Support-free robust topology optimization based on pseudo-inverse stiffness matrix and eigenvalue analysis

Abstract

Current finite element analysis (FEA) and optimizations require boundary conditions, i.e., constrained nodes. These nodes represent structural supports. However, many realistic structures do not have such concrete supports. In a robust optimization, i.e., optimization for uncertain load inputs, it is desirable to involve support uncertainty. However, such a robust optimization has not been available since constrained nodes are required to convert the stiffness matrix to an invertible matrix. This paper demonstrates a quite simple robust optimization based on a pseudo-inverse stiffness matrix and eigenvalue analysis that successfully creates optimal design without constrained nodes. The optimization strategy is to minimize the largest eigenvalue of the pseudo-inverse matrix. It was found that optimization for multiple eigenvalues, i.e., multiple load inputs, is required as the nature of the minimax problem. The created structures are capable of carrying multiple load inputs—bending, torsion, and more complex loads. Configurations created in rectangular design domains exhibited hollow monocoque structures.

Appendix A: An example of a continuum elastic body

Consider a one-dimensional elastic body. The equilibrium equation and the formula of strain tensors in the body are as follows:

$$ \frac{d \sigma(x)}{d x} = f(x) $$

(29)

and

$$ \varepsilon = \frac{d u(x)}{d x} $$

(30)

where f(x) is the force per volume.

In the same way as the finite element model, (29) and (30) can be reduced by Hook’s law as follows:

$$ E \frac{d^{2} u(x)}{d x^{2}} = f(x). $$

(31)

The solution of this equation is as follows:

$$ u(x) = \iint \frac{f(x)}{E} dx^{2} + C_{1} x + C_{2} $$

(32)

where C₁ and C₂ are the constants of integration, and C₁ corresponds to an effect of uniform background pressure which should be neglected. The constant C₂ determines the displacement at certain position, and denotes a rigid body motion.

When the problem is support-free and the mean value of the input force is not zero, the natural boundary condition (zero stresses at the free ends of the body) cannot be satisfied.

The force input can be regarded as the sum of the balanced component f_b and the unbalanced component f_R, i.e., the mean value of f(x).

$$ f(x) = f_{b}(x) + f_{R}. $$

(33)

Substituting (33)–(31), the following relationship can be obtained.

$$ E \frac{d^{2} u(x)}{d x^{2}} - f(x) = -f_{R}. $$

(34)

This relationship seems equivalent with (8).

To cope with the difficulty of an unbalanced load, an additional weak constraint k is introduced as follows:

$$ E \frac{d^{2} u(x)}{d x^{2}} - kE\frac{d u(x)}{d x} = f(x). $$

(35)

The exact solution of this equation is as follows:

$$ u(x) = \iint \frac{f_{b}(x)}{E} dx^{2} + C_{3} e^{qx} x + C_{4} e^{-qx}, $$

(36)

and

$$ q = k^{1/2}. $$

(37)

C₃ and C₄ are constants. The natural boundary conditions determine the constants. At the limit of small k, the constants are as follows:

$$ C_{3} = C_{4} = \frac{L f_{R}}{4k} $$

(38)

where L is the length of the body. Thus, the unbalanced component yields infinite rigid-body motion.