Welcome to the second part of our optimization series. Before we begin explaining the mathematical engines behind these tools, we strongly recommend reading our foundational guide. If you need to review the basic concepts of structural and parametric workflows, please review The Complete Guide to Optimization in ANSYS before continuing.
Figure 1: Optimization basics and workflow in ANSYS
What is DOE and Why Do We Sample?
Entering the world of Parametric Optimization requires generating data. When you set up a project in ANSYS Workbench, you define a range of numbers to test, such as different pressures or geometric sizes. Testing every possible combination manually is impossible. Doing so would require thousands of heavy solver runs and take months or even years to finish. To solve this problem, engineers rely on the Design of Experiments, commonly known as DOE.
Essentially, the Design of Experiments is the science of smart sampling. Rather than testing combinations randomly, the software applies strict mathematical rules to place sample points strategically across the entire design space. By doing so, it extracts the maximum amount of physical information using the absolute minimum number of actual simulations. This approach saves massive amounts of computation time while still capturing the true behavior of your product accurately.
Figure 2: Design of Experiments (DOE) prevents wasted time by replacing chaotic guessing with highly structured, mathematically proven sampling techniques.
Despite its power, there is a strict limitation you must follow to keep this process efficient. DOE methods work best when you have fewer than 20 input variables. If you attempt to optimize more than 20 parameters simultaneously, you will encounter what engineers call the curse of dimensionality. The required number of samples will increase exponentially, making the computation incredibly slow and heavy. For this reason, you should always run a sensitivity analysis first to eliminate unimportant variables. Once you reduce the problem to only the most essential parameters, the sampling process will operate flawlessly.
Deterministic Schemes: The ANSYS DesignXplorer Models
When configuring a Design of Experiments (DOE) in ANSYS Workbench, the standard DesignXplorer interface provides a specific dropdown menu containing highly specialized sampling algorithms. Deterministic schemes use fixed, geometric mathematical rules to place design points inside your parameter boundaries. While classical statistical methods like Full Factorial, Koshal, D-Optimal, or Taguchi are famous in engineering, you will not see them listed in this menu. ANSYS natively focuses on automated space-filling and quadratic calibration models. If you specifically need classical models, you must use the Custom import feature. Let us comprehensively examine the exact deterministic models provided in the software, how they pick their points, and their strict mathematical limits.
Figure 3: The Design of Experiments Type dropdown menu in ANSYS Workbench. Selecting the right model defines how efficiently the software explores your design space.
Central Composite Design (CCD)
The Central Composite Design (CCD) is the default and most powerful native option for calibrating full quadratic response models. Geometrically, it picks design points by combining a center point, standard factorial corner points, and extended star points placed strictly along the main axes. Because of this structure, it can perfectly capture complex curved behaviors. The ANSYS CCD engine supports a strict maximum of 20 Input Parameters. The number of generated cases depends on the input count. For example, with 3 input variables, a standard CCD defines 15 cases (8 corner points + 6 star points + 1 center point). For higher dimensions, ANSYS automatically switches to a fractional factorial base to prevent the computational time from exploding.
Figure 4: Central Composite Design utilizes center, corner, and star points to capture quadratic behaviors.
Inside the software, you can strictly control how CCD picks its star points by selecting different design types:
- Face-Centered: This is a three-level design where the star points rest exactly on the boundary faces (the alpha value equals 1). This guarantees that no test points fall outside your specified minimum and maximum limits.
- Rotatable: This is a five-level design that scales the outer star points mathematically. It ensures that the prediction variance remains exactly the same regardless of the direction from the center.
- VIF-Optimality: This five-level design minimizes mathematical non-orthogonality by using the Variance Inflation Factor.
- G-Optimality: This variation specifically minimizes the maximum expected error in your predictions over the entire region of interest.
- Auto-Defined: This is the recommended default. The software intelligently switches to G-Optimality if you have exactly 5 variables, and uses VIF-Optimality for any other number of inputs. You can also apply a Standard or Enhanced template to force the algorithm to calculate a better mathematical fit for the response surfaces.
Figure 5: Design points distribution as a function of different CCD design types
Box-Behnken Design (BBD)
The Box-Behnken Design (BBD) is an independent three-level quadratic design that supports a maximum of 12 Input Parameters. Instead of placing sample points at the extreme corners of your parameter box, it locates them exactly at the midpoints of the boundary edges, plus one center point. For example, if you define 3 input variables, the BBD algorithm will generate exactly 13 cases (12 mid-edge points + 1 center point), making it slightly more efficient than the 15 cases of a CCD for the same inputs.
Figure 6: BBD algorithm picks midpoints and ignore the extreme corners
Actively avoiding the extreme corners is a massive physical advantage for fluid and structural projects. Combining absolute maximum or minimum parameter values simultaneously often causes CAD geometry to fail or the solver mesh to collapse. Because the BBD lacks extreme corners, it significantly reduces the risk of simulation update failures. It also requires fewer total design points than a full factorial CCD. The only recognized disadvantage of BBD is that prediction accuracy exactly at the absolute corners of your design space remains poor, as no sampling occurs there.
Sparse Grid Initialization
The Sparse Grid Initialization provides a highly adaptive, hierarchical mathematical grid. It must be used when you intend to build a Sparse Grid response surface later. When you select this option, the software begins with a simple, sparse matrix consisting of one center point and points strictly along the axes.
Its greatest strength is intelligent adaptation driven by requested accuracy. The algorithm evaluates the initial results and automatically refines the matrix of design points exclusively in locations where the relative error of the output parameter is high. Because it refines the grid only in necessary directions rather than universally filling the box, it requires significantly fewer points to achieve a high-quality surface. This method is exceptionally effective at handling discontinuities or highly nonlinear behaviors in your engineering results.
Figure 7: The sparse grid combination method with piece-wise linear basis
Custom & Custom + Sampling
Finally, ANSYS provides the Custom and Custom + Sampling options to give you absolute manual control. This is the location where you can utilize classical statistical methods. If your project strictly requires a Full Factorial matrix, a Koshal saturated design, a D-Optimal subset or a Taguchi Orthogonal Array, you must generate that table externally (like in Microsoft Excel) and import it here as a CSV file.
- The Custom type allows you to define your matrix by editing values directly, adding editable rows, or importing your external CSV file. If you previously solved a DOE using another algorithm, those points are safely retained.
- The Custom + Sampling option takes this a step further. It allows you to import your specific fixed design points (like your Taguchi array) and then asks the software algorithm to automatically generate extra random samples to fill the remaining empty areas of your design space efficiently.
To make your software setup effortless, use this comprehensive reference table to understand exactly how each native ANSYS model behaves and its specific input limitations.
| ANSYS DOE Type | Maximum Inputs | Point Selection Logic | Example Case Count (3 Inputs) | Best Used When… |
| Central Composite (CCD) | 20 | Center, Corners, and Star Points. | 15 Cases | You need a highly reliable quadratic map; offers Auto-Defined optimality. |
| Box-Behnken (BBD) | 12 | Midpoints of edges and Center. | 13 Cases | Maximum parameter limits frequently crash your CAD geometry or mesh. |
| Sparse Grid Initialization | Dependent | Hierarchical axes points; refines based on error. | Adaptive | Your physical model has sharp discontinuities or highly nonlinear zones. |
| Custom (+ Sampling) | Unlimited | Manual entry, CSV import, or mixed sampling. | User Defined | You want to import Full Factorial or Taguchi arrays from Excel. |
Random Space-Filling Schemes (LHS & OSF)
While deterministic schemes build strict geometric shapes like grids and stars, random sampling methods aim to scatter points throughout the entire design space. Classical random methods like pure Monte Carlo often create clusters of points or leave massive empty holes in the mathematical boundary. To prevent this, ANSYS DesignXplorer strictly provides two highly advanced, constrained random methods in its dropdown menu: Latin Hypercube Sampling (LHS) and Optimal Space-Filling (OSF). These methods are specifically designed to distribute points evenly and gather the maximum amount of insight with the fewest number of solver evaluations.
Latin Hypercube Sampling (LHS)
Latin Hypercube Sampling (LHS) is a highly advanced statistical form of the Monte Carlo method. Instead of placing points purely randomly, the LHS algorithm actively avoids clustering. It achieves this by dividing the design space into a square grid. The points are randomly generated inside this grid, but the software enforces a strict mathematical rule: no two points can share the same parameter value. Visually, this means no single point shares a row or a column with any other point in the grid.
Figure 8: Latin Hypercube Sampling (LHS) prevents severe clustering
In the ANSYS software properties, LHS provides several critical settings to control how it calculates the number of design points. This is controlled by the Samples Type property:
- CCD Samples (Default): This generates the exact same number of samples that a Central Composite Design (CCD) would generate for your specific number of inputs. This allows you to create a random space-filling design that has the exact same computational cost as a standard CCD. Note: If this option is selected, ANSYS strictly supports a maximum of 20 input parameters.
- Linear / Pure Quadratic / Full Quadratic Samples: The software automatically calculates and generates the number of samples required to build a linear, pure quadratic, or full quadratic metamodel.
- User-Defined Samples: This enables the Number of Samples option, allowing you to manually type the number of solver evaluations you want to run.
Figure 9: Different Sampling types for LHS scheme
To ensure reproducibility, ANSYS uses a Random Generator Seed. Although LHS generates starting points randomly, the internal mathematical sequence is controlled by this numerical seed (default is 0). If you keep the seed value identical, the software will reliably regenerate the same random sampling matrix. If you want a completely new random distribution, you simply change the seed value. The main disadvantage of standard LHS is that it does not necessarily cover the absolute extreme corners of your design space, and choosing too few points can result in poor response prediction.
Optimal Space-Filling Design (OSF)
The Optimal Space-Filling (OSF) design is arguably the most powerful random sampling method in ANSYS. Essentially, OSF is a standard Latin Hypercube Sampling (LHS) matrix that has been heavily upgraded with mathematical post-processing. The OSF algorithm initializes a standard LHS grid, but then it runs internal optimization loops to push the points further apart. It remains a valid LHS matrix (no points share rows or columns), but it achieves a vastly superior uniform distribution by maximizing the physical distance between all points. Because computer simulations do not suffer from physical measurement noise, OSF distributes parameters equally throughout the interior space. This makes OSF the absolute best choice when you plan to use complex, non-parametric metamodels later, such as Kriging or Neural Networks.
ANSYS provides three specific Design Type criteria to dictate how the OSF algorithm pushes the points apart:
- Max-Min Distance (Default): This strategy mathematically maximizes the minimum distance between any two points. It ensures no two points are ever too close together. For a small number of samples, this algorithm generally places points on the exterior boundaries and fills the interior as you increase the sample count. This is generally the fastest algorithm.
- Centered L2: This method minimizes the “centered L2-discrepancy.” In simple terms, it calculates the difference between your empirical sampling and a mathematically perfect uniform distribution, forcing the points to be highly uniform. It is computationally faster than Maximum Entropy.
- Maximum Entropy: This complex algorithm maximizes the determinant of the covariance matrix. Its goal is to minimize uncertainty in empty, unobserved locations. It provides excellent results for highly correlated spaces. However, the calculation cost increases non-linearly with the number of inputs, so it is strictly recommended only for small parametric problems.
Figure 10: Different design types provided for Optimal Space-Filling Design scheme
Plus, OSF requires you to define the Maximum Number of Cycles. This determines exactly how many optimization loops the algorithm runs to push the points apart. A larger number of cycles makes the distribution highly uniform but slows down the generation process significantly. ANSYS sets the default to 10 cycles, which is generally excellent for up to 20 variables. Just like LHS, OSF allows you to control the point count using the Samples Type (CCD, Linear, Quadratic, User-Defined) and ensures repeatability using the Random Generator Seed.
To conclude:
| ANSYS DOE Type | Max Inputs | Point Selection Logic | Key Software Settings | Best Used When… |
| Latin Hypercube (LHS) | 20 (if CCD samples used) | Random grid; no shared rows/columns. | Random Generator Seed; Samples Type selection. | You need a fast, non-clustered random distribution to explore general trends. |
| Optimal Space-Filling (OSF) | 20 (if CCD samples used) | LHS upgraded with post-processing. | Design Types: Max-Min, Centered L2, Max Entropy; Max Cycles. | You plan to build highly complex metamodels like Kriging or Neural Networks. |
Deterministic vs. Random: Which to Choose?
You now know the sampling models available in ANSYS. But when setting up your project, which one should you actually click? The decision always comes down to the number of input parameters and the complexity of your physics.
As a rule of thumb:
- Choose Deterministic (CCD, BBD) when you have a small number of variables (typically 5 or fewer) and you want to build a highly reliable, classic quadratic mathematical model.
- Choose Random Space-Filling (OSF, LHS) when you have many variables (more than 5), highly non-linear physics, or when you plan to use advanced metamodels like Kriging or Neural Networks.
To make this perfectly clear, let us look at two real-world engineering examples.
Example 1: The Deterministic Approach: Imagine you are optimizing a mixing chamber. You only have three geometric parameters: inlet angle, pipe diameter, and chamber length. Because the variable count is low and fluid mixing often follows smooth, curved mathematical behaviors, you do not need random scattering. A Central Composite Design (CCD) will perfectly capture the physics with absolute mathematical certainty. See this in action: Mixing Chamber Optimization using RSM & DOE in ANSYS.
Example 2: The Space-Filling Approach Now, let us look at a much more complex scenario: Heat Sink Design Optimization. You have a heat sink design, but you don’t know the optimal fin thickness, height, or spacing to maximize cooling while simultaneously minimizing material weight. Because this is a multi-objective problem with multiple interacting geometric variables, a simple grid will fail. Instead, you use an Optimal Space-Filling (OSF) design to explore the entire parameter volume.
As a result, the software simulates 20+ variations automatically across the space. You get a response surface plot that instantly reveals the sweet spot where cooling is maximized and material is minimized, saving you days of manual trial-and-error. Learn to turn ANSYS into an automated design engine here: Heat Sink Fins MOGA Optimization: ANSYS CFD Tutorial.
Figure 11: Two great examples from our website to learn choosing the correct DOE model
Response Surface Methodology (RSM)
After the Design of Experiments (DOE) calculates your design points, you only have separated data dots. A response surface is a continuous mathematical map that connects all these dots together. In engineering, we call this a metamodel or surrogate model.
Instead of running the heavy CFD or mechanical solver for every new geometry change, the software uses this mathematical map to predict the result instantly. Without this surface, calculating thousands of design variations for an optimization algorithm would take weeks of computer time. With a response surface, it takes seconds.
Figure 12: A 3D Response Surface maps the outputs across the entire parameter space. It allows you to predict results instantly without running the actual solver.
The RSM Model Library
Based strictly on our ANSYS DesignXplorer 2026 panel, we have exactly six metamodel options. Each model uses a entirely different mathematical approach to draw the map between your DOE points. Here is how they work and when to use them.
Figure 13: Six RSM metamodels in ANSYS panel
Genetic Aggregation (Default)
This is the most advanced and highly recommended model in ANSYS. Instead of forcing you to choose one specific mathematical equation, Genetic Aggregation builds them all. It generates multiple different models (like Kriging, Polynomials, and Moving Least Squares) in the background. It then uses a genetic algorithm to mutate, cross-over, and combine them to find the absolute best fit for your specific data. It often creates an ensemble map, which is a weighted average of different models.
- Best used for: Almost every project. It provides the highest accuracy because it automatically adapts to your data.
- Drawback: Because it builds and tests multiple models simultaneously, it takes more computational time to generate than standard models.
Standard Response Surface – Full 2nd Order Polynomials
This is the classic, traditional regression model. It attempts to fit a smooth, global quadratic curve (a polynomial equation of order 2) through your data points. It uses the method of least squares to minimize the error. It`s simple, smooth physical behaviors with a small number of variables (fewer than 5). It pairs perfectly with deterministic DOE grids like the Central Composite Design (CCD) or Box-Behnken. However, it cannot capture highly complex, wavy, or erratic behaviors.
Kriging
Kriging is a powerful spatial interpolation method. Unlike polynomials that draw a smooth line near your points, a standard Kriging model passes exactly through every single DOE point. It combines a global polynomial model with local deviations. A major mathematical advantage of Kriging is that it calculates an error predictor, meaning it tells you where its own map is uncertain.
- Best used for: Highly non-linear behaviors and spaces with many variables. It is the mandatory choice if you plan to run the Adaptive Single Objective (ASO) or Adaptive Multiple Objective (AMO) optimization algorithms later.
Figure 14: Smoothing Kriging approximation for noisy data
Non-Parametric Regression
While polynomial models force your data to fit a specific equation (like a parabola), Non-Parametric Regression does not assume any predefined equation shape. Internally, ANSYS usually powers this using Support Vector Regression (SVR). It uses hyperplanes and kernel functions to separate data and find a smooth boundary. It can be used for extremely non-linear, unpredictable results where classic mathematical formulas fail completely.
Neural Network
This uses a deep learning approach called a Feedforward Neural Network. It consists of an input layer, hidden layers of neurons, and an output layer. It learns the relationship between your parameters by adjusting the internal weights of these neurons during a training phase. Best for Massive DOE tables. Neural networks require a very large number of training samples to be accurate. If you only have 20 or 30 DOE points, a neural network will perform poorly. If you have hundreds of points, it becomes highly effective.
Figure 15: A Feedforward Neural Network uses hidden layers of mathematical neurons to learn and map highly complex relationships from large datasets.
Sparse Grid
This model must be used together with the Sparse Grid Initialization DOE. It builds a hierarchical mathematical grid. Its greatest feature is self-adaptation. After the first calculation, the Sparse Grid mathematically detects where the error is highest. It then automatically commands the solver to run more CFD or mechanical simulations only in that specific area to fix the map. Use it for Physical models with sharp discontinuities, sudden drops, or highly non-linear zones that require localized, automatic refinement.
Figure 16: The Sparse Grid metamodel automatically refines itself only in complex regions, saving solver time by ignoring simple, flat areas of the design space.
Assessing Quality: CoD, RMSE, and CoP
After the software builds your surface, you cannot trust it blindly. You must check its mathematical accuracy before running an optimization. ANSYS provides specific quality metrics to verify your surface.
- Root Mean Square Error (RMSE): This calculates the average distance between your actual solver data and the predicted surface. A value close to 0 indicates a highly accurate model with very little error.
- Coefficient of Determination (CoD or R2): This parameter measures how well the surface fits your training points. A value of 1 is a perfect fit. However, CoD can be misleading. If your sample size is too small, it might show a falsely high score.
- Coefficient of Prognosis (CoP): This is the most reliable parameter. It uses a cross-validation method. The software hides a portion of your data, builds the surface, and then tries to predict the hidden points. This prevents false scores and shows the true real-world prediction quality of your model.
Figure 17: A residual plot comparing original values to approximated values. Points perfectly on the diagonal line indicate an excellent Coefficient of Prognosis (CoP).
Conclusion
Creating a successful optimization project requires smart data collection and fast mathematical mapping. First, you use DOE to sample your design space efficiently. You choose deterministic schemes for small variable counts and space-filling schemes for highly complex physics. Next, you use RSM models like Kriging or Genetic Aggregation to build an instant prediction map over those points. Finally, you verify the quality using CoP or RMSE.
At CFDLAND, we specialize in turning complicated engineering challenges into simple solutions. You can simply leave your optimization project to our experts. By handing off your project to us, you gain more than just efficiency. You gain the peace of mind that comes with knowing your business is in capable hands. But before that, do not forget to check our full ANSYS optimization training library where you might find what you need.