radiant smile; contented smile; beaming smile
Describes a smile that spreads across the whole face, expressing deep satisfaction or happiness. Often used in literary or descriptive contexts.
He accepted the certificate with a beaming smile.
Seeing her grandchild's face, the grandmother broke into a radiant smile.
満面の笑み just means 'a broad smile' with no hint of smugness, while 得意満面 specifically implies a triumphant, self-satisfied expression.
The phrase combines 満面 (whole face) and 笑み (smile). The exact historical derivation is uncertain, but it is a natural collocation in Japanese.