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- finite-volume method (2)
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- Liquid Polymer Moulding (1)
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In this paper we consider numerical algorithms for solving a system of nonlinear PDEs arising in modeling of liquid polymer injection. We investigate the particular case when a porous preform is located within the mould, so that the liquid polymer flows through a porous medium during the filling stage. The nonlinearity of the governing system of PDEs is due to the non-Newtonian behavior of the polymer, as well as due to the moving free boundary. The latter is related to the penetration front and a Stefan type problem is formulated to account for it. A finite-volume method is used to approximate the given differential problem. Results of numerical experiments are presented. We also solve an inverse problem and present algorithms for the determination of the absolute preform permeability coefficient in the case when the velocity of the penetration front is known from measurements. In both cases (direct and inverse problems) we emphasize on the specifics related to the non-Newtonian behavior of the polymer. For completeness, we discuss also the Newtonian case. Results of some experimental measurements are presented and discussed.

This work presents a proof of convergence of a discrete solution to a continuous one. At first, the continuous problem is stated as a system
of equations which describe filtration process in the pressing section of a
paper machine. Two flow regimes appear in the modeling of this problem.
The model for the saturated flow is presented by the Darcy’s law and the mass conservation. The second regime is described by the Richards approach together with a dynamic capillary pressure model. The finite
volume method is used to approximate the system of PDEs. Then the existence of a discrete solution to proposed finite difference scheme is proven.
Compactness of the set of all discrete solutions for different mesh sizes is
proven. The main Theorem shows that the discrete solution converges
to the solution of continuous problem. At the end we present numerical
studies for the rate of convergence.

The performance of oil filters used in the automotive industry can be significantly improved, especially when computer simulation is an essential component of the design process. In this paper, we consider parallel numerical algorithms for solving mathematical models describing the process of filtration, filtering out solid particles from liquid oil. The Navier-Stokes-Brinkmann system of equations is used to describe the laminar flow of incompressible isothermal oil. The space discretization in the complicated filter geometry is based on the finite-volume method. Special care is taken for an accurate approximation of velocity and pressure on the interface between the fluid and the porous media. The time discretization used here is a proper modification of the fractional time step discretization (cf. Chorin scheme) of the Navier-Stokes equations, where the Brinkmann term is considered at both, prediction and correction substeps. A data decomposition method is used to develop a parallel algorithm, where the domain is distributed among processors by using a structured reference grid. The MPI library is used to implement the data communication part of the algorithm. A theoretical model is proposed for the estimation of the complexity of the given parallel algorithm and a scalability analysis is done on the basis of this model. Results of computational experiments are presented, and the accuracy and efficiency of the parallel algorithm is tested on real industrial geometries.