Global system resolution ============================ The introduction of a new equation does not unduly disturb the method of solving the nonlinear system. In fact, we proceed as in [:ref:`R5.03.01 `], i.e. the resolution is incremental. Note the time step :math:`i` as a subscript and the Newton iteration :math:`n` as an exponent. The non-linear problem is then solved in two stages: * A prediction phase that gives a first estimate of the movements and the Lagrange multipliers noted :math:`(\Delta {u}_{i}^{0},\Delta {\lambda }_{i}^{0})`; * A Newton :math:`(\delta {u}_{i}^{n},\delta {\lambda }_{i}^{n})` correction phase that corrects this first estimate; With a sufficient number :math:`{n}_{\mathrm{CV}}` of Newton iterations, we get a converged result [1] _ : .. math:: : label: eq-5 \ {\ begin {array} {} {u} _ {i} _ {i} ^ {\ text {converged}}} = {u} _ {i-1} +\ Delta {u} _ {i} ^ {0} +\ sum _ {j=1} _ {j=1}} ^ {1} ^ {0} +\ sum _ {j=1}} ^ {0} +\ sum _ {j=1} ^ {0} +\ sum _ {j=1}} ^ {1} ^ {0} +\ sum _ {j=1}} ^ {1} ^ {j} ^ {j} ^ {j} ^ {j} ^ {j} ^ {j} ^ 1} ^ {1} ^ {j} ^ {j} ^ {j} ^ {j} ^ 1} ^ {1} ^ _ {i} ^ {\ text {converged}} = {\ lambda}} = {\ lambda} _ {i-1} +\ Delta {\ lambda} _ {i} ^ {j=1} ^ {j=1} ^ {n= {n}} _ {\ mathrm {n}} _ {\ mathrm {CV}}}}\ delta {\ lambda} _ {i} ^ {j}\ end {array}}\ end {array}} The principle is therefore to linearize the system () that we write in time step :math:`i`: .. math:: :label: eq-6 \ {\ begin {array} {cc} {L} {L} _ {i} _ {i} ^ {i} ^ {int}} + {B} ^ {T}\ mathrm {.} {\ lambda} _ {i} &\ text {=} {=} {L} _ {\ text {tax}, i} ^ {\ text {meca}}} + {\ eta} _ {i}\ mathrm {.} {L} _ {\ text {pilot}, i}} ^ {\ text {mecha}}\\ B\ mathrm {.} {u} _ {i} &\ text {=} {u} _ {\ text {impo}, i} ^ {d} + {\ eta} _ {i}\ mathrm {.} {u} _ {\ text {pilot}, i} ^ {d}\\ P (\ Delta {u} _ {i}) &\ text {=}\ Delta {\ tau} _ {i} _ {i}\ end {array} There will be no linearization with respect to the control variable :math:`{\eta }_{i}`. In this way, all the methodology for updating the tangent operator already implemented for calculations without control is preserved. In addition, the "band" structure of the tangent matrix is maintained. The resolution scheme is therefore no longer strictly a Newton method (which is not a problem because we will see that the control equation is, at most, of degree two). In prediction, if we linearize the system () without the control equation, with respect to the time around :math:`({u}_{i-1},{\lambda }_{i-1})`, we get (see all the development in [:ref:`R5.03.01 `]): .. math:: :label: eq-7 \ mathrm {\ {}\ begin {array} {cc} {\ mathrm {K}} _ {i\ mathrm {-} 1}\ mathrm {.} 1}\ mathrm {.} \ Delta {\ mathrm {u}} _ {i}} ^ {0} + {\ mathrm {B}} ^ {T}\ mathrm {.} \ Delta {\ lambda} _ {i} ^ {0} &\ text {0} &\ text {0} &\ text {0} &\ text {0}} _ {\ text {impo}, i} ^ {\ text {meca}}} + {\ eta}} + {\ eta}} _ {i}\ mathrm {.} {\ mathrm {L}} _ {\ text {pilo}, i}, i}, i} ^ {\ text {mecha}}\ mathrm {-}} {\ mathrm {Q}}} _ {i\ mathrm {-}} _ {i\ mathrm {-} 1} ^ {T}\ mathrm {.} {\ sigma} _ {i\ mathrm {-} 1}} +\ Delta {\ mathrm {L}} _ {i} ^ {\ text {varc}}\\\ mathrm {B}\ mathrm {B}\ mathrm {.} \ Delta {\ mathrm {u}} _ {i}} ^ {0} &\ text {0} &\ text {=} {\ mathrm {u}}} _ {\ text {impo}, i} ^ {d} + {\ d} + {\ eta} ^ {d} + {\ eta} ^ {d} + {\ eta}} ^ {d} + {\ eta}} _ {i}\ mathrm {.} {\ mathrm {u}} _ {\ text {pilo}, i} ^ {d}\ mathrm {-}\ mathrm {B}\ text {.} {\ mathrm {u}} _ {i\ mathrm {-} 1}\ end {array} As a correction, we linearize the system () still without the control equation, with respect to time, but around :math:`({u}_{i}^{n},{\lambda }_{i}^{n})`, we have: .. math:: :label: eq-8 \ mathrm {\ {}\ begin {array} {cc} {\ mathrm {K}}} _ {i} ^ {n\ mathrm {-} 1}\ mathrm {-} 1}\ mathrm {.} \ delta {\ mathrm {u}} _ {i}} ^ {n} + {\ mathrm {B}} ^ {T}\ mathrm {.} \ delta {\ lambda} _ {i} ^ {n} &\ text {n} &\ text {=} {\ mathrm {L}} _ {\ text {impo}, i} ^ {\ text {meca}}} + {\ text {meca}}} + {\ eta}} + {\ eta}} _ {i}\ mathrm {.} {\ mathrm {L}} _ {\ text {pilo}, i}, i}, i} ^ {\ text {mecha}}\ mathrm {-} {\ mathrm {L}}} _ {i} ^ {\ text {int}, n\ mathrm {int}}, n\ mathrm {-} 1}\ mathrm {-}}\ mathrm {-}} {\ mathrm {L}}} _ {i} ^ {\ text {int}, n\ mathrm {int}, n\ mathrm {-} 1}\ mathrm {-} 1}\ mathrm {-}} {\ mathrm {B}}} ^ {T}\ mathrm {int}, n\ mathrm {int}, {\ lambda} _ {i} ^ {n\ mathrm {-} 1}\\ mathrm {B}\ mathrm {.} \ delta {\ mathrm {u}} _ {i}} ^ {n} &\ text {=} {\ eta} _ {i}\ mathrm {.} {\ mathrm {u}} _ {\ text {pilot}, i} ^ {d}\ end {array} We will combine the two systems in a common script, in order to simplify the presentation. The system to be solved is finally written: .. math:: :label: eq-9 \ left [\ begin {array} {cc} {K} _ {k} _ {i} ^ {n-1} & {B} ^ {T}\\ B& 0\ end {array}\\ right]\ mathrm {.} \ left\ {\ begin {array} {} {\ delta u} {\ delta u} _ {i} ^ {n}\\ {\ delta\ lambda} _ {i} ^ {n}\ end {array}\ right\} =\ left\ {\ delta u}} =\ left\ {\ begin {array}}} _ {\ begin {array} {}} _ {\ begin {array} {}} _ {\ begin {array}} {array} {}} _ {\ begin {array} {}} {array} {}} _ {\ begin {array} {} {array} {}} _ {\ begin {array} {}} {array} {}} _ {\ begin {array} {} {array} {}} _ {\ begin {array} {} {array}} {\ array}} i}\ end {array}\ right\} + {\ eta} _ {i}\ mathrm {.} \ left\ {\ begin {array} {} {L} _ {\ text {pilo}, i}\\ {u} _ {\ text {pilo}, i}\ end {array} {L} _ {\ text {pilo}, i}\ end {array} {L} Several remarks can be made: * The :math:`{K}_{i}^{n-1}` matrix depends both on the current time step and possibly on the previous Newton iteration. The different ways to build it (quasi-Newton, elastic, secant, secant, coherent, tangent in speed, etc.) are described in [:ref:`R5.03.01 `]; * External loads were assumed to be linear (they only depend on the time step). We therefore do not consider follower loadings such as pressure or centrifugal force, although from a theoretical point of view, this does not pose any difficulties. On the other hand, the material can be described with non-linear behavior, which implies that :math:`{\mathrm{L}}_{i}^{\text{int},n\mathrm{-}1}` depends on Newton's iteration (result of the linearization of internal forces). * The Dirichlet boundary conditions are always linear, which allows matrix :math:`B` to be constant over the entire transient. * With the right choice of the tangent matrix and the second member, formally, there is equivalence between :math:`(\delta {u}_{i}^{n=0},\delta {\lambda }_{i}^{n=0})` and the increment in prediction :math:`(\Delta {u}_{i}^{0},\Delta {\lambda }_{i}^{0})`. We can now express the corrections of movements :math:`\delta {u}_{i}^{n}` and Lagrange multipliers :math:`\delta {\lambda }_{i}^{n}` as a function of :math:`{\eta }_{i}` by resolving the linear system () with respect to each of the two second members. In other words, we separate the two solutions: .. math:: :label: eq-10 \ left\ {\ begin {array} {} {\ delta u} {\ delta u} _ {i} ^ {n}\\ {\ delta\ lambda} _ {i} ^ {n}\ end {array}\ right\} =\ left\ {\ delta u}} {\ delta u} _ {\ delta u} _ {\ delta u} _ {\ text {impo}, i} ^ {n}\\ {\ delta\ lambda}\\ {\ delta\ lambda}} =\ left\ {\ delta\ lambda}} =\ left\ {\ delta\ lambda}} =\ left\ {\ delta {array}}} {\ delta\ lambda}} =\ left\ {\ delta {array}}} {\ delta {array}}} {\ delta\ lambda}} {impo}, i} ^ {n}\ end {array}\ right\} + {\ eta} _ {i}\ mathrm {.} \ left\ {\ begin {array} {} {\ delta u} {\ delta u} _ {\ text {pilo}, i} ^ {n}\\ {\ delta\ lambda}} _ {\ text {pilo}, i}, i}, i} ^ {n}\ end {array}\ right\} These two solutions correspond to the decoupling of the two loads: .. math:: :label: eq-11 \ left\ {\ begin {array} {} {L} _ {i}} _ {i} ^ {n-1}\\ {u} _ {i}\ end {array}\ right\} =\ left\ {\ begin {array} {} {array} {} {L} _ {} {L} _ {\ text {array}} {L} _ {\ text {array} {L} {L} _ {\ text {impo}, i} {} {L} _ {\ text {array} {L}} _ {\ text {impo}, i} {} {L} _ {\ text {array} {L}} _ {\ text {impo}, i}\ end {array} {L}} _ {\ text {array} {L}} _ {\ text {impo}, i\} + {\ eta} _ {i}\ mathrm {.} \ left\ {\ begin {array} {} {L} _ {\ text {pilo}, i}\\ {u} _ {\ text {pilo}, i}\ end {array} {L} _ {\ text {pilo}, i}\ end {array} {L} We solve independently [2] _ : .. math:: :label: eq-12 \ left [\ begin {array} {cc} {K} _ {k} _ {i} ^ {n-1} & {B} ^ {T}\\ B& 0\ end {array}\\ right]\ mathrm {.} \ left\ {\ begin {array} {} {\ delta u} {\ delta u} _ {\ text {impo}, i} ^ {n}\\ {\ delta\ lambda} _ {\ text {impo}, i} ^ {n} ^ {n}}\ end {\ delta u}}} _ {\ text {impo}, i} ^ {n}}}\ end {array}\ right\}} =\ left\ {\ begin {array} {\ array} {\ lambda} _ {\ text {impo}, i} ^ {n}} {n}}\ end {array}\ right\} =\ left\ {\ begin {array} {L} _ {\ text {cst}, i} ^ {n}} {n}}\ end {array}\ right\}\\ {u} _ {\ text {cst}, i}\ end {array}\ right\} And: .. math:: :label: eq-13 \ left [\ begin {array} {cc} {K} _ {k} _ {i} ^ {n-1} & {B} ^ {T}\\ B& 0\ end {array}\\ right]\ mathrm {.} \ left\ {\ begin {array} {} {\ delta u} {\ delta u} _ {\ text {pilo}} _ {\ text {pilo}, i} ^ {n}\ end {array}\ n}\ end {array}\ right\} _ {\ text {pilo}, i} ^ {n}}}\ end {array}\ right\}} =\ left\ {\ begin {array} {\ array} {\ lambda} _ {\ text {pilo}, i} ^ {n}}}\ end {array}\ right\}} =\ left\ {\ begin {array} {L} _ {\ text {pilo}, i} ^ {n}} {n}}\ end {array}\ right\}} =\ left\ {\ begin {array} {L}\\ {u} _ {\ text {pilot}, i}\ end {array}\ right\} We can now substitute displacement correction :math:`\Delta {u}_{i}^{n}` into the system control equation; the result is a scalar equation in :math:`{\eta }_{i}`: .. math:: :label: eq-14 \ tilde {P} ({\ eta} _ {i})\ underset {\ text {def.}} {\ mathrm {=}} P (\ Delta {\ mathrm {u}}} _ {i}} ^ {n\ mathrm {-} 1} +\ delta {\ mathrm {u}}} _ {\ text {impo}}} _ {\ text {impo}}, i}, i} ^ {n} ^ {n} ^ {n} ^ {n} _ {i}\ mathrm {u}}} _ {\ text {impo}, i}, i}, i} ^ {n} + {\ eta} _ {i}\ mathrm {u}}} _ {\ text {impo}, i}, i}, i} \ delta {\ mathrm {u}} _ {\ text {pilo}, i} ^ {n})\ mathrm {=}\ Delta {\ tau}} _ {i} The method for solving this equation depends on the nature of the control control adopted cf. [ยง :ref:`7 `]. Finally, all that remains is to update the unknown Lagrange movements and multipliers: .. math:: :label: eq-15 \ {\ begin {array} {}\ Delta {u} _ {i} _ {i} ^ {n} =\ Delta {u} _ {i} ^ {n-1} +\ delta {u} _ {\ text {impo} _ {\ text {impo}, i} _ {impo}, i}, i} ^ {n} + {\ eta} _ {i}\ mathrm {.} \ delta {u} _ {\ text {pilo}, i}} ^ {n} ^ {n}\\ {\ lambda} _ {i} ^ {n} = {\ lambda} _ {i} ^ {n-1} +\ delta {\ lambda} +\ delta {\ lambda}} +\ delta {\ lambda}} + {\ eta} _ {i} ^ {n-1} +\ delta {\ lambda} +\ delta {\ lambda}} +\ delta {\ lambda}} +\ delta {\ lambda}} + {\ eta} _ {i} ^ {n-1} +\ delta {\ lambda} +\ delta {\ lambda}} +\ delta {\ lambda}} +\ delta {\ lambda}} \ delta {\ lambda} _ {\ text {pilo}, i} ^ {n}\ end {array} .. _RefNumPara__5953_2146975687: .. _Ref498169557: .. [1] The concept of a "converged" result is more fully detailed in [R5.03.01]. .. [2] In practice, the factorization of the matrix is done only once and the two systems are solved simultaneously.