Transient convergence solutions

Verify that DC convergence has been achieved. View the error statements in the text editor to verify that the convergence problem pertains exclusively to the transient simulation. 

Verify that the time step provides an appropriate resolution. The time step must be small enough to provide appropriate resolution of the switching waveforms generated by the simulation. The time step should be assigned to an order of magnitude smaller than the shortest period in the simulation. For example, in a 100 kHz oscillator, the period is 10 /is. The time step should be set to 1 /is. 

Other factors such as the on time or the duty cycle should be considered when determining the time step. Once convergence has been achieved, this value can be maximized to reduce simulation time. 

Add UIC (Use Initial Conditions) to the. TRAN statement. This statement causes SPICE to bypass the DC operating point analysis. Initial conditions should be placed on capacitors at their expected operating voltage. Just as with the use of incorrect nodesets, incorrect initial condition values can produce incorrect solutions or nonconvergence. Results should be verified for validity. 

Set ITL4 = 500 in the. OPTIONS statement. This statement increases the number of iterations performed by SPICE, before a nonconvergence warning is issued and the simulation is aborted. 

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Although the fundamental equations are written for unsteady state conditions, the computational method presented here is not concerned with a quantitative prediction of the transient performance of the column. The unsteady state analysis is merely used to define a convergence path that corresponds to the transition from unsteady- to steady-state conditions. As the column moves toward steady state, the terms on the right-hand side of Equations 13.68 and 13.69 approach zero and the equations reduce to steady-state relationships. Thus, reaching steady state is equivalent to reaching a converged solution. This is commonly referred to as the relaxation method. 

A common feature in the models reviewed above was to calculate pressure and temperature distributions in a sequential procedure so that the interactions between temperature and other variables were ignored. It is therefore desirable to develop a numerical model that couples the solutions of pressure and temperature. The absence of such a model is mainly due to the excessive work required by the coupling computations and the difficulties in handling the numerical convergence problem. Wang et al. [27] combined the isothermal model proposed by Hu and Zhu [16,17] with the method proposed by Lai et al. for thermal analysis and presented a transient thermal mixed lubrication model. Pressure and temperature distributions are solved iteratively in a iterative loop so that the interactions between pressure and temperature can be examined. 

The time step Tstep = lOu determines each point in time starting from zero that the transient solver will calculate a solution. A safe estimation of the time step is an order of magnitude less than the period of a switching waveform. For example, the time step for a 100 kHz oscillator (period = 10 /xs) should be approximately 1 /xs. Tmax, the maximum time step, can be left out (at default) or specified to increase (decrease TMAX) or decrease (increase TMAX) simulation accuracy. This allows the simulator to take larger steps when the voltage levels in the circuit experience little change. A transient time domain analysis can prove to be the most difficult to get to converge. 

In Fig. 15.8 notice that during the time integration, the steady-state residuals increased for a period as the transient solution trajectory climbed over a hill and into the valley where the solution lies. This behavior is quite common in chemically reacting flow problems, especially when the initial starting estimates are poor. In fact it is not uncommon to see the transient solution path climb over many hills and valleys before coming to a point where the Newton method will begin to converge to the desired steady-state solution. 

From these examples it is apparent that one needs to be cautious when using steady-state methods and continuation procedures near turning points. While the solutions may converge rapidly and even appear to be physically reasonable, there can be significant errors. Fortunately, a relatively simple time-stepping procedure can be used to identify the nonphysical solutions. Beginning from any of the solutions that are shown in Fig. 15.9 as shaded diamonds, a transient stirred-reactor model can be solved. If the initial solution (i.e., initial condition for the transient problem) is nonphysical, the transient procedure will march toward the physical solution. If the initial condition is the physical solution, the transient computational will remain stationary at the correct solution. 

Introduction of the compressible-flow formulation, together with numerical implementation, leads to robust simulations for extremely fast transients. The time steps reduce appropriately to capture high-frequency details of the solution. Moreover there are essentially no convergence failures, indicating that the numerical method remains well conditioned even for extremely small time steps. This behavior demonstrates in practical terms that the system has been successfully reduced to index-one, confirming the analytical result. 

Here, x2 is the slow transient of X2 and x2 is the fast transient. Note that, for the corrected approximation in Equation (2.18) to converge rapidly to the slow approximative solution (2.15), the term x2 must decay as t —> oo to an 0(e) quantity. In the slow time scale t, this decay is fast, since... 

Recently, Douarche et al verified the transient ES FR and steady state ES FR for a harmonic oscillator (a brass pendulum in a water-glycerol solution, that is driven out of equilibrium by an applied torque). They also developed a steady state relation for a system with a sinusoidal forcing, and showed that the convergence time was considerably longer in this case. 

In this section we present said discuss a few numerical results for the two problems considered, transient flow said hmsient convection in microchannels, which were respectively handled by the full and the partial integral hmsformation approaches. The aim is to demonshate file convergence behavior within each strategy and to illushate some physical aspects on file fiansient phenomena at the micro-scale. Although the developed solutions sae readily applicable to dififerent physical situations of eifiier hquid or gas flow, we here concentrate om illustiation of results on typical ex ples of lamina gas slip flow. 

Figure 4.14 shows the transient solution for the two initial conditions Cao - 0 and caq = 2 moi/L. Notice both solutions converge to the same steady-state solution as time increases even though the starting conditions are quite different. We will see in Chapter 6 that the nonisothermal reactor behavior can be much more complex than that shown in Figure 4.14. ... 

The solution of the above equations can necessitate, for reasons of numerical convergence, the study of an associated transient problem. 

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