# Responses

The following responses can be found in this section:

## Internal Responses

OptiStruct allows the use of numerous structural responses, calculated in a finite element analysis, or combinations of these responses to be used as objective and constraint functions in a structural optimization.

Responses are defined using DRESP1 Bulk Data Entries. Combinations of responses are defined using either DRESP2 entries, which reference an equation defined by a DEQATN Bulk Data Entry, or DRESP3 entries, which make use of user-defined external routines identified by the LOADLIB I/O Options Entry. Responses are either global or subcase (loadstep, load case) related. The character of a response determines whether or not a constraint or objective referencing that particular response needs to be referenced within a subcase.

### Subcase Independent

#### Mass and Volume

Both are global responses that can be defined for the whole structure, for individual properties (components) and materials, or for groups of properties (components) and materials.

It is not recommended to use mass and volume as constraints or objectives in a topography optimization. Neither is very sensitive towards design modifications made in a topography optimization.

In order to constrain the mass or volume for a region containing a number of properties (components), the SUM function can be used to sum the mass or volume of the selected properties (components), otherwise, the constraint is assumed to apply to each individual property (component) within the region. Alternatively, a DRESP2 equation needs to be defined to sum the mass or volume of these properties (components). This can be avoided by having all properties (components) use the same material and applying the mass or volume constraint to that material.

`RTYPE`=VOLUME and
`ATTA`=ENCLOSED can be used to create an
enclosed volume response for Shape, Free-Shape, and Topography optimization. The
ENCLOSED option indicates that this response is an enclosed
volume defined by a closed 2D mesh (free-edges are not supported). For this enclosed
volume response type, `ATTI` field(s) should be set to
PID. Multiple `ATTi` are allowed, as long as
the mesh defined by the PID's is fully enclosed. The property type
should be PSHELL, PCOMP, or
PSHEAR only. The normals of the 2D mesh should be oriented
consistently and the mesh should be sufficiently fine to capture the volume data.
The volume is calculated based on the mesh and it is not recommended to use a coarse
mesh.

#### Fraction of Mass and Fraction of Design Volume

Both are global responses with values between 0.0 and 1.0. They describe a fraction of the initial design space in a topology optimization. They can be defined for the whole structure, for individual properties (components) and materials, or for groups of properties (components) and materials.

The difference between the mass fraction and the volume fraction is that the mass fraction includes the non-design mass in the fraction calculation, whereas the volume fraction only considers the design volume.

Volume fraction = (total volume at current iteration - initial non-design volume)/initial design volume

Mass fraction = total mass at current iteration/initial total mass

If, in addition to the topology optimization, a size and shape optimization is performed, the reference value for the volume fraction (the initial design volume) is not altered by size and shape changes. This can, on occasion, lead to negative values for this response. Therefore, if size and shape optimization is involved, it is recommended to use the Volume responses instead of the Volume Fraction response.

These responses can only be applied to topology design domains. OptiStruct will terminate with an error if this is not the case.

#### Center of Gravity

This is a global response that may be defined for the whole structure, for individual properties (components) and materials, or for groups of properties (components) and materials.

By default, the center of gravity response is calculated with respect to the basic coordinate system. Refer to PARAM, UCORD to specify a user-defined coordinate system for the calculations.

#### Moments of Inertia

This is a global response that may be defined for the whole structure, for individual properties (components) and materials, or for groups of properties (components) and materials.

By default, the moment of inertia response is calculated with respect to the basic coordinate system. Refer to PARAM, UCORD to specify a user-defined coordinate system for the calculations.

#### Weighted Compliance

The weighted compliance is a method used to consider multiple subcases (loadsteps, load cases) in a classical topology optimization. The response is the weighted sum of the compliance of each individual subcase (loadstep, load case).

This is a global response that is defined for the whole structure.

#### Weighted Reciprocal Eigenvalue (Frequency)

The weighted reciprocal eigenvalue is a method to consider multiple frequencies in a classical topology optimization. The response is the weighted sum of the reciprocal eigenvalues of each individual mode considered in the optimization.

This is done so that increasing the frequencies of the lower modes will have a larger effect on the objective function than increasing the frequencies of the higher modes. If the frequencies of all modes were simply added together, OptiStruct would put more effort into increasing the higher modes than the lower modes. This is a global response that is defined for the whole structure.

#### Combined Compliance Index

The combined compliance index is a method to consider multiple frequencies and static subcases (loadsteps, load cases) combined in a classical topology optimization. The index is defined as:

This is a global response that is defined for the whole structure.

The normalization factor, $NORM$
, is used to
normalize the contributions of compliances and eigenvalues. A typical structural
compliance value is of the order of 1.0e4 to 1.0e6. However, a typical inverse
eigenvalue is on the order of 1.0e-5. If `NORM` is not used, the
linear static compliance requirements dominate the solution.

The quantity $NORM$ is typically computed using the formula:

Where, ${C}_{\mathrm{max}}$ is the highest compliance value in all subcases (loadsteps, load cases) and ${\lambda}_{\mathrm{min}}$ is the lowest eigenvalue included in the index.

In a new design problem, you may not have a close estimate for $NORM$ . If this happens, OptiStruct automatically computes the $NORM$ value based on compliances and eigenvalues computed in the first iteration step.

#### Bead Discreteness Fraction

This is a global response for topography design domains. This response indicates the amount of shape variation for one or more topography design domains. The response varies in the range 0.0 to 1.0 (0.0 < BEADFRAC < 1.0), where 0.0 indicates that no shape variation has occurred, and 1.0 indicates that the entire topography design domain has assumed the maximum allowed shape variation.

#### Random Response Analysis

##### PSD and RMS Responses

PSD displacement, PSD velocity, PSD acceleration, PSD acoustic pressure, PSD stress, PSD strain, RMS displacement, RMS velocity, RMS acceleration, RMS acoustic pressure, RMS stress, and RMS strain responses are available as global, as well as subcase-dependent responses (this information is also available in Subcase Dependent).

### Subcase Dependent

#### Linear Static Analysis

The compliance $C$ is calculated using the following relationship:

Where, $Ku=f$

or

For a structure with an applied forces $f$ subcase, the compliance $C$ can be considered a reciprocal measure of the stiffness $K$ :

Where, $\frac{1}{2}{f}^{2}$ is constant.

For maximum stiffness $K$ , the compliance $C$ can be minimized.

For a structure with an applied displacements $u$ subcase, the compliance $C$ can be considered a direct measure of the stiffness $K$ :

Where, $\frac{1}{2}{u}^{2}$ is constant.

#### Elements in the Compliance

Compliance can be defined for the whole structure, for individual properties (components) and materials, or for groups of properties (components) and materials. The compliance must be assigned to a linear static subcase (loadstep, load case).

In order to constrain the compliance for a region containing a number of properties (components), the SUM function can be used to sum the compliance of the selected properties (components), otherwise, the constraint is assumed to apply to each individual property (component) within the region. Alternatively, a DRESP2 equation needs to be defined to sum the compliance of these properties (components). This can be avoided by having all properties (components) use the same material and applying the compliance constraint to that material.

#### Static Displacement

Displacements are the result of a linear static analysis. Nodal displacements can be selected as a response. They can be selected as vector components or as absolute measures. They must be assigned to a linear static subcase.

#### Static Stress of Homogeneous Material

Different stress types can be defined as responses. They are defined for components, properties, or elements. Element stresses are used, and constraint screening is applied. This is a linear static subcase (loadstep, load case) related response.

Cluster Size of elements to consider for stress response calculation is supported via
`ATTB` field. 32

Corner Stresses are available by setting
`ATTB`=CORNER. Corner stress responses are
supported only for Shape, Sizing, and Free-Shape optimization.

#### Stress Responses for Topology and Free-Size Optimization

#### Norm-based Approach (Topology and Free-size Optimization)

The Norm-based approach
is the default method for handling stress responses for Topology and Free-Size
Optimization. This method is used when corresponding stress response
`RTYPE`’s on the DRESP1 Bulk Data
Entry are input.

The Response-NORM aggregation is internally used to calculate the stress responses for groups of elements in the model. Solid, shell, bar stresses and solid corner stresses are supported with the Response-Norm aggregation approach (Free-size Optimization is only supported for shell stresses). Refer to NORM Method for more information.

#### Augmented Lagrange Method (ALM) (Topology Optimization only)

The Augmented Lagrange Method (ALM) is an alternative method for handling stress responses for Topology Optimization. It can be activated using DOPTPRM,ALMTOSTR,1 when DRESP1 Bulk Data Entry is used to specify local stress responses.

ALM is also an alternative method to efficiently solve topology optimization problems with local stress constraints, which is stated in Equation 9.

$\mathrm{min}f\left(X\right)$

${g}_{j}\left(X\right)\le 0,j={m}_{1}+1,\text{\hspace{0.17em}}\dots ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{m}_{1}$

- $X$
- Vector of topology design variables
- $f(X)$
- Objective function
- ${g}_{j}\left(X\right)$
- j
^{th}constraint - ${m}_{1}$
- Number of local stress constraints
- $m$
- Total number of constraints

Typically, ${m}_{1}$ is equivalent to the number of elements and is a large number. ALM converts the optimization Equation 9 to the following format:

- ${h}_{j}\left(X\right)=max\left[{g}_{j}\left(X\right),-\frac{{\lambda}_{j}}{\mu}\right]$
- ${g}_{j}\left(X\right)={E}_{j}\left(\frac{{\sigma}_{j}}{{\sigma}_{lim}}-1\right)\left[1+{\left(\frac{{\sigma}_{j}}{{\sigma}_{lim}}-1\right)}^{2}\right]$
- $,j=1,\dots ,{m}_{1}$
- ${E}_{j}=\epsilon +\left(1-\epsilon \right){\left[\frac{tanh\left(\beta \eta \right)+tanh\left(\beta \left({\rho}_{j}-\eta \right)\right)}{tanh\left(\beta \eta \right)+tanh\left(\beta \left(1-\eta \right)\right)}\right]}^{p}$
- ${\rho}_{j}$
- Element density
- ${\sigma}_{j}$
- Element von Mises stress
- ${\sigma}_{\mathrm{lim}}$
- stress upper bound
- ${\lambda}_{j}$
- Lagrange multiplier estimator
- $\mu >0$
- Quadratic penalty factor

The Lagrange multiplier estimators are updated as ${\lambda}_{j}^{\left(k+1\right)}={\lambda}_{j}^{\left(k\right)}+{\mu}^{\left(k\right)}{h}_{j}\left(X\right)$ .

In general, the number of local stress constraints ${m}_{1}$ is very large. Directly solving Equation 9 is computationally time consuming. By penalizing the stress constraints onto the objective function, the total number of constraints can be significantly reduced. As a result, the optimization problem can be efficiently solved.

The default Stress-norm
(P-norm) method continues to efficiently solve Topology Optimization problems with
local stress constraints. The ALM ^{1} is a good alternative for such models.

#### Global von Mises Stress Response (Topology and Free-size Optimization)

`STRESS`optional continuation line on the DTPL or the DSIZE card. There are a number of restrictions with this constraint:

- The definition of stress constraints is limited to a single von Mises permissible stress. The phenomenon of singular topology is pronounced when different materials with different permissible stresses exist in a structure. Singular topology refers to the problem associated with the conditional nature of stress constraints, that is, the stress constraint of an element disappears when the element vanishes. This creates another problem in that a huge number of reduced problems exist with solutions that cannot usually be found by a gradient-based optimizer in the full design space.
- Stress constraints for a partial domain of the structure are not allowed because they often create an ill-posed optimization problem since elimination of the partial domain would remove all stress constraints. Consequently, the stress constraint applies to the entire model when active, including both design and non-design regions, and stress constraint settings must be identical for all DSIZE and DTPL cards.
- The capability has built-in intelligence to filter out artificial stress concentrations around point loads and point boundary conditions. Stress concentrations due to boundary geometry are also filtered to some extent as they can be improved more effectively with local shape optimization.
- Due to the large number of elements with active stress constraints, no element stress report is given in the table of retained constraints in the .out file. The iterative history of the stress state of the model can be viewed in HyperView or HyperMesh.
- Stress constraints do not apply to 1D elements.
- Stress constraints may not be used when enforced displacements are present in the model.

The buckling factor can be constrained for shell topology optimization problems with a base thickness not equal to zero. Constraints on the buckling factor are not allowed in any other cases of topology optimization.

Composite Stress | Composite Strain | Composite Failure Criterion |

Frequency Response Stress | Frequency Response Strain | Frequency Response Force |

#### Grid Point Stress of Homogeneous Material

Different grid point stress types can be defined as responses. They are defined for components, properties, or elements. Grid point stresses are used, and constraint screening is applied. This is currently supported only for Shape, Sizing, and Free-Shape Optimization.

The `RTYPE` field should be set to GPSTR to
activate Grid Point Stress response.

Grid
Point Stress Responses for Topology Optimization are available through
corresponding GPSTR response `RTYPE`’s on the
DRESP1 Bulk Data Entry.

The Response-NORM aggregation is internally used to calculate the failure Responses for groups of grids in the model. Refer to NORM Method for more information.

#### Static Strain of Homogeneous Material

Different strain types can be defined as responses. They are defined for components, properties, or elements. Element strains are used, and constraint screening is applied. It is also not possible to define strain constraints in a topology design space. This is a linear static subcase (loadstep, load case) related response.

Corner Strains are available by setting
`ATTB`=CORNER. Corner strain responses are
supported only for Shape, Sizing, and Free-Shape optimization.

Strain
Responses for Topology and Free-Size Optimization are available through
corresponding Strain response `RTYPE`’s on the
DRESP1 Bulk Data Entry.

The Response-NORM aggregation is internally used to calculate the Strain Responses for groups of elements in the model. Solid, Shell, Bar Stresses and Solid Corner Strains are supported with the Response-Norm aggregation approach (Free-size optimization is only supported for shell strains). Refer to NORM Method for more information.

#### Static Stress of Composite Lay-up

Different composite stress types can be defined as responses. They are defined for PCOMP(G), PCOMPP components or elements, or PLY type properties. Ply level results are used, and constraint screening is applied. It is also possible to define composite stress constraints in a topology and/or free-size design space. This is a linear static subcase (loadstep, load case) related response.

Stress
Responses for Topology and Free-Size Optimization are available through
corresponding Stress response `RTYPE`’s on the
DRESP1 Bulk Data Entry.

The Response-NORM aggregation is internally used to calculate the Stress Responses for groups of elements in the model. Refer to NORM Method for more information.

#### Static Strain of Composite Lay-up

Different composite strain types can be defined as responses. They are defined for PCOMP(G), PCOMPP components or elements, or PLY type properties. Ply level results are used, and constraint screening is applied. It is also possible to define composite strain constraints in a topology and/or free-size design space. This is a linear static subcase (loadstep, load case) related response.

Strain
Responses for Topology and Free-Size Optimization are available through
corresponding Strain response `RTYPE`’s on the
DRESP1 Bulk Data Entry.

The Response-NORM aggregation is internally used to calculate the Strain Responses for groups of elements in the model. Refer to NORM Method for more information.

#### Static Failure in a Composite Lay-up

Different composite failure criterion can be defined as responses. They are defined for PCOMP(G) components or elements, or PLY type properties. Ply level results are used, and constraint screening is applied. It is also possible to define composite failure criteria constraints in a topology design space. This is a linear static subcase (loadstep, load case) related response.

Composite Failure Responses for Topology and Free-Size
Optimization are available through corresponding Failure response
`RTYPE`’s on the DRESP1 Bulk Data
Entry.

The Response-NORM aggregation is internally used to calculate the failure Responses for groups of elements in the model. Refer to NORM Method for more information.

#### Static Failure

Different failure criteria can be defined as responses. They are supported for solid and shell elements only. Failure criteria can be defined on the MAT1 or MATT1 entries.

#### Static Force

Different force types can be defined as responses. They are defined for components, properties, or elements. Constraint screening is applied. It is also not possible to define force constraints in a topology design space. This is a linear static subcase (loadstep, load case) related response.

#### Single Point Force at a Constrained Grid Point

This response can be defined using the DRESP1 Bulk Data Entry
(`RTYPE`=SPCFORCE). This response is defined
for constrained grid points. Constraint screening is applied to this response. This
is a linear static subcase (loadstep, load case) related response.

#### Grid Point Force

This response can be defined using the DRESP1 Bulk Data Entry
(`RTYPE`=GPFORCE). This response defines the
contribution to a specific grid point force component from a non-rigid element
(which is connected to that grid). Constraint screening is applied to this response.
If `ATTi` specify multiple elements, then multiple responses will
be generated; where, each response calculates a specified element's contribution to
the grid point force component at the specified grid. This is a linear static
subcase (loadstep, load case) related response.

#### Nodal Electric Potential

Nodal Electric Potential can be used as an internal response when running a linear
steady-state electrical conduction analysis with optimization. Nodal Electric
Potential response is activated using the
`RTYPE`=ELPOT option. The
`ATTi` field(s) can be used to identify the GRID ID at which
the electric potential is calculated for the optimization response.

#### Global Electric Compliance

Global Electric Compliance can be used as an internal response when running a linear
steady-state electrical conduction analysis with optimization. Global Electric
Compliance response is activated using the
`RTYPE`=ELCOMP option.

#### Bore Distortion

`RTYPE`=BOREDST option. The

`ATTA`field can be used to identify the ORDER number or can be set to CIRCUL (circularity). The

`ATTi`field(s) can be used to identify the ID of BORE Bulk Data Entries. If ORDER is defined by setting the

`ATTA`field to an integer value, then the bore distortion response of the specified order is used for optimization. One bore distortion response is created for each ring of the bore. Similarly, if CIRCUL is defined, then circularity response is created for each ring of the bore.

Stress Responses for Topology and Free-Size optimization can be defined via the DRESP1 Bulk Data Entry. The Stress Responses are internally aggregated using the Stress-NORM approach to maintain the number of created responses at a reasonable number.

### Heat Transfer Analysis

#### Temperature

Temperatures are the result of a heat transfer analysis and must be assigned to a heat transfer subcase (loadstep, load case). Temperature response cannot be used in composite topology and free-size optimization.

#### Thermal Compliance

Thermal compliance $C$ is calculated using:

- $T$
- Temperature vector
- $f$
- Power vector
- ${K}_{C}$
- Conduction matrix
- $H$
- Convection matrix

When minimizing thermal compliance, temperatures at grids of flux input are minimized. The optimum structure provides maximum conduction of thermal energy. Since thermal compliance is a smooth convex function, it is typically more efficient to minimize thermal compliance than to perform a min-max of the temperature of the entire structure. During thermal-structural optimization, the temperature field is updated for each iteration. Additionally, for thermal-structural topology optimization, the density design variable affects the conductivity, and the conductivity is also penalized.

Thermal Compliance Response can only be used in Sizing, Shape, Topology, and Free-Size Optimization.

#### Nodal Flow Pressure (Darcy Flow Analysis)

For a Darcy Flow Analysis nodal flow pressures can be used as an optimization
response. The `RTYPE` should be set as FLOWPRES
and the `ATTi` fields should point to the Grid IDs of the node of
interest. This is currently only supported for Topology Optimization with
Steady-State Heat Transfer analysis (for which Darcy flow analysis is active).

### Normal Modes Analysis

#### Frequency

Natural frequencies are the result of a normal modes analysis, and must be assigned to the normal modes subcase (loadstep, load case). It is recommended to constrain the frequency for several of the lower modes, not just of the first mode.

- Set the
`ATTA`field on DRESP1 entry equal to the Normal Mode number(s) that you want to use as a response. In this process, you will identify and provide the mode number(s) that are to be optimized. - You can also source the mode number of the mode with largest modal
participation at the peak loading frequency by adding a separate Modal
Frequency Response Subcase, in addition to the existing Normal Modes
subcase. The
`ATTA`field should identify the number of modes and the`ATTi`field should be set equal to the subcase ID of the Modal Frequency Response subcase. The Modal Frequency Response Analysis subcase should contain a PEAKOUT entry that identifies peak identification criteria. OptiStruct will internally identify the modes to be optimized based on Modal Frequency Response Analysis, by picking the modes (`ATTA`) for each peak response which contributes the most to the corresponding response. Therefore, the maximum number of modes identified for the optimization is equal to: number of modes (`ATTA`) *`NPEAK`(on PEAKOUT).

#### Mode Shape

Mode shapes are the result of a normal modes analysis. Mode shapes can be selected as a response. They can be selected as vector components or as absolute measures. They must be assigned to a normal modes subcase.

### Linear Buckling Analysis

The buckling factor is the result of a buckling analysis, and must be assigned to a buckling subcase (loadstep, load case).

#### Buckling Factor

A typical buckling constraint is a lower bound of 1.0, indicating that the structure is not to buckle with the given static load. It is recommended to constrain the buckling factor for several of the lower modes, not just of the first mode.

### Linear Transient Analysis

#### Transient Compliance

Transient compliance, ${\tau}_{2}$ , is defined as:

- $f{\left(t\right)}^{T}$
- Transpose of the time dependent applied force
- $u\left(t\right)$
- Transient displacement
- ${T}_{f}$
- Final time of the analysis

- Adjoint method for sensitivity calculation is implemented.
- Enforce displacement load (SPCD) is not supported.
- Only MTRAN analysis is support.
- Currently, size, shape and topology optimization are supported.

#### Transient Displacement

Transient displacement, ${\tau}_{3}$ is defined as:

- $u\left(t\right)$
- Transient displacement
- $\text{\delta}\left(\text{t}-{t}_{i}\right)$
- Applied excitation

- Direct and adjoint method for sensitivity calculation are implemented.
- SPCD is not supported.
- MTRAN and DTRAN analysis are supported.
- Currently, size, shape and topology optimization are supported.

#### Mean Square of Displacement Difference

Mean Square of Displacement Difference Response, ${\tau}_{4}$ is defined as:

Where, ${u}_{i}\left(t\right)$ and ${u}_{j}\left(t\right)$ are the transient displacement of GRID ID $i$ and $j$ , respectively.

- Adjoint method for sensitivity calculation is implemented.
- Only MTRAN analysis is supported.
- Currently, size, shape and topology optimization are supported.

### Frequency Response Function (FRF Analysis)

#### Frequency Response Displacement

Displacements are the result of a frequency response analysis. Nodal displacements, that is,
translational, rotational and normal ^{1}, can be selected as a response. They can be
selected as vector components in real/imaginary or magnitude/phase form. They must
be assigned to a frequency response subcase (loadstep, load case).

#### Frequency Response Velocity

Velocities are the result of a frequency response analysis. Nodal velocities, that is,
translational, rotational and normal ^{2}, can be selected as a response. They can be
selected as vector components in real/imaginary or magnitude/phase form. They must
be assigned to a frequency response subcase (loadstep, load case).

#### Frequency Response Acceleration

Accelerations are the result of a frequency response analysis. Nodal accelerations, that is,
translational, rotational and normal ^{3}, can be selected as a response. They can be
selected as vector components in real/imaginary or magnitude/phase form. They must
be assigned to a frequency response subcase (loadstep, load case).

#### Frequency Response Stress

Different stress types can be defined as responses. They are defined for components, properties, or elements. Element stresses are not used in real/imaginary or magnitude/phase form, and constraint screening is applied. The von Mises stress for solids and shells can also be defined as direct responses. This is a frequency response subcase (loadstep, load case) related response.

FRF
Stress Responses for Topology and Free-Size Optimization are available through
corresponding Frequency-response Stress response `RTYPE`’s on
the DRESP1 Bulk Data Entry.

The Response-NORM aggregation is internally used to calculate the Stress Responses for groups of elements in the model. Solid and Shell FRF Stresses are supported with the Response-Norm aggregation approach (Free-size optimization is only supported with Shell FRF Stresses). Refer to NORM Method for more information.

#### Frequency Response Strain

Different strain types can be defined as responses. They are defined for components, properties, or elements. Element strains are used in real/imaginary or magnitude/phase form, and constraint screening is applied. The von Mises strain for solids and shells can also be defined as direct responses. This is a frequency response subcase (loadstep, load case) related response.

FRF
Strain Responses for Topology and Free-Size Optimization are available through
corresponding Frequency-response Strain response `RTYPE`’s on
the DRESP1 Bulk Data Entry.

The Response-NORM aggregation is internally used to calculate the Strain Responses for groups of elements in the model. Solid and Shell FRF Strains are supported with the Response-Norm aggregation approach (Free-size optimization is only supported with Shell FRF Strains). Refer to NORM Method for more information.

#### Frequency Response Force

Different force types can be defined as responses. They are defined for components, properties, or elements in real/imaginary or magnitude/phase form. Constraint screening is applied. It is also not possible to define force constraints in a topology design space. This is a frequency response subcase (loadstep, load case) related response.

#### Powerflow

Powerflow can be defined as a Response by setting
`RTYPE` field to FRFLOW. Calculated for a
SECTION entry identified via the `ATTi`
field. Constraint screening is applied. It is supported for Topology, Sizing, and
Shape Optimization.

### Random Response Analysis

#### PSD and RMS Responses

PSD displacement, PSD velocity, PSD acceleration, PSD acoustic pressure, PSD stress, PSD strain, RMS displacement, RMS velocity, RMS acceleration, RMS acoustic pressure, RMS stress and RMS strain responses are available as global as well as subcase-dependent responses (this information is also available in the Subcase Independent responses section above).

### Coupled FRF Analysis on a Fluid-structure Model (Acoustic Analysis)

#### Acoustic Pressure

Acoustic pressures are the result of a coupled frequency response analysis on a fluid-structure model. This response is available for fluid grids. It must be assigned to a coupled frequency response subcase (loadstep, load case) on a fluid-structure model.

### Multibody Dynamics Analysis

#### Flexible Body Responses

For Multibody Dynamics problems, the Mass, Center of gravity, and Moment of Inertia of one or more flexible bodies are available as responses. This is in addition to other usual structural responses.

#### MBD Displacement

MBD displacements are the result of a multibody dynamics analysis. They must be assigned to a multibody dynamics subcase (loadstep, load case).

#### MBD Velocity

MBD velocities are the result of a multibody dynamics analysis. They must be assigned to a multibody dynamics subcase (loadstep, load case).

#### MBD Acceleration

MBD acceleration are the result of a multibody dynamics analysis. They must be assigned to a multibody dynamics subcase (loadstep, load case).

#### MBD Force

MBD forces are the result of a multibody dynamics analysis. They must be assigned to a multibody dynamics subcase (loadstep, load case).

#### MBD Expression

MBD expression responses are the result of a multibody dynamics analysis. They are the result of the evaluation of an expression. They must be assigned to a multibody dynamics subcase (loadstep, load case).

### Fatigue

#### Life/Damage/Factor of Safety

`RTYPE`=FATIGUE. Life, Damage, and Factor of Safety (FOS) are
results of a fatigue analysis. They must be assigned to a Fatigue subcase.

#### Grid Point based Factor of Safety

Factor of Safety results when `RTYPE`=GPFAT. They
must be assigned to a Fatigue subcase.

### Dynamic/Nonlinear Analysis

#### Equivalent Plastic Strain

Equivalent plastic strain can be used as an internal response when a nonlinear response optimization is run using the equivalent static load method. This is made possible through the use of an approximated correlation between linear strain and plastic strain, which are calculated in the inner and outer loops respectively, of the ESL method.

#### ESL Velocity

ESL velocity can be used as an internal response, when a nonlinear response optimization is run,
using the equivalent static load method. The
`RTYPE`=ESLV option can be used in conjunction
with the velocity component value and a GRID ID specification.

#### Gasket Pressure

Gasket Pressure can be used as an internal response when a nonlinear static analysis
with optimization is run. Gasket pressure response is activated using
`RTYPE`=GSKP option. The
`PTYPE` field references PGASK and the
`ATTi` field references the corresponding
PGASK
`ID`(s).

#### Contact Pressure

Contact Pressure can be used as an internal response when a model with contact and
optimization is run. Contact pressure response is activated using
`RTYPE`=CNTP option. The
`PTYPE` should be set to CONTACT and the
corresponding CONTACT Bulk Data `ID`(s) can be
referenced on the `ATTi` field. For Shape, Free-Shape, and Sizing
optimization, the design variables can be part of the contact region.

#### Contact Force

Contact Force can be used as an internal response when a model with contact and
optimization is run. Contact force response is activated using
`RTYPE`=CNTF option. The
`PTYPE` should be set to CONTACT, the
`ATTA` field should point to the Contact Force item code of
interest, and the corresponding CONTACT Bulk Data ID(s) can be
referenced on the `ATTi` field. For Shape, Free-Shape, and Sizing
optimization, the design variables can be part of the contact region.

### User Responses

#### Functions

A function response is one that uses a mathematical expression to combine design variables, grid point locations, responses, and/or table entries. Whether the function is subcase (loadstep, load case) related or global, is dependent on the response types used in the equation. Function responses are available via DRESP1, DRESP2, and DRESP3 entries. For DRESP3, the functions are defined using external responses.

#### External

An external response is one that uses an external user-defined routine to combine design variables, grid point locations, eigenvectors, responses, and/or table entries. Whether the function is subcase (loadstep, load case) related or global is dependent on the response types used in the routine. Refer to External Responses for more information.

### Appendix

#### NORM Method

The NORM method is used to approximately calculate the maximum value of a particular response of all the grids/elements included in a particular response. This is also scaled with the response bounds specified for each grid/element. Therefore, to minimize the maximum response in a particular grid/element set, the resulting NORM value ( ${R}_{NORM}$ ) is internally constrained to a value lower than 1.0.

- ${R}_{NORM}$
- Response NORM value
- $n$
- Number of elements
- ${R}_{i}$
- Individual response value of grid/element “ $i$ "
- ${R}_{bound}$
- Response bound for each grid/element
- $p$
- Penalty (power) value (default = 6.0)

The Penalty or Power value ( $p$ ) can be modified using the parameter DOPTPRM, PNORM. The value $p$ = 6.0 is the default and higher values of $p$ ( $p$ → $\infty $ ) increases the accuracy of the norm function ( ${R}_{NORM}$ → $\left(\frac{{R}_{\mathrm{max}}}{{R}_{bound}}\right)$ ), which can lead to instability of the optimization run. Values lower than 6 ( $p$ → 1) moves the response norm function closer to the average ratio ( ${R}_{NORM}$ → $\frac{1}{n}{\displaystyle \sum _{i=1}^{n}\left(\frac{{R}_{i}}{{R}_{bound}}\right)}$ ). The default value is a reasonable approximation of the maximum ratio value and reduces instability.

## External Responses

The DRESP3 Bulk Data Entry, in combination with the LOADLIB I/O Option Entry, allows for the definition of responses through user-defined external functions.

The external functions may be written in Compose Open Matrix Language (OML), Fortran, C or a Microsoft Excel workbook. The resulting libraries and files should be accessible by OptiStruct regardless of the coding language, providing that consistent function prototyping is respected, and adequate compiling and linking options are used.

### Write External Functions for Optimization Responses

The OptiStruct installation provides
barebone functions for Fortran
(`dresp3_barebone.F`

) and for C (`dresp3_barebone.c`

) with
proper function definition, arguments, and compilation directives.

These files can be used as starting points to write your own functions. Refer to Microsoft Excel Example for information on response definition through a user-defined Microsoft Excel workbook.

```
integer function myfunct(iparam, rparam, nparam,
iresp, rresp, dresp, nresp, isens
userdata)
```

```
integer function myfunct(iparam, rparam, nparam,
iresp, rresp, dresp, nresp, isens
userdata)
```

```
character*32000 userdata
integer nparam, nresp
integer iparam(nparam), iresp(nresp)
double precision param(nparam), rresp(nresp), dresp(nparam,nresp)
```

```
int myfunct(int* iparam, double* rparam, int* nparam,
int* iresp, double* rresp, double* dresp, int* nresp, int*
isens, char* userdata)
```

- Function names should be written using either all lower-case or all upper-case characters
- Only alphanumeric characters should be used
- Underscore characters are prohibited
- Names cannot be longer than eight characters

#### Function Return Values

External functions should return 0 or 1 for successful completion, where 1 indicates that a user-defined information message should be output by OptiStruct. External functions should return -1 in case of fatal error, in which case OptiStruct will terminate after outputting a user-defined error message. See below for more information about error and information messages.

#### Function Arguments

Argument | Type | Input/Output | Description |
---|---|---|---|

`iparam` |
integer (table) | Input | Input parameters types (optional use) |

`rparam` |
double (table) | Input | Input parameters values |

`nparam` |
integer | Input | Number of parameters |

`iresp` |
integer (table) | Input | Output responses requests (optional use) |

`rresp` |
double (table) | Output | Output responses values |

`dresp` |
double (table) | Output | Output sensitivity values |

`nresp` |
integer | Input | Number of responses |

`isens` |
integer | Input | Sensitivity output flag |

`userdata` |
string | Input/Output | User data/Error or information message |

`nparam`

is the number of input parameters that were defined on the DRESP3 card.`rparam(nparam)`

contains the values of the input parameters as evaluated by OptiStruct.`iparam(nparam)`

indicates the types of the input parameters as described below.

Parameter values are passed in the exact order in which they were defined on the DRESP3 card, regardless of their type. Using the Parameter Types table is optional, for instance to perform verifications or code-branching.

**Parameter Type**`iparam`

Value- DESVAR
- 1
- DTABLE
- 2
- DGRID/DGRIDB
- 3
- DRESP1
- 4
- DRESP2
- 5
- DRESP1L
- 6
- DRESP2L
- 7
- DVPREL1
- 8
- DVPREL2
- 9
- DVMREL1
- 10
- DVMREL2
- 11
- DVCREL1
- 12
- DVCREL2
- 13
- DVMBRL1
- 14
- DVMBRL2
- 15
- DEIGV
- 16
- DGRIDB
- 18

`nresp`

is the maximum number of responses which the function is able to compute, as defined on the`MAXRESP`field of the DRESP3 card.`rresp(nresp)`

returns the values of the responses as evaluated by the external function.`iresp(nresp)`

contains the responses requests as described below.The responses requests table indicates which of the available responses are actually needed by OptiStruct. Entries in

`iresp(nresp)`

are flagged as 1 for requested responses and as 0 otherwise. Using that information is optional and allows for saving computational effort by not evaluating responses which OptiStruct does not need.

#### Userdata String

Upon entering the function, the `userdata`

string contains data as defined in the
`USRDATA` field of the DRESP3 card. It
provides a convenient mechanism to pass constants or any other relevant information
to the function. There are no restrictions regarding the contents of the string, but
its length must not exceed 32,000 characters.

Upon exiting the function, the string may contain a user-defined error or information message. The updated string is then returned to OptiStruct, where it is printed to the standard output (.out file and/or screen). Here again, the contents of the string are not restricted as long as its length does not exceed 32,000 characters.

The error or information messages may be formatted by using the character "|" as a line-break indicator. Standard C escape sequences are supported, as well. It is advised, but not necessary, to format messages in such a way that each line does not exceed 80 characters, since the same convention is used in OptiStruct output files.

#### Sensitivity Flag

`isens`

indicates whether sensitivities are requested in the code. It
is recommended to skip the calculation of sensitivities when `isens`

is turned off. This will avoid unnecessary computations.

### Build External Libraries for Optimization Responses

Allows building shared libraries on Windows or Linux.

Refer to Build External Libraries for more information.

### External Responses using Altair Compose

Invoking Compose to specify the installation path.

If Altair Compose needs to be invoked during the OptiStruct job in case of external responses (DRESP3), the installation path for Compose must to be specified by the user.

- Options field

For further details, refer to `-compose`

in Run Options.

Regarding the implementation of external, user-defined routines using Open Matrix Language (OML), refer to the documentation for Compose OML for writing scripts.

Compose is supported on Windows and Linux operating systems.

### External Optimization Responses (DRESP3) Example

#### Fortran and C-Shared Library

All files referenced here are located in the Altair HyperWorks installation directory under <install_directory>/hwsolvers/demos/optistruct. Copy the files mentioned below to your working directory to run the following examples.

The files
dresp3_simple.F and
dresp3_simple.c contain source code for simple examples
of external functions written in Fortran and C,
respectively. Both functions are named `mysum`

and compute two
responses - the sum of the parameters and the averaged sum of the
parameters.

```
LOADLIB DRESP3 FLIB dresp3_simple_f.dll
LOADLIB DRESP3 CLIB dresp3_simple_c.dll
```

```
DRESP3 6 SUMF FLIB MYSUM 1 2
+ DRESP1 2 3
DRESP3 7 AVGF FLIB MYSUM 2 2
+ DRESP1 2 3
DRESP3 8 SUMC CLIB MYSUM 1 2
+ DRESP1 2 3
DRESP3 9 AVGC CLIB MYSUM 2 2
+ DRESP1 2 3
```

```
DEQATN 1 F(x,y) = x+y
DEQATN 2 F(x,y) = avg(x,y)
DRESP2 4 SUME 1
+ DRESP1 2 3
DRESP2 5 AVGE 2
+ DRESP1 2 3
```

Running this input deck through OptiStruct shows that the Fortran external functions, the C external functions and the internal equations always return the same values, and are updated simultaneously throughout the optimization process.

The file dresp3_advanced.F contains the Fortran source code of the second example, in which you are making use of advanced features of the DRESP3 functionality.

- The
`USRDATA`string is parsed to determine whether stresses or strains are requested, and an error message is returned otherwise. - The number of parameters is used to determine whether a shell or solid element is treated, and an error message is returned if that number is not equal to 3 or 6.
- An error message is returned if the parameters are not of type DRESP1 or DRESP1L, since stress or strain components are expected.
- Even though the function is able to compute two different responses, only the response(s) actually requested by OptiStruct are computed when the function is called.
- An information message is returned indicating which responses were evaluated.

The input deck dresp3_advanced.fem gives a simple example of problem making use of this external function, for analysis only.

```
DRESP1 10 SXX2D STRESS ELEM SX1 100
DRESP1 11 SYY2D STRESS ELEM SY1 100
DRESP1 12 SXY2D STRESS ELEM SXY1 100
DRESP1 13 SXX3D STRESS ELEM SXX 50
DRESP1 14 SYY3D STRESS ELEM SYY 50
DRESP1 15 SZZ3D STRESS ELEM SZZ 50
DRESP1 16 SXY3D STRESS ELEM SXY 50
DRESP1 17 SXZ3D STRESS ELEM SXZ 50
DRESP1 18 SYZ3D STRESS ELEM SYZ 50
DRESP1 20 SVM2D-1 STRESS ELEM SVM1 100
DRESP1 21 SMP2D-1 STRESS ELEM SMP1 100
DRESP1 22 SVM3D-1 STRESS ELEM SVM 50
DRESP1 23 SMP3D-1 STRESS ELEM SMP 50
```

```
DRESP3 30 SVM2D-3 STRLIB GETSTR 1 2
+ DRESP1 10 11 12
+ USRDATA STRESS
DRESP3 31 SMP2D-3 STRLIB GETSTR 2 2
+ SECOND 30
DRESP3 32 SVM3D-3 STRLIB GETSTR 1 2
+ DRESP1 13 14 15 16 17 18
+ USRDATA STRESS
DRESP3 33 SMP3D-3 STRLIB GETSTR 2 2
+ SECOND 32
```

### Microsoft Excel Example

Microsoft Excel workbooks can be referenced via the LOADLIB entry to define user-defined responses.

Both Implicit and Explicit options are available and are defined here.

```
LOADLIB DRESP3 ELIB dresp3_excel.xlsx
DRESP3 10 SUM ELIB MYSUM
+ DRESP1 5 6
```

```
LOADLIB DRESP3 ELIB dresp3_excel.xlsx
DRESP3 20 FUNC ELIB MYFUNC
+ DRESP1 5 6 7 8
+ DESVAR 1
+ CELLIN B3 THRU B6
+ CELLIN C10
+ CELLOUT E10
```

^{1}The normal at a grid point is calculated based on the normals of the surrounding elements. The normal frequency response displacement at a grid point can be selected as a response and it is the displacement in the normal's direction. The normals are also updated when shape changes occur during shape optimization.

^{2}The normal at a grid point is calculated based on the normals of the surrounding elements. The normal frequency response velocity at a grid point can be selected as a response and it is the velocity in the normal's direction. The normals are also updated when shape changes occur during shape optimization.

^{3}The normal at a grid point is calculated based on the normals of the surrounding elements. The normal frequency response acceleration at a grid point can be selected as a response and it is the acceleration in the normal's direction. The normals are also updated when shape changes occur during shape optimization.