References
- Aggoune , W. , Zarhouni , H. and Potier-Ferry , M. 2004 . A high-order prediction–correction algorithms for unilateral contact problems . J. Comput. Appl. Math. , 168 : 1 – 9 .
- Allgower , E. L. and Georg , K. 1980 . Simplicial and continuation methods for approximations, fixed points and solutions to systems of equations . SIAM Rev. , 22 : 28 – 85 .
- Allgower , E. L. and Georg , K. 1990 . Numerical Continuation Methods: An Introduction , New York : Springer .
- R. Arquier, B. Cochelin, and C. Vergez, Manlab: Logiciel de continuation interactif, user manual, 2005. Available at http://manlab.lma.cnrs-mrs.fr/
- B.M. Averick, R.G. Carter, J.J. Moré, and G.L. Xue, The minpack-2 test problem collection, Preprint MCS-P153-0694, Argonne National Laboratory, 1992
- Azrar , L. , Cochelin , B. , Damil , N. and Potier-Ferry , M. 1993 . An asymptotic-numerical method to compute the postbuckling behavior of elastic plates and shells . Int. J. Numer. Meth. Eng. , 36 : 1251 – 1277 .
- Azrar , L. , Boutyour , E. H. and Potier-Ferry , M. 2002 . Non-linear forced vibrations of plates by an asymptotic numerical method . J. Sound Vibrat. , 252 : 657 – 674 .
- Baker , G. A. and Graves-Morris , P. 1996 . Padé Approximants, Encyclopedia of Mathematics and its Applications , Vol. 59 , Cambridge : Cambridge University Press .
- F. Bechet, A. Lejeune, and M. Potier-Ferry, Taylor series to solve friction problems, C.R. Mecanique 338 (2010), 327–332
- Berz , M. and Makino , K. 1998 . Verified integration of ODEs and flows using differential algebraic methods on high-order Taylor models . Reliab. Comput. , 4 : 361 – 369 .
- Berz , M. , Makino , K. , Shamseddine , K. , Hoffstäter , G. H. and Wan , W. 1996 . COSY INFINITY and its Applications in Nonlinear Dynamics, in Computational Differentiation: Techniques, Applications and Tools , Edited by: Berz , M. , Bischof , C. , Corliss , G. and Griewank , A. 363 – 365 . Philadelphia , PA : SIAM .
- Bilasse , M. , Charpentier , I. , Daya , E. M. and Koutsawa , Y. 2009 . A generic approach for the solution of nonlinear residual equations. Part II: Homotopy and complex nonlinear eigenvalue method . Comput. Method. Appl. , 198 : 3999 – 4004 .
- Bischof , C. H. , Bücker , H. M. , Lang , B. , Rasch , A. and Vehreschild , A. Combining source transformation and operator overloading techniques to compute derivatives for MATLAB programs . Proceedings of the Second IEEE International Workshop on Source Code Analysis and Manipulation, IEEE . pp. 65 – 72 . Los Alamitos , CA
- Boudaoud , H. , Belouettar , S. , Daya , E. M. and Potier-Ferry , M. 2009 . A numerical method for nonlinear complex modes with application to active-passive damped sandwich structures . Eng. Struct. , 31 : 284 – 291 .
- H. Boudaoud, A. Lejeune, M. Potier-Ferry, H. Zahrouni, and I. Charpentier, Différentiation automatique et MAN: Application au contact unilatéral, in Actes du 9ième colloque national en Calcul des structures, Vol. 2, Giens, 2009, 281–286
- Chang , Y. F. and Corliss , G. F. 1982 . Solving ordinary differential equations using taylor series . ACM Trans. Math. Softw. , 8 : 114 – 144 .
- Charpentier , I. 2008 . Sensitivity of solutions computed through the asymptotic numerical method . C. R. Mecanique , 336 : 788 – 793 .
- Charpentier , I. and Potier-Ferry , M. 2008 . Différentiation automatique de la méthode asymptotique numérique typée: l'approche Diamant . C. R. Mecanique , 336 : 336 – 340 .
- Charpentier , I. and Utke , J. 2009 . Fast higher-order derivative tensors with Rapsodia . Optim. Method. Softw. , 24 : 1 – 14 .
- Charpentier , I. , Lejeune , A. and Potier-Ferry , M. 2008 . The Diamant Approach for an Efficient Automatic Differentiation of the Asymptotic Numerical Method, in Advance in Automatic Differentiation, Lecture Notes in Computational Science and Engineering , Edited by: Bischof , C. H. , Bücker , H. M. , Hovland , P. , Naumann , U. and Utke , J. Vol. 64 , 139 – 149 . Springer .
- Chow , S. N. , Malletparret , J. and Yorke , J. A. 1978 . Finding zeros of maps – homotopy methods that are constructive with probability one . Math. Comput. , 32 : 887 – 899 .
- Cochelin , B. , Damil , N. and Potier-Ferry , M. 1994 . Asymptotic-numerical methods and Padé approximants for non-linear elastic structures . Int. J. Numer. Meth. Eng. , 37 : 1187 – 1213 .
- Cochelin , B. , Damil , N. and Potier-Ferry , M. 2007 . Méthode Asymptotique Numérique , France : Hermes Science Publications .
- Damil , N. , Potier-Ferry , M. , Najah , A. , Chari , R. and Lahmam , H. 1999 . An iterative method based upon Padé approximants . Commun. Numer. Meth. Eng. , 15 : 701 – 708 .
- Daya , E. M. and Potier-Ferry , M. 2001 . A numerical method for nonlinear eigenvalue problems application to vibrations of viscoelastic structures . Comput. Struct. , 79 : 533 – 541 .
- Doedel , E. , Keller , H. and Kernevez , J. P. 1991 . Numerical analysis and control of bifurcation problems (i) bifurcation in finite dimensions . Int. J. Bifurc. Chaos , 1 : 493 – 520 .
- I. El Khaldi, I. Charpentier, and E.M. Daya, A gradient method for viscoelastic behaviour identification of damped sandwich structures, submitted
- Griewank , A. and Corliss , G. Proceedings of the SIAM Workshop on the Automatic Differentiation of Algorithms . Automatic Differentiation: Theory, Implementation, and Application , Edited by: Griewank , A. and Corliss , G. Philadelphia , PA : SIAM .
- Griewank , A. and Walther , A. 2008 . Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation , 2 , Philadelphia , PA : SIAM . in Other Titles in Applied Mathematics
- Griewank , A. , Juedes , D. and Utke , J. 1996 . ADOL–C, a package for the automatic differentiation of algorithms written in C/C++ . ACM T. Math. Softw. , 22 : 131 – 167 .
- Griewank , A. , Utke , J. and Walther , A. 2000 . Evaluating higher derivative tensors by forward propagation of univariate Taylor series . Math. Comput. , 69 : 1117 – 1130 .
- Guillaume , Ph. 1999 . Nonlinear eigenproblems . SIAM J. Matrix Anal. A , 20 : 575 – 595 .
- Guillaume , Ph. and Masmoudi , M. 1994 . Computation of high order derivatives in optimal shape design . Numer. Math. , 67 : 231 – 250 .
- Harnau , M. , Konyukhov , A. and Schweizerhof , K. 2005 . Algorithmic aspects in large deformation contact analysis using solid-shell elements . Comput. Struct. , 83 : 1804 – 1823 .
- L. Hascoët and V. Pascual, TAPENADE 2.1 user's guide, Rapport technique 300, INRIA, Sophia Antipolis, 2004
- He , J. H. 2000 . A coupling method of homotopy technique and perturbation technique for nonlinear problems . Int. J. Nonlinear Mech. , 35 : 37 – 43 .
- Hoefkens , J. , Berz , M. and Makino , K. 2002 . Efficient High-Order Methods for ODEs and DAEs, in Automatic Differentiation 2000: From Simulation to Optimization , Edited by: Corliss , G. , Faure , C. , Griewank , A. , Hascoët , L. and Naumann , U. 343 – 348 . New York , NY : Springer .
- Jamal , M. , Braikat , B. , Boutmir , S. , Damil , N. and Potier-Ferry , M. 2002 . A high order implicit algorithm for solving nonlinear problems . Comput. Mech. , 28 : 375 – 380 .
- Kalaba , R. and Tesfatsion , L. 1991 . Solving nonlinear equations by adaptive homotopy continuation . Appl. Math. Comput. , 41 : 99 – 115 .
- Keller , H. B. 1977 . Numerical solution of bifurcation and nonlinear eigenvalue problems, in Applications of Bifurcation Theory , Edited by: Rabinowitz , P. 359 – 384 . New York : Academic Press .
- Keller , H. B. 1978 . Global homotopies and Newton method, in Recent Advances in Numerical Analysis , Edited by: Golub , G. H. and de Boor , C. 73 – 94 . New York : Academic Press .
- Keller , H. B. 1987 . Lectures on Numerical Methods in Bifurcation Problems , Berlin : Springer .
- Van Keulen , F. , Haftka , R. T. and Kim , N. H. 2005 . Review of options for structural design sensitivity analysis. Part I: Linear systems . Comput. Method. Appl. Mech. , 194 : 3213 – 3243 .
- Koepf , W. 1994 . Taylor polynomials of implicit functions, of inverse functions, and of solutions of ordinary differential equations . Compl. Var. Ell. Eq. , 25 : 23 – 33 .
- Koutsawa , Y. , Charpentier , I. , Daya , E. M. and Cherkaoui , M. 2008 . A generic approach for the solution of nonlinear residual equations. Part I: The Diamant toolbox . Comput. Method. Appl. Mech. , 198 : 572 – 577 .
- Lahman , H. , Cadou , J. M. , Zahrouni , H. , Damil , N. and Potier-Ferry , M. 2002 . High-order predictor-corrector algorithms . Int. J. Numer. Meth. Eng. , 55 : 685 – 704 .
- Lampoh , K. , Charpentier , I. and Daya , E. M. 2011 . A generic approach for the solution of nonlinear residual equations. Part III: Sensitivity computations . Comput. Method. Appl. , 200 : 2983 – 2990 .
- S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, PhD diss., Shanghai Jiao Tong University, 1992
- Liao , S. J. 2004 . On the homotopy analysis method for nonlinear problems . Appl. Math. Comput. , 147 : 499 – 513 .
- Ma , B. A. and He , J. F. 1992 . A finite element analysis of viscoelastically damped sandwich plates . J. Sound Vib. , 152 : 107 – 123 .
- Makino , K. and Berz , M. 2002 . New applications of Taylor model methods, in Automatic Differentiation 2000: From Simulation to Optimization , Edited by: Corliss , G. , Faure , C. , Griewank , A. , Hascoët , L. and Naumann , U. 359 – 364 . New York , NY : Springer .
- Mallil , E. , Lahmam , H. , Damil , N. and Potier-Ferry , M. 2000 . An iterative process based on homotopy and perturbation techniques . Comput. Method. Appl. Mech. , 190 : 1845 – 1858 .
- Mohammadi , B. and Pironneau , O. 2001 . Applied Shape Optimization for Fluids , 2 , New York : Oxford University Press .
- Nedialkov , N. S. and Pryce , J. D. 2005 . Solving differential-algebraic equations by Taylor series (I): Computing Taylor coefficients . BIT , 45 : 561 – 591 .
- Nedialkov , N. S. and Pryce , J. D. 2008 . Solving differential-algebraic equations by Taylor series (III): The DAETS code . J. Numer. Anal. Indust. Appl. Math. , 3 : 61 – 80 .
- Pryce , J. D. 2001 . A simple structural analysis method for DAEs . BIT , 41 : 364 – 384 .
- J. Pryce and J. Reid, AD01, Fortran 90 code for automatic differentiation, Technical Report RAL-TR-1998-057, Rutherford Appleton Laboratory, 1998
- Seydel , R. 2009 . Practical Bifurcation and Stability Analysis, Interdisciplinary Applied Mathematics , 3 , Vol. 5 , Berlin : Springer .
- Tesfatsion , L. 1992 . Nonlocal automated comparative static analysis . Comput. Econ. , 5 : 313 – 331 .
- Thompson , J. M.T. and Walker , A. 1968 . The nonlinear perturbation analysis of discrete structural systems . Int. J. Solids Struct. , 4 : 757 – 768 .
- Watson , L. T. 1979 . Algorithm that is globally convergent with probability one for a class of non-linear 2-point boundary-value problems . SIAM J. Numer. Anal. , 16 : 394 – 401 .
- Watson , L. T. 1989 . Globally convergent homotopy method – a tutorial . Appl. Math. Comput. , 31 : 369 – 396 .
- Watson , L. , Billups , S. and Morgan , A. 1987 . Hompack: A suite of codes for globally convergent homotopy algorithms . ACM Trans. Math. Softw. , 13 : 281 – 310 .
- Watson , L. , Sosonkina , M. , Melville , R. , Morgan , A. and Walker , H. 1997 . Algorithm 777: Hompack90: A suite of fortran 90 codes for globally convergent homotopy algorithms . ACM Trans. Math. Softw. , 23 : 514 – 549 .
- Wengert , R. E. 1964 . A simple automatic derivative evaluation program . Comm. ACM , 7 : 463 – 464 .
- Zangwill , W. and Garcia , C. 1981 . Pathways to Solutions, Fixed Points, and Equilibria , Englewood Cliff , NJ : Prentice-Hall .