t red ig 0 for i = 1..g, (9) where the (r x r) matrix red K can be written as red T K ĭKĭ, (10) the (r x 1) vector red f as red T f ĭ f (11) and red ig as ª º ¬ ¼ red i i IJ,M,i IJ,S,i g Q ĭ ĭ q. (12) The (3 x r) matrixes IJ,M,i ĭ and IJ,S,i ĭ denote the rows which belong to the master and slave surface node number i and the (1 x 3) transformation matrix iQ is the corresponding sub-matrix of Q. The right subscript ‘red’ indicates, that a quantity is computed based on generalized coordinates according to equation (12). For the common case, that g > r the system (8) and (9) has more constraints as DOF and can be therefore over constrained. For the further considerations, the joint surfaces will be subdivided into b mutual exclusive subareas about the same size. The vector of master surface DOF can then be written as ª º ¬ ¼ " " T T T IJ,M IJ,M,1 IJ,M,i IJ,M,b x x x x (13) where the (mi x 1) vector IJ,M,i x contains the DOF of the i-th subsurface. The corresponding vector of relative normal joint displacements of the master FE nodes takes on the form ª º ª º ª º ¬ ¼ ¬ ¼ « » ¬ ¼ " " T T T red red red red 1 i b g g g g (14) where the (gi x 1) vector red ig holds the relative normal joint displacements of the master FE nodes in subsurface number i. Finally the vector of FE node related areas is subdivided similarly into ª º ¬ ¼ " " T T T 1 i b a a a a (15) where the (gi x 1) vector ia holds the master FE node related areas in subsurface number i. A possibility to overcome the over constraining issue of equations (8) and (9) is an averaged formulation of the non penetration constraints. Applying this idea to equations (8) and (9) the minimization problem takes on the form ª¬ º¼ T T red red 1 min( ) 2 x q K q f q (16) so that t * ig 0 for i = 1..b, (17) where ¦ ¦ i i g * red i i,j i,j g j 1 i,j j 1 1 g g a a (18) and ai,j represents the j-th entry of ia and red i,j g holds the j-th entry of red ig . Note, when b is equal to the number of FE nodes in the master surface, that latter formulation converges to the FEM type formulation of equation (6). 32
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