# The Adaptation of Two-Dimensional Multiblock Structured Grids Using a PDE-Based Method

• D. Catherall
Chapter
Part of the Notes on Numerical Fluid Mechanics (NNFM) book series (NNFM, volume 44)

## Summary

In previous work it has been shown that solution-adapted grids may be obtained by solving Poisson equations for the node ordinates with the control functions evaluated from a solution obtained on the original grid. However, this can lead to a situation where both the elliptic terms in the grid equations, and the equidistribution mechanism implied in the evaluation of the control functions, attract grid nodes to certain regions, so producing an ‘overkill’.

In the work described here a new, but related, method, termed the LPE method, is introduced to overcome this limitation. In this approach each grid equation is formed from
1. 1.

an inverted Laplace equation (the L in LPE) which, if used in isolation, maximises smoothness and orthogonality.

2. 2.

an inverted Poisson equation (the P in LPE) with control functions evaluated from the original grid — if used in isolation the original grid is regenerated.

3. 3.

an Equidistribution equation (the E in LPE) which, if used in isolation, distributes nodes along each grid line so that node separations are inversely proportional to the local value of a sensor function formed from solution gradients.

Control is effected through choosing the relative weights to be placed on these three constituents of the grid equations. Grid control in the region of trailing edges is obtained separately through the use of source terms placed there.

Other features of the method described here include enforced grid orthogonality at grid boundaries and implementation on multiblock structured grids.

## Keywords

Mach Number Grid Line Flow Solver Inviscid Flow Solution Gradient
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 
Brackbill, J.U. and Saltzman, J.S.: “Adaptive zoning for singular problems in two dimensions”, Journal of Computational Physics, 46, p342, 1982.
2. 
Jacquotte, 0.-P. and Cabello, J.: “A variational method for the optimisation and adaptation of grids in Computational Fluid Dynamics”, in Numerical Grid Generation in CFD ‘88 (Ed. Sengupta, S., Haüser, J., Eiseman, P.R., and Thompson, J.F.), pp 405 to 414, Pineridge Press, Swansea, U.K., 1988.Google Scholar
3. 
Nakahashi, K. and Deiwert, G.S.: “A self-adaptive-grid method with application to airfoil flows”, AIAA Paper 85–1525, 1985.Google Scholar
4. 
Catherall, D.: “Solution-adaptive grids for transonic flows”, in Numerical Grid Generation in CFD ‘88 (Ed. Sengupta, S., Haüser, J., Eiseman, P.R., and Thompson, J.F.), pp 329 to 338, Pineridge Press, Swansea, U.K., 1988.Google Scholar
5. 
Catherall, D.: “The adaptation of structured grids to numerical solutions for transonic flow”, International Journal for Numerical Methods in Engineering. Vol 32, No 4, pp 921 to 937, 1991.Google Scholar
6. 
Thompson, Joe F., Warsi, Z.U.A. and Mastin, C. Wayne: “Numerical grid generation - foundations and applications”, North-Holland, 1985.Google Scholar
7. 
Hagmeijer, R.: “Adaptive grid generation development, a starting point”, NLR CR 91138 L, 1991.Google Scholar
8. 
Catherall, D.: “The adaptation of two-dimensional multiblock structured grids using a pde-based method -final report for Euromesh project”, 1992.Google Scholar
9. 
Shaw, J.A., Forsey, C.R., Weatherill, N.P. and Rose, K.E.: “A block structured mesh generation technique for aerodynamic geometries”, in Proceedings of the first International Conference on Numerical Grid Generation in CFD (Ed Haüser, J. and Taylor, C. ), Pineridge Press, 1986.Google Scholar
10. 
Thomas, P.D. and Middlecoff, J.F.: “Direct control of the grid point distribution in meshes generated by elliptic equations”, AIAA Journal, Vol 18, p652, 1980.
11. 
Hall, M.G.: “Cell-vertex multigrid schemes for solution of the Euler equations”, in Numerical methods for Fluid Dynamics II (Ed Morton, K.W. and Baines, M.J. ), Oxford University Press, 1986.Google Scholar
12. 
Paisley, M.F.: “Developments in shock fitting with the Euler equations”, RAE TR 88075, 1988.Google Scholar
13. 
Pulliam, T.H. and Barton, J.T.: “Euler computations of AGARD Working Group 07 airfoil test cases”, AIAA Paper 85–0018, 1985.Google Scholar 