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

Abstract

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.

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Algorithm

In mathematics and computer science, an algorithm (ˈælɡərɪðəm) is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning), achieving automation eventually.

Microstructure

Microstructure is the very small scale structure of a material, defined as the structure of a prepared surface of material as revealed by an optical microscope above 25× magnification. The microstructure of a material (such as metals, polymers, ceramics or composites) can strongly influence physical properties such as strength, toughness, ductility, hardness, corrosion resistance, high/low temperature behaviour or wear resistance. These properties in turn govern the application of these materials in industrial practice.

Geometry

Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.

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