arXiv subjects

Ilario Mazzieri

Publications and source records attributed to Ilario Mazzieri.

26 records · Page 2Linked to original sources

arXiv2021

A Virtual Element Method for the wave equation on curved edges in two dimensions

In this work we present an extension of the Virtual Element Method with curved edges for the numerical approximation of the second order wave equation in a bidimensional setting. Curved elements are used to describe the domain boundary, as well as internal interfaces corresponding to the change of some mechanical parameters. As opposite to the classic and isoparametric Finite Element approaches, where the geometry of the domain is approximated respectively by piecewise straight lines and by higher order polynomial maps, in the proposed method the geometry is exactly represented, thus ensuring a highly accurate numerical solution. Indeed, if in the former approach the geometrical error might deteriorate the quality of the numerical solution, in the latter approach the curved interfaces/boundaries are approximated exactly guaranteeing the expected order of convergence for the numerical scheme. Theoretical results and numerical findings confirm the validity of the proposed approach.

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arXiv2021

A high-order discontinuous Galerkin method for the poro-elasto-acoustic problem on polygonal and polyhedral grids

The aim of this work is to introduce and analyze a finite element discontinuous Galerkin method on polygonal meshes for the numerical discretization of acoustic waves propagation through poroelastic materials. Wave propagation is modeled by the acoustics equations in the acoustic domain and the low-frequency Biot's equations in the poroelastic one. The coupling is realized by means of (physically consistent) transmission conditions, imposed on the interface between the domains, modeling different pore configurations. For the space discretization, we introduce and analyze a high-order discontinuous Galerkin method on polygonal and polyhedral meshes, which is then coupled with Newmark-$β$ time integration schemes. Stability analysis for both the continuous and semi-discrete problem is presented and error estimates for the energy norm are derived for the semi-discrete one. A wide set of numerical results obtained on test cases with manufactured solutions are presented in order to validate the error analysis. Examples of physical interest are also presented to investigate the capability of the proposed methods in practical scenarios.

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arXiv2021

The conforming virtual element method for polyharmonic and elastodynamics problems: a review

In this paper, we review recent results on the conforming virtual element approximation of polyharmonic and elastodynamics problems. The structure and the content of this review is motivated by three paradigmatic examples of applications: classical and anisotropic Cahn-Hilliard equation and phase field models for brittle fracture, that are briefly discussed in the first part of the paper. We present and discuss the mathematical details of the conforming virtual element approximation of linear polyharmonic problems, the classical Cahn-Hilliard equation and linear elastodynamics problems.

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arXiv2019

A high-order discontinuous Galerkin method for nonlinear sound waves

We propose a high-order discontinuous Galerkin scheme for nonlinear acoustic waves on polytopic meshes. To model sound propagation with and without losses, we use Westervelt's nonlinear wave equation with and without strong damping. Challenges in the numerical analysis lie in handling the nonlinearity in the model, which involves the derivatives in time of the acoustic velocity potential, and in preventing the equation from degenerating. We rely in our approach on the Banach fixed-point theorem combined with a stability and convergence analysis of a linear wave equation with a variable coefficient in front of the second time derivative. By doing so, we derive an a priori error estimate for Westervelt's equation in a suitable energy norm for the polynomial degree $p \geq 2$. Numerical experiments carried out in two-dimensional settings illustrate the theoretical convergence results. In addition, we demonstrate efficiency of the method in a three-dimensional domain with varying medium parameters, where we use the discontinuous Galerkin approach in a hybrid way.

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arXiv2019

Simulation of 3D elasto-acoustic wave propagation based on a Discontinuous Galerkin Spectral Element method

In this paper we present a numerical discretization of the coupled elasto-acoustic wave propagation problem based on a Discontinuous Galerkin Spectral Element (DGSE) approach in a three-dimensional setting. The unknowns of the coupled problem are the displacement field and the velocity potential, in the elastic and the acoustic domains, respectively, thereby resulting in a symmetric formulation. After stating the main theoretical results, we assess the performance of the method by convergence tests carried out on both matching and non-matching grids, and we simulate realistic scenarios where elasto-acoustic coupling occurs. In particular, we consider the case of Scholte waves and the scattering of elastic waves by an underground acoustic cavity. Numerical simulations are carried out by means of the code SPEED, available at http://speed.mox.polimi.it.

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arXiv2018

A high-order discontinuous Galerkin approach to the elasto-acoustic problem

We address the spatial discretization of an evolution problem arising from the coupling of viscoelastic and acoustic wave propagation phenomena by employing a discontinuous Galerkin scheme on polygonal and polyhedral meshes. The coupled nature of the problem is ascribed to suitable transmission conditions imposed at the interface between the solid (elastic) and fluid (acoustic) domains. We state and prove a well-posedness result for the strong formulation of the problem, present a stability analysis for the semi-discrete formulation, and finally prove an a priori $hp$-version error estimate for the resulting formulation in a suitable (mesh-dependent) energy norm. We also discuss the time integration scheme employed to obtain the fully discrete system. The convergence results are validated by numerical experiments carried out in a two-dimensional setting.

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arXiv2017

Insight into the modeling of seismic waves for detection of underground cavities

Motivated by the need to detect an underground cavity within the procedure of an On-Site-Inspection (OSI), of the Comprehensive Nuclear Test Ban Treaty Organization, the aim of this paper is to present results on the comparison of our numerical simulations with an analytic solution. The accurate numerical modeling can facilitate the development of proper analysis techniques to detect the remnants of an underground nuclear test. The larger goal is to help set a rigorous scientific base of OSI and to contribute to bringing the Treaty into force. For our 3D numerical simulations, we use the discontinuous Galerkin Spectral Element Code SPEED jointly developed at MOX (The Laboratory for Modeling and Scientific Computing, Department of Mathematics) and at DICA (Department of Civil and Environmental Engineering) of the Politecnico di Milano.

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arXiv2013

Stability Analysis for Discontinuous Galerkin approximations of the elastodynamics problem

We consider semi-discrete discontinuous Galerkin approximations of a general elastodynamics problem, in both {\it displacement} and {\it displacement-stress} formulations. We present the stability analysis of all the methods in the natural energy norm and derive optimal a-priori error estimates. For the displacement-stress formulation, schemes preserving the total energy of the system are introduced and discussed. We include some numerical experiments in three dimensions to verify the theory.

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