AI 中文总结
针对动态限应变弹性力学中拟线性双曲方程大应变区域丧失椭圆性的挑战，本文提出结合HHT-α格式的全离散连续Galerkin有限元框架，验证了其收敛性与鲁棒性，为非线性限应变波传播模拟提供了高效工具。
AI 中文摘要
本文研究了一类源于动态限应变弹性力学的拟线性双曲方程的数值近似问题，这类问题的非线性本构定律使应力与应变呈非线性关联。此类问题在大应变区域可能丧失椭圆性，给稳定且精确的数值方法设计带来重大挑战。为解决这些难题，本文开发了一种全离散连续Galerkin有限元框架，将用于空间离散的连续线性有限元与Hilber-Hughes-Taylor（HHT-α）时间积分格式相结合；该公式中引入了升力技术以处理非齐次Dirichlet边界条件，并采用一致Newton迭代高效求解生成的非线性方程组。HHT-α方法引入的算法耗散增强了鲁棒性，可抑制退化区域附近的非物理高频振荡，同时保持一致性与精度。在非线性本构系数满足适当结构假设的条件下，于合适的能量空间中建立了弱形式。通过全面的数值实验验证了：L²范数下理论二阶收敛、H¹范数下一阶收敛、Newton迭代具有接近网格无关的快速非线性收敛、波速演化符合物理规律、能量耗散稳定；结果还证实，该框架为模拟非线性限应变波传播提供了精确、稳定且计算高效的工具，为后续扩展至多维非线性弹性动力学、自适应有限元方法及断裂与损伤力学奠定了坚实基础。
英文摘要
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.
