Shape and material optimization of problems with dynamically evolving interfaces applied to solid rocket motors

Abstract

This paper studies design problems where the performance is dominated by the dynamic evolution of interfaces due to chemical processes. Considering the representative example of a solid rocket motor, the shape of the interface between the solid fuel and the gas inside the combustion chamber at the beginning of the burn process and the reference burn rate of a functionally graded propellant are optimized to achieve a desired thrust over time. The initial fuel–gas interface is described by a level set function parameterized by geometric primitives and B-splines. The reference burn rate distribution is discretized by multi-variate B-splines. The thrust is predicted by a semi-analytical approach that requires modeling the recession of the fuel–gas interface. To this end, a stabilized finite element formulation of the Hamilton–Jacobi equation is used to describe the evolution of the level set function during the burn process. The optimization problem is solved by a nonlinear programming method, and the design sensitivities are evaluated by the adjoint method. The proposed optimization approach is studied with numerical examples in 2D and 3D, involving configurations with more than $6 \times 10^{4}$ optimization variables and $12 \times 10^{6}$ state variables. The optimization results show that this approach provides a promising design tool for problems with dynamically evolving interfaces due to surface reactions. However, the results also reveal that the simplicity of the recession and thrust models requires limiting the design freedom through a carefully chosen design parameterization. Furthermore, additional constraints need to be imposed to prevent unphysical designs.

Appendices

Appendix 1: Verification example for Hamilton–Jacobi transport model

To verify the accuracy of the HJ transport model presented in Sect. 3.1 and to identify important model and simulation parameters, we consider the evolution of a level set field with an initially circular zero isocontour in 2D and an initially spherical zero isocontour in 3D for a uniform speed field. For these problems, trivial analytical solutions exist with the radii increasing linearly in time.

The configurations are shown in Fig. 28. In the 2D case, the computational domain is a square of $0.2 \times 0.2$ m. The initial solid–gas interface is circular. In the 3D case, the computational domain is a cube of $0.1 \times 0.1 \times 0.1$ m. The level set function is initialized to describe a spherical solid–gas interface. In both cases, the level set field evolves in time due to a uniform speed field with a reference burn rate of 0.005 m/s. Thus, the solution of the speed field extension model is constant across the computational domain.

Fig. 28

HJ transport model verification problems

With these configurations, we study the accuracy of the simulation with respect to mesh size, time step size, and the algorithmic parameters defined in Sect. 3.1. Note that the dimensions and burn rates are representative of the rocket engine configurations studied in Sects. 6 and 7. The accuracy of the numerical simulations is measured by comparing the surface area of the solid–gas interface against that of the analytical solution. In the finite element model, the interface area is computed as follows:

$$S = \int \limits _{\varOmega} {f_{\text {drc}} \left( {\varphi ,\nabla \varphi } \right) \;{\text {d}}V} ,$$

(46)

where the smoothed Dirac function, $f_{\text {drc}}$, is computed by (17).

For the 2D case, the baseline configuration uses a $128 \times 128$ mesh and a time step size of $\Updelta t=0.05$ s. For the 3D case, the baseline configuration uses a $25 \times 25 \times 25$ mesh and a time step size of $\Updelta t = 0.1$ s. In both cases, the sharpness parameter to compute the interface area with (46) and (17) is $\lambda _{\text {drc}}=1/h_{\text {ele}}$. The Eikonal stabilization is set to $\beta _{\text {eik}}=1.0$ and a uniform target gradient norm of $\Vert \varphi \Vert _{\text {trg}}=1.0$ is used.

The influences of the mesh size, time step size, the sharpness parameter $\lambda _{\text {drc}}$, and the Eikonal stabilization $\beta _{\text {eik}}$ on the accuracy of the solid–gas interface area prediction are studied. The numerical results suggest that for the circular and spherical geometries considered here the influence of $\beta _{\text {eik}}$ is negligible. In contrast, the other parameters have a noticeable influence on the numerical results, which are summarized in Table 2 for the 2D case and Table 3 for the 3D case.

Table 2 Accuracy of solid–gas interface area prediction for 2D case

Table 3 Accuracy of solid–gas interface area prediction for 3D case

These results show that a sufficient mesh refinement is crucial to obtain accurate results. This effect can be clearly seen from the 3D mesh refinement study, where the baseline configuration has a rather coarse mesh. For the examples considered here, the dependence on the time step size is small and the error is dominated by the mesh resolution error. For problems with solid–gas interfaces merging during the burn process, a stronger dependence of the time step size on the accuracy was observed. This study also illustrates the importance of using a proper sharpness parameter for the smoothed Dirac function. The error can significantly increase if the sharpness parameter is too large. This effect is even more pronounced if more complex shapes are considered.

Appendix 2: Verification example for detecting disconnected volumes of gas

The ability of the approach presented in Sect. 3.2 to detect volumes of gas disconnected from the nozzle is illustrated with an example shown in Fig. 29. The computational domain of size $100 \times 100 \times 633$ mm is discretized by a $15 \times 15 \times 95$ uniform mesh. This results in an element size of $h_{\text {ele}} = 6.67$ mm. The reference value is set to ${{\bar{T}}}_{1} = 100$, the source value to $T_{1}^{\text {src}} = -100$, the convection coefficient parameter to $\alpha _{T1} = 10^{ -4}$, and the sharpness parameter to $\lambda _{\text {dif}} = 5/h_{\text {ele}}$ . The cylindrical volumes of gas have a diameter of 10 mm. The gap between the cylinders is placed at one and two thirds of the $x_{3}$-dimension of the computational domain.

Fig. 29

Numerical example to illustrate the ability of identifying volumes of gas connected to and disconnected from the nozzle through the solution of an auxiliary diffusion field

First, a gap size of 25 mm is considered. The scalar field contour plot on the right side of Fig. 29 shows that for this gap size the scalar field value in the volume of gas connected to the nozzle is approximately $T_{1} \approx 100$, while the values in the volumes of gas disconnected from the nozzle are $T_{1} \approx - 100$. Thus, using the scalar field values, one can easily distinguish between volumes of gas connected to the nozzle and disconnected from the nozzle.

In Fig. 30, we show scalar field contour plots for varying gap sizes on a slice through the center of the computational domain in the $x_{1}-x_{3}$ plane. The gap size is altered by elongating the axial length of the cylindrical gas volumes. The results indicate that one can distinguish between connected and disconnected volumes of gas until the gap size is approximately smaller than two times the element size.

Fig. 30

Study of the resolution of the auxiliary diffusion model of identifying volumes of gas disconnected from the nozzle

Appendix 3: Verification example for detecting unburnt propellant

Figure 31 shows a numerical example with two isolated spherical pieces of unburnt fuel within a volume of fuel that is connected to the casing. As in the example of Section B, the computational domain of size $100 \times 100 \times 633$ mm is discretized by a $15 \times 15 \times 95$ uniform mesh. Here, the reference value is set to ${{\bar{T}}}_{2} = - 100$, the source value to $T_{2}^{\text {src}} = 100$, the convection coefficient parameter to $\alpha _{T1} = 10^{ -4}$, and the sharpness parameter to $\lambda _{\text {dif}} = 5/h_{\text {ele}}$. The spherical pieces of propellant have a radius of 15 mm and are contained within a cylindrical volume of gas with a radius of 60 mm. The solution of the auxiliary diffusion problem allows the identification of the isolated piece of unburnt fuel. Their scalar field value is $T_{2} \approx 100$, while the scalar field in the fuel connected to the casing is $T_{2} \approx -100.$

Fig. 31

Numerical example with two isolated spherical pieces of unburnt fuel

To study the resolution of this model, we vary the radius, $R_{\text {inner}}$, of the spherical piece of unburnt fuel. The radius, $R_{\text {outer}}$ of the outer cylindrical cavity remains constant. In Fig. 32, the scalar field values are shown on a slice through the center of the upper sphere for increasing radii. The circular solid lines represent the solid–gas interfaces. These results suggest that one can identify pieces of unburnt fuel if the distance between fuel connected to and disconnected from the casing is more than twice the element size of the mesh. This observation is consistent with the one for the resolution for detecting connected and disconnected volumes of gas in “Appendix 2”.

Fig. 32

Study of the resolution of the auxiliary diffusion model for identifying pieces of unburnt propellant

Appendix 4: Refined axial parameterization of 3D optimization problem

To explore the influence of refining the parameterization in axial direction, we consider a 4-petal configuration with a radial B-spline discretization of the burn rate field, similar to the 8-petal configuration studied in Sect. 7.3.1. However, we increase the number of B-spline basis functions in axial direction from 23 to 63. To capture the large spatial variability in axial direction, we also refine the mesh in axial direction and use $40 \times 40 \times 120$ elements, yielding 406,802 level set and speed field degrees of freedom.

The optimization results are shown in Figs. 33 and 34. The thrust profile is given in Fig. 35. Comparing the optimized design with the one obtained with a coarser axial parameterization in Sect. 7.3.1 suggests that allowing for a larger spatial variability in axial direction leads to a distinctly different geometry of the initial solid–gas interface. Instead of forming distinct petals, the initial interface area is increased by an undulation in axial direction. However, the thrust profile of the optimized design does not match the target thrust.

Fig. 33

Gas phase at start of burn process of 4-petal design with refined parameterization in axial direction

Fig. 34

Contour plots of reference burn rate distribution with gas phase (gray) of 4-petal design with refined parameterization in axial direction

Fig. 35

Comparison of thrust profile of 4-petal design with refined parameterization in axial direction against target thrust