Lifetime-Oriented Structural Design Concepts- P16 pptx

408 4 Methodological Implementation is obtained. Since the linear system of equations (4.81) is non-symmetric and looses, furthermore, the band structure of the generalized tangent matrix, it is solved by applying the partitioning technique [91]. Therefore, the partial incremental solutions Δu r and Δu λ are calculated in advance. K −1 (u k n+1 ) Δu r = λ k n+1 r − r i (u k n+1 ), K −1 (u k n+1 ) Δu λ = r (4.83) Afterwards the increments Δu = Δu r + Δu λ Δλ (4.84) and Δλ = − f(u k n+1 ,λ k n+1 )+f ,u (u k n+1 ,λ k n+1 ) · Δu r f ,u (u k n+1 ,λ k n+1 ) · Δu λ + f ,λ (u k n+1 ,λ k n+1 ) (4.85) are computed. Since this procedure is restricted to the corrector iteration, a specialized predictor step, adopting an user defined step length s,isimple- mented. As shown in Figure 4.20, the load factor is increased by one and the resulting displacement increment Δu λ and step length s 0 are calculated. Δu λ = K −1 (u n ) r,s 0 =  Δu λ · Δu λ + 1 (4.86) Afterwards the increments of the displacement vector and the load factor are scaled, such that the user defined step length s is obtained. Δλ = s s 0 1 Δu = s s 0 Δu λ (4.87) Selected constraints within the framework of the present generalized arc-length method are summarized in Table 4.5. It is worth to mention that the standard control algorithms used in the present book, namely the displacement and load controlled analyses, are also included in Table 4.5 and the load controlled Newton-Raphson scheme has already been discussed in Section 4.2.5.2. As a particular example of the generalized path following method, the algorithmic set-up of the arc-length controlled Newton-Raphson scheme is given in Figure 4.21. 4.2.6 Temporal Discretization Methods Authored by Detlef Kuhl and Sandra Krimpmann The present section is concerned with the numerical methods for the time integration of non-linear multiphysics problems by means of Newmark- α methodsaswellasdiscontinuousandcontinuousGalerkin schemes. Newmark-α time integration methods are using the semidiscrete balance equation evaluated at one selected time instant within a time step and finite 4.2 Numerical Methods 409 Table 4.5. Constraints and load factor increments of selected arc-length methods (f = f(u k n+1 ,λ k n+1 ), [817]) method constraint f(u k+1 n+1 ,λ k+1 n+1 ) increment Δλ(u k n+1 ,λ k n+1 ) load control f = λ k+1 n+1 −c −f displacement control f =u k+1 l n+1 −c u k+1 n+1 =[···u k+1 l n+1 ···] T − f +Δu l r Δu l λ initial normal plane f =[u 1 n+1 −u n ]·[u k+1 n+1 −u k n+1 ] +[λ 1 n+1 −λ n ][λ k+1 n+1 −λ k n+1 ] −[u 1 n+1 −u n ] ·Δu r [u 1 n+1 −u n ] ·Δu λ +λ 1 n+1 −λ n current normal plane f =[u k n+1 −u n ]·[u k+1 n+1 −u k n+1 ] +[λ k n+1 −λ n ][λ k+1 n+1 −λ k n+1 ] −[u k n+1 −u n ] ·Δu r [u k n+1 −u n ] ·Δu λ +λ k n+1 −λ n closest point projection f =Δu λ ·[u k+1 n+1 −u k n+1 ] +1 [λ k+1 n+1 −λ k n+1 ] −Δu λ · Δu r Δu λ · Δu λ +1 sphere radius s f + s=s k+1 =  u k+1 n+1 −u n  2 +ψ 2 [λ k+1 n+1 −λ n ] 2 −f[f +s]−[u k n+1 −u n ]·Δu r [u k n+1 −u n ]·Δu λ +ψ 2 [λ k n+1 −λ n ] cylinder radius s f =u k+1 n+1 −u n −s − f[f +s]+[u k n+1 −u n ]·Δu r [u k n+1 −u n ]·Δu λ difference approximations of the state variables within a typical time interval. These classical methods are second order accurate and exhibit controllable numerical dissipation. In contrast to this, for discontinuous Galerkin meth- ods the semidiscrete balance and the continuity of the primary variables are weakly formulated within time steps and between time steps, respectively. Continuous Galerkin methods are obtained by the strong enforcement of the continuity condition as special case. The introduction of a natural time coordinate allows for the application of standard higher order temporal shape functions of the p-Lagrange type and the well known Gauß-Legendre quadrature of associated time integrals. It is shown that arbitrary order ac- curate integration schemes can be developed within the framework of the proposed temporal p-Galerkin methods. 4.2.6.1 Introduction For the integration of time dependent durability problems either Newmark type finite difference methods or Galerkin type finite element methods are 410 4 Methodological Implementation loop over arc-length steps n =0,NT −1 n +1−→ n loop over iteration steps k =1, ··· k +1−→ k predictor tangent stiffness matrix K(u n ) predictor increment Δu λ = K −1 (u n ) r scaling displacement and load factor increment Δu and Δλ update displacements and load factor u 1 n+1 and λ 1 n+1 internal force vector r i (u k n+1 ) tangent stiffness K(u k n+1 ) partial incremental solutions Δu r and Δu λ constraint f(u k n+1 ,λ k n+1 ) increment of load factor Δλ displacement increment Δu = Δu r + Δu λ Δλ update displacements and load factor u k+1 n+1 and λ k+1 n+1 check for convergence, e.g. η k+1 u ≤ η u Fig. 4.21. Algorithmic set-up of the arc-length controlled Newton-Raphson scheme used. Galerkin integration schemes can be furthermore classified in discon- tinuous and continuous formulations. Main advantages of Newmark schemes are the small numerical effort, the second order accuracy and the controllable numerical dissipation. Continuous Galerkin schemes are decorated by an arbitrary order of accuracy combined with a moderate numerical effort. Dis- continuous Galerkin schemes allow for integrations with an arbitrary order of accuracy and numerically dissipative integrations. 4.2.6.1.1 Motivation Durability of concrete structures is limited by damage caused by external loading and its interaction with environmentally induced deterioration mech- anisms (see Sections 3.1.2 and 3.3.2). Model based prognoses of the degrada- tion of such structures are, in general, adapted from coupled damage models, accounting for the transport of moisture, heat and aggressive substances and the various interactions with diffuse or localized damage. Accurate numerical methods for the time integration of these kind of processes are indispensable for successful and reliable simulation based predictions of environmentally induced aging of structures. Standard time integration schemes of the finite difference or Newmark type (see e.g. [569, 198] and [409]) are not well suited 4.2 Numerical Methods 411 for non-smooth Dirichlet boundary conditions and pronounced changes of source terms typically arising in this class of parabolic differential equations (see e.g. [260, 415]). Since the order of accuracy of these algorithms according to the Dahlquist theorem [226] is restricted by two, adaptively controlled Newmark schemes or alternative time integration schemes are important ingredients of an efficient numerical strategy for the solution of multifield problems arising in durability oriented structural analyses. 4.2.6.1.2 Newmark-α Time Integration Schemes Newmark-α time stepping schemes represent a family of algorithms using finite difference based classical Newmark approximations of velocities and displacements and the strongly fulfilled semidiscrete algorithmic balance equa- tion. The algorithmic balance equation is characterized by two time instants within a typical time step where selected terms of the balance equation are evaluated. This generalized family of Newmark type integration schemes collects the most popular integration schemes in industrial applications and engineering science: The classical Newmark method [568, 569], the Hilber-α method [368] and the Bossak-α method [854]. The generalized Newmark-α method is identical to the combination of the famous Hilber-α and Bossak-α methods. In the paper [198] this method is developed, the numerical proper- ties are investigated and the algorithm is denoted as generalized-α method. In the present book we are using the denotation Newmark-α in honor of the great idea of Nathan Mortimore Newmark which represents still the main ingredient of the modern algorithm. The Newmark-α method is characterized by second order accurate in- tegrations and controllable numerical dissipation. For linear applications it is unconditionally stable and in the non-linear case it can be simply modi- fied to an energy conserving/decaying integration scheme [748, 61, 456, 461]. These positive features of the algorithm combined with its incomplex imple- mentation give reasons for its popularity in science and engineering. However, the Dahlquist theorem [226] anticipates the boundless achievement of the Newmark-α scheme. 4.2.6.1.3 Galerkin Time Integration Schemes Galerkin time integration schemes are based on the temporal weak formu- lation of the ordinary differential equation and the finite element approxima- tions of the state variables and the weight function. According to the weak and strong fulfillment of the continuity of the primary variable in-between two time steps, Galerkin methods are distinguished in their discontinuous and continuous versions, respectively. Historically, first ideas of Galerkin time in- tegration schemes have been published at the end of the 1960’s. In particular, [55, 292, 589] have proposed the temporal weak formulation of semidiscrete balance laws. [399, 56] have presented the continuous Galerkin method for 412 4 Methodological Implementation the discretization of systems of first order differential equations. The accu- racy of these methods has been improved by [616, 375] by using higher order polynomials analogous to the spatial p-finite element method [72]. Discontinuous Galerkin methods have been introduced as spatial dis- cretization techniques by [663] and [480]. Later, the idea of the weak formu- lation of the continuity condition of primary variables has been applied by [408, 261] for the development of discontinuous Galerkin time integration schemes. In [204] a review on the development of discontinuous Galerkin methods is presented. Furthermore, the textbooks by [415, 260] include nu- merous applications of discontinuous and continuous Galerkin methods. In the present section discontinuous and continuous Galerkin time in- tegration schemes for the solution of non-linear semidiscrete multiphysics problems are developed within a generalized framework. This generalized for- mulation is specialized to the discontinuous Bubnov-Galerkin method and the continuous Petrov-Galerkin method. For the temporal approximation of the state variables and the weight function Lagrange shape functions of arbitrary polynomial degree p in terms of the natural time coordinate ξ t ∈ [−1, 1] are used. Furthermore, in Section 4.2.8.2 the Galerkin time integration schemes are enriched by error estimates of the h-andp-method in order to obtain information on the accuracy of the investigated methods. Adaptive time stepping schemes, presented in Section 4.2.8.2, complete the collection of numerical methods for the efficient numerical analysis of highly non-linear initial value problems. 4.2.6.2 Newmark-α Time Integration Schemes Originally the present Newmark methods have been designed for the in- tegration of linear structural dynamics [569, 368, 854, 198]. However, the advantageous properties of these methods can also be transfered to first or- der semidiscrete balance equations of durability mechanics. For the sake of generality the Newmark-α method is presented for second order non-linear balance equations. A version for first order differential equations is simply obtained as special case by cancelation of the second time derivatives and the associated generalized tangent mass matrix [409, 455]. In the present section a brief summary of the Newmark-α method based on the papers [568, 569, 368, 854, 377, 378, 376, 198] and the textbooks [53, 54, 90, 102, 106, 225, 395, 396, 853, 855, 870] in the context of linear and non-linear structural dynamics is given. 4.2.6.2.1 Non-linear Semidiscrete Initial Value Problem The starting point for the development of Newmark-α integration schemes is the non-linear semidiscrete initial boundary value problem which is given in terms of the semidiscrete balance equation and initial conditions 4.2 Numerical Methods 413 r i ( ¨ u, ˙ u, u)=r(t), ¨ u(t = t 0 )= ¨ u 0 , ˙ u(t = t 0 )= ˙ u 0 , u(t = t 0 )=u 0 (4.88) of the state variables ¨ u, ˙ u and u. r i and r are the generalized internal and external force vectors, respectively. Linearization of equation (4.88) with respect to the state variables defines the generalized tangent matrices ∂r i ( ¨ u, ˙ u, u) ∂ ¨ u = M( ¨ u, ˙ u, u) ∂r i ( ¨ u, ˙ u, u) ∂ ˙ u = D ( ¨ u, ˙ u, u) ∂r i ( ¨ u, ˙ u, u) ∂u = K( ¨ u, ˙ u, u) (4.89) denoted as generalized tangent mass matrix M, generalized tangent damping matrix D and generalized tangent stiffness matrix K. 4.2.6.2.2 Numerical Concept of Newmark-α Time Integration Schemes Figure 4.22 summarizes the main development steps of Newmark-α schemes for integrating non-linear first and second order initial value problems: 1. Subdivision of the time interval of interest [t 0 ,T]intimestepsΔt and consideration of a representative time step [t n ,t n+1 ]. 2. Approximation of state variables u, ˙ u and ¨ u using Newmark approxi- mations [569] and generalized mid-point approximations [198]. 3. Evaluation of the semidiscrete balance equation (4.88) at generalized mid- points t n+1−α of the representative time interval [198]. 4. Iterative Newton-Raphson solution of the resulting effective balance equation. 5. Repetition of steps 2 4. for the successive solution of durability mechanics within the time interval [t 0 ,T]. 1. subdivision of time interval in time steps Δt 2. time approximations 3. time t n+1−α 4. N-R-iteration 5. repetition of steps 2 4. t 0 t NT t n t n+1 t 0 T t Δt t n t n+1 t u n+1 ˙u n+1 ¨u n+1 u(t) ˙u(t) ¨u(t) u n ˙u n ¨u n t n t n+1 t t n t n t n t n t n t n t n t n t n t n+1 t n+1 t n+1 t n+1 t n+1 t n+1 t n+1 t n+1 t n+1 t n+1−α t n+1−α t n+1−α t n+1−α t n+1−α t n+1−α t n+1−α t n+1−α t n+1−α α ∈ [0, 1] t n+1−α u n u n+1 r n r n+1 KK KK KK KK K t NT t n t n+1 t 0 t Fig. 4.22. Design of Newmark type time integration schemes 414 4 Methodological Implementation 4.2.6.2.3 Time Discretization As a basis of the numerical integration, the time interval of interest [t 0 ,T]is subdivided in NT time steps Δt. [t 0 ,T]= NT −1  n=0 [t n ,t n+1 ] Δt = t n+1 − t n (4.90) The state variables at the beginning of the time step u n = u(t n ), ˙ u n = ˙ u(t n ) and ¨ u n = ¨ u(t n ) are given and the state variables at the end of the time step u n+1 , ˙ u n+1 and ¨ u n+1 should be determined by the time stepping scheme. 4.2.6.2.4 Approximation of State Variables The approximation of state variables is realized by the combination of New- mark [569] and generalized mid-point approximations [198]. Newmark ap- proximations are based on the assumption of linear varying accelerations ¨ u and the inclusion of Newmark time integration parameters γ and β. ¨ u(τ)= ¨ u n + 2γ Δt [ ¨ u n+1 − ¨ u n ][t − t n ] ¨ u(τ)= ¨ u n + 6β Δt [ ¨ u n+1 − ¨ u n ][t − t n ] (4.91) Single and double integrations of the linear acceleration ansatz, evaluation of the resulting velocity, displacement approximations at the time t = t n+1 and solution of the resulting equations for ˙ u n+1 and u n+1 yields the well known Newmark approximations, compare Figure 4.23. mid-point approximation Newmark approximation second derivative ¨u first derivative ˙u primary variable u t n t n+1 t ¨u n+1 ¨u(t) ¨u n t n t n+1 t ˙u n+1 ˙u(t) ˙u n t n t n+1 t u n+1 u(t) u n t n t n+1−α m t n+1 t ¨u n+1 ¨u n+1−α m ¨u n+1−α m ¨u n+1−α m ¨u n+1−α m ¨u n+1−α m ¨u n+1−α m ¨u n+1−α m ¨u n+1−α m ¨u n+1−α m ¨u n t n t n+1−α d t n+1 t ˙u n+1 ˙u n+1−α d ˙u n+1−α d ˙u n+1−α d ˙u n+1−α d ˙u n+1−α d ˙u n+1−α d ˙u n+1−α d ˙u n+1−α d ˙u n+1−α d ˙u n t n t n+1−α f t n+1 t u n+1 u n+1−α f u n+1−α f u n+1−α f u n+1−α f u n+1−α f u n+1−α f u n+1−α f u n+1−α f u n+1−α f u n Fig. 4.23. Illustration of Newmark and generalized mid-point approximations of the Newmark-α method 4.2 Numerical Methods 415 ˙ u n+1 (u n+1 )= γ βΔt [u n+1 − u n ] − γ −β β ˙ u n − γ −2β 2β Δt ¨ u n ¨ u n+1 (u n+1 )= 1 βΔt 2 [u n+1 − u n ] − 1 βΔt ˙ u n − 1 − 2β 2β ¨ u n (4.92) Generalized mid-point approximations, expressed in terms of state vari- ables at times t n and t n+1 , external loads and the time integration parameters α m and α f complete the set of approximations, compare Figure 4.23. ¨ u n+1−α m ( ¨ u n+1 (u n+1 )) = [1−α m ] ¨ u n+1 (u n+1 )+α m ¨ u n ˙ u n+1−α f ( ˙ u n+1 (u n+1 )) = [1−α f ] ˙ u n+1 (u n+1 )+α f ˙ u n u n+1−α f (u n+1 )=[1−α f ] u n+1 + α f u n r n+1−α f =[1−α f ] r n+1 + α f r n (4.93) 4.2.6.2.5 Algorithmic Semidiscrete Balance Equation The algorithmic balance equation is obtained by applying the state variables at different time instants within the time interval [t n ,t n+1 ] characterized by time integration parameters α m and α f [368, 854, 198]. r i ( ¨ u n+1−α m , ˙ u n+1−α f , u n+1−α f )=r n+1−α f (4.94) Equation (4.94) represents a non-linear algebraic equation for the solu- tion of the end-point displacements u n+1 , compare equations (4.92) and (4.93). 4.2.6.2.6 Effective Balance Equation The consistent linearization of equation (4.94), including approximations (4.92) and (4.93), with respect to the end-point displacements u n+1 yields the effective balance equation for the iterative Newton-Raphson solution. K eff (u k n+1 ) Δu = r eff (u k n+1 ) (4.95) 416 4 Methodological Implementation Herein the effective tangent matrix, K eff (u k n+1 )=M( ¨ u n+1−α m , ˙ u n+1−α f , u n+1−α f ) 1−α m βΔt 2 + D( ¨ u n+1−α m , ˙ u n+1−α f , u n+1−α f ) γ[1−α f ] βΔt + K( ¨ u n+1−α m , ˙ u n+1−α f , u n+1−α f )[1−α f ] (4.96) the effective right hand side r eff (u k n+1 )=r n+1−α f − r i ( ¨ u n+1−α m , ˙ u n+1−α f , u n+1−α f ) (4.97) and the Newton correction are used. u k+1 n+1 = u k n+1 + Δu (4.98) Convergence criteria discussed in Section 4.2.5.2 for static analyses can also be applied for dynamics in order to judge the quality of the iterative solution. 4.2.6.2.7 Newmark-α Algorithm Figure 4.24 shows the algorithmic set-up of the Newmark-α method for non-linear analyses of first and second order durability problems. It is worth to mention that this algorithmic set-up already includes parts for the error based time step control explained in Section 4.2.8.2.4. Links to the calculation of element quantities and the update of history variables are marked on the right hand side by small rectangles. 4.2.6.3 Discontinuous and Continuous Galerkin Time Integration Schemes For the higher order accurate time integration of non-linear semidiscrete ini- tial value problems continuous and discontinuous Galerkin methods in time have been developed (see e.g. [399, 56, 408, 261, 415, 260]). Since the dis- continuous version of Galerkin integration schemes includes the continuous Galerkin method as a special case, the development of both methods will be described by means of the discontinuous Galerkin method of arbitrary polynomial degree p. Subsequently, the generalized method will be specialized to the continuous Galerkin method. The application of discontinuous and continuous Galerkin time in- tegration schemes is investigated by means of the non-linear first order semidiscrete initial value problem. 4.2 Numerical Methods 417 loop over time steps n n +1→ n loop over iteration steps steps k k +1→ k generate initial conditions u 0 , ˙ u 0 , ¨ u 0 generate (or read) external loads r n+1−α f incremental update velocities and accelerations ˙ u n+1 (u k n+1 ), ¨ u n+1 (u k n+1 ) state variables at t n+1−α ¨ u n+1−α m , ˙ u n+1−α f , u n+1−α f calculation internal forces r i (◦), κ f calculation tangential stiffness matrix M(◦), D(◦), K(◦) calculation effective right hand side r eff (u k n+1 ) calculation effective tangent K eff (u k n+1 ) solution effective iterative structural equation K eff Δu n+1 = r eff Newton correction current displacements u k+1 n+1 = u k n+1 + Δu n+1 convergence check η k+1 u ≤ η u final update velocities and accelerations ˙ u n+1 (u n+1 ), ¨ u n+1 (u n+1 ) final update internal variables κ f estimation/indication error e error check ν 1 η ≤ e ≤ ν 2 η retain Δt calculation time step Δt new check (y|n) ν 1 η>e next time interval [t n+1 ,t n+2 ] −→ [t n ,t n+1 ] retry [t n ,t n+1 ] Fig. 4.24. Algorithmic set-up of Newmark-α schemes including error controlled adaptive time stepping [461] r i ( ˙ u, u)=ru(t 0 )=u 0 , ˙ u 0 , ¨ u 0 (4.99) It is worth to mention that non-linear second order semidiscrete initial value problems of type (4.32) can be transformed into first order semidiscrete initial boundary value problems and additional constraints.  r i ( ˙ v, v, u) ˙ u −v  =  r 0  r  i ( ˙ u  , u  )=r  (4.100) [...]... different models However, the element formulation is constructed in such a way that it includes both the continuum and the structural level This can be explained as follows Using one element over the thickness practically means to work with a one-directior shell theory We are then at the structural level The division of such an element into several elements over the thickness (to be performed in the context... generalized modeling of deterioration problems of concrete materials and structures, spatial and temporal discretization methods and iterative solution methods, a modular numerical solution method is designed The strict separation of the algorithmic, element and model levels allows for an effective and reliable computational implementation of time dependent and stationary durability mechanics models... of the hanging node context) provides a classical continuum modelling of the considered domain In contrast to many earlier approaches we do not have to neglect the stress in thickness direction at the structural level [773], [774] and [776] propose an implicit residual error estimate of equilibrated local Neumann type, see in this context also the contributions of [416] and [39] A similar view point, . Newmark-α Time Integration Schemes Originally the present Newmark methods have been designed for the in- tegration of linear structural dynamics [569, 368, 854, 198]. However, the advantageous properties. the continuum and the structural level. This can be explained as follows. Using one element over the thickness practically means to work with a one-directior shell theory. We are then at the structural. textbooks [53, 54, 90, 102, 106, 225, 395, 396, 853, 855, 870] in the context of linear and non-linear structural dynamics is given. 4.2.6.2.1 Non-linear Semidiscrete Initial Value Problem The starting