Skip to main content
SpringerLink
Log in
Book cover

Domain Decomposition Methods in Science and Engineering XX pp 443–450Cite as

Coupling Geometrically Exact Cosserat Rods and Linear Elastic Continua

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 91))

Summary

We consider the mechanical coupling of a geometrically exact Cosserat rod to a linear elastic continuum. The coupling conditions are formulated in the nonlinear rod configuration space. We describe a Dirichlet–Neumann algorithm for the coupled system, and use it to simulate the static stresses in a human knee joint, where the Cosserat rods are models for the ligaments.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. P.-A. Absil, C. G. Baker, and K. A. Gallivan. Trust-region methods on Riemannian manifolds. Found. Comput. Math., 7(3):303–330, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  2. P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn, R. Kornhuber, M. Ohlberger, and O. Sander. A generic interface for parallel and adaptive scientific computing. Part II: Implementation and tests in DUNE. Computing, 82(2–3):121–138, 2008.

    Google Scholar 

  3. P. J. Blanco, M. Discacciati, and A. Quarteroni. Modeling dimensionally-heterogenenous problems: analysis, approximation and applications. Technical Report 14, MATHICSE, 2010.

    Google Scholar 

  4. P. G. Ciarlet, H. LeDret, and R. Nzengwa. Junctions between three-dimensional and two-dimensional linearly elastic structures. J. Math. Pures Appl., 68:261–295, 1989.

    MathSciNet  MATH  Google Scholar 

  5. C. Gräser, U. Sack, and O. Sander. Truncated nonsmooth Newton multigrid methods for convex minimization problems. In Proc. of DD18, LNCSE, pages 129–136. Springer, 2009.

    Google Scholar 

  6. S. Kehrbaum. Hamiltonian Formulations of the Equilibrium Conditions Governing Elastic Rods: Qualitative Analysis and Effective Properties. PhD thesis, University of Maryland, 1997.

    Google Scholar 

  7. J. Lagnese, G. Leugering, and E. Schmidt. Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures. Birkhäuser, 1994.

    Google Scholar 

  8. D. J. Monaghan, I. W. Doherty, D. M. Court, and C. G. Armstrong. Coupling 1D beams to 3D bodies. In Proc. 7th Int. Meshing Roundtable. Sandia National Laboratories, 1998.

    Google Scholar 

  9. A. Quarteroni and A. Valli. Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications, 1999.

    Google Scholar 

  10. O. Sander. Multidimensional Coupling in a Human Knee Model. PhD thesis, Freie Universität Berlin, 2008.

    Google Scholar 

  11. O. Sander. Geodesic finite elements for Cosserat rods. Int. J. Num. Meth. Eng., 82(13):1645–1670, 2010.

    MathSciNet  MATH  Google Scholar 

  12. T. I. Seidman and P. Wolfe. Equilibrium states of an elastic conducting rod in a magnetic field. Arch. Rational Mech. Anal., 102:307–329, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Wächter and L. T. Biegler. On the implementation of a primal–dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Progr., 106(1):25–57, 2006.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Mathematik und Informatik, Freie Universität Berlin, Arnimallee 6, Berlin, Germany

    Oliver Sander

Authors
  1. Oliver Sander
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Oliver Sander .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, University of California, San Diego, Gilman Drive 9500, La Jolla, 92093-0112, California, USA

    Randolph Bank

  2. , Department of Mathematics, University of California, San Diego, Gilman Drive 9500, La Jolla, 92093-0112, California, USA

    Michael Holst

  3. , Courant Institute of Mathematical, New York University, Mercer Street 251, New York, 10012-1185, New York, USA

    Olof Widlund

  4. , Department of Mathematics, Pennsylvania State University, State College, 16802, Pennsylvania, USA

    Jinchao Xu

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Sander, O. (2013). Coupling Geometrically Exact Cosserat Rods and Linear Elastic Continua. In: Bank, R., Holst, M., Widlund, O., Xu, J. (eds) Domain Decomposition Methods in Science and Engineering XX. Lecture Notes in Computational Science and Engineering, vol 91. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35275-1_52

Download citation

Publish with us

Policies and ethics

Search

Navigation