Abstract
The ‘Schur after MOR’ method has proved successful in obtaining stable reduced piezoelectric device models. Even though the method is already used in industry, the stability preservation of ‘Schur after MOR’ is still mathematically unproven. In this work, we show that the involved quasi-Schur transformation indeed does efficiently re-stabilize the aforementioned reduced piezoelectric energy harvester models. The transformation is only quasi-Schur as the unstable reduced systems require eigenspace projection and approximation to become Schur-transformable. During the transformation, the negative eigenvalues are eliminated from the reduced stiffness matrix and the system is stabilized. Further, we compare ‘Schur after MOR’ to another recently presented stabilization method: ‘MOR after Implicit Schur’. We show that the computational effort is significantly reduced.
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Notes
- 1.
\(\mathbf {\widetilde {K}_{r,(1:I,1:I)}}\) is the submatrix consisting of the first I rows and columns of \(\mathbf {\widetilde {K}_{r}}\), and \(\mathbf {\widetilde {K}_{r,(I:p,I:p)}}\) consists of the the rows and columns I to p of \(\mathbf {\widetilde {K}_{r}}\).
- 2.
Expansion point for MOR is set to s 0 = 0 as obtaining the smallest possible accurate reduced model was not the main concern of this experiment. We rather want to verify the equivalence of considered subspaces. With the optimal choice of expansion points, one can obtain smaller models with acceptable accuracy.
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Hu, S., Yuan, C., Bechtold, T. (2020). Quasi-Schur Transformation for the Stable Compact Modeling of Piezoelectric Energy Harvester Devices. In: Nicosia, G., Romano, V. (eds) Scientific Computing in Electrical Engineering. SCEE 2018. Mathematics in Industry(), vol 32. Springer, Cham. https://doi.org/10.1007/978-3-030-44101-2_25
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DOI: https://doi.org/10.1007/978-3-030-44101-2_25
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