4.3 ANFIS 135 - tuprints

4.3 ANFIS 135

Figure 4.30: SA with 215 samples on12 design variables.

Figure 4.31: 170 training, 245 check-ing samples

Figure 4.32: 250 training, 165 check-ing samples. Small cluster radius

Figure 4.33: 190 training, 30 check-ing samples.

Figure 4.34: 220 training, 28 check-ing samples. Small cluster radius

Figure 4.35: 170 training, 78 check-ing samples.

136 4. Approximation Models

Figure 4.36: Surface plot between input variable 1 and 6 and output variable (objective function).

The presented results can be compared to the former section where Bayesian regular-ization networks were applied to the same channel junction optimization problem. Ineither case, networks employing advanced features like regularization or a cluster radius,can considerably improve the approximation capabilities and by using very few functionevaluations also reduce computing time in order to depict the functional coherence of acomplex optimization problem.

From our investigations we can conclude that the ANFIS training with a small networkyields more satisfiable results than a large network. It is recommended that the numberof training samples is at least as large (or better larger) as the number of unknowns, i.e.number of parameters in the network. All our results imply that if this rule is violated,the network tends to perform poorly. By lowering the cluster radius and so working ona number of parameters larger than the number of training samples, the ANFIS is notanymore capable to depict the right functional coherence between input parameters andobjective function. For the largest networks, we chose 30 000 training epochs which tookseveral hours for network training. Still, the smaller the number of parameters, the betterthe network performance. Obviously, the cluster radius and validation technique are aconsiderable improvement for ANFIS approximation abilities.

Figure (4.36) indicates the surface plot between input variables 1 and 6 (also see figure(2.12)) and the pressure drop value (denoted as ‘out1’), which indicates the trained ANFISinput-output coherence. As can be seen, the ANFIS learned that the parameters 1 and6 need to move towards positive values. By both moving into the negative direction, theobjective value massively increases. Such an insight into parameter interdependence is acontribution which is easily provided through the ANFIS algorithm and can hardly beobtained otherwise without employing massive computations.

4.3 ANFIS 137

4.3.4 Conclusions

We presented Bayesian regularization and also adaptive neuro-fuzzy inference systemnetworks to render shape optimization problems for fluid flow processes. After training thenetwork with a sufficient number of samples obtained from a Monte Carlo simulation, andevaluating those with a finite volume solver, we applied an evolutionary strategy which ledclose to the optimum shape. We furthermore applied a specialized evolutionary strategy,i.e. the simulated annealing approach, which also served as an optimization strategy. Theapplication of neural networks for parameter optimization problems is quite recent andhas yet only been investigated by a few authors. Function surrogate models become anincreasingly important ingredient in the context of fluid flow optimization problems. Upto today, these optimization approaches were mostly applied in the context of structuralmechanics. Fluid dynamics are more demanding from a computational point of view andso heuristic methods are most appropriate in this context.

The salient advantage of the considered methods is to save computation time by requir-ing less function evaluations from the finite-volume solver than a conventional optimizerneeds. By considering a representative example, it was shown that, as in case of the chan-nel junction with 12 design variables, only about the half number of function evaluationscompared to a more classical derivative-free optimization method is needed to gain anoptimization result that comes close to the global optimum. Due to Bayesian regulariza-tion networks, which are a major improvement over networks not utilizing regularization,we were able to obtain optimization results with high computational efficiency. In com-parison to the Bayesian regularization networks, the ANFIS approach demonstrated toprovide excellent approximation capabilities. Even without generalization, the networkproved to render a highly complex function approximation tasks. The outstanding fea-ture here was clearly the use of a cluster radius parameter which allowed to control thenetwork complexity. The computations also show that in a few cases, the network failedto give reasonable results. These discrepancies indicate that the network quickly fails togive reasonable answers if the network design and parameters are not properly chosen. InANFIS, for instance, it can be seen that the cluster radius strongly influences the numberof parameters which again determines the overall network performance. Thus, settingup the network is crucial for successful usage; the approximation model approach hencerequires an experienced user.

However, there is still a demand to improve approximation capabilities and to designnetworks which are even more powerful in approximating higher dimensional parameteroptimization problems. For the employment of networks in the function surrogate modelcontext, we here provided an interesting alternative to consider.

Chapter 5

Summary and Outlook

This dissertation is about the application of soft computing methods for shape opti-mization purposes for fluid flow regions in engineering relevant applications. Where softcomputing methods like neural networks and evolutionary algorithms are usually analyzedin computer science and also mathematics, these concepts are not yet thoroughly investi-gated in mechanical engineering aiming at the improvement of real-world applications.

Optimization is difficult when instead of an analytical function a numerical solution de-scribes the objective. This objective can be, for instance, the pressure drop or heatingperformance of a technical device where flow processes take place. In these, the objectivefunction is not expressed analytically in the design variables. This circumstance does notallow for the execution of gradient based methods, neither are methods based on approxi-mation to or substitutes of the derivative considered to be an efficient adoption. The taskwas to provide optimization techniques that support the design of complex systems basedon simulations.

Evolutionary algorithms belong to the class of heuristic methods for which only very fewconvergence proofs have yet been found. This drawback may be neglected when consid-ering practical optimization problems for which scarcely any other methods are suitable.This is because evolutionary algorithms realize a number of advantages over conventionalsearch methods and yield satisfyingly results for practical engineering problems. The mostimportant advantage is here the global search property realized by stochastic based searchand parameter spread operators. EA are the basic ingredient for producing parametersamples that capture a maximum amount of information from the functional coherence ofnumerical solver calls. These function properties have to be discovered since no functioninformation is available in closed form. We emphasize on the fact, that there is virtuallyno a priori information of the original mapping available. So it is of interest if the ob-jective function behaves convex, multi-modal, nonlinear, or may possess other functionalattributes which are important for the optimization model. EA are able to capture highlycomplex problems and can be used to discover these functional attributes. Deterministicmethods in contrast, do not offer any ability to independently detect functional attributes.We demonstrated a variant of EA, the simulated annealing, in structural mechanics de-

138

139

sign improvement and also employed this method for comparing simulated annealing withapproximation models used for the shifted channel junction problem.

Approximation models are taken into consideration since they allow for a rapid functionevaluation instead of employing the computational expensive finite volume solver. Neuralnetworks are in particular able to depict functional coherence with very low computationalcomplexity. The contemporary literature took the network approach into considerationbut has not yet considered progressive networks. This work presents the application ofa Bayesian regularization network which realizes improved generalization features. Thecomputations demonstrated that for the staggered channel, an improvement concerningthe minimization of pressure drop was reached until close to the global minimum. Thereby,the overall computing process required a less number of finite volume evaluations than aconventional sequentiell quadratic programming approach did.

The adaptive neuro-fuzzy inference system (ANFIS) is a further network which allows fora local approximation in the functional coherence surface. The employed cluster radiustechnique is similar to radial basis function networks which also found application in theapplied shape optimization scientific community. The cluster radius essentially determinesthe number of membership functions and thus the total number of parameters in thenetwork system. Our computations showed that the smaller, less complex networks, realizebetter generalization properties than the more complex networks did. This is because acomplex network virtually interpolates data, a characteristic that is not desired: The errormeasure of network output on unseen parameter samples then increases with advancingdata interpolation. The cluster radius parameter can be adjusted such that the mostintensive approximation is done in the most critical regions. Thereby, as the Bayesianregularization network, the ANFIS network also needed less numerical solver calls thanthe derivative-free optimizer did. Both networks succeeded to ameliorate the objectivefunction close to the global minimum.

Neural networks have been used in the past as approximation models, most often radialbasis function network are considered. Although they have mostly found applicationfor rather simple problems as in reliability analysis for two-dimensional storeys, theybecame more and more popular throughout the last decade and were already consideredfor shape optimization. The contribution of this dissertation was to show the abilityof the proposed network models to successfully render demanding shape optimizationproblems. In these, conventional network models completely failed to give a reasonableoptimization result. As our computations discovered, it is generalization that makes anetwork usable for these purposes. Regularization and network complexity are found tobe crucial for successful optimization with approximation models. Since they have notyet found widespread perception, it is due to this work that optimization procedures cannow be improved for challenging optimization problems.

In most real world optimization problems more than one objective has to be considered,multi-objective optimization problems are common and even appear in simple problems.This fact has for long time not be taken into account. We widely discussed that conven-tional methods to approximate the Pareto front quickly prove to be insufficient. It wasconcluded that the Pareto dominance concept has to be taken into consideration for al-

140 5. Summary and Outlook

gorithms in this class. Obviously, evolutionary algorithm operators allow for the accountof the Pareto dominance concept. This Pareto set comprehends design alternatives whichare equally worth. EA first serve as search algorithms - to find the Pareto front - and alsofor decision support - depicted by the spread along the Pareto front.

Our computations for multi-objective problems involved two configurations of heat ex-change units subjected to optimize according to heat increase, pressure drop, covered flowarea and flow velocity. The calculations show a clearly emerging Pareto front betweenpressure drop and covered flow area and also between heat increase and pressure drop.From these observations, the user can choose from a set of alternatives the correspond-ing design with desired features. Knowing the fact about coherence of objectives is alsovaluable, there again is no a priori knowledge about the existence or shape of the Paretofront. Moreover, since each of the proposed designs were evaluated by the numericalsolver, these can be investigated regarding the flow field. This may be interesting inaspects of pre-evaluation of designs in order to learn more about further characteristicssuch as, for instance, stability properties. The employed heat exchange units are exem-plary: In industry, one is interested in the application of more complex designs such as3-D airfoils. From a numerical point of view, the respective scientific fields may alreadyprovide a successful simulation of the flow field for airfoil configurations. However, in arealistic optimization environment they are optimized under many linear and nonlinearconstraints which can make it impossible to find a global optimum [41]. Another exam-ple is the design of a paper machine headbox [151] involving fluid-structure interaction,multiple objectives and nonlinear constraints.

A major aim of numerical simulation is to eventually optimize real-world problems ina computational environment. However, optimization is again a self-contained scientificfield. It is obvious that real-world optimization is usually represented by a very demandingoptimization model for which only specific algorithms can be used. Our contribution hereis to provide engineering science with an insight into only a few problems that may arisewith the optimization task. For the structural mechanics computation, we needed severalthousands of function evaluations until a Pareto front became visible. This is practicallyimpossible to cope for optimizing flow fields since each flow evaluation takes an enormousamount of computation time. We demonstrated the necessity to use evolutionary multi-objective algorithms when confronted with a multi-objective problem. In these, specificoperators are used to ensure a sufficient spread along the Pareto front allowing the engineerin practice to choose from a number of equally worth solutions. This choosing betweenseveral conflicting objectives has up to now only been a matter of intuition. The proposedalgorithms, on the other hand, have been shown to properly detect and describe a Paretofront. In the configurations which were investigated, we explicitly gave the diverse designsand discussed their distinctive features. Thereby, it became obvious how the designsemulate the different objectives. Our computations provided the insight in how far theobjectives influence the final designs. Moreover, we employed a comparison of differentEMO algorithms and also different parameter settings for the heat exchange units. Wediscovered a discontinuous and a non-convex Pareto front and discussed the appropriateoptimization strategy for each problem setting.

141

A future research task is to provide engineering practicioneers with EA operators, EMOalgorithms and approximation models which are designed for the particular purpose athand. Specific features of engineering applications are a long function evaluation time aswell as very sparse information about the functional coherence between design parametersand objective function. Thus, efficient schemes and maximizing exploration of the searchspace have to be prioritized when setting up real-world optimization algorithms. Usually,research in the mentioned fields do not sufficiently take these properties into account. It isnow on the edge of current research to provide algorithms which are specifically designedfor these particular optimization model properties. Furthermore, it is desired to developan integrated optimization environment where real-world problems could be optimizedand evaluated in an easy and automated manner. Clearly, this is a multi-disciplinarytask and incorporates the contribution from diverse scientific fields. Research in thisaspect is just at its beginning and, due to its practical importance, may gather increasedimportance during the next decades.

142 5. Summary and Outlook

List of Figures

2.1 Mapping from decision space into objective space . . . . . . . . . . . . . . 10

2.2 Emerging Pareto-front. Both objectives (f1 and f2) are to be maximized. . 19

2.3 Pareto Front according to a numerical example (cf. section 2.3.1.5), [135] . 20

2.4 Flowchart Optimization Process . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Evolutionary Algorithm with three generations moving towards the Paretofront . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.6 Flowchart Evolutionary Algorithm . . . . . . . . . . . . . . . . . . . . . . 30

2.7 One-Bit Mutation of a Bitstring . . . . . . . . . . . . . . . . . . . . . . . . 30

2.8 One-Point Crossover of Bitstrings . . . . . . . . . . . . . . . . . . . . . . . 31

2.9 Network Training. The upper three operations are independent from eachother. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.10 After training the model is used in an evolutionary strategy. . . . . . . . . 37

2.11 Staggered Channel Junction with six deflection points . . . . . . . . . . . . 43

2.12 Effect of design parameters on shape variation (case with 6 deflection points). 44

2.13 Test example: Tensile bar with centrical hole . . . . . . . . . . . . . . . . . 50

2.14 Optimization runs with different initial geometries . . . . . . . . . . . . . . 51

2.15 2-D Scatter plot: Principle Idea of an Evolutionary Algorithm. Here: Thepopulation moves towards a physical limit, the Pareto front. . . . . . . . . 52

2.16 Comparison of 3 optimization runs with different parameter setup . . . . . 53

2.17 Scatter plot: Identical simulation parameters, varied initial geometries . . . 53

2.18 Optimization runs with different parameter setup (initial geometry pc1,1) . . 54

2.19 Derived geometries using different process configurations. We indicate thecorresponding final designs. . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.20 Weighted objectives to picture the Pareto front . . . . . . . . . . . . . . . 55

143

144 LIST OF FIGURES

2.21 Chain link optimization setup . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.22 Results of the chain link optimization . . . . . . . . . . . . . . . . . . . . . 57

2.23 Example: Optimization of a thin-walled tube . . . . . . . . . . . . . . . . . 57

2.24 Optimization results: Thin-walled tube with torsion load . . . . . . . . . . 58

3.1 1) Approximation to and 2) diversity on Pareto front . . . . . . . . . . . . 61

3.2 These three individuals are excluded from archive with size 5. . . . . . . . 61

3.3 Distance assignment in NSGA-II . . . . . . . . . . . . . . . . . . . . . . . 66

3.4 Cutout of Flow Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.5 3-D scatter plot, scenario 1/1 . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.6 Scatter plot surface area-temperature, scenario 1/1 . . . . . . . . . . . . . 73

3.7 Scatter plot surface area-pressure drop, scenario 1/1 . . . . . . . . . . . . . 73

3.8 Scatter plot pressure drop-temperature, scenario 1/1 . . . . . . . . . . . . 73

3.9 Temperature distribution for individual a3, scenario 1/1 . . . . . . . . . . . 74


3.11 Temperature distribution for individual a2 = b1, scenario 1/1 . . . . . . . . 74

3.12 Temperature distribution for individual b2 = b3, scenario 1/1 . . . . . . . . 74

3.13 3-D scatter plot, scenario 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.14 Scatter plot surface area-temperature, scenario 1/2 . . . . . . . . . . . . . 75

3.15 Scatter plot surface area-pressure drop, scenario 1/2 . . . . . . . . . . . . . 75

3.16 Scatter plot pressure drop-temperature, scenario 1/2 . . . . . . . . . . . . 75

3.17 Temperature distribution for individual b3, scenario 1/2 . . . . . . . . . . . 76

3.18 Temperature distribution for individual a2 = a3, scenario 1/2 . . . . . . . . 76


3.20 Temperature distribution for individual a1 = b2, scenario 1/2 . . . . . . . . 76

3.21 3-D scatter plot for scenario 1/3 . . . . . . . . . . . . . . . . . . . . . . . . 77

3.22 Scatter plot surface area-temperature for scenario 1/3 . . . . . . . . . . . . 77

3.23 Scatter plot surface area-pressure drop for scenario 1/3 . . . . . . . . . . . 77

3.24 Scatter plot pressure drop-temperature for scenario 1/3 . . . . . . . . . . . 77

3.25 Temperature distribution: individual a1, scenario 1/3 . . . . . . . . . . . . 78

LIST OF FIGURES 145








3.33 Temperature distribution: individual a1, scenario 2/1 . . . . . . . . . . . . 79



3.36 Temperature distribution for individual b2 and b3, scenario 2/1 . . . . . . . 79









3.45 Temperature distribution for a2, b1 and b3, scenario 2/2 . . . . . . . . . . . 82

3.46 Temperature distribution for a1, a3 and b2, scenario 2/2 . . . . . . . . . . . 82


3.48 Temperature distribution for individual a2, a3, scenario 2/3 . . . . . . . . . 82

3.49 Cutout of Flow Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.50 18 design variables as indicated. . . . . . . . . . . . . . . . . . . . . . . . . 84

3.51 3-D scatter plot scenario 1, volume, pressure drop and temperature . . . . 86

3.52 3-D scatter plot scenario 1, pressure drop, outflow velocity and temperature 86


3.54 3-D scatter plot scenario 1, pressure drop, outflow velocity and temperature 86

146 LIST OF FIGURES


3.56 3-D scatter plot scenario 1, volume, outflow velocity and temperature . . . 87



3.59 Scatter plot scenario 1, volume vs. pressure drop . . . . . . . . . . . . . . 88

3.60 Scatter plot scenario 1, volume vs. temperature . . . . . . . . . . . . . . . 88

3.61 Scatter plot scenario 1, volume vs. outflow velocity . . . . . . . . . . . . . 88

3.62 Scatter plot scenario 1, pressure drop vs. temperature . . . . . . . . . . . 88

3.63 Scatter plot scenario 1, pressure drop vs. outflow velocity . . . . . . . . . . 89

3.64 3-D scatter plot scenario 2: volume, pressure drop and temperature . . . . 89




3.68 3-D scatter plot scenario 2: pressure drop, outflow velocity and temperature 90

3.69 Scatter plot scenario 2: volume vs. pressure drop . . . . . . . . . . . . . . 91

3.70 Scatter plot scenario 2: volume vs. temperature . . . . . . . . . . . . . . . 91

3.71 Scatter plot scenario 2: volume vs. outflow velocity . . . . . . . . . . . . . 91

3.72 Scatter plot scenario 2: pressure drop vs. temperature . . . . . . . . . . . 91

3.73 Scatter plot scenario 2: pressure drop vs. outflow velocity . . . . . . . . . . 92

3.74 Scatter plot scenario 2: volume vs. pressure drop . . . . . . . . . . . . . . 92

3.75 3-D scatter plot: SPEA2 on both scenarios . . . . . . . . . . . . . . . . . . 92

3.76 3-D scatter plot: NSGA-II on both scenarios . . . . . . . . . . . . . . . . . 92

3.77 3-D scatter plot: Femo on both scenarios . . . . . . . . . . . . . . . . . . . 93

3.78 3-D scatter plot: SPEA2 on both scenarios . . . . . . . . . . . . . . . . . . 93

3.79 Scatter plot: SPEA2 both scenarios, volume vs. pressure drop . . . . . . . 93

3.80 Scatter plot: SPEA2 both scenarios, volume vs. outflow velocity . . . . . . 93

3.81 Scatter plot: NSGA-II both scenarios, pressure drop vs. temperature . . . 94

3.82 Scatter plot: NSGA-II both scenarios, pressure drop vs. outflow velocity . 94

3.83 Temperature distribution for individual volmin. . . . . . . . . . . . . . . . . 95

LIST OF FIGURES 147

3.84 Flow velocity distribution for individual volmin. . . . . . . . . . . . . . . . . 95

3.85 Temperature distribution for individual pdmin = volmax. . . . . . . . . . . . 96

3.86 Flow velocity distribution for individual pdmin = volmax. . . . . . . . . . . . 96

3.87 Temperature distribution for individual tempmin. . . . . . . . . . . . . . . . 97

3.88 Flow velocity distribution for individual tempmin. . . . . . . . . . . . . . . 97

3.89 Temperature distribution for individual tempmax = pdmax. . . . . . . . . . 98

3.90 Flow velocity distribution for individual tempmax = pdmax. . . . . . . . . . 98

4.1 Sigmoid activation function. . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.2 Multilayer Perceptron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.3 An Artificial Neuron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.4 Adjustment of α, β. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.5 Channel junction (for better illustration we give cut-outs of geometries). . 114

4.6 Optimized channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.7 Network Performance for 16/16 network on 415 samples. . . . . . . . . . . 119

4.8 Network Performance for 32/24 network on 415 samples. . . . . . . . . . . 119

4.9 SA with 196 samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.10 Best result by varying parameters. Pressure drop at 5.7698. . . . . . . . . 120

4.11 12-Neuron Network fed with 243 samples. . . . . . . . . . . . . . . . . . . 120






4.17 SA with 215 samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121



4.20 12-Neuron Network with 248 training samples. . . . . . . . . . . . . . . . . 121

4.21 Gaussian initial membership functions for an inference system. . . . . . . . 124

4.22 A fuzzy inference system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

148 LIST OF FIGURES

4.23 A Sugeno type fuzzy inference system with two rules for each input. . . . . 128

4.24 Best result by varying parameters. Pressure drop at 5.7698. . . . . . . . . 134

4.25 SA with 196 samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

4.26 243 samples, large cluster radius. . . . . . . . . . . . . . . . . . . . . . . . 134

4.27 243 samples, small cluster radius. . . . . . . . . . . . . . . . . . . . . . . . 134

4.28 96 samples, small cluster radius. . . . . . . . . . . . . . . . . . . . . . . . . 134

4.29 60 training, 36 checking samples. . . . . . . . . . . . . . . . . . . . . . . . 134

4.30 SA with 215 samples on 12 design variables. . . . . . . . . . . . . . . . . . 135

4.31 170 training, 245 checking samples . . . . . . . . . . . . . . . . . . . . . . 135

4.32 250 training, 165 checking samples. Small cluster radius . . . . . . . . . . . 135


4.34 220 training, 28 checking samples. Small cluster radius . . . . . . . . . . . 135


4.36 Surface plot between input variable 1 and 6 and output variable (objectivefunction). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

List of Tables

2.1 Simulation parameters of three optimization runs . . . . . . . . . . . . . . 52

3.1 Scenario Simulation Settings . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.2 Parameters for simulation runs for scenarios 1 and 2 . . . . . . . . . . . . . 72

3.3 Scenario Simulation Settings . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.4 Parameters for simulation runs for scenarios 1 and 2 . . . . . . . . . . . . . 85

4.1 DFO Algorithm Performance . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.2 Methods used in this numerical example . . . . . . . . . . . . . . . . . . . 115

4.3 One Layer Network, NF Data Set . . . . . . . . . . . . . . . . . . . . . . . 117

4.4 Two Layer Network, NF Data Set . . . . . . . . . . . . . . . . . . . . . . . 117

4.5 Data Sets used for 6 Design Variables . . . . . . . . . . . . . . . . . . . . . 117

4.6 Data Sets used for Test Problem with 12 Design Variables . . . . . . . . . 118

4.7 One Layer Network, NF algorithm . . . . . . . . . . . . . . . . . . . . . . . 118

4.8 Two Layer Network, NF algorithm . . . . . . . . . . . . . . . . . . . . . . 118

4.9 ENF - One Layer Network . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.10 ENF - Two Layer Network . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.11 Methods used for numerical example . . . . . . . . . . . . . . . . . . . . . 131

4.12 Data Sets used for 6 Design Variables . . . . . . . . . . . . . . . . . . . . . 131

4.13 ANFIS performance for 243 training samples from Network Feeding . . . . 132

4.14 ANFIS performance for 96 training samples from Network Feeding . . . . . 133

4.15 Data Sets used for Test Problem with 12 Design Variables . . . . . . . . . 133

4.16 ANFIS performance for training samples from NF process . . . . . . . . . 133

4.17 ANFIS performance for training samples from ENF process . . . . . . . . . 133

149

150 LIST OF TABLES

Glossary

Optimization:

A(x) Gradient of constraints IE Index set of equality conditionsB∆(x) A set of values in the II Index set of inequality conditions

neighborhood of x λ Lagrange parameterc(x) Constraints mk(p) Local approximation at iterate xkCp Space of p-times differentiable P∗ Pareto front

functions PS∗ Pareto optimal set

IA Index set of active indices σ Standard Deviation<p Pareto dominance operator

Evolutionary Strategies:

i Individual E Evaluationi′ Offspring individual M Mutationλ Offspring population R Recombinationµ Parent population S SelectionPg Population in generation g V VariationPg,. Pg subjected to an operation z Gaussian deviation vector

Multiobjective Optimization:

F Pareto front R(i) Raw fitnessF (i) Final fitness S(i) Strength valueNarc Archive size σki Distance from individualP ∗ External archive i to individual kri Pareto rank <p Distance operator

Neural Networks:

E Network error η Learning parameterEW Weight decay regularizer µ Membership functionH Hessian matrix νA Indicator functionJ Jacobian matrix w Vector of network weigths

151

152 LIST OF TABLES

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Lebenslauf

Personliche DatenName Kai HirschenGeburtsdatum 30.11.1976Familienstand ledigStaatsangehorigkeit deutsch

Hochschulausbildung

seit 03/2003 TU Darmstadt, Fachbereich Maschinenbau,Fachgebiet fur Numerische Berechnungsverfahrenim Maschinenbau,Stipendiat im Graduiertenkolleg 853 “Modellierung,Simulation und Optimierung von Ingenieur-anwendungen”

10/1999 - 03/2003 Universitat Hannover,Studium der Mathematik und OkonomieAbschluß als Diplom-Mathematiker

09/2001 - 09/2002 Brunel University, West-London, UKStudiengang “Computational Mathematicswith Modelling”Abschluß als Master of Science

07/1999 Erwerb der fachgebundenen Hochschulreife, Vordiplom

09/1997 - 09/1999 Universitat Siegen,Studium der Mathematik und Okonomie

Berufliche Tatigkeiten

08/1995 - 08/1997 AMEFA GmbH, Kriftel, MedizinproduktegroßhandelTatigkeit als Einkaufssachbearbeiter

08/1993 - 07/1995 Buderus Guss GmbH, Staffel/Lahn,Ausbildung zum Industriekaufmann

Wehrdienst

09/1996 - 06/1997 Transportbataillon 370 Diez/Lahn

Schulausbildung

07/1995 Erwerb der Fachhochschulreife

08/1995 - 07/1997 Fachoberschule, Abendstufe,Friedrich-Dessauer Schule Limburg/Lahn

08/1987 - 07/1993 Realschule, Leo-Sternberg Schule, Limburg/Lahn

08/1983 - 07/1987 Grundschule Limburg-Offheim

4.3 ANFIS 135 - tuprints

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