Discrete grid generation on plane regions has been our interest since the late 80's, our group, UNAMALLA, has developed methods that are effective even on quite irregular regions, our techniques have a sound theoretic basis, however when we were trying to apply our ideas to orthogonal grid generation, we found that something was missing, we think we have now a developed robust automatic grid generators, smooth or orthogonal, that are very efficient we will present our theoretical formulation and our implementation on our grid package UNAMALLA.

A method for regular 2D grid generation is suggested that combines the known grid generator [1] and a simple quasi-1D grid whose nodes are located along straight lines of one family in according to a given law. As a result, the allocation law is approximately satisfied even for very curved lines while the convexity of all grid cells is ensured at each iteration step practically for any distortion of internal and external boundary lines. An iteration procedure on subsets of nodes which reduces considerably the computational cost of the method is also considered. Numerical tests and examples of grids from computations of shock-induced time-dependent flows with moving boundaries are presented.

[1] S.A.Ivanenko and A.A.Charakhch'yan, Soviet Math. Dokl. 36, 51 (1988); USSR Comput. Math. Math. Phys. 28, 126 (1988); J. Comput. Phys. 136, 385 (1997).

A singularly perturbed reaction-diffusion problem with homogeneous Dirichlet boundary condition is considered. It exhibits in general a solution with strong boundary or/and interior layers. This anisotropy is reflected in the finite element discretization by using meshes with anisotropic elements.

Adaptive methods are well established to produce (quasi) optimal meshes and discretizations. A key ingredient of such adaptive methods are a posteriori error estimators. The equilibrated fluxes method and its modification for the singular perturbed case, done by Ainsworth and Oden, are discussed with respect to the application on anisotropic meshes. Further modifications of this error estimator for the anisotropic case are proposed.

A method of a second-order interpolation of the function values on the set of arbitrary points in a finite-dimensional Euclidean space is constructed. The second-order interpolation is based on determining the point neighbors through decomposition into the Dirichlet cells by Delaunay triangulation [1, 2] and used formulas of the first-order non-Sibsonian (harmonic) interpolation. The properties of this method are described, and the results of its application and comparison with the first-order Sibson interpolation [3, 4] and the non-Sibsonian (harmonic) interpolation [5-9] are presented.

REFERENCES

[1] George P.-L., Borouchaki H. (1998) Delaunay Triangulation and Meshing. Application to Finite Elements, Hermes, Paris.

[2] Baker T.J. (1999) Delaunay-Voronoi methods, Chapter 16 in Handbook of Grid Generation, J.F. Thompson, B.K. Soni and N.P. Weatherill (Eds.), CRC, Boca Raton, FL.

[3] Sibson R. (1980) A vector identity for the Dirichlet tessellation, Math. Proc. Cambridge Philos. Soc., Vol.87, No.1, 151-155.

[4] Sibson R. (1981) A brief description of the natural neighbour interpolant, in Interpreting Multivariate Data, V. Barnett (Ed.), 21-36, John Wiley, Chichester, U.K.

[5] Belikov V.V., Ivanov V.D., Kontorovich V.K., Korytnik S.A., Semenov A.Yu. (1997) The non-Sibsonian interpolation: A new method of interpolation of the values of a function on an arbitrary set of points, Comput. Maths Math. Phys., Vol.37, No.1, 9-15.

[6] Belikov V.V., Semenov A.Yu. (1997) New non-Sibson interpolation on arbitrary system of points in Euclidean space, in 15th IMACS World Congress on Scientific Computation, Modelling and Applied Mathematics, Berlin, August 1997, Vol.2: Numerical Mathematics, A. Sydow (Ed.), 237-242, Wissenshaft und Technik Verlag, Berlin.

[7] Belikov V.V., Semenov A.Yu. (1998) Non-Sibsonian interpolation on arbitrary system of points in Euclidean space and adaptive generating isolines algorithm, in Numerical Grid Generation in Computational Field Simulations, M. Cross et al. (Eds.), 277-286, Proc. 6th Int. Conf., July 6-9, 1998, University of Greenwich, London.

[8] Belikov V. V., Semenov A. Yu. (2000) Non-Sibsonian interpolation on arbitrary system of points in Euclidean space and adaptive isolines generation, Appl. Numer. Math., Vol.32, No.4, 371-387.

[9] Sukumar N., Moran B., Semenov A.Yu., Belikov V.V. (2001) Natural neighbour Galerkin methods, Int. J. Numer. Meth. Eng., Vol.50, No.1, 1-27.

The approach to anisotropic adaptation of unstructured volume grids to numerical solutions of external problems for convection-diffusion, Euler and Navier-Stokes equations is developed. The grid adaptation is based on definition of three directions of adaptation, refinement of edges aligned to these directions and reconnection. The weighted sum of the first and second order derivatives along the edge is utilized as interpolation based edge error indicator. The grid adaptation technique was applied to convection-diffusion problem modeling the 3D shear layer and to transonic Euler flow calculation around isolated wing.

The Discontinuous Galerkin method with high order basis functions (k=0,1,2...) was employed in numerical solution of convection-diffusion problem. The comparison of convergence properties for uniform, isotropic and anisotropic adaptations are made. The investigated grid adaptation convergences for Discontinuous Galerkin schemes of different order of accuracy have good correlation with the theoretically predicted values.

The developed code can employ the body geometry provided with conventional CAD systems such as CADKEY, CATIA, or Unigrafics. The capabilities of these CAD systems to create body surfaces were investigated for isolated wing and wing with winglet configurations. The CAD systems were applied to generation of isotropic volume and surface grids around complex configurations that were used as starting grids in adaptation sequences.

The techniques of adaptive mesh generation are of particular interest to mathematicians and engineers. Significant improvement of the accuracy of approximation through the adaptive nodes distribution rather than increasing the number of mesh nodes, is the most attractive feature of adaptivity. During the past decade the most popular were the regular conformal triangulations whose mesh size is governed by some a posteriori error estimators. The error estimators take into account both the discrete solution and the type of the problem to be solved.

In the lecture, an alternative approach is discussed. The basic idea is that a quasi-uniform (in a metric based on the solution Hessian) mesh is to be generated. Since the solution and its Hessian are unknown, the metric is generated by the discrete Hessian recovered from the discrete solution. Thus, the mesh generator takes on input a mesh and respective discrete solution and outputs a new mesh which is more adapted to the solution. The output is obtained by a sequence of local modifications of the current mesh in such a way that the modified mesh be more quasi-uniform in the given metric. Besides the problem independence, adaptive methods based on a Hessian recovery produce meshes with a prescribed number of elements distributing them in an almost optimal way. Furthermore, the methods may be implemented as a "black box" regardless of the discrete problem generation and solver.

References

(a) Yu.Vassilevski, K.Lipnikov, An Adaptive Algorithm for Quasioptimal Mesh Generation, Computational Mathematics and Mathematical Physics, Vol.39,No.9,1999,1468-1486.

(b) A.Agouzal, K.Lipnikov, Yu.Vassilevski, Adaptive Generation of Quasi-optimal Tetrahedral Meshes, East-West Journal, Vol.7,No.4,1999,223-244.