Piezoelectric Tensor Eigenvalues Include Set And Maximum Eigenvalue Calculations

The C-eigenvalue of the piezoelectric tensor plays an important role in the piezo-electric effect and inverse piezoelectric effect.The maximum C-eigenvalue and the corresponding C-eigenvector can be obtained by solving the optimal rank-one approximation of the piezoelectric type tensor.The maximum C-eigenvalue of the piezoelectric tensor determines the coupling parameters in the piezoelectric effect.Firstly,a new approximate alternating minimum algorithm is proposed to calculate the maximum C-eigenvalue of the piezoelectric tensor,and the global convergence of the algorithm is analyzed.In general,it is a NP-hard problem to find all eigenvalues of a piezoelectric tensor.Therefore,it is meaningful if one can find an inclusion interval including all potential eigenvalues.A new inclu-sion set of eigenvalue is established in this paper.Compared with the existing inclusion sets,it is proved that the new interval is more compact.Finally,the validity of the proposed algorithm and the inclusion interval is verified by using the piezoelectric tensor derived from the actual problems,The main contents of this paper include the following two aspects:Firstly,the equivalence conditions between the maximum C-eigenvalue opti-mization model and the multi-linear optimization problem are established.Then,a kind of proximal alternating minimum algorithm is proposed,and it is proved that the generated objective value is monotonically decreasing.Finally,the glob-al convergence of the algorithm is established and the validity and accuracy of the algorithm are verified by numerical experiments.By the entries of the given tensor,a new eigenvalue inclusion interval is proposed.Compared with the existing results,it is proved that the proposed eigenvalue inclusion set is more compact.