Geodesic
On the convergence of a gradient method for the minimization of functionals in finite deformation elasticity theory and for the minimization of barrier grid functionals
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 8, pp. 1450-1465 
V. A. Garanzha; I. E. Kaporin. On the convergence of a gradient method for the minimization of functionals in finite deformation elasticity theory and for the minimization of barrier grid functionals. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 8, pp. 1450-1465. http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_8_a9/
@article{ZVMMF_2005_45_8_a9,
     author = {V. A. Garanzha and I. E. Kaporin},
     title = {On the convergence of a gradient method for the minimization of functionals in finite deformation elasticity theory and for the minimization of barrier grid functionals},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1450--1465},
     year = {2005},
     volume = {45},
     number = {8},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_8_a9/}
}
TY  - JOUR
AU  - V. A. Garanzha
AU  - I. E. Kaporin
TI  - On the convergence of a gradient method for the minimization of functionals in finite deformation elasticity theory and for the minimization of barrier grid functionals
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2005
SP  - 1450
EP  - 1465
VL  - 45
IS  - 8
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_8_a9/
LA  - ru
ID  - ZVMMF_2005_45_8_a9
ER  - 
%0 Journal Article
%A V. A. Garanzha
%A I. E. Kaporin
%T On the convergence of a gradient method for the minimization of functionals in finite deformation elasticity theory and for the minimization of barrier grid functionals
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2005
%P 1450-1465
%V 45
%N 8
%U http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_8_a9/
%G ru
%F ZVMMF_2005_45_8_a9

Voir la notice de l'article provenant de la source Math-Net.Ru

Gradient descent methods are examined for the minimization of barrier-type polyconvex functionals arising in finite-deformation elasticity theory and grid optimization. The minimum of a functional is sought in the class of continuous piecewise affine deformations that preserve orientation. Sufficient conditions are found for a sequence of iterative approximations to belong to the feasible set and for the norm of the gradient of the functional to converge to zero on this set. As the functional, one can use a measure of the deformation of a grid, for instance, a grid formed of triangles or tetrahedra.

[1] Ball J. M., “Convexity conditions and existence theorems in nonlinear elasticity”, Archives for Rational Mech. Analys., 63 (1977), 337–403 | DOI | MR | Zbl

[2] Syarle F., Matematicheskaya teoriya uprugosti, Mir, M., 1992

[3] Garanzha B. A., “Barernyi metod postroeniya kvaziizometricheskikh setok”, Zh. vychisl. matem. i matem. fiz., 40:11 (2000), 1685–1705 | MR | Zbl

[4] Blatz P. J., “On the thermostatic behaviour of elastomers”, Polymer Networks, Structure and Mechanical properties, Plenum Press, New York, 1971, 23–45

[5] Ciarlet P. G., Geymonat G., “Sur les lois de comportement en elasticite non-lineaire compressible”, C.R. Acad. Sci. Paris Ser. II, 295 (1982), 423–426 | MR | Zbl

[6] Ogden R. W., “Large deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubber-like solids”, Proc. Roy. Soc. London A, 328, 567–583 | Zbl

[7] Garanzha B. A., Kaporin I. E., “Regulyarizatsiya barernogo variatsionnogo metoda postroeniya raschetnykh setok”, Zh. vychisl. matem. i matem. fiz., 39:9 (1999), 1489–1503 | MR | Zbl

[8] Kaporin I. E., “Ispolzovanie vnutrennikh iteratsii metoda sopryazhennykh gradientov pri reshenii bolshikh razrezhennykh nelineinykh zadach optimizatsii”, Zh. vychisl. matem. i matem. fiz., 43:6 (2003), 802–807 | MR | Zbl