Thomas Wick

Phase-field modeling of a fluid-driven fracture in a poroelastic medium

In this paper, we present a phase field model for a fluid-driven fracture in a poroelastic medium. In our previous work, the pressure was assumed given. Here, we consider a fully coupled system where the pressure field is determined simultaneously with the displacement and the phase field. To the best of our knowledge, such a model is new in the literature. The mathematical model consists of a linear elasticity system with fading elastic moduli as the crack grows, which is coupled with an elliptic variational inequality for the phase field variable and with the pressure equation containing the phase field variable in its coefficients. The convex constraint of the variational inequality assures the irreversibility and entropy compatibility of the crack formation. The phase field variational inequality contains quadratic pressure and strain terms, with coefficients depending on the phase field unknown. We establish existence of a solution to the incremental problem through convergence of a finite dimensional approximation. Furthermore, we construct the corresponding Lyapunov functional that is linked to the free energy. Computational results are provided that demonstrate the effectiveness of this approach in treating fluid-driven fracture propagation.

Long-term simulation of large deformation, mechano-chemical fluid-structure interactions in ALE and fully Eulerian coordinates

In this work, we develop numerical schemes for mechano-chemical fluid-structure interactions with long-term effects. We investigate a model of a growing solid interacting with an incompressible fluid. A typical example for such a situation is the formation and growth of plaque in blood vessels. This application includes two particular difficulties: First, growth may lead to very large deformations, up to full clogging of the fluid domain. We derive a simplified set of equations including a fluid-structure interaction system coupled to an ODE model for plaque growth in Arbitrary Lagrangian Eulerian (ALE) coordinates and in Eulerian coordinates. The latter novel technique is capable of handling very large deformations up to contact. The second difficulty stems from the different time scales: while the dynamics of the fluid demand to resolve a scale of seconds, growth typically takes place in a range of months. We propose a temporal two-scale approach using local small-scale problems to compute an effective wall stress that will enter a long-scale problem. Our proposed techniques are substantiated with several numerical tests that include comparisons of the Eulerian and ALE approaches as well as convergence studies.

Coupling fluid–structure interaction with phase-field fracture

Abstract In this work, a concept for coupling fluid–structure interaction with brittle fracture in elasticity is proposed. The fluid–structure interaction problem is modeled in terms of the arbitrary Lagrangian–Eulerian technique and couples the isothermal, incompressible Navier–Stokes equations with nonlinear elastodynamics using the Saint-Venant Kirchhoff solid model. The brittle fracture model is based on a phase-field approach for cracks in elasticity and pressurized elastic solids. In order to derive a common framework, the phase-field approach is re-formulated in Lagrangian coordinates to combine it with fluid–structure interaction. A crack irreversibility condition, that is mathematically characterized as an inequality constraint in time, is enforced with the help of an augmented Lagrangian iteration. The resulting problem is highly nonlinear and solved with a modified Newton method (e.g., error-oriented) that specifically allows for a temporary increase of the residuals. The proposed framework is substantiated with several numerical tests. In these examples, computational stability in space and time is shown for several goal functionals, which demonstrates reliability of numerical modeling and algorithmic techniques. But also current limitations such as the necessity of using solid damping are addressed.

Discontinuous and Enriched Galerkin Methods for Phase-Field Fracture Propagation in Elasticity

In this work, we introduce discontinuous Galerkin and enriched Galerkin formulations for the spatial discretization of phase-field fracture propagation. The nonlinear coupled system is formulated in terms of the Euler-Lagrange equations, which are subject to a crack irreversibility condition. The resulting variational inequality is solved in a quasi-monolithic way in which the irreversibility condition is incorporated with the help of an augmented Lagrangian technique. The relaxed nonlinear system is treated with Newton’s method. Numerical results complete the present study.

Variational-Monolithic ALE Fluid-Structure Interaction: Comparison of Computational Cost and Mesh Regularity Using Different Mesh Motion Techniques

In this contribution, different mesh motion models for fluid-structure interaction (FSI) are revisited. The FSI problem is formulated by variational-monolithic coupling in the reference configuration employing the arbitrary-Lagrangian Eulerian (ALE) framework. The goal is to further analyze three different mesh motion models; namely nonlinear harmonic, nonlinear elastic, and linear biharmonic. The novelty in this contribution is a detailed computational analysis of the regularity of the ALE mapping and cost complexity for the nonstationary FSI-2 benchmark problem with large solid deformations.

Coupling Fluid-Structure Interaction with Phase-Field Fracture: Modeling and a Numerical Example

In this work, a framework for coupling arbitrary Lagrangian-Eulerian fluid-structure interaction with phase-field fracture is suggested. The key idea is based on applying the weak form of phase-field fracture, including a crack irreversibility constraint, to the nonlinear coupled system of Navier-Stokes and elasticity. The resulting setting is formulated via variational-monolithic coupling and has four unknowns: velocities, displacements, pressure, and a phase-field variable. The inequality constraint is imposed through penalization using an augmented Lagrangian algorithm. The nonlinear problem is solved with Newton’s method. The framework is tested in terms of a numerical example in which computational stability is demonstrated by evaluating goal functionals on different spatial meshes.

Binary-fluid–solid interaction based on the Navier–Stokes–Cahn–Hilliard Equations

We consider amodel for binary-fluid-solid interaction based on a diffuse-interface model for the binary fluid and a hyperelastic-materialmodel for the solid. The diffuse-interface binary-fluidmodel is described by the quasi-incompressible Navier-Stokes-Cahn-Hilliard equations with preferential-wetting boundary conditions at the fluid-solid interface. The fluid traction on the interface includes a capillary-stress contribution in addition to the regular viscous-stress and pressure contributions. The dynamic interface condition comprises the traction exerted by the nonuniform solid-fluid surface tension in accordance with the Young-Laplace law for the solid-fluid interface. The solid is modeled as a hyperelastic material. We present a weak formulation of the aggregated binary-fluid-solid interaction problem, based on an arbitrary Lagrangian-Eulerian formulation of theNavier-Stokes-Cahn-Hilliard equations and a properweak evaluation of the binary-fluid traction and of the solid-fluid surface tension. We also present an analysis of the essential properties of the binary-fluid-solid interaction problem, including a dissipation relation for the complete fluid-solid interaction problem. To validate the presented binary-fluid-solid interactionmodel, we consider numerical simulations for the elasto-capillary interaction of a droplet with a soft solid substrate and present a comparison to corresponding experimental data.

Grundlagen der linearen Algebra

Wir sammeln zunachst einige Definitionen und grundlegende Resultate. Fur ausfuhrliche Darstellungen sei beispielsweise auf den Klassiker [21] verwiesen.

4. Numerical methods for unsteady thermal fluid structure interaction

We discuss thermal fluid-structure interaction processes, where a simulation of the time-dependent temperature field is of interest. Thereby, we consider partitioned coupling schemes with a Dirichlet-Neumann method. We present an analysis of the method on a model problem of discretized coupled linear heat equations. This shows that for large quotients in the heat conductivities, the convergence rate will be very small. The time dependencymakes the use of time-adaptive implicitmethods imperative. This gives rise to the question as to how accurately the appearing nonlinear systems should be solved, which is discussed in detail for both the nonlinear and linear case. The efficiency of the resulting method is demonstrated using realistic test cases. (Less)

Fluid-Structure Interaction: Modeling, Adaptive Discretisations and Solvers

The objective of this work is to examine the potential of isogeometric methods in the context of multidisciplinary shape optimization. We introduce a shape optimization problem based on a coupled fluid-structure system, whose geometry is defined by NURBS (Non-Uniform Rational B-Spline) curves. This shape optimization problem is then solved by using either an isogeometric approach, or a classical grid-based approach. In spite of the fact that optimization results do not show any major differences, conceptional advantages of the new isogeometric method become apparent. In particular, control points of the spline can be directly handled as design variables without the need of a spline-fit and consequently geometry errors can be excluded at every stages of the optimization loop.