Exact global optimization of frame structures for additive manufacturing

Abstract

We consider the problem of designing lightweight load-bearing frame structures with additive manufacturability constraints. Specifically, we focus on mathematical programming approaches to finding exact globally optimal solutions, given a pre-specified discrete ground structure and continuous design element dimensions. We take advantage of stiffness matrix decomposition techniques and expand on some of the existing modeling approaches, including exact mixed-integer nonlinear programming and its mixed-integer linear programming restrictions. We propose a (non-convex) quadratic formulation using semi-continuous variables, motivated by recent progress in state-of-the-art quadratic solvers, and demonstrate how some additive-specific restrictions can be incorporated into mathematical optimization. While we show with numerical experiments that the proposed methods significantly reduce the required solution time for finding global optima compared to other formulations, we also observe that even with these new techniques and advanced computational resources, discrete modeling of frame structures remains a tremendously challenging problem.

Appendix

Using the relations between displacement and external nodal loads from matrix analysis of the frame structures, we can define the stiffness matrix of each beam element in its local coordinate system as follow:

(A.1)

$\mathbf {k}_e \in \mathrm{I\!R}^{6\times 6}$ is a symmetric positive definite matrix in which, $a_e$ and $I_e$ represents the two cross-sectional properties, namely area and moment of inertia, of the element e (Kassimali (2012)). In order to reduce the number of decision variables in our mathematical models, we can relate the moment of inertia for circular beams to their cross-sectional area as follows ($r_e$ is the radius of element e):

$$\begin{aligned} I_e = \frac{\pi r_e^4}{4} = \frac{a_e^2}{4\pi } \end{aligned}$$

(A.2)

Equilibrium state of the structure, where the internal forces and external nodal loads are balancing each other, can be illustrated in the model by using equilibrium equations:

$$\begin{aligned} \mathbf {K}(\mathbf {a},\mathbf {I})\mathbf {u} = \mathbf {P} \end{aligned}$$

(A.3)

Equation (A.3) represents d similar constraints in the mathematical model. In (A.3), $\mathbf {K} \in \mathrm{I\!R}^{d \times d}$ is the global stiffness matrix of the structure. $\mathbf {u}$ and $\mathbf {P}$ are $d\times 1$ vectors representing respectively, displacements and the external nodal loads on all degrees of freedom. Although we will explain it later in detail, it should be noted here that $\mathbf {K}$ is constructed using the members’ stiffness matrices in a global coordinate system. Based on the boundary conditions, for some of the degrees of freedom j, $u_j=0$ and for some degrees of freedom j, $p_j$ is a variable representing reaction forces on the fixed nodes. Based on the mentioned boundary condition effects, we express each equation of (A.3) in the form of the constraints in mathematical models as follow:

$$\begin{aligned} \sum _{j=1}^d \mathbf {K}_{ij}u_j = p_i \qquad \forall i \in \mathbb {S}_{\text {dof}} \end{aligned}$$

(A.4)

We assume that external loads exist only on the end-nodes of the elements not along them. Loads also assumed to be concentrated dead loads which means they are deterministic,given, and do not change during the analysis. $\mathbf {k}_e$ in (A.1), and as a consequence, $\mathbf {K}$ in (A.3), include $a_e$ and $I_e$ which are decision variables in the model. In finite element analysis, to avoid matrix inversion in solving the system of equations in (A.3), usually different decomposition methods such as LU decomposition, Cholesky decomposition, or Singular Value Decomposition are used. Here, following the approach that (Stolpe and Svanberg 2003; Stolpe 2004; Rasmussen and Stolpe 2008; Kanno 2016) suggested, we decompose the stiffness matrices in (A.1). We define the following vectors together with the eigenvalues:

(A.5)

$$\begin{aligned}&k_{e1} = \frac{a_e E}{l_e}, \quad k_{e2} = \frac{3I_e E}{l_e}, \;k_{e3} = \frac{I_e E}{l_e} \end{aligned}$$

(A.6)

The member stiffness matrix in local coordinates can be obtained using the following matrix multiplication:

$$\begin{aligned} \mathbf {k}_e = \sum _{z=1}^{3} k_{ez}\varvec{\hat{b}_{ez}}\big (\varvec{\hat{b}_{ez}^\top }\big ) \end{aligned}$$

(A.7)

where $\varvec{\hat{b}_{ez}}(\varvec{\hat{b}_{ez}^\top })$ represents the outer product of the two vectors $\varvec{\hat{b}_{ez}} \in \mathrm{I\!R}^{6\times 1}$ and $\varvec{\hat{b}_{ez}^\top } \in \mathrm{I\!R}^{1\times 6}$, which results in a matrix in $\mathrm{I\!R}^{6\times 6}$. Using Eq. (A.2) we can replace $I_e$ with $a_e^2/4\pi$ and rewrite the decomposition in (A.7) as:

$$\begin{aligned} \mathbf {k}_e = \frac{a_e E}{l_e}\varvec{\hat{b}_{e1}}\big (\varvec{\hat{b}_{e1}^\top }\big ) + \frac{3a_e^2 E}{4\pi l_e}\varvec{\hat{b}_{e2}}\big (\varvec{\hat{b}_{e2}^\top }\big ) + \frac{a_e^2 E}{4\pi l_e}\varvec{\hat{b}_{e3}}\big (\varvec{\hat{b}_{e3}^\top }\big ) \end{aligned}$$

(A.8)

To be used in the construction of the structure’s stiffness matrix, vectors in (A.5) need to be transformed to global coordinates. Considering the geometry of each beam element, we define the following transformation matrices:

In which $c=cos (\theta _e)$, $s=sin (\theta _e)$. To assemble the global stiffness matrix of the structure, we define the element specified $\mathbf {T}_e \in \mathrm{I\!R}^{6\times d}$ matrices. Each $\mathbf {T}_e$ is a $6\times d$ binary location matrix that relates the element’s degrees of freedom to the degrees of freedom in the structure. Consider element e which connects node 2 (degrees of freedom: $\{4,5,6\}$) and node 4 (degrees of freedom: $\{10,11,12\}$), the degrees of freedom of element e are: $\{4,5,6,10,11,12\}$ and the $\mathbf {T}_e$ matrix for this element can be defined as:

The following transformation of the $\varvec{\hat{b_{e}}}$ matrices are used in the generating of equilibrium equations:

$$\begin{aligned} \mathbf {b_{e}} = \mathbf {T}_e^\top \times \mathbf {Trans}_e^\top \times \varvec{\hat{b}_{e}} \end{aligned}$$

(A.9)

for which as mentioned before, $\mathbf {T}_e^\top \in \mathrm{I\!R}^{d\times 6}$, $\mathbf {Trans}_e^\top \in \mathrm{I\!R}^{6\times 6}$, and $\varvec{\hat{b_e}} \in \mathrm{I\!R}^{6\times 1}$, therefore $\mathbf {b_e} \in \mathrm{I\!R}^{d\times 1}$. Using these equations and values in (3.2), equation (A.4) can be written as (notice that $\mathbf {b_{e1}}\wedge \mathbf {b_{e1}^\top }$ and $\mathbf {b_{e2}}\wedge \mathbf {b_{e2}^\top }$ and $\mathbf {b_{e3}}\wedge \mathbf {b_{e3}^\top } \in \mathrm{I\!R}^{d\times d}$ ):

$$\left( \sum _{e \in \mathbb {S}_e} k_{e1}\mathbf {b_{e1}}(\mathbf {b_{e1}^\top )} + k_{e2}\mathbf {b_{e2}}(\mathbf {b_{e2}^\top )} + k_{e3}\mathbf {b_{e3}}(\mathbf {b_{e3}^\top })\right) _{i} \mathbf {u} = p_i \,\, \forall i \in \mathbb {S}_{\text {dof}}$$

(A.10)

In the above equation, the summation is the global stiffness matrix of the structure that was generated using the relations (A.5) through (A.9).