Construction of Solution Curves for Large Two-Dimensional Problems of Steady-State Flows of Incompressible Fluids

This work represents a step toward advancing classical methods of bifurcation analysis in conjunction with very large-scale scientific computing needed to model realistically processes which involve laminar and steady flows of incompressible fluids. A robust method of analysis based on pseudoarclength continuation, Newton's method, and direct solution of linear systems is proposed and applied to the analysis of a representative system of fluid mechanics, namely the tilted lid driven cavity. Accurate solution curves, possessing simple singular points, were computed for Reynolds numbers varying from 0 to 10,000 and for different angles of tilt. The results demonstrate that two-dimensional models with up to 1,000,000 algebraic equations can be studied feasibly using the methods described here with state-of-the-art vector supercomputers.

[1]  Jeffrey J. Derby,et al.  Massively parallel finite element computations of three-dimensional, time-dependent, incompressible flows in materials processing systems , 1994 .

[2]  M. Gunzburger,et al.  Predictor and steplength selection in continuation methods for the Navier-Stokes equations , 1991 .

[3]  V. Shamanskii.  A modification of Newton's method , 1967 .

[4]  Scott W. Sloan,et al.  A FORTRAN program for profile and wavefront reduction , 1989 .

[5]  H. Schwetlick,et al.  Higher order predictors and adaptive steplength control in path following algorithms , 1987 .

[6]  Jennifer A. Scott,et al.  MA42 - A new frontal code for solving sparse unsymmetric systems , 1993 .

[7]  H. Keller.  Numerical Methods for Two-Point Boundary-Value Problems , 1993 .

[8]  M. Fortin,et al.  A numerical simulation of the transition to turbulence in a two-dimensional flow , 1987 .

[9]  George Karypis,et al.  Graph partitioning and its applications to scientific computing , 1996 .

[10]  Vipin Kumar,et al.  A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs , 1998, SIAM J. Sci. Comput..

[11]  I. Duff,et al.  THE USE OF PROFILE REDUCTION ALGORITHMS WITH A FRONTAL CODE , 1989 .

[12]  R. Brent.  Some Efficient Algorithms for Solving Systems of Nonlinear Equations , 1973 .

[13]  E. Allgower,et al.  Numerical Continuation Methods , 1990 .

[14]  Jeffrey J. Derby,et al.  Large-scale numerical analysis of materials processing systems : high-temperature crystal growth and molten glass flows , 1994 .

[15]  I. Duff,et al.  The Use of multiple fronts in Gaussian elimination , 1994 .

[16]  Jack J. Dongarra,et al.  A proposal for a set of level 3 basic linear algebra subprograms , 1987, SGNM.

[17]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[18]  Randolph E. Bank,et al.  Stepsize selection in continuation procedures and damped Newton's method , 1989 .

[19]  P. Deuflhard,et al.  Efficient numerical path following beyond critical points , 1987 .

[20]  Werner C. Rheinboldt,et al.  Solution Fields of Nonlinear Equations and Continuation Methods , 1980 .

[21]  W. Rheinboldt,et al.  On steplength algorithms for a class of continuation methods siam j numer anal , 1981 .

[22]  M. Fortin,et al.  A generalization of Uzawa's algorithm for the solution of the Navier-Stokes equations , 1985 .

[23]  José Mario Martínez,et al.  Comparing Algorithms for Solving Sparse Nonlinear Systems of Equations , 1992, SIAM J. Sci. Comput..

[24]  T. Chan,et al.  Domain decomposition algorithms , 1994, Acta Numerica.