Discontinuous　deformation　analysis　(DDA)　is　a　numerical　method　for　analyzing　dynamic　behaviors　of　an　assemblage　of　distinct　blocks,　with　the　block　displacements　as　the　basic　variables.　The　contact　conditions　are　approximately　satisfied　by　the　open-close　iteration,　which　needs　to　fix　or　remove　repeatedly　the　virtual　springs　between　blocks　in　contact.　The　results　from　DDA　are　strongly　dependent　upon　stiffness　of　these　virtual　springs.　Excessively　hard　or　soft　springs　all　incur　numerical　problems.　This　is　believed　to　be　the　biggest　obstacle　to　more　extensive　application　of　DDA.　To　avoid　the　introduction　of　virtual　springs,　huge　efforts　have　been　made　with　little　progress　related　to　low　efficiency　in　solution.　In　this　study,　the　contact　forces,　instead　of　the　block　displacements,　are　taken　as　the　basic　variables.　Stemming　from　the　equations　of　momentum　conservation　of　each　block,　the　block　displacements　can　be　expressed　in　terms　of　the　contact　forces　acting　on　the　block.　From　the　contact　conditions　a　finite-dimensional　quasi-variational　inequality　is　derived　with　the　contact　forces　as　the　independent　variables.　On　the　basis　of　the　projection-contraction　algorithm　for　the　standard　finite-dimensional　variational　inequalities,　an　iteration　algorithm,　called　the　compatibility　iteration,　is　designed　for　the　quasi-variational　inequality.　The　main　processes　can　be　highly　parallelized　with　no　need　to　assemble　the　global　stiffness　matrix.　A　number　of　numerical　tests,　including　those　very　challenging,　suggest　that　the　proposed　procedure　has　reached　practical　level　in　accuracy,　robustness　and　efficiency,　and　the　goal　to　abandon　completely　virtual　springs　has　been　reached.　(C)　2016　Elsevier　B.V.　All　rights　reserved.