Revisiting density-based topology optimization for fluid-structure-interaction problems

Abstract

This study revisits the application of density-based topology optimization to fluid-structure-interaction problems. The Navier-Cauchy and Navier-Stokes equations are discretized using the finite element method and solved in a unified formulation. The physical modeling is limited to two dimensions, steady state, the influence of the structural deformations on the fluid flow is assumed negligible, and the structural and fluid properties are assumed constant. The optimization is based on adjoint sensitivity analysis and a robust formulation ensuring length-scale control and 0/1 designs. It is shown, that non-physical free-floating islands of solid elements can be removed by combining different objective functions in a weighted multi-objective formulation. The framework is tested for low and moderate Reynolds numbers on problems similar to previous works in the literature and two new flow mechanism problems. The optimized designs are consistent with respect to benchmark examples and the coupling between the fluid flow, the elastic structure and the optimization problem is clearly captured and illustrated in the optimized designs. The study reveals new features of topology optimization of FSI problems and may provide guidance for future research within the field.

Appendix

1.1 A.1 Details on the derivation of (3)

The Navier-Cauchy equations are given by:

$$\begin{array}{@{}rcl@{}} \frac{\partial{\sigma^{s}_{ij}}}{\partial{x_{j}}} + f_{i} &=& 0 \quad \text{in} \quad {{\Omega}_{S}} \\ \sigma^{s}_{ij} &=& C_{ijkl} \epsilon_{kl} \\ \epsilon^{s}_{kl} &= &\frac{1}{2} \left( \frac{\partial{d_{k}}}{\partial{x_{l}}} + \frac{\partial{d_{l}}}{\partial{x_{k}}} \right) \end{array} $$

(24)

The coupling between the fluid and the structure is given by (Farhat and Roux 1991; Yoon 2014a)

$$ \sigma^{s}_{ij} n_{j} = \sigma^{f}_{ij} n_{j} \quad \text{on} \quad {{\Gamma}_{SF}} $$

(25)

The weak form of (24) is given by:

$$ {\int}_{{{\Omega}_{S}}} {w^{h}_{i}} \frac{\partial{\sigma_{ij}^{s}}}{\partial{x_{j}}} \hspace{2pt}\text{d} V + {\int}_{{{\Omega}_{S}}} {w^{h}_{i}} f_{i} \hspace{2pt}\text{d} V = 0 $$

(26)

where ${w^{h}_{i}}$ is a suitable basis function. Integration by parts of higher dimensions on the first term of (26), yields:

$$\begin{array}{@{}rcl@{}} {\int}_{{{\Gamma}_{SF}}} {w^{h}_{i}} \sigma^{s}_{ij} n_{j} \hspace{2pt}\text{d} S - {\int}_{{{\Omega}_{S}}} \frac{\partial{{w^{h}_{i}}}}{\partial{x_{j}}} \sigma_{ij}^{s} \hspace{2pt}\text{d} V \\ + {\int}_{{{\Omega}_{S}}} {w^{h}_{i}} f_{i} \hspace{2pt}\text{d} V = 0 \end{array} $$

(27)

Shear stresses on the interface between the fluid and the structure are neglected for which reason (25) can be written as $\sigma ^{s}_{ij} n_{j} = -p n_{i}$ on ΓSF, where p is the pressure on the interface surface. Equation (27) is now rewritten as:

$$\begin{array}{@{}rcl@{}} {\int}_{{{\Gamma}_{SF}}} {w^{h}_{i}} p^{h} n_{i} \hspace{2pt}\text{d} S - {\int}_{{{\Omega}_{S}}} \frac{\partial{{w^{h}_{i}}}}{\partial{x_{j}}} \sigma_{ij}^{s} \hspace{2pt}\text{d} V \\ + {\int}_{{{\Omega}_{S}}} {w^{h}_{i}} f_{i} \hspace{2pt}\text{d} V = 0 \end{array} $$

(28)

Integration by parts of higher dimensions on the first term of (28), yields:

$$\begin{array}{@{}rcl@{}} {\int}_{{{\Omega}_{S}}} \frac{\partial{{w^{h}_{i}}}}{\partial{x_{i}}} p^{h} \hspace{2pt}\text{d} V + {\int}_{{{\Omega}_{S}}} {w^{h}_{i}} \frac{\partial{p^{h}}}{\partial{x_{i}}} \hspace{2pt}\text{d} V \\ - {\int}_{{{\Omega}_{S}}} \frac{\partial{{w^{h}_{i}}}}{\partial{x_{j}}} \sigma_{ij}^{s} \hspace{2pt}\text{d} V + {\int}_{{{\Omega}_{S}}} {w^{h}_{i}} f_{i} \hspace{2pt}\text{d} V = 0 \end{array} $$

(29)

Equation (29) may now be rewritten from the segregated domains Ω_S to a unified domain Ω, by introducing a design variable field 0 ≤ ρ ≤ 1 and the following interpolation function:

$$ C_{ijkl} = E(\rho) C^{0}_{ijkl} $$

(30)

Correct integration of the fluid pressure on the elastic structure is ensured by introducing the filter function Ψ(ρ):

$$ {\int}_{{\Omega}_{S}} \square \hspace{2pt}\text{d} V = {\int}_{{\Omega}} {\Psi}(\rho) \square \hspace{2pt}\text{d} V $$

(31)

Ψ(ρ) is a function which is unity for ρ = 1 and zero for ρ = 0. Inserting (30) and (31) into (29), we arrive at the following expression for the Navier-Cauchy equation defined in a unified domain Ω:

$$\begin{array}{@{}rcl@{}} {\int}_{{{\Omega}}} {\Psi}(\rho) \left( \frac{\partial{{w^{h}_{i}}}}{\partial{x_{i}}} p^{h} + {w^{h}_{i}} \frac{\partial{p^{h}}}{\partial{x_{i}}} \right) \hspace{2pt}\text{d} V \\ + {\int}_{{{\Omega}}} {w^{h}_{i}} f_{i} \hspace{2pt}\text{d} V = {\int}_{{{\Omega}}} E(\rho) \frac{\partial{{w^{h}_{i}}}}{\partial{x_{j}}} \sigma_{ij}^{s_{0}} \hspace{2pt}\text{d} V \end{array} $$

(32)

1.2 A.2 Benchmark examples

To demonstrate the features of the present framework, we have revisited the wall flow design problems presented in the works of Yoon (2010) and Picelli et al. (2017) and Jenkins and Maute (2016). These, what we call, benchmark designs problems are solved with the same physical parameters as in the respective papers but with our framework. The design solution, obtained with the framework presented in this study, to the design problem presented in Jenkins and Maute (2016) has been plotted in Fig. 23. The design solution shown in Jenkins and Maute (2016) and our design solution in Fig. 23 are by visual comparison quite similar. The small difference in the design solutions suggests that the internal pressure, the displacement dependency and /or the shear stress may have minor effects for this specific optimization problem. However, more studies and other optimization problems are required to fully understand the influence of the different modeling approaches and assumptions.

Fig. 23

Design solution for the wall flow problem in (Jenkins and Maute 2016) with Re = 10

In Figs. 24 and 25, the Yoon (2010) and Picelli et al. (2017) benchmark design problems, solved by our framework, have been plotted for Re = 0.004 and Re = 12. The designs have been optimized for the full pressure coupling formulation in (3). It is unclear whether the design solutions in Yoon (2010) are solved for pressure-coupling term 1 or pressure-coupling term 1 and 2. In Yoon (2010), it is seen that an increased Reynolds number causes the wall support to move to the downstream part of the design domain. This tendency is conflicting with the tendencies observed in Picelli et al. (2017) and in Figs. 24–25. In these design problems, it is observed that an increased Reynolds number causes the wall support to move to the upstream part of the design domain. As far as we are aware there does not exist any crosschecks between the Reynolds number and the optimized designs in the mentioned papers, for which reason it is challenging to assess how much significance we can attribute to the features of the optimized designs in Yoon (2010) and Picelli et al. (2017). However, please notice that our optimized designs in Figs. 24–25 pass a crosscheck.

Fig. 24

Design solutions for the wall flow problem stated in (Yoon 2010; Picelli et al. 2017) with Re = 0.004

Fig. 25

Design solutions for the wall flow problem in Yoon (2010) and Picelli et al. (2017) with Re = 12

1.3 A.3 Details on the determination of interpolation functions

TO for FSI problems are highly non-linear, ill-posed and non-convex. Several model parameters influences the design processes and the design solutions, which require a significant amount of parameter tuning due to the deepness of the design space. Numerical experiments with the framework presented in this work, have suggested that the design process is highly dependent on the choice of interpolation functions, α(ρ),E(ρ) and Ψ(ρ) (abbreviated: H_{{α, E,Ψ}}), and the choice of interpolation function parameters, p_α, p_E and p_Ψ (abbreviated: p_{{α, E,Ψ}}). It is our experience that adequate choices of p_{{α, E,Ψ}} and H_{{α, E,Ψ}} are critical to obtain well-performing and 0/1 design solutions. As p_{{α, E,Ψ}} and H_{{α, E,Ψ}} are key to carry out successful optimization problems, we will in this section present a methodology which can be used to determine adequate p_{{α, E,Ψ}} and H_{{α, E,Ψ}} and hereby formulate well-posed optimization problems.

IMwEBR are based on shape sensitivities which may be better suited for TO for some multi-physics problems. IMwEBR have a well-defined boundary between the different types of physics, which ensures that the physics are resolved correctly throughout the entire design process, as no sub-domains are dependent on the quality of the interpolation functions of the intermediate design variables. In density-based methods the correctness of the physical modeling rely, among many other aspects, on the choice of p_{{α, E,Ψ}} and H_{{α, E,Ψ}}. The hypothesis is that topology sensitivities, ∂f/∂ρ_d, may obtain the same well-behaving features as shape sensitivities, ∂f/∂ρ_s, if some adequateH_{{α, E,Ψ}} and p_{{α, E,Ψ}} are chosen.

To determine adequate sets of H_{{α, E,Ψ}} and p_{{α, E,Ψ}}, we compare ∂f/∂ρ_s and ∂f/∂ρ_d for two different problems: (1) a purely elastic problem and (2) an FSI problem. The problem layouts and the boundary conditions have been sketched in Fig. 26. To carry out the study, we compare four different objective functions, f_E, f_C, f_T and f_P (abbreviated: f_{{E, C, T, P}}):

• 1. Dissipated energy in the flow, f_E, see (17).

• 2. Structural compliance f_C, see (16)

• 3. The y-displacement of the tip of the beam in point {x, y} = {2, 0.45}:

  $$ f_{T} = \frac{1}{{\int}_{{{\Omega}_{T}}} \hspace{2pt}\text{d} V} {\int}_{{{\Omega}_{T}}} d_{y} \hspace{2pt}\text{d} V $$

  (33)

• 4. The pressure induced y-directional force on the beam:

  $$ f_{P} = {\int}_{{\Gamma}_{B}} p n_{j} \hspace{2pt}\text{d} S $$

  (34)

Fig. 26

Problem layouts and boundary conditions used to compare the topology sensitivities and the shape sensitivities

The relationship between f_{{E, C, T, P}}, ρ_S and ρ_D is determined with basis in a simple problem where a beam separates a channel into two regions of the same size. The problem is modeled in the unified framework. The beam separating the channel, Ω_I, has fixed unity design variables and the tip of the beam is loaded with a force f_y. With reference to Fig. 26, we consider two different problems: (a) An elastic problem where $u_{x}^{*}= 0$ and f_y = 10, and (b) an FSI problem where $u_{x}^{*}= 1$ and f_y = 0.

With reference to Fig. 26a, the topology sensitivities of various objective functions are computed by changing the design variables of the lowest line of elements of the vertical beam. With reference to Fig. 26b, the shape gradients of various objective functions are computed by changing the position of the nodes on the lower boundary of the beam. The position of the boundary is varied over the length of one element, entailing that ∂f/∂ρ_d and ∂f/∂ρ_s are comparable in material usage.

The relationship between f_{{C, P}}, f_{{C, P}}, ρ_d and ρ_s for the elastic problem have been compared in Fig. 27. The relationships are different but can be characterized by the following attribute: f_{{C, T}}(ρ_s) and f_{{C, T}}(ρ_d) are strictly monotonic entailing that ∂f_{{C, T}}/∂ρ_s and ∂f_{{C, T}}/∂ρ_d are strictly monotonic. Well-versed and crisp 0/1 designs and smooth optimization processes are obtained for a large number of TO for elastic problems, see e.g. (Bendsøe and Sigmund 2003). The hypothesis in this study is, that the well-posed properties of linear elastic problems are explained by the strictly monotonic features of the ∂f_C/∂ρ_d.

Fig. 27

The relationship between f_C, f_P and ρ_d/ρ_s for various choices of H_{{α, E,Ψ}} and p_{{α, E,Ψ}}

We now point out attention to the FSI problem, where we investigate the monotonicity of f_{{E, C, T, P}}(ρ_s) and f_{{E, C, T, P}}(ρ_d). The relationships have been plotted in Fig. 28, and are characterized by:

• 1. f_{{E, C, T, P}}(ρ_s) are strictly monotonic in all cases, which entail that ∂f_{{E, C, T, P}}/∂ρ_s are strictly monotonic in all cases.

• 2. f_{{E, C, T, P}}(ρ_d) are strictly monotonic for some choices of p_{{α, E,Ψ}} and H_{{α, E,Ψ}}.

• 3. The relationship between f_{{E, C, T, P}}(ρ_d) seem to be very sensitive with respect to the choice of p_α, as a small change in p_α may disrupts the monotonicity for all f_{{E, C, T, P}}.

Fig. 28

The relationship between f_E, f_C, f_P, f_T and ρ_d/ρ_s for various choices of H_{{α, E,Ψ}} and p_{{α, E,Ψ}}

To demonstrate the importance of the monotonicity of f_{{E, C, T, P}}(ρ) we have included a numerical example where we compare two design optimized for two different p_α. In Fig. 29, f_D have been optimized for the flow obstacle problem (see Section 6.2) for p_α = 0.5 ⋅ 10⁻⁶ and p_α = 10⁻⁶. The design optimized with a strictly monotonic f_D perform much better than the design optimized for non-monotonic f_{D}. We notice that a small change in p_α significantly influences the topology and the performance of the design solutions.

Fig. 29

Optimized designs for the flow obstacle problem in Fig. 12 for two different interpolation function parameters

We suggest that the correlation between the monotonicity of the objective functions and the well-posedness of the design problems is likely to generalize to all multiphysics topology optimization problems.

1.4 A.4 Details on the Brinkman penalization parameter

Numerical experiments with the framework suggested that α_max should be chosen high (e.g. α_max = 10⁹) to model the pressure field correctly. A low α_max, e.g. α_max = 10⁵, provides a more well-posed optimization problem, however the pressure field is not modeled correctly. Modeling the pressure field incorrectly may provide unintuitive and physically meaningless optimized designs, as the coupling from the fluid to the structure is transfered through the pressure field. To demonstrate the relationship between the pressure field and the magnitude of α_max, we have plotted the pressure field along the line {x, y} = {x, 0.34} for the design shown in Fig. 30 in Fig. 31. The pressure fields for the COMSOL composite model and the unified model with α_max = 10⁹ is closely correlated. However, for small α_max, a large difference between the composite model and the unified model is observed. As a final remark, the minor difference between the fields is caused by different finite element discretizations of the segregated and unified models.

Fig. 30

Design used to evaluate the pressure

Fig. 31

The pressure as function of x for various α_max

To demonstrate the occurrence of unintuitive optimized topologies for design problems with too low α_max, we consider two different design problems. The design in Fig. 32a has been optimized for α_max = 10⁵ and the design in Fig. 32b has been optimized for α_max = 10⁹. The designs provide superior performance for the model parameters under which the designs were optimized. However, the design optimizd for α_max = 10⁵ does not performing well in a segregated FSI formulation. The too low α_max causes poor resolvement of the pressure field which causes unintuitive optimized designs. For comparison we have plotted the design optimized for α_max = 10⁹ in Fig. 32b. This design performs well in a segregated model.

Fig. 32

Design solutions for the flow obstacle problem in Fig. 12 for various α_max

The coupling between the structure and the fluid is carried out through the pressure field, for which reason adequate modeling of the pressure field is crucial in FSI problems. Previous work on topology optimization for fluid problems has used magnitudes of α_max which do not resolve the pressure field correctly. Non-intuitive designs may not have been observed in these studies because the pressure fields were not directly related to the objective functions or the constraints of the optimization problems.