Reduced model sensitivities in SOL 200 Model Update solution

Target sensitivities

The target sensitivities, TSij relate changes in the targets Tj, with respect to changes in design variables ΔDVi:

As the design variable change tends to zero, the sensitivity becomes the derivative of the target evaluated at the initial value of the design variable, DV0, as shown in the following graphic:

Finite design variable change

The SOL 200 Model Update solution uses a finite design variable change, ΔDVi, to compute sensitivities. Different design variable changes result in different values of sensitivity, as shown in the following graphic. The red derivative is computed using a very small change around the initial value of the design variable, whereas the blue derivative is based on a large ΔDVi change:

In the SOL 200 Model Update solution, the design variable change is related to the current value, DVi of the design variable, as shown in the following expression:

ΔDVi = DELB x DVi

where DELB is the relative finite difference move parameter which is specified on the DOPTPRM card. By default, the DELB value is set to 0.01%.

Sensitivity of a property

The sensitivity of a physical or material property P with respect to a design variable is expressed in the forward difference scheme as follows:

The SOL 200 Model Update solution uses the forward difference scheme by default. Using the forward difference scheme means that the design variable change is positive from the initial design variable value. If your required design variable change is negative, and corresponds to a design variable factor that is less than one, the calculated sensitivities give inaccurate results. You can set the parameter CDIF to YES, to let the SOL 200 Model Update solution use the centered difference scheme instead. In that case, the sensitivity of property P is expressed as follows:

Reduced matrix sensitivities from physical (Guyan) reduction

For each design variable, the Model Update solution process gets the following reduced mass [ΔMR] and stiffness [ΔKR] matrix sensitivities from the SOL 200 Model Update solution results:

[ΔMR] = [ΔMAA] + [ΔMOA]T [T] + [T]T [ΔMOA] + [T]T [ΔMOO] [T]

[ΔKR] = [ΔKAA] + [ΔKOA]T [T] + [T]T [ΔKOA] + [T]T [ΔKOO] [T]

where:

• [ΔMAA], [ΔMOA], and [ΔMOO] are the changes in the partitioned matrices of the full mass matrix with respect to a design variable.

• [ΔKAA], [ΔKOA], and [ΔKOO] are the changes in the partitioned matrices of the full stiffness matrix with respect to a design variable.

• [T] is the transformation matrix between the omitted (O) DOF set and the retained (A) DOF set.

Reduced matrix sensitivities from modal reduction

For each design variable, the Model Update solution process gets the following modal reduced mass [ΔR] and stiffness [ΔR] matrix sensitivities from the SOL 200 Model Update solution results:

[ΔR] = [Φ']T [ΔM] [Φ']

[ΔR] = [Φ']T [ΔK] [Φ']

where the eigenvectors [Φ'] are the mass normalized initial work mode shapes in physical coordinates and [ΔM] and [ΔK] are the full mass and stiffness matrix sensitivities to a design variable, respectively.

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Model reduction in a SOL 200 Model Update solution

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Reduced model sensitivities in SOL 200 Model Update solution, Simcenter 3D 2021.1 Series

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