Performing simulations

Performing simulations

This help file contains only basic information; consult Sections 3-5 of tutorial if you wish to use all simulation and training capabilities of the code.

In order to start the simulations, you need to specify the near-cathode voltage (in volts) corresponding to the starting point, U₀, the near-cathode voltage (in volts) corresponding to the last point, U_fin, and the number N_U of steps between the starting and last points. You need also to indicate whether you want to use the solution corresponding to the last point as a starting point for further calculations. Go the tab Step 1, press buttons Reset and Submit, then the tab Step 2, set U = 15V, N_s = 2, and press the button Generate starting-point solution automatically. Go to the tab Step 3, and enter U₀ = 15, U_fin = 13, N_U = 2, Yes, respectively. Make sure that there is -1 in the field Number of the bifurcation point at this stage, N_b = -1, and press the button Start simulations. The following information appears in the window Program output:

U(V) I(A) Tcen(K) Tedge(K)

15.0000 20.49 3281 3419

14.0000 24.84 3329 3466

13.0000 31.49 3391 3528

The simulations have been completed.

You can see that there was 1 iteration for U = 15V, 4 iterations for U = 14V and 4 iterations for U = 13V. The arc current is 20.49A at U = 15V, 24.84A at U = 14V and 31.49A at U = 13V. Note that the role of an initial approximation at U = 15V was played by the starting-point solution, i.e., by an exact solution, which is why just one iteration was sufficient for convergence. The role of the initial approximation at each of the subsequent values of U was played by the preceding solution, so several iterations were necessary. The arc current I increases with decreasing U, so the current-voltage characteristic is falling as it should.

The following integral characteristics of the discharge for all values of U for which simulations were performed (15V, 14V, 13V) appear in the window Integral characteristics:

· arc current I,

· heat Qc removed by thermal conduction to the cathode base,

· heat Qrad irradiated by the cathode surface,

· temperatures Tcen and Tedge at the center and at the edge of the front surface of the cathode.

It may be a good idea to copy right now this information to your computer through COPY and PASTE commands.

Press the button View distributions. You can see in the window Distributions distributions along the cathode surface of the following parameters:

· surface temperature T_w,

· density j of the electric current from the plasma to the cathode surface,

· temperature T_e of electrons in the near-cathode plasma layer.

Note that these distributions represent boundary conditions required for differential equations describing the arc bulk and therefore allow you to use the code as a part of a numerical model of the arc on the whole. The distributions correspond to the last value of U calculated and are presented as functions of the distance r+z, where r and z are cylindrical coordinates with the origin at the center of the front surface of the cathode and with the z-axis directed from the front surface into the cathode body; see Figure 1 below. In accord with this choice, {r ≤ R, z = 0} is the (circular) front surface of the cathode while {r = R, z ≥ 0} is the (cylindrical) lateral surface (here R is the radius of the cathode, which is set equal to 1 mm in these calculations). Thus, the range 0 ≤ r+z ≤ R corresponds to the front surface while the range r+z ≥ R corresponds to the lateral surface. If you need these data, you can copy them to your computer through COPY and PASTE commands.

Figure 1. The system of coordinates.

Set the starting-point voltage U₀ equal to 13V, the last-point voltage U_fin equal to 7V, and the number N_U of steps between the starting and last points equal to 60. Press the button Start simulations. You can see that the iterations converged at U ≥ 10.1V and diverged at U = 10.0V. Hence, the minimum of the current-voltage characteristic of the diffuse mode is somewhere around 10.1V. Set U₀ = 13V, U_fin = 10.1V, N_U = 3, and perform the simulation. After it has been completed, an initial approximation for U = 10.1V is available. Now you can start simulations from U = 10.1V with a smaller step in U (0.01V, for example) in order to determine the minimum point with a higher precision.

The above-described simulations refer to the low-current section of the diffuse mode. Simulations on the high-current section are performed in a similar way. Go to the tab Step 2 and generate the starting-point solution for U = 12V on the high-current section with N_s = -20, then go to the tab Step 3, set U₀= 12V, U_fin = 13V, N_U = 1, and perform the simulation. The iterations will diverge. If, however, you set N_U = 4, then the iterations converge. This example shows that the step in U on the high-current section of the diffuse mode must in a general case be lower than on the low-current section. The reason is that a solution on the high-current section is more sensitive with respect to U than a solution on the low-current section.

The most difficult point in simulations of the spot mode is that if iterations diverge, you have no means to know what the reason is: the initial approximation may be inadequate, or the grid may be inadequate, or the spot mode may simply not exist under these conditions. The uncertainty may be reduced if an initial approximation is used that is not defined manually, but rather represents a solution describing the desired mode under close conditions. Then one can ensure that the initial approximation is good enough and can check whether the grid is adequate. If, nevertheless, the iterations do not converge, then the most likely reason is the inexistence of the solution being sought. Consult Section 4 of tutorial for a detailed description of finding multiple solutions describing different modes of current transfer.

In addition to the calculation of different axially symmetric steady-state modes of current transfer, the code offers the possibility of finding bifurcation points at which other modes, in particular, 3D spot modes, branch off from the axially symmetric modes. The importance of information on bifurcation points is two-fold. First, it facilitates finding 3D solutions. Second, a bifurcation point at which the steady-state 3D mode with a spot at the edge of the front surface branches off from the steady-state diffuse mode represents the limit of stability of the diffuse mode: the diffuse mode is stable at arc higher currents and unstable at lower currents.

Up to now, there was -1 in the field Number of the bifurcation point, N_b= -1, which signaled the code that no bifurcation point is sought and the relevant module of the code should not be executed. Now you should set N_b equal to the number of the bifurcation point which you want to find. If you want to find a bifurcation point at which a 3D mode with one, or two, or three etc spots at the edge of the front surface of the cathode branches off from the axially symmetric mode being considered, then you should set N_b equal to 1, or 2, or 3 etc. Set N_b = 0 in the case of a cathode with an electrically and thermally insulated lateral surface if you need to find a bifurcation point at which an axially symmetric spot mode branches off from the (1D) diffuse mode being considered or vice versa. If you are interested only in the limit of stability of the diffuse mode and not in branching of other modes with multiple spots, set N_b = 1.

If N_b = 0, or 1, or 2 etc, the output windows Program output and Integral characteristics comprise, in addition to the above-described information, also parameter Del, which is the determinant of a finite-difference problem corresponding to the differential eigenvalue problem for axially symmetric and 3D perturbations of the axially symmetric solution. The change of sign of Del signals the bifurcation point. Thus, in order to find the bifurcation point you should first choose an interval [U₀, U_fin] wide enough in order to ensure that Del does change sign inside this interval, and then localize this change of sign to a required accuracy. Note that the sign of Del on the diffuse mode normally changes from plus to minus as the arc current decreases. Consult section 5 of tutorial for details.