Simulation and Numerical Analysis in Elastodynamics

瞬态弹性波数值模拟的总体目标是开发鲁棒可靠的计算工具来处理固体中的变形和应力研究。该项目的结果可能会影响广泛的应用，包括地球物理学中的逆问题（地震成像在地震模拟和石油勘探中的应用）、医疗应用的磁共振弹性成像、机电系统控制（利用固体变形来控制电场或反之亦然）以及固体中的裂缝检测。理解描述各种物理现象（固体变形、电磁、声波传播和热扩散）相互作用的复杂模型方程的计算近似对于评估所提出的模拟的质量和有效性非常重要。本项目着重于瞬态弹性波的数值分析与计算，包括瞬态弹性波与声波的相互作用。该提案解决了涉及弹性动力学的多种物理模型，重点是粘弹性行为，压电效应和完全耦合热弹性。该项目建议统一和简化这些模型的理论和数值处理。数值逼近将使用有限元方法和空间变量的杂交不连续伽辽金格式，以及伪装成卷积正交方法的高阶时间步进工具来处理。本研究的预期结果包括从完全离散方法的稳定性和收敛性分析到三维几何中的实际实施。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The numerical simulation of transient elastic waves has the overall goal of developing robust and reliable computational tools to handle the study of deformation and stress in solids. The results of this project could impact a wide range of applications, including inverse problems in geophysics (seismic imaging with applications in earthquake simulation and oil exploration), magnetic resonance elastography for medical applications, control of electromechanical systems (the use of solid deformation to control electric fields or viceversa), and fracture detection in solids. The understanding of the computational approximations to complex model equations describing the interaction of varied physical phenomena (solid deformation, electromagnetism, acoustic propagation, and thermal diffusion) is of great importance to assess the quality and validity of the proposed simulations.The project focuses on the numerical analysis and computation of transient elastic waves, including their interaction with acoustic waves. The proposal addresses multiple physical models where elastodynamics is involved, with an emphasis on viscoelastic behavior, piezoelectric effects, and fully coupled thermoelasticity. The project proposes to unify and simplify the treatment, theoretical and numerical, of several of these models. The numerical approximation will be handled using Finite Element Methods and Hybridizable Discontinuous Galerkin schemes for the space variables, and high order time-stepping tools disguised as Convolution Quadrature methods. The expected outcomes of this proposed research range from stability and convergence analysis for the fully discrete methods to practical implementation in three dimensional geometries.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.