Reduced Basis Methods for the Solution of Parametrized PDEs in Repetitive and Complex Networks with Application to CFD

The objective of this work is to develop a numerical framework to perform rapid and reliable simulations for solving parametric problems in domains represented by networks and to extend the classical reduced basis method. Aimed at this scope, we propose two original methodological approaches for the approximation of partial differential equations in domains made up by repetitive parametrized geometries where topological features are recurrent: the reduced basis hybrid method (RBHM) and the reduced basis-domain decomposition-finite element (RDF) method. The common paradigm of these methods is the consideration that the blocks composing the computational domain are topologically similar to a few reference shapes. On the latter, we compute representative solutions, corresponding to the same governing partial differential equations, but for different values of some parameters of interest and representing, for example, the deformation of the blocks. A new desired solution for a new deformed domain is recovered as projection on the reduced spaces built by the previously precomputed solutions and the continuity of the solution across subdomain interfaces is guaranteed by suitable coupling conditions. The different choices for the reduced spaces and coupling conditions adopted characterize one method with respect to the other one. The geometrical parametrization of the considered domains, by transfinite maps, induces non-affine parameter dependence: an empirical interpolation technique is used to recover an approximate affine parameter dependence and a sub–sequent offline/online decomposition of the reduced basis procedure. This computational decomposition yields a considerable reduction of the problem complexity. Results computed on some combinations of 2D and 3D geometries, representing cardiovascular networks, show the flexibility and the advantages of the proposed methods in terms of reduced computational costs and complexities. The computational time with these new approaches is, in general, much reduced with respect to a classical finite element method on the whole domain but also only marginally slower than a classical reduced basis approach on the whole domain. However, these approaches decrease drastically the offline time to pre-compute the reduced basis by splitting the total number of parameters characterizing the problem into smaller subsets for each reference block, moreover they allow to considerably increase the geometrical flexibility and versatility.

Reduced Basis method for Isogeometric Analysis: application to structural problems

Marina Rinaldi

Isogeometric Analysis (IGA) is a computational methodology for the numerical approximation of Partial Differential Equations (PDEs). IGA is based on the isogeometric concept, for which the same basis functions, usually Non-Uniform Rational B-Splines (NURBS), are used both to represent the geometry and to approximate the unknown solutions of PDEs. Compared to the standard Finite Element method, NURBS-based IGA offers several advantages: ideally a direct interface with CAD tools, exact geometrical representation, simple refinement procedures, and smooth basis functions allowing to easily solve higher order problems, including structural shell problems. In these contexts, repeatedly solving a problem for a large set of geometric parameters might lead to high and eventually prohibitive computational cost. To cope with this problem, we consider in this work the Reduced Basis (RB) method for the solution of parameter dependent PDEs, specifically for which the NURBS representation of the computational domain is parameter dependent. RB refers to a technique that enables a rapid and reliable approximation of parametrized PDEs by constructing low dimensional approximation spaces. In this work, for the construction of the reduced spaces we adopt two different strategies, namely the Proper Orthogonal Decomposition and the greedy algorithm. In this thesis we combine RB and IGA for the efficient solution of parametrized problems for all the possible cases of NURBS geometrical parametrizations, which specifically include the NURBS control points, the weights, and both the control points and weights. In particular, we first focus on the solution of second order PDEs on parametrized lower dimensional manifolds, specifically surfaces in the three dimensional space. We consider geometrical parametrizations that entail a nonaffine dependence of the variational forms on the spatial coordinates and the geometric parameters. Thus, depending on the parametrization at hand and in order to ensure a suitable Offline/Online decomposition between the reduced order model construction and solution, we resort to the Empirical Interpolation Method (EIM) or the Matrix Discrete Empirical Interpolation Method (MDEIM), by comparing their performances. As application, we solve a class of benchmark structural problems modeled by Kirchoff-Love shells for which we consider NURBS geometric parametrizations and we apply the RB method to the solution of this class of fourth order PDEs. We highlight by means of numerical tests, the performance of the RB method applied to standard IGA approximation of parametrized shell geometries.

2015

Computational fluid dynamics for naval engineering problems

Nicola Parolini

The subject of this thesis is the numerical simulation of viscous free-surface flows in naval engineering applications. State-of-the-art numerical methods based on the solution of the Navier-Stokes equations are used to predict the flow around different classes of boats. We investigate the role of the Computational Fluid Dynamics in the design of racing boats, such as America's Cup yachts and Olympic class rowing hull. The mathematical models describing the different aspects of the physical problem, as well as the numerical methods adopted for their solution, are introduced and critically discussed. The different phases of the overall numerical simulation procedure, from grid generation through the solution of the flow equations to the post-processing of the results, are described. We present the numerical simulations that have been performed to investigate the role of different design parameters in the conception of America's Cup yachts and we describe how the results obtained from the simulations are integrated into the overall design process. The free-surface flow around an Olympic rowing boat is also considered. We propose a simplified approach to take into account the effect of the boat dynamics in the prediction of the hydrodynamic forces acting on the boat. Based on the results of the simulations, we propose a new design concept and we investigate its potential benefits on the boat performances. One of the aspects that is found to be not completely satisfactory, within the standard numerical methods adopted, is the modelling of complex free-surface flows. The second part of this thesis is devoted to a more theoretical and methodological investigation of this aspect. In particular, we present and analyse a new numerical method based on the level set approach for the solution of two-fluid flows. The numerical scheme based on a finite element discretization is introduced and different critical aspects of its implementation are discussed. In particular, we present and analyse a new technique for the stabilization of the advection equation associated to the level set problem. Moreover, we propose a new reinitialization procedure for the level set function which plays a crucial role in the accuracy of the algorithm. The convergence properties of this procedure are analysed and comparisons with more standard approaches are presented. Finally, the proposed method has been used to solve a variety of test cases concerning time dependent two-fluid viscous flows. The results of the simulation are presented and discussed.

EPFL2004

Adaptive analysis-aware defeaturing

Ondine Gabrielle Chanon

Removing geometrical details from a complex domain is a classical operation in computer aided design for simulation and manufacturing. This procedure simplifies the meshing process, and it enables faster simulations with less memory requirements. However, depending on the partial differential equation that one wants to solve in the geometrical model of interest, removing some important geometrical features may greatly impact the solution accuracy. For instance, in solid mechanics simulations, such features can be holes or fillets near stress concentration regions. Unfortunately, the effect of geometrical simplification on the accuracy of the problem solution is often neglected, because its analysis is a time-consuming task that is often performed manually, based on the expertise of engineers. It is therefore important to have a better understanding of the effect of geometrical model simplification, also called defeaturing, to improve our control on the simulation accuracy along the design and analysis phase.In this thesis, we formalize the process of defeaturing, and we analyze its impact on the accuracy of solutions of some partial differential problems. To achieve this goal, we first precisely define the error between the problem solution defined in the exact geometry, and the one defined in the simplified geometry. Then, we introduce an a posteriori estimator of the energy norm of this error. This allows us to reliably and efficiently control the error coming from the addition or the removal of geometrical features. We subsequently consider a finite element approximation of the defeatured problem, and the induced numerical error is integrated to the proposed defeaturing error estimator. In particular, we address the special case of isogeometric analysis based on (truncated) hierarchical B-splines, in possibly trimmed and multipatch geometries. In this framework, we derive a reliable a posteriori estimator of the overall error, i.e., of the error between the exact solution defined in the exact geometry, and the numerical solution defined in the defeatured geometry.We then propose a two-fold adaptive strategy for analysis-aware defeaturing, which starts by considering a coarse mesh on a fully-defeatured computational domain. On the one hand, the algorithm performs classical finite element mesh refinements in a (partially) defeatured geometry. On the other hand, the strategy also allows for geometrical refinement. That is, at each iteration, the algorithm is able to choose which missing geometrical features should be added to the simplified geometrical model, in order to obtain a more accurate solution.Throughout the thesis, we validate the presented theory, the properties of the aforementioned estimators and the proposed adaptive strategies, thanks to an extensive set of numerical experiments.

EPFL2022