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We consider in this paper a mathematical and numerical model to design an industrial software solution able to handle real complex furnaces configurations in terms of geometries, atmospheres, parts positioning, heat generators and physical thermal phenomena. A three dimensional algorithm based on stabilized finite element methods (SFEM) for solving the momentum, energy, turbulence and radiation equations is presented. An immersed volume method (IVM) for thermal coupling of fluids and solids is introduced...
We compare numerical experiments from the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method, applied to three benchmark problems based on two different partial differential equations. Both methods are described in detail and we highlight some strengths and weaknesses of each method via the numerical comparisons. The two equations used in the benchmark problems are the viscous Burgers’ equation and the porous medium equation, both...
This paper deals with the mathematical and numerical analysis of a
simplified two-dimensional model for the interaction between the wind
and a sail. The wind is modeled as a steady irrotational plane flow past
the sail, satisfying the Kutta-Joukowski condition. This condition
guarantees that the flow is not singular at the trailing edge of the
sail. Although for the present analysis the position of the sail is
taken as data, the final aim of this research is to develop tools to
compute the sail...
This paper deals with the mathematical and numerical analysis of a
simplified two-dimensional model for the interaction between the wind
and a sail. The wind is modeled as a steady irrotational plane flow past
the sail, satisfying the Kutta-Joukowski condition. This condition
guarantees that the flow is not singular at the trailing edge of the
sail. Although for the present analysis the position of the sail is
taken as data, the final aim of this research is to develop tools to
compute the sail...
We study a two-grid scheme fully discrete in time and
space for solving the Navier-Stokes system. In the first step, the
fully non-linear problem is discretized in space on a coarse grid
with mesh-size H and time step k. In the second step, the
problem is discretized in space on a fine grid with mesh-size h
and the same time step, and linearized around the velocity uH
computed in the first step. The two-grid strategy is motivated by
the fact that under suitable assumptions, the contribution of
uH...
The newly developed unifying discontinuous formulation named the correction procedure via
reconstruction (CPR) for conservation laws is extended to solve the Navier-Stokes
equations for 3D mixed grids. In the current development, tetrahedrons and triangular
prisms are considered. The CPR method can unify several popular high order methods
including the discontinuous Galerkin and the spectral volume methods into a more efficient
differential form....
This paper is concerned with the problem of computing a small number of eigenvalues of large sparse generalized eigenvalue problems. The matrices arise from mixed finite element discretizations of time dependent equations modelling viscous incompressible flow. The eigenvalues of importance are those with smallest real part and are used to determine the linearized stability of steady states, and could be used in a scheme to detect Hopf bifurcations. We introduce a modified Cayley transform of the...
The fluctuation splitting schemes were introduced by Roe in the beginning of the
80's and have been then developed since then, essentially thanks to Deconinck.
In this paper, the fluctuation splitting
schemes formalism is recalled. Then, the hyperbolic/elliptic decomposition of the
three dimensional Euler equations is presented. This decomposition leads to an acoustic
subsystem and two scalar advection equations, one of them being the entropy advection.
Thanks to this decomposition, the two scalar...
A new error correction method for the stationary Navier-Stokes equations based on two local Gauss integrations is presented. Applying the orthogonal projection technique, we introduce two local Gauss integrations as a stabilizing term in the error correction method, and derive a new error correction method. In both the coarse solution computation step and the error computation step, a locally stabilizing term based on two local Gauss integrations is introduced. The stability and convergence of the...
We introduce a new stable MINI-element pair for incompressible Stokes equations on quadrilateral meshes, which uses the smallest number of bubbles for the velocity. The pressure is discretized with the P1-midpoint-edge-continuous elements and each component of the velocity field is done with the standard Q1-conforming elements enriched by one bubble a quadrilateral. The superconvergence in the pressure of the proposed pair is analyzed on uniform rectangular meshes, and tested numerically on uniform...
There are many problems of groundwater flow in a disrupted rock massifs that should be modelled using numerical models. It can be done via “standard approaches” such as increase of the permeability of the porous medium to account the fracture system (or double-porosity models), or discrete stochastic fracture network models. Both of these approaches appear to have their constraints and limitations, which make them unsuitable for the large- scale long-time hydrogeological calculations. In the article,...
A general construction of test functions in the Petrov-Galerkin method is described. Using this construction; algorithms for an approximate solution of the Dirichlet problem for the differential equation are presented and analyzed theoretically. The positive number is supposed to be much less than the discretization step and the values of . An algorithm for the corresponding two-dimensional problem is also suggested and results of numerical tests are introduced.
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