Modeling phase-transitions using a high-performance, isogeometric analysis framework

In this paper, we present a high-performance framework for solving partial differential equations using Isogeometric Analysis, called PetIGA, and show how it can be used to solve phase-field problems. We specifically chose the Cahn-Hilliard equation, and the phase-field crystal equation as test cases. These two models allow us to highlight some of the main advantages that we have access to while using PetIGA for scientific computing. © The Authors. Published by Elsevier B.V

Refined Isogeometric Analysis for a preconditioned conjugate gradient solver

Starting from a highly continuous Isogeometric Analysis (IGA) discretization, refined Isogeometric Analysis (rIGA) introduces C 0 hyperplanes that act as separators for the direct LU factorization solver. As a result, the total computational cost required to solve the corresponding system of equations using a direct LU factorization solver dramatically reduces (up to a factor of 55) (Garcia et al., 2017). At the same time, rIGA enriches the IGA spaces, thus improving the best approximation error. In this work, we extend the complexity analysis of rIGA to the case of iterative solvers. We build an iterative solver as follows: we first construct the Schur complements using a direct solver over small subdomains (macro-elements). We then assemble those Schur complements into a global skeleton system. Subsequently, we solve this system iteratively using Conjugate Gradients (CG) with an incomplete LU (ILU) preconditioner. For a 2D Poisson model problem with a structured mesh and a uniform polynomial degree of approximation, rIGA achieves moderate savings with respect to IGA in terms of the number of Floating Point Operations (FLOPs) and computational time (in seconds) required to solve the resulting system of linear equations. For instance, for a mesh with four million elements and polynomial degree p=3, the iterative solver is approximately 2.6 times faster (in time) when applied to the rIGA system than to the IGA one. These savings occur because the skeleton rIGA system contains fewer non-zero entries than the IGA one. The opposite situation occurs for 3D problems, and as a result, 3D rIGA discretizations provide no gains with respect to their IGA counterparts when considering iterative solvers

Micropolar fluids using B-spline divergence conforming spaces

We discretized the two-dimensional linear momentum, microrotation, energy and mass conservation equations from micropolar fluids theory, with the finite element method, creating divergence conforming spaces based on B-spline basis functions to obtain pointwise divergence free solutions [8]. Weak boundary conditions were imposed using Nitsche's method for tangential conditions, while normal conditions were imposed strongly. Once the exact mass conservation was provided by the divergence free formulation, we focused on evaluating the differences between micropolar fluids and conventional fluids, to show the advantages of using the micropolar fluid model to capture the features of complex fluids. A square and an arc heat driven cavities were solved as test cases. A variation of the parameters of the model, along with the variation of Rayleigh number were performed for a better understanding of the system. The divergence free formulation was used to guarantee an accurate solution of the flow. This formulation was implemented using the framework PetIGA as a basis, using its parallel stuctures to achieve high scalability. The results of the square heat driven cavity test case are in good agreement with those reported earlier. © The Authors. Published by Elsevier B.V

On round-off error for adaptive finite element methods

Round-off error analysis has been historically studied by analyzing the condition number of the associated matrix. By controlling the size of the condition number, it is possible to guarantee a prescribed round-off error tolerance. However, the opposite is not true, since it is possible to have a system of linear equations with an arbitrarily large condition number that still delivers a small round-off error. In this paper, we perform a round-off error analysis in context of 1D and 2D hp-adaptive Finite Element simulations for the case of Poisson equation. We conclude that boundary conditions play a fundamental role on the round-off error analysis, specially for the so-called 'radical meshes'. Moreover, we illustrate the importance of the right-hand side when analyzing the round-off error, which is independent of the condition number of the matrix. © 2012 Published by Elsevier Ltd

Cytoskeleton structure, pattern of mitochondrial activity and ultrastructure of frozen or vitrified sheep embryos.

Even though sheep embryo cryopreservation is a commonly used procedure the survival and pregnancy outcomes can vary greatly. This study investigated whether cryopreservation was causing subtle changes in ultrastructure, mitochondrial activity or cytoskeletal integrity. Sheep embryos were either slow cooled in 1.5 M EG (n = 22), or vitrified in 20% EG + 20% DMSO with 0.5 M sucrose in Open Pulled Straws (OPS) (n = 24). One hour after warming the cryopreserved embryos differed from control embryos in that they had no mitochondrial activity combined with cytoskeletal disorganization and large vesicles. Vitrified embryos also showed many points of cytoskeleton disruption. Ultrastructural alterations resulting from actin filaments disorganization were observed in both cryopreserved groups. This includes areas presenting no cytoplasmic organelles, Golgi complex located far from the nucleus and a decrease of specialized intercellular junctions. Additionally, large vesicles were observed in vitrified morulae and early blastocysts. The alterations after cryopreservation were proportional to embryo quality as assessed using the stereomicroscope. Even in the absence of mitochondrial activity, grade I and II cryopreserved embryos contained mitochondria with normal ultrastructure. Embryos classified as grade I or II in the stereomicroscope revealed mild ultrastructural alterations, meaning that this tool is efficient to evaluate embryos after cryopreservation

Telescopic hybrid fast solver for 3D elliptic problems with point singularities

This paper describes a telescopic solver for two dimensional h adaptive grids with point singularities. The input for the telescopic solver is an h refined two dimensional computational mesh with rectangular finite elements. The candidates for point singularities are first localized over the mesh by using a greedy algorithm. Having the candidates for point singularities, we execute either a direct solver, that performs multiple refinements towards selected point singularities and executes a parallel direct solver algorithm which has logarithmic cost with respect to refinement level. The direct solvers executed over each candidate for point singularity return local Schur complement matrices that can be merged together and submitted to iterative solver. In this paper we utilize a parallel multi-thread GALOIS solver as a direct solver. We use Incomplete LU Preconditioned Conjugated Gradients (ILUPCG) as an iterative solver. We also show that elimination of point singularities from the refined mesh reduces significantly the number of iterations to be performed by the ILUPCG iterative solver

On the thermodynamics of the Swift–Hohenberg theory

We present the microbalance including the microforces, the first- and second-order microstresses for the Swift–Hohenberg equation concomitantly with their constitutive equations, which are consistent with the free-energy imbalance. We provide an explicit form for the microstress structure for a free-energy functional endowed with second-order spatial derivatives. Additionally, we generalize the Swift–Hohenberg theory via a proper constitutive process. Finally, we present one highly resolved three-dimensional numerical simulation to demonstrate the particular form of the resulting microstresses and their interactions in the evolution of the Swift–Hohenberg equation

Método de extração de DNA de folhas de soja adaptado para larga escala.

O presente trabalho tem por objetivo desenvolver um protocolo de extração de DNA de soja de qualidade e quantidade suficientes, baixo custo e adaptado para larga escala