Theta time stepping

To carry out a numerical solution, a single strip of quadrilateral elements is placed along the x-axis, and all nodal temperatures are set Initially to zero. The right-hand boundary is then subjected to a step Increase in temperature (T(H,t) - 1.0), and we seek to compute the transient temperature variation T(x,t). The flow code accomplishes this by means of an unconditionally stable time-stepping algorithm derived from "theta" finite differences a solution of ten time steps required 22 seconds on a PC/AT-compatible microcomputer operating at 6 MHz. 

The code can treat transient problems by means a two-point "theta-method" time-stepping algorithm. The dynamic algorithm is also useful in nonlinear problems, in which the final fluid state may be approached dynamically from an estimated initial state. 

The dynamic equation of motion can be solved by numerical integration. The numerical integration calls for temporal discretization (i.e., system of coupled equations is discretized temporally), hence the term solution in time domain. The dynamic equation is solved by a time-stepping scheme. Examples of time-stepping schemes include Wilson s theta (0) algorithm (Clough and Penzien 1993) and several variations of the Newmark s p algorithms (Newmark 1959). 

In the last line the helper function print output (given below, but in the actual python script it needs to be defined before it is called) is called with the descriptor values and the final coverages. The variable theta is a (txc) matrix with the coverages c for each time step t. Usually, we are only interested in the last row, conveniently accessed by using —1 as index, but the time evolution is available as well for transient studies or verification that steady-state has been ... 

Fig. 3. X-ray diffractogram of Class-F bituminous coal fly ash. Analytical conditions diffraction data were collected using a Philips X-ray powder diffractometer (45 kV/30-40 mA CuKa theta-compensating variable divergence slit diffracted-beam graphite monochromator scintillation detector) automated with an MDI/Radix Databox. The scan parameters were typically 0.02° step size for 1 s count times over a range of 5-60° 2-theta. All data were analysed and displayed using a data reduction and display code (JADE) from Materials Data Inc., livermore, CA.

Recently, much more extensive adsorption rate studies have been carried out onto chrome from cyclohexane at the theta point and from benzene solutions. The times required to attain equilibrium with cyclohexane as the solvent were approximately independent of solution concentration for the high molecular weight material in the concentration range studied. This indicates molecular rearrangement on the surface as the rate controlling step rather than diffusion to the surface. This concept is consistent with the interpretations obtained from ellipsometry studies. 

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