1,720,987 research outputs found
Bricks for the mixed high-order virtual element method: Projectors and differential operators
No full textWe present the essential tools to deal with virtual element method (VEM) for the approximation of solutions of partial differential equations in mixed form. Functional spaces, degrees of freedom, projectors and differential operators are described emphasizing how to build them in a virtual element framework and for a general approximation order. To achieve this goal, it was necessary to make a deep analysis on polynomial spaces and decompositions. We exploit such “bricks” to construct virtual element approximations of Stokes, Darcy and Navier–Stokes problems and we provide a series of examples to numerically verify the theoretical behaviour of high-order VEM
Virtual Element Method and Optimal Shape Design in Magnetics
No full textWe propose an innovative technique for dealing with optimal shape design problems that exploits the flexibility of the virtual element method (VEM) in handling meshes with general polygonal and polyhedral elements. The shape synthesis of a magnetic pole is considered as the case study
Virtual element method and permanent magnet simulations: Potential and mixed formulations
No full textThe methodological background of the virtual element method is presented and applied to model permanent magnets via the Kikuchi formulation, considering both linear and non-linear magnetic permeability of the ferromagnetic regions. The authors examine several study cases: A permanent magnet in free space, a permanent magnet energising a ferromagnetic core, a four-pole permanent-magnet motor. In order to validate the proposed approach, comparisons with both virtual and finite element potential formulations are presented and discussed
A free-cutting mesh strategy for optimal shape synthesis in magnetics
No full textThe authors propose an innovative technique for dealing with optimal shape design problems that exploits the flexibility of the virtual element method in generating meshes composed of general polygonal and polyhedral elements. Virtual element method and finite element method can coexist on the same discretized domain; therefore, the possibility of dealing with hanging nodes and gluing sub-domain meshes is ensured. Accordingly, the shape synthesis of a magnetic pole is considered as the case study. It is shown that the proposed technique is effective in handling the shape variations dictated by an algorithm of evolutionary optimisation
Curvilinear Virtual Elements for 2D solid mechanics applications
Full text linkIn 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
Bend 3d mixed virtual element method for Darcy problems
No full textIn this study, we propose a virtual element scheme to solve the Darcy problem in three physical dimensions. The main novelty is that curved elements are naturally handled without any degradation of the solution accuracy. Indeed, in presence of curved boundaries, or internal interfaces, the geometrical error introduced by planar approximations may dominate the convergence rate limiting the benefit of high-order approximations. We consider the Darcy problem in its mixed form to directly obtain accurate and mass conservative fluxes without any post-processing. An important step to derive the proposed scheme is the integration over curved polyhedrons, here presented and discussed. Finally, we show the theoretical analysis of the scheme as well as several numerical examples to support our findings
A Virtual Element Method for the Wave Equation on Curved Edges in Two Dimensions
Full text linkIn 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
The Stokes complex for Virtual Elements in three dimensions
Full text linkThis 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
Bend 3d mixed virtual element method for Darcy problems
Full text linkIn this study, we propose a virtual element scheme to solve the Darcy problem in three physical dimensions. The main novelty is that curved elements are naturally handled without any degradation of the solution accuracy. Indeed, in presence of curved boundaries, or internal interfaces, the geometrical error introduced by planar approximations may dominate the convergence rate limiting the benefit of high-order approximations. We consider the Darcy problem in its mixed form to directly obtain accurate and mass conservative fluxes without any post-processing. An important step to derive the proposed scheme is the integration over curved polyhedrons, here presented and discussed. Finally, we show the theoretical analysis of the scheme as well as several numerical examples to support our findings
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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