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.