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Abstract: Surface interpolation finds application in many aspects of science and technology. Two specific areas of interest are surface reconstruction techniques for plant architecture and approximating cell face fluxes in the finite volume discretisation strategy for solving partial differential equations numerically. An important requirement of both applications is accurate local gradient estimation. In surface reconstruction this gradient information is used to increase the accuracy of the local interpolant, while in the finite volume framework accurate gradient information is essential to ensure second order spatial accuracy of the discretisation. In this work two different least squares strategies for approximating these local gradients are investigated and the errors associated with each analysed. It is shown that although the two strategies appear different, they produce the same least squares error. Some carefully chosen case studies are used to elucidate this finding. [Copyright &y& Elsevier]

Realistic virtual models of leaf surfaces are important for several applications in the plant sciences, such as modelling agrichemical spray droplet movement and spreading on the surface. In this context, the virtual surfaces are required to be smooth enough to facilitate the use of the mathematical equations that govern the motion of the droplet. Although an effective approach is to apply discrete smoothing D[sup 2]-spline algorithms to reconstruct the leaf surfaces from three-dimensional scanned data, difficulties arise when dealing with wheat (Triticum aestivum L.) leaves, which tendtotwist and bend. To overcome this topological difficulty, we develop a parameterisation technique that rotates and translates the original data, allowing the surface to be fitted using the discrete smoothing D[sup 2]-spline methods in the new parameter space. Our algorithm uses finite element methods to represent the surface as a linear combination of compactly supported shape functions. Numerical results confirm that the parameterisation, along with the use of discrete smoothing D[sup 2]-spline techniques, produces realistic virtual representations of wheat leaves. [ABSTRACT FROM AUTHOR]

The foliage of a plant performs vital functions. As such, leaf models are required to be developed for modeling the plant architecture from a set of scattered data captured using a scanning device. The leaf model can be used for purely visual purposes or as part of a further model, such as a fluid movement model or biological process. For these reasons, an accurate mathematical representation of the surface and boundary is required. This paper compares three approaches for fitting a continuously differentiable surface through a set of scanned data points from a leaf surface, with a technique already used for reconstructing leaf surfaces. The techniques which will be considered are discrete smoothing D²-splines [R. Arcangéli, M. C. Lopez de Silanes, and J. J. Torrens, Multidimensional Minimising Splines, Springer, New York, 2004], the thin plate spline finite element smoother [S. Roberts, M. Hegland, and I. Altas, SIAM J. Numer. Anal., 1 (2003), pp. 208-234], and the radial basis function Clough-Tocher method [M. N. Oqielat, I. W. Turner, and J. A. Belward, Appl. Math. Model., 33 (2009), pp. 2582-2595]. Numerical results show that discrete smoothing D²-splines produce reconstructed leaf surfaces which better represent the original physical leaf. [ABSTRACT FROM AUTHOR]

Free surface flow past a two-dimensional semi-infinite curved plate is considered, with emphasis given to solving the shape of the resulting wave train that appears downstream on the surface of the fluid. This flow configuration can be interpreted as applying near the stern of a wide blunt ship. For steady flow in a fluid of finite depth, we apply the Wiener-Hopf technique to solve a linearised problem, valid for small perturbations of the uniform stream. Weakly nonlinear results obtained by a forced KdV equation are also presented, as are numerical solutions of the fully nonlinear problem, computed using a conformal mapping and a boundary integral technique. By considering different families of shapes for the semi-infinite plate, it is shown how the amplitude of the waves can be minimised. For plates that increase in height as a function of the direction of flow, reach a local maximum, and then point slightly downwards at the point at which the free surface detaches, it appears the downstream wavetrain can be eliminated entirely [ABSTRACT FROM AUTHOR]