On the computation of multi-dimensional solution manifolds of parametrized equations

SummaryA new algorithm is presented for computing vertices of a simplicial triangulation of thep-dimensional solution manifold of a parametrized equationF(x)=0, whereF is a nonlinear mapping fromRn toRm,p=n−m>1. An essential part of the method is a constructive algorithm for computing moving frames on the manifold; that is, of orthonormal bases of the tangent spaces that vary smoothly with their points of contact. The triangulation algorithm uses these bases, together with a chord form of the Gauss-Newton process as corrector, to compute the desired vertices. The Jacobian matrix of the mapping is not required at all the vertices but only at the centers of certain local “triangulation patches”. Several numerical examples show that the method is very efficient in computing triangulations, even around singularities such as limit points and bifurcation points. This opens up new possibilities for determining the form and special features of such solution manifolds.

[1]  A. C. Walker,et al.  A non-linear finite element analysis of shallow circular arches , 1969 .

[2]  M. Spivak.  A comprehensive introduction to differential geometry , 1979 .

[3]  M. Todd.  The Computation of Fixed Points and Applications , 1976 .

[4]  W. Rheinboldt.  Numerical methods for a class of nite dimensional bifur-cation problems , 1978 .

[5]  P. Deuflhard,et al.  Affine Invariant Convergence Theorems for Newton’s Method and Extensions to Related Methods , 1979 .

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

[7]  Erich Bohl,et al.  Finite Modelle gewőhnlicher Randwertaufgaben , 1981 .

[8]  Werner C. Rheinboldt,et al.  Algorithm 596: a program for a locally parameterized , 1983, TOMS.

[9]  Werner C. Rheinboldt,et al.  A locally parameterized continuation process , 1983, TOMS.

[10]  Gene H. Golub,et al.  Matrix computations , 1983 .

[11]  W. Rheinboldt,et al.  Solution manifolds and submanifolds of parametrized equations and their discretization errors , 1984 .

[12]  Danny C. Sorensen,et al.  A note on the computation of an orthonormal basis for the null space of a matrix , 1982, Math. Program..

[13]  Ivo Babuška,et al.  Adaptive Finite Element Processes in Structural Mechanics. , 1984 .

[14]  A. Griewank,et al.  Characterization and Computation of Generalized Turning Points , 1984 .

[15]  M. Heath,et al.  An algorithm to compute a sparse basis of the null space , 1985 .

[16]  E. Allgower,et al.  An Algorithm for Piecewise-Linear Approximation of an Implicitly Defined Manifold , 1985 .

[17]  W. Rheinboldt.  Numerical analysis of parametrized nonlinear equations , 1986 .

[18]  J. P. Fink,et al.  A geometric framework for the numerical study of singular points , 1987 .

[19]  T. Coleman,et al.  The null space problem II. Algorithms , 1987 .