A Reduced Landau-de Gennes Study for Nematic Equilibria in Three-Dimensional Prisms

We model nematic liquid crystal configurations inside three-dimensional prisms, with a polygonal cross-section and Dirichlet boundary conditions on all prism surfaces. We work in a reduced Landau-de Gennes framework, and the Dirichlet conditions on the top and bottom surfaces are special in the sense, that they are critical points of the reduced Landau-de Gennes energy on the polygonal cross-section. The choice of the boundary conditions allows us to make a direct correspondence between the three-dimensional Landau-de Gennes critical points and pathways on the two-dimensional Landau-de Gennes solution landscape on the polygonal cross-section. We explore this concept by means of asymptotic analysis and numerical examples, with emphasis on a cuboid and a hexagonal prism, focusing on three-dimensional multistability tailored by two-dimensional solution landscapes

Hierarchies of Critical Points of a Landau-de Gennes Free Energy on Three-Dimensional Cuboids

We investigate critical points of a Landau-de Gennes (LdG) free energy in three-dimensional (3D) cuboids, that model nematic equilibria. We develop a hybrid saddle dynamics-based algorithm to efficiently compute solution landscapes of these 3D systems. Our main results concern (a) the construction of 3D LdG critical points from a database of 2D LdG critical points and (b) studies of the effects of cross-section size and cuboid height on solution landscapes. In doing so, we discover multiple-layer 3D LdG critical points constructed by stacking 3D critical points on top of each other, novel pathways between distinct energy minima mediated by 3D LdG critical points and novel metastable escaped solutions, all of which can be tuned for tailor-made static and dynamic properties of confined nematic liquid crystal systems in 3D.Comment: 28 pages,10 figure

Nematic liquid crystals in a rectangular confinement : solution landscape, and bifurcation

We study the solution landscape and bifurcation diagrams of nematic liquid crystals confined on a rectangle, using a reduced two-dimensional Landau–de Gennes framework in terms of two geometry-dependent variables: half short edge length λ and aspect ratio b . First, we analytically prove that, for any b with a small enough λ or for a large enough b with a fixed domain size, there is a unique stable solution that has two line defects on the opposite short edges. Second, we numerically construct solution landscapes by varying λ and b , and report a novel X state, which emerges from saddle-node bifurcation and serves as the parent state in such a solution landscape. Various new classes are then found among these solution landscapes, including the X class, the S class, and the L class. By tracking the Morse indices of individual solutions, we present bifurcation diagrams for nematic equilibria, thus illustrating the emergence mechanism of critical points and several effects of geometrical anisotropy on confined defect patterns

Multistability for nematic liquid crystals in cuboids with degenerate planar boundary conditions

We study nematic configurations within three-dimensional (3D) cuboids, with planar degenerate boundary conditions on the cuboid faces, in the Landau–de Gennes framework. There are two geometry-dependent variables: the edge length of the square cross-section, λ, and the parameter h, which is a measure of the cuboid height. Theoretically, we prove the existence and uniqueness of the global minimizer with a small enough cuboid size. We develop a new numerical scheme for the high-index saddle dynamics to deal with the surface energies. We report on a plethora of (meta)stable states, and their dependence on h and λ, and in particular how the 3D states are connected with their two-dimensional counterparts on squares and rectangles. Notably, we find families of almost uniaxial stable states constructed from the topological classification of tangent unit-vector fields and study transition pathways between them. We also provide a phase diagram of competing (meta)stable states as a function of λ and h

Answer Summarization for Technical Queries: Benchmark and New Approach

Prior studies have demonstrated that approaches to generate an answer summary for a given technical query in Software Question and Answer (SQA) sites are desired. We find that existing approaches are assessed solely through user studies. There is a need for a benchmark with ground truth summaries to complement assessment through user studies. Unfortunately, such a benchmark is non-existent for answer summarization for technical queries from SQA sites. To fill the gap, we manually construct a high-quality benchmark to enable automatic evaluation of answer summarization for technical queries for SQA sites. Using the benchmark, we comprehensively evaluate the performance of existing approaches and find that there is still a big room for improvement. Motivated by the results, we propose a new approach TechSumBot with three key modules:1) Usefulness Ranking module, 2) Centrality Estimation module, and 3) Redundancy Removal module. We evaluate TechSumBot in both automatic (i.e., using our benchmark) and manual (i.e., via a user study) manners. The results from both evaluations consistently demonstrate that TechSumBot outperforms the best performing baseline approaches from both SE and NLP domains by a large margin, i.e., 10.83%-14.90%, 32.75%-36.59%, and 12.61%-17.54%, in terms of ROUGE-1, ROUGE-2, and ROUGE-L on automatic evaluation, and 5.79%-9.23% and 17.03%-17.68%, in terms of average usefulness and diversity score on human evaluation. This highlights that the automatic evaluation of our benchmark can uncover findings similar to the ones found through user studies. More importantly, automatic evaluation has a much lower cost, especially when it is used to assess a new approach. Additionally, we also conducted an ablation study, which demonstrates that each module in TechSumBot contributes to boosting the overall performance of TechSumBot.Comment: Accepted by ASE 202