A robust layer adapted difference method for singularly perturbed two-parameter parabolic problems

A finite difference method for a time-dependent singularly perturbed convection-diffusion-reaction problem involving two small parameters in one space dimension is considered. We use the classical implicit Euler method for time discretization and upwind scheme on the Shishkin-Bakhvalov mesh for spatial discretization. The method is analysed for convergence and is shown to be uniform with respect to both the perturbation parameters. The use of the Shishkin-Bakhvalov mesh gives first-order convergence unlike the Shishkin mesh where convergence is deteriorated due to the presence of a logarithmic factor. Numerical results are presented to validate the theoretical estimates obtained.

Uniformly convergent finite element and finite difference methods for singularly perturbed ordinary differential equations

This thesis is concerned with uniformly convergent finite element and finite difference methods for numerically solving singularly perturbed two-point boundary value problems. We examine the following four problems: (i) high order problem of reaction-diffusion type; (ii) high order problem of convection-diffusion type; (iii) second order interior turning point problem; (iv) semilinear reaction-diffusion problem. Firstly, we consider high order problems of reaction-diffusion type and convection-diffusion type. Under suitable hypotheses, the coercivity of the associated bilinear forms is proved and representation results for the solutions of such problems are given. It is shown that, on an equidistant mesh, polynomial schemes cannot achieve a high order of convergence which is uniform in the perturbation parameter. Piecewise polynomial Galerkin finite element methods are then constructed on a Shishkin mesh. High order convergence results, which are uniform in the perturbation parameter, are obtained in various norms. Secondly, we investigate linear second order problems with interior turning points. Piecewise linear Galerkin finite element methods are generated on various piecewise equidistant meshes designed for such problems. These methods are shown to be convergent, uniformly in the singular perturbation parameter, in a weighted energy norm and the usual L2 norm. Finally, we deal with a semilinear reaction-diffusion problem. Asymptotic properties of solutions to this problem are discussed and analysed. Two simple finite difference schemes on Shishkin meshes are applied to the problem. They are proved to be uniformly convergent of second order and fourth order respectively. Existence and uniqueness of a solution to both schemes are investigated. Numerical results for the above methods are presented.

Uniformly convergent finite element methods for singularly perturbed parabolic partial differential equations

This thesis is concerned with uniformly convergent finite element methods for numerically solving singularly perturbed parabolic partial differential equations in one space variable. First, we use Petrov-Galerkin finite element methods to generate three schemes for such problems, each of these schemes uses exponentially fitted elements in space. Two of them are lumped and the other is non-lumped. On meshes which are either arbitrary or slightly restricted, we derive global energy norm and L2 norm error bounds, uniformly in the diffusion parameter. Under some reasonable global assumptions together with realistic local assumptions on the solution and its derivatives, we prove that these exponentially fitted schemes are locally uniformly convergent, with order one, in a discrete L∞norm both outside and inside the boundary layer. We next analyse a streamline diffusion scheme on a Shishkin mesh for a model singularly perturbed parabolic partial differential equation. The method with piecewise linear space-time elements is shown, under reasonable assumptions on the solution, to be convergent, independently of the diffusion parameter, with a pointwise accuracy of almost order 5/4 outside layers and almost order 3/4 inside the boundary layer. Numerical results for the above schemes are presented. Finally, we examine a cell vertex finite volume method which is applied to a model time-dependent convection-diffusion problem. Local errors away from all layers are obtained in the l2 seminorm by using techniques from finite element analysis.

The MESH book : landscape / infrastructure

Resulting from a series of student-run 'Edge' conferences that have been held in Australia and New Zealand (beginning at RMIT in 1983), The Mesh Book is a collection of essays grouped into themes of Invisible Infrastructures (systems of belief), Immanent Infrastructures (natural systems) and Present Infrastructures (roads and services). Ranging from esoteric discussions to analytical case studies, the book assembles a broad spectrum of ideas on the landscape within the context of Australia and a contemporary study of place.

Gateways placement in backbone wireless mesh networks

This paper presents a novel algorithm for the gateway placement problem in Backbone Wireless Mesh Networks (BWMNs). Different from existing algorithms, the new algorithm incrementally identifies gateways and assigns mesh routers to identified gateways. The new algorithm can guarantee to find a feasible gateway placement satisfying Quality-of-Service (QoS) constraints, including delay constraint, relay load constraint and gateway capacity constraint. Experimental results show that its performance is as good as that of the best of existing algorithms for the gateway placement problem. But, the new algorithm can be used for BWMNs that do not form one connected component, and it is easy to implement and use.

A point interpolation method with locally smoothed strain field (PIM-LS2) for mechanics problems using triangular mesh

A point interpolation method with locally smoothed strain field (PIM-LS2) is developed for mechanics problems using a triangular background mesh. In the PIM-LS2, the strain within each sub-cell of a nodal domain is assumed to be the average strain over the adjacent sub-cells of the neighboring element sharing the same field node. We prove theoretically that the energy norm of the smoothed strain field in PIM-LS2 is equivalent to that of the compatible strain field, and then prove that the solution of the PIM- LS2 converges to the exact solution of the original strong form. Furthermore, the softening effects of PIM-LS2 to system and the effects of the number of sub-cells that participated in the smoothing operation on the convergence of PIM-LS2 are investigated. Intensive numerical studies verify the convergence, softening effects and bound properties of the PIM-LS2, and show that the very ‘‘tight’’ lower and upper bound solutions can be obtained using PIM-LS2.

Biomimetic tubular nanofiber mesh and platelet rich plasma-mediated delivery of BMP-7 for large bone defect regeneration.

There is a growing need for successful bone tissue engineering strategies and advanced biomaterials that mimic the structure and function of native tissues carry great promise. Successful bone repair approaches may include an osteoconductive scaffold, osteoinductive growth factors, cells with an osteogenic potential and capacity for graft vascularisation. To increase osteoinductivity of biomaterials, the local combination and delivery of growth factors has been developed. In the present study we investigated the osteogenic effects of calcium phosphate (CaP)-coated nanofiber mesh tube-mediated delivery of BMP-7 from a PRP matrix for the regeneration of critical sized segmental bone defects in a small animal model. Bilateral full-thickness diaphyseal segmental defects were created in twelve male Lewis rats and nanofiber mesh tubes were placed around the defect. Defects received either treatment with a CaP-coated nanofiber mesh tube (n = 6), an un-coated nanofiber mesh tube (n=6) a CaP-coated nanofiber mesh tube with PRP (n=6) or a CaP-coated nanofiber mesh tube in combination with 5 μg BMP-7 and PRP (n = 6). After 12 weeks, bone volume and biomechanical properties were evaluated using radiography, microCT, biomechanical testing and histology. The results demonstrated significantly higher biomechanical properties and bone volume for the BMP group compared to the control groups. These results were supported by the histological evaluations, where BMP group showed the highest rate of bone regeneration within the defect. In conclusion, BMP-7 delivery via PRP enhanced functional bone defect regeneration, and together these data support the use of BMP-7 in the treatment of critical sized defects.

Adaptive finite element analysis of elliptic problems based on bubble-type local mesh generation

A new mesh adaptivity algorithm that combines a posteriori error estimation with bubble-type local mesh generation (BLMG) strategy for elliptic differential equations is proposed. The size function used in the BLMG is defined on each vertex during the adaptive process based on the obtained error estimator. In order to avoid the excessive coarsening and refining in each iterative step, two factor thresholds are introduced in the size function. The advantages of the BLMG-based adaptive finite element method, compared with other known methods, are given as follows: the refining and coarsening are obtained fluently in the same framework; the local a posteriori error estimation is easy to implement through the adjacency list of the BLMG method; at all levels of refinement, the updated triangles remain very well shaped, even if the mesh size at any particular refinement level varies by several orders of magnitude. Several numerical examples with singularities for the elliptic problems, where the explicit error estimators are used, verify the efficiency of the algorithm. The analysis for the parameters introduced in the size function shows that the algorithm has good flexibility.

The microwell-mesh: A novel device and protocol for the high throughput manufacturing of cartilage microtissues

Microwell platforms are frequently described for the efficient and uniform manufacture of 3-dimensional (3D) multicellular microtissues. Multiple partial or complete medium exchanges can displace microtissues from discrete microwells, and this can result in either the loss of microtissues from culture, or microtissue amalgamation when displaced microtissues fall into common microwells. Herein we describe the first microwell platform that incorporates a mesh to retain microtissues within discrete microwells; the microwell-mesh. We show that bonding a nylon mesh with an appropriate pore size over the microwell openings allows single cells to pass through the mesh into the microwells during the seeding process, but subsequently retains assembled microtissues within discrete microwells. To demonstrate the utility of this platform, we used the microwell-mesh to manufacture hundreds of cartilage microtissues, each formed from 5 × 10(3) bone marrow-derived mesenchymal stem/stromal cells (MSC). The microwell-mesh enabled reliable microtissue retention over 21-day cultures that included multiple full medium exchanges. Cartilage-like matrix formation was more rapid and homogeneous in microtissues than in conventional large diameter control cartilage pellets formed from 2 × 10(5) MSC each. The microwell-mesh platform offers an elegant mechanism to retain microtissues in microwells, and we believe that this improvement will make this platform useful in 3D culture protocols that require multiple medium exchanges, such as those that mimic specific developmental processes or complex sequential drug exposures.

Implementation of Art1 and Art2 Artificial Neural Networks on Ring and Mesh Architectures

The Artificial Neural Networks (ANNs) are being used to solve a variety of problems in pattern recognition, robotic control, VLSI CAD and other areas. In most of these applications, a speedy response from the ANNs is imperative. However, ANNs comprise a large number of artificial neurons, and a massive interconnection network among them. Hence, implementation of these ANNs involves execution of computer-intensive operations. The usage of multiprocessor systems therefore becomes necessary. In this article, we have presented the implementation of ART1 and ART2 ANNs on ring and mesh architectures. The overall system design and implementation aspects are presented. The performance of the algorithm on ring, 2-dimensional mesh and n-dimensional mesh topologies is presented. The parallel algorithm presented for implementation of ART1 is not specific to any particular architecture. The parallel algorithm for ARTE is more suitable for a ring architecture.

Mesh-free approximations via the error reproducing kernel method and applications to nonlinear systems developing shocks

Error estimates for the error reproducing kernel method (ERKM) are provided. The ERKM is a mesh-free functional approximation scheme [A. Shaw, D. Roy, A NURBS-based error reproducing kernel method with applications in solid mechanics, Computational Mechanics (2006), to appear (available online)], wherein a targeted function and its derivatives are first approximated via non-uniform rational B-splines (NURBS) basis function. Errors in the NURBS approximation are then reproduced via a family of non-NURBS basis functions, constructed using a polynomial reproduction condition, and added to the NURBS approximation of the function obtained in the first step. In addition to the derivation of error estimates, convergence studies are undertaken for a couple of test boundary value problems with known exact solutions. The ERKM is next applied to a one-dimensional Burgers equation where, time evolution leads to a breakdown of the continuous solution and the appearance of a shock. Many available mesh-free schemes appear to be unable to capture this shock without numerical instability. However, given that any desired order of continuity is achievable through NURBS approximations, the ERKM can even accurately approximate functions with discontinuous derivatives. Moreover, due to the variation diminishing property of NURBS, it has advantages in representing sharp changes in gradients. This paper is focused on demonstrating this ability of ERKM via some numerical examples. Comparisons of some of the results with those via the standard form of the reproducing kernel particle method (RKPM) demonstrate the relative numerical advantages and accuracy of the ERKM.

Multipath Dissemination in Regular Mesh Topologies

Mesh topologies are important for large-scale peer-to-peer systems that use low-power transceivers. The Quality of Service (QoS) in such systems is known to decrease as the scale increases. We present a scalable approach for dissemination that exploits all the shortest paths between a pair of nodes and improves the QoS. Despite th presence of multiple shortest paths in a system, we show that these paths cannot be exploited by spreading the messages over the paths in a simple round-robin manner; nodes along one of these paths will always handle more messages than the nodes along the other paths. We characterize the set of shortest paths between a pair of nodes in regular mesh topologies and derive rules, using this characterization, to effectively spread the messages over all the available paths. These rules ensure that all the nodes that are at the same distance from the source handle roughly the same number of messages. By modeling the multihop propagation in the mesh topology as a multistage queuing network, we present simulation results from a variety of scenarios that include link failures and propagation irregularities to reflect real-world characteristics. Our method achieves improved QoS in all these scenarios.

Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh

A numerical scheme is presented for accurate simulation of fluid flow using the lattice Boltzmann equation (LBE) on unstructured mesh. A finite volume approach is adopted to discretize the LBE on a cell-centered, arbitrary shaped, triangular tessellation. The formulation includes a formal, second order discretization using a Total Variation Diminishing (TVD) scheme for the terms representing advection of the distribution function in physical space, due to microscopic particle motion. The advantage of the LBE approach is exploited by implementing the scheme in a new computer code to run on a parallel computing system. Performance of the new formulation is systematically investigated by simulating four benchmark flows of increasing complexity, namely (1) flow in a plane channel, (2) unsteady Couette flow, (3) flow caused by a moving lid over a 2D square cavity and (4) flow over a circular cylinder. For each of these flows, the present scheme is validated with the results from Navier-Stokes computations as well as lattice Boltzmann simulations on regular mesh. It is shown that the scheme is robust and accurate for the different test problems studied.