Linkage Mechanisms Governed by Integrable Deformations of Discrete Space Curves

A linkage mechanism consists of rigid bodies assembled by joints which can be used to translate and transfer motion from one form in one place to another. In this paper, we are particularly interested in a family of spacial linkage mechanisms which consist of $n$-copies of a rigid body joined together by hinges to form a ring. Each hinge joint has its own axis of revolution and rigid bodies joined to it can be freely rotated around the axis. The family includes the famous threefold symmetric Bricard6R linkage also known as the Kaleidocycle, which exhibits a characteristic "turning over" motion. We can model such a linkage as a discrete closed curve in $\mathbb{R}^3$ with a constant torsion up to sign. Then, its motion is described as the deformation of the curve preserving torsion and arc length. We describe certain motions of this object that are governed by the semi-discrete mKdV equations, where infinitesimally the motion of each vertex is confined in the osculating plane

Strukture polja v aktivnih in pasivnih tekočih kristalih

Field structures are developed in passive and active nematic fluids. These are field profiles that are determined by confinement, particles, flow and external fields. Our central methodological approach is numerical modeling based on free energy minimization with finite difference method and flow modeling with hybrid lattice Boltzmann method. We develop structures by combining concepts of topological defects, external confinement and colloidal particles. Ordering properties of horseshoe nematic colloidal particles with planar degenerate anchoring are investigated with numerical modeling, where we optimize their geometrical parameters such that the particle exhibit attractive interactions and can self assemble into 2D and even 3D colloidal crystals. The metamaterial response of horseshoe colloids that perform as split ring resonators is studied. Optical cloaking is achieved by generating polymer microstructures embedded directly within a electric field switchable liquid crystal device. Using numerical modelling we explore the director field structures forming in the vicinity of composite colloidal particles with specially designed conic anchoring, which are assumed to induce high multipoles. Simple rule that allow predictions of multipolar moment from defect configuration is extracted. Starting with a gyroid structure, which is a photonic crystal by itself, we introduce an achiral and chiral nematic into one labyrinth of channels with homeotropic anchoring. Complexly shaped channels induce both ordered and disordered structures of defects. Simulating the passive nematic flow in porous microchannels we study the formation of individual umbilic defects of various strength and umbilic defect lattices that arise as the consequence of complex velocity field containing both multiple peaks and saddles. We investigate the 3D active turbulence in droplets of active nematic with homeotropic and non slip boundary condition. The transition from the point defect to the active turbulence is studied by analysing both the topological defects and corresponding events as well as flow. More generally, this work is aimed at the development of novel functional soft matter, which can exhibit exciting and unusual material characteristics, including light guiding, topological defect states, photonic bandgaps, metamaterials and optical cloaking.V doktorskem delu smo razvili strukture polja v pasivnih in aktivnih nematskih tekočinah. Ti profili v polju so določeni z ograditvijo, delci, tokom in zunanjimi polji. Osrednji raziskovalni pristop je numerično modeliranje, ki temelji na minimizaciji proste energije z metodo končnih diferenc, in modeliranje toka s hibridno mrežno Boltzmannovo metodo. Ustvarjene strukture so rezultat kombinacije topoloških defektov, zunanje ograditve in koloidnih delcev. Preučevali smo urejanje podkvastih koloidnih delcev s planarnim sidranjem. Geometrijske parametre koloidnega delca smo optimizirali tako, da so delci medsebojno interagirali privlačno in so se lahko sestavili v 2D in tudi 3D koloidne kristale. Študirali smo tudi metamaterialni odziv tovrstnih podkvastih koloidov, ki se obnašajo kot resonatorji. Pokazali smo optično zakrivanje z ustvarjanjem polimernih struktur direktno v tekočekristalni celici, nastavljivi z električnim poljem. S pomočjo numeričnega modeliranja smo raziskali strukture v nematskem polju, ki se formirajo v okolici kompozitnih koloidnih delcev s posebnim koničnim sidranjem in ustvarjajo višje multipolne momente. Predstavimo tudi preprosto pravilo, s katerim lahko napovemo multipolni moment samo z opazovanjem defektnih struktur. V enega od obeh prepletov kanalov, v giroidni strukturi, uvedemo kiralni in nekiralni nematski tekoči kristal. Kompleksna oblika kanalov povzroči nastanek tako urejenih, kot tudi neurejenih defektnih struktur. Simuliramo pasivni nematski tok v poroznih mikrokanalih in študiramo nastanek umbiličnih defektov različnih moči ter regularnih mrež umbiličnih defektov, ki nastanejo zaradi sedelnih in ekstremalnih točk v toku. Preučimo 3D aktivno turbulenco v kapljicah aktivnega nematika s homeotropnimi robnimi pogoji. Študiramo prehod iz točkastega defekta v topološko turbulenco z analizo topoloških defektov in topoloških dogodkov, kot tudi z analizo samega toka. To delo je torej namenjeno razvoju nove funkcionalne mehke snovi, ki ima zanimive lastnosti, kot so na primer vodenje svetlobe, topološka defektna stanja, fotonske reže, metamateriali in optično zakrivanje

Constructing IGA-suitable planar parameterization from complex CAD boundary by domain partition and global/local optimization

In this paper, we propose a general framework for constructing IGA-suitable planar B-spline parameterizations from given complex CAD boundaries consisting of a set of B-spline curves. Instead of forming the computational domain by a simple boundary, planar domains with high genus and more complex boundary curves are considered. Firstly, some pre-processing operations including B\'ezier extraction and subdivision are performed on each boundary curve in order to generate a high-quality planar parameterization; then a robust planar domain partition framework is proposed to construct high-quality patch-meshing results with few singularities from the discrete boundary formed by connecting the end points of the resulting boundary segments. After the topology information generation of quadrilateral decomposition, the optimal placement of interior B\'ezier curves corresponding to the interior edges of the quadrangulation is constructed by a global optimization method to achieve a patch-partition with high quality. Finally, after the imposition of C1=G1-continuity constraints on the interface of neighboring B\'ezier patches with respect to each quad in the quadrangulation, the high-quality B\'ezier patch parameterization is obtained by a C1-constrained local optimization method to achieve uniform and orthogonal iso-parametric structures while keeping the continuity conditions between patches. The efficiency and robustness of the proposed method are demonstrated by several examples which are compared to results obtained by the skeleton-based parameterization approach

Distribution on Warp Maps for Alignment of Open and Closed Curves

Alignment of curve data is an integral part of their statistical analysis, and can be achieved using model- or optimization-based approaches. The parameter space is usually the set of monotone, continuous warp maps of a domain. Infinite-dimensional nature of the parameter space encourages sampling based approaches, which require a distribution on the set of warp maps. Moreover, the distribution should also enable sampling in the presence of important landmark information on the curves which constrain the warp maps. For alignment of closed and open curves in $\mathbb{R}^d, d=1,2,3$, possibly with landmark information, we provide a constructive, point-process based definition of a distribution on the set of warp maps of $[0,1]$ and the unit circle $\mathbb{S}^1$ that is (1) simple to sample from, and (2) possesses the desiderata for decomposition of the alignment problem with landmark constraints into multiple unconstrained ones. For warp maps on $[0,1]$, the distribution is related to the Dirichlet process. We demonstrate its utility by using it as a prior distribution on warp maps in a Bayesian model for alignment of two univariate curves, and as a proposal distribution in a stochastic algorithm that optimizes a suitable alignment functional for higher-dimensional curves. Several examples from simulated and real datasets are provided

Measuring cellular traction forces on non-planar substrates

Animal cells use traction forces to sense the mechanics and geometry of their environment. Measuring these traction forces requires a workflow combining cell experiments, image processing and force reconstruction based on elasticity theory. Such procedures have been established before mainly for planar substrates, in which case one can use the Green's function formalism. Here we introduce a worksflow to measure traction forces of cardiac myofibroblasts on non-planar elastic substrates. Soft elastic substrates with a wave-like topology were micromolded from polydimethylsiloxane (PDMS) and fluorescent marker beads were distributed homogeneously in the substrate. Using feature vector based tracking of these marker beads, we first constructed a hexahedral mesh for the substrate. We then solved the direct elastic boundary volume problem on this mesh using the finite element method (FEM). Using data simulations, we show that the traction forces can be reconstructed from the substrate deformations by solving the corresponding inverse problem with a L1-norm for the residue and a L2-norm for 0th order Tikhonov regularization. Applying this procedure to the experimental data, we find that cardiac myofibroblast cells tend to align both their shapes and their forces with the long axis of the deformable wavy substrate.Comment: 34 pages, 9 figure