NFX|拓扑优化原理
Topology Optimization
Toplogy optimization is a layout optimization problem, which determines the distribution of materials most suitable to a given objective. It is primarily used to produce a fundamental basis for the designer’s engineering judgment at the conceptual design stage or generate ideas for new alternatives. In addition,toplogy optimization can lead to more realistic design in reflection of various constraints associated with the problem at hand.
In order to express the distribution of materials in toplogy optimization, density variables of the finite elements created for analysis are used. The element density of 1 represents a part that requires the element, while 0 represents a part that does not require the element. Unlike general design optimization, the only design variable is the element’s density, which determines whether or not the material is used. As such, the user does not specify separate design variables but composes an optimization problem using only the combinations of objective functions and constraints. This section explains the types of toplogy optimization problem composition and optimization techniques provided by midas NFX.
5.11.1 Objective Functions and Sensitivity
The problem composition method of toplogy optimization is a method of finding design variables that maximize or minimize the objective functions under the specified constraints like general optimization methods. In case an objective function is minimized, the toplogy optimization problem composition can be expressed as Equation (5.11.1).
The types of toplogy optimization problem composition provided in midas NFX can be classified as Table 5.11.1 depending on objective functions.
Static compliance
Static compliance being a function of element density is expressed in the form of global deformation energy as shown in Equation (5.11.2).
The sensitivity of static compliance is expressed as Equation (5.11.3) applying the adjoint method to Equation (5.11.2).
The sensitivity for the loads included in Equation (5.11.3) exists for the loads in the form of body force, which varies with the element density, such as gravity and rotational inertia force.
Dynamic compliance
Dynamic compliance defines the compliance for complex number responses such as frequency responses, which is expressed in the form of the magnitude of a complex number as shown in Equation (5.11.4).
The sensitivity of dynamic compliance uses the adjoint method and introduces the conjugate complex number expression as Equation (5.11.5).
Average eigenvalues
The average value of eigenvalues in a design mode uses the reciprocal formulation of eigenvalues as Equation (5.11.6).
The sensitivity of the average eigenvalue can be expressed using the sensitivity of eigenvalues as Equation (5.11.7).
The sensitivity of eigenvalue for each mode in Equation (5.11.7) is expressed in terms of eigenvector, element matrix and mass matrix of the corresponding mode as shown in Equation (5.11.8) if it is not a duplicate eigenvalue.
Volume fraction
Volume fraction represents the ratio of the volume at the state of finalized material distribution to the volume of the entire design domain. In the case of homogeneous material distribution, the mass ratio and the weight ratio are the same. Since the design variable of toplogy optimization is the density of elements,the sensitivity of volume fraction can be expressed as the ratio of the original volume of the corresponding element to the total volume of the design domain.
5.11.2 Material Interpolation Scheme
Material interpolation schemes are introduced to correct mass, which is dependent on the physical stiffness or element density toplogy of actual material when the density of an element retains a value in between 0 and 1 during the process of toplogy optimization. midas NFX selectively uses the two methods shown in Table
5.11.3 Optimality Criteria
In an optimization problem defined in Equation (5.11.1), optimality criteria (OC)include constraints together with Lagrange multipliers in the objective function thus composing it into an unconstrained optimization problem. The criteria then find the Karush-Kuhn-Tucker (KKT) conditions in which the differential value of the newly constituted objective function becomes 0.
In case only one constraint exists, Bendsøe proposed an empirical design variable update method using the optimality criteria as follows:
