A large number of problems in manufacturing processes, production planning, finance and engineeri... more A large number of problems in manufacturing processes, production planning, finance and engineering design require an understanding of potential sources of variations and quantification of the effect of variations on product behavior and performance. Traditionally, in engineering problems uncertainties have been formulated only through coarse safety factors. Such methods often lead to overdesigned products. Furthermore, the deterministic optimization algorithms tend to push an optimized design towards the boundaries of the design space. This paper reviews theories and methodologies that have been developed to solve optimization problems under uncertainties. In the first part the paper gives an overview over the state of the art in stochastic optimization methods such as robust design and reliability-based design optimization. In addition, global response surface techniques as well as genetic programming in combination with first order reliability methods in reliability-based optimization are discussed. Two numerical examples from structural analysis under static and dynamic loading conditions show the applicability of these concepts. The probabilistic and structural analysis tasks are performed with ANSYS DesignXplorer and OptiSLang software packages.
Die vorliegende Arbeit beschäftigt sich mit der Berechnung der Sicherheit von Strukturen mit sowo... more Die vorliegende Arbeit beschäftigt sich mit der Berechnung der Sicherheit von Strukturen mit sowohl geometrisch als auch physikalisch nichtlinearem Verhalten. Die Berechnung der Versagenswahrscheinlichkeit einer Struktur mit Hilfe von Monte-Carlo-Simulationsmethoden erfordert, dass die Funktion der Strukturantwort implizit berechnet wird, zum Beispiel durch nichtlineare Strukturanalysen für jede Realisation der Zufallsvariablen. Die Strukturanalysen bilden jedoch den Hauptanteil am Berechnungsaufwand der Zuverlässigkeitsanalyse, so dass die Analyse von realistischen Strukturen mit nichtlinearem Verhalten durch die begrenzten Computer-Ressourcen stark eingeschränkt ist. Die klassischen Antwortflächenverfahren approximieren die Funktion der Strukturantwort oder aber die Grenzzustandsfunktion durch Polynome niedriger Ordnung. Dadurch ist für die Auswertung des Versagens-Kriteriums nur noch von Interesse, ob eine Realisation der Basisvariablen innerhalb oder außerhalb des von der Antwor...
Volume 8: Microturbines, Turbochargers, and Small Turbomachines; Steam Turbines, 2018
This work presents a robust multi-objective optimization of a labyrinth seal used in power plants... more This work presents a robust multi-objective optimization of a labyrinth seal used in power plants steam turbines. The conflicting objectives of this optimization are to minimize the mass flow and to minimize the total enthalpy increase in order to increase the performance and to reduce the temperature, which results in elevated component utilization. The focus should be the robustness aspect to be involved into the optimization. So that the final design is not only optimized for its deterministic values but also robust under its uncertainties. To achieve a robust and optimized design, surrogate models are trained and used to replace the computational fluid dynamic solver (CFD), so as to speed up the calculations. In contrast to most techniques used in literature, the robustness criteria are directly involved in the multi-objective optimization. This leads to a more robust Pareto front compared to a purely deterministic one. This method needs many design evaluations, which would be n...
The correlation length-scale next to the noise variance are the most used hyperparameters for the... more The correlation length-scale next to the noise variance are the most used hyperparameters for the Gaussian processes. Typically, stationary covariance functions are used, which are only dependent on the distances between input points and thus invariant to the translations in the input space. The optimization of the hyperparameters is commonly done by maximizing the log marginal likelihood. This works quite well, if the distances are uniform distributed. In the case of a locally adapted or even sparse input space, the prediction of a test point can be worse dependent of its position. A possible solution to this, is the usage of a non-stationary covariance function, where the hyperparameters are calculated by a deep neural network. So that the correlation length scales and possibly the noise variance are dependent on the test point. Furthermore, different types of covariance functions are trained simultaneously, so that the Gaussian process prediction is an additive overlay of differe...
In order to meet the requirements of rising energy demand, a goal in the design process of modern... more In order to meet the requirements of rising energy demand, a goal in the design process of modern steam turbines is to achieve high efficiencies. A large gain in efficiency is expected from the optimization of the last stage and the following diffuser of a low pressure turbine (LP). The aim of such optimization is to minimize the losses due to separations or a inefficient blade or diffuser design. In the usual design process, as it is state of the art in the industry, the last stage of the LP and the diffuser is sequentially designed and optimized. The potential physical coupling effects are not considered. Therefore the aim of this paper is to perform both, a sequential and coupled optimization of a low pressure steam turbine followed by an axial radial diffuser and after that, the comparison of the results. Apart from the flow simulation there is also a mechanical and modal analysis made in order to satisfy the constraints regarding the natural frequencies and stresses. Thereby th...
Optimizing the reliability and the robustness of a design is important but often unaffordable due... more Optimizing the reliability and the robustness of a design is important but often unaffordable due to high sample requirements. Surrogate models based on statistical and machine learning methods are used to increase the sample efficiency. However, for higher dimensional or multi-modal systems, surrogate models may also require a large amount of samples to achieve good results. We propose a sequential sampling strategy for the surrogate based solution of multi-objective reliability based robust design optimization problems. Proposed local Latin hypercube refinement (LoLHR) strategy is model-agnostic and can be combined with any surrogate model because there is no free lunch but possibly a budget one. The proposed method is compared to stationary sampling as well as other proposed strategies from the literature. Gaussian process and support vector regression are both used as surrogate models. Empirical evidence is presented, showing that LoLHR achieves on average better results compared to other surrogate based strategies on the tested examples.
A large number of problems in manufacturing processes, production planning, finance and engineeri... more A large number of problems in manufacturing processes, production planning, finance and engineering design require an understanding of potential sources of variations and quantification of the effect of variations on product behavior and performance. Traditionally, in engineering problems uncertainties have been formulated only through coarse safety factors. Such methods often lead to overdesigned products. First, this paper gives an overview over the state of the art in reliability analysis methods to describe model uncertainties and to calculate reliability and safety. Different methods exist, but they all have a limited area of application. The aim of the first part is to investigate the applicability of certain methods for specific problems and to give a general indication, which method is appropriate for certain problem classes, implemented in optiSLang. In addition, a new adaptive response surface method is introduced to analyse the design reliability with high accuracy and efficiency. Whereby the surrogate model is based on an improved moving least square approximation combined with an adaptive design of experiment. In order to obtain a fast simulation procedure on the response surface an adaptive importance sampling concept is developed. Several numerical examples show the applicability of this concept for highly nonlinear state and limit state functions and multiple design points and separated unsafe domains. Furthermore, within a structural reliability analysis example a discretization of random fields in combination with a robustness evaluation to identify the relevant random variables is introduced. The so-called nonparametric structural reliabilty analysis is a new method to estimate the safety and reliability of finite element structures in cases where a CAD-based parametrization is not possible or not meaningful. The probabilistic and structural analysis tasks are performed with the optiSLang , SoS and SL ang software packages.
Proceedings of the Eleventh International Conference on Civil, Structural and Environmental Engineering Computing
Any mechanical or civil engineering structure possesses some natural randomness in its properties... more Any mechanical or civil engineering structure possesses some natural randomness in its properties which fluctuates over space. These can be modelled as random fields for the purpose of robust optimization or reliability analysis. The high number of random variables required to model a random field often inhibits accurate probabilistic analyses based on Monte Carlo methods. The present paper proposes a method to reduce the random space based on both stochastic criteria and structural performance. This way the most relevant variables are identified in order to be used in a subsequent reliability analysis. For computation of reliability, an adaptive response surface method is introduced, which utilizes improved Moving Least Squares approximations. It is able to model highly nonlinear limit state functions and can be locally refined. The stability of a cylindrical shell with random geometry is studied. A conventional and the proposed new way to reduce the random space are compared. The structure is then analysed by the adaptive response surface method, results are compared to direct directional simulation. The probabilistic and structural analyses are performed with the optiSLang, SoSand SL ang software packages.
In real case applications within the virtual prototyping process, it is not always possible to re... more In real case applications within the virtual prototyping process, it is not always possible to reduce the complexity of the physical models and to obtain numerical models which can be solved quickly. Usually, every single numerical simulation takes hours or even days. Although the progresses in numerical methods and high performance computing, in such cases, it is not possible to explore various model configu- rations, hence efficient surrogate models are required. The paper gives an overview about advanced methods of meta-modeling. In addition, some new aspects are introduced to impove the accuracy and predictability of surrogate models, commonly used in numerical models for automotive applications. Whereby, the main topic is reducing the neccessary number of design evaluations, e.g. finite element analysis within global variance-based sensitivity and robustness studies. In addition, the similar approach can be used to perform optimization and stochastic analysis and to create synt...
In real case applications within the virtual prototyping process, it is not always possible to re... more In real case applications within the virtual prototyping process, it is not always possible to reduce the complexity of the physical models and to obtain numerical models which can be solved quickly. Usually, every single numerical simulation takes hours or even days. Although the progresses in numerical methods and high performance computing, in such cases, it is not possible to explore various model configurations, hence efficient surrogate models are required. The paper gives an overview about advanced methods of meta-modeling. In ad-dition, some new aspects are introduced to impove the accuracy and predictability of surrogate models, commonly used in numerical models for automotive applications. Whereby, the main topic is reducing the neccessary number of design evaluations, e.g. finite element analysis within global variance-based sensitivity and robustness studies. In addition, the similar approach can be used to perform optimization and stochastic analysis and to create synth...
A large number of problems in manufacturing processes, production planning, finance and engineeri... more A large number of problems in manufacturing processes, production planning, finance and engineering design require an understanding of potential sources of variations and quantification of the effect of variations on product behavior and performance. Traditionally, in engineering problems uncertainties have been formulated only through coarse safety factors. Such methods often lead to overdesigned products. Different methods exist to describe model uncertainties and to calculate reliability and safety, but they all have a limited area of application. A new adaptive response surface method is introduced to analyse the design reliability with high accuracy and efficiency. Whereby the surrogate model is based on an improved moving least square approximation combined with an adaptive design of experiment. In order to obtain a fast simulation procedure on the response surface an adaptive importance sampling concept is used. Two numerical examples show the applicability of this concept for highly nonlinear state and limit state functions and multiple design points and separated unsafe domains. The probabilistic analysis tasks are performed with the optiSLang software package.
In reliability and robustness analysis, imperfections of a mechanical or structural system, such ... more In reliability and robustness analysis, imperfections of a mechanical or structural system, such as material properties or geometrical deviations, are modelled as random fields in order to account for their fluctuations over space. A random field normally comprises a huge number of random variables. The present paper proposes a method to reduce the random variables set. This reduction is performed on the basis on a robustness analysis. In this way, numerical difficulties can be avoided and the efficiency of the subsequent reliability analysis is enhanced. As an example, the reliability of a cylindricral shell structure with random imperfections is studied. Within this example, the imperfections are discretized by Stochastic Finite Element methods. it is demonstrated, how robustness analysis is employed in order to identify the most relevant random variables. The probability of failure is computed by Monte Carlo simulation involving Latin Hypercube sampling. The failure criterion is derived from a comparison of the linear buckling loads of the perfect and the imperfect structures. This so-called non-parametric structural reliability analysis is a new method to estimate the safety and reliability of finite element structures in such cases where a CAD-based parametrization is not possible or not meaningful. The probabilistic and structural analysis tasks are performed with the optiSLang, SoS and SL ang software packages.
In this study a new local and likewise global response surface method is proposed. Lengths and an... more In this study a new local and likewise global response surface method is proposed. Lengths and angles of the limit state check point vectors are being used only without any additional geometrical conditions. The so called weighted radii interpolation of convex and concave failure surfaces is intended to provide reasonably accurate estimates of failure probabilities while maintaining computational efficiency. These are especially suitable for the reliability analysis of complex structures, because global polynomial approximation procedures are not sufficiently flexible. They always need a predefined number of limit state support points in unimportant directions in order to avoid any approximation problems. Moreover, the maximum number of limit state check points is limited, too. This is achieved by a combination of random search strategies (based on the adaptive directional sampling approach) as well as deterministic search refinement in combination with local and global interpolation schemes. A numerical example shows application possibilities in the context of geometrically and materially nonlinear static analysis.
To ensure performance, manufacturability and reliability robustness analysis is an essential comp... more To ensure performance, manufacturability and reliability robustness analysis is an essential component of virtual prototyping analysis cycles. Within the robustness analysis the sensitivity of the unavoidable scatter of environmental conditions and their impact to the most important structural responses is evaluated. Especially for nonlinear structural behavior, when reducing hardware cycles or for optimized designs it is mandatory to analyze the robustness with respect to the most important random variations of the design parameters. Practical applications, using Latin Hypercube sampling, correlation and principal component analysis show that also high dimensional problems with more than 100 random variables can be handled with manageable CPU-cost. With Robustness analysis the most important random design parameter with their correlations and their impact on the variation behavior of the important responses can be evaluated. Additional system instabilities or cluster behavior can be identified. After identifying the most important design parameter safety and reliability analysis can be performed in reduced parameter dimensions.
The most general method to solve stochastic problems in structural mechanics is the well establis... more The most general method to solve stochastic problems in structural mechanics is the well established Monte Carlo simulation method. However, the major shortcoming of this approach is its vast need of computational resources (the number of finite elements runs required) which cannot be provided in general situations. Thus approximations become important which can be based e.g. on the response surface method. Unfortunately, the global approximation schemes widely used in the application of the response surface method can be quite misleading due to the lack of information in certain regions of the random variable space. It is therefore required to avoid such undesirable interpolation errors at reasonable computational effort. The polynomial approximations are not quite flexible. They always need a predefined number of limit state check points in unimportant directions in order to avoid any approximation problems. On this account the maximum number of limit state check points is limited, too. In this study some new local-global interpolation strategies for the response surface method are proposed. The so-called polyhedral and weighted radii interpolations of the failure surface are intended to provide reasonably accurate estimates of failure probabilities while maintaining computational efficiency. In particular, these response surfaces can be adaptively refined to consistently increase the accuracy of the estimated failure probability. This is achieved by a combination of random search strategies (based on the adaptive sampling and directional sampling approach) as well as deterministic search refinement together with local and global interpolation schemes. The advantage of these methods is the flexibility for the approximation of highly nonlinear limit state functions. This is especially suitable for the reliability analysis of complex nonlinear structures. An arbitrary number of check points even in high local concentration can be used without approximation problems. In this sense, the proposed method is very robust and efficient. An numerical example from structural analysis under static loading conditions shows the applicability of these concepts. The probabilistic and structural analysis tasks are performed with the SL ang software package.
In this paper an efficient approach is presented to assist reducing the number of design evaluati... more In this paper an efficient approach is presented to assist reducing the number of design evaluations necessary, in particular the number of nonlinear fluid-structure interaction analyses. In combination with a robust estimation of the safety level within the iteration and a final precise reliability analysis, the method presented is particularly suitable for solving reliability-based structural design optimization problems with ever-changing failure probabilities of the nominal designs. The applicability for real case applications is demonstrated through the example of a radial compressor, with a very high degree of complexity and a large number of design parameters and random variables. Note: an extended version of this paper is originally published in Roos et al. (2013).
Within the design development phases the Design for Six Sigma concept optimizes a design such tha... more Within the design development phases the Design for Six Sigma concept optimizes a design such that the products conform to Six Sigma quality. Which means that robustness and reliability are explicit optimization goals even with variations e.g. in manufacturing, design configuration and environment. The application of the reliability-and variance-based robust design optimization results in optimized designs such that they are insensitive to uncertainties up to a six sigma safety level. In this paper an efficient iterative decoupled loop approach is provided for reducing the necessary number of design evaluations. This is exemplary applied to a CAD and CAE parameter-based robust design optimization of an axial turbine, including manufacturing tolerances based on random field modeling. The probabilistic and optimization tasks are performed with the optiSLang, SoS and SL ang software packages. Whereby, the CAE integration is realized by the ANSYS Workbench environment and optiPLug. In addition, the ANSYS Mechanical and CFD software offers a comprehensive solution for structural, thermal and fluid analysis. The software package also includes solutions for both direct and sequentially coupled physics problems including direct coupled-field elements and the ANSYS multi-field solver for supported physics, which is very efficient for tolerance interpolation of measurement data to different finite element meshes.
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Papers by Dirk Roos