Table 1 Parameters of the numerical experiments

space domain

radius of the cylinder Ω

2.5 mm

height of the cylinder Ω

0.5 mm

radius of the laser beam

0.2 mm

equation and boundary conditions

ambience temperature

\(\theta _{\text{amb}} = {295}\) K

convective heat transfer coefficient

\(h = 20\ \mathrm{W}/ \mathrm{m} ^{2}\)

radiative heat transfer coefficient

\(k = 2.26 \cdot 10^{-9}\ \mathrm{W}/ \mathrm{m} ^{2} \mathrm{K} ^{4}\)

objective functional

welding penetration penalty coefficient

\(\beta _{\text{penetration}} = 10^{-2}\)

solidification velocity penalty coefficient

\(\beta _{\text{velocity}} = 1.5 \cdot 10^{-1}\)

welding completeness penalty coefficient

\(\beta _{\text{completeness}} = 10^{-12}\)

energy consumption penalty coefficient

\(\beta _{\text{control}} = 10^{2}\)

target point

\(z_{\text{target}} = {0.125}\) mm

target maximal temperature at the target point

\(\theta _{\text{target}} = {1048}\) K

p-norm in time domain

p = 20

material properties

solidus point

858 K

liquidus point

923 K

enthalpy of fusion

\({397000}\ \mathrm{J}\ \mathrm{kg}^{-1}\)

coefficients s(θ) and κ(θ)

discussed in Sect. 2, see Fig. 2