PSI - Issue 32

Daria Dolgikh et al. / Procedia Structural Integrity 32 (2021) 246–252 D. Dolgikh, M. Tashkinov / Structural Integrity Procedia 00 (2019) 000–000

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The rigid plates through which boundary conditions are set were attached on upper and lower sides to the RVEs of the cellular structure. On the bottom plate, the displacements are limited in all directions. On the upper plate the displacements are limited along the X and Z axes, along the Y axis the sample is stretched by a value of 0.005 mm (Figure 2).

Fig.2. Scheme of loading of a sample of material with random structure

The optimization procedure is an iterative process that allows changing the structure of the object according to the formulated problem. The implemented optimization algorithm is based on minimization of the strain energy values in the nodes of finite element mesh. The morphology of the finite element mesh represents the design domain that will be modified in the optimization process. The optimization functional consists of the volume occupied by the cellular structure as well as the sum of strain energy across all elements in the design domain. The objective function is minimization of the sum of strain energy in the design domain. Structures are optimized so that

maximum stiffness is achieved while maintaining a given material volume factor of 50%. Solving the static problem by the FE analysis method, the following equation is true: [ ] [ ] { } F K u =

(1)

where [ F ] is the force vector, [ K ] is the stiffness matrix, and { u } is the displacement vector (Pang and Fard, 2020). The state of finite elements acts as optimization parameters. 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 elements. To maximize the overall stiffness, it is necessary to minimize the strain energy. The stiffness is varied by the parameter – [ ] e x 0,1 ∈ . Thus, the optimization process generates finite elements that are either empty (their relative density is very close to zero) or solid (their relative density is equal to one). Then the objective function ( ) e C x     can be expressed as: ( ) e C x min    ⇒

( ) { } [ ] { } T u K u  =

(2)

e C x

 



with: