arXiv subjects

Sudeshna Roy

Publications and source records attributed to Sudeshna Roy.

35 records · Page 2Linked to original sources

arXiv2023

Structural fluctuations in thin cohesive particle layers in powder-based additive manufacturing

Producing dense and homogeneous powder layers with smooth free surface is challenging in additive manufacturing, as interparticle cohesion can strongly affect the powder packing structure and therefore influence the quality of the end product. We use the Discrete Element Method to simulate the spreading process of spherical powders and examine how cohesion influences the characteristics of the packing structure with a focus on the fluctuation of the local morphology. As cohesion increases, the overall packing density decreases, and the free surface roughness increases, which is calculated from digitized surface height distributions. Local structural fluctuations for both quantities are examined through the local packing anisotropy on the particle scale, obtained from Vorono\"ı tessellation. The distributions of these particle-level metrics quantify the increasingly heterogeneous packing structure with clustering and changing surface morphology.

cond-mat.mtrl-sci↗

arXiv2023

DEM simulation of the powder application in powder bed fusion

The packing behavior of powders is significantly influenced by various types of inter-particle attractive forces, including adhesion and non-bonded van der Waals forces [1, 2, 3, 4, 5, 6]. Alongside particle size and shape distributions, the inter-particle interactions, in particular frictional and adhesive forces, play a crucial role in determining the flow behavior and consequently the packing density of the powder layer. The impact of various types of attractive forces on the packing density of powders with different materials and particle size distributions remains largely unexplored and requires further investigation. Accurately comprehending these effects through experiments while considering specific particle size distributions and material properties poses significant challenges. To address these challenges, we employ Discrete Element Method (DEM) simulations to characterize the packing behavior of fine powders. We can demonstrate quantitative agreement with experimental results by incorporating the appropriate particle size distribution and using an adequate model of attractive particle interactions. Furthermore, our findings indicate that both adhesion, which is modeled using the Johnson-Kendall-Roberts (JKR) model [7], and van der Waals interactions are crucial factors that must be taken into account in DEM simulations.

cond-mat.soft↗

arXiv2023

Approximate Expressions for the Capillary Force and the Surface Area of a Liquid Bridge between Identical Spheres

We consider a liquid bridge between two identical spheres and provide approximate expressions for the capillary force and the exposed surface area of the liquid bridge as functions of the liquid bridge's total volume and the sphere separation distance. The radius of the spheres and the solid-liquid contact angle are parameters that enter the expressions. These expressions are needed for efficient numerical simulations of drying suspensions.

physics.comp-ph↗

arXiv2023

Computing mixed multiplicities, mixed volumes, and sectional Milnor numbers

This is an expository version of our paper [arXiv:1902.07384]. Our aim is to present recent Macaulay2 algorithms for computation of mixed multiplicities of ideals in a Noetherian ring which is either local or a standard graded algebra over a field. These algorithms are based on computation of the equations of multi-Rees algebras of ideals that generalises a result of Cox, Lin and Sosa. Using these equations we propose efficient algorithms for computation of mixed volumes of convex lattice polytopes and sectional Milnor numbers of hypersurfaces with an isolated singularity.

math.AC↗

arXiv2023

Graded components of local cohomology modules of $\mathfrak{C}$-monomial ideals in characteristic zero

Let $A$ be a commutative Noetherian ring of characteristic zero and $R=A[X_1, \ldots, X_d]$ be a polynomial ring over $A$ with the standard $\mathbb{N}^d$-grading. Let $I\subseteq R$ be an ideal which can be generated by elements of the form $aU$ where $a \in A$ (possibly nonunit) and $U$ is a monomial in $X_i$'s. We call such an ideal as a `$\mathfrak{C}$-monomial ideal'. Local cohomology modules supported on monomial ideals gain a great deal of interest due to their applications in the context of toric varieties. It was observed that for $\underline{u} \in \mathbb{Z}^d$, their $\underline{u}^{th}$ components depend only on which coordinates of $\underline{u}$ are negative. In this article, we show that this statement holds true in our general setting, even for certain invariants of the components. We mainly focus on the Bass numbers, injective dimensions, dimensions, associated primes, Bernstein-type dimensions, and multiplicities of the components. Under the extra assumption that $A$ is regular, we describe the finiteness of Bass numbers of each component and bound its injective dimension by the dimension of its support. Finally, we present a structure theorem for the components when $A$ is the ring of formal power series in one variable over a characteristic zero field.

math.AC↗

arXiv2022

Graded components of local cohomology modules supported on $\mathfrak{C}$-monomial ideals

Let $A$ be a Dedekind domain of characteristic zero such that its localization at every maximal ideal has mixed characteristic with finite residue field. Let $R=A[X_1,\ldots, X_n]$ be a polynomial ring and $I=(a_1U_1, \ldots, a_c U_c)\subseteq R$ an ideal, where $a_j \in A$ (not necessarily units) and $U_j$'s are monomials in $X_1, \ldots, X_n$. We call such an ideal $I$ as a $\mathfrak{C}$-monomial ideal. Consider the standard multigrading on $R$. We produce a structure theorem for the multigraded components of the local cohomology modules $H^i_I(R)$ for $i \geq 0$. We further analyze the torsion part and the torsion-free part of these components. We show that if $A$ is a PID then each component can be written as a direct sum of its torsion part and torsion-free part. As a consequence, we obtain that their Bass numbers are finite.

math.AC↗

arXiv2022

Algorithms for computing mixed multiplicities, mixed volumes and sectional Milnor numbers

We present a package 'MixedMultiplicity' for computing mixed multiplicities of ideals in a Noetherian ring which is either local or a standard graded algebra over a field. This enables us to find mixed volumes of convex lattice polytopes and sectional Milnor numbers of hypersurfaces with an isolated singularity. The algorithms make use of the defining equations of the multi-Rees algebra of ideals, which are obtained by generalising a result of D. Cox. K.-N. Lin, and G. Sosa.

math.AC↗

arXiv2021

Graded components of Local cohomology modules II

Let $A$ be a commutative Noetherian ring containing a field of characteristic zero. Let $R= A[X_1, \ldots, X_m]$ be a polynomial ring and $A_m(A) = A \langle X_1, \ldots, X_m, \partial_1, \ldots, \partial_m \rangle$ be the $m^{th}$ Weyl algebra over $A$, where $\partial_i=\partial/\partial X_i$. Consider both $R$ and $A_m(A)$ as standard graded with $\mathrm{deg}~ z=0$ for all $z \in A$, $\mathrm{deg}~ X_i=1$, and $\mathrm{deg}~ \partial_i =-1$ for $i=1, \ldots, m$. We present a few results about the behavior of the graded components of local cohomology modules $H_I^i(R)$, where $I$ is an arbitrary homogeneous ideal in $R$. We mostly restrict our attention to the Vanishing, Tameness, and Rigidity properties. To obtain this, we use the theory of $D$-modules and show that generalized Eulerian $A_m(A)$-modules exhibit these properties. As a corollary, we further get that components of graded local cohomology modules with respect to a pair of ideals display similar behavior.

math.AC↗

arXiv2021

A direct link between active matter and sheared granular systems

The similarity in mechanical properties of dense active matter and sheared amorphous solids has been noted in recent years without a rigorous examination of the underlying mechanism. We develop a mean-field model that predicts that their critical behavior should be equivalent in infinite dimensions, up to a rescaling factor that depends on the correlation length of the applied field. We test these predictions in 2d using a new numerical protocol, termed `athermal quasi-static random displacement', and find that these mean-field predictions are surprisingly accurate in low dimensions. We identify a general class of perturbations that smoothly interpolate between the uncorrelated localized forces that occur in the high-persistence limit of dense active matter, and system-spanning correlated displacements that occur under applied shear. These results suggest a universal framework for predicting flow, deformation, and failure in active and sheared disordered materials.

cond-mat.soft↗

arXiv2019

Eakin-Sathaye type theorems for joint reductions and good filtrations of ideals

Analogues of Eakin-Sathaye theorem for reductions of ideals are proved for ${\mathbb N}^s$-graded good filtrations. These analogues yield bounds on joint reduction vectors for a family of ideals and reduction numbers for $\mathbb N$-graded filtrations. Several examples related to lex-segment ideals, contracted ideals in $2$-dimensional regular local rings and the filtration of integral and tight closures of powers of ideals in hypersurface rings are constructed to show effectiveness of these bounds.

math.AC↗

arXiv2019

Inferring Concept Prerequisite Relations from Online Educational Resources

The Internet has rich and rapidly increasing sources of high quality educational content. Inferring prerequisite relations between educational concepts is required for modern large-scale online educational technology applications such as personalized recommendations and automatic curriculum creation. We present PREREQ, a new supervised learning method for inferring concept prerequisite relations. PREREQ is designed using latent representations of concepts obtained from the Pairwise Latent Dirichlet Allocation model, and a neural network based on the Siamese network architecture. PREREQ can learn unknown concept prerequisites from course prerequisites and labeled concept prerequisite data. It outperforms state-of-the-art approaches on benchmark datasets and can effectively learn from very less training data. PREREQ can also use unlabeled video playlists, a steadily growing source of training data, to learn concept prerequisites, thus obviating the need for manual annotation of course prerequisites.

cs.CL↗

arXiv2019

Surface flow profiles for dry and wet granular materials by Particle Tracking Velocimetry; the effect of wall roughness

Two-dimensional Particle Tracking Velocimetry (PTV) is a promising technique to study the behaviour of granular flows. The aim is to experimentally determine the free surface width and position of the shear band from the velocity profile to validate simulations in a split-bottom shear cell geometry. The position and velocities of scattered tracer particles are tracked as they move with the bulk flow by analyzing images. We then use a new technique to extract the continuum velocity field, applying coarse-graining with the postprocessing toolbox MercuryCG on the discrete experimental PTV data. For intermediate filling heights, the dependence of the shear (or angular) velocity on the radial coordinate at the free surface is well fitted by an error function. From the error function, we get the width and the centre position of the shear band. We investigate the dependence of these shear band properties on filling height and rotation frequencies of the shear cell for dry glass beads for rough and smooth wall surfaces. For rough surfaces, the data agrees with the existing experimental results and theoretical scaling predictions. For smooth surfaces, particle-wall slippage is significant and the data deviates from the predictions. We further study the effect of cohesion on the shear band properties by using small amount of silicon oil and glycerol as interstitial liquids with the glass beads. While silicon oil does not lead to big changes, glycerol changes the shear band properties considerably. The shear band gets wider and is situated further inward with increasing liquid saturation, due to the correspondingly increasing trend of particles to stick together.

cond-mat.soft↗

arXiv2018

On the defining equations of Rees algebra of a height two perfect ideal using the theory of $D$-modules

Let $k$ be a field of characteristic zero, and $R=k[x_1, \ldots, x_d]$ with $d \geq 3$ be a polynomial ring in $d$ variables. Let $\m=(x_1, \ldots, x_d)$ be the homogeneous maximal ideal of $R$. Let $\mathcal{K}$ be the kernel of the canonical map $α: \sym(I) \rightarrow \R(I)$, where $\sym(I)$ (resp. $\R(I)$) denotes the symmetric algebra (resp. the Rees algebra) of an ideal $I$ in $R$. We study $\mathcal{K}$ when $I$ is a height two perfect ideal minimally generated by $d+1$ homogeneous elements of same degree and satisfies $G_d$, that is, the minimal number of generators of the ideal $I_{\mathfrak{p}}$, $μ(I_{\mathfrak{p}}) \leq \dim R_{\mathfrak{p}}$ for every $\mathfrak{p} \in V(I) \backslash \{\m\}$. We show that \begin{enumerate}[{\rm (i)}] \item $\mathcal{K}$ can be described as the solution set of a system of differential equations, \item the whole bigraded structure of $\mathcal{K}$ is characterized by the integral roots of certain $b$-functions, \item certain de Rham cohomology groups can give partial information about $\mathcal{K}$. \end{enumerate}

math.AC↗

arXiv2017

Graded components of local cohomology modules of invariant rings

Let $A$ be a regular domain containing a field $K$ of characteristic zero, $G$ be a finite subgroup of the group of automorphisms of $A$ and $B=A^G$ be the ring of invariants of $G$. Let $S= A[X_1,\ldots, X_m]$ and $R= B[X_1, \ldots, X_m]$ be standard graded with $\ deg \ A=0$, $\ deg \ B=0$ and $\ deg \ X_i=1$ for all $i$. Extend the action of $G$ on $A$ to $S$ by fixing $X_i$. Note $S^G=R$. Let $I$ be an arbitrary homogeneous ideal in $R$. The main goal of this paper is to establish a comparative study of graded components of local cohomology modules $H_I^i(R)$ that would be analogs to those proven in a previous paper of the first author for $H_J^i(S)$ where $J$ is an arbitrary homogeneous ideal in $S$.

math.AC↗

arXiv2017

On derived functors of Graded local cohomology modules

Let $K$ be a field of characteristic zero and let $R=K[X_1, \ldots,X_n ]$, with standard grading. Let $\mathfrak{m}= (X_1, \ldots, X_n)$ and let $E$ be the $^*$injective hull of $R/\mathfrak{m}.$ Let $A_n(K)$ be the $n^{th}$ Weyl algebra over $K$. Let $I, J$ be homogeneous ideals in $R$. Fix $i,j \geq 0$ and set $M = H^i_I(R)$ and $N = H^j_J(R)$ considered as left $A_n(K)$-modules. We show the following two results for which no analogous result is known in charactersitc $p > 0$. \begin{enumerate} $H^l_\mathfrak{m}(\Tor^R_ν(M, N)) \cong E(n)^{a_{l,ν}}$ for some $a_{l,ν} \geq 0$. For all $ν\geq 0$; the finite dimensional vector space $\Tor^{A_n(K)}_ν( M^\sharp, N)$ is concentrated in degree $-n$ (here $M^\sharp$ is the standard right $A_n(K)$-module associated to $M$). \end{enumerate} We also conjecture that for all $i \geq 0$ the finite dimensional vector space $\Ext^i_{A_n(K)}(M, N)$ is concentrated in degree zero. We give a few examples which support this conjecture.

math.AC↗

arXiv2016

Towards a general(ized) shear thickening rheology of wet granular materials under small pressure

We study the rheology of dry and wet granular materials in the steady quasistatic regime using the Discrete Element Method (DEM) in a split-bottom ring shear cell with focus on the macroscopic friction. The aim of our study is to understand the local rheology of bulk flow at various positions in the shear band, where the system is in critical state. The general(ized) rheology has four dimensionless control parameters that relate the time scales of five significant phenomena, namely, the time scales related to confining pressure $t_p$, shear rate $t_{\dotγ}$, particle stiffness $t_k$, gravity $t_g$ and cohesion $t_c$, respectively. We show that those phenomena collectively contribute to the rheology as multiplicative correction functions. While $t_{\dotγ}$ is large and thus little important for most of the data studied, it can increase the friction of flow in critical state, where the shear gradients are high. $t_g$ and $t_k$ are comparable to $t_p$ in the bulk, but become more or less dominant relative to $t_p$ at the extremes of the free surface and deep inside the bulk, respectively. We also measure the effect of strong wet cohesion on the flow rheology, as quantified by decreasing $t_c$. Furthermore, the proposed rheological model predicts well the shear thinning behavior both in the bulk and near the free surface; shear thinning develops to shear thickening near the free surface with increasing cohesion.

cond-mat.soft↗

arXiv2015

Micro-macro transition and simplified contact models for wet granular materials

Wet granular materials in a quasi-static steady state shear flow have been studied with discrete particle simulations. Macroscopic quantities, consistent with the conservation laws of continuum theory, are obtained by time averaging and spatial coarse-graining. Initial studies involve understanding the effect of liquid content and liquid properties like the surface tension on the macroscopic quantities. Two parameters of the liquid bridge contact model have been studied as the constitutive parameters that define the structure of this model (i) the rupture distance of the liquid bridge model, which is proportional to the liquid content, and (ii) the maximum adhesive force, as controlled by the surface tension of the liquid. Subsequently a correlation is developed between these micro parameters and the steady state cohesion in the limit of zero confining pressure. Furthermore, as second result, the macroscopic torque measured at the walls, which is an experimentally accessible parameter, is predicted from our simulation results as a dependence on the micro-parameters. Finally, the steady state cohesion of a realistic non-linear liquid bridge contact model scales well with the steady state cohesion for a simpler linearized irreversible contact model with the same maximum adhesive force and equal energy dissipated per contact.

cond-mat.soft↗