1,673 research outputs found
Numerical Methods for Two-Dimensional Stem Cell Tissue Growth.
Growth of developing and regenerative biological tissues of different cell types is usually driven by stem cells and their local environment. Here, we present a computational framework for continuum tissue growth models consisting of stem cells, cell lineages, and diffusive molecules that regulate proliferation and differentiation through feedback. To deal with the moving boundaries of the models in both open geometries and closed geometries (through polar coordinates) in two dimensions, we transform the dynamic domains and governing equations to fixed domains, followed by solving for the transformation functions to track the interface explicitly. Clustering grid points in local regions for better efficiency and accuracy can be achieved by appropriate choices of the transformation. The equations resulting from the incompressibility of the tissue is approximated by high-order finite difference schemes and is solved using the multigrid algorithms. The numerical tests demonstrate an overall spatiotemporal second-order accuracy of the methods and their capability in capturing large deformations of the tissue boundaries. The methods are applied to two biological systems: stratified epithelia for studying the effects of two different types of stem cell niches and the scaling of a morphogen gradient with the size of the Drosophila imaginal wing disc during growth. Direct simulations of both systems suggest that that the computational framework is robust and accurate, and it can incorporate various biological processes critical to stem cell dynamics and tissue growth
Improvements to the APBS biomolecular solvation software suite
The Adaptive Poisson-Boltzmann Solver (APBS) software was developed to solve
the equations of continuum electrostatics for large biomolecular assemblages
that has provided impact in the study of a broad range of chemical, biological,
and biomedical applications. APBS addresses three key technology challenges for
understanding solvation and electrostatics in biomedical applications: accurate
and efficient models for biomolecular solvation and electrostatics, robust and
scalable software for applying those theories to biomolecular systems, and
mechanisms for sharing and analyzing biomolecular electrostatics data in the
scientific community. To address new research applications and advancing
computational capabilities, we have continually updated APBS and its suite of
accompanying software since its release in 2001. In this manuscript, we discuss
the models and capabilities that have recently been implemented within the APBS
software package including: a Poisson-Boltzmann analytical and a
semi-analytical solver, an optimized boundary element solver, a geometry-based
geometric flow solvation model, a graph theory based algorithm for determining
p values, and an improved web-based visualization tool for viewing
electrostatics
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Two-phase flow patterns in turbulent flow through a dose diffusion pipe
A numerical investigation is carried out for turbulent particle-laden flow through a dose diffusion pipe for a model reactor system. A Lagrangian Stochastic Monte-Carlo particle-tracking approach and the averaged Reynolds equations with a k-e turbulence model, with a two-layer zonal method in the boundary layer, are used for the disperse and continuous phases. The flow patterns coupled with the particle dynamics are predicted. It is observed that the coupling of the continuous phase with the particle dynamics is important in this case. It was found that the geometry of the throat significantly influences the particle distribution, flow patterns and length of the recirculation region. The accuracy of the simulations depends on the numerical prediction and correction of the fluid phase velocity during a characteristic time interval of the particles. A numerical solution strategy for the computation of two-way momentum coupled flow is discussed. The three test cases show different flow features in the formation of a recirculation region behind the throat. The method will be useful for the qualitative analysis of conceptual designs and their optimisation
Spatially Adaptive Stochastic Methods for Fluid-Structure Interactions Subject to Thermal Fluctuations in Domains with Complex Geometries
We develop stochastic mixed finite element methods for spatially adaptive
simulations of fluid-structure interactions when subject to thermal
fluctuations. To account for thermal fluctuations, we introduce a discrete
fluctuation-dissipation balance condition to develop compatible stochastic
driving fields for our discretization. We perform analysis that shows our
condition is sufficient to ensure results consistent with statistical
mechanics. We show the Gibbs-Boltzmann distribution is invariant under the
stochastic dynamics of the semi-discretization. To generate efficiently the
required stochastic driving fields, we develop a Gibbs sampler based on
iterative methods and multigrid to generate fields with computational
complexity. Our stochastic methods provide an alternative to uniform
discretizations on periodic domains that rely on Fast Fourier Transforms. To
demonstrate in practice our stochastic computational methods, we investigate
within channel geometries having internal obstacles and no-slip walls how the
mobility/diffusivity of particles depends on location. Our methods extend the
applicability of fluctuating hydrodynamic approaches by allowing for spatially
adaptive resolution of the mechanics and for domains that have complex
geometries relevant in many applications
Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential
We present and analyze finite difference numerical schemes for the Allen
Cahn/Cahn-Hilliard equation with a logarithmic Flory Huggins energy potential.
Both the first order and second order accurate temporal algorithms are
considered. In the first order scheme, we treat the nonlinear logarithmic terms
and the surface diffusion term implicitly, and update the linear expansive term
and the mobility explicitly. We provide a theoretical justification that, this
numerical algorithm has a unique solution such that the positivity is always
preserved for the logarithmic arguments. In particular, our analysis reveals a
subtle fact: the singular nature of the logarithmic term around the values of
and 1 prevents the numerical solution reaching these singular values, so
that the numerical scheme is always well-defined as long as the numerical
solution stays similarly bounded at the previous time step. Furthermore, an
unconditional energy stability of the numerical scheme is derived, without any
restriction for the time step size. The unique solvability and the
positivity-preserving property for the second order scheme are proved using
similar ideas, in which the singular nature of the logarithmic term plays an
essential role. For both the first and second order accurate schemes, we are
able to derive an optimal rate convergence analysis, which gives the full order
error estimate. The case with a non-constant mobility is analyzed as well. We
also describe a practical and efficient multigrid solver for the proposed
numerical schemes, and present some numerical results, which demonstrate the
robustness of the numerical schemes
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Hydrodynamic Analysis of Binary Immiscible Metallurgical Flow in a Novel Mixing Process: Rheomixing
This paper presents a hydrodynamic analysis of binary immiscible metallurgical flow by a numerical simulation of the rheomixing process. The concept of multi-controll is proposed for classifying complex processes and identifying individual processes in an immiscible alloy system in order to perform simulations. A brief review of fabrication methods for immiscible alloys is given, and fluid flow aspects of a novel fabrication method – rheomixing by twin-screw extruder (TSE) are analysed. Fundamental hydrodynamic micro-mechanisms in a TSE are simulated by a piecewise linear (PLIC) volume-of-fluid (VOF) method coupled with the continuum surface force (CFS) algorithm. This revealed that continuous reorientation in the TSE process could produce fine droplets and the best mixing efficiency. It is verified that TSE is a better mixing device than single screw extruder (SSE) and can achieve finer droplets. Numerical results show good qualitative agreement with experimental results. It is concluded that rheomixing by a TSE can be successfully employed for casting immiscible engineering alloys due to its unique characteristics of reorientation and surface renewal
Efficient Multigrid Preconditioners for Atmospheric Flow Simulations at High Aspect Ratio
Many problems in fluid modelling require the efficient solution of highly
anisotropic elliptic partial differential equations (PDEs) in "flat" domains.
For example, in numerical weather- and climate-prediction an elliptic PDE for
the pressure correction has to be solved at every time step in a thin spherical
shell representing the global atmosphere. This elliptic solve can be one of the
computationally most demanding components in semi-implicit semi-Lagrangian time
stepping methods which are very popular as they allow for larger model time
steps and better overall performance. With increasing model resolution,
algorithmically efficient and scalable algorithms are essential to run the code
under tight operational time constraints. We discuss the theory and practical
application of bespoke geometric multigrid preconditioners for equations of
this type. The algorithms deal with the strong anisotropy in the vertical
direction by using the tensor-product approach originally analysed by B\"{o}rm
and Hiptmair [Numer. Algorithms, 26/3 (2001), pp. 219-234]. We extend the
analysis to three dimensions under slightly weakened assumptions, and
numerically demonstrate its efficiency for the solution of the elliptic PDE for
the global pressure correction in atmospheric forecast models. For this we
compare the performance of different multigrid preconditioners on a
tensor-product grid with a semi-structured and quasi-uniform horizontal mesh
and a one dimensional vertical grid. The code is implemented in the Distributed
and Unified Numerics Environment (DUNE), which provides an easy-to-use and
scalable environment for algorithms operating on tensor-product grids. Parallel
scalability of our solvers on up to 20,480 cores is demonstrated on the HECToR
supercomputer.Comment: 22 pages, 6 Figures, 2 Table
Computer simulation of glioma growth and morphology
Despite major advances in the study of glioma, the quantitative links between intra-tumor molecular/cellular properties, clinically observable properties such as morphology, and critical tumor behaviors such as growth and invasiveness remain unclear, hampering more effective coupling of tumor physical characteristics with implications for prognosis and therapy. Although molecular biology, histopathology, and radiological imaging are employed in this endeavor, studies are severely challenged by the multitude of different physical scales involved in tumor growth, i.e., from molecular nanoscale to cell microscale and finally to tissue centimeter scale. Consequently, it is often difficult to determine the underlying dynamics across dimensions. New techniques are needed to tackle these issues. Here, we address this multi-scalar problem by employing a novel predictive three-dimensional mathematical and computational model based on first-principle equations (conservation laws of physics) that describe mathematically the diffusion of cell substrates and other processes determining tumor mass growth and invasion. The model uses conserved variables to represent known determinants of glioma behavior, e.g., cell density and oxygen concentration, as well as biological functional relationships and parameters linking phenomena at different scales whose specific forms and values are hypothesized and calculated based on in vitro and in vivo experiments and from histopathology of tissue specimens from human gliomas. This model enables correlation of glioma morphology to tumor growth by quantifying interdependence of tumor mass on the microenvironment (e.g., hypoxia, tissue disruption) and on the cellular phenotypes (e.g., mitosis and apoptosis rates, cell adhesion strength). Once functional relationships between variables and associated parameter values have been informed, e.g., from histopathology or intra-operative analysis, this model can be used for disease diagnosis/prognosis, hypothesis testing, and to guide surgery and therapy. In particular, this tool identifies and quantifies the effects of vascularization and other cell-scale glioma morphological characteristics as predictors of tumor-scale growth and invasion
Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity
Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).Peer ReviewedPostprint (author's final draft
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