solid boundary are obtained using the free-slip condition and the principal curvatures of the solid surface. The proposed numerical method is implemented in the open-source code Gerris. Numerical tests have been carried out to validate our method. The tests confirm the excellent performances of the proposed method. Although our work focuses on the free-slip condition, the extension of the proposed method to more general slip conditions is straightforward.