08:30
MS14: Reduced Order Models for parametric PDEs: special focus on time-dependent phenomena and time-harmonic wave problems (Part 1)
Chair: Gianluigi Rozza
08:30
25 mins
|
Efficient Error Estimation for Model Order Reduction of Linear (Non-) Parametric Systems
Lihong Feng, Peter Benner
Abstract: We propose an error estimator for reduced-order modeling of linear non-parametric or parametric dynamical systems. The error estimator estimates the error of the reduced transfer function in frequency domain and can be easily extended to output error estimation for reduced-order models of steady linear parametric systems. It is sharp and cheap to compute. Using the error estimator, the reduced-order model can be adaptively obtained with high reliability. Numerical results show that the error estimator can accurately estimate the true error even for transfer functions with many resonances. Compared with an existing error bound [1], the proposed error estimator can be orders of magnitudes sharper and needs much less computational time. It also outperforms the output error estimator based on randomized residual proposed in [2].
References
[1] L. Feng, A. C. Antoulas, and P. Benner, “Some a posteriori error bounds for
reduced order modelling of (non-)parametrized linear systems,” ESAIM: M2AN,
vol. 51, pp. 2127–2158, 2017.
[2] K. Smetana, O. Zahm and A. T. Pater. Randomized residual-based error
estimators for parametrized equations. arXiv e-prints, Cornell University. 2018.
https://arxiv.org/abs/1807.10489. math.N.A.
|
08:55
25 mins
|
Matrix oriented reduction of space-time Petrov-Galerkin variational problem
Karsten Urban, Julian Henning, Davide Palitta, Valeria Simoncini
Abstract: It is quite well-known that transport-dominated and wave phenomena are a particular challenge for model reduction. In this talk we show some recent approaches to deal with model reduction for the wave equation: We introduce energy-based error estimators and show their good behavior. This is done both for a second-order formulation and also for the corresponding first-order system.
Finally, we address a very-weak formulation in terms a space-time variational approach. This allows for solutions with the minimal regularity and also built-in stability.
This talk is based upon joint work with Silke Glas, Constantin Greif (Ulm) and Anthony Patera (MIT).
|
09:20
25 mins
|
Model Order Reduction for Time-Dependent Parametric Problems Using a Space-Time FEM Solver
Fabian Key, Francesco Ballarin, Stefanie Elgeti, Gianluigi Rozza
Abstract: When it comes to industrial applications, the challenges for the utilized simulation tools, e.g., computational fluid dynamics (CFD), are getting more demanding quickly. This holds 1.) with respect to the complexity of the physical phenomena under consideration, but also 2.) for the requirements regarding the efficiency of these tools in terms of runtime and computational resources.
Addressing the first point, we make use of the flow solver XNS based on the Deforming-Spatial Domain/Stabilized Space-Time (DSD/SST) finite element framework [1]. It allows us to tackle complex flow problems, which can include, for example, moving boundary problems or multi-phase flows. To reach a satisfying approximation quality of the underlying phenomena, a proper spatial and temporal discretization leads to high computational needs, which often require high-performance computing machines.
The second point gains importance when such a simulation tool should be integrated in a so-called many-query or real-time context. For example, this could be the study of the influence of certain parameters in an industrial process during its design and development.
For this purpose, reduced basis (RB) methods offer an additional tool which is computationally cheaper though still reliable. Building on top of the high-fidelity solver – the finite element solver in this case – it approximates the parametrized partial differential equations in an efficient way based on the concept of offline-online splitting.
Recently, stabilized RB methods for viscous flows have been presented [2] matching the underlying formulation of the stabilized FEM solver. Thus, we present our work on combining the aforementioned FEM solver with the library for reduced order modelling RBniCS [3]. The two different programming languages – Fortran and Python respectively – require the creation of a suitable interface. Starting from a diffusion problem, a validation of the obtained results is performed. Furthermore, the aspects and consequences of stabilization and the space-time approach are discussed.
REFERENCES
[1] T.E. Tezduyar, M. Behr, and J. Liou, “A new strategy for finite element computations involving moving boundaries and interfaces – the deforming-spatial-domain/space-time procedure: I. The concept and the preliminary tests”, Computer Methods in Applied Mechanics and Engineering, vol. 94, pp. 339-351, 1992.
[2] S. Ali., “Stabilized reduced basis methods for the approximation of parametrized viscous flows”, PhD Thesis, SISSA International School for Advance Studies, 2018.
[3] J. S. Hesthaven, G. Rozza, and B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations, SpringerBriefs in Mathematics, Springer International Publishing, 2018.
|
09:45
25 mins
|
Model Order Reduction for Nonlinear Elasticity: Application of the Reduced Basis Method to Finite Deformation
Lorenzo Zanon, Karen Veroy-Grepl, Oliver Röhrle
Abstract: In the context of numerical solid mechanics, we focus on projection-based model order reduction for nonlinear elasticity problems based on an underlying Finite Element (FE) discretization. The literature on this subject is largely related to the Proper Orthogonal Decomposition (POD). This technique, on the one hand, allows to reduce the dimension of highly nonlinear problems (e.g., viscoelasticity, elastoplasticity, Navier-Stokes equations) with great accuracy and substantial savings of computational time ([1], [2]). On the other hand, a multi-dimensional parameter space would be an obstacle to an efficient POD treatment. For this reason, we adopt the Reduced Basis (RB) Method, and extend it to problems in nonlinear elasticity.
Online-efficiency is a fundamental achievement in model order reduction. An impediment, in our model, is the nonlinear, nonaffine PDE system, a consequence of the choice of a complex, realistic material constitutive law such as a hyperelastic law. We therefore propose an RB-EIM model for large deformation problems, which, in the most general way, detects and circumvents both efficiency issues and implementation pitfalls. Numerical results are provided for a parametrized deflected beam in finite deformation regime (Fig. 1), in terms of the approximation error between the RB and the FE approximations, and the CPU computational time. We round off the talk with a comparison between the RB-EIM and the POD-EIM models ([3]). All the results have been obtained in a C++ computing environment, relying on the open-source FE library libMesh.
REFERENCES
[1] A. Radermacher, B. A. Bednarcyk, B. Stier, J. Simon, L. Zhou, S. Reese, “Displacement-based multiscale modeling of fiber-reinforced composites by means of proper orthogonal decomposition”, Advanced Modeling and Simulation in Engineering Sciences, 3(1), 1–23 (2016).
[2] G. Stabile and G. Rozza, “Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier–Stokes equations”, Computers & Fluids, 173, 273–284 (2018).
[3] L. Zanon, “Model Order Reduction for Nonlinear Elasticity: Applications of the Reduced Basis Method to Geometrical Nonlinearity and Finite Deformation”, PhD thesis, RWTH Aachen University (2017).
|
|
