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Adaptive numerical method algorithms are presented for the numerical simulation of the hysteretic behaviour of nonlinear viscous and bilinear oil dampers within a finite element program for nonlinear dynamic analysis of frame structures under earthquake excitations. The adaptive algorithms are applicable for computing high-precision solutions for nonlinear viscous and bilinear oil dampers with valve relief that are typically represented mathematically with a nonlinear Maxwell model. The algorithms presented possess excellent convergence characteristics for viscous dampers with a wide range of velocity exponents and axial stiffness properties. The algorithms are implemented in an open source finite element software, and their applicability and computational efficiency is demonstrated through a number of validation examples with data that involve component experimentation as well as the utilization of full-scale shake table tests of a 5-story steel building equipped with nonlinear viscous and bilinear oil dampers.