# Grid Generation Tools

Unstructured grid generation can mesh any geometry. One of the design criteria for the two tools presented on this page was minimum user input, ie. the user should have as little information to specify as possible. This has been acheived on two levels: the geometry is specified by defining a curve analytically or discretely and by a few points with grid spacing along the segments. The grid can then be created using just these surface points.

## Curve Discretization with ipol

Ipol allows to distribute points on analytic or discrete curves by specifiying point distances at various locations along the curve. A script language takes the curve parameters and program options. Input and output of the discretized point chains on the curves are in a .pts format as used by the triangular mesher.

Ipol allows to:

• read any discretized geometry in a .pts format,
• has a library of analytic shapes such as lines, circles, arcs, polynomials, naca 4 digit airfoils, etc.,
• scaling, rotation and translation transformations can be applied to each segment of the geometry,
• point density can be specified arbitrarily at any locations alongthe surface,
• global and local scaling factors can be applied to adjust the mesh size,
• the point distribution is smoothend by a Gauss-Seidel relaxation,
• output is in a .pts format compatible with delaundo,
Ipol is in the public domain, the latest version 3.0 can be obtained here.

### Delaunay Triangulation with delaundo

Delaundo creates triangular grids based on the Frontal Delaunay Method (Frod). First the set of discretized curves that describe the boundary is triangulated. This initial mesh is suitable for interpolation of a local mesh size throughout the domain after a few modifications in the connections are made by the algorithm. New internal vertices are then created on frontal edges between well-shaped and ill-shaped triangles such that a new triangle with the desired size and a good shape will result.

Thus, the algorithm is similar to the various Delaunay methods in that the resulting triangulation observes a circum-circle criterion. It is also akin to Advancing Front methods in that new vertices are introduced in layers on the boundaries in a very regular fashion. The regularity of the point distribution and thus the element quality is enhanced by an averaging process that tends to choose an equilibrium position between competing edges when the front is refined or coarsenend.

Delaundo can produce stretched grids and has a multi-grid capability that produces a serios of coarsened grid with nested nodes.

A few examples. A straightforward triangulation of the simple NACA 0012 airfoil with the boundary points from above produces the following grid:

Delaundo has also a rudimentary capability to create grids with stretched layers for viscous calculations that works well for moderate stretching factors of up to 100. Due to the simple implementation the stretched layers must completely wrap around a simply connected body such as an airfoil with a wake. It cannot do bump-like cases, where non-stretched boundaries are attached to stretched ones. The following example shows the grid around a three-element airfoil with a stretching of 1:10.

### Literature on ipol and delaundo

• J.-D. Müller, ``On Triangles and Flow'', PhD Thesis, The University of Michigan, 1996.
• J.-D. Müller, ``The Advancing Front Method and the Delaunay Triangulation'', 24th von Karman Institute Lecture Series on Computational Fluid Dynamics, 1994-02, 1994.
• J.-D. Müller, ``Quality Estimates and Stretched Meshes based on Delaunay Triangulations'', AIAA-Journal, Vol. 32, No. 12, December 1994.
• J.-D. Müller, ``Proven Angular Bounds and Stretched Triangulations with the Frontal Delaunay Method'', AIAA-93-3347-CP, 1993.
• J.-D. Müller, P.L. Roe and H. Deconinck, ``A Frontal Approach for Internal Node Generation for Delaunay Triangulations'', Int. J. of Num. Meth. in Fluids, Vol. 17, No. 3, pp 241-56, 1993.
• J.-D. Müller, P.L. Roe and H. Deconinck, ``Delaunay-based triangulations for the Navier-Stokes equations with minimum user input'', Proceedings of the 13th International Conference on Numerical Modelling in Fluid Dynamics, Rome, 1992.
• J.-D. Müller, P.L. Roe and H. Deconinck, ``A Frontal Approach for Node Generation in Delaunay Triangulations'', VKI Lecture Series on Unstructured Grid Methods'', AGARD R-787, 1992.
Delaundo is in the public domain, the latest version 5.4 can be obtained here. The Phd thesis is also available in A4 or US-letter format (1.7M).

A visualisation program that can deal with delaundo's preferred output format .dpl has been written by Darren DeZeeuw of the University of Michigan. You can obtain a copy of the source here, but you also need to get a copy of MIT's grafic, developed mainly by J. Dannenhoffer.

Postprocessing of the grids produced by delaundo is done with hip.