eindringen
to intrude
To gain access to a place by force or without permission.
Die Diebe sind durch ein Kellerfenster in das Haus eingedrungen.
The thieves intruded into the house through a basement window.

to penetrate / intrude
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to intrude
To gain access to a place by force or without permission.
Die Diebe sind durch ein Kellerfenster in das Haus eingedrungen.
The thieves intruded into the house through a basement window.

'Eindringen' (to penetrate / intrude) is a separable verb. The prefix 'ein-' goes at the end in main clauses. In subordinate clauses and in the infinitive, the verb stays together.
Der Dieb drang nachts in das Museum ein.
The thief broke into the museum at night.
The verb consists of the prefix 'ein-' (into) and the verb 'dringen' (to press, push). The literal meaning is therefore to push or force one's way into something, often against resistance.
Der Splitter ist tief in die Haut eingedrungen.
The splinter penetrated deep into the skin.
The preposition changes depending on the meaning. With physical penetration or intrusion (into a place or material), you use 'in' + accusative. When pressuring someone, it is 'auf' + accusative.
Feuchtigkeit ist in die Wand eingedrungen.
Moisture penetrated the wall.
In the context of unauthorized entry, 'eindringen' (to intrude) is often combined with places. Typical phrases are: 'in ein Haus eindringen' (to break into a house), 'in ein fremdes Gebiet eindringen' (to invade foreign territory), or, in a digital sense, 'in ein Netzwerk eindringen' (to penetrate a network).
Hacker sind in die Datenbank des Unternehmens eingedrungen.
Hackers penetrated the company's database.
to occur to / collapse / invade
'einfallen' and 'eindringen' can both mean that someone enters an area or place where they are not allowed to be. 'einfallen' is used especially for military attacks on a country or territory. It suggests a larger, often organized action. 'eindringen' is more general: someone can intrude into a house, a room, or a site, often by force or without permission. The words are confusable because both describe unauthorized entry. For armies and wars, 'einfallen' is more appropriate; for individual people or groups entering a place, 'eindringen' is more appropriate.

In der Nacht sind Unbekannte in das Büro eingedrungen.
Unknown people intruded into the office during the night.
Die Soldaten fielen in das Nachbarland ein.
The soldiers invaded the neighboring country.