Based on the concept of the constitutive relation error along with the residuals of both the origin and the dual problems,a goal-oriented error estimation method with extended degrees of freedom is developed.It leads to the high quality local error bounds in the problem of the direct-solution steady-state dynamic analysis with a frequency-domain finite element,which involves the enrichments with plural variable basis functions.The solution of the steady-state dynamic procedure calculates the harmonic response directly in terms of the physical degrees of freedom in the model,which uses the mass,damping,and stiffness matrices of the system.A three-dimensional finite element example is carried out to illustrate the computational procedures.