Pad\'e approximants and continualization for 1D mass chain

I. V. Andrianov

Matematičeskoe modelirovanie, Tome 18 (2006) no. 1, pp. 43-58.

Résumé

Various continuum models (CM) for 1D discrete media are under consideration. As example we deal with a difference-differential equation, describing the system of connected oscillators. A common opinion is that the corresponding CM (string). String-type approximation justified for low part of frequency spectra, but for forced oscillations the solution of wave and chain equations can be quite different (splash effect). So, more appropriate CM of chain should be found. Intermediate CM (ICM). The difference operator makes analysis difficult due to its non-local form. Approximate equations can be gained by replacing it with a local derivative operator. If we use derivative of more then second order, we have ICM. ICM give possibility to take splash effect into account, but we have a higher order of approximate differential equation. Quasi-continuum approximation. The quasi-continuum approximation technique makes use of higher-order derivatives to form more accurate approximations of the discrete difference operator via one-point Padé approximants (PA). Pseudo-continuum. Two-point PA give the most accurate CM of difference operator. Possibilities of generalizations and applications are discussed.

Bibliographie

[1] Poston T., Styuart I., Teoriya katastrof, Mir, M., 1980, 608 pp. | Zbl

[2] Glannoulis J., Mielke A., “The nonlinear Schrodinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities”, Nonlinearity, 17 (2004), 551–565 | DOI | MR

[3] Aero E. L., “Suschestvenno nelineinaya mikromekhanika sredy s izmenyaemoi periodicheskoi strukturoi”, Uspekhi mekhaniki, 1:3 (2002), 130–176

[4] Brillyuen L., Parodi M., Rasprostranenie voln v periodicheskikh strukturakh, IL, M., 1959, 458 pp.

[5] Born M., Khuan Kun, Dinamicheskaya teoriya kristallicheskikh reshetok, IL, M., 1958, 488 pp.

[6] Maugin G. A., Nonlinear Waves in Elastic Crystals, Oxford UP, New York, Oxford, 1999, 314 pp. | MR | Zbl

[7] Askar A., Lattice Dynamical Foundations of Continuum Theories. Elasticity, Piezoelectricity, Viscoelasticity, Plasticity, World Scientific, Singapore, 1985, 192 pp. | MR

[8] Aifantis E. C., “Gradient deformation models at nano, micro, and macro scales”, ASME J. Engn. Mater. Techn., 121 (1999), 189–202 | DOI

[9] Askes H., Metrikine A. V., “One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure. Part 1: Generic formulation”, Eur. J. Mech. A /Solids, 21 (2002), 555–572 | DOI | Zbl

[10] Peerlings R. H. J., Geers M. G. D., Borst R. de, Brekelmans W. A. M., “A critical comparison of nonlocal and gradientenhanced softening continua”, Int. J. Solids Struct., 38 (2001), 7723–7746 | DOI | Zbl

[11] Askes H., Sluys L. J. “Explicit and implicit gradient series in damage mechanics”, Eur. J. Mech. A /Solids, 21 (2002), 379–390 | DOI | Zbl

[12] Chang C. S., Askes H., Sluys L. J., “Higher-order strain/higher-order stress gradient models derived from a discrete microstructure, with application to fracture”, Eng. Fracture Mech., 69 (2002), 1907–1924 | DOI

[13] Piero Del G., Truskinovsky L., “A one-dimensional model for localized and distributed failure”, J. de Physique IY France 8, 82 (1998), 199–210

[14] Rogers R. C., Truskinovsky L., “Discretization and hysteresis”, Physica B, 233 (1997), 370–375 | DOI

[15] Blanc X., Bris C. Le, Lions P.-L., “From molecular models to continuum mechanics”, Archive for Rational Mech. Anal., 164 (2002), 341–381 | DOI | MR | Zbl

[16] Braun O. M., Kivshar Y. S., The Frenkel-Kontorova Model, Springer-Verlag, New York, 2004, 472 pp. | MR

[17] Fleck N. A., Hutchinson J. W., “A phenomenological theory for gradient effects in plasticity”, J. Mech. Phys. Solids, 41 (1993), 1825–1857 | DOI | MR | Zbl

[18] Eringen A. C., Nonlocal Continuum Field Theories, Springer-Verlag, New York, 2002, 376 pp. | MR

[19] Kunin I. A., Teoriya uprugikh sred s mikrostrukturoi: nelokalnaya teoriya uprugosti, Nauka (SO), M., 1975, 415 pp. | MR

[20] Potapov A. I., “Volny deformatsii v srede s vnutrennei strukturoi”, Nonlinear Waves'2004, eds. A. V. Gaponov-Grekhov, V. I. Nekorkin, Inst. Appl. Physics RAS, N. Novgorod, 2004

[21] Dowell E. H., Tang D., “Multiscale, multiphenomena modelling and simulation at the nanoscale: on constructing reduced-order models for nonlinear dynamical systems with many degrees-of-freedom”, J. Appl. Mech., 70 (2003), 328–338 | DOI | Zbl

[22] Triantafilyllidis N., Bardenhagen S., “On higher order gradient continuum theories in 1-D elasticity. Deviation from and comparison to the corresponding discrete models”, J. Elasticity, 33 (1993), 259–293 | DOI | MR

[23] Friesecke G., Theil F., “Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice”, J. Nonlin. Sciences, 12 (2002), 445–478 | DOI | MR | Zbl

[24] Paroni R., “From discrete to continuum: a Young measure approach”, ZAMP, 54:2 (2003), 328–348 | DOI | MR | Zbl

[25] Pagano S., Paroni R., “A simple model for phase transition: from the discrete to the continuum problem”, Quart. Appl. Math., 61 (2003), 89–109 | MR | Zbl

[26] Charlotte M., Truskinovsky L., “Linear elastic chain with a hyper-pre-stress”, J. Mech. Phys. Solids, 50 (2002), 217–251 | DOI | MR | Zbl

[27] Ilyushina E. A., “Odna iz modelei sploshnoi sredy s uchetom mikrostruktury”, PMM, 33:5 (1969), 917–923

[28] Potapov A. I., Pavlov I. S., Maugin G. A., “Nonlinear wave interaction in 1D crystals with complex lattice”, Wave Motion, 29 (1999), 297–312 | DOI | MR | Zbl

[29] Lizina S. A., Potapov A. I., Nesterenko V. F., “Nelineinaya granulirovannaya sreda s vraschayuschimisya chastitsami: odnomernaya model”, Akusticheskii zh., 47:5 (2001), 685–693

[30] Maslov V. P., Omel'yanov G. A., Geometric Asymptotics for Nonlinear PDE. I, AMS, Providence, R.I., 2001, 285 pp. | MR | Zbl

[31] Deift P., McLaughlin K. T-R., A Continuum Limit of the Toda Lattice, AMS, Providence, R.I., 1998, 216 pp. | MR

[32] Kurchanov P. F., Myshkis A. D., Filimonov A. M., “Kolebaniya zheleznodorozhnogo sostava i teorema Kronekera”, PMM, 55:6 (1991), 989–995 | MR | Zbl

[33] Zhukovskii N. E., “Rabota (usilie) russkogo skvoznogo i amerikanskogo neskvoznykh tyagovykh priborov pri troganii poezda s mesta i v nachale ego dvizheniya”, Poln. sobr. soch., v. 8, Gostekhizdat, M.–L., 1937, 221–255

[34] Filimonov A. M., “Kontinualnye i diskretnye modeli ogranichennykh odnomernykh sred v teorii vyazkouprugosti”, PMM, 61:2 (1997), 285–296 | MR | Zbl

[35] Lure A. I., Operatsionnoe ischislenie i ego prilozhenie k zadacham mekhaniki, Gostekhizdat, M.–L., 1950, 432 pp.

[36] Kurchanov P. F., Myshkis A. D., Filimonov A. M., “Some unexpected results in the classical problem of vibrations of the string with n beads when n is large”, Comptes Rendus Acad. Sci. Paris, serie 1, 313 (1991), 961–965 | MR | Zbl

[37] Filimonov A., “Some unexpected results on the classical problem of the string with N beads. The case of multiple frequencies”, Comptes Rendus Acad. Sci. Paris, serie 1, 315 (1992), 957–961 | MR | Zbl

[38] Filimonov A. M., “Continuous approximations of difference operators”, J. Difference Equations and Applications, 2:4 (1996), 411–422 | DOI | MR | Zbl

[39] Filimonov A. M., Myshkis A. D., “Asymptotic estimate of solution of one mixed difference-differential equation of oscillations theory”, J. Difference Equations and Applications, 4 (1998), 13–16 | DOI | MR | Zbl

[40] Filimonov A. M., Mao X., Maslov S., “Splash effect and ergodic properties of solution of the classic difference–differential equation”, J. Difference Equations and Applications, 6 (2000), 319–328 | DOI | MR | Zbl

[41] Maslov V. P., Operatornye metody, Nauka, M., 1973, 543 pp. | MR

[42] Shokin Yu. I., Yanenko N. N., Metod differentsialnogo priblizheniya. Primenenie k gazovoi dinamike, Nauka (SO), Novosibirsk, 1985, 363 pp. | MR | Zbl

[43] Rosenau Ph., “Dynamics of nonlinear mass-spring chains near the continuum limit”, Physics Letters A, 118:5 (1986), 222–227 | DOI | MR

[44] Rosenau Ph., “Dynamics of dense lattices”, Physical Review B, 36:11 (1987), 5868–5876 | DOI | MR

[45] Rosenau Ph., “Dynamics of dense discrete systems”, Progress of Theoretical Physics, 89:5 (1988), 1028–1042 | DOI

[46] Rosenau Ph., “Hamiltonian dynamics of dense chains near and lattices: or how to correct the continuum”, Physics Letters A, 311:5 (2003), 39–52 | DOI | MR | Zbl

[47] Rubin M. B., Rosenau P., Gottlieb O., “Continuum model of dispersion caused by an inherent materil characteristics length”, J. Appl. Physics, 77:5 (1995), 4054–4063 | DOI

[48] Wattis J. A. D., “Approximations to solitary waves on lattices, II: quasi-continuum approximations for fast and slow waves”, J. Phys. A: Math. Gen., 26 (1993), 1193–1209 | DOI | MR | Zbl

[49] Wattis J. A. D., “Quasi-continuum approximations to lattice equations arising from discrete nonlinear telegraph equation”, J. Phys. A: Math. Gen., 33 (2000), 5925–5944 | DOI | MR | Zbl

[50] Kevrekidis P. G., Kevrekidis I. G., Bishop A. R., Titi E. S., “Continuum approach to discreteness”, Physical Review E, 65:4 (2002), 46613-1–46613-13 | DOI | MR

[51] Beiker Dzh., Greivs-Morris P., Approksimatsii Pade, Mir, M., 1986, 502 pp. | MR

[52] Erofeev V. I., Volnovye protsessy v tverdykh telakh s mikrostrukturoi, MGU, M., 1999, 328 pp.

[53] Grigolyuk E. I., Selezov I. T., Neklassicheskie teorii kolebanii sterzhnei, plastin i obolochek, VINITI, M., 1973, 272 pp. | Zbl

[54] Chen W., Fish J., “A dispersive model for wave propagation in periodic heterogeneous media based on homogenization with multiple spatial and temporal scales”, J. Appl. Mech., 68 (2001), 153–161 | DOI | Zbl

[55] Manevich L. I., Mikhlin Yu. V., Pilipchuk V. N., Metod normalnykh form dlya suschestvenno nelineinykh sistem, Nauka, M., 1989, 216 pp. | Zbl

[56] Vakakis A. F., Manevitch L. I., Mikhlin Yu. V., Pilipchuk V. N., Zevin A. A., Normal Modes and Localization in Nonlinear Systems, Wiley, New York, 1996, 552 pp. | MR | Zbl

[57] Kosevich A. M., Kovalev A. S., Vvedenie v nelineinuyu fizicheskuyu mekhaniku, Naukova Dumka, Kiev, 1989, 304 pp. | MR | Zbl

[58] Kosevich A. M., Crystal Lattice: Phonons, Solitons, Dislocations, Wiley-VCH, Berlin, New York, 1999, 326 pp. | Zbl

[59] Andrianov I. V., “Kontinualnaya approksimatsiya dlya vysokochastotnykh kolebanii tsepochki”, DAN USSR, cer. A, 1991, no. 2, 13–15

[60] Andrianov I. V., “Ob osobennostyakh predelnogo perekhoda ot diskretnoi uprugoi sredy k nepreryvnoi”, PMM, 66:2 (2002), 271–275 | MR | Zbl

[61] Awrejcewicz J., Andrianov I. V., Manevitch L. I., Asymptotic Approaches in Nonlinear Dynamics: New Trends and Applications, Springer-Verlag, Berlin, Heidelberg, 1998, 310 pp. | MR | Zbl

[62] Obraztsov I. F., Andrianov I. V., Nerubailo B. V., “Kontinualnoe priblizhenie dlya vysokochastotnykh kolebanii tsepochki i sostavnye uravneniya”, Dokl. AN SSSR, 319:1 (1991), 111–112 | MR | Zbl

[63] Obraztsov I. F., Nerubailo B. V., Andrianov I. V., Asimptoticheskie metody v stroitelnoi mekhanike tonkostennykh konstruktsii, Mashinostroenie, M., 1991, 416 pp. | MR | Zbl

[64] Andrianov I. V., Awrejcewicz J., “New trends in asymptotic approaches: summation and interpolation methods”, Appl. Mech. Rev., 54:1 (2001), 69–92 | DOI

[65] Andrianov I. V., Awrejcewicz J., Manevitch L. I., Asymptotical Mechanics of Thin-Walled Structures: A Handbook, Springer-Verlag, Berlin, Heidelberg, 2004, 535 pp. | Zbl

[66] Landa P. S., Nelineinye kolebaniya i volny, Nauka, Fizmatgiz, M., 1997, 495 pp. | MR

[67] Andrianov I. V., Matyash M. V., “Ob uluchshenii metoda differentsialnogo priblizheniya”, Problemy prochnosti i plastichnosti, 2002, no. 64, 171–176

[68] Chekmarev D. G., “Postroenie konechnoraznostnykh skhem, ekvivalentnykh chislennym skhemam metoda konechnogo elementa. Prikladnye problemy prochnosti i plastichnosti”, Chislennoe modelirovanie fiziko-mekhanicheskikh protsessov, Tovarischestvo nauchnykh izd. KMK, M., 1999, 129–138

[69] Ulam S., Nereshennye matematicheskie zadachi, Nauka, M., 1964, 168 pp. | Zbl

[70] Agostini L., Bass J., “Les theories de la Turbulence”, Pub. Sci. tech. du ministre de l'ais, 1950, no. 237 | MR

[71] Golovkin K. K., “Novye modelnye uravneniya dvizheniya vyazkoi zhidkosti i ikh odnoznachnaya razreshimost”, Trudy MIAN im. Steklova, 102, 1967, 29–50 | MR

[72] Ladyzhenskaya O. A., “O novykh uravneniyakh dlya opisaniya dvizheniya vyazkikh neszhimaemykh zhidkostei i razreshimosti v tselom dlya nikh kraevykh zadach”, Trudy MIAN im. Steklova, 102, 1967, 85–104 | Zbl

[73] Ladyzhenskaya O. A., “O modifikatsii uravnenii Nave-Stoksa dlya bolshikh gradientov skorostei”, Zapiski LOMI, 7, 1968, 126–154 | Zbl

[74] Ladyzhenskaya O. A., Matematicheskie voprosy dinamiki vyazkoi zhidkosti, Nauka, M., 1970, 280 pp.

[75] Jaffe A., Quinn F., ““Theoretical Mathematics”: toward a cultural synthesis of Mathematics and Theoretical Physics”, Bull. AMS, 29:1 (1993), 1–13 | DOI | MR | Zbl

[76] Andrianov I. V., ““Teoreticheskaya” i “strogaya” matematika: novyi razdel rodilsya?”, Znanie-sila, 1994, no. 5, 08–110 | MR

[77] Rosenau Ph., “Extending hydrodynamics via the regularization of the Chapman-Enskog expansion”, Physical Review A, 40:12 (1989), 7193–7196 | DOI | MR

[78] Kalyakin L. A., “Dlinnovolnovye asimptotiki. Integriruemye uravneniya kak asimptoticheskii predel nelineinykh sistem”, UMN, 44:1 (1989), 5–33 | MR

[79] Pavlov I. S., Potapov A. I., “Nelineinaya dinamika dvumernoi reshetki. Prikladnye problemy prochnosti i plastichnosti”, Chislennoe modelirovanie fiziko-mekhanicheskikh protsessov, Tovarischestvo nauchnykh izd. KMK, M., 1999, 67–74

[80] Manevitch L. I., “The description of localized normal modes in a chain of nonlinear coupled oscillators using complex variables”, Nonlinear Dynamics, 25 (2001), 95–109 | DOI | MR | Zbl

[81] Manevitch L. I., Pervouchine V. P., “Transversal dynamics of one-dimensional chain on nonlinear asymmetric substrate”, Meccanica, 38 (2003), 669–676 | DOI | MR | Zbl

[82] Schraad M. W., “On the macroscopic properties of discrete media with nearly periodic microstructures”, Int. J. Solids Structures, 38 (2001), 7381–7407 | DOI | Zbl

[83] Jensen J. S., “Phononic band gaps and vibrations in one- and two-dimensional mass-spring structures”, J. Sound Vibr., 266 (2003), 1053–1078 | DOI

[84] Martinsson G. P., Movchan A. B., “Vibrations of lattice structures and phononic band gaps”, Quartely J. Mech. Appl. Math., 56:1 (2003), 45–64 | DOI | MR | Zbl