arXiv subjects

Kundan Kumar

Publications and source records attributed to Kundan Kumar.

At least 19 recordsLinked to original sources

arXiv2026

Frequency Domain Biot--Allard Equations for Isotropic and Anisotropic Poroelastic Media: Two-field formulations and iterative splitting

We present a frequency-domain formulation of Biot's dynamic poroelastic equations with frequency-dependent dissipation (Biot--Allard) for anisotropic, heterogeneous media with memory effects. Two equivalent two-field representations--a displacement-pressure and a velocity-pressure-rate formulation--enable stabilized iterative splitting. While coupling operators generally lack an adjoint or skew-adjoint relationship at finite frequencies, the velocity--pressure-rate representation restores a skew-adjoint structure in the quasi-static limit. We prove continuity of the coupling operators and coercivity of the diagonal blocks, essential for convergence of the L-stabilized splitting scheme. The frequency-domain setting eliminates convolutional memory terms, incorporates attenuation and dispersion via complex-valued parameters, and reduces the time-dependent problem to a family of elliptic boundary-value problems suited for parallel computation and multi-frequency inversion. A conforming Galerkin finite element discretization preserves block structure, and numerical experiments confirm robustness and capture frequency-dependent attenuation. To illustrate discretization independence, we include a large-scale wave simulation using a pseudo-spectral method. This work provides a rigorous and efficient framework for modeling wave phenomena in complex porous media.

math.NA↗

arXiv2026

Modelling the onset and evolution of immiscible viscous fingering in porous media

The simulation of viscous fingering in porous media is of direct relevance to displacement processes in petroleum engineering and hydrogeology. Building on recent work proposing a modelling approach for well-defined fingers at very adverse viscosity ratios, we investigate the physical mechanisms behind viscous fingering and the modelling requirements for capturing the finger scales and saturation patterns observed in experiments. We simulate and match a viscous fingering experiment at a viscosity ratio of $\mu_{o}/\mu_{w}{=}2000$, discussing the physical significance of each modelling step. Linear stability analysis is used to characterize the early-stage instability of the displacement. Subsequent numerical simulations show that, for the simulated finger scales to match the experiment, the most unstable wavelength at onset must be several times smaller than the desired finger width---so that, after accounting for shielding and merging in the nonlinear regime, the fingers remain thin. Small-scale channelling effects are also required to disrupt the trailing stable region commonly observed in simulations of viscous fingering in nearly homogeneous media. Finally, we show that including a weakly oil-wet capillary pressure function enables our model to capture the bypassed oil observed in the experiment.

cond-mat.soft↗

arXiv2026

Numerical analysis of the Biot equations coupled to frictional contact mechanics

We consider a mathematical model of a poro-visco-elastic medium subject to frictional contact with a rigid obstacle, and study its numerical approximation. This model couples the Biot equations and contact conditions in the form of normal compliance and Coulomb friction. The resulting variational problem consists of a linear partial differential equation coupled to a nonlinear variational inequality. We propose and analyze a fully discrete numerical scheme for this problem, using conformal finite elements in space and the implicit Euler method in time. Existence and uniqueness of the discrete solution is established, and stability and a priori error estimates are derived. A numerical experiment is performed in which numerical error estimates are computed and compared to the theoretical results.

math.NA↗

arXiv2026

A Bayesian Filtering Approach for Learning Lagrangian Dynamics from Noisy Measurements

This paper proposes a Bayesian filtering-based approach for learning the dynamics of a physical system from partial, noisy measurements. We model the system dynamics using a Lagrangian mechanics formulation. As in Lagrangian neural networks (LNNs), we parameterize the kinetic and potential energies with neural networks. The unknown external forces in the Lagrangian formulation are modeled as white Gaussian noise. The corresponding Euler--Lagrange equations then yield a continuous-time stochastic state-space model (SSM) that describes the system dynamics. The neural network parameters and system states are then jointly learned via a maximum-likelihood method using Gaussian-approximation-based Bayesian filters. The effectiveness of the proposed method is demonstrated on pendulum and Duffing oscillator examples, and its performance is compared with conventional LNNs and with approximate Bayesian filters using known system models.

cs.LG↗

arXiv2026

Linear Stability Analysis of Two-phase, Two-Component Flow in Porous Media

Viscous fingering instabilities during fluid displacement in porous media can compromise the efficiency of applications such as enhanced oil recovery, CO2 sequestration, and groundwater remediation. While extensive research exists on linear stability analysis for fully immiscible and fully miscible displacements, the intermediate case of partially miscible flow with limited mass transfer between phases remains largely unexplored. This study extends linear stability analysis to a two-phase, two-component system that accounts for gravity effects, fractional flow, capillary forces, mechanical dispersion, and interphase mass transfer, focusing on the case where a partially miscible gaseous fluid displaces a liquid. We formulate an eigenvalue problem to characterize instability growth rates and cutoff wavenumbers. The resulting ordinary differential equations have discontinuous coefficients at the transition from two-phase to pure-liquid flow, resulting in discontinuous eigenfunction derivatives. We derive jump conditions for the derivatives at this transition, and solve the eigenvalue problem using the matched initial value problem method. Results demonstrate that mass transfer has a pre-dominantly stabilizing effect by reducing viscosity contrast and altering shock properties at the displacement front. This stabilizing influence is particularly pronounced for high viscosity contrasts and dampens gravity-induced instability in upward displacements. Mass transfer most significantly affects the perturbation growth rate, while its effect on the cutoff wavenumber is less pronounced. We identify a critical value for the dimensionless longitudinal dispersion coefficient where both growth rate and cutoff wavenumber are maximized, suggesting complex interactions between capillary forces and mechanical dispersion.

physics.flu-dyn↗

arXiv2026

Data-driven stress problem under purely normal homogeneous Neumann boundary conditions

Data-Driven Continuum Mechanics -- the continuous counterpart of Data-Driven Computational Mechanics -- is a modern paradigm that enhances classical continuum mechanics by incorporating finite sets of experimental material data directly, avoiding any form of constitutive modeling. Despite recent progress, its analytical foundations remain at an early stage. In this work, we establish a rigorous functional-analytic framework for the data-driven stress problem under purely homogeneous normal Neumann boundary conditions. The problem is formulated as finding a stress field (satisfying the balance of linear and angular momenta and the boundary conditions) that is closest, in an $L^p$-sense, to an auxiliary stress field that is simultaneously sought and locally resembles a finite discrete set of experimental stress states. Our analysis relies on two key ingredients. First, the divergence operator induces a topological isomorphism between the space of symmetric stress fields modulo its kernel and the space of loads balanced by rigid-body motions, ensuring the existence of an equilibrated response. Second, the finiteness of the material data set guarantees proximinality in the stress space, which in turn yields a complete existence and uniqueness theory for solution equivalence classes. Together, these two properties provide a rigorous mathematical foundation for the data-driven stress problem under purely homogeneous normal Neumann boundary conditions.

math.AP↗

arXiv2026

Integrating Lagrangian Neural Networks into the Dyna Framework for Reinforcement Learning

Model-based reinforcement learning (MBRL) is sample-efficient but depends on the accuracy of the learned dynamics, which are often modeled using black-box methods that do not adhere to physical laws. Those methods tend to produce inaccurate predictions when presented with data that differ from the original training set. In this work, we employ Lagrangian neural networks (LNNs), which enforce an underlying Lagrangian structure to train the model within a Dyna-based MBRL framework. Furthermore, we train the LNN using stochastic gradient-based and state-estimation-based optimizers to learn the network's weights. The state-estimation-based method converges faster than the stochastic gradient-based method during neural network training. Simulation results are provided to illustrate the effectiveness of the proposed LNN-based Dyna framework for MBRL.

eess.SY↗

arXiv2026

Convergent adaptive iterative schemes for solving multi-physics problems

In this paper, we derive a practical, general framework for creating adaptive iterative (linearization or splitting) algorithms to solve multi-physics problems. This means that, given an iterative method, we derive \textit{a posteriori} estimators to predict the success or failure of the method. Based on these estimators, we propose adaptive algorithms, including adaptively switching between methods, adaptive time-stepping methods, and the adaptive tuning of stabilization parameters. We apply this framework to two-phase flow in porous media, surfactant transport in porous media, and quasi-static poroelasticity.

math.NA↗

arXiv2026

Parareal algorithm for coupled elliptic-parabolic problems

We present a convergence analysis of the parallel-in-time integration method known as the Parareal algorithm for degenerate differential-algebraic systems arising from quasi-static Biot models, which govern coupled flow and deformation in porous media. The underlying system exhibits a saddle-point structure and degeneracy due to the quasi-static assumption. We extend the Parareal algorithm to this setting and propose three coarse propagators: monolithic, fixed-stress, and multirate fixed-stress schemes. For each, we derive sufficient conditions for convergence and establish explicit time step restrictions that guarantee contractivity of the iteration matrix. Numerical experiments show computational savings accrued by using a parareal solver in multiphysics simulations involving poroelasticity and other coupled systems.

math.NA↗

arXiv2025

Fractured Poroelastic Media in the Limit of Vanishing Aperture

We consider a poroelastic medium with a thin heterogeneity, also referred to as a fracture. Fluid flow and mechanical deformation inside both bulk and fracture are governed by the quasi-static Biot equations. The fracture's material parameters, such as hydraulic conductivity and elasticity, are assumed to scale with powers of the width-to-length ratio $\varepsilon$ of the fracture. Based on a priori estimates, we rigorously derive limit models as $\varepsilon \rightarrow 0$ and identify different limit regimes. We obtain five regimes for the hydraulic conductivity and two for the elasticity. While many cases yield discrete fracture models, others result in two-scale limit problems dominated by normal flow or deformation.

math.AP↗

arXiv2025

Statistical Linear Regression Approach to Kalman Filtering and Smoothing under Cyber-Attacks

Remote state estimation in cyber-physical systems is often vulnerable to cyber-attacks due to wireless connections between sensors and computing units. In such scenarios, adversaries compromise the system by injecting false data or blocking measurement transmissions via denial-of-service attacks, distorting sensor readings. This paper develops a Kalman filter and Rauch--Tung--Striebel (RTS) smoother for linear stochastic state-space models subject to cyber-attacked measurements. We approximate the faulty measurement model via generalized statistical linear regression (GSLR). The GSLR-based approximated measurement model is then used to develop a Kalman filter and RTS smoother for the problem. The effectiveness of the proposed algorithms under cyber-attacks is demonstrated through a simulated aircraft tracking experiment.

eess.SP↗

arXiv2025

Communication-Efficient Distributed Kalman Filtering using ADMM

This paper addresses the problem of optimal linear filtering in a network of local estimators, commonly referred to as distributed Kalman filtering (DKF). The DKF problem is formulated within a distributed optimization framework, where coupling constraints require the exchange of local state and covariance updates between neighboring nodes to achieve consensus. To address these constraints, the problem is transformed into an unconstrained optimization form using the augmented Lagrangian method. The distributed alternating direction method of multipliers (ADMM) is then applied to derive update steps that achieve the desired performance while exchanging only the primal variables. Notably, the proposed method enhances communication efficiency by eliminating the need for dual variable exchange. We show that the design parameters depend on the maximum eigenvalue of the network's Laplacian matrix, yielding a significantly tighter bound compared to existing results. A rigorous convergence analysis is provided, proving that the state estimates converge to the true state and that the covariance matrices across all local estimators converge to a globally optimal solution. Numerical results are presented to validate the efficacy of the proposed approach.

eess.SY↗

arXiv2025

An efficient preconditioner for mixed-dimensional contact poromechanics based on the fixed stress splitting scheme

Numerical simulation of fracture contact poromechanics is essential for various applications, including CO2 sequestration, geothermal energy production and underground gas storage. Modeling this problem accurately presents significant challenges due to the complex physics involved in strongly coupled poromechanics and frictional contact mechanics of fractures. The robustness and efficiency of the simulation heavily depends on a preconditioner for the linear solver, which addresses the Jacobian matrices arising from Newton's method in fully implicit time-stepping schemes. Developing an effective preconditioner is difficult because it must decouple three interdependent subproblems: momentum balance, fluid mass balance, and contact mechanics. The challenge is further compounded by the saddle-point structure of the contact mechanics problem, a result of the Augmented Lagrange formulation, which hinders the direct application of the well-established fixed stress approximation to decouple the poromechanics subproblem. In this work, we propose a preconditioner hat combines nested Schur complement approximations with a linear transformation, which addresses the singular nature of the contact mechanics subproblem. This approach extends the fixed stress scheme to both the matrix and fracture subdomains. We investigate analytically how the contact mechanics subproblem affects the convergence of the proposed fixed stress-based iterative scheme and demonstrate how it can be translated into a practical preconditioner. The scalability and robustness of the method are validated through a series of numerical experiments.

math.NA↗

arXiv2025

Gaussian Integral based Bayesian Smoother

This work introduces the Gaussian integration to address a smoothing problem of a nonlinear stochastic state space model. The probability densities of states at each time instant are assumed to be Gaussian, and their means and covariances are evaluated by utilizing the odd-even properties of Gaussian integral, which are further utilized to realize Rauch-Tung-Striebel (RTS) smoothing expressions. Given that the Gaussian integration provides an exact solution for the integral of a polynomial function over a Gaussian probability density function, it is anticipated to provide more accurate results than other existing Gaussian approximation-based smoothers such as extended Kalman, cubature Kalman, and unscented Kalman smoothers, especially when polynomial types of nonlinearity are present in the state space models. The developed smoothing algorithm is applied to the Van der Pol oscillator, where the nonlinearity associated with their dynamics is represented using polynomial functions. Simulation results are provided to demonstrate the superiority of the proposed algorithm.

eess.SP↗

arXiv2024

Fixed stress splitting approach for contact problems in a porous medium

We consider a poromechanics model including frictionless contact mechanics. The resulting model consists of the Biot equations with contact boundary conditions leading to a variational inequality modelling mechanical deformations coupled to a linear parabolic flow equation. We propose a fully discrete iterative scheme for solving this model. This scheme decoupled the flow and mechanics equations and extends the fixed-stress splitting scheme for the Biot equations. We use finite elements in space and a backward Euler discretization in time. We show that the fixed stress split scheme is a contraction.

math.NA↗

arXiv2024

A History-dependent Dynamic Biot Model

In this work, we consider a fully dynamic Biot model that includes memory effects due to evolving permeability. Time integrals are used to account for the change in structure. We propose an iterative splitting scheme for this model, extending the fixed-stress split for the quasi-static Biot. We use finite elements in space and a backward Euler discretization in time. The performance of the method is demonstrated through a numerical experiment

math.NA↗

arXiv2023

A fixed-stress type splitting method for nonlinear poroelasticity

In this paper we consider a nonlinear poroelasticity model that describes the quasi-static mechanical behaviour of a fluid-saturated porous medium whose permeability depends on the divergence of the displacement. Such nonlinear models are typically used to study biological structures like tissues, organs, cartilage and bones, which are known for a nonlinear dependence of their permeability/hydraulic conductivity on solid dilation. We formulate (extend to the present situation) one of the most popular splitting schemes, namely the fixed-stress split method for the iterative solution of the coupled problem. The method is proven to converge linearly for sufficiently small time steps under standard assumptions. The error contraction factor then is strictly less than one, independent of the Lam\'{e} parameters, Biot and storage coefficients if the hydraulic conductivity is a strictly positive, bounded and Lipschitz-continuous function.

math.NA↗

arXiv2023

Coupling of flow, contact mechanics and friction, generating waves in a fractured porous medium

We present a mixed dimensional model for a fractured poro-elasic medium including contact mechanics. The fracture is a lower dimensional surface embedded in a bulk poro-elastic matrix. The flow equation on the fracture is a Darcy type model that follows the cubic law for permeability. The bulk poro-elasticity is governed by fully dynamic Biot equations. The resulting model is a mixed dimensional type where the fracture flow on a surface is coupled to a bulk flow and geomechanics model. The particularity of the work here is in considering fully dynamic Biot equation, that is, including an inertia term, and the contact mechanics including friction for the fracture surface. We prove the well-posedness of the continuous model.

math.AP↗