The Couette shear of foams - a summary

Sponsorship: EPSRC, British Council Alliance programme

A mathematical theory of stochastic microlensing. II. Random images, shear, and the Kac-Rice formula

Continuing our development of a mathematical theory of stochastic microlensing, we study the random shear and expected number of random lensed images of different types. In particular, we characterize the first three leading terms in the asymptotic expression of the joint probability density function (pdf) of the random shear tensor due to point masses in the limit of an infinite number of stars. Up to this order, the pdf depends on the magnitude of the shear tensor, the optical depth, and the mean number of stars through a combination of radial position and the star's mass. As a consequence, the pdf's of the shear components are seen to converge, in the limit of an infinite number of stars, to shifted Cauchy distributions, which shows that the shear components have heavy tails in that limit. The asymptotic pdf of the shear magnitude in the limit of an infinite number of stars is also presented. All the results on the random microlensing shear are given for a general point in the lens plane. Extending to the general random distributions (not necessarily uniform) of the lenses, we employ the Kac-Rice formula and Morse theory to deduce general formulas for the expected total number of images and the expected number of saddle images. We further generalize these results by considering random sources defined on a countable compact covering of the light source plane. This is done to introduce the notion of global expected number of positive parity images due to a general lensing map. Applying the result to microlensing, we calculate the asymptotic global expected number of minimum images in the limit of an infinite number of stars, where the stars are uniformly distributed. This global expectation is bounded, while the global expected number of images and the global expected number of saddle images diverge as the order of the number of stars. © 2009 American Institute of Physics.

Multiple Dynamic Processes Contribute to the Complex Steady Shear Behavior of Cross-Linked Supramolecular Networks of Semidilute Entangled Polymer Solutions.

Molecular theories of shear thickening and shear thinning in associative polymer networks are typically united in that they involve a single kinetic parameter that describes the network -- a relaxation time that is related to the lifetime of the associative bonds. Here we report the steady-shear behavior of two structurally identical metallo-supramolecular polymer networks, for which single-relaxation parameter models break down in dramatic fashion. The networks are formed by the addition of reversible cross-linkers to semidilute entangled solutions of PVP in DMSO, and they differ only in the lifetime of the reversible cross-links. Shear thickening is observed for cross-linkers that have a slower dissociation rate (17 s(-1)), while shear thinning is observed for samples that have a faster dissociation rate (ca. 1400 s(-1)). The difference in the steady shear behavior of the unentangled vs. entangled regime reveals an unexpected, additional competing relaxation, ascribed to topological disentanglement in the semidilute entangled regime that contributes to the rheological properties.

Mechanism of Shear Thickening in Reversibly Cross-linked Supramolecular Polymer Networks.

We report here the nonlinear rheological properties of metallo-supramolecular networks formed by the reversible cross-linking of semi-dilute unentangled solutions of poly(4-vinylpyridine) (PVP) in dimethyl sulfoxide (DMSO). The reversible cross-linkers are bis-Pd(II) or bis-Pt(II) complexes that coordinate to the pyridine functional groups on the PVP. Under steady shear, shear thickening is observed above a critical shear rate, and that critical shear rate is experimentally correlated with the lifetime of the metal-ligand bond. The onset and magnitude of the shear thickening depend on the amount of cross-linkers added. In contrast to the behavior observed in most transient networks, the time scale of network relaxation is found to increase during shear thickening. The primary mechanism of shear thickening is ascribed to the shear-induced transformation of intrachain cross-linking to interchain cross-linking, rather than nonlinear high tension along polymer chains that are stretched beyond the Gaussian range.

Changes in landing mechanics in patients following anterior cruciate ligament reconstruction when wearing an extension constraint knee brace.

BACKGROUND: Anterior cruciate ligament (ACL) reconstruction is associated with a high incidence of second tears (graft tears and contralateral ACL tears). These secondary tears have been attributed to asymmetrical lower extremity mechanics. Knee bracing is one potential intervention that can be used during rehabilitation that has the potential to normalize lower extremity asymmetry; however, little is known about the effect of bracing on movement asymmetry in patients following ACL reconstruction. HYPOTHESIS: Wearing a knee brace would increase knee joint flexion and joint symmetry. It was also expected that the joint mechanics would become more symmetrical in the braced condition. OBJECTIVE: To examine how knee bracing affects knee joint function and symmetry over the course of rehabilitation in patients 6 months following ACL reconstruction. STUDY DESIGN: Controlled laboratory study. LEVEL OF EVIDENCE: Level 3. METHODS: Twenty-three adolescent patients rehabilitating from ACL reconstruction surgery were recruited for the study. The subjects all underwent a motion analysis assessment during a stop-jump activity with and without a functional knee brace on the surgical side that resisted extension for 6 months following the ACL reconstruction surgery. Statistical analysis utilized a 2 × 2 (limb × brace) analysis of variance with a significant alpha level of 0.05. RESULTS: Subjects had increased knee flexion on the surgical side when they were braced. The brace condition increased knee flexion velocity, decreased the initial knee flexion angle, and increased the ground reaction force and knee extension moment on both limbs. Side-to-side asymmetry was present across conditions for the vertical ground reaction force and knee extension moment. CONCLUSION: Wearing a knee brace appears to increase lower extremity compliance and promotes normalized loading on the surgical side. CLINICAL RELEVANCE: Knee extension constraint bracing in postoperative ACL patients may improve symmetry of lower extremity mechanics, which is potentially beneficial in progressing rehabilitation and reducing the incidence of second ACL tears.

Intracardiac acoustic radiation force impulse (ARFI) and shear wave imaging in pigs with focal infarctions.

Four pigs, three with focal infarctions in the apical intraventricular septum (IVS) and/or left ventricular free wall (LVFW), were imaged with an intracardiac echocardiography (ICE) transducer. Custom beam sequences were used to excite the myocardium with focused acoustic radiation force (ARF) impulses and image the subsequent tissue response. Tissue displacement in response to the ARF excitation was calculated with a phase-based estimator, and transverse wave magnitude and velocity were each estimated at every depth. The excitation sequence was repeated rapidly, either in the same location to generate 40 Hz M-modes at a single steering angle, or with a modulated steering angle to synthesize 2-D displacement magnitude and shear wave velocity images at 17 points in the cardiac cycle. Both types of images were acquired from various views in the right and left ventricles, in and out of infarcted regions. In all animals, acoustic radiation force impulse (ARFI) and shear wave elasticity imaging (SWEI) estimates indicated diastolic relaxation and systolic contraction in noninfarcted tissues. The M-mode sequences showed high beat-to-beat spatio-temporal repeatability of the measurements for each imaging plane. In views of noninfarcted tissue in the diseased animals, no significant elastic remodeling was indicated when compared with the control. Where available, views of infarcted tissue were compared with similar views from the control animal. In views of the LVFW, the infarcted tissue presented as stiff and non-contractile compared with the control. In a view of the IVS, no significant difference was seen between infarcted and healthy tissue, whereas in another view, a heterogeneous infarction was seen to be presenting itself as non-contractile in systole.

Modelling of industrial multi-physics processes—a key role for computational mechanics.

In the analysis of industrial processes, there is an increasing emphasis on systems governed by interacting continuum phenomena. Mathematical models of such multi-physics processes can only be achieved for practical simulations through computational solution procedures—computational mechanics. Examples of such multi-physics systems in the context of metals processing are used to explore some of the key issues. Finite-volume methods on unstructured meshes are proposed as a means to achieve efficient rapid solutions to such systems. Issues associated with the software design, the exploitation of high performance computers, and the concept of the virtual computational-mechanics modelling laboratory are also addressed in this context.

CAPLib - A 'thin layer' message passing library to support computational mechanics codes on distributed memory parallel systems

The Computer Aided Parallelisation Tools (CAPTools) [Ierotheou, C, Johnson SP, Cross M, Leggett PF, Computer aided parallelisation tools (CAPTools)-conceptual overview and performance on the parallelisation of structured mesh codes, Parallel Computing, 1996;22:163±195] is a set of interactive tools aimed to provide automatic parallelisation of serial FORTRAN Computational Mechanics (CM) programs. CAPTools analyses the user's serial code and then through stages of array partitioning, mask and communication calculation, generates parallel SPMD (Single Program Multiple Data) messages passing FORTRAN. The parallel code generated by CAPTools contains calls to a collection of routines that form the CAPTools communications Library (CAPLib). The library provides a portable layer and user friendly abstraction over the underlying parallel environment. CAPLib contains optimised message passing routines for data exchange between parallel processes and other utility routines for parallel execution control, initialisation and debugging. By compiling and linking with different implementations of the library, the user is able to run on many different parallel environments. Even with today's parallel systems the concept of a single version of a parallel application code is more of an aspiration than a reality. However for CM codes the data partitioning SPMD paradigm requires a relatively small set of message-passing communication calls. This set can be implemented as an intermediate `thin layer' library of message-passing calls that enables the parallel code (especially that generated automatically by a parallelisation tool such as CAPTools) to be as generic as possible. CAPLib is just such a `thin layer' message passing library that supports parallel CM codes, by mapping generic calls onto machine specific libraries (such as CRAY SHMEM) and portable general purpose libraries (such as PVM an MPI). This paper describe CAPLib together with its three perceived advantages over other routes: - as a high level abstraction, it is both easy to understand (especially when generated automatically by tools) and to implement by hand, for the CM community (who are not generally parallel computing specialists); - the one parallel version of the application code is truly generic and portable; - the parallel application can readily utilise whatever message passing libraries on a given machine yield optimum performance.

CFD approach for solid mechanics analysis

A cell-centred finite volume(CC-FV) solid mechanics formulation, based on a computational fluid dynamics(CFD) procedure, is presented. A CFD code is modified such that the velocity variable is used as to the displacement variable. Displacement and pressure fields are considered as unknown variables. The results are validated with finite element(FE) and cell-vertex finite volume(CV-FV) predictions based on discretisation of the equilibrium equations. The developed formulation is applicable for both compressible and incompressible solids behaviour. The method is general and can be extended for the simultaneous analysis of problems involving flow-thermal and stress effects.

Multiphase mesh partitioning for parallel computational mechanics codes

We consider the load-balancing problems which arise from parallel scientific codes containing multiple computational phases, or loops over subsets of the data, which are separated by global synchronisation points. We motivate, derive and describe the implementation of an approach which we refer to as the multiphase mesh partitioning strategy to address such issues. The technique is tested on example meshes containing multiple computational phases and it is demonstrated that our method can achieve high quality partitions where a standard mesh partitioning approach fails.

Dynamic solid mechanics using finite volume methods

A procedure for evaluating the dynamic structural response of elastic solid domains is presented. A prerequisite for the analysis of dynamic fluid–structure interaction is the use of a consistent set of finite volume (FV) methods on a single unstructured mesh. This paper describes a three-dimensional (3D) FV, vertex-based method for dynamic solid mechanics. A novel Newmark predictor–corrector implicit scheme was developed to provide time accurate solutions and the scheme was evaluated on a 3D cantilever problem. By employing a small amount of viscous damping, very accurate predictions of the fundamental natural frequency were obtained with respect to both the amplitude and period of oscillation. This scheme has been implemented into the multi-physics modelling software framework, PHYSICA, for later application to full dynamic fluid structure interaction.

Dynamic mesh partitioning and load-balancing for parallel computational mechanics codes

In this Chapter we discuss the load-balancing issues arising in parallel mesh based computational mechanics codes for which the processor loading changes during the run. We briefly touch on geometric repartitioning ideas and then focus on different ways of using a graph both to solve the load-balancing problem and the optimisation problem, both locally and globally. We also briefly discuss whether repartitioning is always valid. Sample illustrative results are presented and we conclude that repartitioning is an attractive option if the load changes are not too dramatic and that there is a certain trade-off between partition quality and volume of data that the underlying application needs to migrate.

A vertex-based finite volume method applied to non-linear material problems in computational solid mechanics

A vertex-based finite volume (FV) method is presented for the computational solution of quasi-static solid mechanics problems involving material non-linearity and infinitesimal strains. The problems are analysed numerically with fully unstructured meshes that consist of a variety of two- and threedimensional element types. A detailed comparison between the vertex-based FV and the standard Galerkin FE methods is provided with regard to discretization, solution accuracy and computational efficiency. For some problem classes a direct equivalence of the two methods is demonstrated, both theoretically and numerically. However, for other problems some interesting advantages and disadvantages of the FV formulation over the Galerkin FE method are highlighted.