Dynamical Behaviour and Strength of Structural Elements with Regeneration Induced Imperfections and Residual Stresses

Abstract

Repair techniques allow to refurbish blades of jet engines and extend the service life of engine components. However, the mechanical properties of the repaired blades commonly differ from those of the nominal blades. The changes in vibration behaviour and stress distribution resulting from the repair processes could significantly impair the structural integrity of blades. This work addresses this evaluation of repair-specific influences on the structural integrity of blades of blade-integrated disks (blisks). Numerical methods are employed to model and analyse the structural changes of different repair designs. Using Finite Element (FE) simulations, the influences of different repair techniques and designs are systematically evaluated and compared. To further improve the maintenance procedures of blisks, a computational scheme is developed, which features the optimization of repair designs according to the inspected damage of the blade. Since the corresponding optimization tasks involve multiple conflicting objectives, multi-objective optimization is carried out to identify Pareto-optimal repair designs.

1 Introduction

The repair or replacement of damaged blades is essential to maintain the structural integrity of jet engines and ensure a safe operation during flight. With the increasing use of integrated part designs, such as blade-integrated disks (blisks), on one hand the replacement of single blades becomes unfeasible. On the other hand, the spare part costs of these complex parts increase rapidly. Therefore, there is growing interest in technologies to repair the damaged blades of the blisk rather than replacing the whole blisk.

Repair technologies developed for blisk blades differ in manufacturing complexity and their application depends on the size of the particular defect (Bussmann and Bayer 2009). A damage pattern typical for blisk blades is depicted in Fig. 1. The blade shown in the front is damaged at the leading edge. Damage patterns like this are most likely caused by particles passing through the engine and are hence known as foreign object damages (FOD).

Fig. 1

Three blade sectors of a blisk with patch repair (left), undamaged blade (middle) and defect at the leading edge (right)

Depending on the size of the defect, different repair techniques are applicable. Small dents or notches induce local stress concentrations, which cause rapid crack initiation. To prevent crack formation, sharp edges are removed by a milling procedure. This relatively simple repair, named blend repair, leads to remaining geometric changes in the repaired blade and is thus limited to small defects. For more severe damage, the geometric modification due to blending would be too large to maintain the functionality of blades. Therefore, patch repairs are developed, as depicted in the background of Fig. 1. Patching involves the removal of the damaged portion, a welding process to mount the patch material, and a final re-contouring process to restore the aerodynamic contour of the repaired blade. In contrast to blending, a patch repair hence rebuilds the original geometry of the blade.

Consequently, both repair techniques lead to changes in the blade compared to the nominal condition, which affect the remaining life of the repaired components. In the case of blend repairs, the change is mainly characterised by a local geometric modification. The influences of geometric deviations to the nominal state were studied in numerous works. In recent literature, mostly the correlations with natural blade frequencies, mode shapes, forced-response, and vibration amplitude magnification due to mistuning effects were investigated. For example, Brown et al. (2003) found that the natural frequencies of fan blades deviated by 1% when manufacturing imperfections are considered. Other studies on different blade geometries (Heinze et al. 2010) showed comparable results. However, the majority of scientific contributions was limited to nonintended variances caused by manufacturing or wear during operation. A work, which specifically focused on geometric changes in terms of blending, was presented by Beck et al. (2017). They addressed the variations in frequencies of blisk blades with two different sizes of blend shapes. An amplification of vibration amplitudes caused by blade-to-blade variations was determined. A more practical approach towards blend repair shapes was followed by Day et al. (2012). The authors investigated blend repairs carried out on the compressor blades of a stationary gas turbine. A parametric FE model was developed and different blend shapes are compared according to the resulting blade frequency and high-cycle fatigue (HCF) properties utilizing the Goodman relation. The developed computational workflow facilitated the subsequent evaluation of nonstandard blends. A contribution, which goes beyond this parametric assessment of different blend sizes, was later presented by Karger and Bestle (2015). Karger and Bestle firstly combined the structural simulation of blended blades with numerical optimization methods. The shape of the blend, which was parameterized by five geometric design variables, was optimized using a multi-objective optimization approach. The two objective functions of the multi-objective optimization task referred to the minimization of the blend in terms of blade mass and the HCF strength computed in terms of amplitude frequency strengths. The evaluation process was automated using a commercial software and an optimization algorithm implemented in the Matlab Global Optimization toolbox was utilized to solve the problem. Pareto-optimal solutions were determined and hence provided new insights into the influences of blend shapes. They demonstrated that a wide range of different blending shapes (small up to large) may be beneficial and the final design decision should be selected out of the Pareto-optimal set accounting for aerodynamic criteria like efficiency and surge margin.

In contrast to blend repairs, patch repairs received less attention in recent research. Due to the narrow field of application only a few publications were concerned with this hightech repair technique. General considerations on the process chain of blisk repairs by patching were presented by Eberlein (2007). The authors exemplarily showed thermal simulations of the welding process that joints the patch to the blade material and conducted residual stress measurements on patched blades. They further concluded that the geometry of the patch and the position of the welding seam between blade and patch should be selected according to the distribution of vibratory stresses in the blade. A more analytical work was published by Schoenenborn and Reile (2005), who performed FE simulations to assess the residual stresses of a patched blade. The authors performed uncoupled thermomechanical FE simulation to analyse the effect of welding and heat treatment in the leading edge region of a blisk blade. They further concluded that numerically predicted stresses could complement experimental investigation and thus decrease the risk of blade failure due to HCF. However, a generalized approach on the numerical assessment of patch repair, which includes automatic evaluation schemes as well as optimization methods, has not been followed yet.

Although, existing multi-objective optimization approaches towards improved turbomachinery design showed great potential (Dornberger et al. 2000; Karger and Bestle 2015; Adjei et al. 2020) and the key idea of multi-objective formulations is present in almost all engineering disciplines (Parejo et al. 2012), to the best of the authors’ knowledge, there is no universal concept to evaluate different repair designs for blisk blades utilizing state-of-the-art optimization algorithms. In this work a new computational scheme is developed, which exactly addresses this decision making process during the maintenance of blisk blades. The key aspects of this scheme for optimized blisk repairs are illustrated in Fig. 2. The optimization process starts by identifying the damaged portion of the blisk. The defect of the blade is described using the damage model described in Sect. 3.1. Based on the nominal geometry of blades, structural models are developed to model the repair-specific influences. In particular, the geometric modifications of blending and the residual stresses introduced during patching are addressed. Multiple FE analyses are used to determine vibration properties like frequencies or stresses due to welding. The implemented models and modelling assumptions are introduced in Sects. 3.2 and 3.3, respectively.

Fig. 2

Concept of numerical optimization of blend and patch repairs of blisk blades

Moreover, the simulation results provide the basis for the optimization of repair designs. In Sect. 4, the design criteria and objectives are formulated and an optimization task is derived from the structural requirements. Finally, optimal repair designs are identified integrating the structural models, analysis procedures and objectives into the optimization framework EngiO. The framework is a customized optimization software, which is specially designed to handle engineering optimization task like blisk repairs with little effort. The design of the framework is illustrated in Sect. 2. Optimization results for blend and patch repairs utilizing EngiO are presented in Sect. 5. Findings, limitations and aspects for future work are summarized in Sect. 6 of this paper.

2 Optimization Framework EngiO

The initial perquisite for the optimization of blade repairs is a framework where the optimization could be organized with little effort. The framework developed is specifically targeted to optimization tasks like repair optimization. However, it is designed to manage various optimization problems, which may arise in engineering research. The framework named EngiO (Engineering Optimization framework) is a Matlab-based code that is assessable under GNU General Public License via its github repository (https://github.com/isd-luh/EngiO). In the next sections, first the basic design principles are introduced and secondly several features are demonstrated using a multi-objective test function.

2.1 Architecture

The software architecture of EngiO follows an object-oriented programming pattern. The basic framework including main functionalities is illustrated in Fig. 3 using an Unified Modelling Language (UML) diagram. The UML diagram shows dependencies between different implemented classes.

Fig. 3

Software architecture of EngiO

The architecture of EngiO reflects the idea that each optimization process can typically be viewed from two perspectives (Bleuer et al. 2003). The first perspective belongs to the development of new optimization algorithms, their implementation, and their comparison with other state-of-the-art algorithms. The second perspective is related to practical engineering optimization. From a practical point of view, the implementation of custom engineering optimization problems in terms of numerical simulations and user-defined objective functions is of high relevance.

These two perspectives are both considered in the architecture of EngiO and the UML diagram in Fig. 3 reflects this separation between algorithm and problem-specific objective functions as well. On the left side of the diagram, exemplarily three different classes (“Genetic”, “NSGA2”, and “GlobalPatternMO”) of algorithms are shown. Each of these classes includes concrete methods, which are characteristic for the specific algorithm’s logic. On the right side, the “ObjectiveFunction” class contains the optimization problem formulation corresponding to the custom optimization task. These objective functions may include analytic expressions or calls of any numerical simulation software like Abaqus or ANSYS.

The central part of EngiO is the “Optimizer” class, which can be seen as the glue code to link the algorithm and the optimization problem part. This “Optimizer” class is the parent class for all optimization algorithms and provides the basic structure for all further implementations. The properties of the class include general optimization settings as well as algorithm specific parameters and states. Further, the class definition includes several abstract and concrete methods. Abstract methods, which are labelled with #, have to be implemented in derived algorithm classes. They are defined according to the observation that all derivative-free optimization processes could be described within three distinct steps. In the first step, the initial samples are evaluated. Subsequently, in each optimization loop, new samples are generated and evaluation results are processed according to the algorithm logic. The algorithm classes inherit these from the parent class for structured implementation of algorithms. Concrete methods, on the contrary, indicated by + centralized routines in the parent class. The most important method implemented in the “Optimizer” class is the method “optimize”, because it includes the main iteration loop of the optimization. Further, concrete methods allow fast sorting of solutions for the computation of Pareto-optimal sets and the evaluation of performance indicators like the hyper-volume metric.

The optimization process is started using a main script on a global level. In this main script, optimization settings, such as boundaries or maximum number of function evaluations are specified.

It should be noted that in Fig. 3 only a part of the implemented routines is shown. Additionally, constrained optimization is used complementing the basic framework by a penalty handling approach. Moreover, functions for visualisation of optimization results and well-known optimization test problems are part of the optimization framework. For more detailed information on the structure of EngiO and its application to different engineering tasks, it is referred to Berger et al. (2021a, b).

2.2 Features and Benefits

According to the software architecture, EngiO has several advantages to solve engineering optimization problems. The architecture follows the concept of object-oriented programming and is completely compatible with Matlab and GNU Octave. Thus, the source code has a modular, comprehensible and user-friendly structure. The “Optimizer” class of EngiO provides a unified interface between optimization algorithms and optimization problems. The strict separation of algorithm and problem formulation prevents programming bugs and the basic functionality given by the parent class facilitates the usage of the framework. As stated previously the abstract class definition of the parent class enforce the user to split the algorithm’s logic in three distinct parts. This approach has the advantage that all algorithms are structured in a common way. This improves the readability of code and reduces code complexity.

A further benefit of the framework is that the software comes with various implementations of optimization algorithms. The algorithms included are adapted to the framework's architecture and are selected according to the state of the art. Optimization algorithms for single- and multi-objective formulations are implemented. The main focus of EngiO is on optimization tasks involving complex engineering simulations. The related optimization problems most often are not convex and derivatives of the objective function are not available. Thus, only local and global derivative-free algorithms are considered in the interface of the framework. Optimization subjected to constraints is facilitated by a constraint interface complementing the basic framework shown in Fig. 3. The constraints are imposed by adding a penalty on the objective function value, while the specific handling technique is derived from the constraint class.

Another key feature of the optimization framework is the inclusion of several analytic test functions. This allows the performance of algorithms to be tested and compared with each other. The benchmarking of algorithms and their parameter settings is conducted and supports the user decision on a suitable algorithm. Further, the user benefits from the common interface and is able to compare the performance of his own user-defined algorithms with established methods.

Moreover, the interface is further designed such that single- as well as multi-objective optimization problems could be solved, allowing for a broad range of engineering applications. The number of objectives is signalled to the optimizer class calling the objective function with an empty argument. Simple scripts for both cases support the beginner to start with his own optimization setups. Since engineering optimization problems often rely on time-consuming numerical computations, the framework allows to store the current state variables of each iteration. This allows to restart and continue the optimization based on previous optimization results. EngiO also features parallelization aspects. The evaluation of objectives can be performed in parallel to speed up optimization of numerically costly optimization problems. The parallel computing is organized in the “Optimizer” class.

Overall, the design of the optimization framework is based on the requirements of an engineer. The strength of the framework therefore lies in its adaptability to individual needs. Since the framework is regarded as a pure research and teaching code, some basic graphic representations of results are provided.

To illustrate the capabilities of EngiO the results of a multi-objective optimization run utilizing a well-known test function are presented below. The test function was introduced by Poloni (1997) and is an unconstrained two-dimensional two-objective formulation. The optimization problem is stated as follows:

$$\begin{aligned}& minimize\quad \begin{array}{l}{F}_{1}=1+{\left({A}_{1}-{B}_{1}\right)}^{2}+{\left({A}_{2}-{B}_{2}\right)}^{2}\\ {F}_{2}={\left({x}_{1}+3\right)}^{2}+{\left({x}_{2}+1\right)}^{2}\end{array} \\ & with \quad \begin{array}{l}{A}_{1}=0.5\text{sin}\left(1\right)-2\text{cos}\left(1\right)+\text{sin}\left(2\right)-1.5\text{cos}\left(2\right)\\ {A}_{2}=1.5\text{sin}\left(1\right)-\text{cos}\left(1\right)+2\text{sin}\left(2\right)-0.5\text{cos}\left(2\right)\\ {B}_{1}=0.5\text{sin}\left({x}_{1}\right)-2\text{cos}\left({x}_{1}\right)+\text{sin}\left({x}_{2}\right)-1.5\text{cos}\left({x}_{2}\right)\\ {B}_{2}=1.5\text{sin}\left({x}_{1}\right)-\text{cos}\left({x}_{1}\right)+2\text{sin}\left({x}_{2}\right)-0.5\text{cos}\left({x}_{2}\right)\end{array} \\ & subject \; to \quad \begin{array}{c}-\pi \le {x}_{1}\le \pi \\ -\pi \le {x}_{2}\le \pi \end{array} \end{aligned}$$

(1)

The optimization problem is solved using the Non-dominated Sorting Genetic Algorithm-II (NSGA-II), which is a state-of-the-art approach in the context of multi-objective optimization (Deb et al. 2000). The optimization process is stopped after 10,000 objective function evaluations are performed. The optimization results of one optimization run are illustrated in Fig. 4.

Fig. 4

Pareto frontier of Poloni test function optimized using NSGA-II

In Fig. 4, all samples generated during the optimization are plotted in the two-dimensional objective value space. The solutions on the Pareto front, which means that they are non-dominated by other solutions, are further highlighted in blue. In this optimization example, the Pareto front is divided into two parts. This discontinuity in the Pareto front is characteristic for the selected test function. The solutions on the front are well distributed and the comparison with optimization results published by other researchers reveals, that the algorithm identified the Pareto-optimal set correctly. It should be noted that the slope of the Pareto front is individual for each optimization problem formulation (Angus and Woodward 2009).

The selected optimization example can be easily recalculated with EngiO, because the algorithm as well as the test function are part of the framework. The main script used to start the optimization comprises about less than 30 lines of code and has a clear structure. This is mainly allowed by the object-oriented schema and the choice of a high level programming language. Examples for main scripts are also part of the software, which facilitate the application of EngiO in an early stage and allows an easy incorporation into the framework, even for inexperienced users.

3 Structural Models

One way to predict the structural integrity of repaired blades is to model changes in the blade properties, re-analyse stresses and vibration frequencies and predict the remaining lifetime of the blade.

To allow for an automated simulation process several aspects have to be described in a standardized way. The first aspect refers to the description of the damaged portion of the blade. This damage model is introduced in Sect. 3.1 and is utilized for blend as well as patch repairs. In Sects. 3.2 and 3.3 the repair-specific models for blend and patch repairs are introduced.

3.1 Damage Model

Defects of blade may have various forms and shapes. To account for this individual patterns, the defect portion of the blade is described using a point cloud. From optical measurements defect areas can be detected and specified using a set of points in a global coordinate system. This universal specification allows for a very flexible description of blade regions to refurbish. However, in most cases, the damage portion is located at the leading edge of the blade, since particles entering the engine lead to so-called foreign object damage. This kind of damage is illustrated by a point cloud in Fig. 5.

Fig. 5

Blade with damaged region at the leading edge. The size and location of the defect is specified using a set of points in global coordinate system (X, Y, Z)

The resulting damage model is utilized to describe the damage portion and facilitates the subsequent search for optimal repair design. Since blend repair as well as patch repair aim to remove the defect completely in each iteration it should be checked whether the repair design fulfils this requirement. In the case of the blend repair, the removed volume should include all damaged points. When a patch repair is conducted the geometry of the patch has to cover the damaged region as well.

3.2 Blending Model

The model developed for blending of blisk blades is based on the assumption that the blending shape has an elliptical contour. To model this contour a parameterized ellipsoidal volume is utilized. The schematic illustration of a blade and the volume is depicted in Fig. 6.

Fig. 6

Blade with blendig shape parameterized by three design variables d, e and r_z and the related ellipsoidal tool

In accordance with typical damaged regions, as shown in Fig. 5, the ellipsoid is located at the leading edge of the blade and the blending shape corresponds to the cut-out enforced by the ellipsoidal volume. The geometric modification of the blade is thus the result of a Boolean operation between ellipsoidal volume and nominal blade volume.

The position and size of the ellipsoidal tool is defined by three scalar design variables. The first variable e specifies the position of the ellipsoid. The variable in particular determines the distance between the tip of the blade and the centre of the elliptical cut-out. Since only blends at the leading edge are considered, e runs along the leading edge of the blade and is normalized according to the total length of the edge, such that 0 ≤ e ≤ 1. The other two variables define the size of the ellipsoid and cut-out, respectively. The depth of the cut-out resulting from the relative position between ellipsoidal tool and blade is stated by d. The length of the vertical principal axis of the ellipsoid is specified by r_z. The other radii of the ellipsoid are set to fixed values prior to the sampling.

Modelling this geometric change in terms of FE analysis, the mesh of the blade has to be adapted to the new blended blade contour. One way is to utilize an automated meshing procedure and mesh the new blade geometry for each computing example. This approach, however, has the drawback that the simulations for each sample are performed with possibly very different meshes, which could introduce numerical errors. Moreover, evaluating results of simulations performed with similar meshes is much more straight-forward, because displacements can be compared directly. To maintain the majority of the FE mesh related to the nominal geometry a local re-meshing procedure was developed. Only the mesh in the proximity of the blend is adjusted to capture the new geometry, while the rest of the mesh remains unchanged.

3.3 Patching Model

From a geometric point of view, the parametric description of patch repairs is not as complex as for blend repairs. Since the blade region, which is replaced by the patch material mainly has a triangular shape the repair is specified by two dimensions only. In Fig. 7 a blade with a patch is illustrated.

Fig. 7

Blade with patch parameterized by two design variables a and b resulting in a triangular patch geometry

As in the previous section, we concentrate on repairs at the leading edge of the blade. Therefore, one variable is again defined in relation to this edge. The variable a determines the height of the patched region and equals the distance from tip to interface between patch and base material. The second variable b corresponds to the width of the patched part and is measured along the chord of the blade. For the purpose of a more intuitive understanding of design variables, a and b are normalised according to the length of the span and chord of the analysed blade.

In Fig. 7, the interface between patch and blade (coloured in red) is idealized as a plane and therefore is of infinitesimal width. In reality, both parts are joined via a welding procedure and, hence, there is the welding bead and the corresponding heat affected zone. The volume of this region depends on the welding process and is strongly influenced by the welding parameters used. To assess these thermal and mechanical effects within this region thermomechanical FE simulations are employed. Firstly, the temperatures caused by the moving heat source are computed via a transient thermal simulation. The heat input of the moving heat source is implemented in a user-defined subroutine. The current position of the heat source is defined using a welding trajectory along the centreline of the weld and a heat flux distribution according to a conical heat source volume. Residual stresses in the blade are computed in subsequent mechanical simulation.

4 Analyses and Objectives

The optimization objectives of the optimization problem formulation result from the design goals of repair procedures. These design goals could further be classified by two groups. The first category refers to structural aspects. In particular, fatigue strength and vibration properties are part of this first category. The second group relates to practical aspects such as manufacturing effort. In the next sections the optimization objectives are introduced and discussed with regard to blending and patching procedures.

4.1 Natural Frequencies and Mode Shapes

One of the most important aspects in the design of rotating machines refers to resonance conditions. During operation of jet engines, compressor blades are subjected to multiple excitation frequencies, which could cause resonances, if the excitation frequency meets one of the natural frequencies of the blisk. This condition is commonly analysed using established concepts like the Campbell diagram or Interference diagram. In the initial design phase the frequencies of the blisk are, hence, tuned such that they do not intersect with the excitation during nominal operation. These vibration properties of blades are predicted utilizing a modal analysis of the geometry. Exemplary results of a modal analysis at standstill are shown in Fig. 8.

Fig. 8

FE mesh (a), mode shape of the 10th eigenmode (b), and related modal vibratory von Mises stresses (c) of blade sector

To reduce computation time, the modal analysis is not performed using the whole blisk structure. Instead one periodic sector is considered and cyclic symmetry constraints are imposed on the cyclic faces of the sector, neglecting mistuning effects. As depicted in Fig. 8a the sector geometry is meshed using two different element types. The blade and the disk region are meshed using hexahedral elements, while the fillet region requires the use of tetrahedral elements due to the more complex geometry. In Fig. 8b the mode shape of the 10th vibration mode, which refers to a frequency of about 9500 Hz, is shown. Due to the high mode, several local vibration maxima are computed in the blade region, which results in the related vibratory stress distribution Fig. 8c.

In the case of blending, however, the geometry differs from the nominal condition and natural frequencies change. To gain first insights into the effect of these local modifications on natural frequencies, different blend repairs are analysed using the blending model introduced in Sect. 3.2. 6859 samples following a grid sampling scheme are generated and modal analysis is performed for each sample. To capture the stiffening effects due to rotation the modal analysis is performed based on a static analysis imposing rotational forces according to nominal operation. The results of the sampling for two natural frequencies are shown in Fig. 9. The reference frequency denoting the simulation result of the nominal blade is marked by the dark blue line.

Fig. 9

Scattering in first (left) and fifth (right) natural frequencies due to performed blend repairs at the leading edge

The histograms indicate that blend repairs affect the tuning of the first as well as the fifth frequency. Concerning the first frequency mainly a slight increase of the frequency is determined, because the removal of the material mainly reduces the vibrating mass of the blade. In contrast, the distribution of the fifth frequency differs significantly from the distribution calculated for the first mode. While almost any blend repair leads to a higher first frequency, the distribution corresponding to the fifth mode also results in more reduced natural frequencies. This effect is caused by the different mode shapes. The lowest natural frequency corresponds to a pure bending mode. A blend therefore increases the frequency, as the mass reduction predominates over the loss of local stiffness in the blending area.

To improve the blend geometries for damaged blisk blades, two design goals are derived from the statistical study on natural frequency tuning. The first objective results from the comparison between nominal and repaired modal properties. To reduce the risk of mistuning, the repaired blade should behave similarly to the nominal one. Therefore, the difference to the nominal (design) frequencies is considered. The maximal deviations from nominal frequencies are calculated as

$${F}^{nom}=\underset{i}{\text{max}}\left(\frac{\left|{f}_{i}\left(x\right)-{f}_{i}^{nom}\left(x\right)\right|}{{\alpha }_{i}}\right)$$

(2)

where α_i is a weighting factor, f_i is the ith natural frequency of the blended blade, ${f}_{i}^{nom}$ is the corresponding ith natural frequency of the nominal geometry, and x is the design variable vector with repair-specific parameters. The formulation in (2), hence computes the maximum deviation of multiple natural frequencies with respect to the nominal case. Since the variation of some frequencies is more critical than others the weighting factor could be adjusted. Below, the weighting is performed using the nominal frequencies.

A further design goal refers to the distance between natural frequencies and excitation frequencies, which quantifies the risk of meeting resonance conditions. The minimal distance to the closest excitation frequency during nominal operation is stated as

$${F}^{exc}= \underset{i}{\text{min}}\left({\beta }_{i }\cdot \left|{f}_{i}\left(x\right)-{f}_{i}^{exc}\left(x\right)\right|\right)$$

(3)

where β_i is a weighting factor, f_i is the ith natural frequency of the blended blade, ${f}_{i}^{exc}$ is the corresponding closest excitation frequency, and x is again the design variable vector with repair-specific parameters. With the weighting factor, different excitations can be penalized differently.

4.2 High-Cycle Fatigue Strength

Although blades are designed such that resonances are preferably avoided, during operation blades are still subjected to vibrations. These vibrations lead to alternating stresses in the blade material and could finally result in fatigue failure. This failure mode, denoted as high-cycle fatigue, therefore, must be considered in the initial blade design as well as with repaired blisks.

The well-established evaluation concept for HCF, which is also used in this work, is illustrated in Fig. 10.

Fig. 10

Constant life diagram with an initial design point and a modified design point corresponding for example to a repaired blade

The diagram shows the dependency between HCF properties and the alternating stress and the mean stress level. The line connecting the endurance limit σ_e and the yield strength σ_y subdivides the diagram in a safe and a failure region. To ensure safe operation, all designs (nominal as well as repaired) have to correspond to the region below this line. Referring to rotating blades, the mean stresses in the initial design are mainly caused by centrifugal forces and the mean stress level depends on the rotational speed of the engine. The alternating stresses originate from blade vibration. Depending on the excited mode, this results in different spatial stress distributions as shown for the 10th mode in Fig. 8.

As indicated by the red arrow in Fig. 10, the stress state in the modified design (dark blue marker) commonly deviates from the nominal state (light blue marker). In the worst case, both stresses are increased and the modified state is located close or even above the constant life line and failure occurs. One reason for higher alternating stress is the amplification of vibration amplitudes and hence stress amplitudes. For the computation of changes in the vibration amplitudes mistuning as well as aerodynamic conditions have to be considered. Since mistuning (see also subprojects C3 and C6) and aerodynamic are not part of this work, it is focused on the second aspect—the mean stress level. In particular, after a patch repair, the medium stresses in the blade are strongly influenced by the repair process. In the nominal case the blade is assumed to be free from residual stresses, but the welding process involved in patching induces residual stresses in the blade material. These residual stresses are determined in thermomechanical simulations and the welding stresses like they are shown in Fig. 11 are computed.

Fig. 11

Residual stresses (von Mises) remaining in the proximity of the weld after patch repair

Despite the post-weld heat treatment, the depicted simulation results show that the welding process leads to a significant increase in stresses in the proximity of the welding path. These stress components suppose on the mean stresses due to centrifugal forces and in most case increase the mean stress level in parts of the blade. One design goal of patch repairs is therefore to place the weld in a blade region where the combination of increased mean stresses and vibratory stresses does not lead to failure. The minimal strength of the joint between patch and weld is determined by

$${F}^{HCF}= \underset{i}{\text{min}}\left(a{f}_{i }\right)= \underset{i}{\text{min}}\left(\frac{{\sigma }_{e}}{{\sigma }_{v,i}}\left(1-\frac{{\sigma }_{m,i}}{{\sigma }_{y}}\right)\right)$$

(4)

where af_i is the amplitude frequency strength of all FE nodes i, and σ_v,i and σ_m,i are the related alternating and mean stresses. The amplitude frequency strength reflects the diagram shown in Fig. 10 and all stresses in (4) are evaluated using von Mises equivalent stresses.

Moreover, the endurance limit of the welded region is significantly lower than for the base material and therefore it is assumed that failure occurs in joint region only.

4.3 Manufacturing Aspects

In practice, a lot of manufacturing aspects drive the optimal repair design of blisk blades. Since the blades of a blisk are closely arranged, there is relatively small space for welding, drilling or any other manufacturing tool. The geometric boundaries of repair design, should therefore be selected in accordance with the particular manufacturing limitations. In addition, different repair designs may vary in manufacturing complexity as well. Considering patch repairs the difficulties involved in welding process also depend on the particular welding path, e.g. the curvature of the blade along the path. A practical design goal for the welding process of patching is hence the length of the weld

$${F}^{length }={{\ell}}^{weld\, path}\left(x\right)$$

(5)

because it corresponds to the manufacturing effort.

5 Optimization and Results

The develop concept for the optimization of blisk repair procedures is exemplarily applied to a real blisk geometry. The blisk geometry was designed at the Institute of Turbomachinery and Fluid Dynamics and operates in an aerodynamic testing facility. The geometry of the blisk and some basic properties are illustrated in Fig. 12.

Fig. 12

Blisk geometry (left) used for all numerical examples and corresponding properties of the blisk and the axial compressor (right)

As marked in Fig. 12, the simulations are carried out on one out of 24 sectors of the compressor blisk using the FE solver of Abaqus. The analyses for the prediction of structural properties are performed according to the considerations presented in Sect. 4. Simulation results are processed in custom routines and related optimization tasks are implemented as objective functions in the optimization framework EngiO. Since multiple contradicting objectives are of similar importance the optimization problems are formulated using a two-objective formulation. Boundaries are specified in the main script according to the repair-specific parameters of the mode (Sect. 3). The constraint interface of EngiO is utilized to impose constraints and ensure valid repair designs only.

The multi-objective optimization problems reflecting the structural requirements for blend and patch repairs are described in Sects. 5.1 and 5.2. The optimization results determined utilizing derivative-free optimization algorithms are visualised in the objective value space and Pareto-optimal repair designs are discussed.

5.1 Optimization Results for Blending

The first optimization refers to the optimization tasks corresponding to blend repair optimization. According to the optimization objectives introduced in Sect. 4.1 the tuning of natural frequencies of the repaired blade are optimized. The two-objective optimization problem is stated as

$$\begin{aligned}& \text{minimize}\quad \begin{array}{c}{F}_{1}={F}^{nom}\left(x\right)\\ {F}_{2}=-{F}^{exc}\left(x\right)\end{array} \\ & with\quad x={\left[e,d,{r}_{z}\right]}^{T} \\ & subject\; to \quad \begin{array}{l}g(x)\le 0\\ 0.05 \le {x}_{1}\le 0.5\\ 0.5\le {x}_{2}\le 5\\ 0.5\le {x}_{3}\le 25\end{array} \end{aligned}$$

(6)

where g(x) ≤ 0 is an inequality constraint, which is violated when the damage is not completely removed by the blend design. The optimization problem in (6) is formulated as a minimization problem, and consequently for the second objective the expression stated in (3) is multiplied by −1. The design variables and upper and lower boundaries are selected according to the blending model introduced in Sect. 3.2.

Figure 13 shows the damage pattern of the computational example and the results of the optimization problem specified in (6) in the objective value space. The optimization is performed using the state-of-the-art algorithm NSGA-II and allowing for 10,000 objective function evaluations with a population size of 100 individuals. The constraint handling scheme implemented in EngiO is used to penalize infeasible design using a linear penalty approach.

Fig. 13

Damage pattern on the leading edge of a bladed blisk sector (left) and the Pareto front determined for the blend repair optimization in one optimization run (right)

All solutions sampled during the optimization process are highlighted in grey while the non-dominated solutions, which form the Pareto front are coloured in light blue. Similar to the Poloni test problem optimized in Sect. 2, the Pareto front consists of two parts. Consequently, there is no feasible solution in between, which at least to an improvement with respect to one of the two objectives. The practical meaning of the optimization results is illustrated in Fig. 14. Six different Pareto-optimal blend designs, which correspond to the non-dominated points labelled with (a) up to (f) in Fig. 13, are visualised. The first three solutions (a–c) represent the upper left part of the Pareto front. The designs in this part of the front are characterised by a relatively small cut-out and the centre of the blend is located next to the damage. Moreover, the three designs indicate, that the blade frequencies with respect to the excitations (objective value 2) improve with larger blends, while the increased deviation from the nominal shape leads to non-favourable deviations from nominal frequencies (objective value 1). Further, the second part of the Pareto front is associated with blend shapes, which are located closer to the tip of the blade (small e). To still remove the damage, the blending depth and width are hence larger than for most designs of the first part of the Pareto front. For example, blend design (f) has the largest depth of 2.63 mm. With regard to aerodynamic performance of blended blades, especially the Pareto-optimal solutions, which refer to large cut-outs, have to be re-evaluated by means of CFD simulations before choosing this type of blend design. In the presented example, the solutions on the first part of the front should be preferred to avoid poor aerodynamic properties.

Fig. 14

Six exemplary blend designs (a)–(f) out of the Pareto-optimal set

5.2 Optimization Results for Patching

The second optimization considers the design of patch repairs accounting the design goals discussed in Sects. 4.2 and 4.3. Based on the patching model a two-objective optimization task is formulated

$$\begin{aligned}& \text{minimize}\quad \begin{array}{c}{F}_{1}={F}^{length}\left(x\right)\\ {F}_{2}=-{F}^{HCF}\left(x\right)\end{array} \\ & \quad with \quad x={\left[a,b\right]}^{T}\\ & subject\; to \quad \begin{array}{l}g(x)\le 0\\ 0.0\le {x}_{1}\le 1\\ 0.0\le {x}_{2}\le 1\end{array}\end{aligned}$$

(7)

where g(x) ≤ 0 is an inequality constraint, which is violated when the damage is not completely removed by the welded patch. Since large HCF strength is favourable and the problem is formulated as a minimization the fatigue strength value is again multiplied by −1. Via the boundaries the two normalized design variables (Sect. 3.3) describing the patch geometry are limited to values between zero and one.

In the left part of Fig. 15 the damage pattern of the computational example is visualised. The same optimization algorithm with the same algorithmic settings as well as the penalty handling approach is used to solve the constrained optimization problem. Moreover, the optimization problem defined in (7) is computed for two different variants of the second objective function. In the first optimization, the fatigue strength corresponding to the first vibration mode is optimized, whereas the second optimization focuses on the fifth vibration mode. In particular, these two modes are studied, because they show an increased risk of high vibrations according to the Campbell diagram. The Pareto front of the first variant is shown in Fig. 15. The non-dominated solutions computed within 10,000 samples show a continuous course. Additionally, it can be seen that no non-dominated point refers solutions with a first objective value greater than 0.4. This indicates, that all patches with welding seams longer than 4 cm are not optimal in any case. This is also reflected by the associated patch geometries, which are shown in Fig. 16.

Fig. 15

Damage pattern on the leading edge of a bladed blisk sector (left) and the Pareto front determined for the patch repair optimization accounting for HCF due to first vibration mode (right)

Fig. 16

Patch designs corresponding to the optimization results shown in Fig. 15

As indicated by the blue area, all Pareto-optimal patches and therefore all welding beads are located in the upper part of the blades. According to their geometry these patches are also classified as short patches. Additionally, one Pareto-optimal design leading to a solution in the middle part of the Pareto front is shown in Fig. 16. The optimization results are reasonable, since the first mode is a bending mode. The bending results in vibratory stresses next to the blade root and thus beads close to the root are not optimal. The results of the second optimization for the fifth mode are depicted in Fig. 17. The Pareto front has again no significant discontinuities like they are determined for the blend repair. The analysis of Pareto-optimal designs, which are illustrated in Fig. 18, however, reveals, that the designs are different to the designs of the previous optimization.

Fig. 17

Pareto front determined for the patch repair optimization accounting for HCF due to fifth vibration mode

Fig. 18

Patch designs corresponding to the optimization results shown in Fig. 17

The change in the second objective function leads to a Pareto-optimal set that includes of solutions with longer welding beads. For further illustration an exemplary design (b) is visualized in the objective values as well as in the design variable space. These difference in optimization results between both tasks relate to the mode shape and the resulting vibratory stresses. Higher modes often lead to multiple local blade regions with high vibratory stresses. Consequently, the Pareto-optimal designs correspond to weld beads that are not placed close to these regions.

6 Conclusions and Outlook

In this work, a concept for the systematic evaluation and optimization of blade repairs of blisks by blending and patching was presented. The numerical concept addresses several aspects. First, the optimization framework EngiO, which specifically targets at engineering optimization tasks, was introduced. The object-oriented architecture was described and the benefits of the design were demonstrated using the two-objective Poloni test function. Finally, the capabilities of EngiO provided the basis for all further developments in the field of blisk repairs. Systematically and automated simulation of the two repair processes was performed by parametrization and custom FE simulation models. A damage model was used for unified description of arbitrary damage patterns. With respect to repair-specific changes design goals and requirements are discussed and established evaluation metrics were used to assess the structural properties of repaired blades. Combining multi-objective optimization methods, repair-specific modelling and design objectives Pareto-optimal repair designs were found automatically based on predefined damage patterns. Depending on the particular optimization tasks the Pareto fronts showed to have different appearances. Continuous as well as discontinuous Pareto fronts are determined. Moreover, the Pareto-optimal set and the corresponding repair design allow new insight in the repair design process. In this context, the choice of the multi-objective formulation of the optimization problem is a great advantage, because the final decision of the patch or blend repair design could be made based on the whole Pareto-optimal set and accounting for additional engineering preferences. The weighting between individual optimization objectives could be done a posteriori to the optimization run.

In future work, the computational scheme developed in this work could be extended by several aspects. One aspect corresponds to the damage model introduced in Sect. 3.1. In this contribution the optimization is carried out exemplarily on few damage patterns only. Moreover, this patterns are generated based on engineering experience but do not correspond to measurements. In the future, it would be beneficial to systematically inspect damaged blisk blades with optical measurement equipment. Using this data multiple optimization runs could be performed and a more generalized conclusion could be drawn. Further, the works does not consider aerodynamic requirements or mistuning effects. Since both aspects are important for the functionality of blades, these aspects should be investigated in future research. Similarly, simulation models e.g. welding simulations could be improved to capture effects on the structure in more detail. However, with more complex numerical simulation the optimization becomes more difficult, since numerical costly simulations slow down optimization processes.