Monte Carlo Simulation — A Fundamental Method for CAE Uncertainty Quantification

Good question. Roughly speaking, it's a method that randomly samples from the probability distributions of input parameters and repeatedly executes FEM or CFD analyses hundreds or thousands of times. The goal is to statistically determine the variation (distribution) of the output.

Yes, that's also the biggest weakness of the Monte Carlo method. However, its greatest strength is that the convergence rate does not change regardless of the dimensionality of the input variables. With sensitivity analysis, if there are 10 variables, the cost increases exponentially, but Monte Carlo converges at the same pace whether it's 10 dimensions or 100 dimensions.

For example, a case evaluating how the variation in material strength (coefficient of variation CoV ≈ 5%) of an automotive component affects its fatigue life. When multiple parameters like Young's modulus, yield stress, plate thickness, load amplitude, etc., vary simultaneously, investigating how their "combined effects" propagate to the output is the forte of the Monte Carlo method.

The core of the Monte Carlo method lies in the Law of Large Numbers and the Central Limit Theorem. We independently sample $N$ samples $\mathbf{X}_1, \mathbf{X}_2, \ldots, \mathbf{X}_N$ from the input distribution and evaluate the model $Y = f(\mathbf{X})$ for each. Then the sample mean is:

The standard error of this sample mean is expressed by the following formula:

Exactly. This is the famous $1/\sqrt{N}$ convergence rule. The important points are:

• To halve the error, you need 4 times the samples
• To reduce the error to 1/10, you need 100 times the samples
• This convergence rate does not depend on the dimensionality $d$ of the input variables (it is not affected by the curse of dimensionality)

Thanks to the Central Limit Theorem, if $N$ is sufficiently large (in practice, from about $N \geq 30$), the sample mean can be approximated by a normal distribution. Therefore, the 95% confidence interval can be written as:

Let's calculate the relative estimation accuracy (Relative Accuracy). If the coefficient of variation is $\text{CoV}_Y = \sigma_Y / \mu_Y$:

For example, when the output's coefficient of variation is $\text{CoV}_Y = 0.10$ (10%):

• $N = 100$: $\text{RA} = 1.96 \times 0.10 / \sqrt{100} = 1.96\%$
• $N = 1{,}000$: $\text{RA} = 1.96 \times 0.10 / \sqrt{1000} \approx 0.62\%$
• $N = 10{,}000$: $\text{RA} \approx 0.20\%$

With 1,000 runs, the width of the 95% confidence interval is about 6% of the mean value (when $\text{CoV}_Y \approx 1.0$) is a practical guideline.

It's completely different depending on the purpose. Roughly summarized, it looks like this:

That's why for rare event estimation, instead of pure Monte Carlo, special techniques like Importance Sampling or Subset Simulation are used. Or, a surrogate model is inserted to reduce computational cost.

The simplest method is yes. Generate pseudo-random numbers independently from the probability distribution of each input variable to create samples. This is called Pure Random Sampling (Crude Monte Carlo / Simple Random Sampling).

Sharp observation. Especially when the sample number is small (around $N < 100$), pure random sampling tends to produce uneven coverage of the input space. For example, with 2 variables and 100 samples, the corners of the distribution domain might get no samples at all. That's where LHS (Latin Hypercube Sampling) comes in.

The idea of LHS is simple. Partition the distribution of each input variable into $N$ equiprobable strata and take exactly one sample from each stratum. Then randomly shuffle the combinations between variables.

That's the right image. Mathematically, for each variable $X_j$ ($j = 1, \ldots, d$), divide the range $[0, 1]$ of its cumulative distribution function $F_j$ into $N$ equal parts:

Summarizing the benefits of LHS:

• Can uniformly cover the input space with a small number of samples
• The marginal distribution of each variable is accurately reproduced
• Variance becomes smaller compared to pure random sampling (though the improvement depends on the problem)
• In CAE practice, sufficient accuracy is often obtained with LHS of around $N = 50\text{ to }200$

There is a group of techniques known as "Variance Reduction Techniques." They effectively reduce the $\sigma_Y$ in the standard error $\text{SE} = \sigma_Y / \sqrt{N}$, or cleverly design the estimator to improve accuracy even with the same $N$.

It can be very effective if the response function is monotonic with respect to the input (e.g., if material strength increases, life also increases). When $\mathbf{U}$ produces a large output, $1-\mathbf{U}$ produces a small one, so averaging the two cancels out variance. However, for non-monotonic functions, it can sometimes have the opposite effect, so caution is needed.

The basic flow is like this:

1. Identify uncertain input parameters — Material properties, geometric dimensions, load conditions, etc.
2. Set probability distribution for each parameter — Normal distribution, lognormal distribution, uniform distribution, etc.
3. Create sampling plan — Pure random or LHS, decide sample number $N$
4. Execute $N$ CAE analyses — Batch processing or parallel execution
5. Statistical processing of results — Calculate mean, standard deviation, histogram, CDF, confidence intervals
6. Convergence check — Plot cumulative mean / cumulative standard deviation to confirm stability

That's the most troublesome part; "Garbage In, Garbage Out" applies most critically to the setting of input distributions. Ideally, you fit them from manufacturer quality data or experimental data, but in practice, the following guidelines are often used: