A new table of seven-place logarithms by Edward Sang

By Edward Sang

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Each of this operations leads to a system of two equations in the following form − x mn,m o x sn, s ⎤ ⎡Δξ mn ⎤ ⎡ x mn,m o x mn,m + x ms o x mn,mm ⎥ ⎥=⎢ ⎢ x mn,m o x sn,s − x sn, s o x sn, s + x ms o x sn,ss ⎥⎦ ⎢⎣ Δξ sn ⎥⎦ ⎢⎣ T ⎛ ⎡− x x mn,m T ⎤ ⎡Δu mn ⎤ ⎡x ms T 0 ⎤ ⎡Δu mn,m ⎤ ⎞⎟ mn,m ⎜ ⎥⎢ ⎢ ⎥⎢ ⎢ − ×⎜ ⎥⎟ , ⎥ T T ⎜ ⎢ − x sn,s T ⎥ u Δ ⎢ x ⎥ ⎢ 0 x sn ⎦ ⎣ sn, s ⎦ ⎣ ms ⎥ ⎦ ⎢⎣ Δu sn,s ⎥⎦ ⎟⎠ ⎝⎣ −1 which can be solved to express the pairs of unknowns. 5 Discretisation of Kinematic Variables 27 where the matrices A, B and C as well as the vector xms = xmn – xsn were introduced to simplify the notation.

6 Residual Vector and Tangent Stiffness Matrix 29 Similarly, the variations of the local co-ordinates are given by ⎡δu M ⎤ ⎡δξ mn ⎤ ⎥. 40) Furthermore, to simplify the notation the following matrix is introduced L = [G mn [ ] − G sn ] + x mn,m − x sn,s F . 5, finally the residual vector and the tangent stiffness matrix for the contact element can be calculated for both formulations: the penalty method and the Lagrange multipliers method. 42) , where the respective notation for the residual vectors in both methods was introduced.

23) provide the residual vector and the tangent stiffness matrix, respectively, for the contact finite element. 6). 24) Δx sn = x sn,s Δξ sn + Δu sn . After some basic operations the necessary kinematic variables for the normal contact can be expressed as (Wriggers and Zavarise, 1997) Δg N = (Δu mn − Δu sn ) o n , δg N = (δu mn − δu sn ) o n , Δδg N = (Δδu mn − Δδu sn ) o n + (δu mn,m Δξ mn − δu sn,s Δξ sn ) o n + + (Δu mn,mδξ mn − Δu sn, s δξ sn ) o n + + (x mn,mm Δξ mnδξ mn − x sn, ss Δξ snδξ sn ) o n + + 1 (δu mn + x mn,mδξ mn − δu sn − x sn,sδξ sn ) gN × (1 − n ⊗ n )(Δu mn + x mn,m Δξ mn − Δu sn − x sn,s Δξ sn ).

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