Model hierarchies and higher-order discretisation of time-dependent thin-film free boundary problems with dynamic contact angle

We present a mathematical and numerical framework for the physical problem of thin-film fluid flows over planar surfaces including dynamic contact angles. In particular, we provide algorithmic details and an implementation of higher-order spatial and temporal discretisation of the underlying free boundary problem using the finite element method. The corresponding partial differential equation is based on a thermodynamic consistent energetic variational formulation of the problem using the free energy and viscous dissipation in the bulk, on the surface, and at the moving contact line. Model hierarchies for limits of strong and weak contact line dissipation are established, implemented and studied. We analyze the performance of the numerical algorithm and investigate the impact of the dynamic contact angle on the evolution of two benchmark problems: gravity-driven sliding droplets and the instability of a ridge

Variational Implementation of Immersed Finite Element Methods

Dirac-delta distributions are often crucial components of the solid-fluid coupling operators in immersed solution methods for fluid-structure interaction (FSI) problems. This is certainly so for methods like the Immersed Boundary Method (IBM) or the Immersed Finite Element Method (IFEM), where Dirac-delta distributions are approximated via smooth functions. By contrast, a truly variational formulation of immersed methods does not require the use of Dirac-delta distributions, either formally or practically. This has been shown in the Finite Element Immersed Boundary Method (FEIBM), where the variational structure of the problem is exploited to avoid Dirac-delta distributions at both the continuous and the discrete level. In this paper, we generalize the FEIBM to the case where an incompressible Newtonian fluid interacts with a general hyperelastic solid. Specifically, we allow (i) the mass density to be different in the solid and the fluid, (ii) the solid to be either viscoelastic of differential type or purely elastic, and (iii) the solid to be and either compressible or incompressible. At the continuous level, our variational formulation combines the natural stability estimates of the fluid and elasticity problems. In immersed methods, such stability estimates do not transfer to the discrete level automatically due to the non- matching nature of the finite dimensional spaces involved in the discretization. After presenting our general mathematical framework for the solution of FSI problems, we focus in detail on the construction of natural interpolation operators between the fluid and the solid discrete spaces, which guarantee semi-discrete stability estimates and strong consistency of our spatial discretization.Comment: 42 pages, 5 figures, Revision

A priori error estimates of regularized elliptic problems

Approximations of the Dirac delta distribution are commonly used to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. In this work we show a-priori rates of convergence of this approximation process in standard Sobolev norms, with minimal regularity assumptions on the approximation of the Dirac delta distribution. The application of these estimates to the numerical solution of elliptic problems with singularly supported forcing terms allows us to provide sharp H1 and L2 error estimates for the corresponding regularized problem. As an application, we show how finite element approximations of a regularized immersed interface method results in the same rates of convergence of its non-regularized counterpart, provided that the support of the Dirac delta approximation is set to a multiple of the mesh size, at a fraction of the implementation complexity. Numerical experiments are provided to support our theories

Adaptive finite element approximations for elliptic problems using regularized forcing data

We propose an adaptive finite element algorithm to approximate solutions of elliptic problems whose forcing data is locally defined and is approximated by regularization (or mollification). We show that the energy error decay is quasi-optimal in two dimensional space and sub-optimal in three dimensional space. Numerical simulations are provided to confirm our findings.Comment: 28 pages, 6 Figure

Error estimates in weighted Sobolev norms for finite element immersed interface methods

When solving elliptic partial differential equations in a region containing immersed interfaces (possibly evolving in time), it is often desirable to approximate the problem using a uniform background discretisation, not aligned with the interface itself. Optimal convergence rates are possible if the discretisation scheme is enriched by allowing the discrete solution to have jumps aligned with the surface, at the cost of a higher complexity in the implementation. A much simpler way to reformulate immersed interface problems consists in replacing the interface by a singular force field that produces the desired interface conditions, as done in immersed boundary methods. These methods are known to have inferior convergence properties, depending on the global regularity of the solution across the interface, when compared to enriched methods. In this work we prove that this detrimental effect on the convergence properties of the approximate solution is only a local phenomenon, restricted to a small neighbourhood of the interface. In particular we show that optimal approximations can be constructed in a natural and inexpensive way, simply by reformulating the problem in a distributionally consistent way, and by resorting to weighted norms when computing the global error of the approximation

Error estimates in weighted Sobolev norms for finite element immersed interface methods

When solving elliptic partial differential equations in a region containing immersed interfaces (possibly evolving in time), it is often desirable to approximate the problem using a uniform background discretisation, not aligned with the interface itself. Optimal convergence rates are possible if the discretisation scheme is enriched by allowing the discrete solution to have jumps aligned with the surface, at the cost of a higher complexity in the implementation. A much simpler way to reformulate immersed interface problems consists in replacing the interface by a singular force field that produces the desired interface conditions, as done in immersed boundary methods. These methods are known to have inferior convergence properties, depending on the global regularity of the solution across the interface, when compared to enriched methods. In this work we prove that this detrimental effect on the convergence properties of the approximate solution is only a local phenomenon, restricted to a small neighbourhood of the interface. In particular we show that optimal approximations can be constructed in a natural and inexpensive way, simply by reformulating the problem in a distributionally consistent way, and by resorting to weighted norms when computing the global error of the approximation

NURBS-SEM: A hybrid spectral element method on NURBS maps for the solution of elliptic PDEs on surfaces

Non Uniform Rational B-spline (NURBS) patches are a standard way to describe complex geometries in Computer Aided Design tools, and have gained a lot of popularity in recent years also for the approximation of partial differential equations, via the Isogeometric Analysis (IGA) paradigm. However, spectral accuracy in IGA is limited to relatively small NURBS patch degrees (roughly p 648), since local condition numbers grow very rapidly for higher degrees. On the other hand, traditional Spectral Element Methods (SEM) guarantee spectral accuracy but often require complex and expensive meshing techniques, like transfinite mapping, that result anyway in inexact geometries. In this work we propose a hybrid NURBS-SEM approximation method that achieves spectral accuracy and maintains exact geometry representation by combining the advantages of IGA and SEM. As a prototypical problem on non trivial geometries, we consider the Laplace\u2013Beltrami and Allen\u2013Cahn equations on a surface. On these problems, we present a comparison of several instances of NURBS-SEM with the standard Galerkin and Collocation Isogeometric Analysis (IGA)

A numerical study of the jerky crack growth in elastoplastic materials with localized plasticity

We present a numerical implementation of a model of quasi-static crack growth in linearly elastic-perfectly plastic materials. We assume that the displacement is antiplane, and that the cracks and the plastic slips are localized on a prescribed path. We provide numerical evidence of the fact that the crack growth is intermittent, with jump characteristics that depend on the material properties.Comment: 13 pages, 7 figure