A grid movement algorithm has been developed for the purpose of adaptively resolving numerical solutions to physical problems and, in addition, for grid clustering on arbitrary surfaces. Both the solutions and the arbitrary surfaces are represented by grid point data with a continuous definition provided by interpolation between points. Movement is applied relative to this representation. The algorithm comes from a local mean value construction to produce a finite difference molecule for movement. The mean value weights are of a general enough nature to provide for a generous number of clustering possibilities. The movement molecule is executed within an interative cycle in the spirit of point Jacobi or Gauss-Seidel, and as a consequence, corresponds to the solution of some elliptic partial differential equation which satisfies a maximum (minimum) principle due to the mean value construction. From this principle, the movement will always preserve nonsingularity for the continuous transformation. For the discrete representation in the form of a grid, local geometric constraints are established to maintain this preservation.

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