Issues to Consider Regarding the Control Variables

The Effect of Discretization

The sensitivity analysis is always performed on the discretized system of equations. As already mentioned, the control variables can be a scalar, vector, or an element in some infinite-dimensional function space. In the latter case, it is represented on the finite element mesh, just like the solution variables, or global scalar quantities. When using a control variable field represented on the finite element mesh, the sensitivities are therefore associated with individual control variable degrees of freedom rather than with the field value at each point. This makes it difficult to interpret the result. For example, if a domain control variable is set up using a first-order Lagrange shape function representation to control the material density in a model, the solution contains the sensitivity of the objective function with respect to the discrete density value at each node point in the mesh. Because each node influences the density in a small surrounding region, the size of which varies from node to node, the individual sensitivities are not directly comparable to each other.

Displaying such domain control variables results in a plot that is not smooth due to the varying element size. It must therefore not be used to draw any conclusions about the physics and the effect of changing the physical field represented by the control variable. Some insight can, however, be gained by looking at the sensitivities divided by the mesh volume scale factor dvol. This makes the sensitivities in the plot comparable between different parts of the surface but still not mathematically well defined. In particular, using discontinuous constant shape functions together with the division by dvol results in a plot that is proportional to the true pointwise sensitivity.

If the plan is to use the sensitivities in an automatic optimization procedure, as is done through the Optimization interface available with the Optimization Module, the discrete nature of the sensitivities causes no additional complication. The optimization solver searches for optimum values of the discrete control variables using the discrete gradient provided by the sensitivity analysis.

Geometrical Sensitivity

You can use the control variables directly to parameterize any aspect of the physics that is controlled by an expression. This applies to material properties, boundary conditions, loads, and sources. However, the shape, size, and position of parts of the geometry cannot be changed as easily at solution time and require special attention.

Control variables cannot be used directly in the geometry description. Instead, the model must be set up using a Deformed Geometry or Moving Mesh interface to control the shape of the geometry. Then use control variables to control the mesh movement, effectively parameterizing the geometry.

See Deformed Geometry and Moving Mesh for details about these interfaces and ALE in general.