The production of prompt D$^{0}$, D$^{+}$, and D$^{*+}$ mesons was measured at midrapidity (|y|< 0.5) in Pb-Pb collisions at the centre-of-mass energy per nucleon-nucleon pair $\sqrt{s_{\rm NN}}$ = 5.02 TeV with the ALICE detector at the LHC. The D mesons were reconstructed via their hadronic decay channels and their production yields were measured in central (0-10%) and semicentral (30-50%) collisions. The measurement was performed up to a transverse momentum ($p_{\rm T}$) of 36 or 50 GeV/$c$ depending on the D meson species and the centrality interval. For the first time in Pb-Pb collisions at the LHC, the yield of D$^0$ mesons was measured down to $p_{\rm T}$ = 0, which allowed a model-independent determination of the $p_{\rm T}$-integrated yield per unit of rapidity (d$N$/d$y$). A maximum suppression by a factor 5 and 2.5 was observed with the nuclear modification factor ($R_{\rm AA}$) of prompt D mesons at $p_{\rm T}$ = 6-8 GeV/$c$ for the 0-10% and 30-50% centrality classes, respectively. The D-meson $R_{\rm AA}$ is compared with that of charged pions, charged hadrons, and J/$\psi$ mesons as well as with theoretical predictions. The analysis of the agreement between the measured $R_{\rm AA}$, elliptic ($v_2$) and triangular ($v_3$) flow, and the model predictions allowed us to constrain the charm spatial diffusion coefficient $D_s$. Furthermore the comparison of $R_{\rm AA}$ and $v_2$ with different implementations of the same models provides an important insight into the role of radiative energy loss as well as charm quark recombination in the hadronisation mechanisms.
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