52 research outputs found
Error analysis of energy-conservative BDF2-FE scheme for the 2D Navier-Stokes equations with variable density
In this paper, we present an error estimate of a second-order linearized
finite element (FE) method for the 2D Navier-Stokes equations with variable
density. In order to get error estimates, we first introduce an equivalent form
of the original system. Later, we propose a general BDF2-FE method for solving
this equivalent form, where the Taylor-Hood FE space is used for discretizing
the Navier-Stokes equations and conforming FE space is used for discretizing
density equation. We show that our scheme ensures discrete energy dissipation.
Under the assumption of sufficient smoothness of strong solutions, an error
estimate is presented for our numerical scheme for variable density
incompressible flow in two dimensions. Finally, some numerical examples are
provided to confirm our theoretical results.Comment: 22 pages, 1 figure
A high-order artificial compressibility method based on Taylor series time-stepping for variable density flow
In this paper, we introduce a fourth-order accurate finite element method for
incompressible variable density flow. The method is implicit in time and
constructed with the Taylor series technique, and uses standard high-order
Lagrange basis functions in space. Taylor series time-stepping relies on time
derivative correction terms to achieve high-order accuracy. We provide detailed
algorithms to approximate the time derivatives of the variable density
Navier-Stokes equations. Numerical validations confirm a fourth-order accuracy
for smooth problems. We also numerically illustrate that the Taylor series
method is unsuitable for problems where regularity is lost by solving the 2D
Rayleigh-Taylor instability problem
Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem
The thermally coupled incompressible inductionless magnetohydrodynamics (MHD) problem models the ow of an electrically charged fuid under the in uence of an external electromagnetic eld with thermal coupling. This system of partial di erential equations is strongly coupled and highly nonlinear for real cases of interest. Therefore, fully implicit time integration schemes are very desirable in order to capture the di erent physical scales of the problem at hand. However, solving the multiphysics linear systems of equations resulting from such algorithms is a very challenging task
which requires e cient and scalable preconditioners. In this work, a new family of recursive block LU preconditioners is designed and tested for solving the thermally coupled inductionless MHD equations. These preconditioners are obtained after splitting the fully coupled matrix into one-physics problems for every variable (velocity, pressure,
current density, electric potential and temperature) that can be optimally solved, e.g., using preconditioned domain decomposition algorithms. The main idea is to arrange the original matrix into an (arbitrary) 2 2 block matrix, and consider a LU preconditioner obtained by approximating the corresponding Schur complement. For every one
of the diagonal blocks in the LU preconditioner, if it involves more than one type of unknown, we proceed the same way in a recursive fashion. This approach is stated in an abstract way, and can be straightforwardly applied to other multiphysics problems. Further, we precisely explain a fexible and general software design for the code implementation of this type of preconditioners.Preprin
An efficient unconditional energy stable scheme for the simulation of droplet formation
We have developed an efficient and unconditionally energy-stable method for
simulating droplet formation dynamics. Our approach involves a novel
time-marching scheme based on the scalar auxiliary variable technique,
specifically designed for solving the Cahn-Hilliard-Navier-Stokes phase field
model with variable density and viscosity. We have successfully applied this
method to simulate droplet formation in scenarios where a Newtonian fluid is
injected through a vertical tube into another immiscible Newtonian fluid. To
tackle the challenges posed by nonhomogeneous Dirichlet boundary conditions at
the tube entrance, we have introduced additional nonlocal auxiliary variables
and associated ordinary differential equations. These additions effectively
eliminate the influence of boundary terms. Moreover, we have incorporated
stabilization terms into the scheme to enhance its numerical effectiveness.
Notably, our resulting scheme is fully decoupled, requiring the solution of
only linear systems at each time step. We have also demonstrated the energy
decaying property of the scheme, with suitable modifications. To assess the
accuracy and stability of our algorithm, we have conducted extensive numerical
simulations. Additionally, we have examined the dynamics of droplet formation
and explored the impact of dimensionless parameters on the process. Overall,
our work presents a refined method for simulating droplet formation dynamics,
offering improved efficiency, energy stability, and accuracy
An Hybrid Finite Volume-Finite Element Method for Variable Density Incompressible Flows
International audienceThis paper is devoted to the numerical simulation of variable density incompressible flows, modeled by the Navier-Stokes system. We introduce an hybrid scheme which combines a Finite Volume approach for treating the mass conservation equation and a Finite Element method to deal with the momentum equation and the divergence free constraint. The breakthrough relies on the definition of a suitable footbridge between the two methods, through the design of compatibility condition. In turn, the method is very flexible and allows to deal with unstructured meshes. Several numerical tests are performed to show the scheme capabilities. In particular, the viscous Rayleigh-Taylor instability evolution is carefully investigate
3-D Computational Investigation of Viscoelastic Biofilms using GPUs
A biofilm is a slimy colony of bacteria and the materials they secrete, collectively called “extracellular polymeric substances (EPS)”. The EPS consists mostly of bio-polymers, which cross link into a network that behave viscoelastically under deformation. We propose a single-fluid multi-component phase field model of biofilms that captures this behavior, then use numerical simulations on GPUs to investigate the biofilm’s growth and its hydrodynamics properties
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