dringen
to penetrate
To force one's way through an obstacle or into something; to enter or pass through.
Das Wasser drang durch die undichte Stelle im Dach.
The water penetrated through the leaky spot in the roof.

to penetrate / to insist
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to penetrate
To force one's way through an obstacle or into something; to enter or pass through.
Das Wasser drang durch die undichte Stelle im Dach.
The water penetrated through the leaky spot in the roof.

The auxiliary verb for the perfect tense of 'dringen' (to penetrate / to insist) depends on the meaning. If it describes movement or entering into something, 'sein' is used. If it means 'auf etwas bestehen' (to insist on something), 'haben' is used.
Rauch ist aus dem Fenster gedrungen.
Smoke came out of the window.
In the meaning 'energisch fordern' (to demand forcefully), 'dringen' (to insist) always takes the preposition 'auf' followed by the accusative. One insists on a thing.
Die Anwohner dringen auf eine schnelle Lösung des Verkehrsproblems.
The residents are insisting on a quick solution to the traffic problem.
When 'dringen' (to penetrate) describes movement, it is often combined with prepositions such as 'in (+Akk)' (into + accusative), 'durch (+Akk)' (through + accusative), or 'bis zu (+Dat)' (up to + dative) to describe the path or destination.
Das Licht der Taschenlampe drang in die dunkle Höhle.
The flashlight's beam penetrated into the dark cave.
to demand / challenge
'fordern' and 'auf etwas dringen' can both mean that someone demands something emphatically. 'fordern' usually directly states what someone wants: more money, an explanation, or a solution. 'auf etwas dringen' emphasizes more strongly that someone does not let up and insists quickly or forcefully on an action. It is often used with abstract things such as 'Klärung' (clarification), 'Entscheidung' (decision), or 'Einhaltung' (compliance). 'fordern' is more general; 'auf etwas dringen' sounds somewhat more formal and more focused on a specific goal.

Nach dem Fehler dringt der Kunde auf eine schnelle Lösung.
After the mistake, the customer insists on a quick solution.
Die Eltern fordern eine Erklärung von der Schule.
The parents are demanding an explanation from the school.