Invariant variational-difference schemes and conservation laws and conservation laws
USSR Computational Mathematics and Mathematical Physics
Equivalent definitions of conservative finite-difference schemes
USSR Computational Mathematics and Mathematical Physics
On the one-dimensional current driven semiconductor equations
SIAM Journal on Applied Mathematics
Computational Mathematics and Mathematical Physics
The convergence of a difference scheme for the two-dimensional equations of gas dynamics
Computational Mathematics and Mathematical Physics
The completely conservative difference schemes for the nonlinear Landau—Fokker—Planck equation
Journal of Computational and Applied Mathematics - Special issue on applied and computational topics in partial differential equations
Journal of Computational Physics
Toward the ultimate conservative scheme: following the quest
Journal of Computational Physics
Noether-type theorems for difference equations
Applied Numerical Mathematics
Journal of Computational Physics
New Locking-Free Mixed Method for the Reissner--Mindlin Thin Plate Model
SIAM Journal on Numerical Analysis
Finite element approach to modelling evolution of 3D shape memory materials
Mathematics and Computers in Simulation
Simulation of phase combinations in shape memory alloys patches by hybrid optimization methods
Applied Numerical Mathematics
Coupling control and human factors in mathematical models of complex systems
Engineering Applications of Artificial Intelligence
MATH'09 Proceedings of the 14th WSEAS International Conference on Applied mathematics
NANOTECHNOLOGY'10 Proceedings of the 2nd WSEAS international conference on Nanotechnology
A dynamic model for phase transformations in 3d samples of shape memory alloys
ICCS'05 Proceedings of the 5th international conference on Computational Science - Volume Part III
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In this paper, we consider a strongly coupled model of nonlinear thermoelasticity describing the dynamics of materials with shape memory. The model is not amenable to analytical treatments and the development, analysis, and applications of effective numerical approximations for this model is in the focus of the present paper. In particular, we discuss a recently proposed fully conservative difference scheme for the solution of the problem. We note that a standard energy inequality technique, applied to the analysis of convergence properties of the scheme, would lead to restrictive assumptions on the grid size and/or excessive smoothness assumptions on the unknown solution. We show how such assumptions can be removed to achieve unconditional convergence of the proposed scheme. Next, we apply the proposed scheme to the analysis of behaviour of a shape memory alloy rod. We demonstrate that the proposed approximation can describe a complete range of behaviour of the shape memory material, including quasiplastic, pseudoelastic, and almost elastic regimes. We discuss the influence of nonlinear effects in each of these regimes focusing on hysteresis effects.