Lifetime-Oriented Structural Design Concepts- P16 pptx

Newmark-α time integration methods are using the semidiscrete balance equation evaluated at one selected time instant within a time step and finite... 4.2.6.1.2 Newmark-α Time Integration

is obtained Since the linear system of equations (4.81) is non-symmetric andlooses, furthermore, the band structure of the generalized tangent matrix, it

is solved by applying the partitioning technique [91] Therefore, the partial

incremental solutions Δur and Δuλare calculated in advance

K−1(uk

n+1 ) Δur = λ k n+1r−ri(uk n+1 ), K−1(uk

n+1 ) Δuλ=r (4.83)Afterwards the increments

and

Δλ = − f (u

k n+1 , λ k n+1 ) + f ,u(uk n+1 , λ k n+1)· Δur

f ,u(uk n+1 , λ k n+1)· Δuλ + f,λ(uk 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, is

imple-mented As shown in Figure 4.20, the load factor is increased by one and the

resulting displacement increment Δuλ and step length s0 are calculated

Δuλ=K−1(u

n)r, s0=

Afterwards the increments of the displacement vector and the load factor are

scaled, such that the user defined step length s is obtained.

4.2.6 Temporal Discretization Methods

Authored by Detlef Kuhl and Sandra Krimpmann

The present section is concerned with the numerical methods for the timeintegration of non-linear multiphysics problems by means of Newmark-

α methods as well as discontinuous and continuous Galerkin schemes.

Newmark-α time integration methods are using the semidiscrete balance

equation evaluated at one selected time instant within a time step and finite

Table 4.5 Constraints and load factor increments of selected arc-length methods

(f = f (uk n+1 , λ k

n+1), [817])

method constraint f (uk+1

n+1 , λ k+1 n+1) increment Δλ(uk

n+1 , λ k n+1)

functions of the p-Lagrange type and the well known Gauß-Legendre

quadrature of associated time integrals It is shown that arbitrary order curate integration schemes can be developed within the framework of the

ac-proposed temporal p-Galerkin methods.

4.2.6.1 Introduction

For the integration of time dependent durability problems either Newmarktype finite difference methods or Galerkin type finite element methods are

loop over arc-length steps n = 0, N T − 1

scaling displacement and load factor increment Δuand Δλ

update displacements and load factor u1n+1 and λ1n+1

n+1)

n+1)partial incremental solutions Δur and Δuλ

discon-of accuracy and numerically dissipative integrations

4.2.6.1.1 Motivation

Durability of concrete structures is limited by damage caused by externalloading 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 andthe various interactions with diffuse or localized damage Accurate numericalmethods for the time integration of these kind of processes are indispensablefor successful and reliable simulation based predictions of environmentallyinduced aging of structures Standard time integration schemes of the finitedifference or Newmark type (see e.g [569, 198] and [409]) are not well suited

for non-smooth Dirichlet boundary conditions and pronounced changes ofsource 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 controlledNewmark schemes or alternative time integration schemes are importantingredients of an efficient numerical strategy for the solution of multifieldproblems 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 anddisplacements and the strongly fulfilled semidiscrete algorithmic balance equa-tion The algorithmic balance equation is characterized by two time instantswithin a typical time step where selected terms of the balance equation areevaluated This generalized family of Newmark type integration schemescollects 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 themain 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 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 theNewmark-α scheme.

modi-4.2.6.1.3 Galerkin Time Integration Schemes

Galerkintime integration schemes are based on the temporal weak lation of the ordinary differential equation and the finite element approxima-tions of the state variables and the weight function According to the weakand strong fulfillment of the continuity of the primary variable in-between twotime steps, Galerkin methods are distinguished in their discontinuous andcontinuous 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 semidiscretebalance laws [399, 56] have presented the continuous Galerkin method for

formu-the discretization of systems of first order differential equations The racy of these methods has been improved by [616, 375] by using higher order

accu-polynomials analogous to the spatial p-finite element method [72].

Discontinuous Galerkin methods have been introduced as spatial 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 integrationschemes In [204] a review on the development of discontinuous Galerkinmethods is presented Furthermore, the textbooks by [415, 260] include nu-merous applications of discontinuous and continuous Galerkin methods

dis-In the present section discontinuous and continuous Galerkin time tegration schemes for the solution of non-linear semidiscrete multiphysicsproblems are developed within a generalized framework This generalized for-mulation is specialized to the discontinuous Bubnov-Galerkin method andthe continuous Petrov-Galerkin method For the temporal approximation

in-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- and p-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 thecollection of numerical methods for the efficient numerical analysis of highlynon-linear initial value problems

4.2.6.2 Newmark-α Time Integration Schemes

Originally the present Newmark methods have been designed for the tegration of linear structural dynamics [569, 368, 854, 198] However, theadvantageous properties of these methods can also be transfered to first or-der semidiscrete balance equations of durability mechanics For the sake of

in-generality the Newmark-α method is presented for second order non-linear

balance equations A version for first order differential equations is simplyobtained as special case by cancelation of the second time derivatives and theassociated 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 linearand 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

ri(¨u, ˙u,u) =r(t), ¨u(t = t0) = ¨u0, u˙(t = t0) = ˙u0, u(t = t0) =u0

(4.88)

of the state variables ¨u, ˙uand u r i and r are the generalized internal and

external force vectors, respectively Linearization of equation (4.88) withrespect to the state variables defines the generalized tangent matrices

for integrating non-linear first and second order initial value problems:

1 Subdivision of the time interval of interest [t0, T ] in time steps Δt and consideration of a representative time step [t n , t n+1]

2 Approximation of state variablesu, ˙u and ¨u using Newmark mations [569] and generalized mid-point approximations [198]

approxi-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 balanceequation

5 Repetition of steps 2.-4 for the successive solution of durability mechanics

within the time interval [t0, T ].

1 subdivision of time interval in time steps Δt

2 time approximations 3.time t n+1 −α 4 N-R-iteration

un+1, ˙un+1and ¨un+1should 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 mark[569] and generalized mid-point approximations [198] Newmark ap-proximations are based on the assumption of linear varying accelerations ¨u

New-and the inclusion of Newmark time integration parameters γ New-and β.

Single and double integrations of the linear acceleration ansatz, evaluation of

the resulting velocity, displacement approximations at the time t = t n+1 andsolution of the resulting equations for ˙un+1 andun+1 yields the well knownNewmarkapproximations, compare Figure 4.23

Fig 4.23 Illustration of Newmark and generalized mid-point approximations of

the Newmark-α method

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, compareFigure 4.23

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]

ri(¨un+1−αm , ˙un+1−αf ,un+1−αf) =rn+1−αf (4.94)

Equation (4.94) represents a non-linear algebraic equation for the tion of the end-point displacementsun+1, compare equations (4.92) and (4.93)

solu-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 un+1

yields the effective balance equation for the iterative Newton-Raphsonsolution

Keff(uk n+1 ) Δu=reff(uk n+1) (4.95)

Herein the effective tangent matrix,

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 errorbased 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 theright 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 tial value problems continuous and discontinuous Galerkin methods in timehave been developed (see e.g [399, 56, 408, 261, 415, 260]) Since the dis-continuous version of Galerkin integration schemes includes the continuousGalerkinmethod as a special case, the development of both methods will

ini-be descriini-bed 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 tegration schemes is investigated by means of the non-linear first ordersemidiscrete initial value problem

in-loop over time steps n

n + 1 → n

loop over iteration steps steps k

k + 1 → k

calculation tangential stiffness matrix M(◦), D(◦), K(◦)calculation effective right hand side reff(uk n+1)calculation effective tangent Keff(uk n+1)solution effective iterative structural equation KeffΔun+1=reff

Newton correction current displacements uk+1 n+1=uk

u ≤ η u

final update velocities and accelerations u˙n+1(un+1),u¨n+1(un+1)

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]

ri( ˙u,u) =r u(t0) =u0, u˙0, ¨u0 (4.99)

It is worth to mention that non-linear second order semidiscrete initialvalue problems of type (4.32) can be transformed into first order semidiscreteinitial boundary value problems and additional constraints

r

i( ˙u ,u) =r (4.100)

It is obvious that this coupled system can also be expressed by thesymbolic equation of a non-linear vector equation Consequently, the presentdiscontinuous and continuous Galerkin time integrations schemes can also

be applied to second oder semidiscrete initial value problems, for details see[460]

The state variables at the time instant t n are known and the state

variables at the time instant t n+1 are to be computed by means of a timeintegration scheme

4.2.6.3.2 Continuity Condition

The continuity of the primary variable uat the boundaries of the

individ-ual time intervals [t n , t n+1] can be enforced by the continuity or the jumpcondition, compare Figure 4.25

Fig 4.25 Galerkin time integration schemes illustrated by means of polynomial

degree p = 3: Approximation of the primary variable u, defintion of the jump

u n

,

illustration of the physical time t, definition of the natural time coordinate ξ t andthe position of Gauß points (GP)

In equation (4.102) the primary variables at the end of the previous timestep and the beginning of the current time step are defined by

4.2.6.3.3 Temporal Weak Form

The semidiscrete equilibrium equation is transformed into the temporal weakform by multiplication with an arbitrary weight function w(t) and integra- tion over the time interval [t n , t n+1] Furthermore, the continuity condition(4.102) is only weakly enforced using the weight w1 = w(t n) The sum

of these integrals defines the weak form of discontinuous Galerkin methods

The linearization of the weak form (4.104) with respect to the unknowns ˙u(t),

u(t) and u1 leads to the linearized weak form of discontinuous Galerkinmethods

gen-4.2.6.3.5 Temporal Galerkin Approximation

The state variables, the weight function and the increments are temporally

approximated by shape functions N i according to temporal nodes i and

associated nodal values (e.g ui) Standard Lagrange polynomials of

the polynomial degree p (see equation (4.20)) as function of the natural coordinate ξ t ∈ [−1, 1] are used as shape functions.

ξ t i= 2 [i − 1]

In accordance with the isoparametric concept, the physical time t and the displacement vector are approximated by using the shape functions N i of

the polynomial degree p Independent approximations of the weight function

wby shape functions ¯N i with the polynomial degree ¯p are assumed.

For equidistant nodal times t j − t j −1 = Δt/p with j ∈ [2, p + 1] the constant

Jacobi transformer J t = Δt/2 is obtained The derivatives of the shape

functions with respect to the natural time coordinate ∂N i (ξ t )/∂ξ t = N i

;t (ξ t)are obtained according to equation (4.21) Substitution of the approximations(4.107) and (4.108) in the linearized weak form (4.105) yields the discretizedlinearized temporal weak form of discontinuous Galerkin methods

are used These integrals are computed numerically by the Gauß-Legendre

integration scheme using standard Gauß points ξ t l with l ∈ [1, NG t] and

weights α l (see e.g [870])

4.2.6.3.6 Discontinuous Bubnov-Galerkin Schemes dG(p)

For arbitrary weight functionswi equation (4.112) can be transformed into

a linear system of equations (i ∈ [2, ¯p + 1] and j ∈ [2, p + 1])

(4.115)

which can be formally written as an effective linearized system of equations

and solved for the increments of the primary variables ΔudG For ¯p = p

and, consequently, for ¯N i = N i equation (4.116) leads to non-singular lutions Accordingly, discontinuous Galerkin time integrations schemes of

so-the polynomial degree p, denoted as dG(p)-methods, are Bubnov-Galerkin

methods

4.2.6.3.7 Continuous Petrov-Galerkin Schemes cG(p)

For continuous Galerkin time integration schemes the continuity condition(4.102) is strongly fulfilled and the primary variable u1 = u0 = u−

For the solution of equation (4.119) ¯p = p − 1 and, consequently, ¯ N i = N i

are required Accordingly, continuous Galerkin time integration schemes of

the polynomial degree p, denoted as cG(p)-methods, are Petrov-Galerkin

up+1

For alternative formulations of convergence criteria see Table 4.3

4.2.6.3.9 Algorithmic Set-Up of Galerkin Schemes

Figure 4.26 shows the algorithmic set-up of continuous and discontinuousGalerkinschemes for the time integration of non-linear first order semidis-crete initial value problems Links to the calculation of element quantities andthe update of history variables are marked on the right hand side by smallrectangles