Analysis of the Solver Performance for Stokes Flow Problems in Glass Forming Process Simulation Models Speaker: Hans Groot Supervisors: Dr. Hegen (TNO Science and Technology) Dr. Giannopapa (TU/e) Prof. dr. Mattheij (TU/e) Dr. Rienstra (TU/e)

Overview Introduction Simulation Models Problem Description Results Conclusions

Glass Manufacturing Glass Melting Glass Conditioning Automatic Inspection Glass Forming Surface Treatment Pressing Press-blowing Blow-blowing Introduction Simulation Models Problem Description Results Conclusions

Pressing Process plunger ring glass mould Introduction Simulation Models Problem Description Results Conclusions

Press-Blowing Process glass mould Pressing stage Blowing stage ring plunger ring preform mould Introduction Simulation Models Problem Description Results Conclusions

Process Simulation Packages At TNO Glass Group (and some of its customers): glass forming process simulation tools Sepran based Sepran finite element library Fortran 77 based originally developed at TU Delft Introduction Simulation Models Problem Description Results Conclusions

Purpose of Process Simulation Analysis current/existing process comparison between results and measurements Optimisation Innovation current/existing/new process Introduction Simulation Models Problem Description Results Conclusions

Characteristics of Glass Forming Models Flow of glass and air Stokes flow problem: Energy exchange in glass, air and equipment Convection diffusion problem: Evolution of glass-air interfaces Convection problem for level sets: Introduction Simulation Models Problem Description Results Conclusions

Finite Element Discretisation Partition domain into triangular Mini-elements Numerical computation solution in nodes . . . Introduction Simulation Models Problem Description Results Conclusions

Glass Pressing Model Temperature Introduction Simulation Models Problem Description Results Conclusions

Increase Solver Iterations (BiCGstab with ILU preconditioning) Iterations energy problem & level set problem Iterations flow problem Accumulative iterations 2 by 2 mesh refinements Introduction Simulation Model Problem Description Results Conclusions

Test Model Pressing time step in rectangle Uniform mesh: V = constant left: symmetry right: free flow top: constant inflow top/bottom: no slip Uniform mesh: half square triangular Mini-elements V = constant y x Introduction Simulation Model Problem Description Results Conclusions

Solver Performance Test Model 34.6 3.61 5.80E-1 1.37E-1 CPU time direct method 1.39E3 8.27 7.73E-1 1.68E-1 CPU time BiCGstab/ILU 7985 153 36 20 iterations 180224 45056 11264 2816 elements Introduction Simulation Model Problem Description Results Conclusions

Possible Causes Solver problem not caused by: discretisation methods choice iterative solver large condition number Suggestions for improvement: reordering unknowns additional fill-in for ILU implement other preconditioner Introduction Simulation Model Problem Description Results Conclusions

CMK, pressure-last/level Reordering Unknowns None Sloan CMK (Cuthill Mc Kee) CMK, pressure-last/level

Additional Fill-In for ILU Zero Fill-In: ILU for non-zero elements only Additional Fill-In: ILU for non-zero and some zero elements Better approximation of LU factorisation Introduction Simulation Model Problem Description Results Conclusions

Performance for Additional Fill-In CMK, p-last/level, extra fill-in CMK, p-last/level Sloan 6.1 8.45E-1 1.50E-1 17.3 8.65E-1 1.40E-1 50.9 1.77 2.04E-1 CPU time 32768 8192 2048 elements Introduction Simulation Model Problem Description Results Conclusions

Improved Pressing Model 180 time steps 900 time steps Introduction Simulation Model Problem Description Results Conclusions

Conclusions and Recommendations Improvement solver performance/reduction CPU time for ILU preconditioning: CMK instead of Sloan pressure-last/level additional fill-in Other preconditioners suggested: multigrid methods domain decomposition methods stabilised/modified ILU Introduction Simulation Models Problem Description Results Conclusions

Questions?

Finite Element Mesh Introduction Simulation Models Problem Description Results Conclusions

Other Preconditioners besides ILU Some other preconditioners tested, e.g. Eisenstat, Gauss-Seidel: solver performance worse than for ILU Multigrid and domain decomposition methods: suitable for Stokes flow problems multigrid computations increase linearly with unknowns not tested due to implementation issues Introduction Simulation Model Problem Description Results Conclusions

Performance for Different Orderings Introduction Simulation Model Problem Description Results Conclusions

Accuracy of Results Introduction Simulation Model Problem Description Conclusions