2009 International Conference on Artificial Intelligence and Computational Intelligence
Topology Optimization of Continuum Structures with Many Subdomains K. Cai*1 J. Shi 1 H.K. Ding 2 1: College of Water Resources and Architectural Engineering, Northwest A&F University, Shaanxi Yangling 712100, China 2: Glycol Factory of Jilin Petrochemical Company, PetroChina Jilin 132022, China *Corresponding author’s Email
[email protected] distributions of a structure with different initial material distributions under the same loading conditions are different, commonly. To find the effects of the initial material distribution on the final topology of structure, a bionics approach is applied.
Abstract The topology optimization of a structure with many subdomains is solved by a bionics approach, called as floating interval of reference strain energy density (SED). In practical engineering, a structure to be optimized may have many subdomains and each one is filled with a type of solid material as an initial design. To solve such kind of structural topology optimization, we present a heuristic approach according to bone remodelling theory. The main ideas of the method are as follows. First, the structure to be optimized is considered as a piece of bone and the optimization process of structure is equivalent to the bone remodelling process, in which the distribution of local material in a bone tissue varies when the deformation of tissue goes beyond a so-called dead zone. Second, based on the concept of dead zone, a SED interval is adopted to control the update of local material in structure. Third, to satisfy the constraint(s) of the optimization problem, the reference interval changes in optimization. Particularly, the update of the reference interval is determined by the active constraint of optimization problem. A numerical example is given to show the significance of this kind of optimization problem, finally.
Two major aspects are included in the current approach. Firstly, a structure to be optimized is considered as a piece of bone and the structural optimization process is equivalent to bone remodeling process [7], and the amount of local material i.e., the design variable, changes according to the reference interval of SED which is equivalent to the dead zone [8] in bone mechanics. Secondly, the reference interval changes according to the active constraint of optimization problem.
2. Material properties The linear elastic properties of anisotropic porous materials characterized by a fourth rank tensor are dependent on both the solid volume fraction of the material and the geometry of the microstructure. In most applications, orthotropic material properties are sufficiently well described by a scalar and a symmetric, traceless second rank fabric tensor [9]. Here a second rank positive and definite fabric tensor is used to express the microstructure properties and is used as design variables of local material in optimization problems.
1. Introduction Topology optimization is called as layout optimization or generalized shape optimization. The importance of topology optimization is to choose the appropriate topology of a structure in its conceptual design phase. Due the complexity, topology optimization becomes an intellectually challenging field and attracts so much attention in the recent years. Those methods can briefly be classified into two types [1]. One is the microstructure method ([2], [3], [4]). The other is the geometry method ([5], [6]).
3
Fabric tensor B = b q ⊗ q ∑ii i
where bi ∈ [δ 1.0 ] , qi (i = 1, 2, 3) are eigenpairs of fabric tensor B with component matrix Bij . Unit vectors qi (i = 1, 2,3) represent three material principal axes. δ is a very small positive scalar to keep B to be positive and definite. Stiffness tensor Dijkl = λ Bijω Bklω + μ ( Bikω Bωjl + Bilω Bωjk )
In practical engineering, a structure may formed by many components and the material in each component may be different. Such characters should be considered in optimization process. The reason is that, for the same optimization problem the final optimal material 978-0-7695-3816-7/09 $26.00 © 2009 IEEE DOI 10.1109/AICI.2009.94
(1)
i =1
(2)
Obviously, Dijkl in Eq. (2) expresses an isotropic material when B is proportional to the second rank identity tensor, i.e. 245
Bij = ρ ⋅ δij Dijkl = ρ 2ω ⎡⎣λ δijδ kl + μ (δik δ jl + δ il δ jk ) ⎤⎦
ref ref ⎡⎣u inf ⎤⎦ is the optimal interval of reference SED. Ω u sup is the design domain.
(3)
where ρ is the eigenvalue of B and δ ij is the component matrix of the second rank identity tensor. Here we set ω = 1.5 . λ and μ are the Lame constant of solid phase.
For example, as the active constraint of the optimization problem is the displacement of a point in design domain, the optimization model can be expressed as follows.
3. Continuum Topology optimization method
Find
{ Bm ( ρ m ) , m ∈ Ω}
to satisfy
ref um ∈ ⎡⎣u inf
3.1. Basic equations
subject to
Where d 0 is the critical value of the displacement of point p.
(4)
ε ij = ( ui , j + u j ,i ) 2
In a simulation, the update of design variables of a material point is determined by the comparison between the local SED and the current reference interval (at k-th iteration). Briefly, if the local SED of m-th point is less than the infimum of the reference interval, the local relative density should be decreased; if the local SED is greater than the supremum of the reference interval, the local relative density should be increased. Otherwise, the local relative density does not alter. Mathematically, the increments of a local relative density can be expressed as following
σ ij ,i + f j = 0 : σ ij ⋅ n j = Fi* ；
Γ u : u = u*
(5)
where σ ij is stress tensor, ε ij is strain tensor, u is displacement vector, f i is body force vector, Fi* is boundary force on the boundary Γ σ with the normal direction n j , u* is the assigned displacement on the boundary Γ u .
Δρ k , m
3.2 Optimization model In the current work, the design variable of a isotropic material point in design domain is the single eigenpair of fabric tensor and represents the relative density of the material point, too. The objective of the optimization of structure is to find the distribution of the relative densities to satisfy the constraints of optimization problem. Mathematically, the formulations are as Find
{ Bm ( ρ m ), m ∈ Ω }
to satisfy
ref um ∈ ⎡⎣u inf
subject to
p∈Ω
3.3. Update of design variables
σ ij = Dijkl : ε kl
σ
d p − d0 = 0,
(7)
ρm ∈ [δ 1.0]
The classical linear theory of elasticity is adopted in this work. The load process is quasi-static and the deformation process takes an isothermal course simultaneously. The basic equations and boundary conditions are summarized as follows
Γ
ref ⎤⎦ u sup
ref ⎤⎦ u sup
⎧ g1 > 0 ⎪ =⎨0 ⎪− g < 0 ⎩ 2
ref ( k ) if uk , m > u sup
others
(8)
ref ( k ) if uk , m < u inf
where g1 and g2 are called as growth speeds. Correspondingly, the update of the local relative density is expressed as
ρ k +1,m
⎧1.0 ⎪ = ⎨ ρ k ,m + Δρ k ,m ⎪δ ⎩
if ( ρi ,k ,m + Δρ k ,m ) ≥ 1.0 others
(9)
if ( ρi ,k ,m + Δρ k ,m ) ≤ δ
The new fabric tensor for next iteration can be expressed as
(6)
φ j ({Bm }) ≤ 0, ( j = 1, 2,")
Bk +1, m,ij = ρk +1,mδ ij
ρm ∈ [δ 1.0]
(10)
The local stiffness tensor is updated according to the new fabric tensor, i.e.
where Bm is the fabric tensor of the m-th material point. φ j is the constraint function, e.g., volume
Dijkl = ρk3+1,m ⎡⎣λ δijδ kl + μ (δik δ jl + δil δ jk ) ⎤⎦
constraint, displacement constraint, etc. u m is the strain energy density (SED) of the m-th material point.
246
(11)
3.4. Update of reference interval
4. Numerical example
To satisfy the active constraint of an optimization problem, the reference interval should be changed during iteration [4]. Commonly, the length of the reference interval is kept to be null to keep SED distributing uniformly in the final structure. Therefore, only the supremum of the interval needs to be updated with the following form
In the example, a uniformly fixed finite element mesh is used to describe the geometry and mechanical response within the entire design domain.
⎧⎪ R β ⋅ u ref if R ≥ 1.0 − η k u ref = k +1 ⎨ γ ref if R < 1.0 − η ⎪⎩ R ⋅ u k Mod ( k , iFEA ) = 0 R=
Hk
(12)
α
H0
where the exponents β ∈ [1.0 2.0] , γ ∈ [10 30] . η is
the algorithm tolerance. Integer iFEA ∈ { 3, 4,5} is adopted. Hk is the current value of the active constraint and H0 is the critical value. If volume constraint is an active constraint, α = 1 ; If displacement constraint is the active one, α = −1 .
Figure 1. Initial design domain with four subdomains Fig. 1 shows a rectangle design domain with size of 2.0 m ×1.25m × 0.01m . The left side is fixed. A concentrated force t P = 1.0 kN is applied on the centre of the right side. The rectangle is divided into four subdomain with same size. Four cases of the initial material distributions in structure are considered, i.e.,
3.5. Optimization procedure S1: Construct the finite element model of structure and initiate parameters of algorithm, let k=1;
(a)
S2: Analyze the structure to obtain the strain energy density field through Eqs. (4) and (5); S3: Update the design variable (Eqs. (9, 10)) of each material point, renew the interval of reference SED (Eq. (12));
(b) No:1 and No:3 subdomains are filled with the same istropic material with E=69 GPa and v=0.3; No: 2 and No:4 subdomains are filled with the same istropic material with E=210 GPa and v=0.3;
S4: Determine iteration criterion: if the convergent conditions (13) are satisfied or k is equal to a given maximum number of iteration, then go to S5, otherwise let k=k+1 and go to S2;
(c)
S5: Stop.
than unity), over the admissible design domain. Commonly, the initial supremum of the reference interval is set to be equal to the average SED of the initial structure under the given loading conditions. The convergent conditions in Step4 are expressed as follows R ∈ {RHj , j = iFEA − M ," , iFEA − 1, iFEA }
No:1 and No:3 subdomains are filled with the same istropic material with E=210 GPa and v=0.3; No: 2 and No:4 subdomains are filled with the same istropic material with E=69 GPa and v=0.3;
(d) No:1 and No:2 subdomains are filled with the same istropic material with E=210 GPa and v=0.3; No: 3 and No:4 subdomains are filled with the same istropic material with E=69 GPa and v=0.3.
For an initial design it is normally to distribute evenly the porous materials, of which the relative densities are initialized to be a positive scalar ( ρ0 a constant no more
R − 1.0 ≤ η
Four subdomains are filled with the same isotropic material with Young’s modulus of E=69 GPa and Poisson’s ratio of v=0.3;
The objective of the optimization problem is to minimize the structural volume ratio while the displacement of the force-added point is no more than 0.002m.
(13)
where integer M is no less than 2.
247
(a) Optimal topology for case (a)
(b) Optimal topology for case (b)
(c) Optimal topology for case (c)
(d) Optimal topology for case (d)
Figure 2. Optimal material distributions in structure for different initial material distributions
In case (a), the structure contents only one type of isotropic material with Young’s modulus of E=69 GPa and Poisson’s ratio of v=0.3. Thus, the final material distribution (See Fig. 2a) is identical to the traditional one which can also be given by other approach, e.g. SIMP method.
Structural Volume Ratio*100%
1
In case (b), the left subdomains (No:1 and No:3) have the same isotropic material which is softer than that in the right subdomains (No:2 and No:4). Obviously, the final material distribution shown in Fig. 2b is different from the result of case (a). Particularly, the material distribution in the right half part of the structure is complex.
Case (a)
0.8
Case (b)
0.6
Case (c) Case (d)
0.4 0.2 0 0
10
20
30
40
50
60
70
Iteration Nubmer
In case (c), the material in the left subdomains (No:1 and No:3) are stiffer than that in the right subdomains (No:2 and No:4), i.e., the Young’s modulus 210GPa of the material in the left subdomains is greater than 69GPa of the material in the right subdomains. Clearly, the final material distribution of case (c) (See Fig. 2c) is different from both of the results of case (a) and (b) (See Fig.2a and Fig. 2b). It implies the different initial material distributions lead to different final material distributions.
Figure 3. Iteration histories of the structural volume ratio with different initial designs
In case (d), the Young’s modulus of the isotropic material in subdomains of No:1 and No:2 is 210GPa, which is greater than that of the material in the subdomains of No:3 and No:4. This kind of initial material distribution leads to an unsymmetrical topology of structure (See Fig. 2d). It also implies that the initial material distribution has an effect on the final structural topology. Fig. 3 shows the iteration histories of the structural volume ratios (the ratio between the volume of the solid material in structure and the volume of the design domain)
248
References
for different cases of initial material distribution. After about 40 to 50 times of iteration, the ratios are convergent. The convergent values of the structural volume ratios are 20.0% for case (a), 21.8% for case (b), 19.4% for case (c) and 18.8% for case (d). From the results of cases (b) and (c), we can find that not only the final material distributions but also the amount of material in the final structures are different as the initial material distributing differently. The ratio 19.4% for case (c) less than 21.8% for case (b) is means the stiffer material distributing adjacent to the left side, where the freedom is fixed, leads to less amount of material in the final structure. The volume ratio of the final structure for case (d) is the smallest (18.8%) among the results of four cases. What we learn from the result is that unsymmetrical material distribution may have less amount of the material to give the same stiffness of the final structure under the same loading conditions.
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5. Conclusion
[7] J. Wolff, Das Gesetz der Transformation der Knochen, Hirschwald, Berlin, 1892, pp.110-157
The topology optimization of a continuum structure with many subdomains is investigated by a bionics method. Numerical results imply the different initial material distributions in subdomains lead to different final material distributions. In practical engineering, if the material in each subdomain is specified, an optimal practical result may be found by altering the positions of the subdomains.
[8] R. Huiskes, R. Ruimerman, G.H. van Lenthe, et al. “Effects of mechanical forces on maintenance and adaptation on form in trabecular bone”, Nature, 2000, Vol 405, pp.704-706 [9] P.K. Zysset and A. Curnier, “An alternative model for anisotropic elasticity based on fabric tensors”, Mech. Mater., Elservier, 1995, Vol 21, pp.243-250
Acknowledgements The Human Resources Foundation of Northwest A&F University (01140407) is greatly acknowledged.
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