40 research outputs found
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
On the Virtual Element Method for Topology Optimization on polygonal meshes: a numerical study
It is well known that the solution of topology optimization problems may be
affected both by the geometric properties of the computational mesh, which can
steer the minimization process towards local (and non-physical) minima, and by
the accuracy of the method employed to discretize the underlying differential
problem, which may not be able to correctly capture the physics of the problem.
In light of the above remarks, in this paper we consider polygonal meshes and
employ the virtual element method (VEM) to solve two classes of paradigmatic
topology optimization problems, one governed by nearly-incompressible and
compressible linear elasticity and the other by Stokes equations. Several
numerical results show the virtues of our polygonal VEM based approach with
respect to more standard methods
A virtual element method for the vibration problem of Kirchhoff plates
The aim of this paper is to develop a virtual element method (VEM) for the
vibration problem of thin plates on polygonal meshes. We consider a variational
formulation relying only on the transverse displacement of the plate and
propose an conforming discretization by means of the VEM which is
simple in terms of degrees of freedom and coding aspects. Under standard
assumptions on the computational domain, we establish that the resulting
schemeprovides a correct approximation of the spectrum and prove optimal order
error estimates for the eigenfunctions and a double order for the eigenvalues.
The analysis restricts to simply connected polygonal clamped plates, not
necessarily convex. Finally, we report several numerical experiments
illustrating the behaviour of the proposed scheme and confirming our
theoretical results on different families of meshes. Additional examples of
cases not covered by our theory are also presented