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Publication# Adaptive analysis-aware defeaturing

Résumé

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.

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Concepts associés (19)

MOOCs associés (78)

Publications associées (18)

Géométrie

La géométrie est à l'origine la branche des mathématiques étudiant les figures du plan et de l'espace (géométrie euclidienne). Depuis la fin du , la géométrie étudie également les figures appartenant à d'autres types d'espaces (géométrie projective, géométrie non euclidienne ). Depuis le début du , certaines méthodes d'étude de figures de ces espaces se sont transformées en branches autonomes des mathématiques : topologie, géométrie différentielle et géométrie algébrique.

Analysis

Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384–322 B.C.), though analysis as a formal concept is a relatively recent development. The word comes from the Ancient Greek ἀνάλυσις (analysis, "a breaking-up" or "an untying;" from ana- "up, throughout" and lysis "a loosening"). From it also comes the word's plural, analyses.

Analyse numérique

L’analyse numérique est une discipline à l'interface des mathématiques et de l'informatique. Elle s’intéresse tant aux fondements qu’à la mise en pratique des méthodes permettant de résoudre, par des calculs purement numériques, des problèmes d’analyse mathématique. Plus formellement, l’analyse numérique est l’étude des algorithmes permettant de résoudre numériquement par discrétisation les problèmes de mathématiques continues (distinguées des mathématiques discrètes).

Mécanique des Fluides

Ce cours de base est composé des sept premiers modules communs à deux cours bachelor, donnés à l’EPFL en génie mécanique et génie civil.

Matlab et Octave pour débutants

Premiers pas dans MATLAB et Octave avec un regard vers le calcul scientifique

Matlab et Octave pour débutants

Premiers pas dans MATLAB et Octave avec un regard vers le calcul scientifique

Modern manufacturing engineering is based on a

`design-through-analysis'' workflow. According to this paradigm, a prototype is first designed with Computer-aided-design (CAD) software and then finalized by simulating its physical behavior, which usually involves the simulation of Partial Differential Equations (PDEs) on the designed product. The simulation of PDEs is often performed via finite element discretization techniques.A severe bottleneck in the entire process is undoubtedly the interaction between the design and analysis phases. The prototyped geometries must undergo the time-consuming and human-involved meshing and feature removal processes to become `

analysis-suitable''. This dissertation aims to develop and study numerical solvers for PDEs to improve the integration between numerical simulation and geometric modeling. The thesis is made of two parts. In the first one, we focus our attention on the analysis of isogeometric methods which are robust in geometries constructed using Boolean operations. We consider geometries obtained via trimming (or set difference) and union of multiple overlapping spline patches. As differential model problems, we consider both elliptic (the Poisson problem, in particular) and saddle point problems (the Stokes problem, in particular). As it is standard, the Nitsche method is used for the weak imposition of the essential boundary conditions and to weakly enforce the transmission conditions at the interfaces between the patches. After proving through well-constructed examples that the Nitsche method is not uniformly stable, we design a minimal stabilization technique based on a stabilized computation of normal fluxes (and on a simple modification of the pressure space in the case of the Stokes problem). The main core of this thesis is devoted to the derivation and rigorous mathematical analysis of a stabilization procedure to recover the well-posedness of the discretized problems independently of the geometric configuration in which the domain has been constructed. In the second part of the thesis, we consider a different approach. Instead of considering the underlying spline parameterization of the geometrical object, we immerse it in a much simpler and readily meshed domain. From the mathematical point of view, this approach is closely related to the isogeometric discretizations in trimmed domains treated in the first part. In this case, we consider the Raviart-Thomas finite element discretization of the Darcy flow. First, we analyze a Nitsche and a penalty method for the weak imposition of the essential boundary conditions on a boundary fitted mesh, a problem that was not studied before, not needed for our final goal, but still interesting by itself. Then, we consider the case of a general domain immersed in an underlying mesh unfitted with the boundary. We focus on the Nitsche method presented for the boundary fitted case and study its extension to the unfitted setting. We show that the so-called ghost penalty stabilization provides an effective solution to recover the well-posedness of the formulation and the well-conditioning of the resulting linear system.Wave phenomena manifest in nature as electromagnetic waves, acoustic waves, and gravitational waves among others.Their descriptions as partial differential equations in electromagnetics, acoustics, and fluid dynamics are ubiquitous in science and engineering. Having numerical methods to solve these problems efficiently is therefore of great importance and value to domains such as aerospace engineering, geophysics, and civil engineering.Wave problems are characterized by the finite speeds at which waves propagate and present a series of challenges for the numerical methods aimed at solving them. This dissertation is concerned with the development and analysis of numerical algorithms for solving wave problems efficiently using a computer. It contains two parts:The first part is concerned with sparse linear systems which stem from discretizations of such problems. An approximate direct solver is developed, which can be computed and applied in quasilinear complexity. As such, it can also be used as a preconditioner to accelerate the computation of solutions using iterative methods. This direct solver is based on structured Gaussian elimination, using a nested dissection reordering and the compression of dense, intermediate matrices using rank structured matrix formats. We motivate the use of these formats and demonstrate their usefulness in our algorithm. The viability of the method is then verified using a variety of numerical experiments. These confirm the quasilinear complexity and the applicability of the method.The second part focuses on the solution of the shallow water equations using the discontinuous Galerkin method. These equations are used to model tsunamis, storm surges, and weather phenomena.We aim to model large-scale tsunami events, as would be required for the development of an early-warning system. This necessitates the development of a well-balanced numerical scheme, which is efficient, flexible, and robust. We analyze the well-balanced property in the context of discontinuous Galerkin methods and how it can be obtained. Another problem that arises with the shallow water equations is the presence of dry areas. We introduce methods to handle these in a well-balanced, and physically consistent manner. The resulting method is validated using tests in one dimension, as well as simulations on the surface of the Earth. The latter are compared to real-world data obtained from buoys and satellites, which demonstrate the applicability and accuracy of our method.

Over the past decades, timber has gained popularity as a sustainable building material because of the rising environmental awareness. Furthermore, the resurgence of timber has also been encouraged by the advent of the digital age during the 21st century. The development of computer-aided design programming and digital fabrication tools has stimulated significant advances in both architecture and engineering. Within this context, researchers have shown a growing interest in wood-wood connections, inspired by traditional woodworking joints. Geometrically complex timber structures assembled with joints integrated in their plates have been developed using algorithmic geometry processing. However, although their design and fabrication have been automated, research focused on automated numerical tools for their structural analysis has been very limited. This thesis is providing a design methodology for the structural analysis of timber plate structures composed of a large number of discrete planar elements and wood-wood connections. A finite element model, in which the semi-rigid behaviour of the connections is implemented using springs, is proposed. It is applied to a specific case study, namely the Annen Plus SA head office project in Manternach, Luxembourg, which consists of a series of double-layered double-curved timber plate shells. A design framework is introduced to automate the generation of the model and integrate structural analysis into the existing design and fabrication workflow. The numerical model was built for both small- and large-scale structures through custom scripts and subsequently assessed through experimental investigations: first, laboratory tests were performed on small assemblies with simplified geometry; secondly, a displacement study was carried out onsite on a 24 m span structure. Results obtained with the semi-rigid spring model were found to be in good agreement with experimental tests. The proposed model was therefore validated for the serviceability limit state. Furthermore, the semi-rigidity of the connections in translation as well as in rotation was shown to highly influence the model and is therefore crucial for the accuracy of the model. Based on experimental tests observations, an alternative structural system was proposed and compared to the initial one through numerical investigations within the proposed design framework. A significant influence on the structureâs performance was found, demonstrating the possibilities for structural optimisation. Finally, a three-dimensional finite element model for wood-wood connections was investigated. It aimed to predict their semi-rigid behaviour, generally characterised through experimental tests, necessary for their implementation in global models. The material model was evaluated based on shear load tests performed on different engineered wood products. Stiffness and load-carrying capacity of the connections were approximated with numerical simulations. However, experimental tests remain necessary to precisely predict the behaviour of the joints. This thesis highlights the importance of adopting an integrated design strategy encompassing engineering and fabrication aspects for geometrically complex timber structures, as well as establishing a link between local behaviour of the connections and global behaviour of the structure. The gained knowledge can facilitate the design and realisation of large-scale freeform timber structures with wood-wood connections.