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Person# Annalisa Buffa

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Finite element method

The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the tr

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 b

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathema

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MATH-468: Numerics for fluids, structures & electromagnetics

The aim of the course is to give a theoretical and practical knowledge of the finite element method for saddle point problems, such as the ones of fluid dynamics, elasticity and electromagnetic problems.

MATH-638: Integral equations methods for exterior problems

I will introduce integral equation formulations for the Laplace, the wave equations and the electromagnetic scattering problem. The wellposedness and the discretization of these problems are discussed.

Pablo Antolin Sanchez, Annalisa Buffa, Margarita Chasapi

This work introduces a reduced order modeling (ROM) framework for the solution of parameterized second-order linear elliptic partial differential equations formulated on unfitted geometries. The goal is to construct efficient projection-based ROMs, which rely on techniques such as the reduced basis method and discrete empirical interpolation. The presence of geometrical parameters in unfitted domain discretizations entails challenges for the application of standard ROMs. Therefore, in this work we propose a methodology based on i) extension of snapshots on the background mesh and ii) localization strategies to decrease the number of reduced basis functions. The method we obtain is computationally efficient and accurate, while it is agnostic with respect to the underlying discretization choice. We test the applicability of the proposed framework with numerical experiments on two model problems, namely the Poisson and linear elasticity problems. In particular, we study several benchmarks formulated on two-dimensional, trimmed domains discretized with splines and we observe a significant reduction of the online computational cost compared to standard ROMs for the same level of accuracy. Moreover, we show the applicability of our methodology to a three-dimensional geometry of a linear elastic problem.

2022, ,

Region extraction is a very common task in both Computer Science and Engineering with several applications in object recognition and motion analysis, among others. Most of the literature focuses on regions delimited by straight lines, often in the special case of intersection detection among two unstructured meshes. While classical region extraction algorithms for line drawings and mesh intersection algorithms have proved to be able to deal with many applications, the advances in Isogeometric Analysis require a generalization of such problem to the case in which the regions to be extracted are bounded by an arbitrary number of curved segments. In this work we present a novel region extraction algorithm that allows a precise numerical integration of functions defined in different spline spaces. The presented algorithm has several interesting applications in contact problems, mortar methods, and quasi-interpolation problems.(c) 2022 Published by Elsevier Ltd.

2023Pablo Antolin Sanchez, Annalisa Buffa, Simone Deparis, Felipe Figueredo Rocha

Effective properties of materials with random heterogeneous structures are typically determined by homogenising the mechanical quantity of interest in a window of observation. The entire problem setting encompasses the solution of a local PDE and some averaging formula for the quantity of interest in such domain. There are relatively standard methods in the literature to completely determine the formulation except for two choices: i) the local domain itself and the ii) boundary conditions. Hence, the modelling errors are governed by the quality of these two choices. The choice i) relates to the degree of representativeness of a microscale sample, i.e., it is essentially a statistical characteristic. Naturally, its reliability is higher as the size of the observation window becomes larger and/or the number of samples increases. On the other hand, excepting few special cases there is no automatic guideline to handle ii). Although it is known that the overall effect of boundary condition becomes less important with the size of the microscale domain, the computational cost to simulate such large problem several times might be prohibitive even for relatively small accuracy requirements. Here we introduce a machine learning procedure to select most suitable boundary conditions for multiscale problems, particularly those arising in solid mechanics. We propose the combination Reduced-Order Models and Deep Neural Networks in an offline phase, whilst the online phase consists in the very same homogenisation procedure plus one (cheap) evaluation of the trained model for boundary conditions. Hence, the method allows an implementation with minimal changes in existing codes and the use of relatively small domains without losing accuracy, which reduces the computational cost by several orders of magnitude. A few test cases accounting for random circular and elliptical inclusions are reported aiming at proving the potentials of the DeepBND method.

2023