Lifetime-Oriented Structural Design Concepts- P21 potx

558 4 Methodological Implementation x β C = 1    j=1 x β P,Rd j . (4.382) Correspondingly, the recombination pattern has to be repeatedly used on all  parent vectors until the set ˆ S g C containing the individuals ˆx g C α ,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 the property that besides the original optimization/design variables, also the step lengthes during the iteration of the optimization are becoming part of the continuous adjustment and adaptation of variables towards optimal quantities. In the most general case, the population-based model allows for an adaptive adjustment of the step lengthes of each corresponding optimization/design variable (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 ˜x =  x Δ  = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ x 1 x 2 . . . x n δ 1 δ 2 . . . δ n ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (4.383) where the components δ i ,i=1, 2, 3, ,n are the step lengths associated to the variables x i ,i=1, 2, 3, ,n. Themutationschematocreatenewchild vectors from parent vectors takes the following form if the i-th optimization variable of all λ child vectors in the set S g C are contemplated exclusively: x g i C 1 = x g i P,R 1 + δ i P,R 1 · ξ i1 · [∼ N(0, 1)] x g i C 2 = x g i P,R 2 + δ i P,R 2 · ξ i2 · [∼ N(0, 1)] . . . x g i C λ = x g i P,R λ + δ i P,R λ · ξ iλ ·[∼ N(0, 1)] . (4.384) Of course, these instructions of generations have to be carried out for all indices i, i =1, 2, 3, ,n. The following explanations clarify the effects of the equations above. Parent variables are the originators of the generation where at the parent individuals are again randomly drawn from the set S g P according to 4.5 Optimization and Design 559 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 distribution which 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 han- dled by means of a so called multiplicative mutation ansatz avoiding negative values and adequate scaling tailored to the convergence needed. Recapitulatorily, the (μ/ + ,λ)-ES exhibits a plethora of powerful mecha- nisms and concepts to circumnavigate the most difficult optimization scenar- ios at reasonable convergence speed. The main benefits can be seen in the robust behavior compared to other competitive methods, in the ability to find a global optimum with a good chance and in the general applicability, particularly, in algorithmically nonlinear structural optimization problems (as mentioned already in Section 4.5.1). A further significant advantage is the fact that the population-based evolution strategies are inherently parallel in their behavior and, therefore, contain numerous opportunities for parallelization. 4.5.4 Parallelization of Optimization Strategies Authored by Dietrich Hartmann and Matthias Baitsch Since numerical optimization algorithms rely on the repeated evaluation of objective and constraint functions, the process of numerical optimization can be very time consuming when function evaluations are costly. Typically, the number of function evaluations using gradient based algorithms is of order of magnitude of 10 2 where evolution strategies typically require up to 10 4 or even more function evaluations. The potential for parallelization and the associated strategies are determined by the type of analysis involved and the type of the optimization methods used. For instance, in multilevel structural optimization, 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 sections cover generally applicable techniques feasible for a wide range of structural optimization problems. 560 4 Methodological Implementation 4.5.4.1 Parallelization with Gradient-Based Algorithms As outlined in Section 4.5.3.1, many gradient based algorithms repeatedly determine a descent direction using the gradients of the objective function and constraints (see e.g. problem (4.371)) and carry out a line search search along this direction to solve the one dimensional problem (4.373). Hence, there are mainly two possibilities for parallelization: The computation of derivatives and the line-search step. For many problems involving numerical simulation, derivatives can only be approximated numerically using either forward differences ∂f ∂x i (x) ≈ f(x + Δe i ) − f(x) Δ (4.386) or central differences ∂f ∂x i (x) ≈ f(x + Δe i ) − f(x − Δe i ) 2Δ , (4.387) i =1, ,n,wheren is the number of design variables and e i is the i-th unit vector. Obviously, either n +1 or 2n independent function evaluations are required which can easily be carried out in parallel. In the line-search step, several points on the one-dimensional search di- rection can be evaluated in parallel which can yield a substantial parallel speed-up.Forexample,in[703]Schittkowski proposes a sequential quadratic programming 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 steps per iteration: One for the gradients and one for the line search. However, both techniques do not involve enough parallel processes to make full 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 with high-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 chosen according 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 strategies are applied to complex engineering problems. On the other side, the large number of designs to be evaluated in each it- eration step allows for an efficient parallelization since the required computa- tions do not depend on each other. Although a straightforward parallelization 4.6 Application of Lifetime-Oriented Analysis and Design 561 Problem Problem Problem MProblem Optimizer Linux-ClusterServerWorkstation . . . . LAN CORBA MPI {x 1 , x 2 , , x m } { f i (x 1 ), , f i (x m )} f i (x 1 ) x 1 f i (x 2 ) x 2 f i (x n ) x n GUI 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 wide variety 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 the software framework shown in Fig. 4.107. Here, the optimizer software component provides a wide variety of optimization algorithms such as evolution strategies and different variants of gradient-based algorithms in a unified fashion [79]. This software component is implemented as a CORBA server such that it can be used remotely over the Internet. The second part is the multi-problem parallelization component which preferably runs on a cluster of Linux computers. This compo- nent receives a set of design vectors from the optimizer and dispatches them to the individual instances of the actual optimization problem running on the com- pute nodes. The overall optimization process is driven from a GUI application running 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 results based on sophisticated long-term research in lifetime-oriented analysis and de- sign is the most essential achievement and the best possible evidence for work 562 4 Methodological Implementation performed. For that reason, a wide variety of highly different application ex- amples are shown in the following chapters ranging from the lifetime-oriented analysis and design of beam-like structures over structural components used in the automobile industry up to concrete as well as steel structures, where particularly bridge systems are dealt with. According to the specific nature of the structural systems considered with respect to material aspects and/or structural behaviors, all relevant concepts and methodologies uncovered in the recent years of research are elucidated. Hereby, eminent importance is put on the 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 a damage in beam-like structures. The publications from [424] and [858] localize a cut damage in a simple beam made of steel. The localization and quantifica- tion of a cut damage in a cantilever beam made of aluminum are announced in [535] and [751, 752, 754, 753]. In the following sections a subspace method (see chapter 4.3.2) is combined with a derivative-based optimization method Fig. 4.108. Experimental setup 4.6 Application of Lifetime-Oriented Analysis and Design 563 Fig. 4.109. Damage equipment (see chapter 4.5.3.1) to identify a cut damage in a cantilever beam made of steel. 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 massive steel 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 struc- tural damage is a cut with a rectangular cross-section of 10 × 5mm. The cut 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 excited structure are 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 experimental setup, the accelerations of the 16 channels are analyzed with the data-driven stochastic subspace identification of chapter 4.3.2 (with (4.308)-(4.311) 564 4 Methodological Implementation 5101520 0 100 200 300 400 500 Fig. 4.110. Singular values f 1 =4.74,h r =15.0mm f 1 =4.60,h r =7.5mm - 0.20 0.00 0.20 0.40 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 Fig. 4.111. 1’st eigenfrequency and mode shape and (4.298)-(4.302)). The first twenty singular values are visualized in Figure (4.110). In Figure (4.111)-(4.114) the frequencies and mode shapes in the undam- aged and damaged state are visualized. The standard deviations of the mode shapes are smaller than of the frequencies. Comparing the undamaged and damaged state, the relative changes in the coordinates of the mode shapes are 4.6 Application of Lifetime-Oriented Analysis and Design 565 f 2 =29.68,h r =15.0mm f 2 =28.45,h r =7.5mm - 0.20 0.00 0.20 0.40 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 Fig. 4.112. 2’nd eigenfrequency and mode shape f 3 =83.04,h r =15.0mm f 3 =80.99,h r =7.5mm - 0.20 0.00 0.20 0.40 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 Fig. 4.113. 3’rd eigenfrequency and mode shape much smaller than the relative frequency changes. In the damaged state, the coordinates of the first mode shape almost do not change. The coordinates of the higher mode shapes show only small changes in the damaged state. 566 4 Methodological Implementation f 4 = 162.58,h r =15.0mm f 4 = 160.52,h r =7.5mm - 0.20 0.00 0.20 0.40 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 Fig. 4.114. 4’th eigenfrequency and mode shape f 1 =4.6414Hz f 2 =29.5777Hz X-Coordinate [ m ] Y-Coordinate [m] 0.43 0.44 0.45 0.46 0.47 -0.0075 0 0.0075 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. A two-dimensional four-node shell element with bilinear ansatz functions and a two-dimensional nine-node shell element with biquadratic ansatz functions are compared by a convergency study. The nine-node shell element with bi- quadratic ansatz functions enables a better approximation of the bending 4.6 Application of Lifetime-Oriented Analysis and Design 567 modes especially in the damaged state with the modeled cut (see Figure (4.115)). Based on the convergency study, six nine-node shell elements over the cross-sectional height and 1296 elements in length direction are used to discretize the cantilever beam structure in the undamaged and damaged state. In Chapter 4.5.3.1, derivative-based methods like the Newton method are explained. In the context of this chapter, the Gauss-Newton method is derived to solve the least squares problem. The sum of squares, which have to be minimized, are the residuals or differences between the experimental measures und numerical calculated modal data (frequencies and mode shapes). Finding the minimum of the objective function f(x)= 1 2 r(x) T r(x) (4.388) of the sum of squares with the residual vector r(x)= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ r 1 (x 1 , ,x n ) r 2 (x 1 , ,x n ) . . . r m (x 1 , ,x n ) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (4.389) is equal to finding the zero point of the first partial derivatives of the object function ∇f(x)=J(x) T r(x) (4.390) with the Jacobian matrix J(x)= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ∂r 1 (x) ∂x 1 ∂r 1 (x) ∂x 2 ∂r 1 (x) ∂x n ∂r 2 (x) ∂x 1 ∂r 2 (x) ∂x 2 ∂r 2 (x) ∂x n . . . . . . . . . . . . ∂r m (x) ∂x 1 ∂r m (x) ∂x 2 ∂r m (x) ∂x n ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ∇r 1 (x) T ∇r 2 (x) T . . . ∇r m (x) T ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . (4.391) The Jacobian matrix includes the transformed gradients of the residuals ∇r j (x) T =  ∂r j (x) ∂x 1 ∂r j (x) ∂x 2 ∂r j (x) ∂x n  (4.392) in each row. Using a second-order Taylor series to approximate the first partial derivatives of the object function [...]... Application of Lifetime-Oriented Analysis and Design 575 Fig 4.120 FE model of the shaker test arrangement 4.6.2.3.1 Structural Analysis Using Time Integration Time integration means that the loading must be given in a discrete fashion for being entered into the computation For example, the Nastran solution routine 112 provides this approach The computation result is a time series of the structural response,... members, the side frame is completely different The missing roof load-path has to be compensated by the reinforced side-frame design and additional torsion bars, which are packaged at several locations in the convertible structure 4.6 Application of Lifetime-Oriented Analysis and Design 573 Fig 4.117 The new 3-series convertible Low-frequency vibration comfort in convertibles is significantly dominated... σyy + 3τxy (4.417) Subsequent to the structural analysis, the Rainflow counting and the damage accumulation calculation is carried out, here by means of Falancs Results are the total damages as defined in eq (4.404) These damages for the calculated time can be extrapolated to damages for the whole test duration of 8 4.6 Application of Lifetime-Oriented Analysis and Design 581 Fig 4.124 Typical stress... · 10−4 at the cut position cp = 13.35mm and the cut deepness cd = 2.50mm (see Figure (4.116) diamond symbol) Another high local minimum with the sum of squares round 4.6 Application of Lifetime-Oriented Analysis and Design 571 Table 4.10 Gauss-Newton (cp /cd =800/1smm) Cut position cp [mm] 1 800.0000 2 326.2682 3 345.2333 4 382.9806 5 428.5175 6 448.3661 7 452.0016 8 450.9110 9 450.6433 10 450.5973... partial damages are summed up and a total damage Dtot is resulting, for example for loading in different directions x, y, z: Dtot = ΔDi = ΔDi + x ΔDi + y ΔDi z (4.405) 4.6 Application of Lifetime-Oriented Analysis and Design Power spectral density SXX(f) 10000 577 Shell 7672094 vonMises-1 Shell 7672094 vonMises-2 Shell 7672154 vonMises-1 Shell 7672154 vonMises-2 Shell 7672171 vonMises-1 Shell 7672171... For a design, the total damage must be smaller than a limit damage which is usually equated with 1: Dtot < DLim (4.406) 4.6.2.4 Approach 2: Power Spectral Density Functions and Calculation of Spectral Moments An alternative way to the approach above uses the power spectral density functions of the load directly The full particulars of this alternative are outlined in the next subchapter 4.6.2.4.1 Structural. .. Δσ · exp − 4λ0 R2 2 + ϑ3 1 · Δσ · exp − 4λ0 2 Δσ √ 2R λ0 Δσ √ 2 λ0 2 Here, the following parameters are introduced: 2 (4.411) Dirlik probability density function f(¢¾) 4.6 Application of Lifetime-Oriented Analysis and Design 579 0,006 Shell 7672094 vonMises-1 Shell 7672094 vonMises-2 Shell 7672154 vonMises-1 Shell 7672154 vonMises-2 Shell 7672171 vonMises-1 Shell 7672171 vonMises-2 Shell 7672467 vonMises-1... minimum are small if the model is a good approximation of the problem Therefore, the Gauss-Newton search direction sGN = − J(xk )T J(xk ) k −1 J(xk )T r(xk ) (4.399) 4.6 Application of Lifetime-Oriented Analysis and Design 569 Table 4.9 Modal Assurance Criterion φf e1 φf e2 φf e3 φf e4 φex1 0.9992 0.0037 0.0077 0.0065 φex2 0.0043 0.9981 0.0013 0.0050 φex3 0.0069 0.0040 0.9974 0.0025 φex1 0.0053 0.0031... instead of PSD functions [820] The presented method avoids the time consuming time history integration Furthermore, it is more precise in the case of flat spectra because no 4.6 Application of Lifetime-Oriented Analysis and Design 583 additional random phenomena appear in measuring the accelerations The method can be applied if natural stochastic processes like wind are considered Rough roads can be another... direction Obviously, the mesh is chosen relatively coarse Therefore, to come to adequate results, the polynomial degrees of the approximation of the displacements u and the 4.6 Application of Lifetime-Oriented Analysis and Design 585 Fig 4.128 Hygro-mechanically loaded concrete shell structure: Finite element mesh of the numerical analysis capillary pressure pc , which is representing the moisture in the . ex- amples are shown in the following chapters ranging from the lifetime-oriented analysis and design of beam-like structures over structural components used in the automobile industry up to concrete. reinforced side-frame design and additional torsion bars, which are packaged at several locations in the convertible structure. 4.6 Application of Lifetime-Oriented Analysis and Design 573 Fig. 4.117 the following subchapter. 4.6 Application of Lifetime-Oriented Analysis and Design 575 Fig. 4.120. FE model of the shaker test arrangement 4.6.2.3.1 Structural Analysis Using Time Integration Time