Filtering structures out of ground structures – a discrete filtering tool for structural design optimization

Abstract

This paper presents an efficient scheme to filter structures out of ground structures, which is implemented using a nested elastic formulation for compliance minimization. The approach uses physical variables and allows control of the minimum ratio between the minimum and maximum areas in the final topology. It leads to a singular problem which is solved using a Tikhonov regularization on the structural problem (rather than on the optimization problem). The filter allows a multiple choice solution in which the user can control the global equilibrium residual in the final structural topology and limit variations of the objective function between consecutive iterations (e.g., compliance). As a result, an unambiguous discrete solution is obtained where all the bars that belong to the topology have well-defined finite areas. This filter feature, with explicit control of member areas, allows the user (e.g., engineer or architect) to play with different alternatives prior to selecting a specific structural configuration. Examples are provided to illustrate the properties of the present approach and the fact that the technique does not always lead to a fully stressed design. The method is efficient in the sense that the finite element solution is computed on the filtered structure (reduced order model) rather than on the full ground structure.

Appendices

Appendix A – Nomenclature

α _i , η, M :

Optimality criteria parameters

α _f :

Filter size

α _Top :

Resolution of the structural topology

ϵ:

Small positive number

γ :

Move size parameter

ρ :

Global equilibrium tolerance

λ, λ _o :

Tikhonov regularization parameters

Ω :

Potential energy of external loads

Π :

Total potential energy

A _n :

Member n incidence kinematic matrix

E :

Elasticity modulus

F :

Force vector

F _Top :

Force vector in the structural topology

C :

Objective function

$\overline{\mathrm{C}}$ :

Normalized objective function

K :

Global stiffness matrix

K _Top :

Global stiffness matrix for the structural topology

k ⁽ⁱ⁾ :

Stiffness matrix for member i in local coordinate

ℓ :

Member length

L :

Vector of the member length

M :

Move limit (optimality criteria)

N :

Number of members

tol :

Tolerance in the OC

ObjTol :

Tolerance in the objective function between iterations

u :

Displacement vector

u _eq :

Displacement vector at the equilibrium configuration

u _n :

Member n displacement vector

u _Top :

Displacement vector in the structural topology

V :

Eigenvectors of K

v _i :

Eigenvector i of K

U :

Strain energy

V _max :

Maximum volume

ℛ:

Range

ℝ:

Set of real numbers

x ^min _j , x ^max _j :

Lower and upper bounds for member cross-sectional area

Appendix B – Global equilibrium error

In this Appendix, we compare the values of the compliance and the global equilibrium error obtained with the Moore-Penrose Inverse, least squares with Tikhonov regularization (λ _o  = 10^− 8), and the total potential energy with Tikhonov regularization (λ _o  = 10^− 8) for solution of ill conditioned systems. The source of ill conditioning here is the occurrence of significantly different cross-sectional areas in the resulting topology, as illustrated by Fig. 24.

Fig. 24

Ill conditioned stiffness matrix for ϵ < 1

Table 7 shows that when ϵ = 0.01, the values of compliance (C) and particular solution (u _p ) for the three approaches are similar. On the other hand, Table 8 shows that when ϵ = 0.001, the values of compliance and particular solution (u _p ) for the pseudo inverse and the potential energy with Tikhonov regularization are similar, however, such is not the case for the least squares solution, which also displays a large equilibrium residual.

Table 7 Nonsingular (ϵ = 0.01; k ₂ = 10kN/m; P = 10 kN)

Table 8 Nonsingular (ϵ = 0.001; k ₂ = 10kN/m; P = 10 kN)

Appendix C – Iterative scheme for solving singular systems of equations

In this Appendix, we present a practical iterative solution for the singular system defined by (10). The ideas presented here are based on Roonau and Parsons (1995). Adding the factor λ u in both sides of (10), we obtain

$$ \left(\boldsymbol{K}+\lambda\ \boldsymbol{I}\right)\ \boldsymbol{u}=\boldsymbol{F}+\lambda\ \boldsymbol{u} $$

(42)

where λ is a small number (e.g., λ _o  = 10^− 4 to 10^− 12 times the mean of the diagonal of K) that removes the singularity of the stiffness matrix K and provides faster convergence. We obtain the solution using an iterative algorithm in the form

$$ \left(\boldsymbol{K}+\lambda\ \boldsymbol{I}\right)\ {\boldsymbol{u}}^{\left(k+1\right)}=\boldsymbol{F}+\lambda\ {\boldsymbol{u}}^{(k)} $$

(43)

Now we obtain a particular solution of the singular system of equations which, from a numerical point of view, is independent of the parameter λ. The convergence is obtained using a tolerance on the displacement, e.g., |u ^(k + 1) − u ^(k)| ≤ 10^− 7 |u ^(k)|. To illustrate the method, Fig. 19b of Example 2 is solved with this approach and the results obtained are presented in Fig. 25 and Table 9. The new results illustrate convergence toward the same topology of Fig. 19b, Moreover, the equilibrium error is relatively small for all values of the parameter λ.

Fig. 25

Dependence of the topology on the Tikhonov parameter using the filter approach on a full ground structure (11 × 11 grid) with α _f  = 0.01 and γ = 2 for both the approach of Section 2.7 with λ _o  = 10^− 12 and the iterative approach with λ _o  = 10^− 4; 10^− 5; 10^− 6; 10^− 8

Table 9 Number of iterations, objective function, and equilibrium error for different values of λ _o

Appendix D – Filtering out mapping example

In this section we show a simple example demonstrating how the force vector and stiffness matrix corresponding to the topology can be mapped from those corresponding to the ground structure, based on the equations outlined in Section 2.3. Figure 26 shows the load and support conditions (a), the ground structure degrees of freedom (b), and the topology degrees of freedom (c).

Fig. 26

a Loading and support conditions; b Ground structure degrees of freedom (Ku = F ); c Topology degrees of freedom (K _Top u _Top  = F _Top ). Note: Dashed lines denote null stiffness

The force vector and the stiffness matrix for the ground structure are given by

$$ \boldsymbol{F}=P\left\{\begin{array}{c}\hfill 0\hfill \\ {}\hfill -1\hfill \\ {}\hfill 1\hfill \end{array}\right\} $$

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$$ \boldsymbol{K}=\left[-\begin{array}{ccc}\hfill {k}_3\hfill & \hfill -{k}_3\hfill & \hfill 0\hfill \\ {}\hfill {k}_3\hfill & \hfill {k}_2+{k}_3\hfill & \hfill -{k}_2\hfill \\ {}\hfill 0\hfill & \hfill -{k}_2\hfill & \hfill {k}_1+{k}_2\hfill \end{array}\right] $$

(45)

The transformation matrix from topology displacements to ground structure displacements (u = T u _Top ) is given by

$$ \boldsymbol{T}=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 1\hfill \\ {}\hfill 1\hfill & \hfill 0\hfill \end{array}\right] $$

(46)

Now we can obtain the force vector and stiffness matrix for the topology using

$$ {\boldsymbol{F}}_{Top}={\boldsymbol{T}}^T\boldsymbol{F}=P\left\{\begin{array}{c}\hfill 1\hfill \\ {}\hfill -1\hfill \end{array}\right\} $$

(47)

$$ {\boldsymbol{K}}_{Top}={\boldsymbol{T}}^T\boldsymbol{K}\boldsymbol{T}=\left[\begin{array}{ll}{k}_2+{k}_1\hfill & -{k}_2\hfill \\ {}-{k}_2\hfill & {k}_2+{k}_3\hfill \end{array}\right] $$

(48)

Considering the null stiffness coefficient k ₁ and k ₃, we obtain the singular stiffness matrix for the topology

$$ {\boldsymbol{K}}_{Top}=\left[\begin{array}{cc}\hfill {k}_2\hfill & \hfill -{k}_2\hfill \\ {}\hfill -{k}_2\hfill & \hfill {k}_2\hfill \end{array}\right] $$

(49)