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Theory
Simcenter STAR-CCM+ simulations are built on numerical algorithms that solve relevant laws of physics according to the conditions that the simulation defines. So that you can identify the physical laws, and understand the methods by which they are solved, Simcenter STAR-CCM+ comes with a Theory Guide.
Design Manager
Design Manager provides an automated approach within Simcenter STAR-CCM+ to run design exploration studies. Design exploration covers both performance assessment and optimization.
Surrogate
A surrogate model indicates an approximation model in which a continuous fitting function is developed from discrete data samples. To achieve the best approximation for a particular scenario, various modeling types and parameters are provided in Simcenter STAR-CCM+.
Cross Validation
In Design Manager, before obtaining the surrogate function for the entire design space, sample data must first be generated by running designs in a design study. From the sample data generated in these real simulations, a surrogate function is calculated using the specified surrogate type. After computing the surrogate, you can use cross validation to assess the quality of the surrogate for predicting unknown response values.

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Theory
Simcenter STAR-CCM+ simulations are built on numerical algorithms that solve relevant laws of physics according to the conditions that the simulation defines. So that you can identify the physical laws, and understand the methods by which they are solved, Simcenter STAR-CCM+ comes with a Theory Guide.
- Fundamental Equations
  Simcenter STAR-CCM+ models a range of physics phenomena including fluid mechanics, solid mechanics, heat transfer, electromagnetism, and chemical reactions. On a macroscopic scale, where the typical lengths are much greater than the inter-atomic distances, the discrete structure of matter can be neglected and materials can be modeled as continua. The mathematical models that describe the physics of continua are derived from fundamental laws that express conservation principles.
- Fluid Flow
  Simcenter STAR-CCM+ can simulate internal and external fluid flow across a wide range of flow regimes, and for a variety of fluid types. It solves the conservation equations for mass, momentum, and energy for general incompressible and compressible fluid flows.
- Numerical Flow Solution
  In the finite volume method, the solution domain is subdivided into a finite number of small control volumes, corresponding to the cells of a computational grid.
- Turbulence
- Transition
  The term transition refers to the phenomenon of laminar to turbulence transition in boundary layers. A transition model in combination with a turbulence model predicts the onset of transition in a turbulent boundary layer.
- Wall Treatment
  Walls are a source of vorticity in most flow problems of practical importance. Therefore, an accurate prediction of the flow across the wall boundary layer is essential.
- Wall Distance
  Wall distance is a parameter that represents the distance from a cell centroid to the nearest wall face with a non-slip boundary condition. Various physical models require this parameter to account for near-wall effects.
- Heat Transfer
  Heat transfer is the exchange of thermal energy between media at different temperatures. Heat transfers from locations of high temperature to locations of low temperature in order to reach an equilibrium state. The three main mechanisms of heat transfer are: conduction, convection, and radiation.
- Porous Media
  Porous media are continua that contain both fluid and fine-scale solid structures, for example: packed-bed chemical reactors, filters, radiators, honeycomb structures, or fibrous materials. The solid geometrical structures are too fine to be individually meshed and fully resolved by a computational grid.
- Species and Passive Scalars
  Simcenter STAR-CCM+ models the transport, the mixing, and the chemical reactions of multi-component liquids or multi-component gases by solving conservation equations for scalar variables that represent the mass fraction of each species in the mixture. This conservation equation allows for convection, diffusion, and optional source effects, and is solved in addition to the global mass continuity equation.
- Multiphase—Eulerian Approach
  Multiphase flows, where several fluids flow in the domain of interest, play an important role in variety of industrial applications. In general, we associate phases with gases, liquids or solids and as such some simple examples of multiphase flows are: air bubbles rising in a glass of water, sand particles carried by wind, rain drops in air. The definition of phase can be generalized and applied to other fluid characteristics such as size, shape, density, and temperature.
- Multiphase—Lagrangian Approach
  Multiphase flows are found in a wide variety of industrial processes, some examples of which are internal combustion engines, liquid, or solid fueled combustors, spray driers, cyclone dust separators, and chemical reactors. Multiphase in this context refers to one thermodynamic phase, be it a solid, a liquid, or a gas, interacting with another distinct phase.
- Reacting Flow
  In reacting flows, chemical species mix with each other and react when conditions allow. To model these flows, Simcenter STAR-CCM+ couples species and energy transport equations with the chemistry solvers that compute the source terms.
- Internal Combustion Engines
  Numerical modeling of internal combustion engines plays an increasingly important role in improving engine design. CFD simulation of such engines can give a comprehensive insight into the wide-ranging physics of the device, such as turbulent mixing between fuel and air, the ignition, and combustion chemistry.
- Electrochemistry
  Electrochemistry is the discipline investigating relationships between electrical currents and chemical composition change in general.
- Plasma
  Plasma is a state of matter similar to a gas that is composed partially or completely of charged particles such as ions and electrons which are not bound to each other.
- Electromagnetism
  Many engineering applications, such as electric motors, electric switches, and transformers, involve electromagnetic phenomena. Electromagnetic phenomena can be modeled based on the classical theory of Electromagnetism, which describes the interaction between electrically charged particles in terms of electric fields, magnetic fields, and their mutual interaction.
- Batteries
- Solid Mechanics
  Solid Mechanics describes the behavior of a solid continuum in response to applied loads. Applied loads include body forces, surface loads, point forces, or thermal loads that result from changes in the solid temperature. Applied loads induce a stress field in the structure and can cause displacement of the structure—from an initial undeformed configuration to a deformed configuration.
- Aeroacoustics
  Computational aeroacoustics (CAA) is a branch of multiphysics modeling and simulation that involves identifying noise sources that are induced by fluid flow and propagation of the sound waves then generated.
- Finite Element Method
  The Finite Element method is a powerful tool for finding approximate solutions to continuous problems. The methodology is similar to other numerical techniques that approximate continuous partial differential equations with discrete algebraic equations.
- Mesh Motion
  In transient simulations, mesh motion is a numerical technique that allows you to update the position of the computational domain while the solvers run.
- 6-DOF Rigid Body Motion
  Simcenter STAR-CCM+ allows you to model the motion of a rigid body in response to applied forces and moments. In a rigid body, the relative distance between internal points does not change. Therefore, it is sufficient to solve the equations of motion for the center of mass of the body.
- Rotating Flow
  Rotating flow usually occurs in rotating machinery such as pumps, propellers, fans, wind turbines, and so on.
- Harmonic Balance
  Harmonic Balance equations describe periodic unsteady flows where the unsteady frequencies are known beforehand. They are suited to modeling periodically repeating flow fields that typically occur in turbomachinery such as compressors, turbines, and fans.
- Adjoint
  The adjoint method is an efficient means to predict the influence of many input parameters on some engineering quantities of interest in a simulation.
- Design Manager
  Design Manager provides an automated approach within Simcenter STAR-CCM+ to run design exploration studies. Design exploration covers both performance assessment and optimization.
  - Sampling Method
    In Design Manager, a sampling method is an algorithm for populating input parameter values for design runs according to parameter range and parameter distributions that you set.
  - Surrogate
    A surrogate model indicates an approximation model in which a continuous fitting function is developed from discrete data samples. To achieve the best approximation for a particular scenario, various modeling types and parameters are provided in Simcenter STAR-CCM+.
    - Kriging
      Kriging is a method of interpolation which is based on a statistical correlation between sample points. It assumes that the distance between sample points reflects a spatial correlation which can be used to explain variations in the surface.
    - Least Squares
      Least squares linear and quadratic models create a polynomial that approximates the data sample globally as a best-fit.
    - Radial Basis Function
      The Radial Basis Function (RBF) surrogate interpolates the data by passing through each sample point. In the presence of sufficient sample data points, RBF can represent complex data accurately.
    - Cross Validation
      In Design Manager, before obtaining the surrogate function for the entire design space, sample data must first be generated by running designs in a design study. From the sample data generated in these real simulations, a surrogate function is calculated using the specified surrogate type. After computing the surrogate, you can use cross validation to assess the quality of the surrogate for predicting unknown response values.
  - Gradient-Based Optimization
    Gradient-based optimization is a mathematical method for finding the local optimum (often minimum) of an objective function, while varying a set of parameters of interest.
- Scalars, Vectors, and Tensors
  Most physical quantities in Simcenter STAR-CCM+ are either scalars, vectors, or 2nd-order tensors.
- Solution Data Interpolation
  Many operations require interpolation of solution data between different sets of mesh points. For example, remeshing operations require interpolation of existing solution data onto a new mesh. Data interpolation also occurs at the interface between regions, where contacting boundaries exchange data. Additionally, configurable data mappers allow you to interpolate field functions and tabular data to specified meshes.
- Idealizations
  In many simulations, it is common practice to reduce the size of the computational domain using planes of symmetry, axes of rotation, periodicity, or by reducing the 3D domain to a 2D domain. However, when you calculate physical quantities using reports you generally want to account for the whole domain. You can obtain report values for the full model using idealizations.

Cross Validation

In Design Manager, before obtaining the surrogate function for the entire design space, sample data must first be generated by running designs in a design study. From the sample data generated in these real simulations, a surrogate function is calculated using the specified surrogate type. After computing the surrogate, you can use cross validation to assess the quality of the surrogate for predicting unknown response values.

Cross validation assesses the prediction accuracy by removing some of the sample data, fitting the remaining data with the surrogate function, and using the removed data as out-of-sample data by which to test the reduced surrogate.

The following table gives an overview of methods for assessing the prediction accuracy of the surrogate function:

Table 1. Surrogate Prediction
Surrogate Type	Assessing the Surrogate Fit of the Known Data	Assessing the Surrogate Fit of the Unknown Data
Kriging RBF	Since the structure of the surrogate function defines the exact fit of the known data, cross validation is the only method by which to assess the prediction accuracy. Cross V PRESS Correlation Coefficient (R2) is always equal to 1 and the Root Mean Squared Error (RMSE) is always equal to 0.	Cross V PRESS
Least Squares	Correlation Coefficient (R2) Root Mean Squared Error (RMSE) Adjusted Correlation Coefficient (R2adj)	Cross V PRESS

For example, if you have $m$ data points and they are divided in $k$ groups, then each group has $n = \frac{m}{k}$ data points. One group is used as training data to validate the fit of the reduced surrogate function, which is calculated from the remaining data points. For the $i^{t h}$ data point, the predictive residual ${c v}_{i}$ is computed as follows:

Figure 1. EQUATION_DISPLAY

{c v}_{i} = y_{i} - {\hat{y}}_{i}

(5173)

where

$y_{i}$ is the true value of the $i^{t h}$ data point.
${\hat{y}}_{i}$ is the predicted value of the $i^{t h}$ data point when evaluated with the reduced surrogate model.

The predictive residual is computed for all the $m$ data points. The Mean Square Error (MSE) is calculated as follows:

Figure 2. EQUATION_DISPLAY

M S E = \frac{1}{m} \sum_{i = 1}^{m} {({c v}_{i})}^{2}

(5174)

The Cross V value of a design study is the Root Mean Squared Error (RMSE):

Figure 3. EQUATION_DISPLAY

{c v}_{s t u d y} = \sqrt{M S E}

(5175)