Castles worked as great protectors of those inside unless attacked by a large and highly technological force.
The walls are the obvious target and various devices that throw stuff to break the walls work.
Encirclement and starvation of those inside also works, but is slower.
Castles' design works against anyone from inside exiting and taking the fight to the enemy outside which puts the castles always on the defensive, a situation often that leads to defeat.
However, an attacker must be steadfast and not get disappointed too easily and leave. For a while, at least, the castle is a good defense and convincing the enemy it is a waste of effort to attack it is part of its effective defense.
The usual method was to starve them into surrender by encirclement. If the food supply to a castle was interrupted the inhabitants had to rely on what was stored in the castle 'keep'. When it ran out they either surrendered or starved to death.
Typically with heavy artillery... but I wouldn't discount an air strike.
There's a reason why you don't see castles used in modern warfare any more.
Be extremely subtle, even to the point of formlessness. Be extremely mysterious, even to the point of soundlessness. Thereby you can be the director of the opponent's fate.
Sun Tzu
To bring the greatest pressure on the weakest point. Till it folds or collapses.
Use a Trebuchet or Cannon depending what era.
Get them to vote UKIP, then watch the walls crumble...
