The criterion calculation:

Given , the volumetric mass of the body, and , the mass formula of the k-th body reads (continuous and discretized):

Where and are the volume and resp. the volumetric mass of the e-th element.

For Center of Gravity, you can choose the component of interest (x, y or z).

Given X = (x, y, z), a material point within the body, the formula reads (continuous):

The sum over the k-th body of interest.

After discretization:

Where:

is the centroid of the e-th element.

other quantities are defined above

Notes:

is given with respect to the global coordinate system.

for a coordinate system: where P is the rotation matrix.

The Moment of Inertia is by nature a tensor. You can choose the component of interest (Ixx, Iyy, Izz, Ixy, Ixz, Iyz).

The full tensor with respect to the global coordinate system reads (continuous formula):

After discretization:

The goal of the User Defined Criterion feature is to define a singular scalar value that measures the mechanical performance of a design as a result of the following three-step reduction process:

Here is an example of the computation process for a displacement base result:

Component reduction returns a spatial scalar field, defined for every excitation frequency:

Spatial reduction returns a singular value, defined for every excitation frequency:

After discretization, generally reads:

Frequency reduction returns a singular scalar value, the criterion:

After discretization, the criterion reads:

where:

NodeSet is the set of nodes defined by the scoping.

FrequencySet is the set of frequencies defined by the Frequency Reduction.

is the Frequency Range of the criterion:

is the excitation frequency.

is the quantity obtained by component reduction.

Considering the displacement as a base result, its complex harmonic amplitude is a complex-valued vector field that reads:

Then the different component reductions are:

is the weight of each node of the NodeSet.

This parameter is related to the following Spatial Reduction when the property is set to Average:

(Based on scoping, can be the volume of influence, the surface of influence or the length of influence).

When the Spatial Reduction property is set to Absolute Maximum, the criterion is defined as:

However, in order for this calculation to be derivable, the max operator is approximated. The calculation uses the absolute values of the result.

The Frequency Reduction property is a read-only property set to Average.

For the Robust Frequencies setting, the criterion reads:

Where:

For the Average reduction frequency, the weight is the same for all frequencies, that is:

For the Smooth Minimum setting, the weight is given by:

This permits to approximate the min value while staying derivable.

The goal of the User Defined Criterion feature is to define a singular scalar value that measures the mechanical performance of a design as a result of the following two-step reduction process:

Here is an example of the computation process for a displacement base result:

After discretization, the criterion generally reads:

Where:

Considering the displacement as a base result, the different component reductions are:

For Spatial Reduction set to Average: . Not available when the Base Result property is set to Reaction Force.

For Spatial Reduction set to Sum:

is the weight of each node of the NodeSet. This parameter is related to the following Spatial Reduction when the property is set to Average:

(Based on scoping, , can be the volume of influence, the surface of influence, or the length of influence).

When the Spatial Reduction property set to Absolute Maximum, the criterion is defined as:

However, in order for this calculation be derived, the maximum (max) criterion is approximated. The calculation uses the absolute values of the result.