OpenFOAM Error: Carrier Phase curlUc Interpolation Object Not Set (SaffmanMeiLiftForce)

 The error message "Carrier phase curlUc interpolation object not set" in OpenFOAM indicates that a required interpolation object for the Saffman Mei lift force calculation is missing. Here's how to address this:

Divergence detected in AMG solver - How to solve this error ?

 The "Divergence detected in AMG solver" error in ANSYS Fluent indicates that the Algebraic Multigrid (AMG) solver is struggling to converge. This can arise from several factors, including:

How to solve below problem in Ansys Fluent Divergence detected in AMG solver: temperature?

 The error message "Divergence detected in AMG solver: temperature" in Ansys Fluent indicates that the solver is having difficulty converging on a solution for the temperature field. Here are some steps you can take to address this issue:

💥💥💥 How to increase precision in Ansys Workbench calculations?

 One way to increase the precision of your calculations in Ansys Workbench is to use the double-precision solver option in Fluent. This option allows you to use more digits after the decimal point and reduce the numerical errors. You can find this option in the Fluent Launcher, under Options, as shown in this tutorial and this one. Alternatively, you can also increase the precision of values or number of decimal places in SpaceClaim by changing the Primary Precision option in File > SpaceClaim Options > Units. You can learn more about this option from this forum post. 

How do I know if my calculations are precise enough?

There are several factors that can affect the precision of your calculations in Ansys Workbench, such as the mesh quality, the solver settings, the convergence criteria, and the material model. You can check the precision of your calculations by comparing the results with analytical solutions, experimental data, or other reliable sources. You can also monitor the residuals and other quantities of interest during the solution process to ensure that they reach a steady or periodic state. Additionally, you can perform a sensitivity analysis or a mesh refinement study to evaluate how the results change with different input parameters or mesh sizes. You can find more information and examples on how to check the precision of your calculations from the following sources:

• How to know the calculation has converged
• Fundamental Calculations of equivalent stress and strain in ANSYS Workbench
• Steady Flow Past Cylinder Physics Setup
• Physics Setup — Lesson 5

Can you explain more about mesh quality and refinement study?

Sure, I can explain more about mesh quality and refinement study in Ansys Workbench. Mesh quality is a measure of how well the mesh elements represent the geometry and the physics of the problem. Mesh refinement is a process of increasing the number of mesh elements or improving their shape to achieve more accurate results.

Mesh quality depends on several factors, such as the element type, the element shape, the element size, and the element alignment. Different types of elements have different advantages and disadvantages for different problems. For example, tetrahedral elements are easy to generate and can fit complex geometries, but they may have lower accuracy and stability than hexahedral elements. The element shape affects how well the element can approximate the curvature of the geometry and the gradients of the solution. The element size determines how fine or coarse the mesh is, and how well it can capture the details of the problem. The element alignment refers to how well the mesh follows the direction of the flow or the stress.

Mesh refinement is a way to improve the mesh quality by increasing the number of elements or changing their shape. Mesh refinement can be done globally or locally. Global mesh refinement applies to the whole domain or a large region, and it increases the resolution of the mesh uniformly. Local mesh refinement applies to a specific area of interest, such as a boundary layer, a corner, or a crack, and it increases the resolution of the mesh selectively. Mesh refinement can also be done adaptively or manually. Adaptive mesh refinement is an automated process that adjusts the mesh based on some criteria, such as the error estimate, the solution gradient, or the user-defined function. Manual mesh refinement is a user-controlled process that modifies the mesh based on some parameters, such as the element size, the inflation factor, or the smoothing factor.

Mesh quality and refinement study is important for obtaining reliable and accurate results in Ansys Workbench. A good quality mesh can reduce the numerical errors and improve the convergence of the solution. A refined mesh can capture the features and phenomena of interest more accurately and resolve the variations of the solution more precisely. However, mesh quality and refinement also have some trade-offs, such as increased computational cost, memory requirement, and solution time. Therefore, it is necessary to balance between mesh quality and refinement and computational efficiency.

You can learn more about mesh quality and refinement study in Ansys Workbench from these sources:

• Generating Mesh in ANSYS Workbench | Part-1
• How to check the MESH Quality in Ansys Workbench | Mesh Metrics | Skewness | Orthogonal Quality
• Tutorial on Meshing in ANSYS Workbench I ANSYS Basic Tutorials Apply various Mesh on geometry
• Ansys sturcutal analysis tutorials 23: mesh convergence and refinement ; Stress history plot
• mesh refinment study - Ansys Learning Forum
• Using Adaptive Convergence with Ansys Mechanical — Lesson 5
• MESH QUALITY AND ADVENCED TOPICS ANSYS WORKBENCH 16.0 - Academia.edu

How do I choose the right element size for my mesh?

Choosing the right element size for your mesh depends on several factors, such as the geometry, the physics, the accuracy, and the computational cost of your problem. There is no universal rule for selecting the optimal element size, but there are some general guidelines and methods that can help you. Here are some of them:

• You can start with a coarse mesh and refine it gradually until you reach a desired level of accuracy or convergence. You can use global or local refinement, adaptive or manual refinement, or a combination of them. You can also use different types of elements, such as tetrahedral, hexahedral, or hybrid elements, depending on the complexity and curvature of your geometry.
• You can use mesh sizing controls to specify the element size for a selected body, face, edge, or vertex. You can also use the sphere of influence option to apply a local element size around a point of interest. You can access these options by right-clicking on the Mesh in the project tree and selecting Insert > Sizing.
• You can use mesh metrics to evaluate the quality of your mesh, such as the skewness, the aspect ratio, the orthogonal quality, or the smoothness. You can check these metrics by right-clicking on the Mesh in the project tree and selecting Mesh Metrics. You can also use mesh statistics to see the number and type of elements in your mesh. You can access these statistics by right-clicking on the Mesh in the project tree and selecting Statistics.
• You can compare your results with analytical solutions, experimental data, or other reliable sources to validate your mesh. You can also monitor the residuals and other quantities of interest during the solution process to ensure that they are within acceptable limits. You can also perform a sensitivity analysis or a mesh convergence study to evaluate how your results change with different input parameters or mesh sizes.

You can learn more about how to choose the right element size for your mesh in Ansys Workbench from these sources:

• Correct mesh size - a quick guide! - Enterfea
• Learn Everything about the Ansys Workbench | Mesh Sizing - Skill-Lync
• Sizing a Mesh | Ansys Innovation Courses
• Defining Element Size Of A Body In ANSYS® Mechanical(Illustrated Expression) - ML
• Changing The Element Size In ANSYS® Meshing(Iluustrated Expression) - ML

How do I know if my element size is too small or too large?

The element size is a key parameter that affects the accuracy and efficiency of your mesh and solution in Ansys Workbench. If your element size is too small, you may have a very fine mesh that can capture the details of your problem, but it will also increase the computational cost, memory requirement, and solution time. If your element size is too large, you may have a very coarse mesh that can reduce the computational cost, memory requirement, and solution time, but it will also introduce numerical errors and affect the convergence of your solution.

To know if your element size is too small or too large, you can use some methods and criteria to evaluate your mesh and solution quality. Here are some of them:

• You can compare your results with analytical solutions, experimental data, or other reliable sources to validate your mesh and solution. You can also monitor the residuals and other quantities of interest during the solution process to ensure that they are within acceptable limits.
• You can use mesh metrics to evaluate the quality of your mesh, such as the skewness, the aspect ratio, the orthogonal quality, or the smoothness. You can check these metrics by right-clicking on the Mesh in the project tree and selecting Mesh Metrics. You can also use mesh statistics to see the number and type of elements in your mesh. You can access these statistics by right-clicking on the Mesh in the project tree and selecting Statistics.
• You can perform a sensitivity analysis or a mesh convergence study to evaluate how your results change with different input parameters or mesh sizes. You can use global or local refinement, adaptive or manual refinement, or a combination of them. You can also use different types of elements, such as tetrahedral, hexahedral, or hybrid elements, depending on the complexity and curvature of your geometry.

You can learn more about how to know if your element size is too small or too large in Ansys Workbench from these sources:

• Sizing on Ansys Workbench 19.2
• Element size too small
• Meshing Element size

Can you explain more about the aspect ratio and orthogonal quality metrics?

Sure, I can explain more about the aspect ratio and orthogonal quality metrics in Ansys Workbench. These are two of the mesh metrics that you can use to evaluate the quality of your mesh elements.

The aspect ratio quantifies the shape of the elements, where 1 is a perfectly shaped element and the element shape is worse with a higher aspect ratio. The aspect ratio is defined as the ratio of the shortest length of the element to the longest length of the element¹. For example, a tetrahedral element has an aspect ratio of 1 if it is a regular tetrahedron, and a higher aspect ratio if it is elongated or flattened. A hexahedral element has an aspect ratio of 1 if it is a cube, and a higher aspect ratio if it is a rectangular prism or a parallelepiped. The aspect ratio affects the accuracy and stability of the solution, as well as the convergence rate. Generally, you should aim for an aspect ratio less than 20 for structural analysis and less than 1000 for fluid analysis².

The orthogonal quality quantifies the alignment of the elements with respect to the geometry and the physics of the problem. The orthogonal quality is defined as the minimum value of the dot product between the element face normal and the vector from the face centroid to the opposite node, for all faces of the element³. The orthogonal quality ranges from 0 to 1, where 1 is the best value and 0 is the worst value. For example, a tetrahedral element has an orthogonal quality of 1 if it is equilateral and orthogonal to the geometry, and a lower orthogonal quality if it is skewed or distorted. A hexahedral element has an orthogonal quality of 1 if it is aligned with the geometry and the flow or stress direction, and a lower orthogonal quality if it is misaligned or twisted. The orthogonal quality affects the accuracy and robustness of the solution, as well as the numerical diffusion and dispersion. Generally, you should aim for an orthogonal quality greater than 0.1 for most problems⁴.

You can check these metrics by right-clicking on the Mesh in the project tree and selecting Mesh Metrics. You can also use mesh statistics to see the number and type of elements in your mesh. You can access these statistics by right-clicking on the Mesh in the project tree and selecting Statistics.

You can learn more about how to improve these metrics by using different meshing methods, such as global or local refinement, adaptive or manual refinement, or different types of elements, such as tetrahedral, hexahedral, or hybrid elements. You can find some tutorials and examples from these sources:

• Explanations Of All Mesh Metric Options In ANSYS® Mechanical
• 网格划分（二）：网格质量评估（基于ANSYS2020）
• Is Your Mesh Good Enough?
• Ansys Workbench Mesh Quality || Skewness || Aspect Ratio || Orthogonal

💥💥💥 How to fix "overflow" error in Ansys CFD?

 The error “overflow” in Ansys CFD means that a number is being generated that is too large to store. It could be related to dividing a number by a very small number. Some possible ways to fix this error are¹:

• Review the boundary conditions to make sure they are reasonable and consistent with the physical problem.
• Run the simulation in double precision mode, which can handle larger numbers and improve the accuracy of the solution.
• Try using a smaller time step or a different time scale factor, which can help avoid numerical instability and convergence issues.
• Try using a different turbulence model, such as SST, which can better capture the flow physics and reduce numerical errors.
• Try using a different advection scheme, such as High Resolution or Upwind, which can reduce numerical diffusion and oscillations.

What is numerical instability?

Numerical instability is a concept that refers to the tendency of some numerical algorithms or procedures to produce inaccurate or unreliable results due to errors in the computation. These errors may be caused by factors such as round-off errors, truncation errors, or ill-conditioning of the problem. Numerical instability can lead to large deviations from the expected or exact solution, and sometimes even to divergence or failure of the algorithm. Numerical stability is a desirable property of numerical algorithms, which means that the errors do not significantly affect the final result. Numerical stability depends on both the algorithm and the problem being solved. Some algorithms are more robust than others, and some problems are more sensitive than others. Numerical stability can be assessed by various criteria, such as forward error, backward error, condition number, or error propagation. For more information and examples of numerical instability and stability, you can refer to the web search results¹²³. 

How can I improve numerical stability in my simulation?

There are several ways to improve numerical stability in your simulation, depending on the cause and nature of the instability. Some general tips are:

• Review your mesh quality and resolution, and make sure they are adequate for capturing the flow features and gradients. A poor mesh can introduce numerical errors and instability. You can use mesh adaptation techniques to refine the mesh in regions of interest¹.
• Choose appropriate boundary conditions and initial conditions that are consistent with the physical problem and the expected solution. Inappropriate or unrealistic boundary conditions can cause spurious oscillations or divergence in the solution¹.
• Use relaxation factors to control the rate of change of the solution variables during the iteration process. Relaxation factors are coefficients that dampen the effect of the new computed values on the old values. A smaller relaxation factor can improve stability, but it can also slow down convergence. A larger relaxation factor can speed up convergence, but it can also cause instability. The optimal relaxation factor depends on the problem and the solver, but a typical range is between 0.15 and 0.7².
• Use a suitable time step size and time integration scheme for transient simulations. A too large time step size can cause numerical instability and loss of accuracy. A too small time step size can increase the computational cost and time. A good practice is to use a time step size that satisfies the Courant-Friedrichs-Lewy (CFL) condition, which relates the time step size to the spatial discretization and the flow velocity. The CFL number is defined as CFL = uδt/δx, where u is the flow velocity, δt is the time step size, and δx is the spatial discretization. A CFL number less than 1 is usually recommended for stability³. The time integration scheme can also affect stability and accuracy. For example, an explicit scheme can be more unstable than an implicit scheme, but it can also be more accurate for some problems³.
• Use a suitable spatial discretization and advection scheme for convective terms. The spatial discretization determines how the solution variables are interpolated or extrapolated at the cell faces from the cell centers. The advection scheme determines how the convective fluxes are calculated at the cell faces from the solution variables. Different schemes have different levels of accuracy, stability, and computational cost. For example, a first-order upwind scheme can be more stable than a second-order central difference scheme, but it can also introduce more numerical diffusion and smearing of sharp gradients³.

These are some of the common ways to improve numerical stability in your simulation. 

Can you explain more about relaxation factors?

Sure, I can explain more about relaxation factors. Relaxation factors are coefficients that control the amount of change in the solution variables during the iterative process of solving a system of equations. They are used to improve the stability and convergence of the solution, especially for nonlinear and complex problems. Relaxation factors can be either under-relaxation or over-relaxation, depending on whether they are less than or greater than one. Under-relaxation means that only a fraction of the predicted change in the solution variable is applied, which can slow down the convergence but increase the stability. Over-relaxation means that more than the predicted change in the solution variable is applied, which can speed up the convergence but decrease the stability. The optimal value of the relaxation factor depends on the problem and the algorithm, and it is usually determined by trial and error. Some common ranges of relaxation factors are between 0.15 and 0.7 for flow variables, and between 0.5 and 1 for turbulence variables¹. Relaxation factors are often used in CFD simulations, where they can affect the accuracy and efficiency of the solution. For example, relaxation factors can help avoid numerical instability and divergence in steady-state simulations, or reduce numerical diffusion and oscillations in transient simulations². Relaxation factors are also related to the time step size and the spatial discretization schemes, which can also influence the stability and accuracy of the solution³. 

How do I choose the optimal relaxation factor?

Choosing the optimal relaxation factor in Ansys CFD depends on the problem and the algorithm you are using, and it is usually determined by trial and error. There is no universal rule for selecting the best value, but some general guidelines are¹²³:

• A relaxation factor that is too small (< 0.15) can slow down the convergence rate and increase the computational cost.
• A relaxation factor that is too large (> 0.7) can cause numerical instability and divergence of the solution.
• A relaxation factor that is close to 1 (no relaxation) can sometimes improve the accuracy of the solution, but it can also introduce numerical errors and oscillations.
• A relaxation factor that is close to 0 (full relaxation) can sometimes improve the stability of the solution, but it can also introduce numerical diffusion and smearing of sharp gradients.

Therefore, a good practice is to start with a moderate value of the relaxation factor (between 0.3 and 0.7) and then adjust it according to the behavior of the solution. You can monitor the residuals, the convergence history, and the physical quantities of interest to see if the solution is stable, accurate, and realistic. You can also use the auto-relaxation feature in Ansys CFD, which can dynamically change the relaxation factor for the flow variables to speed up convergence while trying to keep the solution stable². However, this feature may not work for all problems, and you may need to try different values manually to find the optimal one.

Can you explain more about auto-relaxation feature?

Sure, I can explain more about the auto-relaxation feature in Ansys CFD. Auto-relaxation is a technique that automatically adjusts the relaxation factors for the flow variables during the iterative process of solving the system of equations. Relaxation factors are coefficients that control the rate of change of the solution variables from one iteration to the next. They are used to improve the stability and convergence of the solution, especially for nonlinear and complex problems. A smaller relaxation factor can increase stability, but it can also slow down convergence. A larger relaxation factor can speed up convergence, but it can also cause instability. The optimal relaxation factor depends on the problem and the algorithm, and it is usually determined by trial and error.

The auto-relaxation feature is designed to dynamically change the relaxation factor for each flow variable based on its residual behavior. The residual is a measure of how well the solution satisfies the equation for each variable. A lower residual means a more accurate solution. The auto-relaxation feature tries to find the optimal balance between stability and convergence by increasing or decreasing the relaxation factor according to the residual trend. For example, if the residual is decreasing steadily, the auto-relaxation feature will increase the relaxation factor to speed up convergence. If the residual is oscillating or increasing, the auto-relaxation feature will decrease the relaxation factor to improve stability.

The auto-relaxation feature can be enabled or disabled in Ansys CFD by selecting or deselecting the Auto Relaxation option in the Numerics panel¹. The auto-relaxation feature can sometimes improve the convergence rate and efficiency of the solution, but it may not work for all problems. In some cases, if the auto-relaxation feature causes divergence or instability, it is recommended to try with manual relaxation factors that are fixed over the iterations². You can also monitor and adjust the relaxation factors manually by using the Relaxation Factors option in the Numerics panel¹.

💥💥💥 How to prepare sensitivity analysis in Ansys Structural?

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.

(2) Principles and Methods of Sensitivity Analyses - The National Academies .... https://nap.nationalacademies.org/read/12955/chapter/7.

(3) How to compare qualitative tests – essential tips for ... - Finbiosoft. https://finbiosoft.com/method-comparison-diagnostic-accuracy-sensitivity-specificity/.

(4) 17.4 - Comparing Two Diagnostic Tests | STAT 509 - Statistics Online. https://online.stat.psu.edu/stat509/lesson/17/17.4.

(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.

💥💥💥 What are mesh quality and mesh errors in Ansys Fluent ?

 Mesh quality and mesh errors in Ansys Fluent are related to how well the mesh represents the geometry and the physics of the problem. A good mesh should have the following characteristics:

• It should conform to the shape of the domain and capture the important features of the geometry.
• It should have a smooth distribution of nodes and elements, without sudden changes in size or shape.
• It should have a low skewness and aspect ratio, which measure how far the elements deviate from an ideal shape (such as a square or an equilateral triangle).
• It should have enough resolution in regions where the flow is complex, such as boundary layers, shear layers, shock waves, etc.
• It should be compatible with the solver settings and the turbulence model used.

Mesh errors can occur when the mesh does not meet these criteria, and they can affect the accuracy and stability of the numerical solution. Some common mesh errors are:

• High skewness: This means that some elements are distorted or twisted, which can cause interpolation errors and convergence problems¹.
• Incorrect boundary layer meshing: This means that the mesh near the walls is not fine enough to resolve the viscous effects, which can lead to inaccurate predictions of wall shear stress and heat transfer¹.
• Coarse meshing: This means that the mesh is too coarse to capture the flow features, which can lead to numerical diffusion and loss of accuracy¹.
• A large aspect ratio leading to interpolation error: This means that some elements are stretched or compressed, which can cause errors in calculating the gradients and fluxes².

To check and improve the mesh quality in Ansys Fluent, there are several tools and methods available, such as:

• The Report Quality button in the General task page, which prints a message to the console with information about the mesh quality criteria¹.
• The Mesh Adaptation feature, which allows you to refine or coarsen the mesh based on various parameters, such as gradients, residuals, curvature, etc¹.
• The TUI commands or the Ansys Meshing tool, which allow you to modify or repair the mesh manually or automatically³.

How to check mesh errors and mesh quality in Ansys Fluent ?

There are several ways to check mesh errors and mesh quality in Ansys Fluent. Here are some of the resources that might help you:

• You can watch this YouTube video Ansys Fluent Tutorial - Check mesh quality errors and define mesh adaptation that shows how to use the Fluent interface to check and improve the mesh quality.
• You can also read this forum post When I open fluent and do mesh quality check, I’m that discusses a common warning message and how to fix it using the TUI commands or the Ansys Meshing tool.
• You can refer to this user’s guide ANSYS FLUENT 12.0 User’s Guide - 6.2.2 Mesh Quality that explains how to use the Report Quality button in the General task page and what the output means.

Quick Tip: How to avoid Incorrect Wave Height in Ansys Fluent

According to my video and web search, this is a common issue that many Fluent users face when simulating open channel flow with wave boundary conditions . Some possible causes and solutions are:

• The wave height may be affected by the mesh quality and resolution. You may need to refine your mesh near the free surface and use a finer time step to capture the wave dynamics more accurately .
• The wave height may also depend on the wave theory and spectrum that you choose for your boundary conditions. You may need to adjust the parameters such as wave length, wave period, wave amplitude, wave direction, etc. to match your desired wave characteristics .
• The wave height may be influenced by the numerical schemes and relaxation factors that you use for your solution methods. You may need to use a higher order scheme such as QUICK or MUSCL and lower the relaxation factors for pressure and volume fraction to improve the stability and accuracy of your solution .

You can find more details and examples on how to perform these steps in the following resources:

• A video tutorial on how to model a tank discharge flow using VOF model in Ansys Fluent .
• A video tutorial on how to model an open channel flow with wave using VOF model in Ansys Fluent ³.
• A forum thread on how to avoid incorrect wave height in Ansys Fluent.
• A forum thread on how to use wave period for input in Fluent.
• The Ansys Fluent User’s Guide, especially Chapter 23: Using the VOF Model.
• The Ansys Fluent Theory Guide, especially Chapter 18: Volume of Fluid Model Theory.

I hope this helps you avoid incorrect wave height in Ansys Fluent and achieve your desired simulation results. If you have any other questions, please feel free to ask me. 😊

Quick Tip: How to fix Error: GENERAL-CAR-CDR in Ansys (Fluent)

 The error message Error: GENERAL-CAR-CDR: invalid argument [1]: improper list means that there is a problem with the scheme code or the UDF (user-defined function) in your case or data file . Some possible solutions are:

• Save the case and data file and try to reopen it.
• Reset the Fluent Setup tab and try opening the mesh.
• Remesh the geometry and reopen in Fluent.
• Change the material definition from “Solid” to “Fluid” in Design Modeler.
• Change the transient to steady if you are using evaporation-condensation as the phase interaction mechanism.
• Check the syntax of your UDF and make sure it follows the new rules in Fluent 6.2.16 or later.

I hope this helps you resolve the error. If you need more assistance, please visit the Ansys Learning Forum or contact Ansys Support . Have a nice day! 😊

Sources:

1. smartadm.ru

2. cfd-online.com

3. forum.ansys.com

4. forum.ansys.com

In Polish 

Błąd: GENERAL-CAR-CDR oznacza, że wystąpił problem z kodem Scheme w programie Ansys Fluent. Scheme to język programowania używany do pisania funkcji zdefiniowanych przez użytkownika (UDF) lub makr w Fluent ¹. Błąd ten pojawia się, gdy Fluent oczekuje listy wartości, ale otrzymuje inną wartość, np. fałszywą (#f) ².

Możliwe przyczyny tego błędu to:

• Niepoprawna składnia lub brakujące znaki w kodzie Scheme
• Niezgodność typów danych lub argumentów w funkcjach Scheme
• Nieprawidłowe ustawienia lub opcje w Fluent

Możliwe rozwiązania tego błędu to:

• Sprawdzenie i poprawienie kodu Scheme, upewniając się, że wszystkie nawiasy i cudzysłowy są zamknięte i dopasowane
• Sprawdzenie i poprawienie typów danych i argumentów w funkcjach Scheme, upewniając się, że są zgodne z dokumentacją Fluent
• Zapisanie i ponowne otwarcie pliku przypadku i danych
• Zresetowanie zakładki ustawień Fluent i ponowne otwarcie siatki
• Przemieszanie geometrii i ponowne otwarcie w Fluent

Możesz znaleźć więcej informacji i przykładów na temat tego błędu na stronach internetowych ³⁴. Mam nadzieję, że to ci pomoże. 😊