Discrete-to-continuum limits of planar disclinations

通过 Airy 应力函数方法对楔形向错和刃型位错系统进行半离散建模

Semi-discrete modeling of systems of wedge disclinations and edge dislocations via the Airy stress function method

本文基于 Airy 应力函数与变分框架建立平面楔形向错、向错偶极子和刃型位错模型，并证明芯半径趋于零时二者能量渐近等价。

部分爆破的SU（3）Toda系统的连续解

A continuum of solutions for the SU(3) Toda System exhibiting partial blow-up

本文研究了平面区域上Dirichlet边界条件下的Toda系统。我们证明了连续的解决方案，其中一个组件是爆破在一定数量的点的存在。证明使用奇异扰动方法。

Continuum field theory for the deformations of planar kirigami

Mechanical metamaterials exhibit exotic properties at the system level, that emerge from the interactions of many nearly rigid building blocks. Determining these emergent properties theoretically has remained an open challenge outside of a few select examples. Here, for a large class of periodic and planar kirigami, we provide a coarse-graining rule linking the design of the panels and slits to the kirigami's macroscale deformations. The procedure gives a system of nonlinear partial differential equations (PDE) expressing geometric compatibility of angle functions related to the motion of indi…

$Γ$-convergence analysis of a generalized $XY$ model: fractional vortices and string defects

We propose and analyze a generalized two dimensional $XY$ model, whose interaction potential has $n$ weighted wells, describing corresponding symmetries of the system. As the lattice spacing vanishes, we derive by $Γ$-convergence the discrete-to-continuum limit of this model. In the energy regime we deal with, the asymptotic ground states exhibit fractional vortices, connected by string defects. The $Γ$-limit takes into account both contributions, through a renormalized energy, depending on the configuration of fractional vortices, and a surface energy, proportional to the length of the string…

作为首尾对称自旋系统极限的Q张量连续能

Q-tensor continuum energies as limits of head-to-tail-symmetric spin systems

考虑了一类自旋型离散系统，分析了当晶格间距趋于零时系统的连续极限。在标准的连续性和增长假设以及一个额外的头到尾对称性条件下，我们观察到，这个极限可以方便地写为$Q$ -张量空间中的一个功能。我们进一步表征极限能量密度在几种情况下（无论是在2和3维）。在平面的情况下，我们还开发了一个二阶理论，我们根据所选择的缩放导出梯度或浓度型模型。