Geometrical Treatise on the Modelling of 3D Particulate Inclusion-Matrix Microstructures with an Application to Historical Stone Masonry Walls

As historical stone masonry structures are vulnerable and prone to damage in earthquakes, investigating their structural integrity is important to reduce injuries and casualties while preserving their historical value. Stone masonry is a composite material that is built with stones and binding mortar. Although experimental campaigns are crucial in understanding the structural behaviour of these walls, the wide spectrum of existing stone masonry typologies and the randomness in geometry and material properties render extensive testing campaigns nearly impossible to account for all the uncertainties and variables. Numerical simulations that explicitly represent the microstructure of the wall (i.e., the geometry and arrangement of the stones) can complement experimental studies. However, methods for generating 3D microstructures were lacking.The focus of this dissertation is to study the geometry of the 3D microstructure of stone masonry walls at multiple scales, which opens unprecedented opportunities for studying the micromechanical behaviour of these walls numerically. The first contribution of this thesis was crafting the first 3D virtual microstructure generator that can cover the main typologies of stone masonry that are frequently found in historical buildings. The herein-created microstructure generator borrowed the conventional packing problem analogy to pack the generated stones inside the boundary of the walls following building and physical placement rules to obtain realistic microstructures. The size spectrum, distribution, interlocking and topology of the generated stones were thoroughly investigated. To quickly and efficiently solve the posed packing problem of each stone, we proposed a general-purpose heuristic algorithm that benefits from Pareto's principle which is commonly known as the $80/20$ rule. This heuristic was called the Pareto sequential sampling (PSS) algorithm as it depends on sampling most of the solutions, using any design of experiment (DoE) method, from a domain that is known to be a prominent pool of solutions.The second contribution was related to studying the topology and roughness of stones and surfaces. The traditional spherical harmonics expansion was used to study the morphology of closed surfaces of the nonconvex stones to estimate their fractal dimension and uniformly remesh the triangulated surfaces to be used in numerical simulations. As the roughness of natural stones changes on a single stone, we similarly studied the morphology of locally sampled rough patches. We developed the spherical cap harmonics and disk harmonics spectral approaches to provide us with information on the roughness and fractal dimension of any isotropic self-affine rough surface. To speed up the numerical integration of these spectral expansions, we proposed new algorithms based on the well-known Kaczmarz orthogonal projection in a non-memory-intensive manner and using reasonable computational times. The developed algorithms were hinged on the conjugate symmetry of the harmonic bases and the sparsity of the real signals to lower the dimensionality of the problem and accelerate the rate of convergence. These spectral methods are considered general-purpose and have multiple applications in various fields such as medical imaging and computer graphics. With this dissertation, we put forward geometrical and topological concepts that make detailed micromechanical simulations of stone masonry walls practical.

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

One dimensional and multiscale models for blood flow circulation

The aim of this work is to provide mathematically sound and computationally effective tools for the numerical simulation of the interaction between fluid and structures as occurring, for instance, in the simulation of the human cardiovascular system. This problem is global, in the sense that local changes can modify the solution far away. From the point of view of computing and modelling this calls for the use of multiscale methods, where simplified models are used to treat the global problem leaving to more accurate models the local description. Moreover it is characterised by the appearance of pressure waves inside the vessels. In large arteries the vessel wall dynamics can be described by a thin elastic membrane model (Navier equation) while the fluid motion can be represented by the Navier-Stokes equations for incompressible Newtonian fluids. Unfortunately, given the high levels of details furnished by this model, its computational complexity is dramatically high. Therefore reduced models have been developed. In particular, one-dimensional models, originally introduced by Euler, seem to be appropriate for the study in the time-space domain of pressure wave propagation induced by the interaction between the fluid and the vessel wall in the arterial tree. These reduced models are obtained after integrating the Navier-Stokes equations over a vessel section, supposed to be circular, and assuming an algebraic wall law to describe the relationship between pressure and wall deformation. They can be used in place of the more complex three dimensional fluid-structure models or in cooperation with them (multiscale approach). The first part of this work deals with one dimensional models. A reduced 1D model taken from literature is presented and analysed. Some extensions of the basic model, in particular with respect to vessel wall law (generalised string model) and more complex geometries (bifurcated and curved arteries), are also considered. Numerical schemes are proposed and some numerical results are presented. In the second part of this thesis we focus on a multiscale model. We consider a 55 arterial tree, described by the 1D model, coupled with lumped parameter models for heart and capillaries. In particular, specific attention has been devoted to the coupling between the left ventricle and the arterial system, whose physiopathological relevance is well known. This mathematical model gives good results in numerical tests and is able to describe the relevant features of the pressure wave propagation and reflections within the arterial system.

EPFL2004

Problèmes de cheminements optimaux dans les réseaux avec contraintes associées aux arcs

In this thesis we consider two optimal routing problems in networks. Both are NP-hard and are often used to modelize real life problems of large size. We first present the capacitated arc routing problem (CARP). In this problem, we have to serve a set of customers represented by the arcs of a network with vehicles of restricted capacity. The aim is to design a set of vehicle routes of least total cost. We describe two heuristic methods for the CARP based on local search techniques. The first one is a Tabu Search heuristic using an original post-optimization procedure. Principles appearing in this procedure are used to define a class of neighborhoods we combine in a Variable Neighborhood Descent method. Our algorithms compare favorably with some of her methods. They were initially developed to deal with the undirected CARP. We describe how to transform them so that they can be applied to the directed CARP. We also propose an adaptation to the directed case of a lower bound originally developed for the undirected CARP. This lower bound is used to evaluate the quality of the solutions produced by our algorithms for the directed CARP. Our methods obtain solutions whose value is, on average, close to known lower bounds. We use the CARP to tackle a real life problem of boat routing and scheduling. This problem can be viewed as a directed CARP with time windows. We describe in detail its modeling process and we analyze our first results. We obtain solutions that require less boats than the present schedule. Nevertheless, these solutions are not completely feasible in practice. We suggest some ways to improve them. The second problem treated in this thesis is the shortest capacitated paths problem (SCPP). We present a Tabu Search heuristic and some post-optimization procedures for the SCPP. These procedures are called in our Tabu Search algorithm. Two versions of this algorithm are considered. In the first one, post-optimization procedures are intensively used. In the second one, they are less frequently applied. This second version is faster. Compared with a greedy approach, the two versions of our Tabu Search heuristic obtain solutions which are, on average, of better quality. The SCPP was proposed to us by EDF (Electricité de France) in order to minimize the total length of the paths followed by cables in a nuclear power-station. It can also be viewed as a subproblem of the bandwidth packing problem (BWP) appearing in telecommunications.

EPFL2000