Sensitivity analysis is a method to study how the variation in the output of a model depends on the variation in the input parameters. It can help you to identify the most significant parameters that affect the model response, and to reduce the computational effort in structural optimization ¹.
There are different types of sensitivity analysis, such as local, global, and probabilistic. Local sensitivity analysis evaluates the effect of a small change in one input parameter at a time, while keeping the others fixed. Global sensitivity analysis evaluates the effect of varying all input parameters simultaneously over their entire range of values. Probabilistic sensitivity analysis incorporates uncertainty in the input parameters and outputs ².
To perform sensitivity analysis in Ansys Structural, you can use various tools and methods, depending on your problem and objectives. Some of the possible options are:
- Inserting a convergence object under the stress and entering the allowable change. You can also change the maximum number of refinement loop and depth needed under the Solution tab ³.
- Using DesignXplorer, which is an integrated tool for design exploration and optimization. It allows you to define design variables, objectives, and constraints, and to perform parametric studies, response surface analysis, optimization, and robust design analysis ⁴.
- Using global sensitivity analysis methods such as Sobol indices, correlation analysis, ANOVA, or FAST. These methods can help you to quantify the contribution of each input parameter to the output variance, and to rank them according to their importance
Source:
(1) Global Sensitivity Analysis in Structural Optimization - LSDYNA. https://lsdyna.ansys.com/wp-content/uploads/attachments/f-i-03.pdf.
(2) How to perform mesh sensitivity study for stress - Ansys Knowledge. https://ansyskm.ansys.com/forums/topic/how-to-perform-mesh-sensitivity-study-for-stress/.
(3) What is Sensitivity Analysis? - Corporate Finance Institute. https://corporatefinanceinstitute.com/resources/financial-modeling/what-is-sensitivity-analysis/.
(4) Mesh Sensitivity Study for CFD Simulations - SimScale. https://www.simscale.com/knowledge-base/mesh-sensitivity-cfd/.
What is the difference between local and global sensitivity analysis?
The difference between local and global sensitivity analysis is that local sensitivity analysis evaluates the effect of a small change in one input parameter at a time, while keeping the others fixed. Global sensitivity analysis evaluates the effect of varying all input parameters simultaneously over their entire range of values ¹.
Local sensitivity analysis is easier to perform and requires less computational effort, but it has some limitations. It can only explore the model behavior around a specific point in the input space, and it cannot account for the interactions between the input parameters. It may also produce biased results if the model is nonlinear or has multiple optima ²³.
Global sensitivity analysis is more comprehensive and robust, but it requires more computational resources and sophisticated methods. It can reveal the global effects of each input parameter on the model output, including any interactive effects. It can also handle nonlinear models and uncertainty in the input parameters ¹².
Depending on your problem and objectives, you may choose different types of sensitivity analysis methods. For example, if you want to identify the most important parameters that affect the model output, you can use local sensitivity analysis or global sensitivity methods such as Sobol indices or ANOVA. If you want to study how the model output varies across the entire input space, you can use global sensitivity methods such as response surface analysis or Monte Carlo simulation ²⁴.
Source:
(1) Sensitivity Analysis: The Basics - Addressing Uncertainty in .... https://uc-ebook.org/docs/html/3_sensitivity_analysis_the_basics.html.
(2) Comparative Study of Global Sensitivity Analysis and Local Sensitivity .... https://www.mdpi.com/1996-1073/16/16/5915.
(3) COMPARISON OF SENSITIVITY ANALYSIS TECHNIQUES D. M. Hamby Savannah .... https://web.engr.oregonstate.edu/~hambydm/papers/senscomparison.pdf.
(4) What are global sensitivity and local sensitivity in differential .... https://stats.stackexchange.com/questions/370591/what-are-global-sensitivity-and-local-sensitivity-in-differential-privacy.
(5) Do we need to do both local sensitivity analysis and global sensitivity .... https://stackoverflow.com/questions/67105593/do-we-need-to-do-both-local-sensitivity-analysis-and-global-sensitivity-analysis.
(6) https://doi.org/10.3390/en16165915.
What is the difference between Sobol indices and ANOVA?
The difference between Sobol indices and ANOVA is that Sobol indices are a type of global sensitivity analysis, while ANOVA is a type of local sensitivity analysis ¹.
Sobol indices measure the contribution of each input parameter or a group of input parameters to the variance of the output of a model. They can also capture the interaction effects between the input parameters. Sobol indices are based on the ANOVA decomposition of the model output, but they require sampling the input parameters over their entire range of values ²³.
ANOVA, which stands for analysis of variance, is a statistical method to compare the means of different groups. It can also be used to decompose the output of a model into terms that depend on one or more input parameters. However, ANOVA assumes that the input parameters are fixed at certain levels, and it does not account for the variability or uncertainty in the input parameters. ANOVA can only measure the main effects of each input parameter, but not their interaction effects ¹⁴.
Therefore, Sobol indices and ANOVA have different objectives and assumptions. Sobol indices are more suitable for studying complex and nonlinear models with uncertain inputs, while ANOVA is more suitable for testing hypotheses and comparing groups with fixed inputs ¹².
Source:
(1) Variance-based sensitivity analysis - Wikipedia. https://en.wikipedia.org/wiki/Variance-based_sensitivity_analysis.
(2) anova - Sobol Indices for Discrete Variables - Cross Validated. https://stats.stackexchange.com/questions/190843/sobol-indices-for-discrete-variables.
(3) HigherorderSobol’indices arXiv:1306.4068v1 [math.NA] 18 Jun 2013. https://arxiv.org/pdf/1306.4068.pdf.
(4) What do the specific values of the Sobol' indices mean?. https://stats.stackexchange.com/questions/471636/what-do-the-specific-values-of-the-sobol-indices-mean.
(5) Difference between Sobol indices and total Sobol indices?. https://stats.stackexchange.com/questions/70930/difference-between-sobol-indices-and-total-sobol-indices.
How to calculate Sobol indices in Ansys Structural?
To calculate Sobol indices in Ansys Structural, you can use the DesignXplorer tool, which is integrated with Ansys Workbench. DesignXplorer allows you to perform global sensitivity analysis using various methods, including Sobol indices ¹.
Here are the steps to calculate Sobol indices in Ansys Structural using DesignXplorer:
- Define your design variables, objectives, and constraints in the Workbench project.
- Launch DesignXplorer from the Workbench project page.
- In the Design Exploration tab, select Sensitivity Analysis from the drop-down menu.
- In the Sensitivity Analysis dialog box, select Sobol Indices as the method.
- Specify the number of samples and the seed for the random number generator.
- Select the output parameters that you want to analyze.
- Click Generate Samples to create the input samples for the Sobol indices calculation.
- Click Update Project to run the analysis and update the results in DesignXplorer.
- In the Results tab, select Sobol Indices from the drop-down menu.
- You can view the first-order and total-order Sobol indices for each output parameter in a table or a bar chart. You can also export the results to a CSV file.
Source:
(1) Global Sensitivity Analysis in Structural Optimization - LSDYNA. https://lsdyna.ansys.com/wp-content/uploads/attachments/f-i-03.pdf.
(2) Variance-based sensitivity analysis - Wikipedia. https://en.wikipedia.org/wiki/Variance-based_sensitivity_analysis.
(3) Computing Sobol Sensitivity Indexes - MATLAB Answers - MathWorks. https://www.mathworks.com/matlabcentral/answers/90230-computing-sobol-sensitivity-indexes.
(4) sobol_indices : Computation of Sobol' indices - R Package Documentation. https://rdrr.io/cran/sensobol/man/sobol_indices.html.
(5) A Tutorial on Sobol’ Global Sensitivity Analysis Applied to Biological .... https://link.springer.com/chapter/10.1007/978-3-030-51862-2_6.
How to interpret the results of Sobol indices analysis?
The results of Sobol indices analysis can help you to understand how the input parameters of your model affect the output variability. The Sobol indices are based on the ANOVA decomposition of the model output, but they account for the uncertainty and interactions of the input parameters ¹.
The first-order Sobol indices measure the contribution of each input parameter to the output variance, while keeping the other parameters fixed. They can tell you which parameters have the main effects on the output. The sum of the first-order indices is equal to or less than one ².
The total-order Sobol indices measure the contribution of each input parameter to the output variance, including all the interactions with other parameters. They can tell you which parameters have the most influence on the output, either individually or in combination with others. The sum of the total-order indices is equal to or greater than one ².
To interpret the results of Sobol indices analysis, you can compare the values of the first-order and total-order indices for each parameter. A high first-order index and a low total-order index indicate that the parameter has a strong main effect, but a weak interaction effect. A low first-order index and a high total-order index indicate that the parameter has a weak main effect, but a strong interaction effect. A high first-order index and a high total-order index indicate that the parameter has both a strong main effect and a strong interaction effect ³.
You can also plot the Sobol indices in a bar chart or a spider plot to visualize their relative importance and uncertainty. The confidence intervals of the Sobol indices depend on the number of samples and the bootstrap method used in the analysis. If the confidence intervals are too large or include zero, it means that the estimates are not reliable or significant ²⁴.
Source:
(1) What do the specific values of the Sobol' indices mean?. https://stats.stackexchange.com/questions/471636/what-do-the-specific-values-of-the-sobol-indices-mean.
(2) Interpreting results from Sobol sensitivity analysis in R. https://stats.stackexchange.com/questions/43504/interpreting-results-from-sobol-sensitivity-analysis-in-r.
(3) Difference between Sobol indices and total Sobol indices?. https://stats.stackexchange.com/questions/70930/difference-between-sobol-indices-and-total-sobol-indices.
(4) sensitivity analysis - Intuition for Sobol Indices - Cross Validated. https://stats.stackexchange.com/questions/468869/intuition-for-sobol-indices.
(5) Negative sobol indices · Issue #102 · SALib/SALib · GitHub. https://github.com/SALib/SALib/issues/102.
How to compare the results of different sensitivity analysis methods?
Comparing the results of different sensitivity analysis methods can be challenging, because different methods may have different objectives, assumptions, and limitations. However, some general principles and tips for comparison are ¹²:
- Define the purpose and scope of the sensitivity analysis. What are the research questions and hypotheses that you want to test? What are the sources of uncertainty and variability in your model and data? What are the criteria for evaluating the model performance and robustness?
- Choose appropriate sensitivity analysis methods that match your purpose and scope. For example, if you want to identify the most influential parameters, you can use local or global sensitivity methods such as Sobol indices or ANOVA. If you want to explore the model behavior across the entire input space, you can use global sensitivity methods such as response surface analysis or Monte Carlo simulation.
- Compare the results of different sensitivity analysis methods using graphical and numerical summaries. For example, you can plot the sensitivity indices or measures in a bar chart or a spider plot to visualize their relative importance and uncertainty. You can also calculate the correlation or agreement between different sensitivity indices or measures to quantify their consistency or discrepancy.
- Interpret the results of different sensitivity analysis methods in the context of your research questions and hypotheses. For example, you can assess whether the results support or contradict your expectations, whether they reveal any new insights or surprises, whether they suggest any potential improvements or modifications for your model or data.
- Report the results of different sensitivity analysis methods transparently and comprehensively. For example, you should describe the methods and assumptions used, the input and output parameters considered, the sampling and estimation procedures applied, the uncertainty and variability involved, and the limitations and caveats acknowledged.
Source:
(1) Sensitivity Analysis and Model Validation | SpringerLink. https://link.springer.com/chapter/10.1007/978-3-319-43742-2_17.
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(5) 9.7 Sensitivity analyses - Cochrane. http://handbook-5-1.cochrane.org/chapter_9/9_7_sensitivity_analyses.htm.
What are the advantages and disadvantages of different sensitivity analysis methods?
Sensitivity analysis is a method to study how the variation in the output of a model depends on the variation in the input parameters. It can help you to identify the most significant parameters that affect the model response, and to reduce the computational effort in structural optimization ¹.
There are different types of sensitivity analysis methods, such as local, global, and probabilistic. Each method has its own advantages and disadvantages, depending on the problem and objectives. Here are some of the main points to consider ²³:
- Local sensitivity analysis evaluates the effect of a small change in one input parameter at a time, while keeping the others fixed. It is easy to perform and requires less computational effort, but it can only explore the model behavior around a specific point in the input space, and it cannot account for the interactions between the input parameters. It may also produce biased results if the model is nonlinear or has multiple optima.
- Global sensitivity analysis evaluates the effect of varying all input parameters simultaneously over their entire range of values. It is more comprehensive and robust, but it requires more computational resources and sophisticated methods. It can reveal the global effects of each input parameter on the model output, including any interactive effects. It can also handle nonlinear models and uncertainty in the input parameters.
- Probabilistic sensitivity analysis incorporates uncertainty in the input parameters and outputs. It can provide information on the probability distribution of the model output, and the confidence intervals of the sensitivity measures. It can also perform risk analysis and robust design analysis. However, it requires a large number of samples and a reliable estimation of the input uncertainty.
Depending on your problem and objectives, you may choose different types of sensitivity analysis methods. For example, if you want to identify the most important parameters that affect the model output, you can use local sensitivity analysis or global sensitivity methods such as Sobol indices or ANOVA. If you want to study how the model output varies across the entire input space, you can use global sensitivity methods such as response surface analysis or Monte Carlo simulation.
Source:
(1) Advantages and Disadvantages of Sensitivity Analysis. https://efinancemanagement.com/investment-decisions/advantages-disadvantages-of-sensitive-analysis.
(2) A tutorial on sensitivity analyses in clinical trials: the what, why .... https://bmcmedresmethodol.biomedcentral.com/articles/10.1186/1471-2288-13-92.
(3) Sensitivity Analysis - Definition, Example, Interpretation, Methods | eFM. https://efinancemanagement.com/investment-decisions/sensitivity-analysis.
(4) Sensitivity Analysis - Advantages, Disadvantages, Application, Practices. https://sattvacfo.com/sensitivity-analysis-financial-modelling/.
(5) What Is a Sensitivity Analysis? Definition and Examples. https://www.indeed.com/career-advice/career-development/sensitivities-analysis.
How to choose a suitable sensitivity analysis method for my problem?
Choosing a suitable sensitivity analysis method for your problem depends on several factors, such as the purpose and scope of your analysis, the characteristics and complexity of your model, the availability and quality of your data, and the computational resources and time you have ¹².
Some general steps to help you choose a suitable sensitivity analysis method are:
- Define your research question and hypothesis. What do you want to learn from the sensitivity analysis? What are the expected outcomes and implications of your analysis?
- Identify the input and output parameters of your model. Which parameters are uncertain or variable, and which ones are fixed or constant? Which parameters are you interested in analyzing, and which ones are irrelevant or redundant?
- Assess the type and degree of uncertainty or variability in your input parameters. How can you describe or quantify the uncertainty or variability? What are the sources and distributions of the uncertainty or variability?
- Select a sensitivity analysis method that matches your purpose, model, data, and resources. Consider the advantages and disadvantages of different methods, such as local or global, deterministic or probabilistic, one-at-a-time or all-at-a-time, etc. You can use some criteria or guidelines to compare and evaluate different methods ³⁴.
- Perform the sensitivity analysis using the selected method. Follow the steps and procedures of the method, such as sampling, estimation, calculation, etc. Check the validity and reliability of the results.
- Interpret and report the results of the sensitivity analysis. Explain how the results answer your research question and hypothesis. Discuss the implications and limitations of the results. Provide graphical and numerical summaries of the results.
Source:
(1) Sensitivity Analysis - What Is It, Examples, Formula, How To Do?. https://www.wallstreetmojo.com/sensitivity-analysis/.
(2) 9.7 Sensitivity analyses - Cochrane. https://handbook-5-1.cochrane.org/chapter_9/9_7_sensitivity_analyses.htm.
(3) Sensitivity Analysis | Examples of Sensitivity Analysis - EDUCBA. https://www.educba.com/sensitivity-analysis/.
(4) How to Do Sensitivity Analysis in Excel (3 Easy Methods) - ExcelDemy. https://www.exceldemy.com/how-to-do-sensitivity-analysis-in-excel/.
(5) Which sensitivity analysis method should I use for my agent-based model .... https://research.wur.nl/en/publications/which-sensitivity-analysis-method-should-i-use-for-my-agent-based.
Examples of using different methods for sensitivity analysis in Ansys Structural
There are different types of sensitivity analysis methods, such as local, global, and probabilistic. Each method has its own advantages and disadvantages, depending on the problem and objectives. You can use different methods to compare and evaluate the sensitivity of your model to different input parameters ².
One example of using different methods for sensitivity analysis in Ansys Structural is from a paper by Reuter et al. ¹. They compared the variance-based approach after Sobol, the correlation analysis, the linear and quadratic ANOVA approaches, and the FAST approach for a structural optimization problem of a composite laminate. They used DesignXplorer, which is an integrated tool for design exploration and optimization in Ansys Workbench, to perform the sensitivity analysis.
They found that the Sobol indices and the correlation coefficients gave consistent results for ranking the input parameters according to their importance. The ANOVA approaches and the FAST approach gave similar results for the main effects, but differed for the interaction effects. They also found that some input parameters had negligible effects on the output, and could be eliminated from the optimization problem.
Another example of using different methods for sensitivity analysis in Ansys Structural is from a paper by Camarda ². He used some innovative techniques for sensitivity analysis of discretized structural systems, such as a finite-difference step-size selection algorithm, a method for derivatives of iterative solutions, a Green's function technique for derivatives of transient response, a simultaneous calculation of temperatures and their derivatives, derivatives with respect to shape, and derivatives of optimum designs with respect to problem parameters.
He applied these techniques to various problems, such as thermal buckling of composite plates, transient thermal response of hypersonic vehicles, shape optimization of aircraft wings, and optimum design of space shuttle tiles. He showed that these techniques could improve the accuracy and efficiency of sensitivity analysis, and could provide valuable information for design and optimization.
Source:
(1) Global Sensitivity Analysis in Structural Optimization - LSDYNA. https://lsdyna.ansys.com/wp-content/uploads/attachments/f-i-03.pdf.
(2) Structural sensitivity analysis: Methods, applications, and needs. https://www.academia.edu/48673736/Structural_sensitivity_analysis_Methods_applications_and_needs.
(3) Study on Structure Sensitivity Analysis Using ANSYS PDS. https://www.scientific.net/AMR.243-249.1830.
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