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}:
where f(x) is a singular right hand side,
f(x) = 1 / |x - x_{0}|, and
x_{0} = (0.5, 0.5, 0.5). The solution u possesses
weak anisotropic edge singularities and a strong point singularity
at the reentrant corner x_{0} 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.