1,706 research outputs found
Higher-order finite element methods for elliptic problems with interfaces
We present higher-order piecewise continuous finite element methods for
solving a class of interface problems in two dimensions. The method is based on
correction terms added to the right-hand side in the standard variational
formulation of the problem. We prove optimal error estimates of the methods on
general quasi-uniform and shape regular meshes in maximum norms. In addition,
we apply the method to a Stokes interface problem, adding correction terms for
the velocity and the pressure, obtaining optimal convergence results.Comment: 26 pages, 6 figures. An earlier version of this paper appeared on
November 13, 2014 in
http://www.brown.edu/research/projects/scientific-computing/reports/201
A cut finite element method for coupled bulk-surface problems on time-dependent domains
In this contribution we present a new computational method for coupled
bulk-surface problems on time-dependent domains. The method is based on a
space-time formulation using discontinuous piecewise linear elements in time
and continuous piecewise linear elements in space on a fixed background mesh.
The domain is represented using a piecewise linear level set function on the
background mesh and a cut finite element method is used to discretize the bulk
and surface problems. In the cut finite element method the bilinear forms
associated with the weak formulation of the problem are directly evaluated on
the bulk domain and the surface defined by the level set, essentially using the
restrictions of the piecewise linear functions to the computational domain. In
addition a stabilization term is added to stabilize convection as well as the
resulting algebraic system that is solved in each time step. We show in
numerical examples that the resulting method is accurate and stable and results
in well conditioned algebraic systems independent of the position of the
interface relative to the background mesh
A partition of unity approach to fluid mechanics and fluid-structure interaction
For problems involving large deformations of thin structures, simulating
fluid-structure interaction (FSI) remains challenging largely due to the need
to balance computational feasibility, efficiency, and solution accuracy.
Overlapping domain techniques have been introduced as a way to combine the
fluid-solid mesh conformity, seen in moving-mesh methods, without the need for
mesh smoothing or re-meshing, which is a core characteristic of fixed mesh
approaches. In this work, we introduce a novel overlapping domain method based
on a partition of unity approach. Unified function spaces are defined as a
weighted sum of fields given on two overlapping meshes. The method is shown to
achieve optimal convergence rates and to be stable for steady-state Stokes,
Navier-Stokes, and ALE Navier-Stokes problems. Finally, we present results for
FSI in the case of a 2D mock aortic valve simulation. These initial results
point to the potential applicability of the method to a wide range of FSI
applications, enabling boundary layer refinement and large deformations without
the need for re-meshing or user-defined stabilization.Comment: 34 pages, 15 figur
Multifluid flows with weak and strong discontinuous interfaces using an elemental enriched space
In a previous paper, the authors presented an elemental enriched space to be used in a finite-element framework (EFEM) capable of reproducing kinks and jumps in an unknown function using a fixed mesh in which the jumps and kinks do not coincide with the interelement boundaries. In this previous publication, only scalar transport problems were solved (thermal problems). In the present work, these ideas are generalized to vectorial unknowns, in particular, the incompressible Navier-Stokes equations for multifluid flows presenting internal moving interfaces. The advantage of the EFEM compared with global enrichment is the significant reduction in computing time when the internal interface is moving. In the EFEM, the matrix to be solved at each time step has not only the same amount of degrees of freedom (DOFs) but also the same connectivity between the DOFs. This frozen matrix graph enormously improves the efficiency of the solver. Another characteristic of the elemental enriched space presented here is that it allows a linear variation of the jump, thus improving the convergence rate, compared with other enriched spaces that have a constant variation of the jump. Furthermore, the implementation in any existing finite-element code is extremely easy with the version presented here because the new shape functions are based on the usual finite-element method shape functions for triangles or tetrahedrals, and once the internal DOFs are statically condensed, the resulting elements have exactly the same number of unknowns as the nonenriched finite elements.Peer ReviewedPreprin
A moving control volume approach to computing hydrodynamic forces and torques on immersed bodies
We present a moving control volume (CV) approach to computing hydrodynamic
forces and torques on complex geometries. The method requires surface and
volumetric integrals over a simple and regular Cartesian box that moves with an
arbitrary velocity to enclose the body at all times. The moving box is aligned
with Cartesian grid faces, which makes the integral evaluation straightforward
in an immersed boundary (IB) framework. Discontinuous and noisy derivatives of
velocity and pressure at the fluid-structure interface are avoided and
far-field (smooth) velocity and pressure information is used. We re-visit the
approach to compute hydrodynamic forces and torques through force/torque
balance equation in a Lagrangian frame that some of us took in a prior work
(Bhalla et al., J Comp Phys, 2013). We prove the equivalence of the two
approaches for IB methods, thanks to the use of Peskin's delta functions. Both
approaches are able to suppress spurious force oscillations and are in
excellent agreement, as expected theoretically. Test cases ranging from Stokes
to high Reynolds number regimes are considered. We discuss regridding issues
for the moving CV method in an adaptive mesh refinement (AMR) context. The
proposed moving CV method is not limited to a specific IB method and can also
be used, for example, with embedded boundary methods
Kernel-free boundary integral method for two-phase Stokes equations with discontinuous viscosity on staggered grids
A discontinuous viscosity coefficient makes the jump conditions of the
velocity and normal stress coupled together, which brings great challenges to
some commonly used numerical methods to obtain accurate solutions. To overcome
the difficulties, a kernel free boundary integral (KFBI) method combined with a
modified marker-and-cell (MAC) scheme is developed to solve the two-phase
Stokes problems with discontinuous viscosity. The main idea is to reformulate
the two-phase Stokes problem into a single-fluid Stokes problem by using
boundary integral equations and then evaluate the boundary integrals indirectly
through a Cartesian grid-based method. Since the jump conditions of the
single-fluid Stokes problems can be easily decoupled, the modified MAC scheme
is adopted here and the existing fast solver can be applicable for the
resulting linear saddle system. The computed numerical solutions are second
order accurate in discrete -norm for velocity and pressure as well as
the gradient of velocity, and also second order accurate in maximum norm for
both velocity and its gradient, even in the case of high contrast viscosity
coefficient, which is demonstrated in numerical tests
An adaptive fixed-mesh ALE method for free surface flows
In this work we present a Fixed-Mesh ALE method for the numerical simulation of free surface flows capable of using an adaptive finite element mesh covering a background domain. This mesh is successively refined and unrefined at each time step in order to focus the computational effort on the spatial regions where it is required. Some of the main ingredients of the formulation are the use of an Arbitrary-Lagrangian–Eulerian formulation for computing temporal derivatives, the use of stabilization terms for stabilizing convection, stabilizing the lack of compatibility between velocity and pressure interpolation spaces, and stabilizing the ill-conditioning introduced by the cuts on the background finite element mesh, and the coupling of the algorithm with an adaptive mesh refinement procedure suitable for running on distributed memory environments. Algorithmic steps for the projection between meshes are presented together with the algebraic fractional step approach used for improving the condition number of the linear systems to be solved. The method is tested in several numerical examples. The expected convergence rates both in space and time are observed. Smooth solution fields for both velocity and pressure are obtained (as a result of the contribution of the stabilization terms). Finally, a good agreement between the numerical results and the reference experimental data is obtained.Postprint (published version
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