Microscopic Origin of Macroscopic Strength in Granular Media: A Numerical and Analytical Approach

Constitutive modeling in granular materials has historically been based on macroscopic experimental observations that, while being usually effective at predicting the bulk behavior of these type of materials, suffer important limitations when it comes to understanding the physics behind grain-to-grain interactions that induce the material to macroscopically behave in a given way when subjected to certain boundary conditions.

The advent of the discrete element method (DEM) in the late 1970s helped scientists and engineers to gain a deeper insight into some of the most fundamental mechanisms furnishing the grain scale. However, one of the most critical limitations of classical DEM schemes has been their inability to account for complex grain morphologies. Instead, simplified geometries such as discs, spheres, and polyhedra have typically been used. Fortunately, in the last fifteen years, there has been an increasing development of new computational as well as experimental techniques, such as non-uniform rational basis splines (NURBS) and 3D X-ray Computed Tomography (3DXRCT), which are contributing to create new tools that enable the inclusion of complex grain morphologies into DEM schemes.

Yet, as the scientific community is still developing these new tools, there is still a gap in thoroughly understanding the physical relations connecting grain and continuum scales as well as in the development of discrete techniques that can predict the emergent behavior of granular materials without resorting to phenomenology, but rather can directly unravel the micro-mechanical origin of macroscopic behavior.

In order to contribute towards closing the aforementioned gap, we have developed a micro-mechanical analysis of macroscopic peak strength, critical state, and residual strength in two-dimensional non-cohesive granular media, where typical continuum constitutive quantities such as frictional strength and dilation angle are explicitly related to their corresponding grain-scale counterparts (e.g., inter-particle contact forces, fabric, particle displacements, and velocities), providing an across-the-scale basis for better understanding and modeling granular media.

In the same way, we utilize a new DEM scheme (LS-DEM) that takes advantage of a mathematical technique called level set (LS) to enable the inclusion of real grain shapes into a classical discrete element method. After calibrating LS-DEM with respect to real experimental results, we exploit part of its potential to study the dependency of critical state (CS) parameters such as the critical state line (CSL) slope, CSL intercept, and CS friction angle on the grain's morphology, i.e., sphericity, roundness, and regularity.

Finally, we introduce a first computational algorithm to ``clone'' the grain morphologies of a sample of real digital grains. This cloning algorithm allows us to generate an arbitrary number of cloned grains that satisfy the same morphological features (e.g., roundness and aspect ratio) displayed by their real parents and can be included into a DEM simulation of a given mechanical phenomenon. In turn, this will help with the development of discrete techniques that can directly predict the engineering scale behavior of granular media without resorting to phenomenology.

Subdivision-stabilised immersed b-spline finite elements for moving boundary flows

An immersed finite element method is presented to compute flows with complex moving boundaries on a fixed Cartesian grid. The viscous, incompressible fluid flow equations are discretized with b-spline basis functions. The two-scale relation for b-splines is used to implement an elegant and efficient technique to satisfy the LBB condition. On non-grid-aligned fluid domains and at moving boundaries, the boundary conditions are enforced with a consistent penalty method as originally proposed by Nitsche. In addition, a special extrapolation technique is employed to prevent the loss of numerical stability in presence of arbitrarily small cut-cells. The versatility and accuracy of the proposed approach is demonstrated by means of convergence studies and comparisons with previous experimental and computational investigations.

A subdivision-based implementation of the hierarchical b-spline finite element method

A novel technique is presented to facilitate the implementation of hierarchical b-splines and their interfacing with conventional finite element implementations. The discrete interpretation of the two-scale relation, as common in subdivision schemes, is used to establish algebraic relations between the basis functions and their coefficients on different levels of the hierarchical b-spline basis. The subdivision projection technique introduced allows us first to compute all element matrices and vectors using a fixed number of same-level basis functions. Their subsequent multiplication with subdivision matrices projects them, during the assembly stage, to the correct levels of the hierarchical b-spline basis. The proposed technique is applied to convergence studies of linear and geometrically nonlinear problems in one, two and three space dimensions. © 2012 Elsevier B.V.

Finite element analysis of compressor disc heat transfer using the transient conduction method with sparse boundary collocation points

Surface temperature measurements from two discs of a gas turbine compressor rig are used as boundary conditions for the transient conduction solution (inverse heat transfer analysis). The disc geometry is complex, and so the finite element method is used. There are often large radial temperature gradients on the discs, and the equations are therefore solved taking into account the dependence of thermal conductivity on temperature. The solution technique also makes use of a multigrid algorithm to reduce the solution time. This is particularly important since a large amount of data must be analyzed to obtain correlations of the heat transfer. The finite element grid is also used for a network analysis to calculate the radiant heat transfer in the cavity formed between the two compressor discs. The work discussed here proved particularly challenging as the disc temperatures were only measured at four different radial locations. Four methods of surface temperature interpolation are examined, together with their effect on the local heat fluxes. It is found that the choice of interpolation method depends on the available number of data points. Bessel interpolation gives the best results for four data points, whereas cubic splines are preferred when there are considerably more data points. The results from the analysis of the compressor rig data show that the heat transfer near the disc inner radius appears to be influenced by the central throughflow. However, for larger radii, the heat transfer from the discs and peripheral shroud is found to be consistent with that of a buoyancy-induced flow.

Multi-objective optimisation of horizontal axis wind turbine structure and energy production using aerofoil and blade properties as design variables

The design of wind turbine blades is a true multi-objective engineering task. The aerodynamic effectiveness of the turbine needs to be balanced with the system loads introduced by the rotor. Moreover the problem is not dependent on a single geometric property, but besides other parameters on a combination of aerofoil family and various blade functions. The aim of this paper is therefore to present a tool which can help designers to get a deeper insight into the complexity of the design space and to find a blade design which is likely to have a low cost of energy. For the research we use a Computational Blade Optimisation and Load Deflation Tool (CoBOLDT) to investigate the three extreme point designs obtained from a multi-objective optimisation of turbine thrust, annual energy production as well as mass for a horizontal axis wind turbine blade. The optimisation algorithm utilised is based on Multi-Objective Tabu Search which constitutes the core of CoBOLDT. The methodology is capable to parametrise the spanning aerofoils with two-dimensional Free Form Deformation and blade functions with two tangentially connected cubic splines. After geometry generation we use a panel code to create aerofoil polars and a stationary Blade Element Momentum code to evaluate turbine performance. Finally, the obtained loads are fed into a structural layout module to estimate the mass and stiffness of the current blade by means of a fully stressed design. For the presented test case we chose post optimisation analysis with parallel coordinates to reveal geometrical features of the extreme point designs and to select a compromise design from the Pareto set. The research revealed that a blade with a feasible laminate layout can be obtained, that can increase the energy capture and lower steady state systems loads. The reduced aerofoil camber and an increased L/. D-ratio could be identified as the main drivers. This statement could not be made with other tools of the research community before. © 2013 Elsevier Ltd.

A fixed-grid b-spline finite element technique for fluid-structure interaction

We present a fixed-grid finite element technique for fluid-structure interaction problems involving incompressible viscous flows and thin structures. The flow equations are discretised with isoparametric b-spline basis functions defined on a logically Cartesian grid. In addition, the previously proposed subdivision-stabilisation technique is used to ensure inf-sup stability. The beam equations are discretised with b-splines and the shell equations with subdivision basis functions, both leading to a rotation-free formulation. The interface conditions between the fluid and the structure are enforced with the Nitsche technique. The resulting coupled system of equations is solved with a Dirichlet-Robin partitioning scheme, and the fluid equations are solved with a pressure-correction method. Auxiliary techniques employed for improving numerical robustness include the level-set based implicit representation of the structure interface on the fluid grid, a cut-cell integration algorithm based on marching tetrahedra and the conservative data transfer between the fluid and structure discretisations. A number of verification and validation examples, primarily motivated by animal locomotion in air or water, demonstrate the robustness and efficiency of our approach. © 2013 John Wiley & Sons, Ltd.

喷灌均匀系数的三次样条两次插值计算方法

为提高喷灌水量分布均匀性评价的准确性,当雨量筒径向布置时,为考虑所有测点数据对插值点降水深的影响,采用径向和周向两次的三次样条插值计算出未知点的降水深,从而计算喷灌均匀系数。以美国雨鸟30PSH型喷头雨量筒间隔为1m和2m的喷洒试验数据,计算网格点取1m和0.25m,分别采用三次样条两次插值法和邻近四点距离线性插值法计算了克里斯琴森均匀系数。结果表明,均匀系数由高至低的顺序依次为采样间隔为2m的线性插值、采样间隔为2m的三次样条两次插值、采样间隔为1m的线性插值和采样间隔为1m的三次样条两次插值。采样间隔2m比1m计算出的均匀系数总体高3~4个百分点,三次样条两次插值法比邻近点距离线性插值法略低1个百分点,2种计算网格点间距下的均匀系数差值小于1个百分点。结果证明,采样间距、插值方法、计算网格间距对均匀系数的影响依次降低,三次样条两次插值法可以用来评价喷灌组合均匀系数。

Surface modelling of soil pH

In addition to classical methods, namely kriging, Inverse Distance Weighting (IDW) and splines, which have been frequently used for interpolating the spatial patterns of soil properties, a relatively more accurate surface modelling technique is being developed in recent years, namely high accuracy surface modelling (HASM). It has been used in the numerical tests, DEM construction and the interpolation of climate and ecosystem changes. In this paper, HASM was applied to interpolate soil pH for assessing its feasibility of soil property interpolation in a red soil region of Jiangxi Province, China. Soil pH was measured on 150 samples of topsoil (0-20 cm) for the interpolation and comparing the performance of HASM, kriging. IDW and splines. The mean errors (MEs) of interpolations indicate little bias of interpolation for soil pH by the four techniques. HASM has less mean absolute error (MAE) and root mean square error (RMSE) than kriging, IDW and splines. HASM is still the most accurate one when we use the mean rank and the standard deviation of the ranks to avoid the outlier effects in assessing the prediction performance of the four methods. Therefore, HASM can be considered as an alternative and accurate method for interpolating soil properties. Further researches of HASM are needed to combine HASM with ancillary variables to improve the interpolation performance and develop a user-friendly algorithm that can be implemented in a GIS package. (C) 2009 Elsevier B.V. All rights reserved.

Comparison of wavelet transform methods - The compression of IR spectral data

In this paper, an introduction of wavelet transform and multi-resolution analysis is presented. We describe three data compression methods based on wavelet transform for spectral information,and by using the multi-resolution analysis, we compressed spectral data by Daubechies's compactly supported orthogonal wavelet and orthogonal cubic B-splines wavelet, Using orthogonal cubic B-splines wavelet and coefficients of sharpening signal are set to zero, only very few large coefficients are stored, and a favourable data compression can be achieved.

基于超冗余度机械臂动力学的时间最优轨迹规划

基于超冗余度机械臂的动力学方程 ,提出了一种超冗余度机械臂同时受速度和力矩约束的时间最优轨迹规划方法 .它首先采用 B样条曲线拟合无碰撞离散路径 ,得到由伪位移参数 s表示的超冗余度机械臂连续、光滑运动路径 ,然后对动力学方程和约束方程进行数学变换 ,得到由 s表示的动力学方程和约束方程 ,最后以 s和伪速度 s· 分别作为动态规划的阶段变量和状态变量 ,对超冗余度机械臂进行时间最优轨迹规划 .仿真结果表明 ,所给出的时间最优轨迹规划算法是正确的 ,所采取的解决方法是可行的

A Theory of Networks for Appxoimation and Learning

Learning an input-output mapping from a set of examples, of the type that many neural networks have been constructed to perform, can be regarded as synthesizing an approximation of a multi-dimensional function, that is solving the problem of hypersurface reconstruction. From this point of view, this form of learning is closely related to classical approximation techniques, such as generalized splines and regularization theory. This paper considers the problems of an exact representation and, in more detail, of the approximation of linear and nolinear mappings in terms of simpler functions of fewer variables. Kolmogorov's theorem concerning the representation of functions of several variables in terms of functions of one variable turns out to be almost irrelevant in the context of networks for learning. We develop a theoretical framework for approximation based on regularization techniques that leads to a class of three-layer networks that we call Generalized Radial Basis Functions (GRBF), since they are mathematically related to the well-known Radial Basis Functions, mainly used for strict interpolation tasks. GRBF networks are not only equivalent to generalized splines, but are also closely related to pattern recognition methods such as Parzen windows and potential functions and to several neural network algorithms, such as Kanerva's associative memory, backpropagation and Kohonen's topology preserving map. They also have an interesting interpretation in terms of prototypes that are synthesized and optimally combined during the learning stage. The paper introduces several extensions and applications of the technique and discusses intriguing analogies with neurobiological data.

Modelos animales para caracteres con efectos maternos, madres no identificadas e información incompleta

La evaluación genética para caracteres de crecimiento pre - destete requiere ajustar modelos animales con efectos maternos (MAM). Tanto la estimación paramétrica de la variabilidad como la evaluación genética mediante MAM son realizadas empleando datos de campo, muchos de los cuales no poseen información completa para todas las variables explicativas maternas. Es común no contar con la identificación de madres (biológicas y/o receptoras), de abuelas maternas y, consecuentemente, de la edad de la madre (EM). Este problema es bien marcado en razas compuestas como Brangus y Braford que tienen políticas para registrar animales de pedigrí "abierto". Además, no existe un consenso sobre cuál es el mejor modelo de predicción, y existen interrogantes sobre la magnitud de los componentes de (co) varianza genético-aditivos y ambientales del modelo de evaluación. La primera investigación de esta tesis consistió en la estimación, mediante métodos bayesianos de los parámetros de dispersión en MAMs con distintas estructuras de (co) varianza, para datos de peso al destete de animales Angus de pedigrí. El análisis se caracterizó por la originalidad en los muestreos de las distribuciones marginales posteriores de las covarianzas genéticas aditivas y de la correlación entre los efectos ambientales maternos permanentes de una vaca y sus hijas también madres. Con el objeto de especificar correctamente la fracción aditiva de las (co) varianzas cuando se desconocen las madres y/o abuelas maternas de los animales con datos, en otro capítulo se desarrollaron MAMs equivalentes que no requieren alargar los vectores de los valores de cría con madres o abuelas fantasmas. Finalmente, se desarrolló un modelo mixto que atenúa el sesgo por error de medición clásico en el efecto EM, e introduce splines penalizadas y una estructura de (co) variación autoregresiva de orden 1 para suavizar las covarianzas residuales Este modelo es apropiado para ajustar datos de animales nacidos por transplante embrionario con madres receptoras desconocidas

Reproducir curvas en la pantalla de una calculadora

¿Cómo se logran esas bonitas y suaves curvas en la pantalla de un ordenador? Parece que fluyen suavemente y no tienen ese efecto desigual que sale si dibujas un montón de puntos y los unes con segmentos rectilíneos. La razón es que el software muestrea los dibujos y usa métodos de interpolación suave. A menudo, el método de interpolación es el llamado de los splines cúbicos, que aprovecha inteligentemente ciertos conceptos matemáticos corrientes, como mostraremos a continuación.

Algunos aspectos de polinomios de Bernstein, Bezier y trazadores

Este trabajo consta de dos partes: la primera presenta, de manera elemental, la teoría de los polinomios de Bernstein en una variable; la segunda esta dedicada a curvas de Bezier y q-trazadores ("q-splines"). Nos parece importante el uso que se puede dar del software Mathematica.

B-spline methods in R-matrix theory for scattering in two-electron systems

The use of B-spline basis sets in R-matrix theory for scattering processes has been investigated. In the present approach a B-spline basis is used for the description of the inner region, which is matched to the physical outgoing wavefunctions by the R-matrix. Using B-splines, continuum basis functions can be determined easily, while pseudostates can be included naturally. The accuracy for low-energy scattering processes is demonstrated by calculating inelastic scattering cross sections for e colliding on H. Very good agreement with other calculations has been obtained. Further extensions of the codes to quasi two-electron systems and general atoms are discussed as well as the application to (multi) photoionization.