Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences current issue
http://rspa.royalsocietypublishing.org
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences RSS feed -- current issue1471-2946February, 2017Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences1364-5021<![CDATA[A construction of a large family of commuting pairs of integrable symplectic birational four-dimensional maps]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160535?rss=1
We give a construction of completely integrable four-dimensional Hamiltonian systems with cubic Hamilton functions. Applying to the corresponding pairs of commuting quadratic Hamiltonian vector fields the so called Kahan–Hirota–Kimura discretization scheme, we arrive at pairs of birational four-dimensional maps. We show that these maps are symplectic with respect to a symplectic structure that is a perturbation of the standard symplectic structure on R4, and possess two independent integrals of motion, which are perturbations of the original Hamilton functions and which are in involution with respect to the perturbed symplectic structure. Thus, these maps are completely integrable in the Liouville–Arnold sense. Moreover, under a suitable normalization of the original pairs of vector fields, the pairs of maps commute and share the invariant symplectic structure and the two integrals of motion.
]]>2017-02-15T00:05:25-08:00info:doi/10.1098/rspa.2016.0535hwp:master-id:royprsa;rspa.2016.05352017-02-15Research articles47321982016053520160535<![CDATA[A wave theory of heat transport with applications to Kapitsa resistance and thermal rectification]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160584?rss=1
We develop a theory of thermal transport in nanoscale-layered structures based on wave processes. The theory incorporates two fundamental principles, first, that the spectra of thermally excited waves are determined by the temperature differential and the heat flux, and second, that the wave fields in the heat exchanging domains are coupled. The developed method includes classical theories as special cases that are valid in larger scales, and it naturally explains such phenomena as interface thermal resistance (Kapitsa resistance) and thermal rectification (asymmetry of thermal transport). Numerical examples demonstrate the feasibility of the approach, and they show good agreement with measurements of Kapitsa resistance reported in the literature.
]]>2017-02-15T00:05:25-08:00info:doi/10.1098/rspa.2016.0584hwp:master-id:royprsa;rspa.2016.05842017-02-15Research articles47321982016058420160584<![CDATA[An experimental study of ultrasonic vibration and the penetration of granular material]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160673?rss=1
This work investigates the potential use of direct ultrasonic vibration as an aid to penetration of granular material. Compared with non-ultrasonic penetration, required forces have been observed to reduce by an order of magnitude. Similarly, total consumed power can be reduced by up to 27%, depending on the substrate and ultrasonic amplitude used. Tests were also carried out in high-gravity conditions, displaying a trend that suggests these benefits could be leveraged in lower gravity regimes.
]]>2017-02-15T00:05:25-08:00info:doi/10.1098/rspa.2016.0673hwp:master-id:royprsa;rspa.2016.06732017-02-15Research articles47321982016067320160673<![CDATA[Stability of vertical magnetic chains]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160703?rss=1
A linear stability analysis is performed for a pair of coaxial vertical chains made from permanently magnetized balls under the influence of gravity. While one chain rises from the ground, the other hangs from above, with the remaining ends separated by a gap of prescribed length. Various boundary conditions are considered, as are situations in which the magnetic dipole moments in the two chains are parallel or antiparallel. The case of a single chain attached to the ground is also discussed. The stability of the system is examined with respect to three quantities: the number of balls in each chain, the length of the gap between the chains, and a single dimensionless parameter which embodies the competition between magnetic and gravitational forces. Asymptotic scaling laws involving these parameters are provided. The Hessian matrix is computed in exact form, allowing the critical parameter values at which the system loses stability and the respective eigenmodes to be determined up to machine precision. A comparison with simple experiments for a single chain attached to the ground shows good agreement.
]]>2017-02-08T00:05:14-08:00info:doi/10.1098/rspa.2016.0703hwp:master-id:royprsa;rspa.2016.07032017-02-08Research articles47321982016070320160703<![CDATA[On the role of micro-inertia in enriched continuum mechanics]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160722?rss=1
In this paper, the role of gradient micro-inertia terms {macron}|| u,t||2 and free micro-inertia terms ||P,t||2 is investigated to unveil their respective effects on the dynamic behaviour of band-gap metamaterials. We show that the term {macron}|| u,t||2 alone is only able to disclose relatively simplified dispersive behaviour. On the other hand, the term ||P,t||2 alone describes the full complex behaviour of band-gap metamaterials. A suitable mixing of the two micro-inertia terms allows us to describe a new feature of the relaxed-micromorphic model, i.e. the description of a second band-gap occurring for higher frequencies. We also show that a split of the gradient micro-inertia {macron}|| u,t||2, in the sense of Cartan–Lie decomposition of matrices, allows us to flatten separately the longitudinal and transverse optic branches, thus giving us the possibility of a second band-gap. Finally, we investigate the effect of the gradient inertia {macron}|| u,t||2 on more classical enriched models such as the Mindlin–Eringen and the internal variable ones. We find that the addition of such a gradient micro-inertia allows for the onset of one band-gap in the Mindlin–Eringen model and three band-gaps in the internal variable model. In this last case, however, non-local effects cannot be accounted for, which is a too drastic simplification for most metamaterials. We conclude that, even when adding gradient micro-inertia terms, the relaxed micromorphic model remains the best performing one, among the considered enriched models, for the description of non-local band-gap metamaterials.
]]>2017-02-01T01:05:46-08:00info:doi/10.1098/rspa.2016.0722hwp:master-id:royprsa;rspa.2016.07222017-02-01Research articles47321982016072220160722<![CDATA[Bioinspired turbine blades offer new perspectives for wind energy]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160726?rss=1
Wind energy is becoming a significant alternative solution for future energy production. Modern turbines now benefit from engineering expertise, and a large variety of different models exists, depending on the context and needs. However, classical wind turbines are designed to operate within a narrow zone centred around their optimal working point. This limitation prevents the use of sites with variable wind to harvest energy, involving significant energetic and economic losses. Here, we present a new type of bioinspired wind turbine using elastic blades, which passively deform through the air loading and centrifugal effects. This work is inspired from recent studies on insect flight and plant reconfiguration, which show the ability of elastic wings or leaves to adapt to the wind conditions and thereby to optimize performance. We show that in the context of energy production, the reconfiguration of the elastic blades significantly extends the range of operating regimes using only passive, non-consuming mechanisms. The versatility of the new turbine model leads to a large increase of the converted energy rate, up to 35%. The fluid/elasticity mechanisms involved for the reconfiguration capability of the new blades are analysed in detail, using experimental observations and modelling.
]]>2017-02-15T00:05:25-08:00info:doi/10.1098/rspa.2016.0726hwp:master-id:royprsa;rspa.2016.07262017-02-15Research articles47321982016072620160726<![CDATA[Nonlinear information fusion algorithms for data-efficient multi-fidelity modelling]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160751?rss=1
Multi-fidelity modelling enables accurate inference of quantities of interest by synergistically combining realizations of low-cost/low-fidelity models with a small set of high-fidelity observations. This is particularly effective when the low- and high-fidelity models exhibit strong correlations, and can lead to significant computational gains over approaches that solely rely on high-fidelity models. However, in many cases of practical interest, low-fidelity models can only be well correlated to their high-fidelity counterparts for a specific range of input parameters, and potentially return wrong trends and erroneous predictions if probed outside of their validity regime. Here we put forth a probabilistic framework based on Gaussian process regression and nonlinear autoregressive schemes that is capable of learning complex nonlinear and space-dependent cross-correlations between models of variable fidelity, and can effectively safeguard against low-fidelity models that provide wrong trends. This introduces a new class of multi-fidelity information fusion algorithms that provide a fundamental extension to the existing linear autoregressive methodologies, while still maintaining the same algorithmic complexity and overall computational cost. The performance of the proposed methods is tested in several benchmark problems involving both synthetic and real multi-fidelity datasets from computational fluid dynamics simulations.
]]>2017-02-08T00:05:14-08:00info:doi/10.1098/rspa.2016.0751hwp:master-id:royprsa;rspa.2016.07512017-02-08Research articles47321982016075120160751<![CDATA[Efficient least angle regression for identification of linear-in-the-parameters models]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160775?rss=1
Least angle regression, as a promising model selection method, differentiates itself from conventional stepwise and stagewise methods, in that it is neither too greedy nor too slow. It is closely related to L_{1} norm optimization, which has the advantage of low prediction variance through sacrificing part of model bias property in order to enhance model generalization capability. In this paper, we propose an efficient least angle regression algorithm for model selection for a large class of linear-in-the-parameters models with the purpose of accelerating the model selection process. The entire algorithm works completely in a recursive manner, where the correlations between model terms and residuals, the evolving directions and other pertinent variables are derived explicitly and updated successively at every subset selection step. The model coefficients are only computed when the algorithm finishes. The direct involvement of matrix inversions is thereby relieved. A detailed computational complexity analysis indicates that the proposed algorithm possesses significant computational efficiency, compared with the original approach where the well-known efficient Cholesky decomposition is involved in solving least angle regression. Three artificial and real-world examples are employed to demonstrate the effectiveness, efficiency and numerical stability of the proposed algorithm.
]]>2017-02-01T01:05:46-08:00info:doi/10.1098/rspa.2016.0775hwp:master-id:royprsa;rspa.2016.07752017-02-01Research articles47321982016077520160775<![CDATA[Eshelby's problem of a spherical inclusion eccentrically embedded in a finite spherical body]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160808?rss=1
Resorting to the superposition principle, the solution of Eshelby's problem of a spherical inclusion located eccentrically inside a finite spherical domain is obtained in two steps: (i) the solution to the problem of a spherical inclusion in an infinite space; (ii) the solution to the auxiliary problem of the corresponding finite spherical domain subjected to appropriate boundary conditions. Moreover, a set of functions called the sectional and harmonic deviators are proposed and developed to work out the auxiliary solution in a series form, including the displacement and Eshelby tensor fields. The analytical solutions are explicitly obtained and illustrated when the geometric and physical parameters and the boundary condition are specified.
]]>2017-02-15T01:16:11-08:00info:doi/10.1098/rspa.2016.0808hwp:master-id:royprsa;rspa.2016.08082017-02-15Research articles47321982016080820160808<![CDATA[Methodical fitting for mathematical models of rubber-like materials]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160811?rss=1
A great variety of models can describe the nonlinear response of rubber to uniaxial tension. Yet an in-depth understanding of the successive stages of large extension is still lacking. We show that the response can be broken down in three steps, which we delineate by relying on a simple formatting of the data, the so-called Mooney plot transform. First, the small-to-moderate regime, where the polymeric chains unfold easily and the Mooney plot is almost linear. Second, the strain-hardening regime, where blobs of bundled chains unfold to stiffen the response in correspondence to the ‘upturn’ of the Mooney plot. Third, the limiting-chain regime, with a sharp stiffening occurring as the chains extend towards their limit. We provide strain-energy functions with terms accounting for each stage that (i) give an accurate local and then global fitting of the data; (ii) are consistent with weak nonlinear elasticity theory and (iii) can be interpreted in the framework of statistical mechanics. We apply our method to Treloar's classical experimental data and also to some more recent data. Our method not only provides models that describe the experimental data with a very low quantitative relative error, but also shows that the theory of nonlinear elasticity is much more robust that seemed at first sight.
]]>2017-02-08T00:05:14-08:00info:doi/10.1098/rspa.2016.0811hwp:master-id:royprsa;rspa.2016.08112017-02-08Research articles47321982016081120160811<![CDATA[Challenges to deployment of twenty-first century nuclear reactor systems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160815?rss=1
The science and engineering of materials have always been fundamental to the success of nuclear power to date. They are also the key to the successful deployment and operation of a new generation of nuclear reactor systems and their associated fuel cycles. This article reflects on some of the historical issues, the challenges still prevalent today and the requirement for significant ongoing materials R&D and discusses the potential role of small modular reactors.
]]>2017-02-01T01:05:46-08:00info:doi/10.1098/rspa.2016.0815hwp:master-id:royprsa;rspa.2016.08152017-02-01Perspective47321982016081520160815<![CDATA[Field patterns: a new mathematical object]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160819?rss=1
Field patterns occur in space–time microstructures such that a disturbance propagating along a characteristic line does not evolve into a cascade of disturbances, but rather concentrates on a pattern of characteristic lines. This pattern is the field pattern. In one spatial direction plus time, the field patterns occur when the slope of the characteristics is, in a sense, commensurate with the space–time microstructure. Field patterns with different spatial shifts do not generally interact, but rather evolve as if they live in separate dimensions, as many dimensions as the number of field patterns. Alternatively one can view a collection as a multi-component potential, with as many components as the number of field patterns. Presumably, if one added a tiny nonlinear term to the wave equation one would then see interactions between these field patterns in the multi-dimensional space that one can consider them to live, or between the different field components of the multi-component potential if one views them that way. As a result of PT-symmetry many of the complex eigenvalues of an appropriately defined transfer matrix have unit norm and hence the corresponding eigenvectors correspond to propagating modes. There are also modes that blow up exponentially with time.
]]>2017-02-15T01:16:11-08:00info:doi/10.1098/rspa.2016.0819hwp:master-id:royprsa;rspa.2016.08192017-02-15Research articles47321982016081920160819