75 research outputs found
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
Serendipity Nodal VEM spaces
We introduce a new variant of Nodal Virtual Element spaces that mimics the
"Serendipity Finite Element Methods" (whose most popular example is the 8-node
quadrilateral) and allows to reduce (often in a significant way) the number of
internal degrees of freedom. When applied to the faces of a three-dimensional
decomposition, this allows a reduction in the number of face degrees of
freedom: an improvement that cannot be achieved by a simple static
condensation. On triangular and tetrahedral decompositions the new elements
(contrary to the original VEMs) reduce exactly to the classical Lagrange FEM.
On quadrilaterals and hexahedra the new elements are quite similar (and have
the same amount of degrees of freedom) to the Serendipity Finite Elements, but
are much more robust with respect to element distortions. On more general
polytopes the Serendipity VEMs are the natural (and simple) generalization of
the simplicial case
Adaptive Algorithms
Overwhelming empirical evidence in computational science and engineering proved that self-adaptive mesh-generation is a must-do in real-life problem computational partial differential equations. The mathematical understanding of corresponding algorithms concerns the overlap of two traditional mathematical disciplines, numerical analysis and approximation theory, with computational sciences. The half workshop was devoted to the mathematics of optimal convergence rates and instance optimality of the Dörfler marking or the maximum strategy in various versions of space discretisations and time-evolution problems with all kind of applications in the efficient numerical treatment of partial differential equations
Fast Numerical Integration on Polytopic Meshes with Applications to Discontinuous Galerkin Finite Element Methods
In this paper we present efficient quadrature rules for the numerical approximation of integrals of polynomial functions over general polygonal/polyhedral elements that do not require an explicit construction of a sub-tessellation into triangular/tetrahedral elements. The method is based on successive application of Stokes’ theorem; thereby, the underlying integral may be evaluated using only the values of the integrand and its derivatives at the vertices of the polytopic domain, and hence leads to an exact cubature rule whose quadrature points are the vertices of the polytope. We demonstrate the capabilities of the proposed approach by efficiently computing the stiffness and mass matrices arising from hp-version symmetric interior penalty discontinuous Galerkin discretizations of second-order elliptic partial differential equations
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