Probabilistic finite element analysis using ANSYS
Introduction
Since quite a number of years, methods and tools to quantify the reliability and quality of mechanical products have received an ever-growing interest from industry as well as academia. Driven by the need to simultaneously reduce costs (manufacturing costs, warranty costs, etc.), reduce time-to-market, improve product quality and product reliability, industrial manufacturers find themselves challenged to optimize apparently conflicting technical and financial goals in an environment of ever increasing product complexity. In addition, this challenge is to be met under the existence of randomness and uncertainty, which the products are subjected to, since they are manufactured and operated under real-life conditions. Naturally, optimization is only possible if the optimization goals as well as possible constraints can be quantified. Consequently, finding the right balance between conflicting goals under the existence of uncertainties requires the use of probabilistic tools. In this context, the following analysis types are typically used to address this question:
Deterministic analysis. A deterministic analysis is the transformation function representing the relationship between the input variables influencing the behaviour of a product and the result parameters characterizing the product behaviour. In simple cases the result parameters can be expressed as an analytical function, but in realistic cases the input-output relationship is only given algorithmically for example using finite element program.
Uncertainty analysis. If the input variables influencing the behaviour of a product are uncertain, i.e. are subjected to scatter, then the primary task of an uncertainty analysis is to quantify how much the result parameters characterizing the product behaviour are affected by those uncertainties.
Reliability analysis. In order to quantify the reliability of a product it is useful to calculate the failure probability or non-conformance probability denoted with Pf. The reliability Ps is the probability that the product will survive or conforms to certain requirements, with Ps = 1 − Pf.
Reliability-based optimization. As the name implies, reliability-based optimization tries to optimize the reliability or failure probability. It should be noted, that improving the reliability often conflicts with other technical and financial goals. Hence, the optimization process should try to achieve a reasonable and quantifiable balance between all goals.
Robust design. Engineering products are becoming more and more complex and prone to the effects of uncertainty [1]. Robust design tries to optimize the design to make it less sensitive to unavoidable uncertainties, thereby reducing the variability in the product behaviour and making it more ‘predictable’. Achieving this is an optimization problem using the results of a probabilistic analysis as goals and constraint functions. Measures to quantify robustness (or the lack thereof) are for example the standard deviation or coefficient of variation, kurtosis, signal-to-noise ratios [2], [3], [4], Shannon’s entropy [5] or the failure probability of parameters describing the behaviour of a product.
Design for Six Sigma. The expression ‘Six Sigma’ was first devised by Motorola [6] defining that Six Sigma quality is given if only about 3.4 parts out of 1 million fall outside the limits given by the design requirements. In practice Design for Six Sigma is used either synonymously to robust design or to reliability-based optimization.
Multi-objective optimization. Improving the design of a product very often involves more than one goal. The more objectives are used to formulate the optimization problem then eventually some of them are conflicting, which means that improving one or more goals is only possible at the expense of one or more others.
Trade-off study. A trade-off study is a way to visualize conflicting goals in a multi-objective optimization in form of a Pareto front.
To address also the probabilistic analysis types mentioned above, ANSYS Inc. released two tools, namely the ANSYS Probabilistic Design System and the ANSYS DesignXplorer. Both tools can account for randomness in input variables such as material properties, boundary conditions, loads and geometry. Both tools can handle several of the analysis types mentioned above. In the following, the ANSYS Probabilistic Design System is referred to as ‘the PDS’ and the ANSYS DesignXplorer is denoted as ‘the DesignXplorer’.
This paper describes these tools in terms of the problems that can be addressed, as well as their capabilities and limitations. The capabilities of the tools to characterize the probabilistic problem are outlined. A separate section is dedicated to the probabilistic methods and their underlying algorithms. Here, a special topic of the paper is the discussion and explanation of the Variational Technology, which is offered in both ANSYS tools. Variational Technology is a method to provide accurate, higher-order response surfaces based on a single finite element analysis. The capabilities, strengths and weaknesses of these methods are discussed. For each method the measures to assess the accuracy and validity of the results is given special attention. The computational effort associated with these methods is compared with each other and the possibility to reduce the overall computation time using parallel computing is mentioned. Various capabilities and methods to post-process the probabilistic results are discussed. Finally, the application of the software is illustrated using various example problems.
Section snippets
Probabilistic tools
To address the growing need for stochastic and probabilistic finite element analysis ANSYS Inc. released first version of the ANSYS Probabilistic Design System in the year 2000. In the meantime also the ANSYS DesignXplorer has been released, which is based on a user-friendly interface with easy access to parameters including CAD parameters. Both tools are addressing a different audience and have common as well as differing capabilities. The capabilities of both ANSYS tools are summarized in
Characterizing the probabilistic model
In both ANSYS tools, the randomness of the input variables is characterized by the joint probability density function of the input variables [7]. Mixed moments of order higher than 2 are assumed to be negligible, which means that the joint probability density function can be described by the marginal distribution functions and the correlation structure.
Both ANSYS tools provide several statistical distribution functions to specify marginal distributions as listed in Table 2. The PDS allows the
Probabilistic methods
Both ANSYS tools offer several probabilistic methods. The PDS includes both the Monte-Carlo simulation method as well as response surface methods, while currently the DesignXplorer is based on response surface methods only. Both ANSYS tools allow the generation of response surfaces with traditional Design of Experiments as well as based on high-order derivatives using the Variational Technology. The probabilistic methods of both ANSYS tools are outlined in Table 3.
Probabilistic post-processing
Both ANSYS tools provide several ways to quantify and visualize probabilistic results. Basic statistical properties such mean values, standard deviation, skewness, kurtosis of result parameters and many others are provided to understand and quantify the variability of result parameter. Histogram plots and cumulative distribution function (CDF) plots help to visualize the scatter of a result parameters and to quantify probabilities. To better visualize the tails of the distribution the CDF can
Deterministic model and results
The application example is a probabilistic analysis of the lifetime of a turbine stage with 100 blades mounted on a disk, as illustrated in Fig. 5. This example shows the capabilities of the PDS to address reliability problems. The geometry of the blades was built in ANSYS. All material properties as well as thermal and static boundary conditions are arbitrary numbers, but represent realistic scenarios. The deterministic model includes a thermo-mechanical analysis of a turbine blade plus a
Summary
The current paper outlines, explains and compares the capabilities of the ANSYS Probabilistic Design System and the ANSYS DesignXplorer. The probabilistic methods implemented in these tools are discussed and the advantages and disadvantages are compared. The theoretical background of these methods and their accuracy is given special attention. A special topic in this context is the Variational Technology, which provides high order response surfaces based on a single finite element analysis.
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