Model-Free Data-Driven Viscoelasticity in the Frequency Domain

Abstract

We develop a Data-Driven framework for the simulation of wave propagation in viscoelastic solids directly from dynamic testing material data, including data from Dynamic Mechanical Analysis (DMA), nano-indentation, Dynamic Shear Testing (DST) and Magnetic Resonance Elastography (MRE), without the need for regression or material modeling. The problem is formulated in the frequency domain and the method of solution seeks to minimize a distance between physically admissible histories of stress and strain, in the sense of compatibility and equilibrium, and the material data. We metrize the space of histories by means of the flat-norm of their Fourier transform, which allows consideration of infinite wave trains such as harmonic functions. Another significant advantage of the flat norm is that it allows the response of the system at one frequency to be inferred from data at nearby frequencies. We demonstrate and verify the approach by means of two test cases, a polymeric truss structure characterized by DMA data and a 3D soft gel sample characterized by MRE data. The examples demonstrate the ease of implementation of the Data-Driven scheme within conventional commercial codes and its robust convergence properties, both with respect to the solver and the data.

Application of Data-Driven computing to patient-specific simulation of brain neuromodulation

“…A detailed summary of the method in presented in the SI Appendix. The model-free Data-Driven solver strives to find admissible states of stress and strain in the finite-element model that are closest to the material data set in the sense of a suitable measure of discrepancy [17]. The finite-element model supplies a representation of the geometry of the brain, which is assumed to be free-standing, and the ultrasound pressure loading applied to it.…”

“…Conveniently, the solver can be implemented as a wrapper around conventional FE element programs, including commercially available software. A complete mathematical description of the Data-Driven solver and an assessment of convergence with respect to the data may be found [17].…”

“…For completeness and ease of reference, we summarize the equations that govern the calculations of brain ultrasound stimulation presented in this work. Further detail of the formulation and the solver can be found in [17].…”

“…An appealing alternative is to formulate solvers that effectively enable the utilization of in situ and in vivo imaging data directly in calculations, thus entirely bypassing the intermediate material modeling step. In this work, we demonstrate how one such model-free Data-Driven paradigm [17] can be used to forge a direct connection between in situ and in vivo patient-specific measurements of viscoelastic brain-tissue properties and real-time full-scale calculations of the mechanical response of the human brain to ultrasound stimulation, cf. Section 1.…”

“…For purposes of demonstration and verification, we specifically use a public-domain MRE data set [9] and the corresponding MNI stereotaxic space to develop a finite-element model of the human brain co-registered with the voxel-based MRE data. We then use our model-free Data-Driven solver [17] to compute, directly from the data, wave patterns in the brain resulting from transcranial ultrasound stimulation, cf. Section 2.…”