The compliance of a structure is a measure of its overall flexibility or stiffness and is defined as the sum of the strain energy of all the elements, for linear models, where is the displacement vector and is the global stiffness matrix. Compliance is the reciprocal of stiffness, and minimizing the compliance is equivalent to maximizing the global stiffness. If a load case is driven by forces or pressures, you should choose to minimize the strain energy to maximize the global stiffness. However, if a load case is driven by a thermal field, strain energy decreases when the optimization modifies the structure to make it softer. As a result, you should always choose to maximize the strain energy because attempting to minimize the strain energy can result in a stiff structure. In addition, you should always choose to maximize the strain energy if prescribed displacements are applied to your model.

Topology optimization considers the total strain energy for all of the elements; therefore, if you choose strain energy as an objective function, you must apply the objective to the entire model. You cannot use strain energy as a constraint in your optimization.

The volume is defined as the sum of the volume of the elements in the design area, , where is the element volume. During a topology optimization, the elements are scaled with the current relative density defined in your Abaqus model. For most optimization problems, you must apply a volume constraint. For example, if you are trying to minimize the strain energy (maximize the stiffness) and do not apply a volume constraint, the Abaqus Topology Optimization Module simply fills the entire design area with material.

You can use the center of gravity of a selected region as a design response in an optimization. You can choose the center of gravity in the three principal directions:

For example, you might want to constrain the center of gravity in the Y-direction so that it remains within a minimum and maximum range during the optimization. The design response can consider the center of gravity of the whole model or a region of the model.

If you select a local coordinate system, the Abaqus Topology Optimization Module uses both the direction of the axes and the position of the origin to recalculate the center of gravity. The Abaqus Topology Optimization Module applies the global coordinate system if you do not select a local coordinate system.

When the Abaqus Topology Optimization Module calculates the center of gravity, it treats shell and membrane regions as three-dimensional regions by applying the thickness of the region. The Abaqus Topology Optimization Module calculates the center of gravity using only the element types that are supported by topology optimization. As a result, the center of gravity calculated by the Abaqus Topology Optimization Module might not be the same as the center of gravity calculated by Abaqus/Standard or Abaqus/Explicit; for example, if your model contains wire regions.

In most optimization problems you will use displacement and/or rotation to define your objective function or constraints. For example, the maximum displacement of a vertex could be either an objective or a constraint of an optimization. The performance of the optimization is improved if you apply displacements and rotations to only vertices or to small regions. In addition, performance is improved if you assign regions that are used to define displacements or reactions as frozen regions (the Abaqus Topology Optimization Module will not remove elements from frozen regions during the optimization).

Table 13.2.1–1 lists the available displacement and rotation variables.

Table 13.2.1–1 Displacement and rotation variables for a general topology optimization.

	Displacement	Rotation
i-direction
Absolute
Absolute in i-direction

Modal eigenvalues are the simplest dynamic responses in structural analysis. Typical uses of data from an eigenfrequency analysis during a topology optimization include the following:

• maximize the lowest eigenfrequencies,

• maximize a selected eigenfrequency,

• constrain an eigenfrequency to be higher or lower than a given value,

• maximize or minimize an eigenfrequency at a certain mode, and

• perform a bandgap optimization that force modes away from a certain frequency.

The Abaqus Topology Optimization Module supports two approaches for evaluating the eigenfrequencies:

• single eigenfrequencies from modal analysis and

• the Kreisselmaier-Steinhauser formulation.

The Kreisselmaier-Steinhauser formulation is the more efficient of the two approaches and should be used whenever possible. The only advantage of evaluating single eigenfrequencies is that you can use the sum of the eigenfrequencies as a constraint in a general topology optimization. You cannot use the sum of the eigenfrequencies from the Kreisselmaier-Steinhauser formulation as a constraint in a general topology optimization.

When you are trying to maximize the lowest eigenfrequency, it is recommended that you consider not only the first eigenfrequency but also at least the next two highest natural frequencies. During the optimization, the various natural frequencies are weighted by their distance from the lowest natural frequency—the closer a natural frequency approaches the first natural frequency during the optimization, the more it is weighted. You should use the Kreisselmaier-Steinhauser eigenvalue formulation if you are trying to maximize the lowest eigenfrequency or, in particular, if you are trying to maximize more than one of the lowest eigenfrequencies. You do not need to use mode tracking if you are using the Kreisselmaier-Steinhauser formulation to maximize the lowest eigenfrequency, but you should use mode tracking for the higher modes because the modes might switch. For example, while the model is being optimized, the frequency of the first mode is maximized and the second eigenmode can become the mode with the lowest frequency.

You can use a moment of inertia design response in an optimization to minimize the rotational inertia about a selected axis. You can use the sum of the moment of inertia of the whole model or a region of the model as an objective function or a constraint in a general topology optimization.

You can choose the moment of inertia in the three principal directions or the three principal planes:

If you select a local coordinate system, the Abaqus Topology Optimization Module uses the direction of the axes to recalculate the center of gravity. The Abaqus Topology Optimization Module applies the global coordinate system if you do not select a local coordinate system.

When the Abaqus Topology Optimization Module calculates the moment of inertia, it treats shell and membrane regions as three-dimensional regions by applying the thickness of the region. The Abaqus Topology Optimization Module calculates the moment of inertia using only the element types that are supported by topology optimization. As a result, the moment of inertia calculated by the Abaqus Topology Optimization Module might not be the same as the moment of inertia calculated by Abaqus/Standard or Abaqus/Explicit; for example, if your model contains wire regions.

The moment of inertia with respect to any two orthogonal axes is zero if you have selected either of the axes to be an axis of symmetry.

You can use nodal internal forces and moments of the whole model or a region of the model as an objective function or a constraint in a general topology optimization.

Table 13.2.1–2 lists the available nodal internal force and moment variables.

Table 13.2.1–2 Nodal internal force and moment variables for a general topology optimization.

	Nodal internal force	Nodal internal moment
i-direction
Absolute
Absolute in i-direction

You cannot use a reference coordinate system with absolute internal force or with absolute internal moment. Your structure must have stiffness in the direction of the force used in the optimization; otherwise, the internal force will be zero in this direction.

Nodal reaction forces and moments can be used as a design response only in general topology optimization. As with displacements, the performance of the optimization is improved if you apply reaction forces or moments to only vertices or to small regions and assign those regions as frozen regions (the Abaqus Topology Optimization Module will not remove elements during the optimization).

Table 13.2.1–3 lists the available nodal reaction force and moment variables.

Table 13.2.1–3 Nodal reaction force and moment variables for a general topology optimization.

	Nodal reaction force	Nodal reaction moment
i-direction
Absolute
Absolute in i-direction

You cannot use a reference coordinate system with an absolute reaction force or with an absolute reaction moment. Your structure must have stiffness in the direction of the force used in the optimization; otherwise, the reaction force will be zero in this direction.

The compliance of a structure is a measure of its overall stiffness and is defined as the sum of the strain energy of all the elements, for linear models, where is the displacement vector and is the global stiffness matrix. Compliance is the reciprocal of stiffness, and minimizing the compliance is equivalent to maximizing the global stiffness. If a load case is driven by a thermal field, strain energy decreases when the structure is made softer. As a result, attempting to minimize strain energy can result in a stiff structure. In addition, you should always choose to maximize the strain energy if prescribed displacements are applied to your model.

Topology optimization considers the total strain energy for all of the elements; therefore, if you choose strain energy as an objective function, you must apply the objective to the entire model.

The volume is defined as the sum of the volume of all the elements in the design area, , where is the element volume. For most optimization problems, you must apply a volume constraint. For example, if you are trying to minimize the strain energy (maximize the stiffness) and do not apply a volume constraint, the Abaqus Topology Optimization Module simply fills the design area with material.

The weight is defined as the sum of the weight of all the elements in the design area, , where is the element weight. The Abaqus Topology Optimization Module scales elements using the current relative density. For most optimization problems, you must apply either a volume or a weight constraint. Using weight instead of volume allows you to constrain the optimized model to a specified physical weight. The Abaqus Topology Optimization Module uses only supported element types when calculating the weight.

You should use the Kreisselmaier-Steinhauser formulation of the eigenvalues as an objective function in a shape optimization if you are trying to maximize the first eigenfrequency or, in particular, if you are trying to maximize more than one of the first eigenfrequencies. You do not need to use mode tracking if you are using the Kreisselmaier-Steinhauser formulation of the eigenvalues.

Equivalent stresses are the most commonly used objective function of a shape optimization. All of the stress values that are calculated by the Abaqus Topology Optimization Module, whether nodal or from Gauss points or elements, are interpolated to the nodes. For example, you can try to optimize your model with an objective function that tries to minimize the maximum von Mises stresses in a region with stress concentrations or tries to minimize contact pressure in a region with contact. The Abaqus Topology Optimization Module considers only the maximum value of an equivalent stress within a region. The Abaqus Topology Optimization Module issues warnings for nodes that do not have the appropriate stress values. For example, if you select an objective response of contact stress, the Abaqus Topology Optimization Module issues warnings about nodes of elements that are not in contact. If your Abaqus model contains multiple load cases, the design response is evaluated by summing the stress values from each load case.

You can choose from the following equivalent stresses:

• von Mises

• Maximum principal and absolute maximum principal

• Minimum principal and absolute minimum principal

• Second principal

• Beltrami

• Drucker Prager

• Galilei

• Kuhn

• Mariotte

• Sandel

• Sauter

• Tresca

• Contact stress pressure

• Total shear contact stress

• Shear contact stress in the 1-direction

• Shear contact stress in the 2-direction

• Total contact stress

If your model is undergoing large deformations, a measure of the stress is not always a good indicator of the model's response. For example, a structure undergoing plastic deformation will, for an ideal plastic material, experience a large constant stress over the plastic area. In these circumstances a measure of the strain is a more reliable indicator of the model's response. You can choose from the following equivalent strains:

• Elastic

• Plastic

• Total (the sum of the elastic and plastic)

The nodal strain energy density, , is a local point-wise strain energy that can provide a better representation of failure than stresses in nonlinear materials.

Volume is the only constraint allowed for a shape optimization. The volume is defined as the sum of the volume of all the elements in the design area, , where is the element volume.

For most optimization problems, you must apply a volume constraint to a region of your model. For example, if you are trying to minimize the strain energy (maximize the stiffness) and do not apply a volume constraint, Abaqus simply fills the design area with material.