Fixed Domain Method Research Articles

Prediction of blast furnace hearth condition: part II – a transient state simulation of hearth condition during blast furnace shutdown

ABSTRACT The extent of hearth cooling is critically important in determining the feasible duration of an extended blast furnace maintenance shutdown. Based on a fixed domain method, a transient numerical model has been developed which considers the effect of solidification enthalpy of liquid iron in a coke-free bath, enabling the progress of hearth cooling to be monitored. The model was initially verified by comparing calculations with actual refractory temperatures during shutdowns of shorter duration (typically 1–2 days) at BlueScope’s No. 5 Blast Furnace. It was then applied to estimate the hearth condition during an extended shutdown period (typically, 5–6 days). The calculated refractory temperature differences were reasonably matched with the measured data over the extended shutdown period. This allowed visualization of the temporal variation of refractory temperatures and the extent of liquid bath cooling during extended shutdowns, providing useful guidance for furnace engineers.

Signal transduction in the invadopodia formation using fixed domain method

The degradation of the extracellular matrix is driven by actin-based protrusions known as invadopodia, lead to the cancer cell invasion across the surrounding tissue barriers. Signal transduction through the binding between ligand and membrane associated receptor is important in order to establish the actin polymerization, and consequently push the membrane of migrating cells. In this study, we considered one-dimensional Stefan-like problem of the signal transduction and cell membrane is treated as a free boundary surface to separate any activity that happened on intra- and extra-cellular regions. The velocity concerning the movement of the free boundary is calculated by the decrease of signal gradient on the front. The problem is solved numerically using finite-difference scheme of fixed-domain method. Our results showed that both free boundary positions and signal distributions are increasing as time progresses.

Numerical Model Of Binary Alloys Solidification Basing On The One Domain Approach And The Simple Macrosegregation Models

AbstractIn the paper the thermal processes proceeding in the domain of solidifying binary alloy are considered. The mathematical model of solidification and cooling processes bases on the one domain method (or fixed domain method). In such a model the parameter called a substitute thermal capacity (STC) appears. At the stage of STC construction the macrosegregation process described by the lever arm rule or the Scheil model is taken into account. In this way one obtains the formulas determining the course of STC resulting from the certain physical considerations and this approach seems to be closer to the real course of thermal processes proceeding in domain of solidifying alloy. In the final part the examples of numerical solutions basing on the finite difference method are presented.

A level set simulation of dendritic solidification with combined features of front-tracking and fixed-domain methods

A method combining features of front-tracking methods and fixed-domain methods is presented to model dendritic solidification of pure materials. To explicitly track the interface growth and shape of the solidifying crystals, a front-tracking approach based on the level set method is implemented. To easily model the heat and momentum transport, a fixed-domain method is implemented assuming a diffused freezing front where the liquid fraction is defined in terms of the level set function. The fixed-domain approach, by avoiding the explicit application of essential boundary conditions on the freezing front, leads to an energy conserving methodology that is not sensitive to the mesh size. To compute the freezing front morphology, an extended Stefan condition is considered. Applications to several classical Stefan problems and two- and three-dimensional crystal growth of pure materials in an undercooled melt including the effects of melt flow are considered. The computed results agree very well with available analytical solutions as well as with results obtained using front-tracking techniques and the phase-field method.

Impingement of an inviscid plane liquid jet on a porous wall: a fixed domain method

Impingement of an inviscid plane liquid jet on a porous wall: a fixed domain method

A numerical approach to the asymptotic behavior of solutions of a one-dimensional free boundary problem of hyperbolic type

A free boundary problem which arises from the physical model called “peeling” will be analyzed numerically. To obtain an equation which describes the motion of the free boundary, the fixed domain method is applied. By using the equation, numerical computations are carried out. Our numerical computations suggest that the peeling speed plays an important role in the existence of global solutions.

Optimization methods applied to material and flextensional actuator design using the homogenization method

The design method based on the extended fixed domain method and global-local modeling (or homogenization) theory is applied to the design of elastic, thermoelastic, and piezoelectric composite materials, and flextensional actuators. rights reserved.

Existence of solution for a free boundary problem in a nonlinear piecewise homogeneous medium

A free boundary problem arising from the bidimensional thermal modelling of aluminium electrolytic cells is studied. The medium is assumed piecewise homogeneous and nonlinear. A fixed domain method is proposed which leads to a weak formulation of the problem. Existence of weak solution is proved by regularizing the contact condition between the homogeneous subdomains and passing to the limit.

A Fixed Domain Method for Injection Governed by the Stokes Equations

In this paper, the authors describe a fixed domain method to simulate the injection of a fluid governed by the Stokes equations into a partially filled cell. The Baiocchi transformation of the pressure and velocity results in equations which hold in the entire cell and which implicitly enforce the conditions on the moving boundary. A finite element discretization is applied to this transformed problem, and the resulting discrete system is solved by a successive approximation iteration on the fractional volume content of fluid in each element. For a two-dimensional channel, transient and steadily advancing fluid fronts (known as the “fountain-flow” problem) computed by this method are compared with those computed by some front-tracking algorithms. New examples, not found in the literature, that involve turning and internal obstacles are also presented.

Numerical solution of a three-dimensional thermoelectric problem taking place in an aluminium electrolytic cell

In this paper we solve a free boundary problem for a system of partial differential equations modelling the thermoelectrical behaviour of an aluminium electrolytic cell. A fixed domain method is proposed which relies upon a weak formulation of this system. A discretisation is introduced by using pentahedral finite elements of six degrees of freedom. To solve the discretised system an iterative algorithm is proposed. Numerical results are given both for a test example and a real industrial situation.

Solution of a problem of nonlinear potential theory by the fixed domain method

A solution is derived for a one-dimensional boundary-value problem of nonlinear potential theory with one free end with a supplementary boundary condition. The problem is solved by a variant of the fixed domain method. In each fixed domain, the problem is reduced to the corresponding nonlinear difference problem, which is solved by Newton's method.

Approximation of a nonlinear thermoelastic problem with a moving boundary via a fixed-domain method

The thermoelastic stresses created in a solid phase domain in the course of solidification of a molten ingot are investigated. A nonlinear behaviour of the solid phase is admitted, too. This problem, obtained from a real situation by many simplifications, contains a moving boundary between the solid and the liquid phase domains. To make the usage of standard numerical packages possible, we propose here a fixed-domain approximation by means of including the liquid phase domain into the problem (in this way we get the fixed domain involving the whole ingot) and by replacing the liquid phase with a solid phase having, however, a small shear modulus. The weak $L^2$-convergence of thus approximated stresses in the solid phase domain is demonstrated. Besides, this convergence is shown to be strong on subsets whose closure belongs to the solid phase domain.

The unsteady plane flow of ice-sheets: A parabolic problem with two moving boundaries

Abstract Finite difference algorithms have been developed to solve a one-dimensional non-linear parabolic equation with one or two moving boundaries and to analyse the unsteady plane flow of ice-sheets. They are designed to investigate the response of an ice-sheet to changes in climate, and to reconstruct climatic changes implied by past ice-sheet variations inferred from glacial geological data. Two algorithms are presented and compared. The first, a fixed domain method, replaces time as an independent variable with span. The grid interval in real space is kept constant, and thus the number of grid points changes with span. The second, a moving mesh method, retains time as one of the independent variables, but normalises the spatial variable relative to the span, which now enters the diffusion and advection coeficients in the parabolic equation for the surface profile. Crank-Nicholson schemes for the solution of the equations are constructed, and iterative schemes for the solution of the resulting non-linear equations are considered. Boundary (margin) motion is governed by the surface slope at the margin. Differentiation of the evolution equations results in an evolution equation for the margin slopes. It is shown that incorporation of this evolution equation, while not formally increasing the accuracy of the finite difference schemes, in practice increases accuracy of the solution.

On the solution of transient free-surface flow problems in porous media by a fixed-domain method : Lin, C S; Bruch, J C; Sloss, J M; Comincioli, V J HydrolV71, N3–4, 25 March 1984, P353–376

On the solution of transient free-surface flow problems in porous media by a fixed-domain method : Lin, C S; Bruch, J C; Sloss, J M; Comincioli, V J HydrolV71, N3–4, 25 March 1984, P353–376

On the solution of transient free-surface flow problems in porous media by a fixed-domain method

The numerical model presented here is the solution to the exact initial-boundary-value problems arising in recharge fluid flow through porous media having a free surface. Since the problem is nonlinear and includes a moving boundary, a numerical solution is obtained by using a fixed-domain approach. The Baiocchi transformation and method are used to develop a boundary-value problem which is then solved by an iterative method of successive over-relaxation type. Transient free-surface seepage through a two-dimensional dam with accretion is presented as an example problem. The effects of hydraulic conductivity, specific storativity and accretion on the seepage flow are studied. Two modified cases are satisfactorily compared with published results.

A fixed domain method for diffusion with a moving boundary

The one-dimensional diffusion equation for a region with one fixed boundary and one unknown moving boundary is transformed to a non-linear equation on a fixed region by using the moving boundary position as the time variable. The boundary velocity becomes a second dependent variable, with dependence only on the new time variable. An implicit finite difference scheme, marching in time, is applied to a problem with known analytic solution to demonstrate the computing speed and accuracy of this approach, and also to a problem solved previously by variable time step methods. This transformation reduces any parabolic or elliptic system of equations on a domain with moving boundary, or with unknown free surface in two space variables, to a non-linear fixed domain system which has advantages for computation.