10,540 research outputs found
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
Mechanics-based solution verification for porous media models
This paper presents a new approach to verify accuracy of computational
simulations. We develop mathematical theorems which can serve as robust a
posteriori error estimation techniques to identify numerical pollution, check
the performance of adaptive meshes, and verify numerical solutions. We
demonstrate performance of this methodology on problems from flow thorough
porous media. However, one can extend it to other models. We construct
mathematical properties such that the solutions to Darcy and Darcy-Brinkman
equations satisfy them. The mathematical properties include the total minimum
mechanical power, minimum dissipation theorem, reciprocal relation, and maximum
principle for the vorticity. All the developed theorems have firm mechanical
bases and are independent of numerical methods. So, these can be utilized for
solution verification of finite element, finite volume, finite difference,
lattice Boltzmann methods and so forth. In particular, we show that, for a
given set of boundary conditions, Darcy velocity has the minimum total
mechanical power of all the kinematically admissible vector fields. We also
show that a similar result holds for Darcy-Brinkman velocity. We then show for
a conservative body force, the Darcy and Darcy-Brinkman velocities have the
minimum total dissipation among their respective kinematically admissible
vector fields. Using numerical examples, we show that the minimum dissipation
and total mechanical power theorems can be utilized to identify pollution
errors in numerical solutions. The solutions to Darcy and Darcy-Brinkman
equations are shown to satisfy a reciprocal relation, which has the potential
to identify errors in the numerical implementation of boundary conditions
Overview of the Incompressible Navier-Stokes simulation capabilities in the MOOSE Framework
The Multiphysics Object Oriented Simulation Environment (MOOSE) framework is
a high-performance, open source, C++ finite element toolkit developed at Idaho
National Laboratory. MOOSE was created with the aim of assisting domain
scientists and engineers in creating customizable, high-quality tools for
multiphysics simulations. While the core MOOSE framework itself does not
contain code for simulating any particular physical application, it is
distributed with a number of physics "modules" which are tailored to solving
e.g. heat conduction, phase field, and solid/fluid mechanics problems. In this
report, we describe the basic equations, finite element formulations, software
implementation, and regression/verification tests currently available in
MOOSE's navier_stokes module for solving the Incompressible Navier-Stokes (INS)
equations.Comment: 54 pages, 16 figures, includes peer reviewer revision
Discrete models for fluid-structure interactions: the Finite Element Immersed Boundary Method
The aim of this paper is to provide a survey of the state of the art in the
finite element approach to the Immersed Boundary Method (FE-IBM) which has been
investigated by the authors during the last decade. In a unified setting, we
present the different formulation proposed in our research and highlight the
advantages of the one based on a distributed Lagrange multiplier (DLM-IBM) over
the original FE-IBM
A staggered semi-implicit hybrid FV/FE projection method for weakly compressible flows
In this article we present a novel staggered semi-implicit hybrid
finite-volume/finite-element (FV/FE) method for the resolution of weakly
compressible flows in two and three space dimensions. The pressure-based
methodology introduced in Berm\'udez et al. 2014 and Busto et al. 2018 for
viscous incompressible flows is extended here to solve the compressible
Navier-Stokes equations. Instead of considering the classical system including
the energy conservation equation, we replace it by the pressure evolution
equation written in non-conservative form. To ease the discretization of
complex spatial domains, face-type unstructured staggered meshes are
considered. A projection method allows the decoupling of the computation of the
density and linear momentum variables from the pressure. Then, an explicit
finite volume scheme is used for the resolution of the transport diffusion
equations on the dual mesh, whereas the pressure system is solved implicitly by
using continuous finite elements defined on the primal simplex mesh.
Consequently, the CFL stability condition depends only on the flow velocity,
avoiding the severe time restrictions that might be imposed by the sound
velocity in the weakly compressible regime. High order of accuracy in space and
time of the transport diffusion stage is attained using a local ADER (LADER)
methodology. Moreover, also the CVC Kolgan-type second order in space and first
order in time scheme is considered. To prevent spurious oscillations in the
presence of shocks, an ENO-based reconstruction, the minmod limiter or the
Barth-Jespersen limiter are employed. To show the validity and robustness of
our novel staggered semi-implicit hybrid FV/FE scheme, several benchmarks are
analysed, showing a good agreement with available exact solutions and numerical
reference data from low Mach numbers, up to Mach numbers of the order of unity
Compatible finite element spaces for geophysical fluid dynamics
Compatible finite elements provide a framework for preserving important
structures in equations of geophysical fluid dynamics, and are becoming
important in their use for building atmosphere and ocean models. We survey the
application of compatible finite element spaces to geophysical fluid dynamics,
including the application to the nonlinear rotating shallow water equations,
and the three-dimensional compressible Euler equations. We summarise analytic
results about dispersion relations and conservation properties, and present new
results on approximation properties in three dimensions on the sphere, and on
hydrostatic balance properties.Comment: Revision to version published on DSC
Automatic Variationally Stable Analysis for FE Computations: An Introduction
We introduce an automatic variationally stable analysis (AVS) for finite
element (FE) computations of scalar-valued convection-diffusion equations with
non-constant and highly oscillatory coefficients. In the spirit of least
squares FE methods, the AVS-FE method recasts the governing second order
partial differential equation (PDE) into a system of first-order PDEs. However,
in the subsequent derivation of the equivalent weak formulation, a
Petrov-Galerkin technique is applied by using different regularities for the
trial and test function spaces. We use standard FE approximation spaces for the
trial spaces, which are C0, and broken Hilbert spaces for the test functions.
Thus, we seek to compute pointwise continuous solutions for both the primal
variable and its flux (as in least squares FE methods), while the test
functions are piecewise discontinuous. To ensure the numerical stability of the
subsequent FE discretizations, we apply the philosophy of the discontinuous
Petrov-Galerkin (DPG) method by Demkowicz and Gopalakrishnan, by invoking test
functions that lead to unconditionally stable numerical systems (if the kernel
of the underlying differential operator is trivial). In the AVS-FE method, the
discontinuous test functions are ascertained per the DPG approach from local,
decoupled, and well-posed variational problems, which lead to best
approximation properties in terms of the energy norm. We present various 2D
numerical verifications, including convection-diffusion problems with highly
oscillatory coefficients and extremely high Peclet numbers, up to a billion.
These show the unconditional stability without the need for any upwind schemes
nor any other artificial numerical stabilization. The results are not highly
diffused for convection-dominated problems ...Comment: Preprint submitted to Lecture Notes in Computational Science and
Engineering, Springer Verla
A novel divergence-free Finite Element Method for the MHD Kinematics equations using Vector-potential
We propose a new mixed finite element method for the three-dimensional steady
magnetohydrodynamic (MHD) kinematics equations for which the velocity of the
fluid is given. Although prescribing the velocity field leads to a simpler
model than full MHD equations, its conservative and efficient numerical methods
are still active research topic. The distinctive feature of our discrete scheme
is that the divergence-free conditions for current density and magnetic
induction are both satisfied. To reach this goal, we use magnetic vector
potential to represent magnetic induction and resort to H(div)-conforming
element to discretize the current density. We develop an preconditioned
iterative solver based on a block preconditioner for the algebraic systems
arising from the discretization. Several numerical experiments are implemented
to verify the divergence-free properties, the convergence rate of the finite
element scheme and the robustness of the preconditioner.Comment: 16 pages, 6 figure
Modification to Darcy model for high pressure and high velocity applications and associated mixed finite element formulations
The Darcy model is based on a plethora of assumptions. One of the most
important assumptions is that the Darcy model assumes the drag coefficient to
be constant. However, there is irrefutable experimental evidence that
viscosities of organic liquids and carbon-dioxide depend on the pressure.
Experiments have also shown that the drag varies nonlinearly with respect to
the velocity at high flow rates. In important technological applications like
enhanced oil recovery and geological carbon-dioxide sequestration, one
encounters both high pressures and high flow rates. It should be emphasized
that flow characteristics and pressure variation under varying drag are both
quantitatively and qualitatively different from that of constant drag.
Motivated by experimental evidence, we consider the drag coefficient to depend
on both the pressure and velocity. We consider two major modifications to the
Darcy model based on the Barus formula and Forchheimer approximation. The
proposed modifications to the Darcy model result in nonlinear partial
differential equations, which are not amenable to analytical solutions. To this
end, we present mixed finite element formulations based on least-squares
formalism and variational multiscale formalism for the resulting governing
equations. The proposed modifications to the Darcy model and its associated
finite element formulations are used to solve realistic problems with relevance
to enhanced oil recovery. We also study the competition between the nonlinear
dependence of drag on the velocity and the dependence of viscosity on the
pressure. To the best of the authors' knowledge such a systematic study has not
been performed
A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier-Stokes equations
In this work we present a mimetic spectral element discretization for the 2D
incompressible Navier-Stokes equations that in the limit of vanishing
dissipation exactly preserves mass, kinetic energy, enstrophy and total
vorticity on unstructured grids. The essential ingredients to achieve this are:
(i) a velocity-vorticity formulation in rotational form, (ii) a sequence of
function spaces capable of exactly satisfying the divergence free nature of the
velocity field, and (iii) a conserving time integrator. Proofs for the exact
discrete conservation properties are presented together with numerical test
cases on highly irregular grids
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