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Articular human joint modelling
Copyright @ Cambridge University Press 2009.The work reported in this paper encapsulates the theories and algorithms developed to drive the core analysis modules of the software which has been developed to model a musculoskeletal structure of anatomic joints. Due to local bone surface and contact geometry based joint kinematics, newly developed algorithms make the proposed modeller different from currently available modellers. There are many modellers that are capable of modelling gross human body motion. Nevertheless, none of the available modellers offer complete elements of joint modelling. It appears that joint modelling is an extension of their core analysis capability, which, in every case, appears to be musculoskeletal motion dynamics. It is felt that an analysis framework that is focused on human joints would have significant benefit and potential to be used in many orthopaedic applications. The local mobility of joints has a significant influence in human motion analysis, in understanding of joint loading, tissue behaviour and contact forces. However, in order to develop a bone surface based joint modeller, there are a number of major problems, from tissue idealizations to surface geometry discretization and non-linear motion analysis. This paper presents the following: (a) The physical deformation of biological tissues as linear or non-linear viscoelastic deformation, based on spring-dashpot elements. (b) The linear dynamic multibody modelling, where the linear formulation is established for small motions and is particularly useful for calculating the equilibrium position of the joint. This model can also be used for finding small motion behaviour or loading under static conditions. It also has the potential of quantifying the joint laxity. (c) The non-linear dynamic multibody modelling, where a non-matrix and algorithmic formulation is presented. The approach allows handling complex material and geometrical nonlinearity easily. (d) Shortest path algorithms for calculating soft tissue line of action geometries. The developed algorithms are based on calculating minimum âsurface massâ and âsurface covarianceâ. An improved version of the âsurface covarianceâ algorithm is described as âresidual covarianceâ. The resulting path is used to establish the direction of forces and moments acting on joints. This information is needed for linear or non-linear treatment of the joint motion. (e) The final contribution of the paper is the treatment of the collision. In the virtual world, the difficulty in analysing bodies in motion arises due to body interpenetrations. The collision algorithm proposed in the paper involves finding the shortest projected ray from one body to the other. The projection of the body is determined by the resultant forces acting on it due to soft tissue connections under tension. This enables the calculation of collision condition of non-convex objects accurately. After the initial collision detection, the analysis involves attaching special springs (stiffness only normal to the surfaces) at the âpotentially colliding pointsâ and motion of bodies is recalculated. The collision algorithm incorporates the rotation as well as translation. The algorithm continues until the joint equilibrium is achieved. Finally, the results obtained based on the software are compared with experimental results obtained using cadaveric joints
Finding and evaluating community structure in networks
We propose and study a set of algorithms for discovering community structure
in networks -- natural divisions of network nodes into densely connected
subgroups. Our algorithms all share two definitive features: first, they
involve iterative removal of edges from the network to split it into
communities, the edges removed being identified using one of a number of
possible "betweenness" measures, and second, these measures are, crucially,
recalculated after each removal. We also propose a measure for the strength of
the community structure found by our algorithms, which gives us an objective
metric for choosing the number of communities into which a network should be
divided. We demonstrate that our algorithms are highly effective at discovering
community structure in both computer-generated and real-world network data, and
show how they can be used to shed light on the sometimes dauntingly complex
structure of networked systems.Comment: 16 pages, 13 figure
Automatic Deformation of Riemann-Hilbert Problems with Applications to the Painlev\'e II Transcendents
The stability and convergence rate of Olver's collocation method for the
numerical solution of Riemann-Hilbert problems (RHPs) is known to depend very
sensitively on the particular choice of contours used as data of the RHP. By
manually performing contour deformations that proved to be successful in the
asymptotic analysis of RHPs, such as the method of nonlinear steepest descent,
the numerical method can basically be preconditioned, making it asymptotically
stable. In this paper, however, we will show that most of these preconditioning
deformations, including lensing, can be addressed in an automatic, completely
algorithmic fashion that would turn the numerical method into a black-box
solver. To this end, the preconditioning of RHPs is recast as a discrete,
graph-based optimization problem: the deformed contours are obtained as a
system of shortest paths within a planar graph weighted by the relative
strength of the jump matrices. The algorithm is illustrated for the RHP
representing the Painlev\'e II transcendents.Comment: 20 pages, 16 figure
Exposing Multi-Relational Networks to Single-Relational Network Analysis Algorithms
Many, if not most network analysis algorithms have been designed specifically
for single-relational networks; that is, networks in which all edges are of the
same type. For example, edges may either represent "friendship," "kinship," or
"collaboration," but not all of them together. In contrast, a multi-relational
network is a network with a heterogeneous set of edge labels which can
represent relationships of various types in a single data structure. While
multi-relational networks are more expressive in terms of the variety of
relationships they can capture, there is a need for a general framework for
transferring the many single-relational network analysis algorithms to the
multi-relational domain. It is not sufficient to execute a single-relational
network analysis algorithm on a multi-relational network by simply ignoring
edge labels. This article presents an algebra for mapping multi-relational
networks to single-relational networks, thereby exposing them to
single-relational network analysis algorithms.Comment: ISSN:1751-157
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