Applied Analysis
http://hdl.handle.net/20.500.11824/2
Thu, 18 Apr 2019 14:57:26 GMT2019-04-18T14:57:26ZConvex Integration Arising in the Modelling of Shape-Memory Alloys: Some Remarks on Rigidity, Flexibility and Some Numerical Implementations
http://hdl.handle.net/20.500.11824/964
Convex Integration Arising in the Modelling of Shape-Memory Alloys: Some Remarks on Rigidity, Flexibility and Some Numerical Implementations
Rüland, A.; Taylor J. M.; Zillinger, C.
We study convex integration solutions in the context of the modelling of shape-memory
alloys. The purpose of the article is twofold, treating both rigidity and flexibility prop-
erties: Firstly, we relate the maximal regularity of convex integration solutions to the
presence of lower bounds in variational models with surface energy. Hence, variational
models with surface energy could be viewed as a selection mechanism allowing for
or excluding convex integration solutions. Secondly, we present the first numerical
implementations of convex integration schemes for the model problem of the geomet-
rically linearised two-dimensional hexagonal-to-rhombic phase transformation. We
discuss and compare the two algorithms from Rüland et al. (J Elast. 2019. https://doi.
org/10.1007/s10659-018-09719-3; SIAM J Math Anal 50(4):3791–3841, 2018) and
give a numerical estimate of the regularity attained.
Sat, 30 Mar 2019 00:00:00 GMThttp://hdl.handle.net/20.500.11824/9642019-03-30T00:00:00ZTopological singular set of vector-valued maps, I: application to manifold-constrained Sobolev and BV spaces
http://hdl.handle.net/20.500.11824/962
Topological singular set of vector-valued maps, I: application to manifold-constrained Sobolev and BV spaces
Canevari G.; Orlandi G.
We introduce an operator $\mathbf{S}$ on vector-valued maps $u$ which has the ability to capture the relevant topological information carried by $u$.
In particular, this operator is defined on maps that take values in a closed submanifold $\mathcal{N}$ of the Euclidean space $\mathbb{R}^m$, and coincides with the distributional Jacobian
in case $\mathcal{N}$ is a sphere. More precisely, the range of $\mathbf{S}$ is a set of maps whose values are flat chains with coefficients in a suitable normed abelian group. In this paper, we use $\mathbf{S}$ to characterise strong limits of smooth, $\mathcal{N}$-valued maps with respect to Sobolev norms, extending a result by Pakzad and Rivière. We also discuss applications to the study of manifold-valued maps of bounded variation. In a companion paper, we will consider applications to the asymptotic behaviour of minimisers of Ginzburg-Landau type functionals, with $\mathcal{N}$-well potentials.
Sat, 30 Mar 2019 00:00:00 GMThttp://hdl.handle.net/20.500.11824/9622019-03-30T00:00:00ZOrder Reconstruction for neatics on squares with isotropic inclusions: A Landau-de Gennes study
http://hdl.handle.net/20.500.11824/961
Order Reconstruction for neatics on squares with isotropic inclusions: A Landau-de Gennes study
Wang Y.; Canevari G.; Majumdar A.
e study a modified Landau-de Gennes model for nematic liq- uid crystals, where the elastic term is assumed to be of subquadratic growth in the gradient. We analyze the behaviour of global minimizers in two- and three-dimensional domains, subject to uniaxial boundary conditions, in the asymptotic regime where the length scale of the defect cores is small com- pared to the length scale of the domain. We obtain uniform convergence of the minimizers and of their gradients, away from the singularities of the limiting uniaxial map. We also demonstrate the presence of maximally biaxial cores in minimizers on two-dimensional domains, when the temperature is sufficiently low.
Sat, 30 Mar 2019 00:00:00 GMThttp://hdl.handle.net/20.500.11824/9612019-03-30T00:00:00ZThe excluded volume of two-dimensional convex bodies: shape reconstruction and non-uniqueness
http://hdl.handle.net/20.500.11824/941
The excluded volume of two-dimensional convex bodies: shape reconstruction and non-uniqueness
Taylor J. M.
In the Onsager model of one-component hard-particle systems, the entire phase behaviour is dictated by a function of relative orientation, which represents the amount of space excluded to one particle by another at this relative orientation. We term this function the excluded volume function. Within the context of two-dimensional convex bodies, we investigate this excluded volume function for one-component systems addressing two related questions. Firstly, given a body can we find the excluded volume function? Secondly, can we reconstruct a body from its excluded volume function? The former is readily answered via an explicit Fourier series representation, in terms of the support function. However we show the latter question is ill-posed in the sense that solutions are not unique for a large class of bodies. This degeneracy is well characterised however, with two bodies admitting the same excluded volume function if and only if the Fourier coefficients of their support functions differ only in phase. Despite the non-uniqueness issue, we then propose and analyse a method for reconstructing a convex body given its excluded volume function, by means of a discretisation procedure where convex bodies are approximated by zonotopes with a fixed number of sides. It is shown that the algorithm will always asymptotically produce a best least-squares approximation of the trial function, within the space of excluded volume functions of centrally symmetric bodies. In particular, if a solution exists, it can be found. Results from a numerical implementation are presented, showing that with only desktop computing power, good approximations to solutions can be readily found.
Tue, 05 Feb 2019 00:00:00 GMThttp://hdl.handle.net/20.500.11824/9412019-02-05T00:00:00Z