3D modelling using partial differential equations (PDEs).

Partial differential equations (PDEs) are used in a wide variety of contexts in computer science ranging from object geometric modelling to simulation of natural phenomena such as solar flares, and generation of realistic dynamic behaviour in virtual environments including variables such as motion, velocity and acceleration. A major challenge that has occupied many players in geometric modelling and computer graphics is the accurate representation of human facial geometry in 3D. The acquisition, representation and reconstruction of such geometries are crucial for an extensive range of uses, such as in 3D face recognition, virtual realism presentations, facial appearance simulations and computer-based plastic surgery applications among others. The principle aim of this thesis should be to tackle methods for the representation and reconstruction of 3D geometry of human faces depending on the use of partial differential equations and to enable the compression of such 3D data for faster transmission over the Internet. The actual suggested techniques are based on sampling surface points at the intersection of horizontal and vertical mesh cutting planes. The set of sampled points contains the explicit structure of the cutting planes with three important consequences: 1) points in the plane can be defined as a one dimensional signal and are thus, subject to a number of compression techniques; 2) any two mesh cutting planes can be used as PDE boundary conditions in a rectangular domain; and 3) no connectivity information needs to be coded as the explicit structure of the vertices in 3D renders surface triangulation a straightforward task. This dissertation proposes and demonstrates novel algorithms for compression and uncompression of 3D meshes using a variety of techniques namely polynomial interpolation, Discrete Cosine Transform, Discrete Fourier Transform, and Discrete Wavelet Transform in connection with partial differential equations. In particular, the effectiveness of the partial differential equations based method for 3D surface reconstruction is shown to reduce the mesh over 98.2% making it an appropriate technique to represent complex geometries for transmission over the network

Fourth-order flows in surface modelling

This short article is a brief account of the usage of fourth-order curvature flow in surface modelling

Numerical simulation of conservation laws with moving grid nodes: Application to tsunami wave modelling

In the present article we describe a few simple and efficient finite volume type schemes on moving grids in one spatial dimension combined with appropriate predictor-corrector method to achieve higher resolution. The underlying finite volume scheme is conservative and it is accurate up to the second order in space. The main novelty consists in the motion of the grid. This new dynamic aspect can be used to resolve better the areas with large solution gradients or any other special features. No interpolation procedure is employed, thus unnecessary solution smearing is avoided, and therefore, our method enjoys excellent conservation properties. The resulting grid is completely redistributed according the choice of the so-called monitor function. Several more or less universal choices of the monitor function are provided. Finally, the performance of the proposed algorithm is illustrated on several examples stemming from the simple linear advection to the simulation of complex shallow water waves. The exact well-balanced property is proven. We believe that the techniques described in our paper can be beneficially used to model tsunami wave propagation and run-up.Comment: 46 pages, 7 figures, 7 tables, 94 references. Accepted to Geosciences. Other author's papers can be downloaded at http://www.denys-dutykh.com

Mathematical modelling and numerical simulations of actin dynamics in the eukaryotic cell

The aim of this article is to study cell deformation and cell movement by considering both the mechanical and biochemical properties of the cortical network of actin filaments and its concentration. Actin is a polymer that can exist either in fil- amentous form (F-actin) or in monometric form (G-actin) (Chen et al. 2000) and the filamentous form is arranged in a paired helix of two protofilaments (Ananthakrish- nan et al. 2006). By assuming that cell deformations are a result of the cortical actin dynamics in the cell cytoskeleton, we consider a continuum mathematical model that couples the mechanics of the network of actin filaments with its bio-chemical dy- namics. Numerical treatment of the model is carried out using the moving grid finite element method (Madzvamuse et al. 2003). Furthermore, by assuming slow deforma- tions of the cell, we use linear stability theory to validate the numerical simulation results close to bifurcation points. Far from bifurcation points, we show that the math- ematical model is able to describe the complex cell deformations typically observed in experimental results. Our numerical results illustrate cell expansion, cell contrac- tion, cell translation and cell relocation as well as cell protrusions. In all these results, the contractile tonicity formed by the association of actin filaments to the myosin II motor proteins is identified as a key bifurcation parameter

Modelling cell motility and chemotaxis with evolving surface finite elements

We present a mathematical and a computational framework for the modelling of cell motility. The cell membrane is represented by an evolving surface, with the movement of the cell determined by the interaction of various forces that act normal to the surface. We consider external forces such as those that may arise owing to inhomogeneities in the medium and a pressure that constrains the enclosed volume, as well as internal forces that arise from the reaction of the cells' surface to stretching and bending. We also consider a protrusive force associated with a reaction-diffusion system (RDS) posed on the cell membrane, with cell polarization modelled by this surface RDS. The computational method is based on an evolving surface finite-element method. The general method can account for the large deformations that arise in cell motility and allows the simulation of cell migration in three dimensions. We illustrate applications of the proposed modelling framework and numerical method by reporting on numerical simulations of a model for eukaryotic chemotaxis and a model for the persistent movement of keratocytes in two and three space dimensions. Movies of the simulated cells can be obtained from http://homepages.warwick.ac.uk/maskae/CV_Warwick/Chemotaxis.html

A computational method for the coupled solution of reaction–diffusion equations on evolving domains and manifolds: application to a model of cell migration and chemotaxis

In this paper, we devise a moving mesh finite element method for the approximate solution of coupled bulk–surface reaction–diffusion equations on an evolving two dimensional domain. Fundamental to the success of the method is the robust generation of bulk and surface meshes. For this purpose, we use a novel moving mesh partial differential equation (MMPDE) approach. The developed method is applied to model problems with known analytical solutions; these experiments indicate second-order spatial and temporal accuracy. Coupled bulk–surface problems occur frequently in many areas; in particular, in the modelling of eukaryotic cell migration and chemotaxis. We apply the method to a model of the two-way interaction of a migrating cell in a chemotactic field, where the bulk region corresponds to the extracellular region and the surface to the cell membrane

A Nitsche-based domain decomposition method for hypersingular integral equations

We introduce and analyze a Nitsche-based domain decomposition method for the solution of hypersingular integral equations. This method allows for discretizations with non-matching grids without the necessity of a Lagrangian multiplier, as opposed to the traditional mortar method. We prove its almost quasi-optimal convergence and underline the theory by a numerical experiment.Comment: 21 pages, 5 figure