1,230 research outputs found
Dirichlet conditions in Poincar\'e-Sobolev inequalities: the sub-homogeneous case
We investigate the dependence of optimal constants in Poincar\'e- Sobolev
inequalities of planar domains on the region where the Dirichlet condition is
imposed. More precisely, we look for the best Dirichlet regions, among closed
and connected sets with prescribed total length (one-dimensional Hausdorff
measure), that make these constants as small as possible. We study their
limiting behaviour, showing, in particular, that Dirichler regions homogenize
inside the domain with comb-shaped structures, periodically distribuited at
different scales and with different orientations. To keep track of these
information we rely on a -convergence result in the class of varifolds.
This also permits applications to reinforcements of anisotropic elastic
membranes. At last, we provide some evidences for a conjecture.Comment: arXiv admin note: text overlap with arXiv:1412.294
Asymptotics of an optimal compliance-location problem
We consider the problem of placing n small balls of given radius in a certain
domain subject to a force f in order to minimize the compliance of the
configuration. Then we let n tend to infinity and look at the asymptotics of
the minimization problem, after properly scaling the functionals involved, and
to the limit distribution of the centres of the balls. This problem is both
linked to optimal location and shape optimization problems.Comment: 20 pages with 2 figures; final accepted version (minor changes, some
extra details on the positivity assumption on
Where best to place a Dirichlet condition in an anisotropic membrane?
We study a shape optimization problem for the first eigenvalue of an elliptic
operator in divergence form, with non constant coefficients, over a fixed
domain . Dirichlet conditions are imposed along and,
in addition, along a set of prescribed length (-dimensional
Hausdorff measure). We look for the best shape and position for the
supplementary Dirichlet region in order to maximize the first
eigenvalue. The limit distribution of the optimal sets, as their prescribed
length tends to infinity, is characterized via -convergence of suitable
functionals defined over varifolds: the use of varifolds, as opposed to
probability measures, allows one to keep track of the local orientation of the
optimal sets (which comply with the anisotropy of the problem), and not just of
their limit distribution.Comment: 23 pages, 2 figure
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
On the Scaling of the Cubic-to-Tetragonal Phase Transformation with Displacement Boundary Conditions
We provide (upper and lower) scaling bounds for a singular perturbation model
for the cubic-to-tetragonal phase transformation with (partial) displacement
boundary data. We illustrate that the order of lamination of the affine
displacement data determines the complexity of the microstructure. As in
\cite{RT21} we heavily exploit careful Fourier space localization methods in
distinguishing between the different lamination orders in the data.Comment: 32 pages, 6 figures, comments welcom
Immersogeometric cardiovascular fluid–structure interaction analysis with divergence-conforming B-splines
This paper uses a divergence-conforming B-spline fluid discretization to address the long-standing issue of poor mass conservation in immersed methods for computational fluid–structure interaction (FSI) that represent the influence of the structure as a forcing term in the fluid subproblem. We focus, in particular, on the immersogeometric method developed in our earlier work, analyze its convergence for linear model problems, then apply it to FSI analysis of heart valves, using divergence-conforming B-splines to discretize the fluid subproblem. Poor mass conservation can manifest as effective leakage of fluid through thin solid barriers. This leakage disrupts the qualitative behavior of FSI systems such as heart valves, which exist specifically to block flow. Divergence-conforming discretizations can enforce mass conservation exactly, avoiding this problem. To demonstrate the practical utility of immersogeometric FSI analysis with divergence-conforming B-splines, we use the methods described in this paper to construct and evaluate a computational model of an in vitro experiment that pumps water through an artificial valve
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