In the present work, a numerical study is conducted in order to investigate the unsteady aerodynamics of finite-span flapping rigid wings. The non-dimensional incompressible Navier-Stokes equations are solved in their velocity-pressure formulation using a second-order accurate in space and time finite-difference scheme. To tackle the problem of moving boundaries, the governing equations are solved on overlapping structured grids. The main goal is to investigate the wake topology and aerodynamic performance of low aspect-ratio flapping wings and their dependence on the flapping parameters (such as flapping frequency, flapping amplitude and Reynolds number). Specifically, the root-flapping motion characteristic of flying animals is studied. The numerical simulations are performed at a Reynolds number of Re = 1000 (a Reynolds number value often found in nature) and at different values of Strouhal number St.

In the present work, we have studied the root-flapping motion characteristic of flying animals, a subject virtually unexplored. It was found that, indeed, root-flapping motion produces wake structures similar to those of heaving or coupled heaving-and-pitching motions, but with the difference that the latter motions generate larger vortices and forces than root-flapping motion, presumably because the average velocity is higher across the span; aside from this, similar wake regimes occurs at similar St. In general, it was found that for values less than St < 0.25 we are in the drag production regime, for values approximately equal to St = 0.25 we produce little or no drag (or thrust), whereas for values higher that St > 0.25 we are in the thrust production regime. 

The simulations also show that the wake of thrust producing, rigid finite-span wings is formed by vortex loops close to the wing surface, that slowly convert into vortex rings as they are convected downstream. It was also observed that the vortex rings are themselves inclined with respect to the free-stream; the angle of inclination of the vortex rings is found to be in the direction of their travel and in the streamwise direction for thrust producing configurations, whereas for drag producing configurations the angle of inclination is opposite to the direction of travel.

Additionally, the effect of aspec- ratio AR on the aerodynamic forces of finite-span wings was also assessed.  It was observed that as we increase the wing AR, the aerodynamic forces also increase and this is chiefly attributed to the large area of high aspect ratio wings and to the decrease of three-dimensional effects in long wings.  This observation leads us to think that the assumption of two-dimensionality has some validity for birds and insects, where the wings of many species tend to have relatively large aspect ratio.

All the qualitative and quantitative results found during this research support the hypothesis that: “flying and swimming animals cruise at a Strouhal number tuned for high power efficiency”

The stochastic modelling of biological systems is an informative, and in some cases, very adequate technique, which may however result in being more expensive than other modelling approaches, such as differential equations. Despite that, current stochastic simulation frameworks do not fully exploit the full power of modern multicore platforms. As an example, the StochKit framework, which is a “reference” stochastic simulation framework aiming at making stochastic simulation accessible to practicing biologists and chemists, is coded (in C++) as sequential process exhibiting several non-reentrant functions, and therefore, it could not easily parallelised to exploit the full power of modern multi-core processors (i.e. parallelised in a shared memory fashion).

This project aims to validate a porting methodology able to support the easy parallelisation of complex C/C++ codes for multi-core platforms using the StochKit framework as testbed, which is a complex and third-party code developed at University of Santa Barbara, CA.

The methodology will have two cornerstones: 1) the FastFlow programming framework for multicore architectures (designed and developed by the author of this proposal, http://mc-fastflow.sourceforge.net/ ) and 2) the concept of “selective memory”, i.e. a novel data structure supporting the on-line reduction of trajectory data from different simulations by way of user-defined associative and commutative functions.

The success of the project will be evaluated on both code parallelisation effort and speedup of the parallelised code (i.e. StochKit-FF framework). In particular, the design and development, validation and testing of StochKit-FF will be fully performed during the period of the visit. The validation of the code will be performed during the visit by running a novel stochastic model of HIV infection dynamics developed in cooperation with Dr. Andrea Bracciali, who proposed a companion project in the same period. This model, will be also used to evaluate StochKit-FF speedup on EPCC computing facilities.

The objective of the projects have been fully achieved. StochKit-FF has been designed and developed. It is available under LGPLv3 licence at FastFlow website ( http://mc-fastflow.sourceforge.net/ ), since it represents a bright example of the ability of the FastFlow programming framework to support the easy (yet efficient) shift to multicore for complex legacy applications.

The performance of StochKit-FF has been evaluated on the HIV diffusion case-study, which has been designed in cooperation with Dr. Andrea Bracciali (ISTI-CNR, Italy) and Prof. Pietro Liò (University of Cambridge, UK) in a companion HPC-Europa project.

A single run of the HIV simulation with StochKit averagely produces ~150 millions of simulation points for 4000 days of simulated time, which will turn into about 6 GBytes of generated data; multiple runs of the same simulation will need a linearly greater time and space. These simulations can be parallelised (by way of the FastFlow toolkit) on a multicore platform by running several independent instances, which however, will compete for memory and disk accesses, thus lead to suboptimal performances in the case of high output pressure.

StochKit-FF mainly attacks these costs by online reducing the outputs of simulations, which are run in parallel and on-line combined by means of the selective memory. Overall, selective memory produces a combined simulation that has been adaptively sampled: time intervals exhibiting a higher variability across different simulations exhibit a higher sampling rate. Selective memory effectively mitigates the memory pressure of results logging when running many simulations on a multicore because substantially reduces output size, thus capacity misses and memory bus pressure. Thanks to this, StochKit-FF exhibits a super-linear speedup with respect the original StochKit (i.e. more than n-fold faster than original code when running on a n-core machine).

Two months after the end of the visit the technical advances and achieved results have been described in three research papers:

[1] M. Aldinucci, A. Bracciali, and P. Li ́o. Formal synthetic immunology. ERCIM News, 82:40–41, July 2010.

[2] M. Aldinucci, A. Bracciali, P. Lio`, A. Sorathiya, and M. Torquati. StochKit-FF: Efficient systems biology on multicore architectures. Technical Report TR-10-12, Universit`a di Pisa, Dipartimento di Informatica, Italy, 2010. Submitted to Intl. HiBB 2010 conference.

[3] M. Aldinucci. Efficient Parallel MonteCarlo with FastFlow: Time- and space-efficient parallelisation of the StochKit framework with FastFlow. HPC-Europa2 final paperwork, to appear.

The Finite Element Tearing and Interconnecting method Dual-Primal (FETI-DP) method is one of the non-overlapping domain decomposition methods, which was published by Farhat and his co-workers in the article [FarhatEtAll-01] in 2001. The domain decomposition methods divide the original domain into several smaller subdomains. The FETI-DP method was developed due to problems with singular subdomain matrices in the original FETI method. The FETI-DP method is the combination of the FETI method and Schur complement method.

The FETI-DP method divide unknowns into two categories – corner unknowns and remaining unknowns. The remaining unknowns are further split into remaining interface unknowns and internal unknowns. The continuity conditions are enforced by Langrange multipliers which are defined on the remaining interface unknowns and also by corner nodes. The corner unknowns ensure the non-singularity of the subdomain matrices. The remaining unknowns are eliminated and the coarse problem is after obtained. The matrix of the coarse problem is symmetric and positive-definitive. Therefore the coarse problem can be solved by the conjugate gradient method. More information about FETI-DP method can be found in the article [FarhatEtAll-01] or in the book [Kruis06] by Kruis.

The selection of the the corner nodes where are corner unknowns defined deserves special attention. A definition of the corner nodes was published in the original article [FarhatEtAll-01] by Farhat. The corner nodes are there defined as

D1: Cross-points - It means the nodes which belonged to more than two subdomains

D2: The set of nodes located at the beginning and end of each edge of each subdomain.

But the definition is not suitable for all possible meshes. If the original domain is divided into two subdomains and the first subdomain is surrounded by the second domain, then there is no cross-points and there is not beginning and the end of any edge. Recently, strong influence of the definition of the corner unknowns on the condition number of the subdomain matrix has been observed in the work [KabelikovaEtAll-09] by Kabelíková et all. The large condition numbers of subdomain matrices significantly deteriorate the convergence of the iterative methods used for the solution of the coarse problem. There is a minimal needed number of corner nodes. In the case of two-dimensional meshes, plane strain and plane stress problems require two different nodes, three nodes are better for plate problems. Therefore, the minimum number of needed nodes is three in the case of two-dimensional meshes. In the case of three-dimensional meshes must be selected three non-collinear nodes. Theoretically can be selected all nodes on the subdomain boundaries, but then the FETI-DP method transforms itself to the Schur complement method.

There is no software known to author which can be use for the selection of corner nodes. This was a motivation for a development of an algorithm for the selection such nodes. The algorithm will be developed with help of the graph theory and several heuristic rules.

References:

[FarhatEtAll-01] Farhat, C. and Lesoinne, M. and LeTallec, P. and Pierson, K. and Rixen, D.: “FETI-DP: A Dual-Primal Unified FETI Method-Part I: Faster Alternative to the Two-Level FETI Method” , International Journal for Numerical Methods in Engineering, vol. 50, pages = 1523 – 1544.

[KabelikovaEtAll-09] Kabelíková, P. and Dostál, Z. and Kozubek, T. and Markopoulos, A.: “Generalized inverse matrix evaluation using graph theory”, In Proceedings of the Modelling 2009, Blaheta, R., Starý J. (ed.),Institute of Geonics AS CR, Ostrava, Czech Republic, 2009.

[Kruis-06] Kruis, J.: “Domain Decomposition Methods for Distributed Computing”, Saxe-Coburg Publications, edition 1st, 2006.

There were developed two independent algorithm for selection of the corner nodes. The first algorithm is used for two-dimensional meshes and the second algorithm for three-dimensional meshes. Both algorithm are based on the Graph theory. Therefore there will be defined several necessary terms from the Graph theory. A graph G(V,E) consist of a finite set V of elements called vertices and a finite set E of elements called edges. Nodes of a finite element mesh can be mapped to the set V of graph vertices. Vertices vi and vj are connected by an edge if the appropriate nodes belong to a edge of the finite element. Otherwise, there is no edge between vi and vj. This graph is called nodal graph. The degree dG(v) of vertex v in graph G is the number of edge of G incident with v. A walk in a graph G = (V, E) is a finite sequence of vertices v0 , v1 , . . . , vk such that (vi−1, vi ), 1 ≤ i ≤ k is an edge in the graph G. The nodes v0 and vk are called end vertices of the walk. A walk is a trail if all its edges are distinct.

In the case of the two-dimensional algorithm, the graph B(V,E) is established form boundary nodes between subdomains. The vertex degree is established for each vertex in the graph. The corner nodes is then defined as

vertices which vertex degree is equal to one

vertices which vertex degree is more than two

The algorithm, which selects corner nodes, based on the vertex degree is called minimal number algorithm. A control of chosen nodes must be done after the selection procedure. The number of corner nodes is controlled. The minimum number of needed nodes is three. If there are no enough nodes from the minimal number algorithm then the another nodes must be selected.

The next control is aimed on the geometrical relation between corner nodes. If the selected nodes are too close each other, the subdomain matrix has usually very large condition number. A distance between corner nodes are controlled . The next condition is a non-collinearity condition. This is necessary in order to avoid several nodes in one line which lead also to large condition number of subdomain matrix.

It is possible to add further corner nodes by an extended algorithm for adding corner nodes. The extended algorithm is based on the restriction of the graph B(V,E) into several subgraphs. The subgraphs Sj (V,E) are defined as the open trail between two corner nodes in the graph B(V,E). The further corner nodes can be added as the centre of such subgraph or the walk can be split into k part and the corner node can be added at the end of such part of the walk.

The several numerical test was done with help of SMP computer Ness and supercomputer HECToR.

The following behaviour of the algorithm was observed. Higher number of corner nodes reduces the number of iterations in the conjugate gradient method and therefore also the computational time is reduced. If an optimal number is reached, the number of iterations still decreases but the total computational time starts to grow because time of condensation of the matrix contains entries related to the corner nodes prevails over time saved by the reduced number of iterations. The tests also showed the ability of the proposed algorithm to select the minimum number of corner nodes in the case of very general domains. The FETI-DP method can be therefore used without manual selection of corner nodes.