Lifetime-Oriented Structural Design Concepts- P19 potx

The proposed method is an evolution of the classicalCraig-Bamptonapproach to partially damaged problems and incorporatesdis-an advdis-antageous decomposition of the entire structure into

350 300 250 200 150 100 50 0

Fig 4.94 Numerical investigation of crack propagation of an anchor pull-out test:

Load-displacement curve

displacement reaches u z ≈ 0.21 [mm] the carck propagation nearly stops due

to the fact that most of the load is carried by the area of counter pressure.This is evident from Figure(4.93,right) that shows the strongly increased value

of the stress component σ33at the counter pressure support On the contrary

in the beginning of the cracking process (figure(4.93,left)) there is nearly nocompression at the area of counter pressure because of the load carrying of theconcrete in the vicinity of the anchor plate When the displacement reaches

u z ≈ 0.21 [mm], the load is primarily induced to the counter pressure

sup-ports by the nearly fully separated upper part of the concrete structure Thisbearing behaviour of the system is primarly characterized by the position ofthe crack surface that is running underneath the pressure supports In thecase that the crack reaches the upper surface of the concrete structure being

in front of the supports, there is going to be a structural softening of theconcrete block This interpretation is similiary to the one in [301] where asimiliar numerical test is performed As can be further deduced from Figure4.93 there is a high compression atop of the anchor plate, which would alsolead to compression failure of concrete This does not occur here, because thisanalysis only takes failure of concrete in tension into account

4.2.10 Substructuring and Model Reduction of Partially Damaged

Structures

Authored by Christian Rickelt and Stefanie Reese

The objective of this section is to present an efficient strategy to copewith demanding dynamical simulations of complex structures We are in par-ticular interested in long term calculations like life cycle investigations Our

ansatz emanates from the idea that a structure only comprises locally tributed damaged zones The proposed method is an evolution of the classicalCraig-Bamptonapproach to partially damaged problems and incorporates

dis-an advdis-antageous decomposition of the entire structure into linear dis-and linear segments The former are reduced by model reduction techniques Inthe nonlinear components the deterioration of the material is simulated by

non-a continuum dnon-amnon-age model for ductile dnon-amnon-age phenomennon-a of metnon-als Thismaterial model is included into a new solid beam finite element formulationQ1SPb based on reduced integration with hourglass stabilisation The sectioncloses with an example to validate the proposed strategy and to determine theapproximation error caused by the model reduction of the linear components

4.2.10.1 Motivation and Overview

The importance of simulation tools for the design and lifetime estimation

of complex structures increases noticeably Highly demanding tasks and quirements on the accuracy of numerical computations necessitate powerfulsimulation techniques and more elaborate models Since in general these mod-els cannot be solved analytically the underlying ordinary or partial differentialequations are often discretised and solved by the finite element method.Dynamic problems require beside the spatial discretisation as well a dis-cretisation in time This can be accomplished by implicit or explicit timeintegration schemes While the former lead to large, sparse equation systems,whose solution is memory and time consuming, the disadvantage of the ex-plicit methods is that they are only conditionally stable Accordingly its timestep size is limited by a critical time step As a consequence the latter schemesare not suitable for long-term computations

re-Strategies to solve such complex long-term calculation numerically efficienthave been a topic of research in the field of engineering technologies and math-ematics More related to the former one are the condensation methods and thestaggered schemes The domain decomposition methods are mostly researched

in mathematics while the model reduction techniques are established in bothcommunities

The idea of model reduction is to transform the large equation systeminto a small, dynamically equivalent substitute model Powerful model reduc-tion methods exist for the solution of linear and weakly nonlinear systems.Highly nonlinear problems like the evolution of damage cannot be solved [761]and [808] employ nonlinear model reduction techniques to reduce geometri-cally nonlinear problems and fluid-structure interactions, respectively For anoverview of model reduction methods in general refer to the publications of[111, 761, 49, 50, 651, 288, 76, 51, 451, 287, 236, 583, 485, 813]

Linear model reduction methods can be classified into three differentgroups: the singular value decomposition (SVD)-based methods, the Krylov-based methods and the condensation methods Out of the group of SVD-based model reduction two well-known methods are the Modal Reduction

(MODRED) (see the standard text books [195, 203]) and the Proper onal Decomposition (POD) (see [121, 384, 755]) Two important members ofthe Krylov-based methods are the Load-Dependent Ritz Vectors (LDRV)(see [328, 203, 585]) as well as the Pad´e-Via-Lanczos algorithm (PVL).

Orthog-In this publication a symmetric version for undamped second order systems,called symmPVL, is used (see e.g [288, 76])

The above mentioned group of condensation methods comprise model duction methods which condense the inner degrees-of-freedom (dofs) and con-serve the physical interface dofs This idea can be structured into static as well

re-as dynamic condensation methods According to [620] the latter yield exactlyreduced dynamical systems which even for linear problems result in nonlinearfunctions which depend on the frequency Hence they are not numerically effi-cient In contrast the static condensation of [333] is suitable to only a limitedextent for the model reduction of linear dynamic systems Corresponding tothe text book of [651] component mode synthesis techniques (CMS techniques)establish a significant extension of Guyan’s reduction to methods with hy-brid transformation matrices Initialised by the classical papers of [401] and[216] (see also the review article of [215]), CMS techniques with fixed inter-faces have been developed within the scope of reduction methods for largestructural dynamic models and the design of single dynamical components

of challenging structures These methods are based on the superposition ofdifferent contribution of the deformations Hence they are only valid for lin-ear problems For further information on CMS techniques see the overviewpublications of [217, 206, 498, 722, 511]

Alternatively the objective of staggered algorithms (see for example [548])

is to solve multi-field or dynamic problems which are discretised by differenttime integration schemes (implicit/explicit) and varying time step lengths(subcycling) within each subdomain (see the classical papers of [108] and[107]) An interesting approach of an explicit-implicit multi time step methodfor nonlinear structural dynamics which prescribes the continuity of velocities

at the interface and uses a dual Schur formulation has been published by[323] [274] and [275] extend this ansatz successively to nonmatching meshesand linear as well as nonlinear model reduction techniques

The primary interest of domain decomposition methods, grouped into lapping (”Schwarz methods”) and non-overlapping (”iterative substructur-ing methods”) methods, is the development of highly efficient parallelisediterative solution techniques These usually result in conjugate gradientschemes The latter are subdivided into the Neumann-Neumann- (Bal-anced Domain Decomposition (BDD), [515]), Dirichlet-Neumann- andDirichlet-Dirichlet-Algorithm (Finite Element Tearing and Interconnect-ing Method (FETI), [273]) Most of the publications deal with linear problems.Applications to geometrically nonlinear problems can be found in the papers

over-of [218] and [272]

Structural Dynamics

Modelling (FEM)

Linear Components Nonlinear Components

Model Reduction SubstructureTechnique ModellingDamage

Fig 4.95 Concept for the efficient simulation of dynamic, partially damaged

struc-tures by means of model reduction and substructuring

This ansatz, depicted in Figure 4.95, enables to decompose any discretisedstructure strictly into its linear and nonlinear components In the latter theevolution of ductile damage of metals is considered The damage model isbased on the void growth model of [691, 690] Further we assume that theevolution of damage is only influenced indirectly by the dominating linearsubsystems Hence, to increase the efficiency of the strategy, the latter are re-duced by model reduction methods in conjunction with the Craig-Bamptonmethod – one of the widely used CMS substructure techniques Beside thecommon Modal Reduction, linear model reduction methods of superior accu-racy – the Pad´e-Via-Lanczosalgorithm (PVL), the Load-Dependent RitzVectors (LDRV) and the Proper Orthogonal Decomposition (POD) – are em-ployed Finally the substructuring of the total structure into reduced lin-ear and nonlinear components is exploited Instead of solving a large mono-lithic equation system the system response of the redundant interfaces of thecomponents is computed Subsequently the inner dofs of all components arecalculated

One advantage of the presented concept rests upon the beneficial nation of well-known and robust linear model reduction methods, the CMS

combi-technique, the material as well as the efficient element formulation Anotherimportant aspect is the numerical implementation For this purpose the math-ematical development environment Matlab and the finite element programFeapare linked by the interface Feapmex1.

4.2.10.3 Derivation of a Substructure Technique for Nonlinear

Dynamics

In this section the Craig-Bampton method is summarised Afterwards thelinear model reduction methods and the employed substructure technique arediscussed

4.2.10.3.1 Craig-Bampton Method

The concept of CMS techniques has been developed [401] and later rewrittenand simplified by [216] to analyse complex structural systems decomposedinto interconnected components with fixed interfaces The Craig-Bamptonmethod superposes two different fractions of the motion: the static or con-straint modes Ψic of the matrix ΨT := 

ic IT cc

are defined as the staticdeformation of a structure when a unit displacement is applied to one in-terface dof while the remaining interface dofs are restrained The matrixIcc

herein is an identity matrix of dimension c × c The indices i and c indicate

the inner and the interface dofs, respectively The second fraction are the k (k  i) remaining inner dynamical or so-called normal modes φ j , j = 1, , k

of the fixed subsystem s They are stored column by column into the matrix

The matrixOck is a zero matrix of dimension c × k Usually the modesΦik

are represented by eigenvectors In this contribution the LDRV-, POD- orsymmPVL-vectors of alternative model reduction methods of superior accu-racy, as presented in section 4.2.10.3.2, are utilised With these modes the

physical coordinates u can be transformed by the relation

the relation (4.244) the linear equation of motion of one component s by an

leads for linear undamped systems to a component of smaller dimension in

which the interface dofs u c = q c are conserved in physical coordinates In thelatter equationM,K, b and f (t) are the mass matrix, the stiffness matrix, the

load distribution and the loading function The index r denotes the dimension

of the reduced component

4.2.10.3.2 Model Reduction of Linear Dynamic Structures

Besides our objective to save computational effort by decomposing a structureinto its linear and non-linear parts the linear substructures are approximated

by dynamically equivalent subsystems Within the Craig-Bampton methodthe fixed interface normal modes of each component are defined as a reducedset of modes by restraining all boundary dofs These modesΦik form part ofthe Craig-Bampton transformation matrixVCB (see section 4.2.10.3.1) Inthis contribution we will derive the model reduction of linear dynamic systems

within a general framework For simplicity we leave out the superscript s We

start from the ansatz

4.2.10.3.2.1 Modal Reduction

Modal Reduction, also known as Modal Truncation, is the most simpleand popular model reduction method The idea is to solve a subset of the

generalised eigenvalue problem in whichΦik = [φ1, φ2, · · · , φ k] is the reducedmodal matrix and Λkk = diag

4.2.10.3.2.2 Proper Orthogonal Decomposition

A second possibility is the POD method This method is also known asempirical eigenvectors, Karhunen-Lo`eve expansion, principle componentanalysis, empirical orthogonal eigenvectors, etc An overview of nomenclaturesused in the literature and areas of application are given e.g in [121]

The mathematical basis for the POD method is the spectral theory ofcompact, selfadjoint operators which is explained e.g in the standard textbook of [384] One problem of this ansatz is that even for small systemsthe eigenvectors of a large spatial covariance matrix have to be calculated.One approach to lower the computational costs is known as the “method ofsnapshots” ([755]) In this case each POD basis vector

is generated out of m uncorrelated zero-mean “snapshots” w j In the latter

equation w j = u j − ¯u describes the deviation of the “snapshot” u jfrom theirtemporal mean ¯u β j are unknown coefficients which have to be determined.After some derivations and using the assumption that the investigated pro-cess is ergodic (see e.g [536, 384]) only a reduced eigenvalue problem of di-

mension m × m

1

TWβ l = λ l β l with W= [w1, · · · , w m] , (4.252)

in whichW contains the m zero-mean “snapshots” has to be solved The k

basis vectors of the POD

corresponding to the eigenvalues λ1 > λ2> · · · > λ l > · · · > λ k, result from

a linear combination of the zero-mean “snapshots”

4.2.10.3.2.3 Pad´ e-Via-Lanczos Algorithm

The Pad´e-Via-Lanczos algorithm and the Dual Rational Arnoldimethod belong to the Krylov-based model reduction methods This

system-theoretical approach for first order differential equations can also

be applied to second order systems A differential-algebraic equation system

Kii u i (t) +Mii u¨i (t) = b i f (t) y(t) = c i u i (t) (4.254)

is converted by the Laplace transformation to the transfer function

H(s) = c i [s2Mii+Kii]−1 b

Here the equations are given for a single input single output (SISO) systems

The measurement vector c i of the dimension (1× i) relates the displacement

vector u i (t) to the measured output y(t) of the system.

The transfer function (4.255) is re-written and expanded around an

expan-sion point σ2 into a power series (Laurent or Taylor series)

For the special case c T

i = b i and symmetric, positive definite matrices

Mii and Kii [290] show that the reduced systems are stable According to

[486] for mechanical problems purely imaginary expansion points σ = jω c are

chosen (ω c is the angular frequency in the centre of the interesting frequencyrange) Employing a Cholesky decomposition Kii − σ2Mii =NiiNT

The reduced vector r k = (Φik)T r i is computed according to the projection(4.248) The proposed algorithm for symmetric positive definite system istermed in the following symmPVL.

4.2.10.3.2.4 Load-Dependent Ritz Vectors

The method of Load-Dependent Ritz Vectors (LDRV) is an approach ofstructural dynamics In the special case that the matricesMii and Kii are

symmetric positive definite matrices, the expansion point is zero (σ = 0), the

basis vectors are mass normalised and the input and measuring vectors are

According to [585] the method delivers the following reduced coupled ferential equation system:

dif-Tkk q¨k+Ikk q k={β1, 0, · · · , 0} T f (t) (4.259)Herein the stiffness matrix and the mass matrix are degenerated to an iden-tity matrix Ikk and a tridiagonal matrix Tkk in generalised coordinates,respectively If we assume that the load distribution on the structure is con-stant during the simulation, the projected external load vector reduces to

b k ={β1, 0, · · · , 0} T f (t) The scalar value β1 =

ϕ T

1 Mϕ1 is given by the

first not mass normalised Ritz vector ϕ1

4.2.10.3.3 Substructuring in the Framework of Nonlinear Dynamics

The derivation is based, as displayed in Figure 4.96, on a decomposition ofthe structure into two arbitrary components Only with the assembly and thesolution of the equation system one of the two components is limited to areduced linear subsystem

4.2.10.3.3.1 Discretisation and Linearisation

Starting point is the balance of linear momentum of a subsystem s in the

reference configuration

DivP (s) + ρ0b (s) v − ρ0u¨(s)+F (s) c =0 s = 1, 2 (4.260)Herein isP (s)the first Piola-Kirchhoff stress tensor,b (s) v the volume forcevector, ¨u (s) the acceleration vector and ρ0 the mass density in the referenceconfiguration Additionally interface forcesF (s) c have to be introduced Theseinternal forces only possess non-zero components at the redundant interfaces

Γ c (s) As constraints the equilibrium of the interface forces

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Interface nodeInternal node

since the interface area is of identical size In the field of contact simulations

λ c is named contact pressure If equation (4.263) is inserted into the strong

forms (4.260) and (4.262) in accordance with [473, 856] each subsystem s can

be transformed into the weak form

which refers to the initial configuration Ω (s)0 The last integral in Equation(4.264a) specifies the virtual work of the Lagrange multipliers λ at theinterface Equation (4.264b) enforces the constraint condition in a weak sense.

δu (s) and δ λare the test functions of the independent variablesu (s) and λ

The external boundary of each component Γ (s) = Γ u (s) ∪ Γ (s)

T ∪ Γ (s)

c consists of

the Dirichlet boundary Γ u (s) , the Neumann boundary Γ T (s)and the interface

boundary Γ c (s) The vector ¯T (s)represents the external tensions

Beside a spatial discretisation according to the isoparametric element cept the constraints at the interface are fulfilled in a strong manner Hencethe virtual work of the Lagrange multiplier at the interface (4.264a) andthe compatibility constraint (4.264b)

are transformed into a summation over all interface nodes nn c The interface

force Λ i=λ i ·A i at node i is the product of the Lagrange multiplier λ iand

the corresponding area A i of node i δu (s) i and δΛ i are the test functions of

node i Additionally the position vectors at the corresponding interface nodes

X(1) ≡ X(2) are identical As a result in Equation (4.265b) the difference

between the position vectors of the deformed configuration x (s) is replaced

by the the difference of the displacements u (s) The matrixdCreduces in the

discrete case to a Boolean allocation matrix At this the index d indicates that

the interface constraints are fulfilled in a strong manner

Finally incorporating any time integration like e.g the Newmark method

we result in the fully discretised nonlinear equation system subjected to aconstraint:

n + 1 denotes the current time step which is omitted below R(u) and Pext

are the inner and the external force vectors, respectively d G g (u n+1) and

d G c (u n+1) are the residual vectors

To solve Equation (4.266) by the Newton-Raphson method a consistent

linearisation with respect to the independent variables u and Λ leads to the

linearised and decoupled system

The indices eff and m denote that in the tangential stiffness matrixKT eff

the time discretisation is already included as well as m signifies the number of

iterations Both indices are omitted below in order to improve the readability

4.2.10.3.3.2 Primal Assembly

The objective of this strategy is to solve partially reduced systems withlocal nonlinearities such as material damage behaviour Thus in the follow-ing derivation the components (1) and (2) are regarded to be the nonlinearsubsystem (nl) and subsystem (2) the reduced, but linear subsystem (lin).The (* )-symbol is used to indicate reduced components The transforma-tion of the reduced linear subsystem (2) results from the presented Craig-Bamptontransformation u(2)=VCB(2)q(2) The vectors u(2) and q(2) are thephysical and the generalised coordinates of the linear component (2)

In the following synthesis of the components, which is published by [217]for linear systems, the interface forces are regarded as Lagrange multipli-ers At first the compatibility condition (4.266b) has to be transformed intogeneralised coordinates

into e coordinates which have to be kept and d coordinates which are deleted

(4.274)which is further multiplied byRT, we arrive at the direct assembled globalsystem

The iteration index m + 1 in Equation (4.274) is only applied to underline

that the latter equation depends on the current Lagrange multipliers

4.2.10.3.3.3 Solution of the Decomposed Structure

The most common solution in the literature is the monolithic solution ofthe overall equation system In this work the existing decomposition of thestructure into large linear and small nonlinear components and the modelreduction of the linear subsystems is exploited to substitute the solution ofone monolithic equation system by the efficient solution of a number of smallequation systems On the basis of Equation (4.273) the individual linear andnonlinear components are transformed by means of static condensation intothe local Schur complement systems

is independent of the Lagrange multipliers The global Schur complement

Sg and the modified global residual G gare divided into its linear and nonlinearcomponents:

ma-The complete number of subsystems N = Nlin+ Nnl consists of Nlin linear

components and Nnl nonlinear components

Finally, depending on the solution of the global Schur complement system(4.278) the internal dofs

4.2.10.4 Example: M¨ unster-Hiltruper Road Bridge

This example serves to validate the overall strategy At first the solution ofthe decomposed but unreduced structure is compared to a monolithic solution.Subsequently the influence of the different employed model reduction methods

on the accuracy of the computation is investigated

In this example the bar-like structure of an arched steel bridge is regarded.Its geometry is based on the road bridge in M¨unster-Hiltrup (federal roadB54), which is one of the reference buildings of the Collaboratory ResearchCentre 398 The dimensions, cross sections and parameters are chosen ac-cording to existing mechanical drawings In accordance with the proposedstrategy the structure is subdivided into linear and nonlinear components.The former are additionally reduced to increase the numerical efficiency Theresults of the simulation are compared to the solution of a monolithic transientanalysis

The road bridge, depicted in Figure 4.97, is l = 87.37 m long, b = 17.85 m wide and h = 13.68 m high In the nonlinear substructures we model the

evolution of material deterioration for ductile damage behaviour of metals.For this purpose we extend the material model of Rousselier (see [691]).Compare also [251] where an alternative approach has been chosen The

material parameters read E = 210 000 N/mm2, ν = 0.3, ρ0= 7.85 kg/dm3,

σy0= 400 N/mm2, H = 2100 N/mm2, D = 2.0, σk = 400 N/mm2, f0= 0.01,

fN= 0.25, εN= 0.2 and s = 0.4.