Do you mean how to get the ID or the title of the artist of the current post from the relationship field instead of using a hard coded name like in your example ‘natasha_yudina’ or you want to show only post from ‘natasha_yudina’?
Either way you are using compare the wrong way I guess. This is purely a wordpress question.
Change this:
array(
'key' => 'artist', // name of custom field
'value' => 'name', // matches exaclty "123", not just 123.This prevents a match for "1234"
'compare' => 'natasha_yudina'
)
To this:
array(
'key' => 'artist', // name of custom field
'value' => 'natasha_yudina' // value of custom field
)
