19.3.1 Gradient Adaption Approach

Solution-adaptive mesh refinement is performed to efficiently reduce the numerical error in the digital solution, with minimal numerical cost. Unfortunately, direct error estimation for point-insertion adaption schemes is difficult because of the complexity of accurately estimating and modeling the error in the adapted meshes. A comprehensive mathematically rigorous theory for error estimation and convergence is not yet available for CFD simulations. Assuming that maximum error occurs in high-gradient regions, the readily available physical features of the evolving flow field may be used to drive the mesh adaption process.

Three approaches for using this information for mesh adaption are available in ANSYS FLUENT:

• Gradient approach: In this approach, ANSYS FLUENT multiplies the Euclidean norm of the gradient of the selected solution variable by a characteristic length scale [ 68]. For example, the gradient function in two dimensions has the following form:

  
  

  
  

  where $e_{i1}$ is the error indicator, $A_{\rm cell}$ is the cell area, $r$ is the gradient volume weight, and $\nabla f$ is the Euclidean norm of the gradient of the desired field variable, $f$.

  The default value of the gradient volume weight is unity, which corresponds to full volume weighting. A value of zero will eliminate the volume weighting, and values between 0 and 1 will use proportional weighting of the volume.

  If you specify adaption based on the gradient of a scalar, then the value of $\vert e_{i1}\vert$ is displayed when you plot contours of the adaption function.

  This approach is recommended for problems with strong shocks, e.g., supersonic inviscid flows.

• Curvature approach: This is the equidistribution adaption technique formerly used by ANSYS FLUENT, that multiplies the undivided Laplacian of the selected solution variable by a characteristic length scale [ 368].

  For example, the gradient function in two dimensions has the following form:

  
  

  
  

  where $e_{i2}$ is the error indicator, $A_{\rm cell}$ is the cell area, $r$ is the gradient volume weight, and $\nabla ^2 f$ is the undivided Laplacian of the desired field variable ( $f$).

  The default value of the gradient volume weight is unity, which corresponds to full volume weighting. A value of zero will eliminate the volume weighting, and values between 0 and 1 will use proportional weighting of the volume.

  This approach is recommended for problems with smooth solutions.

• Isovalue approach: This approach is not based on derivatives. Instead, the isovalues of the required field variable $f$, are used to control the adaption. Therefore, the function is of the form:

  
  

  
  

  where $e_{i3}$ is the error indicator. This approach is recommended for problems where derivatives are not helpful. For example, if you want to refine the mesh where the reaction is taking place, you can use the isovalues of the reaction rate and mark for refinement at high reaction rates. This approach also allows you to customize the criteria for controlling the adaption using custom field functions, user-defined scalars, etc.

The length scale is the square (2D) or cube (3D) root of the cell volume. Introducing the length scale allows resolution of both strong and weak disturbances, increasing the potential for more accurate solutions. However, you can reduce or eliminate the volume weighting by changing the gradient Volume Weight in the Mesh Adaption Controls dialog box (see this section in the separate User's Guide for details).

Any of the field variables available for contouring can be used in the gradient adaption function. These scalar functions include, both geometric and physical features of the numerical solution. Therefore, in addition to traditional adaption to physical features, such as the velocity, you may choose to adapt to the cell volume field to reduce rapid variations in cell volume.

In addition to the Standard (no normalization) approach formerly used by ANSYS FLUENT, two options are available for Normalization [ 107]:

• Scale, which scales the values of $e_{i1}$, $e_{i2}$, or $e_{i3}$ by their average value in the domain, i.e.:

  
  

  
  

  when using the Scale option, suitable first-cut values for the Coarsen Threshold and the Refine Threshold are 0.3 to 0.5, and 0.7 to 0.9, respectively. Smaller values will result in larger adapted regions.

• Normalize, which scales the values of $e_{i1}$, $e_{i2}$, or $e_{i3}$ by their maximum value in the domain, therefore always returning a problem-independent range of [0, 1] for any variable used for adaption, i.e.:

  
  

  
  

  when using the Normalize option, suitable first-cut values for the Coarsen Threshold and the Refine Threshold are 0.2 to 0.4, and 0.5 to 0.9, respectively. Smaller values will result in larger adapted regions.