Walter D’Ambrogio and Annalisa Fregolent 2.1 Primal formulation In the primal formulation, a unique set of DoFs is defined: uAB uB = LAB C LB C {q} (7) where {q}is the unique set of DoFs, and[LC] is a localisation matrix similar to[LE] introduced previously. Note that [LC] is a (NAB +NB)×(NAB +NB −NC) matrix. Since there is a unique set of DoFs, {q}, the compatibility condition is satisfied automatically for any set {q}, i.e. BAB C BB C uAB uB = BAB C BB C LAB C LB C {q}={0} ∀{q} (8) Hence, [LC] is the nullspace of [BC] and, viceversa, [BC] T is the nullspace of [LC] T: ⎧⎪ ⎪⎨ ⎪⎪⎩ BAB C BB C LAB C LB C ={0} LAB C LB C T BAB C BB C T ={0} (9) Since the compatibility condition, Eq. (5), is satisfied (see Eq. (8)) by the choice of the unique set {q}, the 3-field formulation reduces to: ⎧⎪ ⎪⎪⎨ ⎪⎪⎪⎪⎩ ZAB [0] [0] − ZB LAB C LB C {q}= f AB f B + gAB gB LAB E LB E T gAB gB ={0} (4∗) (6) Pre-multiplying the equation (4∗) by [LE] T and noting that [LE] T{g}={0}, the formulation reduces to: LAB E LB E T ZAB [0] [0] − ZB LAB C LB C {q}= LAB E LB E T f AB f B (10) from which: {q}=⎛ ⎝ LAB E LB E T ZAB [0] [0] − ZB LAB C LB C ⎞ ⎠ + LAB E LB E T f AB f B (11) where the superscript +denotes the generalized inverse. 52
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