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The rapid advances in 3D scanning and acquisition techniques have given rise to the explosive increase of volumetric digital models in recent years. This dissertation systematically trailblazes a novel volumetric modeling framework to…
Knitting is an effective technique for producing complex three-dimensional surfaces owing to the inherent flexibility of interlooped yarns and recent advances in manufacturing providing better control of local stitch patterns. Fully…
If the Universe has non-trivial spatial topology, observables depend on both the parameters of the spatial manifold and the position and orientation of the observer. In infinite Euclidean space, most cosmological observables arise from the…
Topological defects are ubiquitous on surfaces with orientational order fields. Here, we study equilibrium states generated by the feedback between geometry and nematic order on fluid membranes with an integer topological defect. When the…
We consider membranes as fluid deformable surface and allow for higher order geometric terms in the bending energy. The evolution equations are derived and numerically solved using surface finite elements. The higher order geometric terms…
An $(\alpha,\beta)$-manifold $(M,F)$ is a Finsler manifold with the Finsler metric $F$ being defined by a Riemannian metric $\alpha$ and $1$-form $\beta$ on the manifold $M$. In this paper, we classify $n$-dimensional…
A nematic membrane is a sheet with embedded orientational order, which can occur in biological cells, liquid crystal films, manufactured materials, and other soft matter systems. By formulating the free energy of nematic films using tensor…
A fundamental theorem of Laman characterises when a bar-joint framework realised generically in the Euclidean plane admits a non-trivial continuous deformation of its vertices. This has recently been extended in two ways. Firstly to…
This paper presents a phase-field model for simulating the three-dimensional deformation of vesicle membranes, incorporating area-difference elasticity, with constraints on bulk volume and surface area. We develop efficient numerical…
By using a certain second order differential equation, the notion of adapted coordinates on Finsler manifolds is defined and some classifications of complete Finsler manifolds are found. Some examples of Finsler metrics, with positive…
The paper considers a thermodynamically consistent phase-field model of a two-phase flow of incompressible viscous fluids. The model allows for a non-linear dependence of fluid density on the phase-field order parameter. Driven by…
This paper extends sliding-mode control theory to nonlinear systems evolving on smooth manifolds. Building on differential geometric methods, we reformulate Filippov's notion of solutions, characterize well-defined vector fields on quotient…
In this study, we define the generalized normal ruled surface of a curve in the Euclidean 3-space $E^3$. We study the geometry of such surfaces by calculating the Gaussian and mean curvatures to determine when the surface is flat or minimal…
A systematic investigation is presented of grain boundaries and grain boundary networks in two dimensional flexible membranes with crystalline order. An isolated grain boundary undergoes a buckling transition at a critical value of the…
Shape spaces are fundamental in a variety of applications including image registration, morphing, matching, interpolation, and shape optimization. In this work, we consider two-dimensional shapes represented by triangular meshes of a given…
The geometry of a two-dimensional surface in a curved space can be most easily visualized by using an isometric embedding in flat three-dimensional space. Here we present a new method for embedding surfaces with spherical topology in flat…
The isometric embedding of surfaces in three-dimensional space is fundamental to various physical systems, from elastic sheets to programmable materials. While continuous surfaces typically admit unique solutions under suitable boundary…
Manifold learning is a fundamental task at the core of data analysis and visualisation. It aims to capture the simple underlying structure of complex high-dimensional data by preserving pairwise dissimilarities in low-dimensional…
In this paper we study the general affine differential geometry of surfaces in affine space $A^3$. For a regular elliptical surface we define a moving frame of minimal order and get the complete system of differential invariants. As an…
In this paper, we propose a novel meshfree Generalized Finite Difference Method (GFDM) approach to discretize PDEs defined on manifolds. Derivative approximations for the same are done directly on the tangent space, in a manner that mimics…