Collaborative Research: Multiscale Simulations and Imaging of Viscoelastic Media in Reduced Order Model Framework

粘弹性材料特性的成像和非侵入性评估在医学诊断、地球物理勘探和工程应用中具有极其重要的意义。材料的粘弹性是耗散过程的结果，这些耗散过程可能源于原子晶格中的非线性波能耗散，源于薄膜和多晶材料中的晶界位错，或者源于聚合物和生物材料中微观尺度的损伤、断裂和修复。在生物医学应用中，软组织粘弹性的变化往往是与特定恶性肿瘤相关的重要生物标志物，对动脉硬化、骨质疏松症和各种癌症的检测、定性和监测具有诊断价值。最近在大系统降阶建模方面的进展使我们能够处理许多十年前被视为无法解决的实际问题。然而，粘弹性建模和成像仍然是具有挑战性的问题。波在粘弹性介质中的传播具有衰减和色散效应，这在弹性材料中是不存在的。该项目将开发一种有效的计算方法，用于粘弹性介质成像和衰减波在耗散材料中传播的模拟。该项目将Stieltjes-Krein连续分数降阶模型方法扩展到耗散粘弹性系统。这种非常有效的方法是基于Stieltjes网络实现的非耗散问题，目前不能应用于具有耗散和色散的问题。该项目将消除这一限制，并将该方法扩展到耗散粘弹性系统。该方法的三个主要组成部分是：(1)将矩阵连分式方法推广到Stieltjes-Krein理论之外，从而实现粘弹性问题ROM的高效耗散稀疏网络实现；(2)基于Neumann-to-Dirichlet映射的耗散稀疏网络近似的多尺度模拟框架；(3)将网络嵌入方法扩展到耗散问题和粘弹性介质的直接非线性成像算法。该方法将导致大规模波传播模拟以及基于网络嵌入的散射问题的新型直接成像算法的计算成本降低数量级。该奖项反映了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。

Imaging and noninvasive evaluation of viscoelastic material properties have paramount importance in medical diagnostics, geophysical exploration, and engineering applications. Viscoelastic properties of materials are a result of dissipative processes that could stem from nonlinear wave energy dissipation in atomic lattices, from grain boundaries dislocations in thin films and polycrystalline materials, or from damage, fracturing, and repairing at the microscale in polymers and biological materials. In biomedical applications, changes in viscoelasticity of soft tissues often serve as an important biomarker associated with specific malignancies and have a diagnostic value for the detection, characterization, and monitoring of arteriosclerosis, osteoporosis, and various cancers. Recent advances in reduced order modeling of large scale systems have enabled us to approach many practical problems viewed as unsolvable ten years ago. However, viscoelastic modeling and imaging are still challenging problems. Wave propagation in viscoelastic media is characterized by attenuation and dispersion effects, which are absent in the case of elastic materials. This project will develop an efficient computational method for imaging of viscoelastic media and simulation of attenuating waves propagating in dissipative materials.The project develops an extension of the Stieltjes-Krein continuous fraction reduced order modeling method to dissipative viscoelastic systems. This very efficient method is based on the Stieltjes network realizations of nondissipative problems and currently cannot be applied to problems with dissipation and dispersion. The project will eliminate this limitation and extend the method to dissipative viscoelastic systems. Three main components of the approach are the following: (1) Efficient dissipative sparse network realizations of ROMs for viscoelastic problems by generalizing the matrix continued-fraction approach beyond Stieltjes-Krein theory; (2) Multiscale simulation framework based on dissipative sparse network approximations of Neumann-to-Dirichlet maps; (3) Extension of the network embedding approach to dissipative problems and direct nonlinear imaging algorithms for viscoelastic media. The method will lead to an orders-of-magnitude reduction in the computational cost of large-scale wave propagation simulations as well as novel direct imaging algorithms for scattering problems based on network embedding.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.