VEM and topology optimization on polygonal meshes

Topology optimization is a fertile area of research that is mainly concerned with the automatic generation of optimal layouts to solve design problems in Engineering. The classical formulation addresses the problem of finding the best distribution of an isotropic material that minimizes the work of the external loads at equilibrium, while respecting a constraint on the assigned amount of volume. This is the so-called minimum compliance formulation that can be conveniently employed to achieve stiff truss-like layout within a two-dimensional domain. A classical implementation resorts to the adoption of four node displacement-based finite elements that are coupled with an elementwise discretization of the (unknown) density field. When regular meshes made of square elements are used, well-known numerical instabilities arise, see in particular the so-called checkerboard patterns. On the other hand, when unstructured meshes are needed to cope with geometry of any shape, additional instabilities can steer the optimizer towards local minima instead of the expected global one. Unstructured meshes approximate the strain energy of truss-like members with an accuracy that is strictly related to the geometrical features of the discretization, thus remarkably affecting the achieved layouts. In this paper we will consider several benchmarks of truss design and explore the performance of the recently proposed technique known as the Virtual Element Method (VEM) in driving the topology optimization procedure. In particular, we will show how the capability of VEM of efficiently approximating elasticity equations on very general polygonal meshes can contribute to overcome the aforementioned mesh-dependent instabilities exhibited by classical finite element based discretization technique

A stream virtual element formulation of the Stokes problem on polygonal meshes

In this paper we propose and analyze a novel stream formulation of the virtual element method (VEM) for the solution of the Stokes problem. The new formulation hinges upon the introduction of a suitable stream function space (characterizing the divergence free subspace of discrete velocities) and it is equivalent to the velocity-pressure (inf-sup stable) mimetic scheme presented in [L. Beir\ue3o da Veiga et al., J. Comput. Phys., 228(2009), pp. 7215-7232] (up to a suitable reformulation into the VEM framework). Both schemes are thus stable and linearly convergent but the new method results to be more desirable as it employs much less degrees of freedom and it is based on a positive definite algebraic problem. Several numerical experiments assess the convergence properties of the new method and show its computational advantages with respect to the mimetic one

Numerical performance of discontinuous and stabilized continuous Galerkin methods for convection–diffusion problems

We compare the performance of two classes of numerical methods for the approximation of linear steady–state convection–diffusion equations, namely, the discontinuous Galerkin (DG) method and the continuous stream- line upwind Petrov-Galerkin (SUPG) method. We present a fair comparison of such schemes considering both diffusion–dominated and convection–dominated regimes, and present numerical results obtained on a series of test problems including smooth solutions, and test cases with sharp internal and boundary layers

Matematica applicata ai tessuti: uno stile tutto da dimostrare

Pubblicato sula rivista Newton, n.2 (2010

Finite Element Approximation of the Sobolev Constant

Denoting by S the sharp constant in the Sobolev inequality in W1,2 0 (B), being B the unit ball in ℝ, and denoting by Sh its approximation in a suitable finite element space, we show that Sh converges to S as h {south east double arrow} 0 with a polynomial rate of convergence. We provide both an upper and a lower bound on the rate of convergence, and present some numerical results

Schwarz methods for Interior Penalty approximations to elliptic problems

Electroni

Multiplicative Schwarz algorithms for symmetric discontinuous Galerkin methods

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A Class of Domain Decomposition Preconditioners for hp-Discontinuous Galerkin Finite Element Methods

In this article we address the question of efficiently solving the algebraic linear system of equations arising from the discretization of a symmetric, elliptic boundary value problem using hp-version discontinuous Galerkin finite element methods. In particular, we introduce a class of domain decomposition preconditioners based on the Schwarz framework, and prove bounds on the condition number of the resulting iteration operators. Numerical results confirming the theoretical estimates are also presented