Analytic study of 2D and 3D grid motion using modified Laplacian
The modified Laplacian has been used to move unstructured grids in response to changes in the surface grid for a variety of grid movement applications including store separation, aero-elastic wing deformation and free surface flow simulations. However, the use of the modified Laplacian can result in elements with negative areas/volumes, because it has no inherent mechanism to prevent inversion of elements. In this paper, the use of a modified Laplacian is studied analytically for a two-dimensional problem of deforming the inner circle of two concentric circles and for a three-dimensional problem of deforming the inner sphere of two concentric spheres. By analysing the exact solution for this problem, the amount of translation and deformation of the inner circle that maintains a valid mesh is determined. A general grid movement theorem is presented which determines analytically the maximum allowable deformation before an invalid mesh results. Under certain circumstances, the inner circle and sphere can be expanded until it reaches the outer circle or sphere, while remaining a valid grid, and the inner circle and sphere can be rotated by an extreme amount before failure of the mesh occurs. By choosing the exponent to the modified Laplacian appropriately, extreme deformations for single frequency deformations is possible, although for practical applications where the grid movement has multiple frequencies, choosing the optimal exponent for the modified Laplacian may not be practical or provide much improvement. For grid movement simulations involving rigid body translation and rotation or uniform expansion, the modified Laplacian can yield excellent results, and the optimum choice of the modified Laplacian can be analytically deter-mined for these types of motions, but when there are multiple frequencies in the deformation, the modified Laplacian does not allow much deformation before an invalid grid results. Copyright (c) 2006 John Wiley & Sons, Ltd.
modified laplacian unstructured grids grid movement linear spring analogy