Topology optimization of subsonic compressible flows

Abstract

This work proposes a new topology optimization formulation to subsonic compressible flows. Compressible flows can be seen in many aerodynamic problems and optimization techniques can be used to improve designs of wings, diffusers, and other applications. In this work, the compressible Navier-Stokes equations are used coupled with a density-based material model. The effects of the material model are shown for both the momentum and energy equations. Two objective functions are considered, the entropy variation in a control volume and the entropy generation. The algorithm is implemented by a finite element model to solve the state equations. The pyadjoint libraries are used to perform the automatic sensitivity derivation and an internal point optimizer is used to update the design variable. The algorithm is evaluated by performing the optimization of 2D domains and a 3D diffuser. The results show optimized designs with reduced entropy generation.

Appendices

Appendix A: Weak form derivation

In this appendix, the weak form derivation of the momentum and energy equations are presented.

1.1 A.1 Momentum equation

By using the “weak-form Galerkin” method, (1) is multiplied by a test function v:

$$ \begin{array}{@{}rcl@{}} {\int}_{{\varOmega}} \nabla \cdot (\rho \boldsymbol{u} \boldsymbol{u})\boldsymbol{v} &+& \nabla p \boldsymbol{v} - \nabla \cdot (\mu\nabla\boldsymbol{u}) \boldsymbol{v} - \nabla \cdot (\mu \nabla \boldsymbol{u}^{T}) \boldsymbol{v} \\&-& \lambda \nabla (\nabla \cdot \boldsymbol{u}) \boldsymbol{v} + \kappa(\alpha)\boldsymbol{u}\boldsymbol{v} d{\varOmega} = \boldsymbol{0} \end{array} $$

(24)

The integration by parts on the second term gives:

$$ + {\int}_{{\varOmega}} \nabla p \boldsymbol{v} d{\varOmega} = + {\int}_{{\varGamma}} p\boldsymbol{n} \cdot \boldsymbol{v} d{\varGamma} - {\int}_{{\varOmega}} p \nabla \cdot \boldsymbol{v} d{\varOmega} $$

(25)

The integration by parts on the third term:

$$ \begin{array}{@{}rcl@{}} - {\int}_{{\varOmega}} \nabla \cdot (\mu\nabla\boldsymbol{u}) \boldsymbol{v} &=& - {\int}_{{\varGamma}} (\mu\nabla\boldsymbol{u})\boldsymbol{n} \cdot \boldsymbol{v} d{\varGamma} \\&&+ {\int}_{{\varOmega}} (\mu\nabla\boldsymbol{u}) \cdot \nabla\boldsymbol{v} d{\varOmega} \end{array} $$

(26)

The integration by parts on the fourth term:

$$ \begin{array}{@{}rcl@{}} - {\int}_{{\varOmega}} \nabla \cdot (\mu\nabla\boldsymbol{u}^{T}) \boldsymbol{v} &=& - {\int}_{{\varGamma}} (\mu\nabla\boldsymbol{u}^{T})\boldsymbol{n} \cdot \boldsymbol{v} d{\varGamma} \\&&+ {\int}_{{\varOmega}} (\mu\nabla\boldsymbol{u}^{T}) \cdot \nabla\boldsymbol{v} d{\varOmega} \end{array} $$

(27)

The integration by parts on the fifth term:

$$ \begin{array}{@{}rcl@{}} - {\int}_{{\varOmega}} \lambda \nabla (\nabla \cdot \boldsymbol{u})\boldsymbol{v} &=&\! - {\int}_{{\varGamma}} \lambda (\nabla \cdot \boldsymbol{u}) \boldsymbol{n} \cdot \boldsymbol{v} d{\varGamma} \\&&+ {\int}_{{\varOmega}} \lambda (\nabla \cdot \boldsymbol{u}) (\nabla\cdot\boldsymbol{v}) d{\varOmega} \end{array} $$

(28)

Finally, by substituting all the integration by parts in the original equation the following weak form is obtained:

$$ \begin{array}{@{}rcl@{}} &&+ {\int}_{{\varOmega}} \nabla \cdot (\rho \boldsymbol{u} \boldsymbol{u})\boldsymbol{v} - p \nabla \cdot \boldsymbol{v} + (\mu\nabla\boldsymbol{u}) \cdot \nabla\boldsymbol{v} + (\mu\nabla\boldsymbol{u}^{T}) \cdot \nabla\boldsymbol{v} \\&&+ \lambda (\nabla \cdot \boldsymbol{u}) (\nabla\cdot\boldsymbol{v}) + \kappa(\alpha)\boldsymbol{u}\boldsymbol{v} d{\varOmega} \\&&+ {\int}_{{\varGamma}} p\boldsymbol{n} \cdot \boldsymbol{v} - (\mu\nabla\boldsymbol{u}) \boldsymbol{n} \cdot \boldsymbol{v} - (\mu\nabla\boldsymbol{u}^{T}) \boldsymbol{n} \cdot \boldsymbol{v} \\&&- \lambda (\nabla \cdot \boldsymbol{u}) \boldsymbol{n} \cdot \boldsymbol{v} d{\varGamma} = \boldsymbol{0} \end{array} $$

(29)

1.2 A.2 Energy equation

By using the Galerkin method, (4) is multiplied by a test function v_h:

$$ \begin{array}{@{}rcl@{}} {\int}_{{\varOmega}} &&\nabla \cdot \left[\rho \left( h + \frac{1}{2}\boldsymbol{u} \cdot \boldsymbol{u} \right) \boldsymbol{u} \right] {v_{h}} d{\varOmega} \\&&- {\int}_{{\varOmega}} \nabla \cdot \left( \frac{k}{C_{p}} \nabla h\right) v_{h} d{\varOmega} \\&&-{\int}_{{\varOmega}} \nabla \cdot (\boldsymbol{\tau}_{h} \cdot \boldsymbol{u}) v_{h} d{\varOmega} = \boldsymbol{0} \end{array} $$

(30)

Then, integrating by parts on the second term:

$$ \begin{array}{@{}rcl@{}} -{\int}_{{\varOmega}} \nabla \cdot \left( \frac{k}{C_{p}} \nabla h\right) v_{h} d{\varOmega} &=& -{\int}_{{\varGamma}} \left( \frac{k}{C_{p}} \nabla h \right) \cdot \boldsymbol{n} v_{h} d{\varGamma} \\&&+ {\int}_{{\varOmega}} \left( \frac{k}{C_{p}} \nabla h \right) \cdot \nabla v_{h} d{\varOmega}\\ \end{array} $$

(31)

The integration by parts on the third term:

$$ \begin{array}{@{}rcl@{}} -{\int}_{{\varOmega}} \nabla \cdot \left( \boldsymbol{\tau}_{h} \cdot \boldsymbol{u}\right) v_{h} d{\varOmega} &=& -{\int}_{{\varGamma}} \left( \boldsymbol{\tau}_{h} \cdot \boldsymbol{u} \right) \cdot \boldsymbol{n} v_{h} d{\varGamma} \\&&+ {\int}_{{\varOmega}} \left( \boldsymbol{\tau}_{h} \cdot \boldsymbol{u} \right) \cdot \nabla v_{h} d{\varOmega} \end{array} $$

(32)

Finally, by substituting all the integration by parts in the original equation, the following weak form is obtained:

$$ \begin{array}{@{}rcl@{}} &&{\int}_{{\varOmega}} \nabla \cdot \left[\rho \left( h + \frac{1}{2}\boldsymbol{u} \cdot \boldsymbol{u} \right) \boldsymbol{u} \right] {v_{h}} \\&&+ \left( \frac{k}{C_{p}} \nabla h \right) \cdot \nabla v_{h}+ \left( \boldsymbol{\tau}_{h} \cdot \boldsymbol{u} \right) \cdot \nabla v_{h} d{\varOmega} \\ &&-{\int}_{{\varGamma}} \left( \frac{k}{C_{p}} \nabla h \right) \cdot \boldsymbol{n} v_{h} - \left( \boldsymbol{\tau}_{h} \cdot \boldsymbol{u} \right) \cdot \boldsymbol{n} v_{h} d{\varGamma} = \boldsymbol{0} \end{array} $$

(33)

Appendix B: Finite element matrix derivation

This section shows the matrix assemble by using the finite element method to solve the state equations. This derivation follows the steps shown in Reddy and Gartling (2010). Suppose that the relation between approximated continuous distributions (u_i,p, h) and the nodal values (u_i,P, H) is given by:

$$ \begin{array}{@{}rcl@{}} u_{i}(\mathbf{x}) &\approx&\sum\limits_{n=1}^{N} \psi_{n}(\mathbf{x}) {u_{i}^{n}} =\mathbf{\Psi}^{T}\mathbf{u}_{i} \end{array} $$

(34)

$$ \begin{array}{@{}rcl@{}} p(\mathbf{x}) &\approx& \sum\limits_{l=1}^{L} \xi_{l}(\mathbf{x}) p_{l} = \mathbf{\Xi}^{T}\mathbf{P} \end{array} $$

(35)

$$ \begin{array}{@{}rcl@{}} h(\mathbf{x}) &\approx& \sum\limits_{m=1}^{M} \theta_{m}(\mathbf{x}) h_{m} = \mathbf{\Theta}^{T}\mathbf{H} \end{array} $$

(36)

where Ψ, Θ, and Ξ are vectors of interpolation (or shape) functions, u_i, P, and H are vectors of nodal values of velocity components, pressure, and enthalpy, respectively.

Substitution of (34), (35), and (36) into the state equations weak forms results in the following finite element equations:

Continuity

$$ \begin{array}{@{}rcl@{}} &&\left[ \frac{C_{p}}{R} {\int}_{{\varOmega}^{e}} \mathbf{\Xi} \left( \frac{\partial \mathbf{\Xi}^{T}}{\partial x_{i}}\mathbf{P} (\mathbf{\Theta}^{T} \mathbf{H})^{-1} \mathbf{\Psi}^{T} + \mathbf{\Xi}^{T} \mathbf{P} \frac{\partial (\mathbf{\Theta}^{T} \mathbf{H})^{-1}}{\partial x_{i}} \mathbf{\Psi}^{T} \!\right.\right.\\&&\left.\left.+ \mathbf{\Xi}^{T} \mathbf{P} (\mathbf{\Theta}^{T} \mathbf{H})^{-1} \frac{\partial \mathbf{\Psi}^{T}}{\partial x_{i}} \right) d\mathbf{x} \right] \mathbf{u}_{i} = \mathbf{Q}(\mathbf{P,H}) \mathbf{u} = \boldsymbol{0}\\ \end{array} $$

(37)

The matrix R representing the density is defined as:

$$ \mathbf{R}(\mathbf{P,H}) = \frac{C_{p}}{R} \mathbf{\Xi}^{T} \mathbf{P} (\mathbf{\Theta}^{T} \mathbf{H})^{-1} $$

(38)

Momentum

$$ \begin{array}{@{}rcl@{}} &&{\int}_{{\varOmega}^{e}} \mathbf{\Psi R(P,H) u}_{j} \frac{\partial \mathbf{\Psi}^{T}}{\partial x_{j}} d\mathbf{x} \mathbf{u}_{i} - {\int}_{{\varOmega}^{e}} \frac{\partial \mathbf{\Psi}}{\partial x_{i}}\mathbf{\Xi}^{T} d\mathbf{x} \mathbf{P} \\&&\quad+ {\int}_{{\varOmega}^{e}} \mu \frac{\partial \mathbf{\Psi}}{\partial x_{j}}\frac{\partial \mathbf{\Psi}^{T}}{\partial x_{j}} d\mathbf{x} \mathbf{u_{i}} + {\int}_{{\varOmega}^{e}} \mu \frac{\partial \mathbf{\Psi}}{\partial x_{j}}\frac{\partial \mathbf{\Psi}^{T}}{\partial x_{i}} d\mathbf{x} \mathbf{u_{j}} \\ &&\quad+ {\int}_{{\varOmega}^{e}} \lambda \frac{\partial \mathbf{\Psi}}{\partial x_{i}}\frac{\partial \mathbf{\Psi}^{T}}{\partial x_{i}} d\mathbf{x} \mathbf{u_{i}} + {\int}_{{\varOmega}^{e}} \kappa(\alpha) \mathbf{\Psi}\mathbf{\Psi}^{T} d\mathbf{x} \mathbf{u_{i}} \\&&\quad- {\int}_{{\varGamma}^{e}} \mathbf{\Psi} {T}_{m_{i}} ds = \boldsymbol{0} \end{array} $$

(39)

Energy

$$ \begin{array}{@{}rcl@{}} &&{\int}_{{\varOmega}^{e}} \mathbf{\Theta R(P,H)} \left[ \mathbf{\Psi}^{T} \mathbf{u}_{j} \frac{\partial \mathbf{\Theta}^{T}}{\partial x_{j}} \mathbf{H} + 2 \mathbf{\Psi}^{T} \mathbf{u}_{j} \mathbf{\Psi}^{T} \frac{\partial \mathbf{\Psi}^{T}}{\partial x_{j}} \right] d\mathbf{x} \\&&+ {\int}_{{\varOmega}^{e}} \frac{k}{C_{p}} \frac{\partial \mathbf{\Theta}}{\partial x_{j}}\frac{\partial \mathbf{\Theta}^{T}}{\partial x_{j}} d\mathbf{x} \mathbf{H}\\ &&+ {\int}_{{\varOmega}^{e}} \mu \left( \frac{\partial \mathbf{\Psi}^{T}}{\partial x_{j}}\mathbf{u}_{i} + \frac{\partial \mathbf{\Psi}^{T}}{\partial x_{i}}\mathbf{u}_{j} \right) \mathbf{u}_{i} \frac{\partial \mathbf{\Theta}}{\partial x_{j}} \\&&+ \lambda \frac{\partial \mathbf{\Psi}^{T}}{\partial x_{j}}\mathbf{u}_{j} \mathbf{u_{i}} \frac{\partial \mathbf{\Theta}^{T}}{\partial x_{i}} d\mathbf{x} - {\int}_{{\varGamma}^{e}} \mathbf{\Theta} {T}_{e_{i}} ds = \boldsymbol{0} \end{array} $$

(40)

By replacing the approximation for pressure and velocity fields obtained through finite element interpolation functions in the weak formulation, the algebraic equations for the finite element method are obtained. The matrix coefficients per element are given by:

$$ \begin{array}{@{}rcl@{}} \mathbf{C}(\mathbf{R,u}) &=& {\int}_{{\varOmega}^{e}} \mathbf{\Psi R(P,H) u}_{j} \frac{\partial \mathbf{\Psi}^{T}}{\partial x_{j}} d\mathbf{x}, \\\mathbf{G}&=& {\int}_{{\varOmega}^{e}} \frac{\partial \mathbf{\Psi}}{\partial x_{i}}\mathbf{\Xi}^{T} d\mathbf{x}, \\ \mathbf{K}_{\alpha} &=& {\int}_{{\varOmega}^{e}} \alpha \mathbf{\Psi} \mathbf{\Psi}^{T} dx, \!\!\qquad\mathbf{K}_{ij}(a) = {\int}_{{\varOmega}^{e}} a \frac{\partial \mathbf{\Psi}}{\partial x_{j}} \frac{\partial \mathbf{\Psi}^{T}}{\partial x_{i}} dx \end{array} $$

$$ \begin{array}{@{}rcl@{}} \mathbf{D(R,u)} &=& {\int}_{{\varOmega}^{e}} \mathbf{\Theta R(P,H)} \mathbf{\Psi}^{T} \mathbf{u}_{j} \frac{\partial \mathbf{\Theta}^{T}}{\partial x_{j}}, \qquad \mathbf{K}_{kin}\mathbf{(R,u)} \\&=& 2\mathbf{\Psi}^{T} \mathbf{u}_{j} \mathbf{\Psi}^{T} \frac{\partial \mathbf{\Psi}^{T}}{\partial x_{j}} \\ \mathbf{L} &=& {\int}_{{\varOmega}^{e}} \frac{k}{C_{p}} \frac{\partial \mathbf{\Theta}}{\partial x_{j}}\frac{\partial \mathbf{\Theta}^{T}}{\partial x_{j}} d\mathbf{x}, \\\mathbf{R}(\mathbf{P,H})&=& \frac{C_{p}}{R} \mathbf{\Xi}^{T} \mathbf{P} (\mathbf{\Theta}^{T} \mathbf{H})^{-1} \\ \mathbf{T}_{H_{\mu}}(\mathbf{u}) &=&\! {\int}_{{\varOmega}^{e}} \mu \left( \! \frac{\partial \mathbf{\Psi}^{T}}{\partial x_{j}}\mathbf{u}_{i} + \frac{\partial \mathbf{\Psi}^{T}}{\partial x_{i}}\mathbf{u}_{j}\! \right) \mathbf{u}_{i} \frac{\partial \mathbf{\Theta}}{\partial x_{j}} d\mathbf{x},\\\!\!\!\!\!\! \qquad\mathbf{T}_{H_{\lambda}}(\mathbf{u}) &=& {\int}_{{\varOmega}^{e}} \lambda \frac{\partial \mathbf{\Psi}^{T}}{\partial x_{j}}\mathbf{u}_{j} \mathbf{u_{i}} \frac{\partial \mathbf{\Theta}^{T}}{\partial x_{i}} d\mathbf{x} \\ \mathbf{F_{m}}_{i} &=& {\int}_{{\varGamma}^{e}} \mathbf{\Psi} {T}_{m_{i}} ds\\ \mathbf{F_{e}}_{i} &=& {\int}_{{\varGamma}^{e}} \mathbf{\Theta} {T}_{e_{i}} ds \\ \Bar{\mathbf{K}}_{11} &=& 2 \mathbf{K}_{11}(\mu) + 2 \mathbf{K}_{11}(\lambda) + \mathbf{K}_{22}(\mu), \\\Bar{\mathbf{K}}_{12} &=& \mathbf{K}_{12}(\mu) \\ \Bar{\mathbf{K}}_{22} &=& \mathbf{K}_{11}(\mu) + 2 \mathbf{K}_{22}(\lambda) + 2 \mathbf{K}_{22}(\mu), \\\Bar{\mathbf{K}}_{21} &=& \mathbf{K}_{21}(\mu) \end{array} $$

The global algebraic system, considering two-dimensional case, is defined in the matrix form as:

$$ \begin{array}{@{}rcl@{}} &&\begin{bmatrix} \mathbf{C}(\mathbf{R,u}) + \mathbf{K}_{\alpha} +\Bar{\mathbf{K}} & & -\mathbf{G} & & 0 \\ \mathbf{Q}(\mathbf{P,H}) & & 0 & & 0 \\ 0 & & 0 & & \mathbf{D(R,u)} + \mathbf{L} \end{bmatrix} \left\{\begin{array}{c} \mathbf{u} \\\mathbf{P} \\ \mathbf{H} \end{array}\right\} \\&&= \left\{\begin{array}{c} \mathbf{F_{m}} \\ \mathbf{0} \\ \mathbf{F_{e}}_{i} - \mathbf{K}_{kin}\mathbf{(R,u)} - \mathbf{T}_{H_{\mu}}\mathbf{(u)} -\mathbf{T}_{H_{\lambda}}\mathbf{(u)} \end{array}\right\} \end{array} $$

(41)

where u is the velocity column vector with all directions ($\mathbf {u} = \{\mathbf {u}_{1} \mathbf {u}_{2}\}^{T}$). This resulting system is non-symmetrical and highly non-linear.

Appendix C: Material model assessment

In this section, the behavior of the material model is verified by comparing the flows calculated by a body-fitted mesh case and by an equivalent case with the domain represented by the material model. As explained in Section 3.1, the very low velocities in the solid region reduces the energy equation to a Laplacian equation on enthalpy (see 13). If there are no heat generation and no other source terms in the energy equation, this Laplacian of enthalpy is equal to zero. By using the divergence theorem, this equation can be interpreted as no heat going in or out of the solid region. Thus, the higher the velocity penalization on the momentum equations, the lower are the terms with velocity (u) in the energy equation, and then, the solid region defined by the design variable becomes closer to an adiabatic material.

In Fig. 14, the two domains used for a numerical comparison are presented. The body fitted case with definitions of boundary conditions can be seen in Fig. 14a. These boundary conditions have the same definition as the cases previously presented in the text: specified velocity and enthalpy at the inlet, zero velocity, adiabatic condition at the walls, and specified pressure at the outlet. In Fig. 14b, this same domain is represented by using the material model presented in the paper. The boundary conditions are the same, and the contour is defined by the material model. The boundary conditions at the domain external walls are defined as zero velocity and adiabatic.

Fig. 14

Domains for body fitted case and case with penalization

The black arrow indicates a cut performed in both cases to verify the state variables profile in that section. The body fitted case is calculated without the material model penalization, and the extended case is calculated with the penalization term, with the value for κ_max varying from 10³ to 10⁶.

In Fig. 15, comparisons of the state variables profile at the cut section can be seen. The black solid line indicates the profile in the body fitted case and the colorful lines indicate the cases with penalization. As can be seen, the higher the penalization, the closer to the body fitted case. It is important to notice that the profiles will not match perfectly because velocity will never be exactly zero in the solid region and also the penalized cases do not have a perfectly smooth contour. This contour is jagged and depends on the mesh refinement. Even so, it can be seen that this material model exhibits a good representation of adiabatic boundaries.

Fig. 15

Comparison of state variables for body fitted and penalized cases

Appendix D: Regularization term analysis

In this appendix, the effects of the regularization term J_κ are further analyzed by considering the double channel Stokes problem presented by Borrvall and Petersson (2003) with:

• μ = 1.0[Pa ⋅ s];

• κ_min = 2.5μ10^− 4;

• κ_max = 2.5μ10⁴;

• 1[m] × 1[m] mesh with 100 × 100 triangular elements;

• their energy dissipation objective function without J_κ, i.e. $J_{ed}={\int \limits }_{{\varOmega }} \mu \nabla \boldsymbol {u} : \nabla \boldsymbol {u} d{\varOmega }$,

and by testing different values of q = [1, 10, 100, 1000]. This is done in order to show that increasing the q value without the regularization is not enough to remove the gray regions.

The results are shown in Fig. 16. It is shown that even though the topologies minimize the viscous energy dissipation functional (J_ed), they still present many gray areas. Moreover, even when high values of q are used, the topologies are still blurred by intermediate values of α. These grays areas are removed when the regularization J_κ is used.

Fig. 16

Optimized topologies for Stokes flow with different values of q and without J_κ in the objective function

Appendix E: : Sensitivity verification via finite differences

This section shows the finite difference verification of the automatic differentiated sensitivities calculated by the pyadjoint library. The central finite difference is used, which is given by:

$$ \frac{\partial J}{\partial \alpha} = \frac{J(\alpha + h) - J(\alpha - h)}{2h} $$

(42)

where the perturbation used is h = 1 ⋅ 10^− 6.

This verification considers the domain shown in Fig. 9 and an uniform design variable distribution of α = 0.5. The chosen trial points are presented in Fig. 17. Given that two objective functions are used in this work the results of the sensitivity verification are split into two tables.

Fig. 17

Trial points for FD verification

Table 7 shows the sensitivity comparison between the value calculated by pyadjoint and the value obtained via finite differences for the entropy variation functional (15). The sensitivities are almost identical, and the highest discrepancy occurs for sensitivity values close to zero, such as for trial point 14 which presents a difference of − 3.23%.

Table 7 Sensitivity verification for FD and adjoint approaches for the entropy variation functional

Table 8 shows the sensitivity comparison between the value calculated by pyadjoint and the value obtained by finite differences for the entropy generation functional (16) with the outlet velocity functional (22). The sensitivities are in good agreement and the highest discrepancy occurs for trial point 8 which presents a difference of 0.0085%.

Table 8 Sensitivity verification for FD and adjoint approaches for the entropy generation functional