arXiv subjects

Michele Botti

Publications and source records attributed to Michele Botti.

31 records · Page 2Linked to original sources

arXiv2022

A serendipity fully discrete div-div complex on polygonal meshes

In this work we address the reduction of face degrees of freedom (DOFs) for discrete elasticity complexes. Specifically, using serendipity techniques, we develop a reduced version of a recently introduced two-dimensional complex arising from traces of the three-dimensional elasticity complex. The keystone of the reduction process is a new estimate of symmetric tensor-valued polynomial fields in terms of boundary values, completed with suitable projections of internal values for higher degrees. We prove an extensive set of new results for the original complex and show that the reduced complex has the same homological and analytical properties as the original one. This paper also contains an appendix with proofs of general Poincaré--Korn-type inequalities for hybrid fields.

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arXiv2022

Uncertainty quantification for mineral precipitation and dissolution in fractured porous media

In this work, we present an uncertainty quantification analysis to determine the influence and importance of some physical parameters in a reactive transport model in fractured porous media. An accurate description of flow and transport in the fractures is key to obtain reliable simulations, however, fractures geometry and physical characteristics pose several challenges from both the modeling and implementation side. We adopt a mixed-dimensional approximation, where fractures and their intersections are represented as objects of lower dimension. To simplify the presentation, we consider only two chemical species: one solute, transported by water, and one precipitate attached to the solid skeleton. A global sensitivity analysis to uncertain input data is performed exploiting the Polynomial Chaos expansion along with spectral projection methods on sparse grids.

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arXiv2022

Discontinuous Galerkin approximation of the fully-coupled thermo-poroelastic problem

We present and analyze a discontinuous Galerkin method for the numerical modelling of the non-linear fully-coupled thermo-poroelastic problem. For the spatial discretization, we design a high-order discontinuous Galerkin method on polygonal and polyhedral grids based on a novel four-field formulation of the problem. To handle the non-linear convective transport term in the energy conservation equation we adopt a fixed-point linearization strategy. We perform a robust stability analysis for the linearized semi-discrete problem under mild requirements on the problem data. A priori hp-version error estimates in suitable energy norms are also derived. A complete set of numerical simulations is presented in order to validate the theoretical analysis, to inspect numerically the robustness properties, and to test the capability of the proposed method in a practical scenario inspired by a geothermal problem.

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arXiv2022

A Hybrid High-Order method for creeping flows of non-Newtonian fluids

In this paper, we design and analyze a Hybrid High-Order discretization method for the steady motion of non-Newtonian, incompressible fluids in the Stokes approximation of small velocities. The proposed method has several appealing features including the support of general meshes and high-order, unconditional inf-sup stability, and orders of convergence that match those obtained for scalar Leray-Lions problems. A complete well-posedness and convergence analysis of the method is carried out under new, general assumptions on the strain rate-shear stress law, which encompass several common examples such as the power-law and Carreau-Yasuda models. Numerical examples complete the exposition.

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arXiv2021

On mathematical and numerical modelling of multiphysics wave propagation with polytopal Discontinuous Galerkin methods

In this work we review discontinuous Galerkin finite element methods on polytopal grids (PolydG) for the numerical simulation of multiphysics wave propagation phenomena in heterogeneous media. In particular, we address wave phenomena in elastic, poro-elastic, and poro-elasto-acoustic materials. Wave propagation is modeled by using either the elastodynamics equation in the elastic domain, the acoustics equations in the acoustic domain and the low-frequency Biot's equations in the poro-elastic one. The coupling between different models is realized by means of (physically consistent) transmission conditions, weakly imposed at the interface between the subdomains. For all models configuration, we introduce and analyse the PolydG semi-discrete formulation, which is then coupled with suitable time marching schemes. For the semi-discrete problem, we present the stability analysis and derive a-priori error estimates in a suitable energy norm. A wide set of verification tests with manufactured solutions are presented in order to validate the error analysis. Examples of physical interest are also shown to demonstrate the capability of the proposed methods.

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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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arXiv2019

An abstract analysis framework for monolithic discretisations of poroelasticity with application to Hybrid High-Order methods

In this work, we introduce a novel abstract framework for the stability and convergence analysis of fully coupled discretisations of the poroelasticity problem and apply it to the analysis of Hybrid High-Order (HHO) schemes. A relevant feature of the proposed framework is that it rests on mild time regularity assumptions that can be derived from an appropriate weak formulation of the continuous problem. To the best of our knowledge, these regularity results for the Biot problem are new. A novel family of HHO discretisation schemes is proposed and analysed, and their performance numerically evaluated.

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arXiv2019

Numerical approximation of poroelasticity with random coefficients using Polynomial Chaos and Hybrid High-Order methods

In this work, we consider the Biot problem with uncertain poroelastic coefficients. The uncertainty is modelled using a finite set of parameters with prescribed probability distribution. We present the variational formulation of the stochastic partial differential system and establish its well-posedness. We then discuss the approximation of the parameter-dependent problem by non-intrusive techniques based on Polynomial Chaos decompositions. We specifically focus on sparse spectral projection methods, which essentially amount to performing an ensemble of deterministic model simulations to estimate the expansion coefficients. The deterministic solver is based on a Hybrid High-Order discretization supporting general polyhedral meshes and arbitrary approximation orders. We numerically investigate the convergence of the probability error of the Polynomial Chaos approximation with respect to the level of the sparse grid. Finally, we assess the propagation of the input uncertainty onto the solution considering an injection-extraction problem.

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arXiv2019

A high-order discretization of nonlinear poroelasticity

In this work we construct and analyze a nonconforming high-order discretization method for the quasi-static single-phase nonlinear poroelasticity problem describing Darcean flow in a deformable porous medium saturated by a slightly compressible fluid. The nonlinear elasticity operator is discretized using a Hybrid High-Order method, while the Darcy operator relies on a Symmetric Weighted Interior Penalty discontinuous Galerkin scheme. The method is valid in two and three space dimensions, delivers an inf-sup stable discretization on general meshes including polyhedral elements and nonmatching interfaces, supports arbitrary approximation orders, and has a reduced cost thanks to the possibility of statically condensing a large subset of the unknowns for linearized versions of the problem. Moreover, the proposed construction can handle both nonzero and vanishing specific storage coefficients.

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arXiv2019

A low-order nonconforming method for linear elasticity on general meshes

In this work we construct a low-order nonconforming approximation method for linear elasticity problems supporting general meshes and valid in two and three space dimensions. The method is obtained by hacking the Hybrid High-Order method, that requires the use of polynomials of degree $k\ge1$ for stability. Specifically, we show that coercivity can be recovered for $k=0$ by introducing a novel term that penalises the jumps of the displacement reconstruction across mesh faces. This term plays a key role in the fulfillment of a discrete Korn inequality on broken polynomial spaces, for which a novel proof valid for general polyhedral meshes is provided. Locking-free error estimates are derived for both the energy- and the $L^2$-norms of the error, that are shown to convergence, for smooth solutions, as $h$ and $h^2$, respectively (here, $h$ denotes the meshsize). A thorough numerical validation on a complete panel of two- and three-dimensional test cases is provided.

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arXiv2017

Equilibrated stress tensor reconstruction and a posteriori error estimation for nonlinear elasticity

We consider hyperelastic problems and their numerical solution using a conforming finite element discretization and iterative linearization algorithms. For these problems, we present equilibrated, weakly symmetric, $H(\rm{div)}$-conforming stress tensor reconstructions, obtained from local problems on patches around vertices using the Arnold--Falk--Winther finite element spaces. We distinguish two stress reconstructions, one for the discrete stress and one representing the linearization error. The reconstructions are independent of the mechanical behavior law. Based on these stress tensor reconstructions, we derive an a posteriori error estimate distinguishing the discretization, linearization, and quadrature error estimates, and propose an adaptive algorithm balancing these different error sources. We prove the efficiency of the estimate, and confirm it on a numerical test with analytical solution for the linear elasticity problem. We then apply the adaptive algorithm to a more application-oriented test, considering the Hencky--Mises and an isotropic damage models.

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arXiv2017

A Hybrid High-Order method for nonlinear elasticity

In this work we propose and analyze a novel Hybrid High-Order discretization of a class of (linear and) nonlinear elasticity models in the small deformation regime which are of common use in solid mechanics. The proposed method is valid in two and three space dimensions, it supports general meshes including polyhedral elements and nonmatching interfaces, enables arbitrary approximation order, and the resolution cost can be reduced by statically condensing a large subset of the unknowns for linearized versions of the problem. Additionally, the method satisfies a local principle of virtual work inside each mesh element, with interface tractions that obey the law of action and reaction. A complete analysis covering very general stress-strain laws is carried out, and optimal error estimates are proved. Extensive numerical validation on model test problems is also provided on two types of nonlinear models.

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arXiv2016

A nonconforming high-order method for the Biot problem on general meshes

In this work, we introduce a novel algorithm for the Biot problem based on a Hybrid High-Order discretization of the mechanics and a Symmetric Weighted Interior Penalty discretization of the flow. The method has several assets, including, in particular, the support of general polyhedral meshes and arbitrary space approximation order. Our analysis delivers stability and error estimates that hold also when the specific storage coefficient vanishes, and shows that the constants have only a mild dependence on the heterogeneity of the permeability coefficient. Numerical tests demonstrating the performance of the method are provided.

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