arXiv subjects

Shiva Rudraraju

Publications and source records attributed to Shiva Rudraraju.

At least 19 recordsLinked to original sources

arXiv2026

Numerical modeling of microstructure evolution in nanocrystalline alloys - grain boundary segregation, solute drag and mechanics

Nanocrystalline (NC) alloys hold much promise as structural alloys due to their superior mechanical properties over traditional coarser grained microcrystalline alloys. Strength of metallic alloys is related to the underlying grain size - as represented by the classical Hall-Petch relation. Generally, a metals strength increases with decreasing mean grain size from the micrometer scale to the nanometer scale, until about a mean size of 20 nm. Any further decrease of grain size results in decreasing strength. Thus, there is an optimal range of mean grain size for most metals about which maximum material strength can be obtained. In the context of NC alloys, stabilization of the grain size in this optimal range is one of the primary synthesis challenges. Since nm-scale mean grain sizes are desired, phenomena like GB solute segregation and solute precipitation are utilized during alloy synthesis to mitigate grain growth. Numerical modeling the phenomena of GB-solute interactions and the evolution of these stabilized GBs under mechanical load are of immense interest to the NC alloy community. To enrich the numerical modeling formulations available in this space, we present here a phase-field method based numerical framework to model GB segregation, solute precipitation and effect of external loading on NC alloys. While some of these effects have been modeled in isolation, a unified treatment of the solute and GB segregation with mechanics interactions has not be considered in the literature. We present a 3D FEM finite-strain phase-field formulation for modeling grain evolution and microstructure stabilization. Beyond the formulation, various case studies demonstrate the applicability of this framework. Further, thermodynamic and kinetic arguments are provided based on the evolution of GB energy to explain the effects of solute drag, GB pinning and mechanical deformation.

cond-mat.mtrl-sci↗

arXiv2025

Mechano-chemical modeling of glia initiated secondary injury of neurons under mechanical load

Traumatic Brain Injury (TBI) results from an impact or concussion to the head with the injury being specifically characterized through pathological degradation at various biological length scales. Following injury, various mechanical modeling techniques have been proposed in the literature that seek to quantify neuronal-scale to tissue-scale metrics of brain damage. Broadly, the two categories of degradation encompass physiological deterioration of neurons and upregulation of chemical entities such as neurotransmitters which causes initiation of downstream pathophysiological effects. Despite the many contributing pathways, in this work, we delineate and model a potential glia-initiated injury pathway that leads to secondary injury. The goal of this work is to demonstrate a continuum framework which models the multiphysics of mechano-chemical interactions underlying TBI. Using a coupled PDE (partial differential equation) formulation and FEM (finite element method) discretization, the framework highlights evolution of field variables which spatio-temporally resolve mechanical metrics and chemical species across neuronal clusters. The modeling domain encompasses microglia, neurons and the extracellular matrix. The continuum framework used to model the mechano-chemical interactions assumes a three dimensional viscoelastic network to capture the mechanical response underlying proteins constituting the neuron microstructure and advection-diffusion equations modeling spatio-temporal evolution of chemical species. We use this framework to numerically estimate key concentrations of chemical species produced by the strain field. In this work, we identify key biomarkers within the labyrinth of molecular pathways and build a framework that captures the core mechano-chemical interactions. This framework is an attempt to quantify secondary injury and thus assist in developing targeted TBI treatments.

q-bio.QM↗

arXiv2024

Electro-diffusive modeling and the role of spine geometry on action potential propagation in neurons

Electrical signaling in the brain plays a vital role to our existence but at the same time, the fundamental mechanism of this propagation is undeciphered. Notable advancements have been made in the numerical modeling supplementing the related experimental findings. Cable theory based models provided a significant breakthrough in understanding the mechanism of electrical propagation in the neuronal axons. Cable theory, however, fails for thin geometries such as a spine or a dendrite of a neuron, amongst its other limitations. Recently, the spatiotemporal propagation has been precisely modeled using the Poisson-Nernst-Planck (PNP) electro-diffusive theory in the neuronal axons as well as the dendritic spines respectively. Patch clamp and voltage imaging experiments have extensively aided the study of action potential propagation exclusively for the neuronal axons but not the dendritic spines because of the challenges linked with their thin geometry. Assisted by the super-resolution microscopes and the voltage dyeing experiments, it has become possible to precisely measure the voltage in the dendritic spines. This has facilitated the requirement of a high fidelity numerical frame that is capable of acting as a digital twin. Here, using the PNP theory, we integrate the dendritic spine, soma and the axon region to numerically model the propagation of excitatory synaptic potential in a complete neuronal geometry with the synaptic input at the spines, potential initiating at the axon hillock and propagating through the neuronal axon. The model outputs the forward propagation of the action potential along the neuronal axons as well as the back propagation into the spines. We point out the significance of the intricate geometry of the dendritic spines, namely the spine neck length and radius, and the ion channel density in the axon hillock to the action potential initiation and propagation.

q-bio.NC↗

arXiv2022

A numerical investigation of dimensionless numbers characterizing meltpool morphology of the laser powder bed fusion process

Microstructure evolution in metal additive manufacturing (AM) is a complex multi-physics and multi-scale problem. Understanding the impact of AM process conditions on the microstructure evolution and the resulting mechanical properties of the printed part is an active area of research. At the meltpool scale, the thermo-fluidic governing equations have been extensively modeled in the literature to understand the meltpool conditions and the thermal gradients in its vicinity. In many phenomena governed by partial differential equations, dimensional analysis and identification of important dimensionless numbers can provide significant insights into the process dynamics. In this context, a novel strategy using dimensional analysis and the method of linear least squares regression to numerically investigate the thermo-fluidic governing equations of the Laser Powder Bed Fusion AM process is presented in this work. First, the governing equations are solved using the Finite Element Method, and the model predictions are validated by comparing with experimentally estimated cooling rates, and with numerical results from the literature. Then, through dimensional analysis, an important dimensionless quantity - interpreted as a measure of heat absorbed by the powdered material and the meltpool, is identified. This dimensionless measure of heat absorbed, along with classical dimensionless quantities such as Peclet, Marangoni, and Stefan numbers, is used to investigate advective transport in the meltpool for different alloys. Further, the framework is used to study the variations of thermal gradients and the solidification cooling rate. Important correlations linking meltpool morphology and microstructure evolution related variables with classical dimensionless numbers are the key contribution of this work.

math.NA↗

arXiv2022

Spatio-temporal modeling of saltatory conduction in neurons using Poisson-Nernst-Planck treatment and estimation of conduction velocity

Action potential propagation along the axons and across the dendrites is the foundation of the electrical activity observed in the brain and the rest of the central nervous system. Theoretical and numerical modeling of this action potential activity has long been a key focus area of electro-chemical neuronal modeling. Specifically, considering the presence of nodes of Ranvier along the myelinated axon, single-cable models of the propagation of action potential have been popular. Building on these models, and considering a secondary electrical conduction pathway below the myelin sheath, the double-cable model has been proposed. Such cable theory based treatments have inherent limitations in their lack of a representation of the spatio-temporal evolution of the neuronal electro-chemistry. In contrast, a Poisson-Nernst-Planck (PNP) based electro-diffusive framework accounts for the underlying spatio-temporal ionic concentration dynamics and is a more comprehensive treatment. In this work, a high-fidelity implementation of the PNP model is demonstrated. This model is shown to produce results similar to the cable theory based electrical models, and in addition, the rich spatio-temporal evolution of the underlying ionic transport is captured. Novel to this work is the extension of PNP model to axonal geometries with multiple nodes of Ranvier and multiple variants of the electro-diffusive model - PNP without myelin, PNP with myelin, and PNP with the myelin sheath and peri-axonal space. Further, we apply this spatio-temporal model to numerically estimate conduction velocity in a rat axon. Specifically, saltatory conduction due to the presence of myelin sheath and the peri-axonal space is investigated.

q-bio.QM↗

arXiv2022

Modeling of dendritic solidification and numerical analysis of the phase-field approach to model complex morphologies in alloys

Dendrites are one of the most widely observed patterns in nature and occur across a wide spectrum of physical phenomena. In solidification and growth patterns in metals and crystals, the multi-level branching structures of dendrites pose a modeling challenge, and a full resolution of these structures is computationally demanding. In the literature, theoretical models of dendritic formation and evolution, essentially as extensions of the classical moving boundary Stefan problem exist. Much of this understanding is from the analysis of dendrites occurring during the solidification of metallic alloys. Motivated by the problem of modeling microstructure evolution from liquid melts of pure metals and alloys during MAM, we developed a comprehensive numerical framework for modeling a large variety of dendritic structures that are relevant to metal solidification. In this work, we present a numerical framework encompassing the modeling of Stefan problem formulations relevant to dendritic evolution using a phase-field approach and a finite element method implementation. Using this framework, we model numerous complex dendritic morphologies that are physically relevant to the solidification of pure melts and binary alloys. The distinguishing aspects of this work are - a unified treatment of both pure metals and alloys; novel numerical error estimates of dendritic tip velocity; and the convergence of error for the primal fields of temperature and the order parameter with respect to numerical discretization. To the best of our knowledge, this is a first-of-its-kind study of numerical convergence of the phase-field equations of dendritic growth in a finite element method setting. Further, we modeled various types of physically relevant dendritic solidification patterns in 2D and 3D computational domains.

math.NA↗

arXiv2021

Biomembranes undergo complex, non-axisymmetric deformations governed by Kirchhoff-Love kinematics and revealed by a three dimensional computational framework

Biomembranes play a central role in various phenomena like locomotion of cells, cell-cell interactions, packaging of nutrients, and in maintaining organelle morphology and functionality. During these processes, the membranes undergo significant morphological changes through deformation, scission, and fusion. Modeling the underlying mechanics of such morphological changes has traditionally relied on reduced order axisymmetric representations of membrane geometry and deformation. Axisymmetric representations, while robust and extensively deployed, suffer from their inability to model symmetry breaking deformations and structural bifurcations. To address this limitation, a 3D computational mechanics framework for high fidelity modeling of biomembrane deformation is presented. The proposed framework brings together Kirchhoff-Love thin-shell kinematics, Helfrich-energy based mechanics, and state-of-the-art numerical techniques for modeling deformation of surface geometries. Lipid bilayers are represented as spline-based surfaces immersed in a 3D space; this enables modeling of a wide spectrum of membrane geometries, boundary conditions, and deformations that are physically admissible in a 3D space. The mathematical basis of the framework and its numerical machinery are presented, and their utility is demonstrated by modeling 3 classical, yet non-trivial, membrane problems: formation of tubular shapes and their lateral constriction, Piezo1-induced membrane footprint generation and gating response, and the budding of membranes by protein coats during endocytosis. For each problem, the full 3D membrane deformation is captured, potential symmetry-breaking deformation paths identified, and various case studies of boundary and load conditions are presented. Using the endocytic vesicle budding as a case study, we also present a "phase diagram" for its symmetric and broken-symmetry states.

q-bio.QM↗

arXiv2019

A mechanical model reveals that non-axisymmetric buckling lowers the energy barrier associated with membrane neck constriction

Membrane neck formation is essential for scission, which, as recent experiments on tubules have demonstrated, can be location dependent. The diversity of biological machinery that can constrict a neck such as dynamin, actin, ESCRTs and BAR proteins, and the range of forces and deflection over which they operate, suggest that the constriction process is functionally mechanical and robust to changes in biological environment. In this study, we used a mechanical model of the lipid bilayer to systematically investigate the influence of location, symmetry constraints, and helical forces on membrane neck constriction. Simulations from our model demonstrated that the energy barriers associated with constriction of a membrane neck are location-dependent. Importantly, if symmetry restrictions are relaxed, then the energy barrier for constriction is dramatically lowered and the membrane buckles at lower values of forcing parameters. Our simulations also show that constriction due to helical proteins further reduces the energy barrier for neck formation compared to cylindrical proteins. These studies establish that despite different molecular mechanisms of neck formation in cells, the mechanics of constriction naturally leads to a loss of symmetry that can lower the energy barrier to constriction.

physics.bio-ph↗

arXiv2019

A computational framework for the morpho-elastic development of molluskan shells by surface and volume growth

Mollusk shells are an ideal model system for understanding the morpho-elastic basis of morphological evolution of invertebrates' exoskeletons. During the formation of the shell, the mantle tissue secretes proteins and minerals that calcify to form a new incremental layer of the exoskeleton. Most of the existing literature on the morphology of mollusks is descriptive. The mathematical understanding of the underlying coupling between pre-existing shell morphology, de novo surface deposition and morpho-elastic volume growth is at a nascent stage, primarily limited to reduced geometric representations. Here, we propose a general, three-dimensional computational framework coupling pre-existing morphology, incremental surface growth by accretion, and morpho-elastic volume growth. We exercise this framework by applying it to explain the stepwise morphogenesis of seashells during growth: new material surfaces are laid down by accretive growth on the mantle whose form is determined by its morpho-elastic growth. Calcification of the newest surfaces extends the shell as well as creates a new scaffold that constrains the next growth step. We study the effects of surface and volumetric growth rates, and of previously deposited shell geometries on the resulting modes of mantle deformation, and therefore of the developing shell's morphology. Connections are made to a range of complex shells ornamentations.

q-bio.QM↗

arXiv2019

DFT-FE -- A massively parallel adaptive finite-element code for large-scale density functional theory calculations

We present an accurate, efficient and massively parallel finite-element code, DFT-FE, for large-scale ab-initio calculations (reaching $\sim 100,000$ electrons) using Kohn-Sham density functional theory (DFT). DFT-FE is based on a local real-space variational formulation of the Kohn-Sham DFT energy functional that is discretized using a higher-order adaptive spectral finite-element (FE) basis, and treats pseudopotential and all-electron calculations in the same framework, while accommodating non-periodic, semi-periodic and periodic boundary conditions. We discuss the main aspects of the code, which include, the various strategies of adaptive FE basis generation, and the different approaches employed in the numerical implementation of the solution of the discrete Kohn-Sham problem that are focused on significantly reducing the floating point operations, communication costs and latency. We demonstrate the accuracy of DFT-FE by comparing the energies, ionic forces and periodic cell stresses on a wide range of problems with popularly used DFT codes. Further, we demonstrate that DFT-FE significantly outperforms widely used plane-wave codes---both in CPU-times and wall-times, and on both non-periodic and periodic systems---at systems sizes beyond a few thousand electrons, with over $5-10$ fold speedups in systems with more than 10,000 electrons. The benchmark studies also highlight the excellent parallel scalability of DFT-FE, with strong scaling demonstrated on up to 192,000 MPI tasks.

physics.comp-ph↗

arXiv2018

A numerical study of branching and stability of solutions to three-dimensional martensitic phase transformations using gradient-regularized, non-convex, finite strain elasticity

In the setting of continuum elasticity, phase transformations involving martensitic variants are modeled by a free energy density function that is non-convex in strain space. Here, we adopt an existing mathematical model in which we regularize the non-convex free energy density function by higher-order gradient terms at finite strain and derive boundary value problems via the standard variational argument applied to the corresponding total free energy, inspired by Toupin's theory of gradient elasticity. These gradient terms are to preclude existence of arbitrarily fine microstructures, while still allowing for existence of multiple solution branches corresponding to local minima of the total free energy; these are classified as metastable solution branches. The goal of this work is to solve the boundary value problem numerically in three dimensions, observe solution branches, and assess stability of each branch by numerically evaluating the second variation of the total free energy. We also study how these microstructures evolve as the length-scale parameter, the coefficient of the strain gradient terms in the free energy, approaches zero.

math.NA↗

arXiv2016

A comparison of Redlich-Kister polynomial and cubic spline representations of the chemical potential in phase field computations

Free energies play a central role in many descriptions of equilibrium and non-equilibrium properties of solids. Continuum partial differential equations (PDEs) of atomic transport, phase transformations and mechanics often rely on first and second derivatives of a free energy function. The stability, accuracy and robustness of numerical methods to solve these PDEs are sensitive to the particular functional representations of the free energy. In this communication we investigate the influence of different representations of thermodynamic data on phase field computations of diffusion and two-phase reactions in the solid state. First-principles statistical mechanics methods were used to generate realistic free energy data for HCP titanium with interstitially dissolved oxygen. While Redlich-Kister polynomials have formed the mainstay of thermodynamic descriptions of multi-component solids, they require high order terms to fit oscillations in chemical potentials around phase transitions. Here we demonstrate that high fidelity fits to rapidly fluctuating free energy functions are obtained with spline functions. Spline functions that are many degrees lower than Redlich-Kister polynomials provide equal or superior fits to chemical potential data and, when used in phase field computations, result in solution times approaching an order of magnitude speed up relative to the use of Redlich-Kister polynomials.

cond-mat.mtrl-sci↗

arXiv2016

A variational treatment of material configurations with application to interface motion and microstructural evolution

We present a unified variational treatment of evolving configurations in crystalline solids with microstructure. The crux of our treatment lies in the introduction of a vector configurational field. This field lies in the material, or configurational, manifold, in contrast with the traditional displacement field, which we regard as lying in the spatial manifold. We identify two distinct cases which describe (a) problems in which the configurational field's evolution is localized to a mathematically sharp interface, and (b) those in which the configurational field's evolution can extend throughout the volume. The first case is suitable for describing incoherent phase interfaces in polycrystalline solids, and the latter is useful for describing smooth changes in crystal structure and naturally incorporates coherent (diffuse) phase interfaces. For sharp interfaces that are out-of-equilibrium, the second law of thermodynamics furnishes restrictions on the kinetic law for the interface velocity. The class of problems in which the material undergoes configurational changes between distinct, stable crystal structures are characterized by free energy density functions that are non-convex with respect to configurational strain. For physically meaningful solutions and mathematical well-posedness, it becomes necessary to incorporate interfacial energy. This we have done by introducing a configurational strain gradient dependence in the free energy density function following ideas laid out by Toupin (Arch. Rat. Mech. Anal. 11, 1962, 385-414). The variational treatment leads to a system of partial differential equations governing the configuration that is coupled with the traditional equations of nonlinear elasticity. Numerical examples are presented to demonstrate interface motion as well as evolving microstructures of crystal structures.

cond-mat.soft↗

arXiv2016

Multi-physics simulations of lithiation-induced stress in Li$_{\rm 1+x}$Ti$_2$O$_4$ electrode particles

Cubic spinel Li$_{\rm 1+x}$Ti$_2$O$_4$ is a promising electrode material as it exhibits a high lithium diffusivity and undergoes minimal changes in lattice parameters during lithiation and delithiation, thereby ensuring favorable cycleability. The present work is a multi-physics and multi-scale study of Li$_{\rm 1+x}$Ti$_2$O$_4$ that combines first principles computations of thermodynamic and kinetic properties with continuum scale modeling of lithiation-delithiation kinetics. Density functional theory calculations and statistical mechanics methods are used to calculate lattice parameters, elastic coefficients, thermodynamic potentials, migration barriers and Li diffusion coefficients. These quantities then inform a phase field framework to model the coupled chemo-mechanical evolution of electrode particles. Several case studies accounting for either homogeneous or heterogeneous nucleation are considered to explore the temporal evolution of maximum principle stress values, which serve to indicate stress localization and the potential for crack initiation, during lithiation and delithiation.

cond-mat.mtrl-sci↗

arXiv2016

Mechano-chemical spinodal decomposition: A phenomenological theory of phase transformations in multi-component, crystalline solids

We present a phenomenological treatment of diffusion-driven martensitic phase transformations in multi-component crystalline solids that arise from non-convex free energies in mechanical and chemical variables. The treatment describes diffusional phase transformations that are accompanied by symmetry breaking structural changes of the crystal unit cell and reveals the importance of a mechano-chemical spinodal, defined as the region in strain-composition space where the free energy density function is non-convex. The approach is relevant to phase transformations wherein the structural order parameters can be expressed as linear combinations of strains relative to a high-symmetry reference crystal. The governing equations describing mechano-chemical spinodal decomposition are variationally derived from a free energy density function that accounts for interfacial energy via gradients of the rapidly varying strain and composition fields. A robust computational framework for treating the coupled, higher order diffusion and nonlinear strain gradient elasticity problems is presented. Because the local strains in an inhomogeneous, transforming microstructure can be finite, the elasticity problem must account for geometric nonlinearity. An evaluation of available experimental phase diagrams and first-principles free energies suggests mechano-chemical spinodal decomposition should occur in metal hydrides such as ZrH$_{2-2c}$. The rich physics that ensues is explored in several numerical examples in two and three dimensions and the relevance of the mechanism is discussed in the context of important electrode materials for Li-ion batteries and high temperature ceramics.

physics.chem-ph↗

arXiv2016

Unconditionally stable, second-order accurate schemes for solid state phase transformations driven by mechano-chemical spinodal decomposition

We consider solid state phase transformations that are caused by free energy densities with domains of non-convexity in strain-composition space; we refer to the non-convex domains as mechano-chemical spinodals. The non-convexity with respect to composition and strain causes segregation into phases with different crystal structures. We work on an existing model that couples the classical Cahn-Hilliard model with Toupin's theory of gradient elasticity at finite strains. Both systems are represented by fourth-order, nonlinear, partial differential equations. The goal of this work is to develop unconditionally stable, second-order accurate time-integration schemes, motivated by the need to carry out large scale computations of dynamically evolving microstructures in three dimensions. We also introduce reduced formulations naturally derived from these proposed schemes for faster computations that are still second-order accurate. Although our method is developed and analyzed here for a specific class of mechano-chemical problems, one can readily apply the same method to develop unconditionally stable, second-order accurate schemes for any problems for which free energy density functions are multivariate polynomials of solution components and component gradients. Apart from an analysis and construction of methods, we present a suite of numerical results that demonstrate the schemes in action.

physics.comp-ph↗

arXiv2016

A three dimensional field formulation, and isogeometric solutions to point and line defects using Toupin's theory of gradient elasticity at finite strains

We present a field formulation for defects that draws from the classical representation of the cores as force dipoles. We write these dipoles as singular distributions. Exploiting the key insight that the variational setting is the only appropriate one for the theory of distributions, we arrive at universally applicable weak forms for defects in nonlinear elasticity. Remarkably, the standard, Galerkin finite element method yields numerical solutions for the elastic fields of defects that, when parameterized suitably, match very well with classical, linearized elasticity solutions. The true potential of our approach, however, lies in its easy extension to generate solutions to elastic fields of defects in the regime of nonlinear elasticity, and even more notably for Toupin's theory of gradient elasticity at finite strains(Arch. Rat. Mech. Anal., 11, 385, 1962). In computing these solutions we adopt recent numerical work on an isogeometric analytic framework that enabled the first three-dimensional solutions to general boundary value problems of Toupin's theory (Rudraraju et al. Comp. Meth. App. Mech. Engr., 278, 705, 2014). We first present exhaustive solutions to point defects, edge and screw dislocations, and a study on the energetics of interacting dislocations. Then, to demonstrate the generality and potential of our treatment, we apply it to other complex dislocation configurations, including loops and low-angle grain boundaries.

cond-mat.mtrl-sci↗

arXiv2015

Three-dimensional iso-geometric solutions to general boundary value problems of Toupin's gradient elasticity theory at finite strains

We present, to the best of our knowledge, the first complete three-dimensional numerical solutions to a broad range of boundary value problems for a general theory of finite strain gradient elasticity. We have chosen for our work, Toupin's theory [Arch. Rat. Mech. Anal., 11(1), 385-414, 1962]-one of the more general formulations of strain gradient elasticity. Our framework has three crucial ingredients: The first is iso-geometric analysis [Hughes et al., Comp. Meth. App. Mech. Engrg., 194(39-41), 4135-4195, 2005], which we have adopted for its straightforward and robust representation of C1-continuity. The second is a weak treatment of the higher-order Dirichlet boundary conditions in the formulation, which control the development of strain gradients in the solution. The third ingredient is algorithmic (automatic) differentiation, which eliminates the need for linearization "by hand" of the rather complicated geometric and material nonlinearities in gradient elasticity at finite strains. We present a number of numerical solutions to demonstrate that the framework is applicable to arbitrary boundary value problems in three dimensions. We discuss size effects, the role of higher-order boundary conditions, and perhaps most importantly, the relevance of the framework to problems with elastic free energy density functions that are non-convex in strain space.

math-ph↗