Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. II: The incompressible Navier-Stokes equations

This paper presents the construction of a correct-energy stabilized finite element method for the incompressible Navier-Stokes equations. The framework of the methodology and the correct-energy concept have been developed in the convective--diffusive context in the preceding paper [M.F.P. ten Eikelder, I. Akkerman, Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. I: The convective--diffusive context, Comput. Methods Appl. Mech. Engrg. 331 (2018) 259--280]. The current work extends ideas of the preceding paper to build a stabilized method within the variational multiscale (VMS) setting which displays correct-energy behavior. Similar to the convection--diffusion case, a key ingredient is the proper dynamic and orthogonal behavior of the small-scales. This is demanded for correct energy behavior and links the VMS framework to the streamline-upwind Petrov-Galerkin (SUPG) and the Galerkin/least-squares method (GLS). The presented method is a Galerkin/least-squares formulation with dynamic divergence-free small-scales (GLSDD). It is locally mass-conservative for both the large- and small-scales separately. In addition, it locally conserves linear and angular momentum. The computations require and employ NURBS-based isogeometric analysis for the spatial discretization. The resulting formulation numerically shows improved energy behavior for turbulent flows comparing with the original VMS method.Comment: Update to postprint versio

Thermoforming of foam sheet

Thermoforming is a widely used process for the manufacture of foam sheet products. Polystyrene foam food trays for instance can be produced by first heating the thermoplastic foam sheet, causing the gas contained to build up pressure and expand, after which a vacuum pressure can be applied to draw the sheet in the required form on the mould. This production method appears to be a very sensitive process with respect to e.g. the sheet temperature, the pressures applied and the cooling time. More problems can be foreseen when for environmental reasons the blowing agent will be adapted (for instance replaced by a gas with a lower molecular weight). To gain more insight in the occuring phenomena the large deformations of a foam structure have been analysed using finite element modelling. To this end a constitutive model has to be defined. Starting from the basic theory given by Gibson & Ashby [1], the behaviour of a closed cubic cell has been elaborated for large strains. The total stiffness is then the sum of the contributions of the edges and faces of the cell and the gas contained in it. The large deformations cause anisotropy of the cells [2], which influences their tangential stiffness. The constitutive model developed here includes the effects of internal gas pressure and the evolving anisotropy

A space-time framework for periodic flows with applications to hydrofoils

In this paper we propose a space-time framework for the computation of periodic flows. We employ the isogeometric analysis framework to achieve higher-order smoothness in both space and time. The discretization is performed using residual-based variational multiscale modelling and weak boundary conditions are adopted to enhance the accuracy near the moving boundaries of the computational domain. We show conservation properties and present a conservative method for force extraction. We apply our framework to the computation of a heaving and pitching hydrofoil. Numerical results display very accurate results on course meshes

From Golgi body movement to cellulose microfibril alignment

The shape and strength of plant cells is determined by a combination of turgor pressure and constraining cell wall. The main load bearing structures in the cell wall, cellulose microfibrils (CMFs), are deposited in highly organized textures. For more than 50 years scientists have tried to elucidate how the organized CMF textures are being generated and what role cortical microtubules (CMTs) play in CMF deposition. In 2006 Paredez et al. caused a breakthrough by live imaging of cellulose synthase (CESA) complexes that move along CMTs. However, the mechanism by which CMTs guide CESA complexes is still unknown at the moment of writing this thesis and many questions related to the CMF organization still are unanswered. This thesis illuminates the mechanism behind the highly organized CMF textures. To analyze the positioning and patterning of CESA complexes in the cell we studied the following three aspects: (1) the distribution and delivery (close) to the plasma membrane of CESA complexes via Golgi bodies, (2) the distribution and movement of CESA complexes inside the plasma membrane while producing CMFs and finally (3) their product, the CMF texture of the cell wall. We chose epidermal root cells of Arabidopsis thaliana and compared cells of different growth stages. Chapter 1 is an introduction into cellulose deposition and an outline of this thesis. In chapter 2 the movement and distribution of Golgi bodies is studied in the cortex of cells of different growth stages, early elongation zone compared to late elongation zone, in relation to the configuration of the actin cytoskeleton. Golgi bodies in the cortex of cells in the early elongation zone, where growth accelerates to rapid growth, show slow random oriented movement, called wiggling. In the cortex of cells in the late elongation zone, where cell elongation ceases, they also show a second kind of motility, fast directed movement with velocities of up to 7 µm.s-1, like in cytoplasmic strands in the same cells. The cortical areas where Golgi body movement is slow and random co-localize with fine F-actin, a configuration of single or thin bundles of filaments. On the other hand, areas where Golgi body movement is fast and directed co-localize with thick actin filament bundles. When Golgi bodies enter an area with a different actin cytoskeleton configuration they change their type of motility concomitantly. We conclude that Golgi body dynamics correlate with the actin cytoskeleton organization. CESA complexes are known to run in rows along CMTs in Arabidopsis hypocotyl cells. In chapter 3 we studied the orientation, density, alignment and movement of CMTs and CESA complexes using immunocytochemistry and live cell imaging. Furthermore we studied the orientation and density of the product of the CESA complexes, the CMFs, in the innermost wall layer with Field Emission Scanning Electron Microscopy (FESEM). The CMTs, the tracks of CESA complexes and the innermost CMFs lay in the same orientation, approximately transverse to the elongation axis in both the inner and outer periclinal cell face in the elongation zone and root hair zone, where cell elongation ceases. CESA complexes predominantly move in rows along CMTs in both directions. While the CMFs form a uniform cell wall layer, CESA complexes run one after the other along CMTs that are wider spread from each other than the CMFs and only few CESA complexes move in between the CMTs. To understand how CESA complexes can produce a uniform layer of CMFs, instead of local CMF thickenings, we studied whether the CMTs change position during CMF production. Time lapse movies of CMTs show that CMTs reposition over time, so that CESA complexes produce an even CMF layer. In this way we can understand how the density of CMFs in the nascent cell wall can be higher than that of the CMTs and the moving rows of CMFs in the plasma membrane. CMFs are deposited consecutively next to earlier deposited ones in the same orientation. In chapter 4 we used several different electron microscopy techniques to visualize CMF texture: transmission Electron Microscopy (TEM) of ultrathin sections after mild or complete matrix extraction, TEM of surface preparations and FESEM of surface preparations. We used root hairs of three different species; Arabidopsis thaliana, Medicago truncatula and Vicia sativa. We compare and discuss the results of the techniques for the capacity to measure orientation, density, length and width of the CMFs. In ultrathin sections and surface preparations we observed that the three species studied have root hairs with an axial/helical wall texture. Surface preparations are best suitable for density and orientation measurements of CMFs within the most inner cell wall layer. Ultrathin sections showed that the thickness of CMFs in Arabidopsis is approximately 3 nm. which indicates that these CMFs are produced by single CESA complexes. Chapter 5 is a general discussion of our work in relation to the field. It describes the role of the actin cytoskeleton , Golgi body motility and CMTs in the deposition of an organized texture of CMFs. </p