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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.
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.
Sensitivity with Respect to Boundary Parameters
The optimization of an objective based on a modification of the boundary conditions requires the sensitivity of the objective with respect to the boundary parameters, that is, $\frac{d L}{d Q_{b}}$ .

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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.
  - Sensitivity with Respect to Design Parameters
    The optimization of objectives on the basis of design modifications requires the sensitivities of the objectives with respect to the design parameters, that is $\frac{d L}{d D}$ .
  - Sensitivity with Respect to Boundary Parameters
    The optimization of an objective based on a modification of the boundary conditions requires the sensitivity of the objective with respect to the boundary parameters, that is, $\frac{d L}{d Q_{b}}$ .
  - Surface Sensitivity
    The surface sensitivity allows you to quantify the impact of changes to a part surface on an objective of interest. Cell faces that show large values of surface sensitivity have a large impact on the objective.
  - Adjoint Error Estimation
    Adjoint error estimates give estimates of the impact of local discretization errors in the flow equations on the accuracy of the respective cost function.
  - Adjoint Topology Optimization Model
    The topology optimization model determines the optimal distribution of material within a domain subject. The model is used in conjunction with the Topology Physics Model and the Adjoint model to perform topology optimization.
- 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.
- 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.

Sensitivity with Respect to Boundary Parameters

The optimization of an objective based on a modification of the boundary conditions requires the sensitivity of the objective with respect to the boundary parameters, that is, $\frac{d L}{d Q_{b}}$ .

Consider a cost function $L$ of the form:

Figure 1. EQUATION_DISPLAY

L = L (Q (Q_{b}), Q_{b})

(5100)

where $Q$ is the solution of the following set of nonlinear, discrete equations:

Figure 2. EQUATION_DISPLAY

R (Q (Q_{b}), Q_{b}) = 0

(5101)

$R$ are the residuals of the flow equations and $Q_{b}$ are the imposed boundary conditions.

Since the solution $Q$ to this set of equations depends on the imposed boundary conditions $Q_{b}$ , there is an implicit dependence of $L$ on $Q_{b}$ in addition to the explicit dependence. The objective is to quantify this dependency.

Quantifying the dependency is achieved in a similar manner to the mesh dependency, by differentiating the residual equation and substituting the result into the objective differentiation:

Figure 3. EQUATION_DISPLAY

\frac{d L}{d Q_{b}} = \frac{\partial L}{\partial Q_{b}} - (\frac{\partial L}{\partial Q}) {\frac{\partial R}{\partial Q}}^{- 1} \frac{\partial R}{\partial Q_{b}}

(5102)

This equation can be simplified by using:

Figure 4. EQUATION_DISPLAY

{\frac{\partial R}{\partial Q}}^{- T} (\frac{\partial L}{\partial Q}) = Λ_{Q}

(5103)

where the solution of the flow adjoint $Λ_{Q}$ is obtained from Eqn. (5085).

This simplification leads to:

Figure 5. EQUATION_DISPLAY

\frac{d L}{d Q_{b}} = \frac{\partial L}{\partial Q_{b}} - {Λ_{Q}}^{T} \frac{\partial R}{\partial Q_{b}}

(5104)

From Eqn. (5104) it can be seen that the computation of the cost function sensitivity with respect to boundary conditions involves two steps:

Differentiation of the explicit dependency of the cost function on the imposed boundary conditions.
Accounting for the implicit dependence that is given by the second term in Eqn. (5104), which requires obtaining the global adjoint solution.