Multiscale modeling, discretization, and algorithms: a survey in biomechanics

Multiscale models allow for the treatment of complex phenomena involving different scales, such as remodeling and growth of tissues, muscular activation, and cardiac electrophysiology. Numerous numerical approaches have been developed to simulate multiscale problems. However, compared to the well-established methods for classical problems, many questions have yet to be answered. Here, we give an overview of existing models and methods, with particular emphasis on mechanical and bio-mechanical applications. Moreover, we discuss state-of-the-art techniques for multilevel and multifidelity uncertainty quantification. In particular, we focus on the similarities that can be found across multiscale models, discretizations, solvers, and statistical methods for uncertainty quantification. Similarly to the current trend of removing the segregation between discretizations and solution methods in scientific computing, we anticipate that the future of multiscale simulation will provide a closer interaction with also the models and the statistical methods. This will yield better strategies for transferring the information across different scales and for a more seamless transition in selecting and adapting the level of details in the models. Finally, we note that machine learning and Bayesian techniques have shown a promising capability to capture complex model dependencies and enrich the results with statistical information; therefore, they can complement traditional physics-based and numerical analysis approaches.