General Curvilinear Grids Research Articles

Study of interpolation methods for high-accuracy computations on overlapping grids

Overset strategy can be an efficient way to keep high-accuracy discretization by decomposing a complex geometry in topologically simple subdomains. Apart from the grid assembly algorithm, the key point of overset technique lies in the interpolation processes which ensure the communications between the overlapping grids. The family of explicit Lagrange and optimized interpolation schemes is studied. The a priori interpolation error is analyzed in the Fourier space, and combined with the error of the chosen discretization to highlight the modification of the numerical error. When high-accuracy algorithms are used an optimization of the interpolation coefficients can enhance the resolvality, which can be useful when high-frequency waves or small turbulent scales need to be supported by a grid. For general curvilinear grids in more than one space dimension, a mapping in a computational space followed by a tensorization of 1-D interpolations is preferred to a direct evaluation of the coefficient in the physical domain. A high-order extension of the isoparametric mapping is accurate and robust since it avoids the inversion of a matrix which may be ill-conditioned. A posteriori error analyses indicate that the interpolation stencil size must be tailored to the accuracy of the discretization scheme. For well discretized wavelengthes, the results show that the choice of a stencil smaller than the stencil of the corresponding finite-difference scheme can be acceptable. Besides the gain of optimization to capture high-frequency phenomena is also underlined. Adding order constraints to the optimization allows an interesting trade-off when a large range of scales is considered. Finally, the ability of the present overset strategy to preserve accuracy is illustrated by the diffraction of an acoustic source by two cylinders, and the generation of acoustic tones in a rotor–stator interaction. Some recommandations are formulated in the closing section.

Streamline Tracing on Curvilinear Structured and Unstructured Grids

Summary The streamline method has proven to be extremely successful for reservoir simulation.1 For a Cartesian grid, the standard five-point (seven-point in 3D) discrete pressure equation is employed within the IMPES streamline method. The accuracy of the velocity field plays a critical role in the success of the technique, which relies on precise determination of streamline paths. Modeling complex geology and geometry requires the use of sophisticated grids, and thus the ultimate power of the streamline method will only be realized when the method is extended to general curvilinear grids. In this paper, a generalization of the Cartesian tracing algorithm is developed for general tensor fields on structured (corner-point) grids and unstructured grids. The extension of the method requires the identification of appropriate volumes and fluxes over which tracing equivalent to the Pollock Cartesian method2 can be performed. Essential components of the method include the use of flux-continuous finite-volume schemes for the different grid types coupled with a novel flux recovery technique and a generalization of streamline tracing to polygons.

A nonorthogonal higher-order wavelet-oriented FDTD technique for 3-D waveguide structures on generalized curvilinear grids

A generalized higher-order FDTD rendition of the covariant and contravariant vector component theory for the accurate modeling of complex waveguides in 3-D nonorthogonal curvilinear coordinates, is presented in this paper. The novel algorithm, which postulates conventional and nonstandard concepts, embodies a spatially-localized wavelet-Galerkin formulation in order to efficiently deal with fast field variations in the vicinity of arbitrarily-angled wedges. The proposed method is combined with a pulsed excitation and enhanced unsplit-field PMLs thus, achieving significant accuracy and suppression of all discretization errors with a simultaneous diminishment of computational resources, as indicated by various numerical results.

A three-dimensional coastal barotropic model in generalized curvilinear grids

The method of automatically generating generalized curvilinear meshes has many advantages and is beginning to be used in ocean simulations. This three-dimensional (3-D) coastal barotropic model in generalized curvilinear grids was developed to simulate theM2,S2,K1 andO1 tidal waves in the Bohai Sea, China. The numerical results agreeing with observations showed that the method is an effective tool for improving accuracy of simulations in shallow shelf seas, especially in the near coast region, if the pseudo effect there usually caused by rectangular grids can be removed.

Multigrid calculation of fluid flows in complex 3D geometries using curvilinear grids

A multigrid method is developed for numerical simulation of fluid flows in general curvilinear grids. The numerical formulation is based on a recently developed finite volume method for flows in complex geometries which uses a staggered grid system, grid-oriented velocity unknowns, a direct discretization technique and a coupled solution procedure. The present study reveals that the discrete governing equations on coarse and fine grids can become inconsistent in certain curvilinear grids due to complex flow boundaries, thus reducing the efficiency of the multigrid algorithm. A new method is proposed to solve this problem. Computations using grids with both significant non-orthogonality and strong curvature were carried out and the results show that the developed multigrid algorithm is very efficient: the computation can be hundreds of magnitudes faster when compared with the single-grid method. Comparison between the proposed multigrid method and conventional multigrid methods shows that the proposed formulation for the coarse-grid equations can substantially improve the multigrid efficiency for certain configurations and Reynolds numbers.

A different approach to discretization for the numerical simulation of three-dimensional heat conduction in irregularly shaped materials

A different approach to discretization is described with which complicated three-dimensional heat transfer problems can be solved with a finite volume approach on a general curvilinear grid. It represents an improvement on the existing methods in that it can easily be expanded to three-dimensional problems. A concise explanation of the transformation process is given, together with a discussion of the discretization procedure. The method is evaluated by solving two simple test problems and comparing the results with those of existing methods and the analytical solution. In conclusion it is found that this method yields equally or more accurate results than the existing methods, with the additional advantage of being easily expandable to three-dimensional problems.

A variable boundary model of storm surge flooding in generalized curvilinear grids

AbstractIn this paper, generalized 2D shallow sea dynamic equations in movable curvilinear co‐ordinates are derived. Through a differential co‐ordinate transformation a self‐adaptive grid is proposed to treat a continuously deforming lateral boundary and a kinematical boundary condition is adopted. The self‐adaptive grid method (SAM) is used to simulate numerically the storm surge flooding in the Bohai Sea on 23 April 1969, which was one of the largest storm surge inundations in China.

PRESSURE-BASED VISCOUS FLOW COMPUTATION USING MULTIBLOCK OVERLAPPED CURVILINEAR GRIDS

A pressure-based multiblock computational method is developed for solving the incompressible Navier-Stokes equations in general curvilinear grid systems. A conservative interface scheme is devised with desirable accuracy to handle the information transfer between blocks. The scheme is based on the semiimplicit-type flow solver with the staggered grid. Issues concerning discontinuous grids, global mass conservation, viscous term treatment, and boundary conditions at the grid interface are addressed. The method is tested for two flow problems, a curved channel flow and a bifurcated channel flow. The calculations demonstrate that, besides maintaining desirable solution characteristics across discontinuous grid interfaces, the present multiblock algorithm can achieve convergence rates comparable to that of the single-block algorithm, yielding an improved computational capability for treating complex flow problems.

A Convective-Difference Scheme Using a General Curvilinear Coordinate Grid for Steady Incompressible Viscous Flow Problems.

A numerical scheme for analyzing the steady two-dimensional incompressible viscous flow using a general curvilinear coordinate grid is proposed. In this scheme, the unsteady Navier-Stokes equations are solved by a convective-difference scheme using a staggered square grid in transformed space. An elliptic equation of pressure is solved by the Tchebyscheff SLOR method. The substantial derivative term in the convective-difference scheme is integrated along a path line and the values at the upstream end are interpolated considering a TVD concept. As numerical examples, the backward-facing step duct and U curved duct flows were calculated. The calculated results show that the scheme has good accuracy as a second-order scheme and is speedy in reaching for the steady condition.

Finite-difference schemes for steady incompressible Navier-Stokes equations in general curvilinear coordinates

Abstract Time-marching finite-difference schemes for solving the steady incompressible Navier-Stokes equations are proposed. In these schemes, the fractional-step method and the SMAC method are applied to a general curvilinear coordinates grid, so that the continuity condition can be satisfied identically. The momentum equations of contravariant velocities are newly derived for the accurate and easy treatment of the boundary conditions and are formulated concisely by introducing the contravariant vorticities. Spurious errors and numerical instabilities can be reasonably suppressed by employing the staggered grid and the upwind difference for moderate-to-high Reynolds number flows. Numerical results for two-dimensional laminar duct flow over a backward-facing step are shown, and compared with existing results to assess the reliability of the present schemes.

A Convective-Difference Scheme Using General Curvilinear Coordinate Grid for Incompressible Viscous Flow Problems.

A numerical scheme for analyzing the unsteady two-dimensional incompressible viscous flow using a general curvilinear coordinate grid is proposed. In this scheme, the unsteady Navier-Stokes equations are solved by a convective-difference scheme using a staggered square grid in transformed space and an interpolation formula considering TVD concept. And an elliptic equation of pressure is solved by the iteration scheme. The continuity condition in the scheme is identically satisfied and the spurious errors are completely removed by the similar manner to the MAC scheme. As numerical examples square cavity, U-type duct and backward-facing step duct flows were calculated. The calculated results show that the scheme has a good accuracy as a second-order scheme and is stable for the high Reynolds numbers flows.

4831477 Loading mechanism for a magnetic disk device : Yuz Sakamoto, Hisashi Koga, Shoich Saito, Kohriyama, Japan assigned to Mitsubishi Denki Kabushiki Kaisha

A multigrid method is developed for numerical simulation of fluid flows in general curvilinear grids. The numerical formulation is based on a recently developed finite volume method for flows in complex geometries which uses a staggered grid system, grid-oriented velocity unknowns, a direct discretization technique and a coupled solution procedure. The present study reveals that the discrete governing equations on coarse and fine grids can become inconsistent in certain curvilinear grids due to complex flow boundaries, thus reducing the efficiency of the multigrid algorithm. A new method is proposed to solve this problem. Computations using grids with both significant non-orthogonality and strong curvature were carried out and the results show that the developed multigrid algorithm is very efficient: the computation can be hundreds of magnitudes faster when compared with the single-grid method. Comparison between the proposed multigrid method and conventional multigrid methods shows that the proposed formulation for the coarse-grid equations can substantially improve the multigrid efficiency for certain configurations and Reynolds numbers.

An Integral Approach for Discretizing the Reservoir Flow Equations

Summary An approach for discretizing the integral form of the flow equations in porous media is proposed. The method has the advantage of providing the correct block volume calculations and steady-state flow in curvilinear coordinates. The harmonic mean for permeability is also obtained from the discretization. The equations are developed for a general orthogonal curvilinear grid system in terms of the components of the metric tensor.

COASTAL AND ESTUARINE HYDRODYNAMIC MODELING IN CURVILINEAR GRIDS

A time-dependent three-dimensional numerical model of coastal and estuarine circulation, which allows the use of generalized curvilinear grids to resolve the complex shoreline geometry and bathymetry, has been developed. Model formulation is briefly presented, followed by example model applications to such water bodies as Lake Okeechobee, James River and Chesapeake Bay.

A simple scheme for generating general curvilinear grids

An intuitively simple approach is presented for the computer generation of two-dimensional curvilinear grids suitable for finite difference solutions of problems in the field of continuum dynamics. An iterative process is employed to transform uniform networks of rectangular zones into more complex configurations. Ease of use and optimal adjustment are stressed, and numerous examples are given.