Development of a multiphysics solver for complex coupled problems involving thin shells: fluid-structure-electrophysiology interaction

The present work is devoted to the development of a multiphysics solver for simulating two classes of coupled problems. A computational framework is designed to accurately predict the elastic response of thin shells undergoing large displacements induced by local hydrodynamic forces, as well as to resolve the complex fluid pattern arising from its interaction with an incompressible fluid. Within the context of partitioned algorithms, two different approaches are employed for the fluid and structural domain. The fluid motion is resolved with a pressure projection method on a Cartesian structured grid. The immersed shell is modeled by means of a NURBS surface, and the elastic response is obtained from a displacement-based Isogeometric Analysis relying on the Kirchhoff-Love theory. The two solvers exchange data through a direct-forcing Immersed Boundary approach, where the interpolation/spreading of the variables between Lagrangian and Eulerian grids is implemented with a Moving Least Squares approximation, which has proven to be very eective for moving boundaries. In this scenario, the isoparametric paradigm is exploited to perform an adaptive collocation of the Lagrangian markers, decoupling the local grid density of fluid and shell domains and reducing the computational expense. The convergence rate of the method is verified by refinement analyses, segregating the Eulerian/Lagrangian refinement, which confirms the expected scheme accuracy in space and time. The effectiveness of the method is then verified against different test–cases of engineering and biologic inspiration, involving fundamentally different physical and numerical conditions, namely: i) a flapping flag, ii) an inverted flag, iii) a clamped plate, iv) a buoyant seaweed in a free stream. Both strong and loose coupling approaches are implemented to handle different fluid-to-structure density ratios, providing accurate results. In second instance, we propose an IGA approximation of the system of equations describing the propagation of an electrophysiologic stimulus over a thin cardiac tissue with the subsequent muscle contraction. The underlying method relies on the monodomain model for the electrical sub-problem. This requires the solution of a reaction-diffusion equation over a surface in the three-dimensional space. Exploiting the benefits of the high-order NURBS basis functions within a curvilinear framework, the method is found to reproduce complex excitation patterns with a limited number of degrees of freedom. Furthermore, the curvilinear description of the diusion term provides a flexible and easy-to-implement approach for general surfaces. The electrophysiological stimulus is converted into a mechanical load by means of the wellestablished active strain approach. The multiplicative decomposition of the deformation gradient tensor is grafted into the classical finite elasticity weak formulation, providing the necessary tensor expressions in curvilinear coordinates. The expressions derived provides what is needed to implement the active strain approach in standard finite-element solvers without resorting to dedicated formulations. Such a formulation is valid for general three-dimensional geometries and isotropic hyperelastic materials. The formulation is then restricted to Kirchhoff-Love shells by means of the static condensation of the material tensor. The purely elastic response of the structure is investigated with simple static test-cases of thin shells undergoing different active strain patterns. Eventually, various numerical tests performed with a staggered scheme illustrate that the coupled electromechanical model can capture the excitation-contraction mechanism over thin tissue and reproduce complex curvature variations.