arXiv subjects

Ram Manohar

Publications and source records attributed to Ram Manohar.

7 recordsLinked to original sources

arXiv2026

Space-Time Finite Element Approximation of Quasilinear Hyperbolic Equations Arising in Dynamic Strain-Limiting Elasticity

The numerical approximation of a class of quasilinear hyperbolic equations arising in dynamic strain-limiting elasticity-whose nonlinear constitutive law relates stress and strain nonlinearly-is investigated. Ellipticity may be lost by such problems in regions of large strain, leading to significant challenges in the design of stable and accurate numerical methods. To address these difficulties, a fully discrete continuous Galerkin finite element framework is developed, combining continuous linear finite elements for spatial discretization with the Hilber-Hughes-Taylor (HHT-alpha) time integration scheme. A lifting technique is incorporated within the proposed formulation to treat non-homogeneous Dirichlet boundary conditions, and a consistent Newton iteration is employed for the efficient solution of the resulting nonlinear systems. Robustness is enhanced by the algorithmic dissipation introduced through the HHT-alpha method by suppressing nonphysical high-frequency oscillations near degenerate regions, while consistency and accuracy are preserved. Under suitable structural assumptions on the nonlinear constitutive coefficient, the weak formulation is established in appropriate energy spaces. Theoretical second-order convergence in the L^2-norm and first-order convergence in the H^1-norm, rapid nonlinear convergence with nearly mesh-independent Newton iterations, physically consistent wave-speed evolution, and stable energy dissipation are demonstrated through comprehensive numerical experiments. Furthermore, it is confirmed by the results that an accurate, stable, and computationally efficient framework for simulating nonlinear strain-limiting wave propagation is provided, offering a solid foundation for future extensions to multidimensional nonlinear elastodynamics, adaptive finite element methods, and fracture and damage mechanics.

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arXiv2025

A convergent adaptive finite element method for a phase-field model of dynamic fracture

We propose and analyze an adaptive finite element method for a phase-field model of dynamic brittle fracture. The model couples a second-order hyperbolic equation for elastodynamics with the Ambrosio-Tortorelli regularization of the Francfort-Marigo variational fracture energy, which circumvents the need for explicit crack tracking. Our numerical scheme combines a staggered time-stepping algorithm with a variational inequality formulation to strictly enforce the irreversibility of damage. The mesh adaptation is driven by a residual-based a posteriori-type estimator, enabling efficient resolution of the evolving fracture process zone. The main theoretical contribution is a rigorous convergence analysis, where we prove that the sequence of discrete solutions generated by the AFEM converges (up to a tolerance) to a critical point of the governing energy functional. Numerical experiments for a two-dimensional domain containing an edge-crack under dynamic anti-plane shear loading demonstrate our method's capability of autonomously capturing complex phenomena, including crack branching and tortuosity, with significant computational savings over uniform refinement.

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arXiv2025

Adaptive finite element convergence analysis of AT1 phase-field model for quasi-static fracture in strain-limiting solids

This research rigorously investigates the convergence of adaptive finite element methods for regularized variational models of quasi-static brittle fracture in elastic solids. We specifically examine a novel Ambrosio-Tortorelli (AT1) phase-field model within the framework of elasticity theories, particularly for material models characterized by an algebraically nonlinear stress-strain relationship. Two distinct and novel adaptive mesh refinement algorithms, underpinned by robust local error indicators, were introduced to efficiently solve the underlying nonlinear energy minimization problem. A detailed convergence analysis was conducted on the sequences of minimizers produced by these strategies. Our findings rigorously demonstrate that the minimizer sequences from the first adaptive algorithm achieve convergence to a predefined tolerance. Crucially, the second algorithm is proven to generate inherently convergent sequences, thereby eliminating the need for an explicit stopping criterion. The practical effectiveness of this proposed adaptive framework is thoroughly validated through extensive numerical simulations. A case study involving an edge crack in an elastic body, governed by an algebraically nonlinear strain-limiting relationship and subjected to anti-plane shear-type loading, is presented. Critical comparisons of the energy components-bulk, surface, and total-showcase the superior performance of both adaptive algorithms.

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arXiv2025

Convergence Analysis of Adaptive Finite Element Algorithms for a Regularized Variational Model of Quasi-Static Brittle Fracture in "Strain-Limiting" Elastic Solids

The rigorous convergence analysis of adaptive finite element methods for regularized variational models of quasi-static brittle fracture in strain-limiting elastic solids is presented. This work introduces two novel adaptive mesh refinement algorithms, based on robust local error indicators, designed to solve the underlying energy minimization problem efficiently. A comprehensive convergence analysis is provided for minimizer sequences generated by these distinct adaptive strategies. It is rigorously demonstrated that sequences from the first algorithm converge to a prescribed tolerance. Notably, the second algorithm is proven to yield inherently convergent sequences without requiring an explicit stopping criterion. The practical efficacy of the proposed adaptive framework is validated through extensive numerical simulations, where critical comparisons of energy components (bulk, surface, and total) demonstrate the performance of the two adaptive algorithms in the case of an edge crack in a strain-limiting solid subjected to anti-plane shear-type loading.

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arXiv2025

Optimal control of fractional Poisson equation from non-local to local

In this article, the limiting behavior of the solution $\bar u_s$ of the optimal control problem subjected to the fractional Poisson equation $$(-\Delta)^s u_s(x)=f_s(x), \quad x\in \Omega$$ defined on domain $\Omega$ bounded by smooth boundary with zero exterior boundary conditions $u_s(x)\equiv 0, \quad x \in \Omega^c $ is established. We will prove that $\lim_{s\to 1^-} \bar u_s= \bar u$, where $\bar u$ is a solution of the optimal control problem subjected to classical Poisson equation $-\Delta u(x)=f(x), \quad x \in \Omega$ and $u(x)=0, \quad x\in \partial \Omega.$

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arXiv2025

Adaptive SIPG method for approximations of boundary control problems governed by parabolic PDEs

This study presents an aposteriori error analysis of adaptive finite element approximations of parabolic boundary control problems with bilateral box constraints that act on a Neumann boundary. The control problem is discretized using the symmetric interior penalty Galerkin (SIPG) technique. We derive both reliable and efficient type residual-based error estimators coupling with the data oscillations. The implementation of these error estimators serves as a guide for the adaptive mesh refinement process, indicating whether or not more refinement is required. Although the control error estimator effectively captured control approximation errors, it had limitations in guiding refinement localization in critical cases. To overcome this, an alternative control indicator was used in numerical tests. The results demonstrated the clear superiority of adaptive refinements over uniform refinements, confirming the proposed approach's effectiveness in achieving accurate solutions while optimizing computational efficiency. numerical experiment showcases the effectiveness of the derived error estimators.

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arXiv2024

An $hp$-adaptive discontinuous Galerkin discretization of a static anti-plane shear crack model

We propose an $hp$-adaptive discontinuous Galerkin finite element method (DGFEM) to approximate the solution of a static crack boundary value problem. The mathematical model describes the behavior of a geometrically linear strain-limiting elastic body. The compatibility condition for the physical variables, along with a specific algebraically nonlinear constitutive relationship, leads to a second-order quasi-linear elliptic boundary value problem. We demonstrate the existence of a unique discrete solution using Ritz representation theory across the entire range of modeling parameters. Additionally, we derive a priori error estimates for the DGFEM, which are computable and, importantly, expressed in terms of natural energy and $L^2$-norms. Numerical examples showcase the performance of the proposed method in the context of a manufactured solution and a non-convex domain containing an edge crack.

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