Database
[{"_id":ObjectId("5a934e000102030405000000"),"product_name":"A","product_type":[{"name":"Metal"},{"name":"Wood"},{"name":"Carbon"}]},{"_id":ObjectId("5a934e000102030405000001"),"product_name":"B","product_type":[{"name":"Metal"},{"name":"Iron"}]},{"_id":ObjectId("5a934e000102030405000002"),"product_name":"C","product_type":[{"name":"Metal"},{"name":"Wood"}]},{"_id":ObjectId("5a934e000102030405000003"),"product_name":"D","product_type":[{"name":"Wood"}]}]
Query
db.collection.find({"product_type.name":{$in:["Wood","Carbon"]}})