My answer above is misleading. Simcha is from the shoresh SIN MEM CHES as in שָׂמֵֽחַ and the spelling is שִׂמְחָ֥ה.
Smicha comes from the shoresh of SAMACH MEM CHAF as in וְסָמַ֣ךְ. The first time we see someone receive “smicha” is Yehoshua from Moshe (Bamidbar 27: 23): וַיִּסְמֹ֧ךְ אֶת־יָדָ֛יו עָלָ֖יו וַיְצַוֵּ֑הוּ. I believe that YU on their CHAG HASMICHAH invitations spells it SAMACH MEM YUD CHAF HAY.