Searcharxiv⌕ Search

arXiv subjects

K. B. Nakshatrala

Publications and source records attributed to K. B. Nakshatrala.

At least 19 recordsLinked to original sources
arXiv2026

An Adaptive Machine Learning Framework for Fluid Flow in Dual-Network Porous Media

Porous materials -- natural or engineered -- often exhibit dual pore-network structures that govern processes such as mineral exploration and hydrocarbon recovery from tight shales. Double porosity/permeability (DPP) mathematical models describe incompressible fluid flow through two interacting pore networks with inter-network mass exchange. Despite significant advances in numerical methods, there remains a need for computational frameworks that enable rapid forecasting, data assimilation, and reliable inverse analysis. To address this, we present a physics-informed neural network (PINN) framework for forward and inverse modeling of DPP systems. The proposed approach encodes the governing equations in mixed form, along with boundary conditions, directly into the loss function, with adaptive weighting strategies to balance their contributions. Key features of the framework include adaptive weight tuning, dynamic collocation point selection, and the use of shared trunk neural architectures to efficiently capture the coupled behavior of the dual pore networks. It is inherently mesh-free, making it well-suited for complex geometries typical of porous media. It accurately captures discontinuities in solution fields across layered domains without introducing spurious oscillations commonly observed in classical finite element formulations. Importantly, the framework is well-suited for inverse analysis, enabling robust parameter identification in scenarios where key physical quantities -- such as the mass transfer coefficient in DPP models -- are difficult to measure directly. In addition, a systematic convergence analysis is provided to rigorously assess the stability, accuracy, and reliability of the method. The effectiveness and computational advantages of the approach are demonstrated through a series of representative numerical experiments.

math.NA↗
arXiv2026

A Machine Learning-Enhanced Hopf-Cole Formulation for Nonlinear Gas Flow in Porous Media

Accurate modeling of gas flow through porous media is critical for many technological applications, including reservoir performance prediction, carbon capture and sequestration, and fuel cells and batteries. However, such modeling remains challenging due to strong nonlinear behavior and uncertainty in model parameters. In particular, gas slippage effects described by the Klinkenberg model introduce pressure-dependent permeability, which complicates numerical simulation and obscures deviations from classical Darcy flow behavior. To address these challenges, we present an integrated modeling framework for gas transport in porous media that combines a Klinkenberg-enhanced constitutive relation, Hopf-Cole-transformed mixed-form linear governing equations, a shared-trunk neural network architecture, and a Deep Least-Squares (DeepLS) solver. The Hopf-Cole transformation reformulates the original nonlinear flow equations into an equivalent linear system closely related to the Darcy model, while the mixed formulation, together with a shared-trunk neural architecture, enables simultaneous and accurate prediction of both pressure and velocity fields. A rigorous convergence analysis is performed both theoretically and numerically, establishing the stability and convergence properties of the proposed solver. Importantly, the proposed framework also naturally facilitates inverse modeling of pressure-dependent permeability and slippage parameters from limited or indirect observations, enabling efficient estimation of flow properties that are difficult to measure experimentally. Numerical results demonstrate accurate recovery of flow dynamics and parameters across a wide range of pressure regimes, highlighting the framework's robustness, accuracy, and computational efficiency for gas transport modeling and inversion in tight formations.

math.NA↗
arXiv2025

Flow Through Porous Media: A Hopf-Cole Transformation Approach for Modeling Pressure-Dependent Viscosity

Most organic liquids exhibit a pressure-dependent viscosity, making it crucial to consider this behavior in applications where pressures significantly exceed ambient conditions (e.g., geological carbon sequestration). Mathematical models describing flow through porous media while accounting for viscosity-pressure dependence are nonlinear (e.g., the Barus model). This nonlinearity complicates mathematical analysis and makes numerical solutions more time-intensive and prone to convergence issues. In this paper, we demonstrate that the Hopf-Cole transformation, originally developed for Burgers' equation, can recast the governing equations -- describing flow through porous media with pressure-dependent viscosity -- into a linear form. The transformed equations, resembling Darcy's equations in the transformed variables, enable (a) systematic mathematical analysis to establish uniqueness and maximum principles, (b) the derivation of a mechanics-based principle, and (c) the development of efficient numerical solutions using solvers optimized for Darcy equations. Notably, many properties of the linear Darcy equations naturally extend to nonlinear models that depend on pressure. For example, solutions to these nonlinear models adhere to a reciprocal relation analogous to that observed in Darcy's equations.

math.NA↗
arXiv2024

Shapley values and machine learning to characterize metamaterials for seismic applications

Given the damages from earthquakes, seismic isolation of critical infrastructure is vital to mitigate losses due to seismic events. A promising approach for seismic isolation systems is metamaterials-based wave barriers. Metamaterials -- engineered composites -- manipulate the propagation and attenuation of seismic waves. Borrowing ideas from phononic and sonic crystals, the central goal of a metamaterials-based wave barrier is to create band gaps that cover the frequencies of seismic waves. The two quantities of interest (QoIs) that characterize band-gaps are the first-frequency cutoff and the band-gap's width. Researchers often use analytical (band-gap analysis), experimental (shake table tests), and statistical (global variance) approaches to tailor the QoIs. However, these approaches are expensive and compute-intensive. So, a pressing need exists for alternative easy-to-use methods to quantify the correlation between input (design) parameters and QoIs. To quantify such a correlation, in this paper, we will use Shapley values, a technique from the cooperative game theory. In addition, we will develop machine learning models that can predict the QoIs for a given set of input (material and geometrical) parameters.

math.NA↗
arXiv2024

Closed-loop geothermal systems: Modeling and predictions

Geothermal energy is a sustainable baseload source recognized for its ability to provide clean energy on a large scale. Advanced Geothermal Systems (AGS) -- offer promising prototypes -- employ a closed-loop vascular layout that runs deep beneath the Earth's surface. A working fluid (e.g., water or supercritical carbon dioxide (sCO2)) circulates through the vasculature, entering the subsurface at the inlet and exiting at the outlet with an elevated temperature. For designing and performing cost-benefit analysis before deploying large-scale projects and maintaining efficiency while enabling real-time monitoring during the operational phase, modeling offers cost-effective solutions; often, it is the only available option for performance assessment. A knowledge gap exists due to the lack of a fast predictive modeling framework that considers the vascular intricacies, particularly the jumps in the solution fields across the channel. Noting that the channel diameter is considerably smaller in scale compared to the surrounding geological domain, we develop a reduced-order modeling (ROM) framework for closed-loop geothermal systems. This ROM incorporates the jump conditions and provides a quick and accurate prediction of the temperature field, including the outlet temperature, which directly correlates with the power production capacity and thermal draw-down. We demonstrate the predictive capabilities of the framework by establishing the uniqueness of the solutions and reporting representative numerical solutions. The modeling framework and the predictions reported in this paper benefit the closed-loop geothermal community, enabling them to determine the system's performance and optimal capacity.

cs.CE↗
arXiv2024

Effect of temperature-dependent material properties on thermal regulation in microvascular composites

Fiber-reinforced composites (FRC) provide structural systems with unique features that appeal to various civilian and military sectors. Often, one needs to modulate the temperature field to achieve the intended functionalities (e.g., self-healing) in these lightweight structures. Vascular-based active cooling offers one efficient way of thermal regulation in such material systems. However, the thermophysical properties (e.g., thermal conductivity, specific heat capacity) of FRC and their base constituents depend on temperature, and such structures are often subject to a broad spectrum of temperatures. Notably, prior active cooling modeling studies did not account for such temperature dependence. Thus, the primary aim of this paper is to reveal the effect of temperature-dependent material properties -- obtained via material characterization -- on the qualitative and quantitative behaviors of active cooling. By applying mathematical analysis and conducting numerical simulations, we show this dependence does not affect qualitative attributes, such as minimum and maximum principles (in the same spirit as \textsc{Hopf}'s results for elliptic partial differential equations). However, the dependence slightly affects quantitative results, such as the mean surface temperature and thermal efficiency. The import of our study is that it provides a deeper understanding of thermal regulation systems under practical scenarios and can guide researchers and practitioners in perfecting associated designs.

math.NA↗
arXiv2024

Dynamic properties of double porosity/permeability model

Understanding fluid movement in multi-pored materials is vital for energy security and physiology. For instance, shale (a geological material) and bone (a biological material) exhibit multiple pore networks. Double porosity/permeability models provide a mechanics-based approach to describe hydrodynamics in aforesaid porous materials. However, current theoretical results primarily address steady-state response, and their counterparts in the transient regime are still wanting. The primary aim of this paper is to fill this knowledge gap. We present three principal properties -- with rigorous mathematical arguments -- that the solutions under the double porosity/permeability model satisfy in the transient regime: backward-in-time uniqueness, reciprocity, and a variational principle. We employ the ``energy method'' -- by exploiting the physical total kinetic energy of the flowing fluid -- to establish the first property and Cauchy-Riemann convolutions to prove the next two. The results reported in this paper -- that qualitatively describe the dynamics of fluid flow in double-pored media -- have (a) theoretical significance, (b) practical applications, and (c) considerable pedagogical value. In particular, these results will benefit practitioners and computational scientists in checking the accuracy of numerical simulators. The backward-in-time uniqueness lays a firm theoretical foundation for pursuing inverse problems in which one predicts the prescribed initial conditions based on data available about the solution at a later instance.

math.NA↗
arXiv2023

CoolPINNs: A Physics-informed Neural Network Modeling of Active Cooling in Vascular Systems

Emerging technologies like hypersonic aircraft, space exploration vehicles, and batteries avail fluid circulation in embedded microvasculatures for efficient thermal regulation. Modeling is vital during these engineered systems' design and operational phases. However, many challenges exist in developing a modeling framework. What is lacking is an accurate framework that (i) captures sharp jumps in the thermal flux across complex vasculature layouts, (ii) deals with oblique derivatives (involving tangential and normal components), (iii) handles nonlinearity because of radiative heat transfer, (iv) provides a high-speed forecast for real-time monitoring, and (v) facilitates robust inverse modeling. This paper addresses these challenges by availing the power of physics-informed neural networks (PINNs). We develop a fast, reliable, and accurate Scientific Machine Learning (SciML) framework for vascular-based thermal regulation -- called CoolPINNs: a PINNs-based modeling framework for active cooling. The proposed mesh-less framework elegantly overcomes all the mentioned challenges. The significance of the reported research is multi-fold. First, the framework is valuable for real-time monitoring of thermal regulatory systems because of rapid forecasting. Second, researchers can address complex thermoregulation designs inasmuch as the approach is mesh-less. Finally, the framework facilitates systematic parameter identification and inverse modeling studies, perhaps the current framework's most significant utility.

math.NA↗
arXiv2023

Thermal regulation in thin vascular systems: A sensitivity analysis

One of the ways natural and synthetic systems regulate temperature is via circulating fluids through vasculatures embedded within their bodies. Because of the flexibility and availability of proven fabrication techniques, vascular-based thermal regulation is attractive for thin microvascular systems. Although preliminary designs and experiments demonstrate the feasibility of thermal modulation by pushing fluid through embedded micro-vasculatures, one has yet to optimize the performance before translating the concept into real-world applications. It will be beneficial to know how two vital design variables -- host material's thermal conductivity and fluid's heat capacity rate -- affect a thermal regulation system's performance, quantified in terms of the mean surface temperature. This paper fills the remarked inadequacy by performing adjoint-based sensitivity analysis and unravels a surprising non-monotonic trend. Increasing thermal conductivity can either increase or decrease the mean surface temperature; the increase happens if countercurrent heat exchange -- transfer of heat from one segment of the vasculature to another -- is significant. In contrast, increasing the heat capacity rate will invariably lower the mean surface temperature, for which we provide mathematical proof. The reported results (a) dispose of some misunderstandings in the literature, especially on the effect of the host material's thermal conductivity, (b) reveal the role of countercurrent heat exchange in altering the effects of design variables, and (c) guide designers to realize efficient microvascular active-cooling systems. The analysis and findings will advance the field of thermal regulation both on theoretical and practical fronts.

math.NA↗
arXiv2022

How to pose material design problems for flow through porous media applications?: Sensitivity of dissipation rate to medium's permeability holds the key

Recent studies have advocated using the total dissipation rate under topology optimization to realize material designs involving the flow of fluids through porous media. However, these studies decided how to pose the design problem, such as maximizing the total dissipation rate for some situations while minimizing for others, by solving one-dimensional problems and justifying their choices using numerical experiments. The rigor is lacking -- a bottleneck for further scientific advancements to computational material design. This paper provides the missing theoretical justification. We identify four classes of boundary value problems using the adjoint state method and analytically calculate the sensitivity of the total dissipation rate to the permeability field. For two of those classes in which the flow of fluids is pressure-driven, the sensitivity is positive -- the total dissipation rate increases if the medium's permeability increases. While for the other two classes, in which the flow is velocity-driven, the trend is the opposite. These sensitivities provide rigorous answers to the central question: how to pose a material design problem for flow through porous media applications. The impact of our work is multi-fold. First, this study further elevates the role of the dissipation rate in posing well-posed material design problems using topology optimization. Second, besides the theoretical significance, the results benefit computational scientists and practitioners to realize optimal designs. Third, given their simplicity yet far-reaching impact, both the approach and results possess immense pedagogical value.

math.NA↗
arXiv2021

On optimal designs using topology optimization for flow through porous media applications

Topology optimization (TopOpt) is a mathematical-driven design procedure to realize optimal material architectures. This procedure is often used to automate the design of devices involving flow through porous media, such as micro-fluidic devices. TopOpt offers material layouts that control the flow of fluids through porous materials, providing desired functionalities. Many prior studies in this application area have used Darcy equations for primal analysis and the minimum power theorem (MPT) to drive the optimization problem. But both these choices (Darcy equations and MPT) are restrictive and not valid for general working conditions of modern devices. Being simple and linear, Darcy equations are often used to model flow of fluids through porous media. However, two inherent assumptions of the Darcy model are: the viscosity of a fluid is a constant, and inertial effects are negligible. There is irrefutable experimental evidence that viscosity of a fluid, especially organic liquids, depends on the pressure. Given the typical small pore-sizes, inertial effects are dominant in micro-fluidic devices. Next, MPT is not a general principle and is not valid for (nonlinear) models that relax the assumptions of the Darcy model. This paper aims to overcome the mentioned deficiencies by presenting a general strategy for using TopOpt. First, we will consider nonlinear models that take into account the pressure-dependent viscosity and inertial effects, and study the effect of these nonlinearities on the optimal material layouts under TopOpt. Second, we will explore the rate of mechanical dissipation, valid even for nonlinear models, as an alternative for the objective function. Third, we will present analytical solutions of optimal designs for canonical problems; these solutions not only possess research and pedagogical values but also facilitate verification of computer implementations.

math.NA↗
arXiv2021

Effect of viscous shearing stresses on optimal material designs for flow of fluids through porous media

Topology optimization offers optimal material layouts, enabling automation in the design of devices. Given the recent advances in computer technology and additive manufacturing, topology optimization is increasingly being used to design complex porous structures, for example, microfluidic devices. For the flow of fluids in such miniature-sized porous structures, viscous shearing stress will be significant. But the Darcy model -- the most popular mathematical model describing the flow of a single-phase incompressible fluid in rigid porous media -- neglects the internal friction arising from viscous shearing stress. We will therefore develop a material design framework under the topology optimization based on the Darcy-Brinkman model -- a mathematical model for the flow of fluids through porous media that accounts for internal friction besides the drag considered in the Darcy model. The proposed framework uses the total rate of mechanical dissipation -- a physical quantity -- for the objective function. To understand the effect of viscous shearing stress on the design, we will compare the optimal material layouts provided by the Darcy and Darcy-Brinkman models under topology optimization. In particular, we show, using analytical solutions corroborated by numerical simulations, that the optimal material layouts are identical for the class of problems exhibiting axisymmetry, for which viscous shearing stress vanishes. These analytical solutions will be valuable to check the veracity of numerical simulators. For those problems with dominant viscous shearing stresses (e.g., flow in a backward-facing step domain), we show, using numerical solutions, that the optimal material layouts under the Darcy and Darcy-Brinkman models are very different. Also, the associated solution fields (i.e., velocity and pressure) are qualitatively different under their respective optimal layouts for these two models.

physics.flu-dyn↗
arXiv2021

A deep learning modeling framework to capture mixing patterns in reactive-transport systems

Prediction and control of chemical mixing are vital for many scientific areas such as subsurface reactive transport, climate modeling, combustion, epidemiology, and pharmacology. Due to the complex nature of mixing in heterogeneous and anisotropic media, the mathematical models related to this phenomenon are not analytically tractable. Numerical simulations often provide a viable route to predict chemical mixing accurately. However, contemporary modeling approaches for mixing cannot utilize available spatial-temporal data to improve the accuracy of the future prediction and can be compute-intensive, especially when the spatial domain is large and for long-term temporal predictions. To address this knowledge gap, we will present in this paper a deep-learning (DL) modeling framework applied to predict the progress of chemical mixing under fast bimolecular reactions. This framework uses convolutional neural networks (CNN) for capturing spatial patterns and long short-term memory (LSTM) networks for forecasting temporal variations in mixing. By careful design of the framework -- placement of non-negative constraint on the weights of the CNN and the selection of activation function, the framework ensures non-negativity of the chemical species at all spatial points and for all times. Our DL-based framework is fast, accurate, and requires minimal data for training.

cs.LG↗
arXiv2020

On boundedness and growth of unsteady solutions under the double porosity/permeability model

There is a recent surge in research activities on modeling the flow of fluids in porous media with complex pore-networks. A prominent mathematical model, which describes the flow of incompressible fluids in porous media with two dominant pore-networks allowing mass transfer across them, is the double porosity/permeability (DPP) model. However, we currently do not have a complete understanding of unsteady solutions under the DPP model. Also, because of the complex nature of the mathematical model, it is not possible to find analytical solutions, and one has to resort to numerical solutions. It is therefore desirable to have a procedure that can serve as a measure to assess the veracity of numerical solutions. In this paper, we establish that unsteady solutions under the transient DPP model are stable in the sense of Lyapunov. We also show that the unsteady solutions grow at most linear with time. These results not only have a theoretical value but also serve as valuable a posteriori measures to verify numerical solutions in the transient setting and under anisotropic medium properties, as analytical solutions are scarce for these scenarios under the DPP model.

math.NA↗
arXiv2020

A modeling framework for coupling plasticity with species diffusion

This paper presents a modeling framework---mathematical model and computational framework---to study the response of a plastic material due to the presence and transport of a chemical species in the host material. Such a modeling framework is important to a wide variety of problems ranging from Li-ion batteries, moisture diffusion in cementitious materials, hydrogen diffusion in metals, to consolidation of soils under severe loading-unloading regimes. The mathematical model incorporates experimental observations reported in the literature on how (elastic and plastic) material properties change because of the presence and transport of a chemical species. Also, the model accounts for one-way (transport affects the deformation but not vice versa) and two-way couplings between deformation and transport subproblems. The resulting coupled equations are not amenable to analytical solutions; so, we present a robust computational framework for obtaining numerical solutions. Given that popular numerical formulations do not produce nonnegative solutions, the computational framework uses an optimized-based nonnegative formulation that respects physical constraints (e.g., nonnegative concentrations). For completeness, we will also show the effect and propagation of the negative concentrations, often produced by contemporary transport solvers, into the overall predictions of deformation and concentration fields. Notably, anisotropy of the diffusion process exacerbates these unphysical violations. Using representative numerical examples, we will discuss how the concentration field affects plastic deformations of a degrading solid. Based on these numerical examples, we also discuss how plastic zones spread because of material degradation. To illustrate how the proposed computational framework performs, we report various performance metrics such as optimization iterations and time-to-solution.

math.NA↗
arXiv2019

On interface conditions for flows in coupled free-porous media

Many processes in nature (e.g., physical and biogeochemical processes in hyporheic zones, and arterial mass transport) occur near the interface of free-porous media. A firm understanding of these processes needs an accurate prescription of flow dynamics near the interface which (in turn) hinges on an appropriate description of interface conditions along the interface of free-porous media. Although the conditions for the flow dynamics at the interface of free-porous media have received considerable attention, many of these studies were empirical and lacked a firm theoretical underpinning. In this paper, we derive a complete and self-consistent set of conditions for flow dynamics at the interface of free-porous media. We first propose a principle of virtual power by incorporating the virtual power expended at the interface of free-porous media. Then by appealing to the calculus of variations, we obtain a complete set of interface conditions for flows in coupled free-porous media. A noteworthy feature of our approach is that the derived interface conditions apply to a wide variety of porous media models. We also show that the two most popular interface conditions -- the Beavers-Joseph condition and the Beavers-Joseph-Saffman condition -- are special cases of the approach presented in this paper. The proposed principle of virtual power also provides a minimum power theorem for a class of flows in coupled free-porous media, which has a similar mathematical structure as the ones enjoyed by flows in uncoupled free and porous media.

cs.CE↗
arXiv2019

On numerical stabilization in modeling double-diffusive viscous fingering

A firm understanding and control of viscous fingering (VF) and miscible displacement will be vital to a wide range of industrial, environmental, and pharmaceutical applications, such as geological carbon-dioxide sequestration, enhanced oil recovery, and drug delivery. We restrict our study to VF, a well-known hydrodynamic instability, in miscible fluid systems but consider double-diffusive (DD) effects---the combined effect of compositional changes because of solute transport and temperature. One often uses numerical formulations to study VF with DD effects. The primary aim of the current study is to show that popular formulations have limitations to study VF with DD effect. These limitations include exhibiting node-to-node spurious oscillations, violating physical constraints such as the non-negativity of the concentration field or mathematical principles such as the maximum principle, and suppressing physical instabilities. We will use several popular stabilized finite element formulations---the SUPG formulations and three modifications based on the SOLD approach---in our study. Using representative numerical results, we will illustrate two critical limitations. First, we document that these formulations do not respect the non-negative constraint and the maximum principle for the concentration field. We will also show the impact of these violations on how viscous fingers develop. Second, we show that these stabilized formulations, often used to suppress numerical instabilities, may also suppress physical instabilities, such as viscous fingering. Our study will be valuable to practitioners who use existing numerical formulations and to computational mathematicians who develop new formulations.

math.NA↗