98 research outputs found
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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
Reduced Lagrange multiplier approach for non-matching coupling of mixed-dimensional domains
Many physical problems involving heterogeneous spatial scales, such as the
flow through fractured porous media, the study of fiber-reinforced materials,
or the modeling of the small circulation in living tissues -- just to mention a
few examples -- can be described as coupled partial differential equations
defined in domains of heterogeneous dimensions that are embedded into each
other. This formulation is a consequence of geometric model reduction
techniques that transform the original problems defined in complex
three-dimensional domains into more tractable ones. The definition and the
approximation of coupling operators suitable for this class of problems is
still a challenge. We develop a general mathematical framework for the analysis
and the approximation of partial differential equations coupled by non-matching
constraints across different dimensions, focusing on their enforcement using
Lagrange multipliers. In this context, we address in abstract and general terms
the well-posedness, stability, and robustness of the problem with respect to
the smallest characteristic length of the embedded domain. We also address the
numerical approximation of the problem and we discuss the inf-sup stability of
the proposed numerical scheme for some representative configuration of the
embedded domain. The main message of this work is twofold: from the standpoint
of the theory of mixed-dimensional problems, we provide general and abstract
mathematical tools to formulate coupled problems across dimensions. From the
practical standpoint of the numerical approximation, we show the interplay
between the mesh characteristic size, the dimension of the Lagrange multiplier
space, and the size of the inclusion in representative configurations
interesting for applications. The latter analysis is complemented with
illustrative numerical examples.Comment: 41 pages, 11 Figure
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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)
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