The underlying mechanism responsible for the technologically important martensitic transformations found in shape memory alloys is the change in stability of the alloy's crystal lattice. This change in stability results in a transformation between an austenite (high symmetry) phase and a martensite (low symmetry) phase as a critical transition temperature is crossed.
Therefore, when studying these materials, it is critically important to determine the stability of the crystal structure. A tempting approach, based on continuum mechanics, is to consider the crystal stable if it is an energy minimizer with respect to uniform deformations. This is, in fact, inappropriate since the crystal's discrete lattice spacing dictates that its stability properties depend on the wavelength of the perturbation considered. It is imperative, then, to investigate all modes of perturbation accessible to the crystal.
The direct approach for conducting this type of stability calculation results in a numerical algorithm with a time complexity of order N cubed, where N is the number of different perturbations investigated. In the problems of interest, the use of a cubic algorithm results in prohibitively long computation times. Fortunately, the inherent translational symmetry of the crystal structure may be used to our advantage. Indeed, the application of the Fourier transform results in a block-diagonalized system of equations that are solvable in linear time.
The derivation of this fast algorithm will be summarized, and a particular numerical example will be presented that illustrates the importance of considering perturbations of all wavelengths when investigating the stability of crystal structures.
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