Layout and material gradation in topology optimization of functionally graded structures: a global–local approach

Abstract

By means of continuous topology optimization, this paper discusses the influence of material gradation and layout in the overall stiffness behavior of functionally graded structures. The formulation is associated to symmetry and pattern repetition constraints, including material gradation effects at both global and local levels. For instance, constraints associated with pattern repetition are applied by considering material gradation either on the global structure or locally over the specific pattern. By means of pattern repetition, we recover previous results in the literature which were obtained using homogenization and optimization of cellular materials.

Nomenclature

a :

length of the pattern in X direction

b :

length of the pattern in Y direction

B :

strain-displacement matrix

c :

objective function representing the compliance of the structure

C ₀ :

constitutive matrix of the solid phase of the reference material

d :

nodal density

d :

set of nodal densities

d ¹ :

primary nodal densities

d ² :

secondary densities

e :

index identifying the element

E ^H :

Young’s modulus of the homogeneous material

$E_i^e$ :

Young’s modulus of element e at node i

E _s :

Young’s modulus of the solid material

E ₀ :

reference Young’s modulus

E ₁ :

Young’s modulus at the end of the gradation curve

f :

global nodal force vector

H :

height of the structure

K :

global stiffness matrix

K _e :

element stiffness matrix

L :

length of the structure

m :

number of patterns in X direction

n :

number of patterns in Y direction

nnodes :

total number of nodes

$N_i^e$ :

shape function of element e related to node i

p :

penalization factor of the SIMP and the FGM-SIMP models

P :

point load

q :

uniform load

$r_j^n$ :

distance between the nodes n and j

r _min :

radius of the projection region

S _i :

set of elements sharing node i

$S_d^k$ :

set of nodal densities d assigned to the design variable y _k

$S_w^n$ :

set of nodes included in the projection region of node n

u :

global displacement vector

u _e :

element displacement vector

V _s :

maximum volume of structural material

w :

weight-function

x :

coordinates of a point

X and Y:

Cartesian coordinates of position x

X _m and Y _m :

coordinates of symmetry axes

X^∗ and Y^∗:

coordinates of position x in the local pattern system

y :

design variable

y :

set of design variables

α :

coefficient that defines the change of material property in the direction of axis X

β :

coefficient that defines the change of material property in the direction of axis Y

ρ :

material density

$\rho _i^e$ :

nodal density and projected density at node i of element e

ρ _n :

projected nodal density for node n

ΔV _i :

volume around node i

Ω:

extended domain

Ω _e :

domain of element e

$\Omega _w^n$ :

projection region of node n