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Lifetime-Oriented Structural Design Concepts- P15

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Lifetime-Oriented Structural Design Concepts- P15: At the beginning of 1996, the Cooperative Research Center SFB 398 financially supported by the German Science Foundation (DFG) was started at Ruhr-University Bochum (RUB). A scientists group representing the fields of structural engineering, structural mechanics, soil mechanics, material science, and numerical mathematics introduced a research program on “lifetimeoriented design concepts on the basis of damage and deterioration aspects”. | 378 4 Methodological Implementation represents all state variables u p u p and spatial gradients VuNP . On the basis of equation 4.14 the weak form of the coupled multiphysics system 8W is generated in equation 4.15 by the weighted summation of the individual weak forms 5Wf whereby Af is introduced to adapt physical units and dimensions of coupled fields. W Af W 0 f 1 4.15 4.2.3.2 Linearized Weak Form of Coupled Balance Equations In order to prepare the weak form for the numerical solution with the Newton-Raphson scheme the weak form is expanded in a Taylor series about the trial solution of all state variables 5W k 1 5W k A8W k 0 4.16 and spatial gradients characterized by k. A8W A8W Vunf Unf Unf unf Af ASWf 0 f 1 4.17 The increment of weak forms is generated by summation of individual portions in terms of the increments of field variables Aug and gradients of field variables V Au g with g G 1 NF . ASWf f dW AVug d-W Aug d-W Aug _dyug dug dug 4.18 It is worth to mention that the derivative with respect to gradient Vug is performed explicitly in oder to obtain an advantageous format for 4.2 Numerical Methods 379 the finite element discretization discussed in Section 4.2.4.2. The individual terms in equation 4.18 are expressed as follows dsWf r - o AVug ouf o dV Ug J . n 9Ugf Aug Ouf 96Wf A x O Aug Ouf O dug J d - f . f d f A ug dV 6Vuf o O Adu dV d Vug J dVug N Q . Aug dV d f . d f . - o Aug dV SVuf o - o Aug dV dug J Oug ft 4.19 4.2.4 Spatial Discretization Methods Authored by Detlef Kuhl and Christian Becker Within the framework of the semdiscretization technique applied to solve durability single- and multiphysics problems the spatial discretization is realized by the finite element method see e.g. 90 106 223 224 870 871 . The scientific and industrial oriented literature documents the broad range of applications of this method for the spatial discretization of differential equations. Highly non-linear problems stationary and transient problems as well as .