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ABSTRACT. IT IS WELL KNOWN THAT THE MATRIX H OF CINEMATIC TRANSFORMATIONS ONE ARRIVES AT IN THE CONVENTIONAL BOUNDARY ELEMENT METHOD IS SINGULAR, FOR A FINITE DOMAIN, AS A CONSEQUENCE OF THE FACT THAT RIGID BODY DISPLACEMENTS DO NOT PRODUCE DEFORMATIONS IN AN ELASTIC BODY. A LITTLE MORE THAN A DECADE AGO, THE AUTHOR INTRODUCED THE "HYBRID BOUNDARY ELEMENT METHOD", IN WHICH A SYMMETRIC, POSITIVE SEMI-DEFINITE FLEXIBILITY MATRIX F IS OBTAINED ON THE BASIS OF A SOUND VARIATIONAL PRINCIPLE. ALONG WITH THIS NOVEL FORMULATION, CONCEPTS OF GENERALIZED INVERSE MATRICES HAD TO BE CONSIDERED FOR THE ADEQUATE UNDERSTANDING AND MANIPULATION OF THE MATRIX SINGULARITIES INVOLVED. MOTIVATED BY THESE ACCOMPLISHMENTS, DE FIGUEIREDO DERIVED A FEW YEARS LATER THE "HYBRID DISPLACEMENT BOUNDARY ELEMENT METHOD", A VARIATIONAL COUNTERPART THAT GIVES RISE TO THE SAME MATRIX F, BUT THIS TIME IN RELATION TO THE MATRIX G OF THE CONVENTIONAL BOUNDARY ELEMENT METHOD. QUITE RECENTLY, THE AUTHOR DECIDED TO INVESTIGATE THE SPECTRAL PROPERTIES OF THIS FLEXIBILITY-LIKE MATRIX G, IN ORDER DO DEMONSTRATE THAT, ADEQUATELY OBTAINED, THIS IS ALSO A SINGULAR MATRIX (AS ONLY RELATED TO FORCES IN BALANCE). SUCH A SINGULARITY IS ALSO A WELCOME FEATURE ONE COULD AND SHOULD TAKE ADVANTAGE OF. THE PRESENT PAPER IS AN ATTEMPT OF ASSESSING THE SPECTRAL PROPERTIES OF THE MATRICES H, G AND F, ACCORDING TO THE ROLES THEY PLAY IN THE DIFFERENT BOUNDARY ELEMENT FORMULATIONS. IT IS DEMONSTRATED THAT SUCH SPECTRAL PROPERTIES ARE INTERRELATED IN THE FORMULATIONS OUTLINED. A NOT UNREMARKABLE CONSEQUENCE OF THIS COMPARED STUDY IS THE CONCLUSION THAT BOTH THE CONVENTIONAL AND THE HYBRID DISPLACEMENT BOUNDARY ELEMENT METHODS NEED SOME CONCEPTUAL IMPROVEMENTS IN THEIR FORMULATIONS, IN ORDER TO BECOME COMPLETELY CONSISTENT.
KEY WORDS: BOUNDARY ELEMENT METHODS, VARIATIONAL METHODS, GENERALIZED INVERSE MATRICES.
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