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Derivatives of fourth order Kronecker power systems with applications in nonlinear elasticity. (English) Zbl 1433.74098

Summary: A natural way to describe systems with polynomial nonlinearities is using the Kronecker product. Particularly, third-order Kronecker power systems can express a wide range of systems from electronic engineering to nonlinear elasticity. But such development (e.g., equations of motions of elastic structures from nonlinear strain energy) requires standard formulation for the derivative of the Kronecker power of vectors with respect to the same vector. Such standard way cannot be found in literature. This paper presents a method to obtain the derivatives of Kronecker powers of vectors with respect to itself up to a power of four and also third-order Kronecker power systems containing those terms in a concise matrix form. The matrix expression of these systems provides new approach for efficient numerical implementation, organized analysis and linearization. To demonstrate the strength of this method, an example of application for a finite element nonlinear Euler beam is also presented.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74B20 Nonlinear elasticity

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[1] Brewer, J., Kronecker products and matrix calculus in system theory, IEEE Trans. Circuits Syst., 25, 9, 772-781 (1978) · Zbl 0397.93009
[2] Bodewig, E., Matrix Calculus (2014), Elsevier · Zbl 0086.32501
[3] Neudecker, H., Some theorems on matrix differentiation with special reference to Kronecker matrix products, J. Am. Stat. Assoc., 64, 327, 953-963 (1969) · Zbl 0179.33102
[4] Vetter, W. J., Matrix calculus operations and Taylor expansions, SIAM Rev., 15, 2, 352-369 (1973) · Zbl 0254.65033
[5] 10158, 1982, 130.
[6] Henderson, H. V.; Searle, S. R., The vec-permutation matrix, the vec operator and Kronecker products: A review, Linear Multilinear Algebra, 9, 4, 271-288 (1981) · Zbl 0458.15006
[7] Zhang, H.; Ding, F., On the Kronecker products and their applications, J. Appl. Math., 2013 (2013) · Zbl 1275.15019
[8] Huhtanen, M., Real linear Kronecker product operations, Linear Algebra Appl., 418, 1, 347-361 (2006) · Zbl 1104.15021
[9] Fa-Jun, Y.; Hong-Qing, Z., Constructing new discrete integrable coupling system for soliton equation by Kronecker product, Commun. Theor. Phys., 50, 3, 561 (2008) · Zbl 1392.37061
[10] Regalia, P. A.; Sanjit, M. K., Kronecker products, unitary matrices and signal processing applications, SIAM Rev., 31, 4, 586-613 (1989) · Zbl 0687.15010
[11] Amabili, M., A comparison of shell theories for large-amplitude vibrations of circular cylindrical shells: Lagrangian approach, J. Sound Vibr., 264, 5, 1091-1125 (2003)
[12] Amabili, M.; Pellicano, F., Multimode approach to nonlinear supersonic flutter of imperfect circular cylindrical shells, J. Appl. Mech., 69, 2, 117-129 (2001) · Zbl 1110.74313
[13] doi:10.1142/S0219455412500782. 1250078 (20 pp.). · Zbl 1274.74180
[14] Selmane, A.; Lakis, A. A., Dynamic analysis of anisotropic open cylindrical shells, Comput. Struct., 62, 1, 1-12 (1997) · Zbl 0899.73240
[15] Rossikhin, Y. A.; Shitikova, M. V., Nonlinear dynamic response of a fractionally damped cylindrical shell with a three-to-one internal resonance, Appl. Math. Comput., 257, 498-525 (2015) · Zbl 1338.74085
[16] Mahmoudkhani, S., A semi-analytical method for calculation of strongly nonlinear normal modes of mechanical systems, J. Comput. Nonlinear Dyn., 13, 4, Article 041005-041005-12 (2018)
[17] Ghayesh, M. H.; Farokhi, H.; Alici, G., Internal energy transfer in dynamical behavior of slightly curved shear deformable microplates, J. Comput. Nonlinear Dyn., 11, 4, 041002 (2015)
[18] Amabili, M., A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells, J. Sound Vibr., 332, 19, 4620-4640 (2013)
[19] Amabili, M.; Pellicano, F.; Paidoussis, M. P., Nonlinear vibrations of simply supported, circular cylindrical shells, coupled to quiescent fluid, J. Fluids Struct., 12, 7, 883-918 (1998)
[20] Amabili, M.; Reddy, J. N., A new non-linear higher-order shear deformation theory for large-amplitude vibrations of laminated doubly curved shells, Int. J. Non-Linear Mech., 45, 4, 409-418 (2010)
[21] Kurylov, Y.; Amabili, M., Polynomial versus trigonometric expansions for nonlinear vibrations of circular cylindrical shells with different boundary conditions, J. Sound Vibr., 329, 9, 1435-1449 (2010)
[22] Pilgun, G.; Amabili, M., Non-linear vibrations of shallow circular cylindrical panels with complex geometry. Meshless discretization with the R-functions method, Int. J. Non-Linear Mech., 47, 3, 137-152 (2012)
[23] Selmane, A.; Lakis, A. A., Influence of geometric non-linearities on the free vibrations of orthotropic open cylindrical shells, Int. J. Numer. MethodsEng., 40, 6, 1115-1137 (1997) · Zbl 0892.73029
[24] Van Khang, N., Kronecker product and a new matrix form of Lagrangian equations with multipliers for constrained multibody systems, Mech. Res. Commun., 38, 4, 294-299 (2011) · Zbl 1272.70041
[25] Zienkiewicz, O. C.; Taylor, R. L., The finite element method for solid and structural mechanics (2005), Butterworth-Heinemann · Zbl 1084.74001
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