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The scaled-boundary finite element method is a novel semi-analytical technique, combiningthe advantages of the finite element and the boundary element methods with unique properties of its own.The method works by weakening the governing differential equations in one coordinate direction throughthe introduction of shape functions, then solving the weakened equations analytically in the other (radial)coordinate direction. These coordinate directions are defined by the geometry of the domain and a scalingcentre. This paper presents a general development of the scaled boundary finite-element method for two-dimensional problems where two boundaries of the solution domain are similar. Unlike three-dimensionaland axisymmetric problems of the same type, the use of logarithmic solutions of the weakened differentialequations is found to be necessary. The accuracy and efficiency of the procedure is demonstrated throughtwo examples. The first of these examples uses the standard finite element method to provide acomparable solution, while the second combines both solution techniques in a single analysis. Onesignificant application of the new technique is the generation of transition super-elements requiring fewdegrees of freedom that can connect two regions of vastly diferent levels of discretisation.