The mountain pass theorem is an important tool in the calculus of variations and in finding solutions to nonlinear PDEs in general. The mountain pass structure can be exploited numerically, as well. We explain the main ideas on an example of buckling of a cylindrical shell. First, we prove existence of an MP-solution for almost all values of a given load parameter. Then, we find a numerical approximation of such a solution. Finally, we compare the results of a numerical continuation in the load parameter with results of physical experiments and make a few comments about further numerical investigations of the problem. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

References

[1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), pp. 349–381.
10.1016/0022-1236(73)90051-7
Google Scholar
[2] W. Ballerstedt and H. Wagner, Versuche über die Festigkeit dünner unversteifter Zylinderunter Schub- und Längskräften, Luftfahrtforschung, 13 (1936), pp. 309–312.
Google Scholar
[3] F. J. Bridget, C. C. Jerome, and A. B. Vosseller, Some new experiments on buckling of thin-wall construction, Trans. ASME Aero. Eng., 56 (1934), pp. 569–578.
Google Scholar
[4] Y. S. Choi and P. J. McKenna, A mountain pass method for the numerical solution of semilinear elliptic problems, Nonlinear Anal., 20 (1993), pp. 417–437.
10.1016/0362-546X(93)90147-K
Web of Science® Google Scholar
[5] L. H. Donnell, A new theory for buckling of thin cylinders under axial compression and bending, Trans. ASME Aero. Eng., AER-56-12 (1934), pp. 795–806.
Google Scholar
[6] J. Horák, Constrained mountain pass algorithm for the numerical solution of semilinear elliptic problems, Numer. Math., 98 (2004), pp. 251–276.
10.1007/s00211-004-0544-7
Web of Science® Google Scholar
[7] J. Horák, G. J. Lord, and M. A. Peletier, Cylinder buckling: the mountain pass as an organizing center. To apper in SIAM J. Appl. Math.
Google Scholar
[8] J. Horák, G. J. Lord, and M. A. Peletier, Numerical variational methods applied to cylinder buckling. In preparation.
Google Scholar
[9] J. Horák and P. J. McKenna, Traveling waves in nonlinearly supported beams and plates, in Nonlinear equations: methods, models and applications (Bergamo, 2001), vol. 54 of Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 2003, pp. 197–215.
Google Scholar

Citing Literature

Volume6, Issue1

Special Issue:GAMM Annual Meeting 2006 – Berlin

December 2006

Pages 7-10

Numerical Mountain Pass and its Applications

Abstract

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Numerical Mountain Pass and its Applications

Abstract

References

Citing Literature

References

Related

Information