[en] This paper offers a non-linear iterative process which can determine the exact boundary conditions for the artificial boundary without recourse to analytical results. It describes the process and exhibits its effectiveness with illustrative problems. The process involves an adaptive evaluation of boundary conditions using a relaxation solution process. It establishes conditions along the artificial boundary such that decay of deformations within the truncated region is consistent with their vanishing at infinity. Iteration is performed by relaxing displacements at each boundary point in turn. A second analysis with a larger domain produces absolute measures of analysis error. Illustrations include linear systems in one, two and three dimensional space: Winkler's beam-foundation problem and Boussinesq's. Numerical analysis uses finite element models and produces data for comparisons with the exact solutions. In each case, the exact responses are predicted, within the limitations of the finite element articulation and computer manipulation and process errors. When accuracy of results in the usual approach is unacceptable, the same truncated region produces results of negligible error using the iteration. Analysis running times are less than four times those of the usual approach. Adding condensation logic could reduce this time factor to less than 1.50. The procedure is directly applicable to finite difference or boundary integral modeling. It can encompass certain types of nonlinear systems