On non-reflecting boundary conditions in unbounded elastic solids

Abstract

Problems in unbounded domains can be solved using domain-based computation by introducing an artificial boundary, and then selecting appropriate boundary conditions. The DtN method, which specifies such boundary conditions, is investigated in this work for wave problems in elastic solids. The DtN method defines an exact relation between the displacement field and its normal and tangential tractions on an artificial boundary. This relation is expressed in terms of an infinite series. The DtN boundary conditions are shown to be non-reflective, thus uniqueness of the solution is guaranteed. For practical purposes the full DtN operator is truncated. The truncated DtN operator fails to completely inhibit reflections of higher modes, resulting in loss of uniqueness at characteristic wave numbers of higher harmonics. Guidelines for determining a sufficient number of terms in the truncated operator to retain uniqueness of the solution at any given wave number are derived. The validity of these guidelines is examined and verified by numerical examples. Local DtN boundary conditions are also investigated, and it is shown that local boundary conditions guarantee uniqueness of the solution for all wave numbers, regardless of the number of terms in the operator. This property is used here to modify the truncated DtN operator and to enhance its capability to retain uniqueness of solutions. A modified DtN operator, combining the truncated operator with the local one, is introduced. The modified DtN operator is shown to retain uniqueness of solutions regardless of the number of terms and regardless of the wave number.