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

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

[no abstract available

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

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

Adjoint-based methods for optimization and goal-oriented error control applied to fluid-structure interaction: implementation of a partition-of-unity dual-weighted residual estimator for stationary forward FSI problems in deal.II

[EN] In this work, we implement goal-oriented error control and spatial mesh adaptivity for stationary fluid-structure interaction (FSI). The a posteriori error estimator is accomplished using the dual-weighted residual method in which the adjoint equation arises. The fluid-structure interaction problem is formulated within a variational-monolithic framework using arbitrary Lagrangian-Eulerian coordinates. The overall problem is nonlinear and solved with Newton’s method. We specifically consider the FSI-1 benchmark problem in which quantities of interest include the elastic beam displacements, drag, and lift. The implementation is based on the deal.II finite element library and provided open-source published on github https://github.com/tommeswick/goal-oriented-fsi. Possible extensions are discussed in the source code and in the conclusions of this paper.This work is supported by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy within the cluster of Excellence PhoenixD (EXC 2122, Project ID 390833453).Wick, T. (2022). Adjoint-based methods for optimization and goal-oriented error control applied to fluid-structure interaction: implementation of a partition-of-unity dual-weighted residual estimator for stationary forward FSI problems in deal.II. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 257-266. https://doi.org/10.4995/YIC2021.2021.12332OCS25726

Interfaces

In this course, coupled problems with interfaces are considered. Some applications and examples are discussed first. Then, interfaces are defined and classified into three categories. Numerical modeling of interfaces is a central aspect in this presentation. These theoretically-oriented parts are followed by numerical simulations using an open-source fluid-structure interaction benchmark code based on the finite element library deal.II. For joint coding, a docker image was installed on qarnot and repl.it for cloud computing.Course held at the CSMA Junior section workshop ahead of the 14th WCCM & EDDOMAS Congress 202