The first time I connected to the revamped Knowbrainer website, I found that it did not recognise my existing password for my username.
"ERROR: your login credentials were invalid" is the message.

So I clicked on "Forgot login details", a new one was sent to my e-mail address, and I logged in successfully with it. I carefully kept a record of it. Next time I logged in, with my username and the new password, same story. And the next time, and the next time, ad nauseam.

What must I do to get the website to retain my password? There is no prompt that says you have to change the artificially generated password.

I haven't experienced this with any other users. Your password is encrypted and secured in the database, so there should be no reason for it to change unless you force a change for it. There is no requirement to change the temporary password that the site generates for you. You may want to consider deleting cookies in your browser if you have the browser set to remember your login.

Regards, Webmaster

No, I don't have the browser (IE9) set to remember passwords.

Despite what you say about there being no need to change the automatically generated password, I have just found the following.  With my latest new password, I went to Profile, edited in a new password of my own, logged out, and voila!  I successfully logged in again with my own (new) password.  Weird.