Chrome Shelled Regios is set on Earth After the End. There the remaining population survives inside domed, mobile, cities called Regios. These are needed to protect from the horribly polluted atmosphere and giant man-eating Filth Monsters.The series protagonist Layfon Alseif, arrives at the academy city of Zuellni with the intent to leave his past behind him and start over as a general studies student. However, he demonstrates his true abilities saving a girl who got caught up in the destruction of a fight between two military arts students. Despite not wanting to fight he is forced into the military arts department which is used for defense of the city as well as competing in intercity matches for valuable resources.A series of Light Novels that has 4 (Yes, 4!) Manga adaptations, 2 novel spin-offs (a prequel and a sidestory) and an Anime adaption.
"Trope Shelled Regios":
A-Cup Angst: Felli appears to suffer from this to an extent, showing annoyance at Nina's size in episode 15.
Elopement: Happens to Layfon and Felli at the end of the series, with Layfon asking Felli to Walking the Earth with him. Hillariously, Layfon had no idea what he was asking her to do until she tells him.
I Did What I Had to Do - Layfon's justification for participating in underground matches and attempting to kill the person who was blackmailing him in a Heaven's Blade selection match. Thankfully Nina was able understand and forgive Layfon on what he did. The anime on the other hand she forgave him by the next episodes end.
Katanas Are Just Better - Played straight and subverted. Almost every single character used a weapon other than a katana, and they aren't really being shown as inferior to those (few) that do use katanas. However, the protagonist, Layfon Alseif, who has been using a standard sword the entirety of the series, gets a big scene after fusing two different kinds of Infinity Plus One Swords together. Partially justified in that all of his training focused on katana-using, and it was just a personal vow of repentance that kept him from using it the whole time.
Rebellious Princess - Nina is actually the heir to the city of Sheniebel. But she felt her freedom was restricted by her parents since she wanted to do more to protect people rather than just being the next heir. When her parents wouldn't allow her freedom, she ran away from home to study at Zuellni which her parents in turn disown her.
Sempai/Kohai - Felli is Layfon's sempai, but she hates being called that.
Serious Business - Conflicts between cities are settled with games of what very strongly resembles capture the flag, which decides whether the opposing cities get to keep the other city's Selenium mines (the source of energy for the cities) or not.
It's also only the Academy Cities that have the capture the flag rules- to hold down the body count. It's implied other city types are more... unrestrained in fighting.
Supernatural Aid - With a bit of a twist as Layfon has lost his Heaven's Blade, the anime then plays it straight when it is returned and combined with his katana.
Takehito Koyasu - Karian Loss, Student Council President and Felli's older brother.
Thanks for the Mammary - A funny version where the female doesn't punch the guy. Layfon and Naruki have a sparring match where he accidentally falls on top of her. Of course he touches her boobs. Naruki is not angry about it but she is annoyed that Layfon didn't realize he was feeling her big breasts as if they were flat in the first place!