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Realistic modeling of a granular mass Dr. Guilhem MOLLON Prof. ZHAO Jidong Hong Kong, December 2011

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-Early attempts of discrete modeling of granular materials used assemblies of circles or spheres. -There is a growing interest in assessing the effect of the particle shapes on the behavior of a granular mass. Ellipses and ellipsoids are often used to introduce shape anisotropy. -However, such a modeling can hardly account for the wide variety of grain shapes that actually exist, especially in natural materials such as sands. Introduction 2 Cho et al. (2006) Azéma et al.(2009) Ng (2009) PFC 2D

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Introduction 3 Meanwhile, many researchers are working on a precise characterization of the shapes of sand grains. Due to the variety of the possible shapes, a large number of shape descriptors have been proposed to describe grain properties such as Elongation, Angularity, Sphericity, Regularity, etc. Specific definitions of some of these properties were chosen in this study: -Aspect ratio (A) - related to the extreme dimensions of the grain -Roundness (B) - related to the average radius of the corners (Wadell, 1932) -Sphéricity (C) - related to the radius of the inscribed and circumscribed circles (Riley, 1941) -Regularity (D) - related to the convex and actual perimeters of the grain Blott and Pye (2008)

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Introduction 4 Purposes of the study : 1. Generate randomly some grain shapes fulfilling some target properties of aspect ratio, roundness, circularity, and regularity. 2. Implement a general method to introduce these shapes in any code of discrete modeling. 3. Develop a method to pack these grains in any container shape, respecting a target size distribution and/or grain orientation.

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1. Grain generation 5 Fourier Discrete transform for shape description A method of description of the shape of the 2D contour of a sand grain was proposed in Bowman et al. (2000), and in Das (2007): -Discretize the contour by constant angular sectors -Evaluate the corresponding radius (distances to the centre) -Submit this series (θ i, R i ) to a FFT (Fast Fourier Transform) to obtain its DFT (Discrete Fourier Transform) -The modulus of the corresponding complex spectrum is then used as a descriptor of the grain shape Das (2007)

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1. Grain generation 6 Fourier Discrete transform for shape description These researchers observed that: -the few first modes define the overall shape of the grain. -the statistics of these first modes are a good description of a specific sand. -the next modes define the roughness of the grain surface. -the amplitudes of these modes decrease with a logarithmic law depending on only one parameter, typical of each sand. Bowman et al. (2000) Das (2007)

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1. Grain generation 7 Is it possible to reverse the operation ? Since each spectrum is typical of a given sand, why not use this tool to generate random grains ? -mode 0: average radius (equal to 1) -mode 1: shift from the centre (equal to 0) -mode 2: elongation of the particle -modes 3-7: shape of the particle -modes>7: roughness of the particle To simplify the spectrum generation, 5 descriptors are chosen: -D2 -D3 -Decay1 (from D3 to D7) -D8 -Decay2 (from D8 to D…)

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1. Grain generation 8 How to introduce the randomness ? For a given spectrum, it is desirable to be able to define a large number of different grain shapes respecting the same properties. The randomness may be introduced using the phase delay of each mode, even if they have a constant amplitude. Each mode is therefore assigned a random phase (between –π and π). An inverse FFT leads to a random discrete signal R i (θ i ). By transposing it in Cartesian coordinates, we obtain a random grain shape that respects the target spectrum.

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1. Grain generation 9 Random grain generation The grain properties (Aspect ratio, Roundness, Circularity, Regularity) can be computed quite easily after programming algorithms of determination of the inscribed and circumscribed circles and of the convex envelope. The grain aspects are very well correlated with the chosen shape descriptors: D2=0 D3=0 Decay1=-1 D8=0 Decay2=-1 D2=0.2 D3=0 Decay1=-1 D8=0 Decay2=-1 D2=0 D3=0.05 Decay1=-1 D8=0 Decay2=-1 D2=0 D3=0.08 Decay1=-0.5 D8=0 Decay2=-1 D2=0 D3=0 Decay1=-1 D8=0.03 Decay2=-0.8 D2=0.2 D3=0.1 Decay1=-0.8 D8=0.03 Decay2=-0.8

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2. Introduction into a DEM code 10 Existing methods to introduce shapes in DEM codes -Sphero-polyedra -> not efficient for very complex shapes -Potential particles -> does not work for concave particles -Overlapping Discrete Element Clusters -> ODECs seem to fulfill all the conditions Das (2007) Mollon et al. (2011) Houlsby (2009) Ferellec and McDowell (2010)

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2. Introduction into a DEM code 11 Overlapping Discrete Element Clusters (ODECs) Ferellec and McDowell (2010) The principle of ODECs is to fill a particle of complex shape with overlapping circles (2D) or spheres (3D). Several algorithms exist for this filling, and the most recent one was proposed by Ferellec and McDowell (2010). -Pick a point randomly -Find the largest circle tangent to the contour at this point -Define the covered points -Start again from any uncovered point -Stop when a target proportion of the points are covered

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3. Efficient packing of complex particles 12 Packing strategies Fu and Dafalias(2011) PFC 2D -Packing by gravity -> Suitable but long -Particle expansion/compression -> Difficult for complex shapes -Proposition: why not try to use Voronoi diagrams for efficient packing ?

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3. Efficient packing of complex particles 13 Limitation of the Voronoi Packing A classical Voronoi diagram does not completely cover a closed domain, because of open cells -> Need for a modified Voronoi algorithm: -Localize the problematic cells (i.e. the ones with at least one point outside of the domain) -Localize the corresponding points -Define their symmetric with respect to the domain boundary -Start again the Voronoi tesselation -Keep the cells of the initial points only

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3. Efficient packing of complex particles 14 Voronoi Packing How to tailor the distribution of a cloud of points in order for the Voronoi Diagram to match some target properties, e.g. size distribution ? The Inverse Monte-Carlo (IMC) method was proposed in Gross and Li (2002). The principle is to choose a point and move it randomly to a new position. If the statistics of the Voronoi diagram are improved (with respect to a target distribution), the new position is kept. The process is iterated until a satisfying distribution is achieved.

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3. Efficient packing of complex particles 15 IMC improvement The Inverse Monte-Carlo method has one main drawback: its proposed formulation requires a new Voronoi tessellation at each algorithm cycle. For large numbers of cells, this is extremely long. -> A new algorithm is needed to speed up the computations. -> A work on the convergence speed is necessary

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3. Efficient packing of complex particles 16 Cell filling Algorithm of cell filling of a Voronoi diagram by a circle, based on optimization: maximize the radius, respecting the constraint that no point of the perimeter of the circle should be outside of the cell -> use of a penalization function. This algorithm is rather easy to transpose to complex shapes, but is not optimal yet in terms of computational efficiency.

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Conclusion 17 Preliminary results Packing of 500 particles in a square box: -> Low size dispersion-> High size dispersion-> Elongated particles-> Very irregular shapes-> Specific particle orientation

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Conclusion 18 Next stages -Build up a method to choose the Fourier descriptors matching with the target values of Aspect ratio, Roundness, Circularity, and Regularity -Improve the convergence speed of the Inverse Monte-Carlo method -Improve the efficiency of the cell-filling algorithm -Publish these results -Use it ! -Extend it to 3D. Fu and Dafalias (2011) Cho and Santamarina (2006)

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Thank you for your attention Dr. Guilhem MOLLON Prof. ZHAO Jidong Hong Kong, December 2011

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