The usage of 'know' with object and object complement

Designers draw on their experience of design when approaching a new project.

This includes the use of previous designs that they know work ― both designs that they have created themselves and those that others have created.

Is this something like know A B (like "I know him(A) a doctor(B)" ?

Solution 1:

No.

• *I know him a doctor.

is ungrammatical. Not all verbs take part in this construction, 2-place with B-Raising (see John Lawler's article). To illustrate:

[I consider him] [He is a friend]. The object of the first statement twins as the (suppressed) subject of the second statement: B-raising (... to subject). [I consider him a friend]. (The object of the second simple statement becomes an object complement in the process.)

Note that 'regard' doesn't have this property: an as is required.

As seen, one that does is consider

• I consider him a friend, a grammatical deletion of
• I consider him to be a friend.

But 'know' needs the 'to be', undeleted.

The basic structure you're asking about in your first example is a noun with a defining relative clause

• designs that / which they know work.

where the relativiser 'that' is often deleted, especially in conversation

• designs they know work.