NOMENCLATURE

Abbreviations

DSE

design space exploration

FE

finite element

2D

two-dimensional

3D

three-dimensional

Symbols

$r$, $\varphi $, $z$

components of the cylindrical coordinate system

H

flange height selected for parametric variation

P

flange positioning selected for parametric variation

p

pressure

T

sink temperature

W

flange width selected for parametric variation

α

heat transfer coefficient

μ

mean value

$ \sigma$

standard deviation

Subscripts

a

surface of the outer side of the casing

c

stator cavity surfaces wetted with leakage flow

i

surface of the inner side of the casing

m

surface of the inner casing structures

p

peripheral surfaces of the casing

2.0. SURROGATE MODELLING

Detailed finite element analysis of complete compressor casing assemblies is very time-consuming. Design Space Exploration (DSE) of the structural response across different parametric combinations can help to understand the correlation of design variables and optimise a design, achieved through the required large number of analyses. Compared with conventional finite element analysis and especially when linked with DSE, surrogate modelling represents an attractive alternative. Based on training data stored from previously analysed designs, surrogate models are data-driven mathematical approximations for numerically inexpensive result reproduction of new designs. Consequently, to enable the prediction of reliable results for new designs with negligible additional effort, the initial expense of surrogate modelling lies in creating the training data for specific designs.^{(Reference Bunell, Thelin, Gorell, Salmon, Ruoti and Hepworth8)}

The number of reports dealing with the application of surrogate modelling in turbomachinery engineering indicates increasing interest from both industry and academia. Recent work^{(Reference Bunell, Thelin, Gorell, Salmon, Ruoti and Hepworth8-Reference Walther and Nadarajah12)} utilised this technique to predict structural responses from finite element analysis. In 2017, for instance, Zhang et al.^{(Reference Zhang, Guo, Li, Yang and Yue9)} and Geller et al.^{(Reference Geller, Schemmann and Kluck10)} explored and optimised the design of various turbomachinery engine components. Concentrating on the maximum stress as a single structural value that can be rapidly emulated by a surrogate model, the geometries considered were a turbine disk^{(Reference Zhang, Guo, Li, Yang and Yue9)} and compressor impellers^{(Reference Geller, Schemmann and Kluck10)}. Both publications mentioned an accurate estimation at negligible computational cost in comparison with finite element simulation results. Furthermore, Heap et al.^{(Reference Heap, Hepworth and Jensen11)} and Walther et al.^{(Reference Walther and Nadarajah12)} derived innovative methods for the expansion from single- towards multi-value result prediction. By representing the stress at each node by a unique surrogate model, Heap et al.^{(Reference Heap, Hepworth and Jensen11)} found that the entire stress distribution can be predicted. Concurrently, this method required that the nodes of the mesh structure specified for reconstruction of the FE result via surrogate models shared an identical relative position across each of the FE models used during training. The use of a parametric mesh limited that study to simple parts. Next, Walther et al.^{(Reference Walther and Nadarajah12)} reported that surrogate models permit accurate predictions of nodal locations of very complex geometries. Instead of setting up a parametric mesh, as done by Heap et al.^{(Reference Heap, Hepworth and Jensen11)}, mesh morphing was applied. This meshing strategy enabled Walther et al.^{(Reference Walther and Nadarajah12)} to predict more complex geometries. In 2018, Bunnel et al.^{(Reference Bunell, Thelin, Gorell, Salmon, Ruoti and Hepworth8)} combined the possibilities identified by Heap et al.^{(Reference Heap, Hepworth and Jensen11)} and Walther et al.^{(Reference Walther and Nadarajah12)}, developing a method for the reproduction of the entire stress distribution and geometric shape of turbomachinery engine components. Thus, each nodal stress value and position was emulated using a unique surrogate model. By means of mesh morphing, the requirement for an identical FE model mesh structure was satisfied. An essential motivation was to perform DSE on compressor blades, seeking to understand their full structural response across different parametric combinations. For visualisation, Fig. 2 compares the von Mises stress distribution at the suction side of a compressor blade calculated via surrogate models versus FE simulation. The colour scale is matched. X indicates the maximum stress position, and O the minimum stress position. Referring to Bunnel et al.^{(Reference Bunell, Thelin, Gorell, Salmon, Ruoti and Hepworth8)}, the FE results and surrogate model predictions exhibited comparable geometric shape, and stress distributions and magnitudes. The reconstruction of the FE result via the surrogate models correctly predicted a region of high stress located at the root of the blade at the middle of the chord. Less than $5\% $ error was achieved on all the tested blades.

The procedure applied for surrogate model generation in this work is illustrated in Fig. 3. Following the definition of the variable design parameters, the Monte Carlo sampling method is chosen for extensive database creation. It relies on random sampling to explore the design space. Full factorial, fractional factorial and basic Latin hypercube are well-known alternative sampling methods, but are not recommended in literature^{(Reference Zimmermann and Krischer13)}. Next, FE models of the new design parameter combinations obtained from the Monte Carlo sampling method were set up and solved using the commercial Abaqus FEA software. Evaluation of the simulation results at specified nodes supplied the database with response values for given sets of design parameter combinations. Provisionally, Gaussian process regression was selected as the algorithm for generating the surrogate models. The Gaussian process regression is also known in the geostatistics field as kriging, representing a non-parametric kernel-based probabilistic approach that is widely used in applied mathematics, statistics and machine learning for constructing surrogate models^{(Reference Zimmermann and Krischer13-Reference Yang, Tartakovsky and Tartakovsky15)}. Trained on the information stored in the database, the resulting surrogate models allow for rapid predictions of the results for new design parameter combinations. This procedure was automated using the commercial software MATLAB by MathWorks.

Figure 2.

von Mises stress distributions on the suction side of a compressor blade published by Bunnel et al.^{(Reference Bunell, Thelin, Gorell, Salmon, Ruoti and Hepworth8)}.

Figure 3.

Procedure for surrogate model generation.

The implementation of alternative options for surrogate model algorithms should be tested in future investigations, with a focus on training time, robustness, predictive speed and accuracy. According to literature^{(Reference Bunell, Thelin, Gorell, Salmon, Ruoti and Hepworth8,Reference Jin, Chen and Simpson16)} , the radial basis function is considered to be the most reliable method in terms of accuracy and robustness, performing well with sparse training data in design spaces that include many parameters. In contrast, the first studies applied to the case presented herein demonstrated much better results in terms of accuracy and time consumption when implementing the Gaussian process regression. A brief summary of selected surrogate model algorithms extracted from literature^{(Reference Bunell, Thelin, Gorell, Salmon, Ruoti and Hepworth8,Reference Zimmermann and Krischer13,Reference Rasmussen and Williams14,Reference Jin, Chen and Simpson16,17)} is given in Annex B.

3.0. COMPRESSOR CASING DESIGN

Traditionally, designs of compressor casings are highly coupled to the technology available at the time of design, depending on technology-determining characteristics such as the materials used or the manufacturing methods applied. Bretschneider et al.^{(Reference Bretschneider, Rothe, Rose and Staudacher18)} identified a high level of similarity among compressor casing designs disclosed by aero-engine manufacturers. This tendency is a direct consequence of the use of engine family concepts based on similar, scaled or further developed high-pressure systems with high commonality, driven by saving development costs and simultaneously implementing mature, successfully marketed component technologies^{(Reference Bretschneider, Rothe, Rose and Staudacher18)}.

In practice, mixed architectures between the four types of axial compressor casing design illustrated in Fig. 4 are prevalent. Most current aero-engine axial compressor casings feature an axial split in the front stages and change to a combination of double-walled ring style in the rear section. The axial split casing design provides the required access for assembling variable stator vanes and the corresponding actuation mechanism. The radial split casing is preferred when short local casing elements are preferred per stage. Well-defined axial load transmission linked with the provision of a very high casing stiffness is one of the main roles of the double-walled casing design, besides the accommodation of bleed air off-take in different stages. Frequently, the inner casing wall is thermally designed to allow tip clearance optimisation in stages where rotor blades are physically small and the stage efficiency is very sensitive to changes in tip clearance.^{(Reference Bretschneider, Rothe, Rose and Staudacher18)}

Figure 4.

General types of axial compressor casings^{(Reference Bretschneider, Rothe, Rose and Staudacher18,19)} : (a) integral case, (b) radial split case, (c) axial split case and (d) double-walled case.

For reasons of detailed categorisation, literature^{(Reference Bretschneider, Rothe, Rose and Staudacher18)} differentiates between mid-rotor split and inter-stage flange designs as derived sub-elements. The mid-rotor split flange design is characterised by splitting the casing above the rotor and offers advantages in terms of stiffness as well as blade-loss containment ability. The concept of the inter-stage flange design extends over a full stage, covering a stator and a rotor (or vice versa). Thus, it incorporates the rub-in liner for rotor blades as well as the T-Slot mount for stator vanes. A mix between both casing flange design shapes is possible but rarely found in existing aero-engine arrangements.^{(Reference Bretschneider, Rothe, Rose and Staudacher18)}

Figure 5.

High-pressure compressor casing geometry (left) and schematic of the boundary and load conditions implemented for the compressor casing assembly (right).

4.0. MODEL SETUP

This section describes the 2D axisymmetric FE model setup for the multi-stage high-pressure axial compressor casing assembly conceived for training database generation. Figure 5 visualises the main architecture and a schematic of the boundary and load conditions implemented. Created with reference to the rear section of the general arrangement illustrated in Fig. 1, it represents a double-walled casing construction, combining radial split as well as inter-stage flange designs. With respect to future work involving the extension of a given casing arrangement, a standardised component design process was established to allow the addition of more casing rings and stators. Detail A defines the geometric parameters (height $H$, width $W$ and positioning $P$) selected for the modification of each flange within the ranges listed in Table 1, and the red-coloured arrows indicate the corresponding directions of geometry variation. To avoid unfeasible geometry modifications, linear inequality constraints must be imposed between the flange widths and positions. If not other specified, the flange dimensions used in the figures of this section correspond to the default values listed in Table 1.

Using the commercial Abaqus FEA software, coupled thermo-mechanical implicit simulations were carried out. As mentioned above, Fig. 5 graphically summarises the boundary and load conditions implemented, as well. Index $a$ describes the outer side of the casing, index $i$ the inner side of the casing exposed to the compressed core flow and index $m$ the surface of the inner casing structures wetted with bleed air. Considering leakage, index $c$ characterises those surfaces that are in contact with the stator cavity leakage flow driven by the pressure gradient across the stator row. Index $p$ designates the peripheral surfaces of the model. T symbolises the temperature, p the pressure and $\alpha$ the heat transfer coefficient. The corresponding set of parameter values used are presented in Table 2. Regarding the definition of contacts between the components included in the casing assembly, the implementation was configured consistently with previous work^{(Reference Schmidt, Gümmer and Konle4-Reference Schmidt, Hupfer, Gümmer and Konle6)}. For the heat transfer analysis, tie constraints were introduced, and their subsequent substitution by surface-to-surface contact interactions permitted thermo-mechanical analysis with allowed part separation. Moreover, an additional fixation is installed to lock the degree of freedom in the $\varphi $–$z$ direction.

Provisional material selection for the complete casing assembly resulted in high-temperature-resistant Inconel 718. The isotropic material model implemented contains temperature-dependent data for the Young’s modulus, Poisson’s ratio, density, specific heat capacity, thermal expansion coefficient and thermal heat conductance, sourced from refs. ^(20-22). Validation attempts confirmed the correct reproduction of the material characteristics, while an upgrade to a multi-material definition will be part of future work.

Table 1

Geometric flange parameter variation

The parameter settings presented in Table 2 are considered to be steady state. They first define the case for an unloaded structure (initial condition). The subsequent initial load condition represents the load setting resulting from operation at the design point of a modern high-pressure compressor. The core flow pressure ${p_{i,I}}$ and temperature ${T_{i,I}}$ were defined as $z$-dependent functions, both increasing linearly in the flow direction. Compared with ${p_{i,I}}$ and ${T_{i,I}}$, the transition to ${T_{i,II}}$ and ${p_{i,II}}$ constitutes the end of the compression induced by the rotor. Consequently, ${T_{i,II}}$ and ${p_{i,II}}$ remain constant. In accordance with previous work^{(Reference Schmidt, Gümmer and Konle5,Reference Schmidt, Hupfer, Gümmer and Konle6)} , even pressure-gradient-driven leakage flows through the stator cavities were considered via ${T_c}$ and ${p_c}$. The corresponding pressure compensations from the cavity inlets towards their outlets were configured by setting ${p_c}$ equal to the expression used for ${p_{i,I}}$. Constant values calculated for ${T_{i,I}}$ at the respective $z$-positions of the cavity inlets were taken for ${T_c}$.

Table 2

Parameter settings applied on the surfaces of the generic high-pressure compressor casing structure

As an example of the hybrid computational grids generated and used in this work, Fig. 6 shows the mesh of the 2D axisymmetric FE model of the compressor casing assembly for the case using the default flange dimensions. It consists of approximately 12,800 second-order elements, and the mesh resolution is capable of providing smooth and converged computational solutions. Moreover, attention was paid to each contact surface to ensure node-to-node matching between the part instances involved. The area within the red frame shows a magnified detailed view of the meshed geometry.

Figure 6.

Generated mesh of the 2D axisymmetric high-pressure compressor casing assembly with detail view.

Figure 7.

Measurement sites selected for FE simulation result evaluation.

The expansion behaviour of the inner casing surface during engine operation is particularly relevant for an adequate tip clearance assessment. With a focus on the radial casing displacement, the measurement sites coloured in red and displayed in Fig. 7 were considered for to evaluate the FE simulation and supplement the training data. R represents the centre position immediately above the rotor blades, and S the centre position of the stator vane shrouds. In this way, the resulting tip clearance formed by the rotor blade tips and the compressor casing can be calculated at the measurement sites R1, R2, R3, R4 and R5. Meanwhile, the clearances described by the vane tips radially inboard of the stator vanes and the rotor drum will be calculated at the measurement sites S1, S2, S3 and S4.