arXiv subjects

Matteo Cicuttin

Publications and source records attributed to Matteo Cicuttin.

4 recordsLinked to original sources

arXiv2021

Hybrid high-order methods. A primer with application to solid mechanics

This book is organized into eight chapters. The first three gently introduce the basic principles of hybrid high-order methods on a linear diffusion problem, the key ideas underlying the mathematical analysis, and some useful variants of the method as well as links to other methods from the literature. The following four present various challenging applications to solid mechanics, including linear elasticity and hyperelasticity, elastodynamics, contact/friction, and plasticity. The last chapter reviews implementation aspects. This book is primarily intended for graduate students, researchers (in applied mathematics, numerical analysis, and computational mechanics), and engineers working in related fields of application. Basic knowledge of the devising and analysis of finite element methods is assumed. Special effort was made to streamline the presentation so as to pinpoint the essential ideas, address key mathematical aspects, present examples, and provide bibliographic pointers.

math.NA↗

arXiv2017

Spectral approximation of elliptic operators by the Hybrid High-Order method

We study the approximation of the spectrum of a second-order elliptic differential operator by the Hybrid High-Order (HHO) method. The HHO method is formulated using cell and face unknowns which are polynomials of some degree $k\geq0$. The key idea for the discrete eigenvalue problem is to introduce a discrete operator where the face unknowns have been eliminated. Using the abstract theory of spectral approximation of compact operators in Hilbert spaces, we prove that the eigenvalues converge as $h^{2t}$ and the eigenfunctions as $h^{t}$ in the $H^1$-seminorm, where $h$ is the mesh-size, $t\in [s,k+1]$ depends on the smoothness of the eigenfunctions, and $s>\frac12$ results from the elliptic regularity theory. The convergence rates for smooth eigenfunctions are thus $h^{2k+2}$ for the eigenvalues and $h^{k+1}$ for the eigenfunctions. Our theoretical findings, which improve recent error estimates for Hybridizable Discontinuous Galerkin (HDG) methods, are verified on various numerical examples including smooth and non-smooth eigenfunctions. Moreover, we observe numerically in one dimension for smooth eigenfunctions that the eigenvalues superconverge as $h^{2k+4}$ for a specific value of the stabilization parameter.

math.NA↗

arXiv2017

A Hybrid High-Order method for highly oscillatory elliptic problems

We devise a Hybrid High-Order (HHO) method for highly oscillatory elliptic problems that is capable of handling general meshes. The method hinges on discrete unknowns that are polynomials attached to the faces and cells of a coarse mesh; those attached to the cells can be eliminated locally using static condensation. The main building ingredient is a reconstruction operator, local to each coarse cell, that maps onto a fine-scale space spanned by oscillatory basis functions. The present HHO method generalizes the ideas of some existing multiscale approaches, while providing the first complete analysis on general meshes. It also improves on those methods, taking advantage of the flexibility granted by the HHO framework. The method handles arbitrary orders of approximation $k\geq 0$. For face unknowns that are polynomials of degree $k$, we devise two versions of the method, depending on the polynomial degree $(k-1)$ or $k$ of the cell unknowns. We prove, in the case of periodic coefficients, an energy-error estimate of the form $\left(\varepsilon^{\frac{1}{2}}+H^{k+1}+(\varepsilon/H)^{\frac{1}{2}}\right)$, and we illustrate our theoretical findings on some test-cases.

math.NA↗

arXiv2015

Modeling of anechoich chambers with equivalent materials and equivalent sources

Numerical simulation of anechoic chambers is a hot topic since it can provide useful data about the performance of the EMC site. However, the mathematical nature of the problem, the physical dimensions of the simulated sites and the frequency ranges pose nontrivial challenges to the simulation. Computational requirements in particular will quickly become unmanageable if adequate techniques are not employed. In this work we describe a novel approach, based on equivalent elements, that enables the simulation of large chambers with modest computational resources. The method is then validated against real measurement results.

cs.CE↗