@article {2022, title = {Driving bifurcating parametrized nonlinear PDEs by optimal control strategies: application to Navier{\textendash}Stokes equations with model order reduction}, journal = {ESAIM: M2AN}, volume = {56}, year = {2022}, month = {2022///}, pages = {1361 - 1400}, url = {https://doi.org/10.1051/m2an/2022044}, author = {Federico Pichi and Maria Strazzullo and F. Ballarin and Gianluigi Rozza} } @article {2022, title = {Model order reduction for bifurcating phenomena in fluid-structure interaction problems}, journal = {International Journal for Numerical Methods in FluidsInternational Journal for Numerical Methods in FluidsInt J Numer Meth Fluids}, volume = {n/a}, year = {2022}, month = {2022/05/23}, abstract = {

Abstract This work explores the development and the analysis of an efficient reduced order model for the study of a bifurcating phenomenon, known as the Coand? effect, in a multi-physics setting involving fluid and solid media. Taking into consideration a fluid-structure interaction problem, we aim at generalizing previous works towards a more reliable description of the physics involved. In particular, we provide several insights on how the introduction of an elastic structure influences the bifurcating behavior. We have addressed the computational burden by developing a reduced order branch-wise algorithm based on a monolithic proper orthogonal decomposition. We compared different constitutive relations for the solid, and we observed that a nonlinear hyper-elastic law delays the bifurcation w.r.t.\ the standard model, while the same effect is even magnified when considering linear elastic solid.

}, keywords = {Bifurcation theory, Coand{\u a} effect, continuum mechanics, fluid dynamics, monolithic method, parametrized fluid-structure interaction problem, Proper orthogonal decomposition, reduced order modeling}, isbn = {0271-2091}, url = {https://doi.org/10.1002/fld.5118}, author = {Moaad Khamlich and Federico Pichi and Gianluigi Rozza} } @unpublished {2021, title = {An artificial neural network approach to bifurcating phenomena in computational fluid dynamics}, year = {2021}, author = {Federico Pichi and Francesco Ballarin and Gianluigi Rozza and Jan S Hesthaven} } @article {2021, title = {Efficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element method}, journal = {Advances in Computational Mathematics}, volume = {47}, year = {2021}, abstract = {

The majority of the most common physical phenomena can be described using partial differential equations (PDEs). However, they are very often characterized by strong nonlinearities. Such features lead to the coexistence of multiple solutions studied by the bifurcation theory. Unfortunately, in practical scenarios, one has to exploit numerical methods to compute the solutions of systems of PDEs, even if the classical techniques are usually able to compute only a single solution for any value of a parameter when more branches exist. In this work, we implemented an elaborated deflated continuation method that relies on the spectral element method (SEM) and on the reduced basis (RB) one to efficiently compute bifurcation diagrams with more parameters and more bifurcation points. The deflated continuation method can be obtained combining the classical continuation method and the deflation one: the former is used to entirely track each known branch of the diagram, while the latter is exploited to discover the new ones. Finally, when more than one parameter is considered, the efficiency of the computation is ensured by the fact that the diagrams can be computed during the online phase while, during the offline one, one only has to compute one-dimensional diagrams. In this work, after a more detailed description of the method, we will show the results that can be obtained using it to compute a bifurcation diagram associated with a problem governed by the Navier-Stokes equations.\ 

}, doi = {10.1007/s10444-020-09827-6}, author = {Moreno Pintore and Federico Pichi and Martin W. Hess and Gianluigi Rozza and Claudio Canuto} } @article {13850, title = {Efficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element method}, journal = {Advances in Computational Mathematics}, year = {2020}, abstract = {

The majority of the most common physical phenomena can be described using partial differential equations (PDEs). However, they are very often characterized by strong nonlinearities. Such features lead to the coexistence of multiple solutions studied by the bifurcation theory. Unfortunately, in practical scenarios, one has to exploit numerical methods to compute the solutions of systems of PDEs, even if the classical techniques are usually able to compute only a single solution for any value of a parameter when more branches exist. In this work we implemented an elaborated deflated continuation method, that relies on the spectral element method (SEM) and on the reduced basis (RB) one, to efficiently compute bifurcation diagrams with more parameters and more bifurcation points. The deflated continuation method can be obtained combining the classical continuation method and the deflation one: the former is used to entirely track each known branch of the diagram, while the latter is exploited to discover the new ones. Finally, when more than one parameter is considered, the efficiency of the computation is ensured by the fact that the diagrams can be computed during the online phase while, during the offline one, one only has to compute one-dimensional diagrams. In this work, after a more detailed description of the method, we will show the results that can be obtained using it to compute a bifurcation diagram associated with a problem governed by the Navier-Stokes equations.

}, url = {https://arxiv.org/abs/1912.06089}, author = {Moreno Pintore and Federico Pichi and Martin W. Hess and Gianluigi Rozza and Claudio Canuto} } @article {2020, title = {A reduced order modeling technique to study bifurcating phenomena: Application to the gross-pitaevskii equation}, journal = {SIAM Journal on Scientific Computing}, year = {2020}, abstract = {

We propose a computationally efficient framework to treat nonlinear partial differential equations having bifurcating solutions as one or more physical control parameters are varied. Our focus is on steady bifurcations. Plotting a bifurcation diagram entails computing multiple solutions of a parametrized, nonlinear problem, which can be extremely expensive in terms of computational time. In order to reduce these demanding computational costs, our approach combines a continuation technique and Newton{\textquoteright}s method with a reduced order modeling (ROM) technique, suitably supplemented with a hyperreduction method. To demonstrate the effectiveness of our ROM approach, we trace the steady solution branches of a nonlinear Schr{\"o}dinger equation, called the Gross{Pitaevskii equation, as one or two physical parameters are varied. In the two-parameter study, we show that our approach is 60 times faster in constructing a bifurcation diagram than a standard full order method.

}, doi = {10.1137/20M1313106}, url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85096768803\&doi=10.1137\%2f20M1313106\&partnerID=40\&md5=47d6012d10854c2f9a04b9737f870592}, author = {Federico Pichi and Annalisa Quaini and Gianluigi Rozza} } @article {2020, title = {A Reduced Order technique to study bifurcating phenomena: application to the Gross-Pitaevskii equation}, journal = {SIAM Journal on Scientific Computing}, year = {2020}, abstract = {

We propose a computationally efficient framework to treat nonlinear partial differential equations having bifurcating solutions as one or more physical control parameters are varied. Our focus is on steady bifurcations. Plotting a bifurcation diagram entails computing multiple solutions of a parametrized, nonlinear problem, which can be extremely expensive in terms of computational time. In order to reduce these demanding computational costs, our approach combines a continuation technique and Newton{\textquoteright}s method with a Reduced Order Modeling (ROM) technique, suitably supplemented with a hyper-reduction method. To demonstrate the effectiveness of our ROM approach, we trace the steady solution branches of a nonlinear Schr{\"o}dinger equation, called Gross-Pitaevskii equation, as one or two physical parameters are varied. In the two parameter study, we show that our approach is 60 times faster in constructing a bifurcation diagram than a standard Full Order Method.

}, doi = {https://doi.org/10.1137/20M1313106}, url = {https://arxiv.org/abs/1907.07082}, author = {Federico Pichi and Annalisa Quaini and Gianluigi Rozza} } @article {2019, title = {Reduced Basis Approaches for Parametrized Bifurcation Problems held by Non-linear Von K{\'a}rm{\'a}n Equations}, journal = {Journal of Scientific Computing}, volume = {81}, year = {2019}, pages = {112-135}, abstract = {

This work focuses on the computationally efficient detection of the buckling phenomena and bifurcation analysis of the parametric Von K{\'a}rm{\'a}n plate equations based on reduced order methods and spectral analysis. The computational complexity{\textemdash}due to the fourth order derivative terms, the non-linearity and the parameter dependence{\textemdash}provides an interesting benchmark to test the importance of the reduction strategies, during the construction of the bifurcation diagram by varying the parameter(s). To this end, together the state equations, we carry out also an analysis of the linearized eigenvalue problem, that allows us to better understand the physical behaviour near the bifurcation points, where we lose the uniqueness of solution. We test this automatic methodology also in the two parameter case, understanding the evolution of the first buckling mode.

}, doi = {10.1007/s10915-019-01003-3}, url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85068973907\&doi=10.1007\%2fs10915-019-01003-3\&partnerID=40\&md5=a09af83ce45183d6965cdb79d87a919b}, author = {Federico Pichi and Gianluigi Rozza} } @article {2019, title = {Reduced basis approaches for parametrized bifurcation problems held by non-linear Von K{\'a}rm{\'a}n equations}, volume = {81}, year = {2019}, pages = {112{\textendash}135}, abstract = {

This work focuses on the computationally efficient detection of the buckling phenomena and bifurcation analysis of the parametric Von K{\'a}rm{\'a}n plate equations based on reduced order methods and spectral analysis. The computational complexity - due to the fourth order derivative terms, the non-linearity and the parameter dependence - provides an interesting benchmark to test the importance of the reduction strategies, during the construction of the bifurcation diagram by varying the parameter(s). To this end, together the state equations, we carry out also an analysis of the linearized eigenvalue problem, that allows us to better understand the physical behaviour near the bifurcation points, where we lose the uniqueness of solution. We test this automatic methodology also in the two parameter case, understanding the evolution of the first buckling mode. journal = Journal of Scientific Computing

}, doi = {10.1007/s10915-019-01003-3}, url = {https://arxiv.org/abs/1804.02014}, author = {Federico Pichi and Gianluigi Rozza} } @inbook {2018, title = {Reduced Basis Approximation and A Posteriori Error Estimation: Applications to Elasticity Problems in Several Parametric Settings}, booktitle = {Numerical Methods for PDEs}, volume = {15}, number = {SEMA SIMAI}, year = {2018}, doi = {https://doi.org/10.1007/978-3-319-94676-4_8}, url = {https://link.springer.com/chapter/10.1007/978-3-319-94676-4_8}, author = {Huynh, D. B. P. and Federico Pichi and Gianluigi Rozza} } @article {2018, title = {Reduced Basis Approximation and A Posteriori Error Estimation: Applications to Elasticity Problems in Several Parametric Settings}, journal = {SEMA SIMAI Springer Series}, volume = {15}, year = {2018}, pages = {203-247}, abstract = {

In this work we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for elasticity problems in affinely parametrized geometries. The essential ingredients of the methodology are: a Galerkin projection onto a low-dimensional space associated with a smooth {\textquotedblleft}parametric manifold{\textquotedblright}{\textemdash}dimension reduction; an efficient and effective greedy sampling methods for identification of optimal and numerically stable approximations{\textemdash}rapid convergence; an a posteriori error estimation procedures{\textemdash}rigorous and sharp bounds for the functional outputs related with the underlying solution or related quantities of interest, like stress intensity factor; and Offline-Online computational decomposition strategies{\textemdash}minimum marginal cost for high performance in the real-time and many-query (e.g., design and optimization) contexts. We present several illustrative results for linear elasticity problem in parametrized geometries representing 2D Cartesian or 3D axisymmetric configurations like an arc-cantilever beam, a center crack problem, a composite unit cell or a woven composite beam, a multi-material plate, and a closed vessel. We consider different parametrization for the systems: either physical quantities{\textemdash}to model the materials and loads{\textemdash}and geometrical parameters{\textemdash}to model different geometrical configurations{\textemdash}with isotropic and orthotropic materials working in plane stress and plane strain approximation. We would like to underline the versatility of the methodology in very different problems. As last example we provide a nonlinear setting with increased complexity.

}, doi = {10.1007/978-3-319-94676-4_8}, url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85055036627\&doi=10.1007\%2f978-3-319-94676-4_8\&partnerID=40\&md5=e9c07038e7bcc6668ec702c0653410dc}, author = {D.B.P. Huynh and Federico Pichi and Gianluigi Rozza} }