ADAPTIVE GRID GENERATION FROM HARMONIC MAPS ON RIEMANNIAN MANIFOLDS

THE RESULTS OBTAINED TO DATE BY VARIOUS AUTHORS INDICATE CLEARLY THAT NUMERICAL SOLUTIONS ARE MORE ACCURATE AND RE QUIRE MUCH LESS COMPUTER TIME AND STORAGE WHEN THE GRID POINTS ARE PROPERLY LOCATED. THE DEVELOPMENT OF THE SOLU- TION OVER A REAL TIME PATH OR ITERATIVE PROCESS (FALSE TIME PATH) REQUIRES THE MESH NODES TO MOVE DYNAMICALLY IN THE PROPER POSITION AT EACH ITERATION OR INSTANCE OF TIME. AL- THOUGH A NUMBER OF METHODS HAVE BEEN PROPOSED TO GENERATE SOLUTION ADAPTIVE MESHES, EACH OF THEM HAS AT LEAST ONE OF THE FOLOWING SHORTCOMINGS: (1) POSSIBILITY OF GRID ENTANGLE-MENT EVEN IN 2-D, (2) LACK OF GRID CONTROL PRECISION, OR (3)BASICALLY 1-D CHARACTER OF GRID ADAPTION. THUS, INNOVATIVE GRID GENERATORS ARE NEEDED. THE PROPOSED APPROACH TO ADA- TIVE GRID GENERATION IS BASED ON A NOVEL IDEA WHICH HAS A SOLID MATHEMATICAL FOUNDATION. THE INTERDEPENDENCE OF THE SPACE MATRIC STRUCTURE AND SOME MEASURE OF NUMERICAL ERROR IN THIS SPACE IS ASSUMED. THE POWERFUL MATHEMATICAL APPARATUS OF HARMONIC MAPS PROVIDES THE GRID GENERATING EQUATIONS AMENABLE TO EFFICENT NUMERICAL SOLUTION. PHASE I CON-SISTS OF THEORETICAL ANALYSIS AND NUMERICAL TESTS TO DETER- MINE THE FEASIBILITY OF THE ABOVE APPROACH TO ADAPTIVE GRID GENERATION.