Abstract
One of the major challenges of coupled problems is to manage nonconforming meshes at the interface between two models and/or domains, due to different numerical schemes or domain discretizations employed. Moreover, very often complex submodels depend on (e.g., physical or geometrical) parameters, thus making the repeated solutions of the coupled problem through high-fidelity, full-order models extremely expensive, if not unaffordable. In this paper, we propose a reduced order modeling (ROM) strategy to tackle parametrized one-way coupled problems made by a first, master model and a second, slave model; this latter depends on the former through Dirichlet interface conditions. We combine a reduced basis method, applied to each subproblem, with the discrete empirical interpolation method to efficiently interpolate or project Dirichlet data across either conforming or non-conforming meshes at the domains interface, building a low-dimensional representation of the overall coupled problem. The proposed technique is numerically verified by considering a series of test cases involving both steady and unsteady problems, after deriving a posteriori error estimates on the solution of the coupled problem in both cases. This work arises from the need to solve staggered cardiac electrophysiological models and represents the first step towards the setting of ROM techniques for the more general two-way Dirichlet-Neumann coupled problems solved with domain decomposition sub-structuring methods, when interface non-conformity is involved.
References
Bazilevs, Y., Takizawa, K., Tezduyar, T.: Computational Fluid-Structure Interaction: Methods and Applications. https://doi.org/10.1002/9781118483565 (2013)
Discacciati, M., Quarteroni, A.: Navier-Stokes/Darcy coupling: modeling, analysis, and numerical approximation. Revista Matemática Complutense 22. https://doi.org/10.5209/rev_REMA.2009.v22.n2.16263 (2009)
Korvink, J., Paul, O.: MEMS: A Practical Guide to Design, Analysis and Applications. https://doi.org/10.1007/978-3-540-33655-6 (2005)
Piersanti, R., Regazzoni, F., Salvador, M., Corno, A., Dede’, L., Vergara, C., Quarteroni, A.: 3D-0D closed-loop model for the simulation of cardiac biventricular electromechanics. arXiv:2108.01907 (2021)
Quarteroni, A., Dede, L., Manzoni, A., Vergara, C.: Mathematical Modelling of the Human Cardiovascular System: Data, Numerical Approximation, Clinical Applications. https://doi.org/10.1017/9781108616096 (2019)
Wong, J., Göktepe, S., Kuhl, E.: Computational modeling of chemo-electro-mechanical coupling: a novel implicit monolithic finite element approach. International Journal for Numerical Methods in Biomedical Engineering 29. https://doi.org/10.1002/cnm.2565 (2013)
Zhao, Y., Su, X.: Computational Fluid-Structure Interaction: Methods, Models, and Applications. Academic Press, New York (2018)
Bonomi, D., Manzoni, A., Quarteroni, A.: A matrix DEIM technique for model reduction of nonlinear parametrized problems in cardiac mechanics. Comput. Methods Appl. Mech. Eng. 324. https://doi.org/10.1016/j.cma.2017.06.011 (2017)
Forti, D., Rozza, G.: Efficient geometrical parametrisation techniques of interfaces for reduced-order modelling: application to fluid-structure interaction coupling problems. International Journal of Computational Fluid Dynamics 28 (3-4), 158–169 (2014). https://doi.org/10.1080/10618562.2014.932352
Fresca, S., Manzoni, A., Dede, L., Quarteroni, A.: POD-enhanced deep learning-based reduced order models for the real-time simulation of cardiac electrophysiology in the left atrium. Front. Physiol. 12. https://doi.org/10.3389/fphys.2021.679076 (2021)
Geneser, S., Kirby, R., MacLeod, R.: Application of stochastic finite element methods to study the sensitivity of ECG forward modeling to organ conductivity. IEEE Transaction on Biomedical Engineering 55(1), 31–40 (2008). https://doi.org/10.1109/TBME.2007.900563
Pacciarini, P., Rozza, G.: Reduced basis approximation of parametrized advection-diffusion PDEs with High Péclet Number. In: Numerical Mathematics and Advanced Applications-ENUMATH 2013, pp 419–426. Springer, Lausanne (2015)
Pagani, S., Manzoni, A., Quarteroni, A.: Numerical approximation of parametrized problems in cardiac electrophysiology by a local reduced basis method. Comput. Methods Appl. Mech. Eng. 340. https://doi.org/10.1016/j.cma.2018.06.003(2018)
Swenson, D., Geneser, S., Stinstra, J., Kirby, R., MacLeod, R.: Cardiac position sensitivity study in the electrocardiographic forward problem using stochastic collocation and boundary element methods. Ann. Biomed. Eng. 39, 2900 (2011). https://doi.org/10.1007/s10439-011-0391-5
Bernardi, C., Maday, Y., Rapetti, F.: Basics and some applications of the mortar element method. GAMM-Mitteilungen 28. https://doi.org/10.1002/gamm.201490020 (2005)
Chan, T., Smith, B., Zou, J.: Overlapping schwarz methods on unstructured meshes usingnon-matching coarse grids. Numer. Math. 73, 149–167 (1996). https://doi.org/10.1007/s002110050189
Deparis, S., Forti, D., Gervasio, P., Quarteroni, A.: INTERNODES: an accurate interpolation-based method for coupling the Galerkin solutions of PDEs on subdomains featuring non-conforming interfaces. Computers & Fluids 141. https://doi.org/10.1016/j.compfluid.2016.03.033 (2016)
Gervasio, P., Quarteroni, A.: The INTERNODES method for non-conforming discretizations of PDEs. Communications on Applied Mathematics and Computation 1, 361–401 (2019). https://doi.org/10.1007/s42967-019-00020-1
Hesch, C., Gil, A., Arranz Carreño, A., Bonet, J., Betsch, P.: A mortar approach for fluid-structure interaction problems: immersed strategies for deformable and rigid bodies. Comput. Methods Appl. Mech. Eng. 278. https://doi.org/10.1016/j.cma.2014.06.004 (2014)
Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (1999)
Ripepi, M., Verveld, M., Karcher, N., Franz, T., Abu-Zurayk, M., Görtz, S., Kier, T.: Reduced-order models for aerodynamic applications, loads and MDO. CEAS Aeronaut. J. 9. https://doi.org/10.1007/s13272-018-0283-6 (2018)
Amsallem, D., Cortial, J., Farhat, C.: Toward real-time computational-fluid-dynamics-based aeroelastic computations using a database of reduced-order information. AIAA J. 48, 2029–2037 (2010)
Ballarin, F., Rozza, G., Maday, Y.: Reduced-order semi-implicit schemes for fluid-structure interaction problems. In: Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K. (eds.) Model Reduction of Parametrized Systems, pp 149–167. Springer, Cham (2017)
Lassila, T., Quarteroni, A., Rozza, G.: A reduced basis model with parametric coupling for fluid-structure interaction problems. SIAM Journal on Scientific Computing 34(2). https://doi.org/10.1137/110819950 (2012)
Lassila, T., Manzoni, A., Quarteroni, A., Rozza, G.: A reduced computational and geometrical framework for inverse problems in hemodynamics. International Journal for Numerical Methods in Biomedical Engineering 29(7), 741–776 (2013). https://doi.org/10.1002/cnm.2559
Ballarin, F., Rozza, G.: POD-Galerkin monolithic reduced order models for parametrized fluid-structure interaction problems: POD-galerkin monolithic ROM for parametrized FSI problems. Int. J. Numer. Methods Fluids 82. https://doi.org/10.1002/fld.4252 (2016)
Benner, P., Ohlberger, M., Cohen, A., Willcox, K.: Model Reduction and Approximation: Theory and Algorithms Society for Industrial and Applied Mathematics, Philadelphia, PA. https://doi.org/10.1137/1.9781611974829 (2017)
Hesthaven, J., Rozza, G., Stamm, B.: Certified Reduced Basis Methods for Parametrized Partial Differential Equations. https://doi.org/10.1007/978-3-319-22470-1 (2016)
Quarteroni, A., Manzoni, A., Negri, F.: Reduced Basis Methods for Partial Differential Equations. An Introduction. Springer, Cham (2016)
Løvgren, A.E., Maday, Y., Rønquist, E.M.: A reduced basis element method for the steady Stokes problem. ESAIM: Mathematical Modelling and Numerical Analysis 40(3), 529–552 (2006). https://doi.org/10.1051/m2an:2006021
Iapichino, L., Quarteroni, A., Rozza, G.: A reduced basis hybrid method for the coupling of parametrized domains represented by fluidic networks. Comput. Methods Appl. Mech. Eng. 221–222, 63–82 (2012). https://doi.org/10.1016/j.cma.2012.02.005
Iapichino, L., Quarteroni, A., Rozza, G.: Reduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries. Computers & Mathematics with Applications 71(1), 408–430 (2016). https://doi.org/10.1016/j.camwa.2015.12.001
Pegolotti, L., Pfaller, M.R., Marsden, A.L., Deparis, S.: Model order reduction of flow based on a modular geometrical approximation of blood vessels. Comput. Methods Appl. Mech. Eng. 380, 113762 (2021). https://doi.org/10.1016/j.cma.2021.113762
Dal Santo, N., Deparis, S., Manzoni, A., Quarteroni, A.: Multi space reduced basis preconditioners for large-scale parametrized PDEs. SIAM J. Sci. Comput. 40(2), 954–983 (2018)
Eftang, J., Patera, A.: A port-reduced static condensation reduced basis element method for large component-synthesized structures: approximation and a posteriori error estimation. Advanced Modeling and Simulation in Engineering Sciences 1, 3 (2014). https://doi.org/10.1186/2213-7467-1-3
P, H., Bao, D., Knezevic, D.J., Patera, A.T.: A static condensation reduced basis element method : approximation and a posteriori error estimation. ESAIM: Mathematical Modelling and Numerical Analysis 47(1), 213–251 (2013). https://doi.org/10.1051/m2an/2012022
Barrault, M., Maday, Y., Nguyen, N.C., Patera, A.T.: An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. Acad. Sci. Paris 339(9), 667–672 (2004)
Chaturantabut, S., Sorensen, D.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010). https://doi.org/10.1137/090766498
Grepl, M., Maday, Y., Nguyen, N., Patera, A.: Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations/ ESAIM: Mathematical Modelling and Numerical Analysis 41. https://doi.org/10.1051/m2an:2007031 (2007)
Maday, Y., Nguyen, N., Patera, A., Pau, G.S.H.: A general multipurpose interpolation procedure: The magic points. Communications on Pure and Applied Analysis 8. https://doi.org/10.3934/cpaa.2009.8.383 (2008)
Negri, F., Manzoni, A., Amsallem, D.: Efficient model reduction of parametrized systems by matrix discrete empirical interpolation. J. Comput. Phys. 303, 431–454 (2015). https://doi.org/10.1016/j.jcp.2015.09.046
Boulakia, M., Cazeau, S., Fernández, M., Gerbeau, J.F., Zemzemi, N.: Mathematical modeling of electrocardiograms: a numerical study. Ann. Biomed. Eng. 38, 1071–97 (2010). https://doi.org/10.1007/s10439-009-9873-0
Bjørstad, P.E., Brenner, S.C., Halpern, L., Kim, H.H., Kornhuber, R., Rahman, T., Widlund, O.B.: Domain Decomposition Methods in Science and Engineering XXIV. Lecture Notes in Computational Science and Engineering. Springer, Cham (2018)
Bernardi, C., Maday, Y., Patera, A.T.: A new non conforming approach to domain decomposition: the mortar element method. A New Non Conforming Approach to Domain Decomposition: The Mortar Element Method 13–51 (1994)
Gervasio, P., Quarteroni, A.: INTERNODES for heterogeneous couplings. In: Bjørstad, P.E., Brenner, S.C., Halpern, L., Kim, H.H., Kornhuber, R., Rahman, T., Widlund, O.B. (eds.) Domain Decomposition Methods in Science and Engineering XXIV, pp 59–71. Springer, Cham (2018)
Gervasio, P., Quarteroni, A.: Analysis of the INTERNODES method for non-conforming discretizations of elliptic equations. Comput. Methods Appl. Mech. Eng. 334. https://doi.org/10.1016/j.cma.2018.02.004 (2018)
Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics. Springer, Berlin (2008). https://doi.org/10.1007/978-3-540-85268-1
Schroeder, W., Martin, K., Lorensen, B.: The Visualization Toolkit (4th Ed.) (2006)
Deparis, S., Forti, D., Quarteroni, A.: A rescaled localized radial basis function interpolation on non-cartesian and nonconforming grids. SIAM J. Sci. Comput. 36 (2014)
Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Classic in applied mathematics. SIAM, Philadelphia (1995)
Kreiss, O., Ortiz, O.E.: Introduction to Numerical Methods for Time Dependent Differential Equations. Wiley, Hoboken (2014)
Mckay, M., Beckman, R., Conover, W.: A comparison of three methods for selecting vales of input variables in the analysis of output from a computer code. Technometrics 21, 239–245 (1979). https://doi.org/10.1080/00401706.1979.10489755
Iman, R., Helton, J.: An investigation of uncertainty and sensitivity analysis techniques for computer-models. Risk Anal. 8, 71–90 (2006). https://doi.org/10.1111/j.1539-6924.1988.tb01155.x
Farhat, C., Grimberg, S., Manzoni, A., Quarteroni, A.: Algorithms computational bottlenecks for PROMs: precomputation and hyperreduction. In: Benner, P., Grivet-Talocia, S., Quarteroni, A., Rozza, G., Schilders, W.H.A., Silveira, L.M. (eds.) Snapshot-Based Methods and Algorithms, pp 181–244. De Gruyter, Berlin (2020)
Haasdonk, B., Ohlberger, M.: Efficient reduced models for parametrized dynamical systems by offline/online decomposition. Math. Comput. Model. Dyn. Syst. 17(2), 145–161 (2009). https://doi.org/10.1080/13873954.2010.514703
Wirtz, D., Sorensen, D.C., Haasdonk, B.: A posteriori error estimation for DEIM reduced nonlinear dynamical systems. SIAM J. Sci. Comput. 36. https://doi.org/10.1137/120899042 (2012)
Africa, P.C., Piersanti, R., Fedele, M., Dede, L., Quarteroni, A.: Lifex–heart module: a high-performance simulator for the cardiac function Package 1: Fiber generation. arXiv:2201.03303 (2022)
Arndt, D., Bangerth, W., Blais, B., Fehling, M., Gassmöller, R., Heister, T., Heltai, L., Köcher, U., Kronbichler, M., Maier, M., Munch, P., Pelteret, J.P., Proell, S., Simon, K., Turcksin, B., Wells, D., Zhang, J.: The deal.II library, version 9.3. Journal of Numerical Mathematics (2021)
Quarteroni, A., Veneziani, A., Zunino, P.: Mathematical and numerical modeling of solute dynamics in blood flow and arterial walls. SIAM J. Numer. Anal. 39, 1488–1511 (2002). https://doi.org/10.1137/S0036142900369714
Karner, G., Perktold, K., Zehentner, H.P.: Computational modeling of macromolecule transport in the arterial wall. Comput. Methods Biomech. Biomed. Engin. 4(6), 491–504 (2001). https://doi.org/10.1080/10255840108908022
Funding
Open access funding provided by Politecnico di Milano within the CRUI-CARE Agreement.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no competing interests.
Additional information
Communicated by: Olga Mula
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix : A: A posteriori error estimator for unsteady reduced basis models
Appendix : A: A posteriori error estimator for unsteady reduced basis models
To find an a posteriori error estimate for the ROM approximation in a time-dependent case, according to [55], we can start by considering the following parametrized linear dynamical system for a vector \(\mathbf {u}(t;\boldsymbol {\mu }) \in \mathbb {R}^{n}\):
Here the matrix \(\mathbb {A}_{N}(t;\boldsymbol {\mu }) \in \mathbb {R}^{N \times N}\) and the vector \(\mathbf {f}_{N}(t;\boldsymbol {\mu }) \in \mathbb {R}^{N}\), where N denotes the dimension of the reference FOM space, are μ-dependent. Moreover, we define the projection matrix \(\mathbb {V} \in \mathbb {R}^{N\times n}\) defined through RB methods, where n ≤ N is the ROM dimension. Then, the reduced dynamical system is:
where \(\mathbb {A}_{n}(t;\boldsymbol {\mu }) = \mathbb {V}^{T}\mathbb {A}_{N}(t;\boldsymbol {\mu })\mathbb {V}\), \(\mathbf {f}_{n}(t;\boldsymbol {\mu }) = \mathbb {V}^{T}\mathbf {f}_{N}(t;\boldsymbol {\mu }) \mathbb {V}\), un(t; μ) is the reduced approximation, i.e., \(\mathbf {u}_{N}(t;\boldsymbol {\mu }) \approx \mathbb {V}\mathbf {u}_{n}(t;\boldsymbol {\mu })\), and un,0(μ) is the projection of u0(μ) onto the reduced space.
Let us now denote the error and the residual as
respectively; given a symmetric positive definite matrix \(\mathbf {G} \in \mathbb {R}^{N \times N}\), let us denote by 〈⋅,⋅〉G the induced inner product, and the induced norm as \(\| \mathbf {u}\|_{\mathbf {G}} := \sqrt {\langle \mathbf {u},\mathbf {u}\rangle _{\mathbf {G}}}\) on RN. Similarly, \(\| \mathbb {A}\|_{\mathbf {G}} := \textup {sup}_{\| \mathbf {u}\|_{\mathbf {G}}}\| \mathbb {A}\mathbf {u}\|_{\mathbf {G}}\), for \(\mathbb {A} \in \mathbb {R}^{N\times N}\). For example, if \(\mathbf {G} = \mathbb {I}_{N \times N}\), i.e., it is the identity matrix, than we obtain the simple 2-norm used in this work. Then, the following a posteriori error estimate can be stated:
Proposition 1 (A posteriori error estimate)
Assuming that \(\mathbb {A}_{N} (t;\boldsymbol {\mu }) = \mathbb {A}_{N}(\boldsymbol {\mu })\) is time-invariant and has eigenvalues with negative real part for all \(\boldsymbol {\mu } \in {\mathscr{P}}\), than the solution is bounded by
where C1(μ) is a computable constant. Then, the following error estimates holds:
Proof
From the residual definition, we obtain that
Subtracting this equation from the original system, we get the evolution system
for the error, that admits the explicit solution
The thesis follows thanks to the assumption \(\| \exp (\mathbb {A}_{N}(\boldsymbol {\mu })s)\|_{\mathbf {G}} \leq C_{1}(\boldsymbol {\mu })\) for \(s \in \mathbb {R}^{+}\). â–¡
Error relations similar to (A2) can also be found for time dependent systems, meaning when \(\mathbb {A}_{N}(t;\boldsymbol {\mu })\) depends on time, by a suitable modification of C1(μ). To do this, we first point out that the error evolution system (A3) holds also for time-variants systems. Then, integrating, we get
Denoting by Φ(t) := ∥e(t; μ)∥G, \(\boldsymbol {\alpha }(t):= \| \mathbf {e}(0;\boldsymbol {\mu })\|_{\mathbf {G}} + {{\int \limits }_{0}^{T}} \| \mathbf {r}(\tau ;\boldsymbol {\mu })\|_{\mathbf {G}}d\tau \) and \(\boldsymbol {\beta }(t) := \| \mathbb {A}_{N}(\tau ;\boldsymbol {\mu })\|_{\mathbf {G}}\), we can obtain
Moreover, assuming an upper bound \(\| \mathbb {A}_{N}(t;\boldsymbol {\mu })\|_{\mathbf {G}} \leq C_{3}(\boldsymbol {\mu })\) for t ∈ [0,T], \(\boldsymbol {\mu } \in {\mathscr{P}}\), using the Gronwall inequality, we can write
Then, equation (A2) can be found denoting by \(C_{1} := 1 + C_{3}(\boldsymbol {\mu })T \exp (C_{3} T)\).
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Zappon, E., Manzoni, A. & Quarteroni, A. Efficient and certified solution of parametrized one-way coupled problems through DEIM-based data projection across non-conforming interfaces. Adv Comput Math 49, 21 (2023). https://doi.org/10.1007/s10444-022-10008-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10444-022-10008-w
Keywords
- Coupled problems
- Reduced order models
- Proper orthogonal decomposition
- Discrete empirical interpolation
- Interface non-conformity
- A posteriori error estimates
Mathematics Subject Classification (2010)
- 65M60
- 65N99
- 68U99