Computer Experiments & Simulation Studies

Computer Experiments involve the strategic design and analysis of simulation studies to understand complex systems. Unlike physical experiments, computer experiments are deterministic (no random error) and can explore many factor combinations efficiently using space-filling designs like Latin Hypercube Sampling.

Computer Experiment Overview & Design Principles

What Computer Experiments Accomplish:

• Efficiently explore high-dimensional factor spaces
• Build surrogate models for expensive simulations
• Perform sensitivity analysis and uncertainty quantification
• Optimize complex systems with many parameters
• Validate and calibrate mathematical models

Key Advantages:

• No experimental error (deterministic outcomes)
• Can explore extreme or dangerous conditions safely
• Highly efficient space-filling designs
• Suitable for high-dimensional problems
• Reproducible and controllable conditions

Design Considerations:

• Model validity depends on simulation accuracy
• Computational cost can still be significant
• Requires different statistical approaches
• May need specialized sampling methods

Space-Filling Design & Surrogate Model Formulas

Latin Hypercube Sampling (LHS) stratifies each of the p input dimensions into n equal-probability intervals and permutes them independently, so every interval of every factor is sampled exactly once.

Latin Hypercube coordinate for run i, factor j

xi,j

= scaled coordinate (0 to 1) for run i on factor j

πj(i)

= random permutation of {1,dots,n} for factor j

ui,j

= uniform random draw on (0, 1)

= number of design runs

Because simulations are deterministic, a Gaussian Process (Kriging) surrogate treats the unknown response as a correlated random field so it can predict untested points and quantify prediction uncertainty.

Gaussian Process surrogate prediction

ˆy(x)

= predicted response at new point x

= estimated process mean

r(x)

= vector of correlations between x and the n design points

= n x n correlation matrix of the design points

= vector of observed simulation outputs

R is built from a correlation kernel (commonly Gaussian) whose parameters are fit by maximum likelihood.

Key Assumptions

• • The simulator is deterministic — repeated runs at the same input return the same output
• • The response surface is reasonably smooth so a correlation-based surrogate is appropriate
• • Design points adequately fill the factor space relative to response complexity
• • Extrapolation outside the sampled region is unreliable and should be avoided

Reading the Computer Experiment (Space-Filling) Design Design Layout

Example Output & Sensitivity Result Interpretation

Analysis Techniques:

Sensitivity Analysis:

• Sobol indices for global sensitivity

• Main effect and interaction quantification

• Factor screening and ranking

Uncertainty Quantification:

• Propagation of input uncertainties

• Output distribution characterization

• Confidence and prediction intervals

Optimization:

• Multi-objective optimization

• Robust optimization under uncertainty

• Pareto frontier analysis

Worked Example:

In the thermal simulation study below (see Implementation), a fitted Sobol sensitivity analysis attributes 62% of heat-transfer output variance to Temperature, 24% to Flow Rate, and 14% to the remaining three factors combined — indicating Temperature and Flow Rate should be prioritized for tighter process control, while Material K, Geometry, and Pressure can be screened out of subsequent, more expensive experiments.

Model Validation:

• Cross-validation for surrogate models

• Prediction accuracy assessment

• Residual analysis and diagnostics

• Comparison with experimental data (when available)

• Model refinement and improvement

Decision Rules:

• Rank factors by sensitivity index; retain only high-impact factors for further study

• If cross-validated prediction error is unacceptably high, add design points before trusting the surrogate

• Prefer robust (low-variance) operating regions over marginally optimal but sensitive ones

• Flag any recommended operating point that falls outside the sampled design space

sixsigmacenter.com Implementation Guide

When to Use Computer Experiments

Ideal Applications:

Engineering Simulation:

• Finite element analysis (FEA)

• Computational fluid dynamics (CFD)

• Structural optimization

• Thermal analysis

Process Modeling:

• Chemical reaction modeling

• Supply chain optimization

• Manufacturing system design

• Quality prediction models

Risk Assessment:

• Monte Carlo simulations

• Reliability analysis

• Safety system evaluation

• Environmental impact studies

Design Method Selection:

Latin Hypercube Sampling (LHS): Efficient space-filling

Sobol Sequences: Low-discrepancy sampling

Orthogonal Arrays: Balanced factor level combinations

Optimal Designs: Customized for specific objectives

Adaptive Designs: Sequential improvement strategies

Example Data Layout for Latin Hypercube Sampling Design (5 factors, 20 runs)

Example computer experiment design for a thermal simulation study. Five factors (temperature, pressure, flow rate, material properties, geometry) are studied using Latin Hypercube Sampling for efficient space coverage. Sample size typically 10×(number of factors) for initial exploration. Use adaptive sampling to refine important regions of the design space.

This is how your spreadsheet or CSV file should be structured before you upload it. Match the column names, keep one row per observation, and leave the analysis to the platform.