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ABSTRACT The isolation of structures from ground transmitted waves by open and infilled trenches in a three-dimensional context is numerically studied. The soil medium is assumed to be elastic or viscoelastic, homogeneous and isotropic. Waves generated by the harmonic motion of a surface rigid machine foundation are considered in this work. The formulation and solution of the problem is accomplished by the boundary element method in the frequency domain. The infinite space fundamental solution is used requiring discretization of the trench surface, the soil-foundation interface and some portion of the free soil surface. The proposed methodology is first tested for accuracy by solving three characteristic wave propagation problems with known solutions and then applied to several vibration isolation problems involving open and concrete infilled trenches. Three-dimensional graphic displays of the surface displacement pattern around the trenches are also presented.