The kratos requires the elements to return A RESIDUA as RHS. This may not represent the natural choice for linear problems but allows to mix linear and non linear elements.
The residua is defined as RHS = dWext - dWint (note that indeed this corresponds to the standard definition changed of sign)
For linear problems, if the vector of external "forces" fext is known, The RHS can be calculated as RHS=fext-K*xp Where K is the "standard" stiffness matrix and xp is a vector containing the last known values of the dofs (note that the prescribed values are all known here)
Clearly the residual form is natural for non linear problems and requires no further explanation. The "tangent" stiffness matrix, called here LHS should be introduce with its natural sign. Note that for linear elements the "tangent" coincides with the "standard" stiffness matrix.