Accuracy evaluation of newmark method referring to theoretical solutions

Current accuracy evaluation of a step-by-step integration method is generally based on the numerical solution itself and the exact solution is not referred to except for period distortion. Furthermore, the numerical accuracy associated with external force is generally not considered. In this study, an evaluation technique is proposed to assess the numerical accuracy, including the accuracy associated with a linear step loading type, of the Newmark method by comparing the numerical solution to the exact solution. No errors are found in the steady-state response. Period distortion in the cosine term of the transient response is different from that in the sine term except for the subfamily of gamma = 1/2. A new measure of amplitude distortion is defined that incorporates the error in amplitude relative to the exact solution. This measure can reflect amplitude distortion more realistically in a numerical solution when compared to the current measures of numerical dissipation. In general, period and amplitude distortions are cumulative except that there is no cumulation in amplitude distortion for the subfamily of gamma = 1/2. It is also shown that numerical damping is ineffective for the steady-state response.