119 research outputs found
Fast Isogeometric Boundary Element Method based on Independent Field Approximation
An isogeometric boundary element method for problems in elasticity is
presented, which is based on an independent approximation for the geometry,
traction and displacement field. This enables a flexible choice of refinement
strategies, permits an efficient evaluation of geometry related information, a
mixed collocation scheme which deals with discontinuous tractions along
non-smooth boundaries and a significant reduction of the right hand side of the
system of equations for common boundary conditions. All these benefits are
achieved without any loss of accuracy compared to conventional isogeometric
formulations. The system matrices are approximated by means of hierarchical
matrices to reduce the computational complexity for large scale analysis. For
the required geometrical bisection of the domain, a strategy for the evaluation
of bounding boxes containing the supports of NURBS basis functions is
presented. The versatility and accuracy of the proposed methodology is
demonstrated by convergence studies showing optimal rates and real world
examples in two and three dimensions.Comment: 32 pages, 27 figure
Extending BACOLI to solve multi-scale problems
The BACOLI package is a numerical software package for solving parabolic partial differential
equations in one spatial dimension. It implements a B-spline collocation method for the spatial
discretization of a system of partial differential equations. The resultant ordinary differential equations
together with the boundary conditions form a system of differential-algebraic equations. The
differential-algebraic equations are then solved using the DASSL solver. The BACOLI software package
features adaptive error control in the temporal and spatial domains. The estimate of the temporal
error is controlled through the DASSL solver. The estimate of the spatial error is controlled based
on the difference between two solutions computed in the BACOLI software package. This difference
gives an estimation of the error. If this error estimate does not meet the user-supplied tolerance,
then the spatial mesh is changed.
The BACOLI software package can only solve parabolic partial differential equations that depend
on spatial derivatives. In this thesis, the BACOLI software package is modified to solve a broader
spectrum of problems. In fact, after some modifications, the extended BACOLI software package can
solve systems of parabolic partial differential equations and time-dependent equations that do not
depend on spatial derivatives. We apply this extended software package to solve the monodomain
model of cardiac electrophysiology.
The monodomain model is a multi-scale mathematical model for the evolution of the electrical
potential in cardiac tissue that couples the ionic currents at the cellular scale with their propagation
at the tissue scale. Because of their local nature, the mathematical models of a single cell have no
dependency on spatial derivatives whereas the models at the tissue level do.
The heart models considered in our numerical experiments use various cardiac cell models. We
find that solving the heart models through the extended BACOLI software package, in some cases,
leads to a speed-up in comparison with the Chaste software package, which is a powerful, widely
used, and well-respected software package for heart simulation
ICASE semiannual report, April 1 - September 30, 1989
The Institute conducts unclassified basic research in applied mathematics, numerical analysis, and computer science in order to extend and improve problem-solving capabilities in science and engineering, particularly in aeronautics and space. The major categories of the current Institute for Computer Applications in Science and Engineering (ICASE) research program are: (1) numerical methods, with particular emphasis on the development and analysis of basic numerical algorithms; (2) control and parameter identification problems, with emphasis on effective numerical methods; (3) computational problems in engineering and the physical sciences, particularly fluid dynamics, acoustics, and structural analysis; and (4) computer systems and software, especially vector and parallel computers. ICASE reports are considered to be primarily preprints of manuscripts that have been submitted to appropriate research journals or that are to appear in conference proceedings
Investigation of the use of meshfree methods for haptic thermal management of design and simulation of MEMS
This thesis presents a novel approach of using haptic sensing technology combined with virtual environment (VE) for the thermal management of Micro-Electro-Mechanical-Systems (MEMS) design. The goal is to reduce the development cycle by avoiding the costly iterative prototyping procedure. In this regard, we use haptic feedback with virtua lprototyping along with an immersing environment. We also aim to improve the productivity and capability of the designer to better grasp the phenomena operating at the micro-scale level, as well as to augment computational steering through haptic channels. To validate the concept of haptic thermal management, we have implemented a demonstrator with a user friendly interface which allows to intuitively "feel" the temperature ïŹeld through our concept of haptic texturing. The temperature ïŹeld in a simple MEMS component is modeled using ïŹnite element methods (FEM) or ïŹnite difference method (FDM) and the user is able to feel thermal expansion using a combination of different haptic feedback. In haptic application, the force rendering loop needs to be updated at a frequency of 1Khz in order to maintain continuity in the user perception. When using FEM or FDM for our three-dimensional model, the computational cost increases rapidly as the mesh size is reduced to ensure accuracy. Hence, it constrains the complexity of the physical model to approximate temperature or stress ïŹeld solution. It would also be difïŹcult to generate or reïŹne the mesh in real time for CAD process. In order to circumvent the limitations due to the use of conventional mesh-based techniques and to avoid the bothersome task of generating and reïŹning the mesh, we investigate the potential of meshfree methods in the context of our haptic application. We review and compare the different meshfree formulations against FEM mesh based technique. We have implemented the different methods for benchmarking thermal conduction and elastic problems. The main work of this thesis is to determine the relevance of the meshfree option in terms of ïŹexibility of design and computational charge for haptic physical model
Isogeometric approximation of cardiac electrophysiology models on surfaces: An accuracy study with application to the human left atrium
We consider Isogeometric Analysis in the framework of the Galerkin method for the spatial approximation
of cardiac electrophysiology models defined on NURBS surfaces; specifically, we perform a numerical comparison
between basis functions of degree p â„ 1 and globally C
k
-continuous, with k = 0 or p â 1, to find
the most accurate approximation of a propagating front with the minimal number of degrees of freedom.
We show that B-spline basis functions of degree p â„ 1, which are C
pâ1
-continuous capture accurately the
front velocity of the transmembrane potential even with moderately refined meshes; similarly, we show that,
for accurate tracking of curved fronts, high-order continuous B-spline basis functions should be used. Finally,
we apply Isogeometric Analysis to an idealized human left atrial geometry described by NURBS with
physiologically sound fiber directions and anisotropic conductivity tensor to demonstrate that the numerical
scheme retains its favorable approximation properties also in a more realistic setting
Isogeometric Analysis of the electrophysiology in the human heart: Numerical simulation of the bidomain equations on the atria
We consider Isogeometric Analysis (IGA) for the numerical solution of the electrophysiology of the atria, which in this work is modeled by means of the bidomain equations on thin surfaces. First, we consider the bidomain equations coupled with the RogerâMcCulloch ionic model on simple slabs. Here, our goal is to evaluate the effects of the spatial discretization by IGA and the use of different B-spline basis functions on the accuracy of the approximation, in particular regarding the accuracy of the front velocity and the dispersion error. Specifically, we consider basis functions with high polynomial degree, p, and global high order continuity, C^{pâ1}, in the computational domain: our results show that the use of such basis functions is beneficial to the accurate approximation of the solution. Then, we consider a realistic application of the bidomain equations coupled with the CourtemancheâRamirezâNattel ionic model on the two human atria, which are represented by means of two NURBS surfaces
Natureâs Optics and Our Understanding of Light
Optical phenomena visible to everyone abundantly illustrate important ideas in science and mathematics. The phenomena considered include rainbows, sparkling reflections on water, green flashes, earthlight on the moon, glories, daylight, crystals, and the squint moon. The concepts include refraction, wave interference, numerical experiments, asymptotics, Regge poles, polarisation singularities, conical intersections, and visual illusions
Simulation of pore-scale flow using finite element-methods
I present a new finite element (FE) simulation method to simulate pore-scale
flow. Within the pore-space, I solve a simplified form of the incompressible
Navier-Stokeâs equation, yielding the velocity field in a two-step solution
approach. First, Poissonâs equation is solved with homogeneous boundary
conditions, and then the pore pressure is computed and the velocity field
obtained for no slip conditions at the grain boundaries. From the computed
velocity field I estimate the effective permeability of porous media samples
characterized by thin section micrographs, micro-CT scans and synthetically
generated grain packings. This two-step process is much simpler than solving
the full Navier Stokes equation and therefore provides the opportunity to
study pore geometries with hundreds of thousands of pores in a computationally
more cost effective manner than solving the full Navier-Stokeâs equation.
My numerical model is verified with an analytical solution and validated on
samples whose permeabilities and porosities had been measured in laboratory
experiments (Akanji and Matthai, 2010). Comparisons were also made with
Stokes solver, published experimental, approximate and exact permeability
data. Starting with a numerically constructed synthetic grain packings, I also
investigated the extent to which the details of pore micro-structure affect the
hydraulic permeability (Garcia et al., 2009). I then estimate the hydraulic
anisotropy of unconsolidated granular packings.
With the future aim to simulate multiphase flow within the pore-space, I also compute the radii and derive capillary pressure from the Young-Laplace
equation (Akanji and Matthai,2010
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