Simulation and Numerical Analysis in Elastodynamics

瞬变弹性波的数值模拟的总体目标是开发稳健和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，包括地球物理中的反问题(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、机电系统的控制(使用固体变形来控制电场或相反)，以及固体中的断裂检测。对描述各种物理现象(固体变形、电磁、声传播和热扩散)相互作用的复杂模型方程的计算近似的理解对于评估所提出的模拟的质量和有效性是非常重要的。该项目侧重于瞬时弹性波的数值分析和计算，包括它们与声波的相互作用。该提案涉及涉及弹性动力学的多种物理模型，重点关注粘弹性行为、压电效应和完全耦合的热弹性。该项目建议统一和简化其中几个模型的理论和数值处理。数值逼近将使用空间变量的有限元方法和杂交间断Galerkin格式，以及伪装成卷积求积方法的高阶时间步长工具。这项拟议研究的预期结果从全离散方法的稳定性和收敛分析到三维几何的实际实施。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.