On a Computational Method for the Second Fundamental Tensor and Its Application to Bifurcation Problems.

Abstract

An algorithm is presented for the computation of the second fundamental tensor V of a Riemannian sub-manifold M of R superscript n. From V the Riemann curvature tensor of M is easily obtained. Moreover, V has a close relation to the second derivative of certain functionals on M which, in turn, provides a powerful new tool for the computational determination of multiple bifurcation directions. Frequently, in applications, the manifold M is defined implicitly as the zero set of a submersion F on R superscript n. In this case, the principal cost of algorithm for computing V (p) at a given point p an element of M involves only the decomposition of the Jacobian DF (p) of F at p and the projection of d(d+1) neighboring points onto M by means of a local iterative process using DF(p). Several numerical examples are given which show the efficiency and dependability of the method. (jhd)

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Document Details

Document Type
Technical Report
Publication Date
Feb 01, 1989
Accession Number
ADA207517

Entities

People

  • Patrick J. Rabier
  • Werner Rheinboldt

Organizations

  • University of Pittsburgh

Tags

Communities of Interest

  • Air Platforms

DTIC Thesaurus Topics

  • Algorithms
  • Boundary Value Problems
  • Computational Science
  • Computations
  • Coordinate Systems
  • Curvature
  • Eigenvalues
  • Equations
  • Geometry
  • Mathematics
  • Military Research
  • Quadratic Equations
  • Real Numbers
  • Square Roots
  • Standards
  • Three Dimensional
  • Universities

Fields of Study

  • Mathematics

Readers

  • Finite Element Method (FEM) for solving Partial Differential Equations (PDEs)
  • Graph Algorithms and Convex Optimization.