The diffuse Nitsche method: Dirichlet constraints on phase-field boundaries

We explore diffuse formulations of Nitsche's method for consistently imposing Dirichlet boundary conditions on phase-field approximations of sharp domains. Leveraging the properties of the phase-field gradient, we derive the variational formulation of the diffuse Nitsche method by transferring all integrals associated with the Dirichlet boundary from a geometrically sharp surface format in the standard Nitsche method to a geometrically diffuse volumetric format. We also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem, from which the stabilization parameter can be estimated automatically in each element. We advertise metastable phase-field solutions of the Allen-Cahn problem for transferring complex imaging data into diffuse geometric models. In particular, we discuss the use of mixed meshes, that is, an adaptively refined mesh for the phase-field in the diffuse boundary region and a uniform mesh for the representation of the physics-based solution fields. We illustrate accuracy and convergence properties of the diffuse Nitsche method and demonstrate its advantages over diffuse penalty-type methods. In the context of imaging based analysis, we show that the diffuse Nitsche method achieves the same accuracy as the standard Nitsche method with sharp surfaces, if the inherent length scales, i.e., the interface width of the phase-field, the voxel spacing and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body

Phase-field boundary conditions for the voxel finite cell method: surface-free stress analysis of CT-based bone structures

The voxel finite cell method employs unfitted finite element meshes and voxel quadrature rules to seamlessly transfer CT data into patient-specific bone discretizations. The method, however, still requires the explicit parametrization of boundary surfaces to impose traction and displacement boundary conditions, which constitutes a potential roadblock to automation. We explore a phase-field based formulation for imposing traction and displacement constraints in a diffuse sense. Its essential component is a diffuse geometry model generated from metastable phase-field solutions of the Allen-Cahn problem that assumes the imaging data as initial condition. Phase-field approximations of the boundary and its gradient are then employed to transfer all boundary terms in the variational formulation into volumetric terms. We show that in the context of the voxel finite cell method, diffuse boundary conditions achieve the same accuracy as boundary conditions defined over explicit sharp surfaces, if the inherent length scales, i.e., the interface width of the phase-field, the voxel spacing and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human femur and a vertebral body

Simulations of bulk and confined bubble nucleation

The present thesis investigates, with atomistic simulations, vapor nucleation and liquid dynamics under nanoscale confinement. The main objective of this work is to go beyond the quasi-static classical picture of liquid-vapor phase transition, including kinetic and inertial effects. The performed simulations provide an accurate description of the phenomenon and a framework to interpret experimental observations. The dynamics of vapor nucleation is investigated in the pure bulk liquid and in confined conditions. In the last case, also wetting transition is studied. Particular attention is devoted to surfaces that combine textured geometries with an hydrophobic chemistry. These are able to stabilize vapor phase within surfaces asperities, producing a state in which liquid is suspended above the entrapped vapor pockets. In these conditions, remarkable properties arise that are collectively known as superhydropobicity. In this suspended state, known also as Cassie-Baxter state, the contact area between solid and liquid is reduced with respect to a flat surface and with respect to the textured surface in which the corrugations are flooded with the liquid. Moreover, the liquid presents a higher contact angle (CA), with a lower CA hysteresis and a reduced liquid-solid friction. Due to these properties, superhydrophobic surfaces are suitable for applications such as self-cleaning glass, window, andwallpaint. Theypreventmoistureaccumulation, helpanti-icing, andallowdropwisecondensationtoincreasetheheattransferefficiencyandwaterharvesting. These are all in-air applications. However, the presence of a large shear free liquid/gas interface suggested that super-hydrophobic surfaces can be used in many submerged applications, e.g. drag reduction, anti-friction, anti-adhesive, anti-corrosion, and boiling heat transfer. Cassie-Baxter state can be destabilized by changes in pressure and temperature, that produce the intrusion of the liquid within surface defects. The corresponding state in which the surface is completely wetted is known as Wenzel state. The loss of super-hydrophobic properties (Cassie-Baxter to Wenzel transition) has proved to be experimentally irreversible. It is therefore crucial to characterize both wetting and recovery mechanisms in order understand how to design surfaces supporting a robust Cassie-Baxter state, i.e. a suspended state that can resist to temperature and pressure fluctuations. Wetting transition and recovery of superhydrophobic state take place via vapor/liquid and liquid/vapor phase transitions occurring under confinement at the nanoscale within geometric defects. Over the last decades, a significant amount of experimental and theoretical work has been devoted to the study of confined liquidvapor transition. In spite of this, not much is known yet about the kinetics of the process. The contribution to the topic obtained during the three years of my PhD is presented in this thesis. The first part of the work has been devoted to develop and test Molecular Dynamics and Monte Carlo methods able to properly simulate multiphase systems. Indeed, it has been demonstrated that serious issues arise when the standard global barostats, developed to simulate bulk systems, are straightforwardly applied to systems with subdomains at different pressures, e.g. liquid and vapor domains during nucleation. A solution to overcome these artifacts has been proposed, consisting in the implementation of a local barostat that imposes a local force balance between a piston and the contacting liquid. With this approach, a more accurate prediction of the vapor nucleation barrier in a super-heated liquid has been obtained. Secondly, the simulation techniques developed at the first stage of my PhD work have been employed to study homogeneous bubble nucleation. At the liquid pressure andtemperaturehereinvestigated, thisphenomenonisarareevent: thewaitingtime to observe the inception of vapor formation is order of magnitude longer than the typical time that can be explored by atomistic simulations. This issue, that causes waste of computational resources, has been tackled by carefully selecting special techniques able to preserve kinetic and inertial effects during bubbles growth. With this approach, “dynamical” quantities have been estimated, e.g. the nucleation rate. Other two essential aspects have been addressed: the limits of theoretical expressions routinely used to evaluate the kinetic prefactor in Eyring equation for vapor nucleation; the relation between successful nucleation events and relevant observables, such as temperature and liquid velocity, at beginning and during bubble expansion. The last section of this thesis is focused on heterogeneous nucleation and wetting of super-hydrophobic surfaces. Recent theoretical and experimental studies have produced conflicting results in the characterization of the pathways by which liquid intrudes in pores. The disagreement resides, specifically, in the symmetry properties expected for the advancing meniscus shape. Experiments show a symmetric pathways, in which the liquid penetrates in the surface pores with an essentially flat meniscus, while quasi-static theories predict that the asymmetric pathway is more probable, in which the liquid entering in the surface cavities bend forming a bubble in a corner. My simulations have proved that inertial effects change the wetting and recovery path with respect the predictions of quasi-static approaches. This reconcile theory and experiments: when the transition is barrierless, as expected in experimental conditions in which only nearly spontaneous processes can be addressed, the more complete theory developed here predicts a symmetric wetting as observed in the experiments

Geometric partial differential equations: Surface and bulk processes

The workshop brought together experts representing a wide range of topics in geometric partial differential equations ranging from analyis over numerical simulation to real-life applications. The main themes of the conference were the analysis of curvature energies, new developments in pdes on surfaces and the treatment of coupled bulk/surface problems

A Metalearning Approach for Physics-Informed Neural Networks (PINNs): Application to Parameterized PDEs

Physics-informed neural networks (PINNs) as a means of discretizing partial differential equations (PDEs) are garnering much attention in the Computational Science and Engineering (CS&E) world. At least two challenges exist for PINNs at present: an understanding of accuracy and convergence characteristics with respect to tunable parameters and identification of optimization strategies that make PINNs as efficient as other computational science tools. The cost of PINNs training remains a major challenge of Physics-informed Machine Learning (PiML) - and, in fact, machine learning (ML) in general. This paper is meant to move towards addressing the latter through the study of PINNs on new tasks, for which parameterized PDEs provides a good testbed application as tasks can be easily defined in this context. Following the ML world, we introduce metalearning of PINNs with application to parameterized PDEs. By introducing metalearning and transfer learning concepts, we can greatly accelerate the PINNs optimization process. We present a survey of model-agnostic metalearning, and then discuss our model-aware metalearning applied to PINNs as well as implementation considerations and algorithmic complexity. We then test our approach on various canonical forward parameterized PDEs that have been presented in the emerging PINNs literature

Analysis of Coarsening of Complex Structures.

Coarsening is an ubiquitous phenomenon that alters the microstructure of the material and its properties. While coarsening of spherical particles has been extensively studied over the last half century, the understanding of coarsening of complex microstructures is still at an early stage. The complex morphology and topology pose difficulty in establishing a theory of coarsening of such microstructures. In an effort to elucidate the dynamics of coarsening, we examine the morphological evolution of bicontinuous structures simulated using the phase-field method. To improve the accuracy of the calculation of interfacial characteristics of the simulated structures, we develop a smoothing algorithm termed ``level-set smoothing.'' We employ statistical analyses to uncover correlations between interfacial characteristics and their rate of changes. As the framework for the coarsening theory development, we propose to consider the evolution as a consequence of (i) the interfacial velocity induced by diffusion and (ii) the resulting evolution of the interfacial curvatures. As a first step, we examine the evolution of a bicontinuous structure simulated via nonconserved dynamics, in which the interfacial velocity is proportional to the local mean curvature, in order to focus on the second aspect of the evolution (ii). We find that, while the interfacial velocity is locally determined, the evolution of mean curvature is nonlocal and depends on the curvatures of the nearby interfaces. As a second step, we examine the evolution of bicontinuous structures simulated via conserved dynamics to investigate both aspects of the evolution, (i) and (ii). Here, we find that the interfacial velocity is correlated with both the mean curvature and the surface Laplacian of mean curvature. Based on these correlations, we employ a semi-analytical approach to predict the average rate of change of mean curvature, which is found to be consistent with the simulation results. Lastly, in an effort to develop a theory of coarsening of complex microstructures, we derive a general continuity equation of interfacial area to predict the evolution of the overall morphology of a microstructure undergoing coarsening. Simulation of rods undergoing pinching is also conducted to provide insights into the source term arising from topological singularity.PhDMaterials Science and EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/133292/1/challan_1.pd

Proceedings of the FEniCS Conference 2017

Proceedings of the FEniCS Conference 2017 that took place 12-14 June 2017 at the University of Luxembourg, Luxembourg