Numerical simulations of possible finite time singularities in the incompressible Euler equations: comparison of numerical methods

The numerical simulation of the 3D incompressible Euler equation is analyzed with respect to different integration methods. The numerical schemes we considered include spectral methods with different strategies for dealiasing and two variants of finite difference methods. Based on this comparison, a Kida-Pelz like initial condition is integrated using adaptive mesh refinement and estimates on the necessary numerical resolution are given. This estimate is based on analyzing the scaling behavior similar to the procedure in critical phenomena and present simulations are put into perspective.Comment: Euler equations: 250 years o

A Three-dimensional Deformable Brain Atlas for DBS Targeting. I. Methodology for Atlas Creation and Artifact Reduction.

BackgroundTargeting in deep brain stimulation (DBS) relies heavily on the ability to accurately localize particular anatomic brain structures. Direct targeting of subcortical structures has been limited by the ability to visualize relevant DBS targets.Methods and resultsIn this work, we describe the development and implementation, of a methodology utilized to create a three dimensional deformable atlas for DBS surgery. This atlas was designed to correspond to the print version of the Schaltenbrand-Bailey atlas structural contours. We employed a smoothing technique to reduce artifacts inherent in the print version.ConclusionsWe present the methodology used to create a three dimensional patient specific DBS atlas which may in the future be tested for clinical utility

Hierarchical Large-scale Volume Representation with 3rd-root-of-2 Subdivision and Trivariate B-spline Wavelets

Multiresolution methods provide a means for representing data at multiple levels of detail. They are typically based on a hierarchical data organization scheme and update rules needed for data value computation. We use a data organization that is based on what we call $\sqrt[n]{2}$ subdivision. The main advantage of $\sqrt[n]{2}$ subdivision, compared to quadtree (n=2) or octree (n=3) organizations, is that the number of vertices is only doubled in each subdivision step instead of multiplied by a factor of four or eight, respectively. To update data values we use n-variate B-spline wavelets, which yield better approximations for each level of detail. We develop a lifting scheme for n=2 and n=3 based on the $\sqrt[n]{2}$-subdivision scheme. We obtain narrow masks that provide a basis for out-of-core techniques as well as view-dependent visualization and adaptive, localized refinement

Design and Topology Optimisation of Tissue Scaffolds

Tissue restoration by tissue scaffolding is an emerging technique with many potential applications. While it is well-known that the structural properties of tissue scaffolds play a critical role in cell regrowth, it is usually unclear how optimal tissue regeneration can be achieved. This thesis hereby presents a computational investigation of tissue scaffold design and optimisation. This study proposes an isosurface-based characterisation and optimisation technique for the design of microscopic architecture, and a porosity-based approach for the design of macroscopic structure. The goal of this study is to physically define the optimal tissue scaffold construct, and to establish any link between cell viability and scaffold architecture. Single-objective and multi-objective topology optimisation was conducted at both microscopic and macroscopic scales to determine the ideal scaffold design. A high quality isosurface modelling technique was formulated and automated to define the microstructure in stereolithography format. Periodic structures with maximised permeability, and theoretically maximum diffusivity and bulk modulus were found using a modified level set method. Microstructures with specific effective diffusivity were also created by means of inverse homogenisation. Cell viability simulation was subsequently conducted to show that the optimised microstructures offered a more viable environment than those with random microstructure. The cell proliferation outcome in terms of cell number and survival rate was also improved through the optimisation of the macroscopic porosity profile. Additionally artificial vascular systems were created and optimised to enhance diffusive nutrient transport. The formation of vasculature in the optimisation process suggests that natural vascular systems acquire their fractal shapes through self-optimisation

Simulating cosmic metal enrichment by the first galaxies

We study cosmic metal enrichment via AMR hydrodynamical simulations in a (10 Mpc/h)$^3$ volume following the Pop III-Pop II transition and for different Pop III IMFs. We have analyzed the joint evolution of metal enrichment on galactic and intergalactic scales at z=6 and z=4. Galaxies account for <9% of the baryonic mass; the remaining gas resides in the diffuse phases: (a) voids, i.e. regions with extremely low density ($\Delta$<1), (b) the true intergalactic medium (IGM, 1<$\Delta$<10) and (c) the circumgalactic medium (CGM, 10<$\Delta<10^{2.5}$), the interface between the IGM and galaxies. By z=6 a galactic mass-metallicity relation is established. At z=4, galaxies with a stellar mass $M_*=10^{8.5}M_\odot$ show log(O/H)+12=8.19, consistent with observations. The total amount of heavy elements rises from $\Omega^{SFH}_Z=1.52\, 10^{-6}$ at z=6 to 8.05 $10^{-6}$ at z=4. Metals in galaxies make up to ~0.89 of such budget at z=6; this fraction increases to ~0.95 at z=4. At z=6 (z=4) the remaining metals are distributed in CGM/IGM/voids with the following mass fractions: 0.06/0.04/0.01 (0.03/0.02/0.01). Analogously to galaxies, at z=4 a density-metallicity ($\Delta$-Z) relation is in place for the diffuse phases: the IGM/voids have a spatially uniform metallicity, Z~$10^{-3.5}$Zsun; in the CGM Z steeply rises with density up to ~$10^{-2}$Zsun. In all diffuse phases a considerable fraction of metals is in a warm/hot (T>$10^{4.5}$K) state. Due to these physical conditions, CIV absorption line experiments can probe only ~2% of the total carbon present in the IGM/CGM; however, metal absorption line spectra are very effective tools to study reionization. Finally, the Pop III star formation history is almost insensitive to the chosen Pop III IMF. Pop III stars are preferentially formed in truly pristine (Z=0) gas pockets, well outside polluted regions created by previous star formation episodes.Comment: 23 pages, 18 figures, 3 tables, Accepted for publication in MNRA

Design and Topology Optimisation of Tissue Scaffolds

Tissue restoration by tissue scaffolding is an emerging technique with many potential applications. While it is well-known that the structural properties of tissue scaffolds play a critical role in cell regrowth, it is usually unclear how optimal tissue regeneration can be achieved. This thesis hereby presents a computational investigation of tissue scaffold design and optimisation. This study proposes an isosurface-based characterisation and optimisation technique for the design of microscopic architecture, and a porosity-based approach for the design of macroscopic structure. The goal of this study is to physically define the optimal tissue scaffold construct, and to establish any link between cell viability and scaffold architecture. Single-objective and multi-objective topology optimisation was conducted at both microscopic and macroscopic scales to determine the ideal scaffold design. A high quality isosurface modelling technique was formulated and automated to define the microstructure in stereolithography format. Periodic structures with maximised permeability, and theoretically maximum diffusivity and bulk modulus were found using a modified level set method. Microstructures with specific effective diffusivity were also created by means of inverse homogenisation. Cell viability simulation was subsequently conducted to show that the optimised microstructures offered a more viable environment than those with random microstructure. The cell proliferation outcome in terms of cell number and survival rate was also improved through the optimisation of the macroscopic porosity profile. Additionally artificial vascular systems were created and optimised to enhance diffusive nutrient transport. The formation of vasculature in the optimisation process suggests that natural vascular systems acquire their fractal shapes through self-optimisation

Real-Time High-Quality Image to Mesh Conversion for Finite Element Simulations

Technological Advances in Medical Imaging have enabled the acquisition of images accurately describing biological tissues. Finite Element (FE) methods on these images provide the means to simulate biological phenomena such as brain shift registration, respiratory organ motion, blood flow pressure in vessels, etc. FE methods require the domain of tissues be discretized by simpler geometric elements, such as triangles in two dimensions, tetrahedra in three, and pentatopes in four. This exact discretization is called a mesh . The accuracy and speed of FE methods depend on the quality and fidelity of the mesh used to describe the biological object. Elements with bad quality introduce numerical errors and slower solver convergence. Also, analysis based on poor fidelity meshes do not yield accurate results specially near the surface. In this dissertation, we present the theory and the implementation of both a sequential and a parallel Delaunay meshing technique for 3D and ---for the first time--- 4D space-time domains. Our method provably guarantees that the mesh is a faithful representation of the multi-tissue domain in topological and geometric sense. Moreover, we show that our method generates graded elements of bounded radius-edge and aspect ratio, which renders our technique suitable for Finite Element analysis. A notable feature of our implementation is speed and scalability. The single-threaded performance of our 3D code is faster than the state of the art open source meshing tools. Experimental evaluation shows a more than 82% weak scaling efficiency for up to 144 cores, reaching a rate of more than 14.3 million elements per second. This is the first 3D parallel Delaunay refinement method to achieve such a performance, on either distributed or shared-memory architectures. Lastly, this dissertation is the first to develop and examine the sequential and parallel high-quality and fidelity meshing of general space-time 4D multi-tissue domains

Wavelet representation of contour sets

Journal ArticleWe present a new wavelet compression and multiresolution modeling approach for sets of contours (level sets). In contrast to previous wavelet schemes, our algorithm creates a parametrization of a scalar field induced by its contours and compactly stores this parametrization rather than function values sampled on a regular grid. Our representation is based on hierarchical polygon meshes with subdivision connectivity whose vertices are transformed into wavelet coefficients. From this sparse set of coefficients, every set of contours can be efficiently reconstructed at multiple levels of resolution. When applying lossy compression, introducing high quantization errors, our method preserves contour topology, in contrast to compression methods applied to the corresponding field function. We provide numerical results for scalar fields defined on planar domains. Our approach generalizes to volumetric domains, time-varying contours, and level sets of vector fields