Soft wetting with (a)symmetric Shuttleworth effect

The wetting of soft polymer substrates brings in multiple complexities as compared to the wetting on rigid substrates. The contact angle of the liquid is no longer governed by Young's law, but is affected by the substrate's bulk and surface deformations. On top of that, elastic interfaces exhibit a surface energy that depends on how much they are stretched -- a feature known as the Shuttleworth effect (or as surface-elasticity). Here we present two models by which we explore the wetting of drops in the presence of a strong Shuttleworth effect. The first model is macroscopic in character and consistently accounts for large deformations via a neo-Hookean elasticity. The second model is based on a mesoscopic description of wetting, using a reduced description of the substrate's elasticity. While the second model is more empirical in terms of the elasticity, it enables a gradient dynamics formulation for soft wetting dynamics. We provide a detailed comparison between the equilibrium states predicted by the two models, from which we deduce robust features of soft wetting in the presence of a strong Shuttleworth effect. Specifically, we show that the (a)symmetry of the Shuttleworth effect between the "dry" and "wet" states governs horizontal deformations in the substrate. Our results are discussed in the light of recent experiments on the wettability of stretched substrates

Image-based goal-oriented adaptive isogeometric analysis with application to the micro-mechanical modeling of trabecular bone

Isogeometric analysis (IGA) of geometrically complex three-dimensional objects is possible when used in combination with the Finite Cell method (FCM). In this contribution we propose a computational methodology to automatically analyze the effective elastic properties of scan-based volumetric objects of arbitrary geometric and topological complexity. The first step is the reconstruction of a smooth geometry from scan-based voxel data using a B-spline level set function. The second step is a goal-oriented adaptive isogeometric linear elastic analysis. Elements are selected for refinement using dual-weighted residual shape function indicators, and hierarchical splines are employed to construct locally refined spline spaces. The proposed methodology is studied in detail for various numerical test cases, including the computation of the effective Young's modulus of a trabecular bone micro-structure reproduced from μCT-scan data

Efficient Earthquake Inversion using the Finite Element Method

A vital component in the management of seismic hazard is the study of past seismic events. Classically, this has been the domain of seismology, which studies the dynamic manifestations of the event to infer properties such as epicenter and moment magnitude. More recently it has become possible to perform similar analyses on the basis of the static consequences of a seismic event, as satellite borne Synthetic Aperture Radar (SAR) data allows us to compare the local surface geometries before and aftera seismic event. The locality of the deformation data promises reconstructions with greater detail and subject to fewer model uncertainties.With current technology, it is not possible to use SAR to their full potential. The non-linearity of the static dislocation problem that links faulting mechanisms to observed deformations causes any inverse method to require many evaluations of the forward model. This poses limits on the permissible cost of solving the dislocation problem, restricting most approaches to simplified model assumptions such as material homogeneity and absence of topography. In situations where more accurate information is available, this presents a clear opportunity for improvement by accelerating the computational methods instead.This thesis presents the Weakly-enforced Slip Method (WSM), a modification of the Finite Element Method (FEM), as a fast approach for solving static dislocation problems. While the computational cost of the WSM is similar to that of the FEM for single dislocations, the WSM is significantly faster when many different dislocation geometries are considered, owing to the reuse of computationally expensive components such as matrix factors. This property makes the method ideally suited for inverse settings, opening the way to incorporating all available in situ data in a forward model that is simultaneously flexible and cheaply evaluable. Moreover, we prove that the WSM retains the essential convergence properties of the FEM.A limitation of the WSM is that it produces continuous displacement fields, which implies a large error local to the dislocation. We show that this error decreases rapidly with distance, and that in a typical scenario the majority of deformation data has a discretization error that is smaller than observational noise, particularly when a fault is buried. In the case of shallow or rupturing faults, neighbouring data needs to be discarded from the analysis to avoid disruption. With this measure in place, we show via Bayesian inference of synthesized datasets that the discretization errors of the WSM do not significantly affect the inverse problem

Efficient Earthquake Inversion using the Finite Element Method

A vital component in the management of seismic hazard is the study of past seismic events. Classically, this has been the domain of seismology, which studies the dynamic manifestations of the event to infer properties such as epicenter and moment magnitude. More recently it has become possible to perform similar analyses on the basis of the static consequences of a seismic event, as satellite borne Synthetic Aperture Radar (SAR) data allows us to compare the local surface geometries before and aftera seismic event. The locality of the deformation data promises reconstructions with greater detail and subject to fewer model uncertainties.With current technology, it is not possible to use SAR to their full potential. The non-linearity of the static dislocation problem that links faulting mechanisms to observed deformations causes any inverse method to require many evaluations of the forward model. This poses limits on the permissible cost of solving the dislocation problem, restricting most approaches to simplified model assumptions such as material homogeneity and absence of topography. In situations where more accurate information is available, this presents a clear opportunity for improvement by accelerating the computational methods instead.This thesis presents the Weakly-enforced Slip Method (WSM), a modification of the Finite Element Method (FEM), as a fast approach for solving static dislocation problems. While the computational cost of the WSM is similar to that of the FEM for single dislocations, the WSM is significantly faster when many different dislocation geometries are considered, owing to the reuse of computationally expensive components such as matrix factors. This property makes the method ideally suited for inverse settings, opening the way to incorporating all available in situ data in a forward model that is simultaneously flexible and cheaply evaluable. Moreover, we prove that the WSM retains the essential convergence properties of the FEM.A limitation of the WSM is that it produces continuous displacement fields, which implies a large error local to the dislocation. We show that this error decreases rapidly with distance, and that in a typical scenario the majority of deformation data has a discretization error that is smaller than observational noise, particularly when a fault is buried. In the case of shallow or rupturing faults, neighbouring data needs to be discarded from the analysis to avoid disruption. With this measure in place, we show via Bayesian inference of synthesized datasets that the discretization errors of the WSM do not significantly affect the inverse problem.Mathematical Geodesy & Positionin

An adaptive isogeometric analysis approach to elasto-capillary fluid-solid interaction

We present an adaptive isogeometric-analysis approach to elasto-capillary fluid-solid interaction (FSI), based on a diffuse-interface model for the binary fluid and an Arbitrary-Lagrangian-Eulerian formulation for the FSI problem. We consider approximations constructed from adaptive high-regularity truncated hierarchical splines, as employed in the isogeometric analysis (IGA) paradigm. The considered adaptive strategy comprises a two-level hierarchical a posteriori error estimate. The hierarchical a posteriori error estimate directs a support-based refinement procedure. To attain robustness of the solution procedure for the aggregated binary-fluid-solid-interaction problem, we apply a fully monolithic solution procedure and we introduce a continuation process in which the diffuse interface of the binary fluid is artificially enlarged on the coarsest levels of the adaptive-refinement procedure. To assess the capability of the presented adaptive IGA method for elasto-capillary FSI, we conduct numerical computations for a configuration pertaining to a sessile droplet on a soft solid substrate

A finite-element/boundary-element method for three-dimensional, large-displacement fluid-structure-interaction

This paper presents a hybrid finite-element/boundary-element method for fluid-structure-interaction simulations of inflatable structures. The flow model consists of the steady Stokes equation, which admits a boundary-integral formulation. The structure is represented by a Kirchhoff-Love shell. The boundary-element approximation of the Stokes equation reduces the flow problem to an integral equation on the actual structure configuration, thus obviating the need for volumetric meshing of the strongly deforming fluid domain. The Stokes model moreover exhibits a lubrication effect that acts as an intrinsic mechanism to treat the ubiquitous self-contact that occurs in inflation problems. The aggregated fluid-structure-interaction problem, composed of the boundary-integral equation and the Kirchhoff-Love shell connected by dynamic and kinematic interface conditions, is approximated by means of isogeometric discretizations to accommodate the smoothness requirements on the approximation spaces imposed by the flexural rigidity in the Kirchhoff-Love shell and to provide an accurate and smooth representation of the boundary for the boundary-element method. Auxiliary results presented in this paper are: (1) a parametrization-free Kirchhoff-Love formulation; (2) establishment of a cubic relationship between distance and tractions due to the lubrication effect; and (3) the interpretation of the Lagrange multiplier pertaining to fluid incompressibility as the total excess pressure

Stabilized second-order convex splitting schemes for Cahn-Hilliard models with application to diffuse-interface tumor-growth models

We present unconditionally energy-stable second-order time-accurate schemes for diffuse-interface (phase-field) models; in particular, we consider the Cahn–Hilliard equation and a diffuse-interface tumor-growth system consisting of a reactive Cahn–Hilliard equation and a reaction–diffusion equation. The schemes are of the Crank–Nicolson type with a new convex–concave splitting of the free energy and an artificial-diffusivity stabilization. The case of nonconstant mobility is treated using extrapolation. For the tumor-growth system, a semi-implicit treatment of the reactive terms and additional stabilization are discussed. For suitable free energies, all schemes are linear. We present numerical examples that verify the second-order accuracy, unconditional energy-stability, and superiority compared with their first-order accurate variants

A robust and accurate adaptive approximation method for a diffuse-interface model of binary-fluid flows

We present an adaptive simulation framework for binary-fluid flows, based on the Abels–Garcke–Grün Navier–Stokes–Cahn–Hilliard (AGG NSCH) diffuse-interface model. The adaptive-refinement procedure is guided by a two-level hierarchical a-posteriori error estimate, and it effectively resolves the spatial multiscale behavior of the diffuse-interface model. To improve the robustness of the solution procedure and avoid severe time-step restrictions for small-interface thicknesses, we introduce an ɛ-continuation procedure, in which the diffuse interface thickness (ɛ) are enlarged on coarse meshes, and the mobility is scaled accordingly. To further accelerate the computations and improve robustness, we apply a modified Backward Euler scheme in the initial stages of the adaptive-refinement procedure in each time step, and a Crank–Nicolson scheme in the final stages of the refinement procedure. To enhance the robustness of the nonlinear solution procedure, we introduce a partitioned solution procedure for the linear tangent problems in Newton's method, based on a decomposition of the NSCH system into its NS and CH subsystems. We conduct a systematic investigation of the conditioning of the monolithic NSCH tangent matrix and of its NS and CH subsystems for a representative 2D model problem. To illustrate the properties of the presented adaptive simulation framework, we present numerical results for a 2D oscillating water droplet suspended in air, and we validate the obtained results versus those of a corresponding sharp-interface model

Soft wetting with (a)symmetric Shuttleworth effect

The wetting of soft polymer substrates brings in multiple complexities when compared with the wetting on rigid substrates. The contact angle of the liquid is no longer governed by Young's Law, but is affected by the substrate's bulk and surface deformations. On top of that, elastic interfaces exhibit a surface energy that depends on how much they are stretched - a feature known as the Shuttleworth effect (or as surface-elasticity). Here, we present two models through which we explore the wetting of drops in the presence of a strong Shuttleworth effect. The first model is macroscopic in character and consistently accounts for large deformations via a neo-Hookean elasticity. The second model is based on a mesoscopic description of wetting, using a reduced description of the substrate's elasticity. While the second model is more empirical in terms of the elasticity, it enables a gradient dynamics formulation for soft wetting dynamics. We provide a detailed comparison between the equilibrium states predicted by the two models, from which we deduce robust features of soft wetting in the presence of a strong Shuttleworth effect. Specifically, we show that the (a)symmetry of the Shuttleworth effect between the 'dry' and 'wet' states governs horizontal deformations in the substrate. Our results are discussed in the light of recent experiments on the wettability of stretched substrates