31,255 research outputs found
Curvilinear Virtual Elements for 2D solid mechanics applications
No full textIn the present work we generalize the curvilinear Virtual Element technology, introduced for a simple linear scalar problem in a previous work, to generic 2D solid mechanic problems in small deformations. Such generalization also includes the development of a novel Virtual Element space for displacements that contains rigid body motions. Our approach can accept a generic black-box (elastic or inelastic) constitutive algorithm and, in addition, can make use of curved edges thus leading to an exact approximation of the geometry. Rigorous theoretical interpolation properties for the new space on curvilinear elements are derived. We develop an extensive numerical test campaign, both on elastic and inelastic problems, to assess the behavior of the scheme. The results are very promising and underline the advantages of the curved VEM approach over the standard one in the presence of curved geometries
Vorticity-stabilized virtual elements for the Oseen equation
No full textIn this paper, we extend the divergence-free VEM of [L. Beiraõ da Veiga, C. Lovadina and G. Vacca, Virtual elements for the Navier-Stokes problem on polygonal meshes, SIAM J. Numer. Anal. 56 (2018) 1210-1242] to the Oseen problem, including a suitable stabilization procedure that guarantees robustness in the convection-dominated case without disrupting the divergence-free property. The stabilization is inspired from [N. Ahmed, G. R. Barrenechea, E. Burman, J. Guzman, A. Linke and C. Merdon, A pressure-robust discretization of Oseen's equation using stabilization in the vorticity equation, SIAM J. Numer. Anal. 59 (2021) 2746-2774] and includes local SUPG-like terms of the vorticity equation, internal jump terms for the velocity gradients, and an additional VEM stabilization. We derive theoretical convergence results that underline the robustness of the scheme in different regimes, including the convection-dominated case. Furthermore, as in the non-stabilized case, the influence of the pressure on the velocity error is moderate, as it appears only through higher-order terms
A family of three-dimensional virtual elements with applications to magnetostatics
No full textWe consider, as a simple model problem, the application of Virtual Element Methods (VEM) to the linear Magnetostatic three-dimensional problem in the formulation of F. Kikuchi. In doing so, we also introduce new serendipity VEM spaces, where the serendipity reduction is made only on the faces of a general polyhedral decomposition (assuming that internal degrees of freedom could be more easily eliminated by static condensation). These new spaces are meant, more generally, for the combined approximation of -conforming (0-forms), -conforming (1-forms), and -conforming (2-forms) functional spaces in three dimensions, and they could surely be useful for other problems and in more general contexts
A denoising tool for the reconstruction of cortical geometries from MRI
No full textThe reconstruction of individual geometries from medical imaging is quite a standard in the framework of patient-specific medicine. A major drawback in such a context is represented by noise inherent to the data acquisition. Low signal-to-noise ratios can negatively impact extraction algorithms, and result in artefacts or poor quality of the reconstructed meshes. Direct application of numerical methods on such meshes can yield misleading results. Indeed, artefacts and badly shaped elements may corrupt numerical simulations or induce relevant errors in the computation of meaningful geometrical quantities, such as the curvature or the geodesic surface distance. In this paper, we propose a denoising procedure to remove artefacts from a triangular mesh of a three-dimensional closed surface which represents a brain cortex. For this purpose, we combine a smoothing technique (i.e., the Taubin or the HC-Laplacian smoothing) with an edge-flipping algorithm. To control the denoising procedure, we introduce a stopping criterion that takes into account both the improvement of the mesh quality and the loss of volume enclosed by the surface. On a brain cortical surface reconstructed from Magnetic Resonance Imaging (MRI) data, we first perform a tuning analysis of the parameters involved in the smoothing algorithm, then we investigate the effectiveness of the denoising procedure. Finally, as an example of relevant geometrical feature, we study the improvement generated by the proposed algorithm on the computation of the cortical curvature
SUPG-stabilized virtual elements for diffusion-convection problems: A robustness analysis
No full textThe objective of this contribution is to develop a convergence analysis for SUPG-stabilized Virtual Element Methods in diffusion-convection problems that is robust also in the convection dominated regime. For the original method introduced in [Benedetto et al., CMAME 2016] we are able to show an "almost uniform"error bound (in the sense that the unique term that depends in an unfavourable way on the parameters is damped by a higher order mesh-size multiplicative factor). We also introduce a novel discretization of the convection term that allows us to develop error estimates that are fully robust in the convection dominated cases. We finally present some numerical result
Lowest order Virtual Element approximation of magnetostatic problems
No full textWe give here a simplified presentation of the lowest order Serendipity Virtual Element method, and show its use for the numerical solution of linear magneto-static problems in three dimensions. The method can be applied to very general decompositions of the computational domain (as is natural for Virtual Element Methods) and uses as unknowns the (constant) tangential component of the magnetic field H on each edge, and the vertex values of the Lagrange multiplier p (used to enforce the solenoidality of the magnetic induction B). In this respect the method can be seen as the natural generalization of the lowest order Edge Finite Element Method (the so-called "first kind Nedelec'' elements) to polyhedra of almost arbitrary shape, and as we show on some numerical examples it exhibits very good accuracy (for being a lowest order element) and excellent robustness with respect to distortions
Pressure robust SUPG-stabilized finite elements for the unsteady Navier–Stokes equation
No full textIn the present contribution, we propose a novel conforming finite element scheme for the time-dependent Navier–Stokes equation, which is proven to be both convection quasi-robust and pressure robust. The method is built combining a "divergence-free" velocity/pressure couple (such as the Scott-Vogelius element), a discontinuous Galerkin in time approximation and a suitable streamline upwind Petrov–Galerkin-curl stabilization. A set of numerical tests, in accordance with the theoretical results, is included
The Stokes complex for Virtual Elements in three dimensions
No full textThis paper has two objectives. On one side, we develop and test numerically divergence-free Virtual Elements in three dimensions, for variable "polynomial" order. These are the natural extension of the two-dimensional divergence-free VEM elements, with some modification that allows for a better computational efficiency. We test the element's performance both for the Stokes and (diffusion dominated) Navier-Stokes equation. The second, and perhaps main, motivation is to show that our scheme, also in three dimensions, enjoys an underlying discrete Stokes complex structure. We build a pair of virtual discrete spaces based on general polytopal partitions, the first one being scalar and the second one being vector valued, such that when coupled with our velocity and pressure spaces, yield a discrete Stokes complex
Robust Finite Elements for Linearized Magnetohydrodynamics
No full textWe introduce a pressure robust Finite Element Method for the linearized Magnetohydrodynamics equations in three space dimensions, which is provably quasi-robust also in the presence of high fluid and magnetic Reynolds numbers. The proposed scheme uses a non-conforming BDM approach with suitable DG terms for the fluid part, combined with an H1-conforming choice for the magnetic fluxes. The method introduces also a specific CIP-type stabilization associated to the coupling terms. Finally, the theoretical result are further validated by numerical experiments
Pressure and convection robust finite elements for magnetohydrodynamics
No full textWe propose and analyze two convection quasi-robust and pressure robust finite element methods for a fully nonlinear time-dependent magnetohydrodynamics problem. Both methods employ the conforming BDM element coupled with an appropriate pressure space guaranteeing the exact diagram for the fluid part, and the conforming Lagrange element for the approximation of the magnetic fluxes, and make use of suitable DG upwind terms and CIP stabilizations to handle the fluid and magnetic convective terms. The main difference between the two approaches here proposed (labeled as three-field scheme and four field-scheme respectively) lies in the strategy adopted to enforce the divergence-free condition of the magnetic field. The three-field scheme implements a grad-div stabilization, whereas the four-field scheme introduces a suitable Lagrange multiplier and additional stabilization terms in the formulation. The developed error estimates for the two schemes are uniform in both diffusion parameters and optimal with respect to the diffusive norm. Furthermore, in the convection dominated regime, being k the degree of the method and h the mesh size, we are able to prove and pre-asymptotic error reduction rate for the three-field scheme and four-filed scheme respectively. A set of numerical tests support our theoretical findings
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