For Adoptees Only?

I adopted twins and stay in contact with the birth mother. She sees the babies and hugs and kisses them, but is very happy they are with us (she is a single mom and barely gets by with her children). Here is my question...Do you think this will be good for my babies to know their birthmother and two siblings, or upsetting? Should I have them call her Aunt or Miss or something like that? I do not and don't plan on inviting her to family functions (christening, birthday parties, holidays). Sometimes I am worried it will be hard for them to know her and know how much she loves them, but I feel so strongly that I shouldn't keep anything from them (I will ofcourse explain things age appropriately to them). Am I naive to think that we can all just get along and grow up happy and healthy babies.