in order to update the Computed Weight automatically,  you need to activate “Refresh Characteristics during PLM update” preference.

With this preference activated, the Weight update is now integrated in the PLM update. After changing the geometry and/or the material, update the computed weight by clicking the PLM Update icon in the Action Bar instead of selecting B.I. Essentials/Weight Definition.

To ensure the updated weight is being saved, activate the preference “Allow you to update detected out-of-date data before save”

Weight needs to be already computed on the part to be modified.

In conjunction with the previous preference "Refresh characteristics during PLM Update" (active by default), every time the user tries to save a part with modified geometry, it will ask whether to update it or not. Clicking Yes, will update the part along with its weight before it gets saved.