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

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.