## Generators of anisotropic adaptive grids

Authors: Konstantin Lipnikov & Yuri Vasilevski

Principle ideas of our methodology.
• Black-box FORTRAN-77 routines.
• Parallel mesh generation based on inertia bisection algorithms.
• Mesh adaptation by a sequence of robust local mesh modifications.
• A posteriori error estimators on anisotropic meshes (know-how).
• Control of the mesh adaptation (know-how).
• Mesh adaptation in domains with discrete boundaries (know-how).

Flexibility and robustness of the mesh generators.
Both 2D and 3D mesh generators are implemented as black-box FORTRAN-77 routines. The input information includes: an initial coarse grid, the required number of triangles (or tetrahedra) and a metric field. The metric field is defined either analytically or by the Hessian of a solution. In the first case, the user should write short subroutines that define the metric field. In the second case, the user should supply a solution.

In the model example shown below the initial grid has 4 triangles. Suppose that the desired grid should have approximately 600 triangles and the metric field (with an anisotropy in x-direction) is given analytically. The resulting grid is in the first row on the right. Let us take this grid as the initial grid and change the metric field. The output of the generator is the grid shown in the second row on the left. Again, let us take this grid as the initial grid and replace the metric field so that the generator gives the forth grid in the figure.

Both sequential 2D and parallel 3D codes are available for research purposes (details). The research work is conducted during our spare time using home computers.

Example I: Diffusion problem.

Example II: 2D fully potential transonic flow.
The 2D example is the calculation of the compressible irrotational isotropic adiabatic flow of an ideal gas around a wing:

In order to compute a solution we use approximately 6000 triangles and 30 adaptive iterations. The starting point for a new iteration is the finite element solution computed on the previous iteration. First, we generate a mesh adapted to this solution. Second, we solve the problem by a nonlinear conjugate gradient method and repeat the process. The evolution of the adaptive mesh is demonstrated with the following snapshots:

The pressure profiles on both sides of the wing computed on the 1st and 10th (red curve) iterations are shown in the next figure. It is clear to observe that a sharp shock has already started to form.

Example III: 3D convection-diffusion problem.
The 3D example is the singularly perturbed convection diffusion problem in the back-step domain,

subject to the homogeneous Dirichlet boundary condition. The solution possesses a severe exponential boundary layer at x=1 and parabolic boundary layers at y=0, y=1, z=0, and z=1, as well as a weak interior layer at y=0.5. The next figure shows edges of the anisotropic tetrahedral mesh adapted to the solution and colored according to the solution value. Almost 90% of the tetrahedra are concentrated at the boundary layer at x=1.

Control of the mesh adaptation.
The optimal adaptive anisotropic mesh may not be appropriate for some engineering applications due to strong size-disproportionality of neighboring mesh elements. For instance, this disproportionality may increase the LBB constant in mixed finite element discretizations of the Stokes problem. This constant is used to evaluate the difference between the approximation and interpolation errors in the pressure. The more this difference, the less accurate may be the metric-based approach for resolving pressure singularities. One of the solutions is to modify (for example, to smooth) the metric.

A properly chosen modification which preserves the main properties of the original metric defines success of many engineering simulations. The modified metric allows us to enforce the mesh adaptation in regions of physical interest. Similar objectives are typical for the goal oriented adaptive methods. The main advantage of the metric-based methods is a simple control of metric properties and properties of the resulting meshes.

As an illustrative example, we consider the homogeneous Dirichlet boundary value problem for the Poisson equation in the domain &Omega with one reentrant corner, &Omega = (0,1)3 / [0,0.5]3:

-&Delta u = f
where f(x) is a singular right hand side, f(x) = 1 / |x - x0|, and x0 = (0.5, 0.5, 0.5). The solution u possesses weak anisotropic edge singularities and a strong point singularity at the reentrant corner x0 due to the singular right hand side. The left picture below shows the quasi-optimal adaptive mesh which is highly anisotropic. The other two pictures show the results of the controlled mesh adaptation. The picture in the middle shows the mesh adapted to the maximal solution curvature. On the right picture, the mesh anisotropy has been relaxed and the adaptation has been enforced along domain edges.

References.
• (2005) V.Dyadechko, K.Lipnikov, and Yu.Vassilevski.
Hessian based anisotropic mesh adaptation in domains with discrete boundaries.
to appear in Russian J. Numer. Analysis Math. Modelling.

• (2005) Yu.Vassilevski and K.Lipnikov.
Error estimates for controled mesh adaptation.
submitted to Computational Mathematics and Mathematical Physics.

• (2004) K.Lipnikov and Yu.Vassilevski.
On control of adaptation in parallel mesh generation.
Engrg. Computers, V.20, pp.193-201.

• (2004) K.Lipnikov and Yu.Vassilevski.
On a parallel algorithm for controlled Hessian-based mesh adaptation.
Proceedings of the 3rd Conf. Appl. Geometry, Mesh Generation and High Performance Computing,
Moscow, June 28 - July 1, Comp. Center RAS, Vol.1, pp.154-166.

• (2003) K.Lipnikov and Yu.Vassilevski.
Optimal triangulations: existence, approximation and double differentiation of \$P_1\$ finite element functions.
Computational Mathematics and Mathematical Physics, Vol.43, No.6, pp. 827-835.

• (2003) K.Lipnikov and Y.Vassilevski.
Parallel adaptive solution of 3D boundary value problems by Hessian recovery.
Comput. Methods Appl. Mech. Engrg., Vol.192, pp. 1495-1513.

• (1999) A.Agouzal, K.Lipnikov and Y.Vasilevskii.
Adaptive generation of quasi-optimal tetrahedral meshes.
East-West Journal, Vol.7, No.4, pp. 223-244.

• (1999) Y.Vasilevskii and K.Lipnikov.
An adaptive algorithm for quasioptimal mesh generation.
Computational Mathematics and Mathematical Physics, Vol.39, No.9, pp.1468-1486.

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