How do you select and generate appropriate stochastic inputs and parameters for SFEM models?

1 Random variables

A random variable is a quantity that can take different values depending on the outcome of a random event. For example, the load on a beam, the material properties of a soil, or the temperature of a fluid are random variables. To model random variables in SFEM, you need to assign them a probabilistic distribution that describes their likelihood of occurrence. For example, you can use a normal distribution, a lognormal distribution, a uniform distribution, or any other suitable distribution that fits your data and assumptions. You can also use statistical methods, such as regression, correlation, or Bayesian inference, to estimate the parameters of the distribution, such as the mean, the standard deviation, or the shape.

2 Random fields

A random field is a function that assigns a random variable to each point in a spatial domain. For example, the porosity of a rock, the stiffness of a composite, or the pressure of a fluid are random fields. To model random fields in SFEM, you need to define their spatial characteristics, such as their mean, their variance, their covariance, or their correlation length. You can also use spatial statistics, such as geostatistics, kriging, or spectral methods, to generate random fields from data or assumptions. You can also use different types of random fields, such as Gaussian, non-Gaussian, homogeneous, heterogeneous, isotropic, or anisotropic, depending on your problem and objectives.

3 Stochastic inputs

Stochastic inputs are the sources of uncertainty and variability that affect your system or phenomenon. For example, the boundary conditions, the initial conditions, the external forces, or the environmental factors are stochastic inputs. To model stochastic inputs in SFEM, you need to identify and quantify them using random variables or random fields. You can also use different methods to generate stochastic inputs, such as Monte Carlo simulation, Latin hypercube sampling, polynomial chaos expansion, or stochastic collocation, depending on your computational resources and accuracy requirements.

4 Stochastic parameters

Stochastic parameters are the coefficients or functions that appear in your governing equations or constitutive models. For example, the Young's modulus, the Poisson's ratio, the thermal conductivity, or the permeability are stochastic parameters. To model stochastic parameters in SFEM, you need to express them using random variables or random fields. You can also use different methods to solve your stochastic equations or models, such as perturbation methods, stochastic Galerkin methods, stochastic finite differences, or stochastic meshfree methods, depending on your mathematical formulation and numerical scheme.

5 Stochastic outputs

Stochastic outputs are the quantities of interest or performance measures that you want to evaluate or optimize. For example, the displacement, the stress, the temperature, or the flux are stochastic outputs. To model stochastic outputs in SFEM, you need to compute them using your stochastic inputs and parameters. You can also use different methods to analyze your stochastic outputs, such as statistical moments, probability density functions, cumulative distribution functions, reliability indices, or sensitivity indices, depending on your engineering criteria and decision making.

6 Stochastic validation

Stochastic validation is the process of verifying and validating your stochastic models and simulations. For example, you can check the consistency, the convergence, the accuracy, or the robustness of your stochastic results. To perform stochastic validation in SFEM, you need to compare your stochastic outputs with experimental data, analytical solutions, or other numerical methods. You can also use different methods to assess your stochastic validation, such as error analysis, uncertainty quantification, model calibration, or model updating, depending on your data availability and quality.

7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?