19,697 research outputs found
A consistent approach for fluid-structure-contact interaction based on a porous flow model for rough surface contact
Simulation approaches for fluid-structure-contact interaction, especially if
requested to be consistent even down to the real contact scenarios, belong to
the most challenging and still unsolved problems in computational mechanics.
The main challenges are twofold - one is to have a correct physical model for
this scenario, and the other one is to have a numerical method that is capable
of working and being consistent down to a zero gap. And when analyzing such
challenging setups of fluid-structure interaction that include contact of
submersed solid components it gets obvious that the influence of surface
roughness effects is essential for a physical consistent modeling of such
configurations. To capture this system behavior, we present a continuum
mechanical model which is able to include the effects of the surface
microstructure in a fluid-structure-contact interaction framework. An averaged
representation for the mixture of fluid and solid on the rough surfaces, which
is of major interest for the macroscopic response of such a system, is
introduced therein. The inherent coupling of the macroscopic fluid flow and the
flow inside the rough surfaces, the stress exchange of all contacting solid
bodies involved, and the interaction between fluid and solid is included in the
construction of the model. Although the physical model is not restricted to
finite element based methods, a numerical approach with its core based on the
Cut Finite Element Method (CutFEM), enabling topological changes of the fluid
domain to solve the presented model numerically, is introduced. Such a CutFEM
based approach is able to deal with the numerical challenges mentioned above.
Different test cases give a perspective towards the potential capabilities of
the presented physical model and numerical approach.Comment: 33 pages, 23 figure
An overlapping decomposition framework for wave propagation in heterogeneous and unbounded media: Formulation, analysis, algorithm, and simulation
A natural medium for wave propagation comprises a coupled bounded
heterogeneous region and an unbounded homogeneous free-space. Frequency-domain
wave propagation models in the medium, such as the variable coefficient
Helmholtz equation, include a faraway decay radiation condition (RC). It is
desirable to develop algorithms that incorporate the full physics of the
heterogeneous and unbounded medium wave propagation model, and avoid an
approximation of the RC. In this work we first present and analyze an
overlapping decomposition framework that is equivalent to the full-space
heterogeneous-homogenous continuous model, governed by the Helmholtz equation
with a spatially dependent refractive index and the RC. Our novel overlapping
framework allows the user to choose two free boundaries, and gain the advantage
of applying established high-order finite and boundary element methods (FEM and
BEM) to simulate an equivalent coupled model.
The coupled model comprises auxiliary interior bounded heterogeneous and
exterior unbounded homogeneous Helmholtz problems. A smooth boundary can be
chosen for simulating the exterior problem using a spectrally accurate BEM, and
a simple boundary can be used to setup a high-order FEM for the interior
problem. Thanks to the spectral accuracy of the exterior computational model,
the resulting coupled system in the overlapping region is relatively very
small. Using the decomposed equivalent framework, we develop a novel
overlapping FEM-BEM algorithm for simulating the acoustic or electromagnetic
wave propagation in two dimensions. Our FEM-BEM algorithm for the full-space
model incorporates the RC exactly. Numerical experiments demonstrate the
efficiency of the FEM-BEM approach for simulating smooth and non-smooth wave
fields, with the latter induced by a complex heterogeneous medium and a
discontinuous refractive index.Comment: 32 pages, 5 figures, 4 table
An inverse problem formulation of the immersed boundary method
We formulate the immersed-boundary method (IBM) as an inverse problem. A
control variable is introduced on the boundary of a larger domain that
encompasses the target domain. The optimal control is the one that minimizes
the mismatch between the state and the desired boundary value along the
immersed target-domain boundary. We begin by investigating a na\"ive problem
formulation that we show is ill-posed: in the case of the Laplace equation, we
prove that the solution is unique but it fails to depend continuously on the
data; for the linear advection equation, even solution uniqueness fails to
hold. These issues are addressed by two complimentary strategies. The first
strategy is to ensure that the enclosing domain tends to the true domain as the
mesh is refined. The second strategy is to include a specialized parameter-free
regularization that is based on penalizing the difference between the control
and the state on the boundary. The proposed inverse IBM is applied to the
diffusion, advection, and advection-diffusion equations using a high-order
discontinuous Galerkin discretization. The numerical experiments demonstrate
that the regularized scheme achieves optimal rates of convergence and that the
reduced Hessian of the optimization problem has a bounded condition number as
the mesh is refined
Spectral method for matching exterior and interior elliptic problems
A spectral method is described for solving coupled elliptic problems on an
interior and an exterior domain. The method is formulated and tested on the
two-dimensional interior Poisson and exterior Laplace problems, whose solutions
and their normal derivatives are required to be continuous across the
interface. A complete basis of homogeneous solutions for the interior and
exterior regions, corresponding to all possible Dirichlet boundary values at
the interface, are calculated in a preprocessing step. This basis is used to
construct the influence matrix which serves to transform the coupled boundary
conditions into conditions on the interior problem. Chebyshev approximations
are used to represent both the interior solutions and the boundary values. A
standard Chebyshev spectral method is used to calculate the interior solutions.
The exterior harmonic solutions are calculated as the convolution of the
free-space Green's function with a surface density; this surface density is
itself the solution to an integral equation which has an analytic solution when
the boundary values are given as a Chebyshev expansion. Properties of Chebyshev
approximations insure that the basis of exterior harmonic functions represents
the external near-boundary solutions uniformly. The method is tested by
calculating the electrostatic potential resulting from charge distributions in
a rectangle. The resulting influence matrix is well-conditioned and solutions
converge exponentially as the resolution is increased. The generalization of
this approach to three-dimensional problems is discussed, in particular the
magnetohydrodynamic equations in a finite cylindrical domain surrounded by a
vacuum
An Operator-Based Local Discontinuous Galerkin Method Compatible With the BSSN Formulation of the Einstein Equations
Discontinuous Galerkin Finite Element (DGFE) methods offer a mathematically
beautiful, computationally efficient, and efficiently parallelizable way to
solve hyperbolic partial differential equations. These properties make them
highly desirable for numerical calculations in relativistic astrophysics and
many other fields. The BSSN formulation of the Einstein equations has
repeatedly demonstrated its robustness. The formulation is not only stable but
allows for puncture-type evolutions of black hole systems. To-date no one has
been able to solve the full (3+1)-dimensional BSSN equations using DGFE
methods. This is partly because DGFE discretization often occurs at the level
of the equations, not the derivative operator, and partly because DGFE methods
are traditionally formulated for manifestly flux-conservative systems. By
discretizing the derivative operator, we generalize a particular flavor of DGFE
methods, Local DG methods, to solve arbitrary second-order hyperbolic
equations. Because we discretize at the level of the derivative operator, our
method can be interpreted as either a DGFE method or as a finite differences
stencil with non-constant coefficients.Comment: Incorporated referee comments during peer-review. Added several
references pointed out to us. Added a figure and discussion comparing
pointwise error for 4th- and 5th-order stencils for wave equatio
A Nystr\"om-based finite element method on polygonal elements
We consider families of finite elements on polygonal meshes, that are defined
implicitly on each mesh cell as solutions of local Poisson problems with
polynomial data. Functions in the local space on each mesh cell are evaluated
via Nystr\"om discretizations of associated integral equations, allowing for
curvilinear polygons and non-polynomial boundary data. Several experiments
demonstrate the approximation quality of interpolated functions in these
spaces
Spectral Methods for Partial Differential Equations in Irregular Domains: The Spectral Smoothed Boundary Method
In this paper, we propose a numerical method to approximate the solution of
partial differential equations in irregular domains with no-flux boundary
conditions by means of spectral methods. The main features of this method are
its capability to deal with domains of arbitrary shape and its easy
implementation via Fast Fourier Transform routines. We discuss several examples
of practical interest and test the results against exact solutions and standard
numerical methods.Comment: 15 pages, 11 figure
The topology of Helmholtz domains
The goal of this paper is to describe and clarify as much as possible the
3-dimensional topology underlying the Helmholtz cuts method, which occurs in a
wide theoretic and applied literature about Electromagnetism, Fluid dynamics
and Elasticity on domains of the ordinary space. We consider two classes of
bounded domains that satisfy mild boundary conditions and that become "simple"
after a finite number of disjoint cuts along properly embedded surfaces. For
the first class (Helmholtz), "simple" means that every curl-free smooth vector
field admits a potential. For the second (weakly-Helmholtz), we only require
that a potential exists for the restriction of every curl-free smooth vector
field defined on the whole initial domain. By means of classical and rather
elementary facts of 3-dimensional geometric and algebraic topology, we give an
exhaustive description of Helmholtz domains, realizing that their topology is
forced to be quite elementary (in particular, Helmholtz domains with connected
boundary are just possibly knotted handlebodies, and the complement of any
non-trivial link is not Helmholtz). The discussion about weakly-Helmholtz
domains is a bit more advanced, and their classification appears to be a quite
difficult issue. Nevertheless, we provide several interesting characterizations
of them and, in particular, we point out that the class of links with
weakly-Helmholtz complements eventually coincides with the class of the
so-called homology boundary links, that have been widely studied in Knot
Theory.Comment: 39 pages, 14 figure
Fluid-structure interaction with -conforming finite elements
In this paper a novel application of the (high-order)
-conforming Hybrid Discontinuous Galerkin finite element method
for monolithic fluid-structure interaction (FSI) is presented. The Arbitrary
Lagrangian Eulerian (ALE) description is derived for -conforming
finite elements including the Piola transformation, yielding exact divergence
free fluid velocity solutions. The arising method is demonstrated by means of
the benchmark problems proposed by Turek and Hron [50]. With hp-refinement
strategies singularities and boundary layers are overcome leading to optimal
spatial convergence rates
Unstable Spiral Waves and Local Euclidean Symmetry in a Model of Cardiac Tissue
This paper investigates the properties of unstable single-spiral wave
solutions arising in the Karma model of two-dimensional cardiac tissue. In
particular, we discuss how such solutions can be computed numerically on
domains of arbitrary shape and study how their stability, rotational frequency,
and spatial drift depend on the size of the domain as well as the position of
the spiral core with respect to the boundaries. We also discuss how the
breaking of local Euclidean symmetry due to finite size effects as well as the
spatial discretization of the model is reflected in the structure and dynamics
of spiral waves. This analysis allows identification of a self-sustaining
process responsible for maintaining the state of spiral chaos featuring
multiple interacting spirals.Comment: 17 pages, 14 figure
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