A phase-field model for fractures in incompressible solids

Within this work, we develop a phase-field description for simulating fractures in incompressible materials. Standard formulations are subject to volume-locking when the solid is (nearly) incompressible. We propose an approach that builds on a mixed form of the displacement equation with two unknowns: a displacement field and a hydro-static pressure variable. Corresponding function spaces have to be chosen properly. On the discrete level, stable Taylor-Hood elements are employed for the displacement-pressure system. Two additional variables describe the phase-field solution and the crack irreversibility constraint. Therefore, the final system contains four variables: displacements, pressure, phase-field, and a Lagrange multiplier. The resulting discrete system is nonlinear and solved monolithically with a Newton-type method. Our proposed model is demonstrated by means of several numerical studies based on two numerical tests. First, different finite element choices are compared in order to investigate the influence of higher-order elements in the proposed settings. Further, numerical results including spatial mesh refinement studies and variations in Poisson's ratio approaching the incompressible limit, are presented

Adaptive Finite Elements for Monolithic Fluid-StructureInteraction on a Prolongated Domain: Applied to an Heart Valve Simulation

In this work, we apply a fluid-structure interaction method to a long axis heart valve simulation. Our method of choice is based on a monolithic coupling scheme for fluid-structure interaction, where the fluid equations are rewritten in the arbitrary Lagrangian Eulerian' framework. To prevent back-flow of waves in the structure due to its hyperbolic nature, a damped structure equation is solved on an artificial layer that prolongates the computational domain. This coupling is stable on the continuous level. To reduce the increased computational cost in presence of the artificial layer, we refine the mesh only regions of interest. To this end, a stationary version of goal-oriented mesh refinement is part of our numerical tests. The results show that heart valve dynamics can be simulated with our proposed model

Solving Monolithic Fluid-Structure Interaction Problems in Arbitrary Lagrangian Eulerian Coordinates with the deal.II Library

We briefly describe a setting of a non-linear fluid-structure interaction problem and its solution in the finite element software package deal.II. The fluid equations are transformed via the ALE map (Arbitrary Lagrangian Eulerian framework) to a reference configuration. The mapping is constructed using the biharmonic operator. The coupled problem is defined in a monolithic framework and serves for unsteady (or quasi-stationary) configurations. Different types of time stepping schemes are implemented. The non-linear system is solved by a Newton method. Here, the Jacobian matrix is build up by exact computation of the directional derivatives. The implementation serves for the computation of the fluid-structure benchmark configurations proposed by J. Hron and S. Turek

Sergey I. Repin, Stefan A. Sauter: “Accuracy of Mathematical Models” : EMS, 2020, 317 pp.

[no abstract available

Modeling, Discretization, Optimization, and Simulation of Multiphysics Problems (IIT Indore)

The goal of this winter school is to give an introduction to numerical modeling of multiphysics problems. These are nonstationary, nonlinear, coupled partial differential equations. The philosophy of this school is to provide a mixture of very basic techniques that are immediately applied to `complicated' practical and/or current research problems