Shifted Inverse Power Method for Computing the Smallest M-Eigenvalue of a Fourth-Order Partially Symmetric Tensor

The strong ellipticity condition (abbr. SE-condition) of the displacement equations of equilibrium for general nonlinearly elastic materials plays an important role in nonlinear elasticity and materials. Qi et al. (Front Math China 4(2):349–364, 2009) pointed out that the SE-condition of the displacement equations of equilibrium can be equivalently transformed into the SE-condition of a fourth-order real partially symmetric tensor \({\mathcal {A}}\) $\mathcal{A}$ , and that the SE-condition of \({\mathcal {A}}\) $\mathcal{A}$ holds if and only if the smallest M-eigenvalue of \({\mathcal {A}}\) $\mathcal{A}$ is positive. In order to judge the strong ellipticity of \({\mathcal {A}}\) $\mathcal{A}$ , we propose a shifted inverse power method for computing the smallest M-eigenvalue of \({\mathcal {A}}\) $\mathcal{A}$ and give its convergence analysis. And then, we borrow and fine-tune an existing initialization strategy to make the sequence generated by the shifted inverse power method rapidly converge to a good approximation of the smallest M-eigenvalue of \({\mathcal {A}}\) $\mathcal{A}$ . Finally, we by numerical examples illustrate the effectiveness of the proposed method in computing the smallest M-eigenvalue of \({\mathcal {A}}\) $\mathcal{A}$ and judging the SE-condition of the displacement equations of equilibrium.