TY - JOUR
T1 - Lagrange multipliers in intrinsic elasticity
AU - Ciarlet, Philippe G.
AU - Ciarlet Jr., Patrick
AU - Iosifescu, Oana
AU - Sauter, Stefan
AU - Zou, Jun
PY - 2011/4
Y1 - 2011/4
N2 - In an intrinsic approach to three-dimensional linearized elasticity, the unknown is the linearized strain tensor field (or equivalently the stress tensor field by means of the constitutive equation), instead of the displacement vector field in the classical approach. We consider here the pure traction problem and the pure displacement problem and we show that, in each case, the intrinsic approach leads to a quadratic minimization problem constrained by Donati-like relations (the form of which depends on the type of boundary conditions considered). Using the Babuka-Brezzi inf-sup condition, we then show that, in each case, the minimizer of the constrained minimization problem found in an intrinsic approach is the first argument of the saddle-point of an ad hoc Lagrangian, so that the second argument of this saddle-point is the Lagrange multiplier associated with the corresponding constraints. Such results have potential applications to the numerical analysis and simulation of the intrinsic approach to three-dimensional linearized elasticity. © 2011 World Scientific Publishing Company.
AB - In an intrinsic approach to three-dimensional linearized elasticity, the unknown is the linearized strain tensor field (or equivalently the stress tensor field by means of the constitutive equation), instead of the displacement vector field in the classical approach. We consider here the pure traction problem and the pure displacement problem and we show that, in each case, the intrinsic approach leads to a quadratic minimization problem constrained by Donati-like relations (the form of which depends on the type of boundary conditions considered). Using the Babuka-Brezzi inf-sup condition, we then show that, in each case, the minimizer of the constrained minimization problem found in an intrinsic approach is the first argument of the saddle-point of an ad hoc Lagrangian, so that the second argument of this saddle-point is the Lagrange multiplier associated with the corresponding constraints. Such results have potential applications to the numerical analysis and simulation of the intrinsic approach to three-dimensional linearized elasticity. © 2011 World Scientific Publishing Company.
KW - Babuška-Brezzi inf-sup condition
KW - constrained quadratic optimization
KW - intrinsic elasticity
KW - Lagrange multipliers
KW - Linearized elasticity
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UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-79955502049&origin=recordpage
U2 - 10.1142/S0218202511005167
DO - 10.1142/S0218202511005167
M3 - RGC 21 - Publication in refereed journal
VL - 21
SP - 651
EP - 666
JO - Mathematical Models and Methods in Applied Sciences
JF - Mathematical Models and Methods in Applied Sciences
SN - 0218-2025
IS - 4
ER -