Topology optimization for design-dependent hydrostatic pressure loading via the level-set method

Consider a hydrostatic fluid domain Ω_f illustrated in Fig. 1 governed by Laplace’s equation

where P_f is the fluid pressure and ∇² is the Laplacian differential operator. Boundary conditions include an imposed pressure P₀ on a portion Γ_p of the fluid boundary as

and the hard wall condition

where n is the outward unit normal vector to the fluid and ∇ is the gradient vector operator, imposed on the portion Γ_w of the total fluid boundary Γ_f = Γ_f ∪Γ_w. The discretization of (1) via the finite element method (FEM) yields

where P_f is the vector of pressures and f_f is the equivalent load to enforce the Dirichlet boundary condition from (2) computed, for instance, using a penalty method (Bathe 2006). The global fluid stiffness matrix K_f is computed as

where nef is the total number of fluid elements and N_f is the vector with shape functions of the fluid element Ω_fe. The solution of (4) provides the pressure field used in the following structural analysis.

The considered structural domain Ω_s shown in Fig. 1 bounded by Γ_s = Γ_d ∪Γ_n ∪Γ_h is governed by the linear elasticity equation expressed, disregarding body loads, as

where σ_s (u_s) is the Cauchy stress tensor and u_s is the structural displacement field. The right-hand side of (6) includes the loads f_s applied on the structure. Dirichlet boundary conditions are applied on the portion Γ_d of the structural boundary as

where u₀ is the vector of constrained displacements. Neumann boundary conditions are applied by assembling f_p on the boundary portion Γ_n of the structure. The boundary Γ_h is the portion with no boundary conditions and only the subset Γ_n ∪Γ_h is subjected to optimization. A void domain Ω_v represents holes inside the design region. In the FEM discretization, (6) becomes

where \(\tilde {\mathbf {u}}_{s}\) is the vector of structural displacements and K_s is the structural stiffness matrix computed as

with B being the strain-displacement matrix, D the constitutive matrix, nes the total number of finite structural elements in the mesh, and Ω_se indicates the element domain. In this work, the applied loads are only in the form of distributed pressure loads, represented by f_p. These governing equations are general and describe the fluid and structural analyses for 2D and 3D.

In the fixed grid level-set method, the structural boundaries intersect the elements and the pressure load f_p cannot be directly applied at those boundaries. As illustrated in 2D in Fig. 2, work equivalent nodal forces f_e can be computed by integrating along the cutting boundary the matrix product of the element shape functions N_s and the distributed pressure loading acting on the element as

with ns being the total number pressure load segments and

where L_k is the k th segment defined by the cutting boundary Γ_n and f_pk is the pressure loading acting on that element (Cook et al. 2002).

The integral from (11) can be computed via Gaussian quadrature; therefore, it can then be rewritten as

where the integral interval over the line L_k is changed to [− 1,1] and the Gaussian points are modified to reflect the element local coordinates, herein represented by g_mod. We have implemented the quadrature using one Gauss point, i.e., the midpoint of the line L. For this, the modified Gauss point coordinates are

where {x_g,y_g} and {x_c,y_c} are the global coordinates of the Gauss point and the element centroid, respectively. The terms a and b are, respectively, half of length and height of the element (see Fig. 11). Considering that the distributed pressure is expressed by

Equation (12) is rewritten as

where,

w = 2 being the weight of the Gaussian quadrature for one point and J the Jacobian given, in this case, as half of the length of L_k. This classic FE procedure is a much simpler alternative to XFEM and mixed models.

Fluid flooding as first called by Chen and Kikuchi (2001) is used to track the fluid-structure interfaces. The process consists in defining an initial fixed fluid region that is propagated to the adjacent neighbor elements or nodes until it finds a structural boundary. In the FEM framework, this can be done by creating a list of element types, verifying the neighboring elements of each fluid and applying the following update rules while looping through the elements:

These rules should be repeated until there is no more change in the status of each element type. Once this procedure is finished, the pressure segments can be identified as the ones located between fluids and completely solid elements and in cut elements with fluid status. The interface tracking procedure is illustrated in Fig. 3.

where x is any point inside the design region Ω_d that contains the structural domain Ω_s. The boundary of the structure can be changed implicitly through ϕ (x). This way of representing the structure allows its boundary to morph by solving the following Hamilton-Jacobi equation,

where V_n is the normal velocity and t is a pseudo time domain in which the level set evolves.

Equation (18) is discretized and solved numerically as

where i is a discrete point in the domain and |∇ϕ_i (t)| is computed using the Hamilton-Jacobi WENO (weighted essentially non-oscillatory) scheme (Osher and Fedkiw 2003). The update of (19) it is conveniently restricted to nodes within a narrow band close to the boundary. Periodically, the level-set function ϕ_i is reinitialized to a signed distance function in order to maintain the stability of the method. The reinitialization and velocity extension are done via the fast-marching method (Sethian 1996).

The velocity function required to change the implicit function ϕ_j in (19) can be obtained by solving the following linearized problem,

where f is the objective function, g^m is the m th constraint with constrained value \(\bar {g}^{m}\), \({V_{n}^{j}} {\Delta } t = {\Delta } \text {x}_{j}\), and d_cfl is the maximum distance the boundary point can move along the normal direction inside the design region, i.e., the Courant-Friedrichs-Lewy (CFL) stability condition.

The integrals in (20) can be estimated as,

and,

where l_j is the discrete length of the boundary around the boundary point j, S_f and S_g are vectors containing the product of boundary lengths and sensitivities, and V_n is the vector of normal velocities. With (21) and (22), V_n can be found in order to minimize the objective function, given any constraints. This provides a numerical implementation of the steepest-descent of the objective function for the set of V_n. However, we switch the problem to a variable time step that is optimized according to the values of the sensitivities as

where λ_f and \(\lambda _{g^{1}} + \lambda _{g^{2}} + ... + \lambda _{g^{m}}\) are the optimization variables to be determined. We then reformulate (20) as

where d_cfl includes the CFL condition and any other possible bounds for the movement of each boundary point, e.g., to restrict a point from moving out of the design domain. The term \({\Delta } \bar {g}\) is the allowed change of each constraint function at every iteration.

In summary, the optimization problem from (24) is solved for λ_f and \(\lambda _{g^{m}}\), therefore considerably reducing the number of optimization variables from the number of boundary points to the number of objective and constraint functions. For instance, when doing compliance minimization subject to volume constraints, the amount of optimization variables is only two, independently of the boundary size. This procedure enforces the velocity field to be a linear combination of the sensitivity fields, leading to a smooth propagation of the structural boundaries. This is also advantageous when handling a high number of constraints, as shown in Dunning et al. (2016). More details of this optimization formulation can be found in Picelli et al. (2018a) and Hedges et al. (2017).

Compliance minimization of structures under design-dependent pressure loads subject to a volume constraint is considered. This optimization problem, P₁, can be written as

where V_s is the volume fraction that the structure occupies regarding the design domain and \(\bar {V}\) is the maximum allowed volume fraction.

For the LSTO method, regularization techniques (e.g., length scale and perimeter control) have been often used to reduce the number of local minima and make the problem well-posed. We acknowledge that the use of perimeter control, for instance, mitigates the dependency on the mesh size and on the initial level-set function of the problem, i.e., the initial hole configuration. We, therefore, also formulate the following optimization problem, P₂, including a perimeter term in the objective. The needs and effects of the perimeter regularization term will further be discussed in Section 4.

where μ is a fixed positive parameter and χ_s is the perimeter of the structure given by

This work applies shape sensitivities to morph the structural boundaries towards an optimal solution. Shape sensitivities for a structural compliance function considering design-dependent pressure loads have been studied in Allaire et al. (2004) and Xia et al. (2015). Including the perimeter parameter, the shape sensitivity can be written as,

where p₀ is the pressure load, u is the displacement, Dε (u) is the stress tensor, ε (u) is the strain tensor, and κ is the curvature of the boundary.

Equation (28) represents the shape sensitivity in the continuum space for the objective in P₂. One advantage of employing shape sensitivities is that it allows the use of any analysis method (FEM or others) to compute the terms in sensitivity function, possibly allowing it to be directly associated with black-box commercial packages. In this case, (28) indicates that the sensitivity of a boundary point depends on the divergence of the pressure times displacement, stress times strain, and lastly, the curvature of the boundary. In this work, the FEM is employed to compute (28) as

In the fixed grid LSTO, the stiffness of the elements cut by the structural boundaries is weighted by the area fraction of those elements. This is popularly known as the Ersatz material interpolation. Another advantage of the present LSTO method is that a least-squares interpolation is used to recover the continuity of the structural response at the boundary points. This allows a fixed grid to be used with desired accuracy and showed to produce smooth topologies in our stress-based optimization work (Picelli et al. 2018a). This procedure is used here. First, the terms in (29) that depend on displacements are computed at the Gauss points of the elements, where the FEM gives high accuracy when the element is not distorted (Zienkiewicz and Taylor 2005). Each term is then interpolated via least squares at the boundary point x_j. For the free boundaries Γ_h, the terms depending on P_f in (28) and (29) are null. For P₁, in which no perimeter function is considered, the term μκ is zero. For P₂, this term is computed via the finite difference method. One can efficiently compute the perimeter sensitivities by the local difference in the length of each point segment with a small perturbation of each boundary point coordinate in the normal direction. The least squares and perimeter sensitivities can be computed using our open source code available at http://​m2do.​ucsd.​edu/​software/​.

Figure 4 provides the flowchart of the present optimization procedure. The list of steps is as follows:

The schematic piston design problem from Fig. 5 is considered for optimization. To the best of the authors’ knowledge, this problem was first explored by Bourdin and Chambolle (2003) and later used by several other works for a range of different methods (Sigmund and Clausen 2007; Lee and Martins 2012; Picelli et al. 2015b; Emmendoerfer et al. 2018). This schematic problem is explored here to verify the topology solution offered by our proposed method.

The left half of the model in Fig. 5 is solved here with 156 × 104 (156 × 100 for the structure) quadrilateral finite elements in a regular fixed mesh. The problem P₁ from (25) without a regularization is solved for final volume fraction \(\bar {V} = 30 \%\). Boundary conditions include the symmetry at the center line and an imposed fluid pressure P₀ = 1.0 per unit length at the top edge of the fluid domain. Figure 6 presents the mirrored solution of the piston design problem with the proposed LSTO method.

The topology solution from Fig. 6 agrees well with the existing solutions, e.g., by Lee and Martins (2012) and Wang et al. (2016) using the SIMP method and Picelli et al. (2015b) using the BESO method. They have the central holes converging to the bottom center support, convex shapes of the top structural members (resembling seashells), and the curved shapes of the structures touching the lateral walls. They mainly differ between each other in the amount of internal structural members. The piston design by Sigmund and Clausen (2007) is less similar to the seashell shapes and the solutions are dependent on the control of the fluid volume. Regarding other level-set methods, the solutions presented in the literature like in Emmendoerfer et al. (2018) and Xia et al. (2015) are noticeably different. The overall looks of the piston design from Emmendoerfer et al. (2018) and this work are similar, but the shapes and position of the central and top structural members in Emmendoerfer et al. (2018) are somehow altered. The solution obtained by Xia et al. (2015) is evidently less similar than Emmendoerfer et al. (2018) and this work, presenting different shapes and holes configuration. Bourdin and Chambolle (2003) applied a phase field to represent the fluid-solid-void of the piston design. Their solution is similar to the ones obtained by the other methods only when a perimeter penalization is high enough. When such penalization is lowered, a different set of central structural members are obtained. Figure 7 presents snapshots of our LSTO solution. The initial solution chosen is similar to the one presented by Xia et al. (2015).

The piston design problem demonstrated that the solution of the proposed LSTO method converges to the ones in the literature without regularization. This following example explores the perimeter integral in the objective function and compares problems P₁ and P₂ from (25) and (26), respectively. Herein, the model depicted in Fig. 8 is considered for optimization. A fixed grid with 104 × 104 quadrilateral finite elements is used, with the structural design domain of 100 × 100. Both problems P₁ and P₂ are solved for comparison, with the volume constraint \(\bar {V} = 30 \%\) and μ = 1.0. The fluid pressure P₀ = 1.0 per unit length is applied. The solutions are shown in Fig. 9.

Figure 9 a presents the solution without the perimeter regularization, and the solution has small holes in the arch structure. A close investigation of the strain energy and sensitivities revealed that their distributions are near uniform and the volume constraint is satisfied. This means that small holes have little effects in the compliance function. A similar effect was observed in Picelli et al. (2017a). In such cases, these holes remain in the final solution when solving P₁. Solving problem P₂ penalizes the perimeter and the small holes do not remain in the final solution as shown in Fig. 9 b. The difference in the compliance function values between these two solutions is 4.6%. Figure 9 a shows the convergence history of the structural compliance function when solving P₁. Sharp changes are observed due to the fact that several holes are merging and structural members are breaking during optimization. Also, due to the reinitialization of the level set, tiny holes are formed in the solution as it can be noticed in the converged structural topology. Figure 9 b, however, presents smooth convergence with less structural members breaking during optimization, as indicated by the snapshots of the solution (see Fig. 10).

The pressurized chamber design problem was solved in the pioneering works by Hammer and Olhoff (2000) and Chen and Kikuchi (2001). Besides these two, only Zhang et al. (2008) and Picelli et al. (2015b) solved this problem using the SIMP and the BESO methods, respectively. Therefore, this example is explored less in the literature and to the best of the authors’ knowledge, it has not been solved with the LSTO method.

Herein, the schematic problem from Fig. 11 a is solved with 120 × 76 quadrilateral elements. The initial fluid region is kept fixed during the optimization and used in the fluid flooding process. Two 40 × 6 parts of the structure are considered as non-design domain, as depicted in Fig. 11 a. An inlet pressure P_f = P_in is imposed at the right edge of the initial fluid domain and an outlet pressure P_out = 1.0 per unit length is imposed at the entire bottom edge of the pressurized chamber. The initial solution from Fig. 11 b is considered for optimization. Problem P₂ is solved with μ = 50.0. Figure 12 presents the final solutions for different values of P₀.

The chamber topologies from Fig. 12 resemble the ones obtained by the SIMP and the BESO methods (Zhang et al. 2008; Picelli et al. 2015b). One advantage of using the present LSTO technique is to obtain smooth and very well-defined structural boundaries with a relative small grid. For instance, Picelli et al. (2015b) obtained a crisp chamber topology visually similar to the present LSTO solution when using 57000 finite elements, while here we use 9120. Although there is not any other LSTO solution in the literature for this problem, we can point out that the pressure modeling via work equivalent integrals represents a quite simple alternative to other techniques, e.g., XFEM and mixed models. Figure 13 shows the snapshots of the chamber optimization iterations with P₀ = 10.0 per unit length, converged to the final compliance of 9.09× 10⁵ as seen in Fig. 14.

When deriving the compliance function, a term that accounts for the changes in the load due to a change in the design variable should appear in the sensitivity function when considering the design-dependency of the load. For a pressure loading problem, the change in the compliance due to a change in the pressure distribution is represented by the term 2div (p₀u) in (28) (Allaire et al. 2004).

For a full solid pressure chamber, Fig. 15 presents the magnitude of the sensitivity function values for all boundary points separated by its different terms when P_in = 1.0 and P_in = 10.0 per unit length. In this case, these values were computed with the sensitivity function discretized by FEM and expressed in (29). Figure 15 shows that the pressure term \( 2 \left (P_{f} \mathbf {B} \tilde {\mathbf {u}}_{s} \cdot \left \{1, 1, 0 \right \} + \tilde {\mathbf {u}}_{s} \cdot \nabla \mathbf {N}_{f} P_{f} \right )\) from (29) is generally much lower when compared with the stress × strain term \(\mathbf {D} \mathbf {B} \tilde {\mathbf {u}}_{s} \cdot \mathbf {B}\tilde {\mathbf {u}}_{s}\). Pressure P_f is only non-zero for boundary points located at the fluid-structure interface and when the pressure values are the same at all points, the gradient term ∇N_fP_f is null. This study shows that the sensitivity of the pressure change in this design optimization problem has a negligible effect. Figure 16 presents the final design solutions when evaluating the completed sensitivity function and when computing sensitivities without the pressure term.

The last example considers the beam support design inside a fluid flow channel. The works by Jenkins and Maute (2016), Yoon (2010), Picelli et al. (2017b), and Lundgaard et al. (2018) describe the interaction between the fluid flow governed by the incompressible steady-state Navier-Stokes equations with the linear elastic structures. As proposed by Zhao et al. (2018), an approximate linear model can be used to simplify the flow based on a Darcy potential flow model. Borrowing this idea, herein we rewrite (1) in the form of Poisson’s equation as

where v_f is the fluid velocity. The pressure boundary condition is applied as (2) and a velocity profile is imposed as

with v₀ being the velocity on a portion of the boundary Γ_f.

The design problem in Fig. 17 is considered for optimization. A distributed inlet velocity profile v₀ = 1.0 is applied at the left edge of the fluid domain. An outlet pressure P₀ = 0.0 is imposed at the right edge. Problem P₂ is solved with μ = 1.0. The 10 × 100 beam in the black region in Fig. 17 is specified as the non-design domain. The mesh of solid finite elements including the design and non-design domains is 60 × 100. For a compliance minimization problem with volume constraint \(\bar {V} \leq 30\%\), Fig. 18 presents the initial and the structural topology solution.

The final solution follows the tendency of obtaining curved structural members similarly to the previous examples of this work. This is expected for a hydrostatic fluid pressure field as arch-like structures are known to bear pressure loads well. The pressure drop in the channel is relatively smooth. If the Navier-Stokes equations were used, the pressure drop would be possibly higher specially for higher Reynolds number, which can significantly change the solution patterns. For future works, this methodology can be extended to different types of fluid-structure interaction.