Freeform shape optimization of a compact dc photoelectron gun using isogeometric analysis

Compact dc high-voltage photoelectron guns are able to meet the sophisticated demands of high-current applications such as energy recovery linacs. A main design parameter for such sources is the electric field strength, which depends on the electrode geometry and is limited by the field emission threshold of the electrode material. In order to minimize the maximum field strength for optimal gun operation, isogeometric analysis (IGA) can be used to exploit the axisymmetric geometry and describe its cross section by nonuniform rational B-splines, the control points of which are the parameters to be optimized. This computationally efficient method is capable of describing CAD-generated geometries using open source software (geopdes, nlopt, octave) and it can simplify the step from design to simulation. We will present the mathematical formulation, the software workflow, and the results of an IGA-based shape optimization for a planned high-voltage upgrade of the dc photogun teststand Photo-CATCH at TU Darmstadt. The software builds on a general framework for isogeometric analysis and allows for easy adaptations to other geometries or quantities of interest. Simulations assuming a bias voltage of $-300\mathrm{kV}$ yielded maximum field gradients of $9.06\mathrm{MV}{\mathrm{m}}^{-1}$ on the surface of an inverted insulator electrode and below $3\mathrm{MV}{\mathrm{m}}^{-1}$ on the surface of the photocathode.

Figure 1

Basic components of the IIGG design in a longitudinal cross section of the vacuum chamber.

Figure 2

Different types of design optimization. From left to right: parameter, shape, and topology optimization.

Figure 3

Exemplary B-spline curves and their basis functions. The original knot vector is $\mathbf{\Xi }=({\mathbf{0}}_{1\times 3},0.3,0.5,{\mathbf{1}}_{1\times 3})$ and the control points are ${\mathbf{P}}_1=(0,1)$, ${\mathbf{P}}_2=(1,-1)$, ${\mathbf{P}}_3=(3,2)$, ${\mathbf{P}}_4=(5,4)$ and ${\mathbf{P}}_5=(7,1)$. (a) Original curve and basis functions. (b) Curve and basis functions after elevating the degree by 1. (c) Curve and basis functions after inserting knots at 0.5 and 0.7.

Figure 4

Original geometry and boundary conditions of the domain ${\mathbf{\Omega }}^{2\mathrm{D}}$. Grey lines indicate patch boundaries.

Figure 5

Electric field magnitude for the original [Fig. 5] and optimized [Fig. 5 according to (7) and Fig. 5 according to (9)] geometries. The plot representation uniformly divides each patch into 8 elements per coordinate direction ($n_{\mathrm{sub}}=8$) and the pop-outs zoom in on the triple point. Computed using geopdes.

Figure 6

Original curve and the least squares fit serving as the initial shape for the optimization.

Figure 7

Curves obtained from optimizations employing isres, cobyla and the modified objective function (9) respectively.

Figure 8

Potential distribution and magnitude of the electric field gradient along the outer insulator.

Figure 9

Convergence of maximum electric field magnitude and volume constraint with respect to curve degree and number of internal knots.

Figure 10

Comparison of electric field solutions computed using IGA (left) and linear tetrahedral finite elements (right). The results clearly showcase the advantages of IGA for obtaining smooth fields for tracking purposes.

Figure 11

Initial spatial distribution of the $N_{\mathrm{p}}=2^{11}$ macroparticles to be emitted from the cathode. The positions are sampled from a measurement of the diode laser in use at Photo-CATCH.

Figure 12

RMS beam widths and RMS beam length for the original and optimized geometries as obtained from astra using the reference parameters.

Figure 13

RMS beam widths in relation to part of the geometry, compare Fig. 4. The values are scaled up by a factor of 10 to indicate that the focussing effect of the electric field is sufficiently strong.