Lifetime-Oriented Structural Design Concepts- P21 pptx

scenar-4.5.4 Parallelization of Optimization Strategies Authored by Dietrich Hartmann and Matthias Baitsch Since numerical optimization algorithms rely on the repeated evaluation ofobjec

Correspondingly, the recombination pattern has to be repeatedly used on

all $ parent vectors until the set ˆ S gC containing the individuals ˆx gCα, α =

1, 2, 3, , $, is determined.

The mutation mechanisms in the (μ/$+, λ)-ES is the heart of the strategy

and the most vigorous optimization force At this, a specific strength is theproperty that besides the original optimization/design variables, also the steplengthes during the iteration of the optimization are becoming part of thecontinuous adjustment and adaptation of variables towards optimal quantities

In the most general case, the population-based model allows for an adaptiveadjustment of the step lengthes of each corresponding optimization/designvariable (called anisotropic mutative step length control) This necessitates

to expand the original optimization vector x by the step lengths or strategy parameters, concentrated in the vector Δ, leading to the new vector

The mutation schema to create new child vectors from parent vectors takes

the following form if the i-th optimization variable of all λ child vectors in the set SCg are contemplated exclusively:

R j = random of{1, 2, 3, , μ} , j = 1, 2, 3, , λ , λ  μ (4.385)The increments added to the parent components are also random quantities.According to the nature of a mutation they are mainly driven by Gauss-distributed values (large changes are rare, small changes are more frequent!).This behavior is assured by computing a new Gauss-normally distributed ran-dom number from the Intervall [0,1], indicated by [∼ N(0, 1)], for each equa-

tion of the above generation instruction Multiplying such random numbers

by appropriate scalars yield the standard deviation of the Gauss distributionwhich can be interpreted as a step length to navigate the optimization process.The two factors in front of [∼ N(0, 1)] both together represent the standard

deviation without going into the details (for an accurate derivation see e.g

[607, 340]), It should be only mentioned here that the ξ-quantities are

provid-ing that the step lengthes or standard deviations are adapted continuously due

to the current topology of the optimization domain The adaptation is dled by means of a so called multiplicative mutation ansatz avoiding negativevalues and adequate scaling tailored to the convergence needed

han-Recapitulatorily, the (μ/$+, λ)-ES exhibits a plethora of powerful

mecha-nisms and concepts to circumnavigate the most difficult optimization ios at reasonable convergence speed The main benefits can be seen in therobust behavior compared to other competitive methods, in the ability tofind a global optimum with a good chance and in the general applicability,particularly, in algorithmically nonlinear structural optimization problems (asmentioned already in Section 4.5.1) A further significant advantage is the factthat the population-based evolution strategies are inherently parallel in theirbehavior and, therefore, contain numerous opportunities for parallelization

scenar-4.5.4 Parallelization of Optimization Strategies

Authored by Dietrich Hartmann and Matthias Baitsch

Since numerical optimization algorithms rely on the repeated evaluation ofobjective and constraint functions, the process of numerical optimization can

be very time consuming when function evaluations are costly Typically, thenumber of function evaluations using gradient based algorithms is of order

of magnitude of 102 where evolution strategies typically require up to 104

or even more function evaluations The potential for parallelization and theassociated strategies are determined by the type of analysis involved and thetype of the optimization methods used For instance, in multilevel structuraloptimization, the original optimization problem is decomposed into a number

of smaller non-interacting subproblems coupled on a coordination level [802]

In contrast to such highly specialized schemes, the following two sectionscover generally applicable techniques feasible for a wide range of structuraloptimization problems

4.5.4.1 Parallelization with Gradient-Based Algorithms

As outlined in Section 4.5.3.1, many gradient based algorithms repeatedlydetermine a descent direction using the gradients of the objective functionand constraints (see e.g problem (4.371)) and carry out a line search searchalong this direction to solve the one dimensional problem (4.373) Hence, thereare mainly two possibilities for parallelization: The computation of derivativesand the line-search step

For many problems involving numerical simulation, derivatives can only beapproximated numerically using either forward differences

required which can easily be carried out in parallel

In the line-search step, several points on the one-dimensional search rection can be evaluated in parallel which can yield a substantial parallelspeed-up For example, in [703] Schittkowski proposes a sequential quadraticprogramming algorithm with distributed and non-monotone line search Com-bining the parallel approximation of gradients and a parallel line search, gra-dient based optimization requires in the ideal case two computational stepsper iteration: One for the gradients and one for the line search

di-However, both techniques do not involve enough parallel processes to makefull use of modern cluster computers with more than 150 CPUs Therefore,the described techniques can been combined with a parallel structural analysis

in order to save more computing time (see e.g [78] for an application withhigh-order finite element methods)

4.5.4.2 Parallelization Using Evolution Strategies

Population-based evolution strategies as introduced in Section 4.5.3.2 require

λ function evaluations in each optimization step where λ is the population

size (number of children in one generation) The population size is chosenaccording to the type and size of problem at hand and typically ranges from

50 to 200 Taking into account that up to 400 iteration steps might be required,

it becomes obvious that parallelization is mandatory when evolution strategiesare applied to complex engineering problems

On the other side, the large number of designs to be evaluated in each eration step allows for an efficient parallelization since the required computa-tions do not depend on each other Although a straightforward parallelization

Problem

Problem MProblem

Optimizer

Linux-Cluster Server

Fig 4.107 Parallel software framework

scheme such as the manager-worker approach often renders good performance,further improvements can be achieved if the communication overhead is re-duced by applying packeting or load balancing mechanisms [326]

4.5.4.3 Distributed and Parallel Software Architecture

There are basically two demands for a parallel optimization software: (i) a widevariety of optimization algorithms have to be readily available in order to en-able the designer to choose a suitable method for the problem at hand and (ii)the parallel part of the software should be isolated as much as possible in order

to facilitate software development These requirements are accomplished by thesoftware framework shown in Fig 4.107 Here, the optimizer software componentprovides a wide variety of optimization algorithms such as evolution strategiesand different variants of gradient-based algorithms in a unified fashion [79] Thissoftware component is implemented as a CORBA server such that it can be usedremotely over the Internet The second part is the multi-problem parallelizationcomponent which preferably runs on a cluster of Linux computers This compo-nent receives a set of design vectors from the optimizer and dispatches them tothe individual instances of the actual optimization problem running on the com-pute nodes The overall optimization process is driven from a GUI applicationrunning on the user’s workstation or laptop computer

4.6 Application of Lifetime-Oriented Analysis and

Design

Authored by Dietrich Hartmann and Detlef Kuhl

The successful application as well as the practical implementation of resultsbased on sophisticated long-term research in lifetime-oriented analysis and de-sign is the most essential achievement and the best possible evidence for work

performed For that reason, a wide variety of highly different application amples are shown in the following chapters ranging from the lifetime-orientedanalysis and design of beam-like structures over structural components used

ex-in the automobile ex-industry up to concrete as well as steel structures, whereparticularly bridge systems are dealt with According to the specific nature

of the structural systems considered with respect to material aspects and/orstructural behaviors, all relevant concepts and methodologies uncovered in therecent years of research are elucidated Hereby, eminent importance is put onthe verification and the validation of theoretical findings

4.6.1 Testing of Beam-Like Structures

Authored by Stefanie Reese and Andreas S Kompalka

In the literature a couple of publications focus on the identification of adamage in beam-like structures The publications from [424] and [858] localize

a cut damage in a simple beam made of steel The localization and tion of a cut damage in a cantilever beam made of aluminum are announced

quantifica-in [535] and [751, 752, 754, 753] In the followquantifica-ing sections a subspace method(see chapter 4.3.2) is combined with a derivative-based optimization method

Fig 4.108 Experimental setup

Fig 4.109 Damage equipment

(see chapter 4.5.3.1) to identify a cut damage in a cantilever beam made ofsteel

4.6.1.1 Experimental Setup

The experimental setup is a clamped cantilever beam with a length of 1.62m

and a rectangular cross-section of 40× 15mm made of steel The cantilever

beam is fixed with several clamps and a steel bar (HEB-100) at a massivesteel plate (1000× 800 × 100mm) on a vibration decoupled foundation (see Figure (4.108)) The used measurement technology from Hottinger & Baldwin

consists of 16 micro-mechanic accelerometers and two amplifiers The tural damage is a cut with a rectangular cross-section of 10× 5mm The cut

struc-is realized by a milling machine and a cross-support (see Figure (4.109)) The

central position of the cut is 450.00mm from the clamping The system is

ex-cited by a static displacement Three measurements of the exex-cited structureare recorded in the undamaged and damaged state

4.6.1.2 Identification of Modal Data

To obtain the modal data (frequencies and mode shapes) of the experimentalsetup, the accelerations of the 16 channels are analyzed with the data-drivenstochastic subspace identification of chapter 4.3.2 (with (4.308)-(4.311)

5 10 15 20 0

100 200 300 400 500

Fig 4.110 Singular values

Fig 4.111 1’st eigenfrequency and mode shape

and (4.298)-(4.302)) The first twenty singular values are visualized inFigure (4.110)

In Figure (4.111)-(4.114) the frequencies and mode shapes in the aged and damaged state are visualized The standard deviations of the modeshapes are smaller than of the frequencies Comparing the undamaged anddamaged state, the relative changes in the coordinates of the mode shapes are

Fig 4.113 3’rd eigenfrequency and mode shape

much smaller than the relative frequency changes In the damaged state, thecoordinates of the first mode shape almost do not change The coordinates ofthe higher mode shapes show only small changes in the damaged state

Fig 4.115 Cut modelling

4.6.1.3 Updating of the Finite Element Model

The experimental setup is discretized by means of a finite element model Atwo-dimensional four-node shell element with bilinear ansatz functions and

a two-dimensional nine-node shell element with biquadratic ansatz functionsare compared by a convergency study The nine-node shell element with bi-quadratic ansatz functions enables a better approximation of the bending

modes especially in the damaged state with the modeled cut (see Figure(4.115)) Based on the convergency study, six nine-node shell elements overthe cross-sectional height and 1296 elements in length direction are used todiscretize the cantilever beam structure in the undamaged and damaged state.

In Chapter 4.5.3.1, derivative-based methods like the Newton methodare explained In the context of this chapter, the Gauss-Newton method isderived to solve the least squares problem The sum of squares, which have

to be minimized, are the residuals or differences between the experimentalmeasures und numerical calculated modal data (frequencies and mode shapes).Finding the minimum of the objective function

∇f(x) = ∇f(x0) +∇2f (x0)(x − x0) (4.393)leads to the Newton method

parameter x and x0by the incremental sizes x k+1 and x k leads to the toniteration

Fur-s GN =−CJ(xk) TJ(xk)D−1

Table 4.9 Modal Assurance Criterion

with the search direction sk and the stepsize parameter αk The control of the

stepsize is important for the updating algorithm One of the first publications

of Gauss-Newton iteration with line search approach was given by [233] A

simple choice for the stepsize is the exponential ansatz αk = 0.5 m k Depending

on the sum of squares, the integer mk reduces or enlarges the stepsize Themethod uses one information of the object function to control the stepsize Astepsize control with a quadratic-cubic ansatz is published by [284] or [582].The method uses a quadratic or cubic extrapolation based on three evaluations

of the object function to estimate the optimum stepsize Here, the iterationstops if the changes in the model parameter (cut position and cut deepness)

are smaller than the realization of the cut which is 1/10mm.

Prerequisite for a successfull updating of the finite element model is themode pairing and the mode scaling The “Modal Assurance Criterion”

M AC ij = (φ

T

ex i φ f e j)2

(φ T ex i φ ex i )(φ T f e j φ f e j) (4.401)

was introduced by [43] and compares the experimental identified and the

numerical calculated eigenvectors φ ex

i and φ f e

i The MAC values are sorted

in a table where values close to 1 denote a good agreement of the measuredand modeled data and values close to 0 denote a bad agreement In Tab (4.9)the correct pairing of the modal data is verified MAC values close to 100percentage indicate a good approximation of the experimental setup with thefinite element model The “Coordinate Modal Assurance Criterion”

In [293] and [511] other mode pairing criterions are mentioned as e.g the

Fig 4.116 Optimization topology

“Orthogonality Check” or the “Normalized Cross Orthogonality” The vantge of the mentioned methods is that the mass matrix has to be assumedand the results depend on the accuracy of these assumptions After pairing

disad-of the modes it is important to scale the eigenvectors before calculating theresiduals The “Modal Scale Factor”

scal-In Figure (4.116) the sums of squares with the first four frequencies are

plotted in an optimization topology The plot is cut off at a value of 0.0015

visualized with white arrays The optimization topology shows a global

mini-mum with the sum of squares of 3.2 · 10 −7 at the cut position c

p = 450.60mm and the cut deepness c d = 4.95mm (see Figure (4.116) circle symbol) This

is very close to the damage in the experimental setup with the cut position

c p = 450.00mm and the cut deepness c d = 5.00mm There is a deep local minimum with the sum of squares round about 1.6 · 10 −4 at the cut position

c p = 13.35mm and the cut deepness c d = 2.50mm (see Figure (4.116)

dia-mond symbol) Another high local minimum with the sum of squares round

-Table 4.11 Gauss-Newton iteration (c p /c d=1400/1mm)

Cut position Cut deepness Residual Gradient Stepsize

-about 3.9 · 10 −4 lies at the cut position c

p = 1198.35mm and the cut ness c d = 3.83mm (see Figure (4.116) square symbol) The Gauss-Newton iteration with the start values c p /c d =800/1mm and c p /c d =1400/1mm are vi-

deep-sualized in Figure (4.116) and summarized in Tab (4.10) and (4.11) In theneighborhood of small gradients the Gauss-Newton search direction is notrectangular to the contour line and the stepsize control is required

Finally we can state, minimizing the sum of squares of the modal data withthe Gauss-Newton method it is possible to identify the cut position and the

cut deepness in a cantilever beam (length 1.62m) with an accuracy less than

1mm

4.6.2 Lifetime Analysis for Dynamically Loaded Structures at BMW AG

Authored by Dietrich Hartmann, Heiner Weber and Gero Pflanz

The design of the car body-in-white using CAE-technology is an iterativeprocess which can be described as an extensive optimization process A desir-able target of such an optimization can be the minimum weight of the result-ing structure taking into account prescribed boundary conditions Boundaryconditions may include cost, geometrical guidelines, production engineeringdemands and functional specifications

For instance, geometrical guidelines are the length of the car, the height

or width of the door sill Production engineering demands to consider thedeep drawing process or the thickness Characteristic functional specificationscomprise crash behaviour, static and dynamic stiffness and acoustics as well

as strength and durability requirements

Static stiffness requirements include both the torsional stiffness of the in-white as well as several bending and transverse stiffness cases For each ofthese load cases target values are defined for the body-in-white componentwhich have to comply with certain handling performance or further desiredproperties of the whole car Dynamic stiffness requirements include the fre-quency range of certain global eigenmodes of the structure, for example, toavoid the excitation of an eigenmode of the car body while the engine isrunning in idle-speed Additional dynamic stiffness targets are necessary toguarantee the desired vibration comfort Strength requirements define themaximum tolerable plastic deformation for a particular load case, e.g towingthe car onto a tow-track Durability requirements are set up to guarantee that

body-no cracks in sheet-metal parts or failure in weld-spots occur within a givenmileage under certain statistical loading conditions

Plenty of months before the first prototypes of a car are built, the functionalrequirements are analysed and iteratively improved in virtual prototypes bymeans of the finite element method The finite element model is hereby gen-erated from early CAD data Typically, those virtual prototypes have a size

of a few million degrees of freedom

4.6.2.1 Works for the New 3-Series Convertible

The 3-series convertible (see Figure 4.117) is the fourth member of the rent BMW 3-series along with sedan, station wagon and coup´e While thefrontal, middle and rear part of the body-in-white are similar or identical tothe other model members, the side frame is completely different The miss-ing roof load-path has to be compensated by the reinforced side-frame designand additional torsion bars, which are packaged at several locations in theconvertible structure