Abstract
Having the application in structural health monitoring in mind, we propose reduced port spaces that exhibit an exponential convergence for static condensation procedures on structures with changing geometries for instance induced by newly detected defects. Those reduced port spaces generalize the port spaces introduced in [K. Smetana and A.T. Patera, SIAM J. Sci. Comput., 2016] to geometry changes and are optimal in the sense that they minimize the approximation error among all port spaces of the same dimension. Moreover, we show numerically that we can reuse port spaces that are constructed on a certain geometry also for the static condensation approximation on a significantly different geometry, making the optimal port spaces well suited for use in structural health monitoring.
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- 1.
Note, that this may restrict the applicability of the spectral greedy to training sets \(\varXi _{PDE}\) of moderate cardinality.
- 2.
We note that in order to prove (22) it is necessary to define the lifting inner product on the ports for one reference parameter \(\bar{\varvec{\mu }} \in \mathscr {P}_{Geo}\) and use the equivalence of the norm induced by the lifting inner product and the \(H^{1/2}\)-norm on the ports. Exploiting that the latter is the same for all considered geometries allows switching between the geometries in the proof.
- 3.
As indicated above it is necessary to slightly modify the spectral greedy algorithm to prove convergence.
- 4.
Note that the values of the random Dirichlet boundary conditions do not belong to the parameter set.
References
Babuška, I., Huang, X., Lipton, R.: Machine computation using the exponentially convergent multiscale spectral generalized finite element method. ESAIM Math. Model. Numer. Anal. 48(2), 493–515 (2014)
Babuška, I., Lipton, R.: Optimal local approximation spaces for generalized finite element methods with application to multiscale problems. Multiscale Model. Simul. 9(1), 373–406 (2011)
Bampton, M., Craig, R.: Coupling of substructures for dynamic analyses. AIAA J. 6(7), 1313–1319 (1968)
Bourquin, F.: Component mode synthesis and eigenvalues of second order operators: discretization and algorithm. RAIRO Modél. Math. Anal. Numér. 26(3), 385–423 (1992)
Buhr, A., Engwer, C., Ohlberger, M., Rave, S.: ArbiLoMod, a simulation technique designed for arbitrary local modifications. SIAM J. Sci. Comput. 39(4), A1435–A1465 (2017)
Buhr, A., Smetana, K.: Randomized local model order reduction. SIAM J. Sci. Comput. 40(4), A2120–A2151 (2018)
Eftang, J.L., Patera, A.T.: Port reduction in parametrized component static condensation: approximation and a posteriori error estimation. Int. J. Numer. Meth. Eng. 96(5), 269–302 (2013)
Eigen: a C++ linear algebra library: http://eigen.tuxfamily.org/
Fehr, J., Holzwarth, P., Eberhard, P.: Interface and model reduction for efficient explicit simulations—a case study with nonlinear vehicle crash models. Math. Comput. Model. Dyn. Syst. 22(4), 380–396 (2016)
Hetmaniuk, U., Lehoucq, R.B.: A special finite element method based on component mode synthesis. ESAIM Math. Model. Numer. Anal. 44(3), 401–420 (2010)
Hughes, T.J.R., Engel, G., Mazzei, L., Larson, M.G.: The continuous Galerkin method is locally conservative. J. Comput. Phys. 163(2), 467–488 (2000)
Hurty, W.C.: Dynamic analysis of structural systems using component modes. AIAA J. 3(4), 678–685 (1965)
Huynh, D.B.P., Knezevic, D.J., Patera, A.T.: A static condensation reduced basis element method: approximation and a posteriori error estimation. ESAIM Math. Model. Numer. Anal. 47(1), 213–251 (2013)
Iapichino, L., Quarteroni, A., Rozza, G.: Reduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries. Comput. Math. Appl. 71(1), 408–430 (2016)
Jakobsson, H., Bengzon, F., Larson, M.G.: Adaptive component mode synthesis in linear elasticity. Int. J. Numer. Meth. Eng. 86(7), 829–844 (2011)
Kirk, B.S., Peterson, J.W., Stogner, R.H., Carey, G.F.: libMesh: a C++ library for parallel adaptive mesh refinement/coarsening simulations. Eng. Comput. 22(3–4), 237–254 (2006)
Knezevic, D.J., Kang, H., Sharma, P., Malinowski, G., Nguyen, T.T., et al.: Structural integrity management of offshore structures via RB-FEA and fast full load mapping based digital twins. In: The 28th International Ocean and Polar Engineering Conference. International Society of Offshore and Polar Engineers (2018)
Knezevic, D.J., Peterson, J.W.: A high-performance parallel implementation of the certified reduced basis method. Comput. Meth. Appl. Mech. Eng. 200(13–16), 1455–1466 (2011)
Kolmogoroff, A.: Über die beste Annäherung von Funktionen einer gegebenen Funktionenklasse. Ann Math. (2) 37(1), 107–110 (1936)
Martini, I., Rozza, G., Haasdonk, B.: Reduced basis approximation and a-posteriori error estimation for the coupled Stokes-Darcy system. Adv. Comput. Math. 41(5), 1131–1157 (2015)
Pinkus, A.: \(n\)-widths in Approximation Theory, vol. 7. Springer-Verlag, Berlin (1985)
Quarteroni, A., Valli, A.: Domain decomposition methods for partial differential equations. In: Numerical Mathematics and Scientific Computation. The Clarendon Press, Oxford University Press, New York, reprint (2005)
Smetana, K.: A new certification framework for the port reduced static condensation reduced basis element method. Comput. Meth. Appl. Mech. Eng. 283, 352–383 (2015)
Smetana, K., Patera, A.T.: Optimal local approximation spaces for component-based static condensation procedures. SIAM J. Sci. Comput. 38(5), A3318–A3356 (2016)
Smetana, K., Patera, A.T.: Fully localized a posteriori error estimation for the port reduced static condensation reduced basis element method (2018+). In preparation
Taddei, T., Patera, A.T.: A localization strategy for data assimilation; application to state estimation and parameter estimation. SIAM J. Sci. Comput. 40(2), B611–B636 (2018)
Veroy, K., Prud’homme, C., Rovas, D.V., Patera, A.T.: A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In: Proceedings of the 16th AIAA Computational Fluid Dynamics Conference, vol. 3847 (2003)
Acknowledgements
I would like to thank Prof. Dr. Anthony Patera for many fruitful discussions and comments on the content of this paper. This work was supported by OSD/AFOSR/MURI Grant FA9550-09-1-0613 and ONR Grant N00014-11-1-0713.
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Smetana, K. (2020). Static Condensation Optimal Port/Interface Reduction and Error Estimation for Structural Health Monitoring. In: Fehr, J., Haasdonk, B. (eds) IUTAM Symposium on Model Order Reduction of Coupled Systems, Stuttgart, Germany, May 22–25, 2018. IUTAM Bookseries, vol 36. Springer, Cham. https://doi.org/10.1007/978-3-030-21013-7_1
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