You need to set $subModel->isNewRecord = true
in each iteration and also unset (or manually set to the value of your choice) the model's id field. It would look something like this:
for ($i=0; $i<=sizeof($keyword)-1; $i++)
{
// if $subModel->name is not the primary key for that model, unset the PK here:
// $subModel->id = null
$subModel->name = $keyword[$i];
$subModel->isNewRecord = true;
$subModel->save();
}