Wind load modeling for topology optimization of continuum structures

Wind typically blows in many directions at varying speeds. Ideally multiple load cases need to be considered in the topology optimization process to account for the consequent variabilities in wind forces acting on a structure. However, in this paper only one load case is considered for simplicity; with uni-directional incoming wind blowing from left to right (see Fig. 1a). This assumption remains valid for any incoming wind directions. Wind loading is included in the formulation using a direct density based method that does not involve explicit construction of the loading surfaces. The wind load is defined at each node of the discretized domain by introducing a loading function g using the standard formula for drag forces where \(F_{Wind_{C}}\) is the wind load value at the node C, Q _∞ is the incoming wind dynamic pressure, C _D is the drag coefficient, and A is the area of the surface side facing the wind in YZ directions. Computation of the area A depends on the node position of C, i.e. whether the node is at a corner, a side, or the middle of the discretized domain. For example for a middle node, in the 2D case, A is equal to a times the thickness of the structure and in the 3D case it is equal to a ² where a is the distance between the nodes of a uniform mesh. The factor g is a peak function interpolation (Yin and Ananthasuresh 2001) given by where ρ _C is the density of the node C and ρ _N is the density of the neighboring node in the direction opposite to wind direction (ρ _N  = ρ _L when the incoming wind is from left to right), and σ is a standard deviation parameter.

Yin and Ananthasuresh (2001) proposed to use the peak function to include multiple materials in the design without increasing the design variables. By adjusting the parameters in the model, multiple peaks can be created, so that, the final solution can be made to posses pure materials by avoiding intermediate values. In this paper, the force \(F_{Wind_{C}}\) is applied at every node of the domain and the peak function g is used to steer its value towards to 0/1 (see Fig. 1b), which ensures that the wind loading is applied only at nodes where a large change in density occurs, indicating a structural surface, and that the loading is unidirectional. In other words, the load is applied only when the node C is in a solid region and the node N is in void. In the present formulation, we are interested in truss like structures. Thus, we make the assumption that no wind shadowing is present.

The sensitivity of the compliance with respect to the design variables, which are the node densities in this case, is obtained from The sensitivity of the displacement in the above equation can be obtained by differentiating the equilibrium equation with respect to the node densities and is expressed as where K is the global stiffness matrix and F _Wind is the wind load vector. Thus, the sensitivity of the compliance is given by As can be observed from (8), the sensitivity of the compliance is decomposed into two terms; the first term is due to the dependence of external forces on the design variables, and the second one is due to the dependence of the stiffness on the same variables. We denote these two terms by ψ and ϕ, respectively. Furthermore, external forces applied to a given structure can be decomposed into two parts, which are the static loads and the incoming wind forces. Dead loads are assumed to be independent of the structural configuration. Thus, the expression of ψ for each node C depends only on wind loading and simplifies to where the subscripts L and R refer to the nodal points to the left and to the right of node C, respectively, and the wind direction is assumed to be from left to right. Using the same notation, u _C and u _R are the horizontal displacement at node C and its right neighbor, respectively.

The optimization problem is solved by the Method of Moving Asymptotes (MMA) (Svanberg 1987). According to Svanberg (1987), the compliance is approximated as where k is the MMA iteration index, n is the total number of nodes of the discretized domain, \(p_{0j}^{(k)}\) and \(q_{0j}^{(k)}\) are determined by a first-order approximation of the first Kuhn–Tucker condition, and \(L_j^{(k)}\) and \(U_j^{(k)}\) are the lower and upper asymptotes, respectively.

It is observed that, in some cases, the optimization process tends to oscillate when the MMA solver tries to update the asymptotes \(L_j^{(k)}\) and \(U_j^{(k)}\). The convergence is stabilized by modifying the lower asymptote according to the continuous optimality criteria interpreted as local Kuhn–Tucker conditions (Abdalla and Gürdal 2002). The update of the upper asymptote is kept the same as suggested by Svanberg (1987). Using the continuous optimality criteria interpreted as local Kuhn–Tucker conditions the objective function is approximated as This approximation can be convexified by linearizing the stiffness coefficient \(\rho_j^p\) around the most recent design point to obtain where \(L_j=(1-\frac{1}{p})\rho_j^{(k-1)}\) is the modified formula for the lower asymptote.

With the modifications discussed above the MMA solver does not always converge monotonously and in some cases it diverges, especially for strong wind loadings. To improve its performance, the calculation of the coefficients \(p_{0j}^{(k)}\) and \(q_{0j}^{(k)}\) defined in Svanberg (1987) is modified by considering the sensitivity contributions from the ϕ and ψ terms independently, Moreover, it is also observed that the objective function does not always decrease after an MMA step and that the constraints are violated in some cases. In an earlier paper, Zillober (1993) demonstrated that, by adding a line search in the optimization process the behavior of the MMA method can be stabilized. Therefore, a line search is used in the present work to ensure satisfaction of the constraints and decrease in the objective value. Let the solution produced by MMA after the k ^th iteration be denoted by ρ ^*. A solution is sought in the form where α ∈ [0,1] is a parameter that needs to be determined such that the constraints be satisfied and the objective function be decreased. The value of α is found by solving a one-dimensional approximate optimization problem constructed from the original problem by using approximate cubic polynomial approximations for the objective and the constraints as functions of the independent variable α.

A 2D problem is first considered to investigate the topology optimization of the wind turbine support structure. The results of this problem were generated with a discretized grid of 15 × 141 nodes in the L and H directions, respectively. The final designs are represented in Fig. 4a and b for the cases with and without wind loading, respectively, while maintaining the hub and rotor weight P _WR , and the rotor thrust F _W for the both cases. It can be observed that the topology without wind is a leaning tower that supports only the dead loads. On the other hand, the topology subject to wind loads has a tie forming at the root. Moreover, it can be seen from Fig. 4a, b that the tower is a narrow cantilever as opposed to having a thick root. This is due to the magnitude difference between the two forces; P _WR is about 20 times larger than F _W . The compliances computed with dead and wind loads for both optimum structures are 0.0229 MNm for the topology without wind loading and 0.0139 MNm for the topology with wind loading, respectively. Thus, including wind loading in the design formulation leads to 40% increase in structural stiffness for the same material volume. If the force F _W were to be applied at the front end of the top surface, the tower would have been leaning more and possibly produce a better optimum.

Figure 5 shows the evolution of the wind turbine support topology. The algorithm converges to a black/white topology in about 884 analysis iterations. It can be observed that the topology starts from a grey column design and ends up with a black/white design with apparition of a tie forming at the root that supports the loads coming from the wind.

Figure 6a shows the distribution of wind loads for the design obtained considering wind loading. It can be observed that the wind loads are uniformly distributed throughout the design domain. In some cases, especially for strong wind (Zakhama et al. 2007), the wind loads may be not uniformly distributed in some parts of the domain. This is due mainly to the way the coefficients \(p_{0j}^{(k)}\) and \(q_{0j}^{(k)}\) of the compliance approximation are updated and the parameter ε _p of the explicit constraint is reduced. Further investigation is needed to determine an appropriate update scheme for these coefficients. Figure 6b shows the convergence history. The convergence is reasonably smooth with jumps in compliance corresponding to tightened tolerance on grey density. Although the introduction of the explicit constraint leads to an increased number of iterations to convergence, the obtained solution is physically more meaningful.

The off-shore wind turbine domain (see Fig. 3) is now discretized with 15 ×15 ×141 nodes. Figure 7 shows the different views of the optimal topology of wind turbine support without and with wind loads. To compare the compliances of the two solutions, the designs obtained with and without wind loads are postprocessed to force exact black/white topologies. The compliance without considering wind loads is 0.0239 MNm and the compliance considering wind loads is 0.0151 MNm, which corresponds to 37% difference between the two solutions.

The topologies obtained for the 3D case are seen to be very similar to the topologies obtained for the 2D case in XZ plane (see Figs. 4 and 7a, b). The topology of the wind turbine support considering wind loading is supported by two frames at the bottom of the structure. Figure 7d shows that, by including wind loading in the formulation, the topology allowed a larger hole facing the wind loads than the topology without wind. In the literature, most techniques for topology optimization with design dependent loads do not allow holes facing a surface loads.