# Partitioned time couplings of an aero-mechanical wind turbine problem.

Caroline Le Guern, Frédéric Blondel, Jean-Lou Pfister, Nicolas Guy, Michel Bergmann

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Caroline Le Guern, Frédéric Blondel, Jean-Lou Pfister, Nicolas Guy, Michel Bergmann. Partitioned time couplings of an aero-mechanical wind turbine problem. Journal of Physics: Conference Series, 2024, 2767 (2), pp.022012. 10.1088/1742-6596/2767/2/022012. hal-04878061

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PAPER • OPEN ACCESS

Partitioned time couplings of an aero-mechanical
wind turbine problem.

To cite this article: Caroline Le Guern et al 2024 *J. Phys.: Conf. Ser.* **2767** 022012

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# Partitioned time couplings of an aero-mechanical wind turbine problem.

Caroline Le Guern¹,², Frédéric Blondel¹, Jean-Lou Pfister¹, Nicolas Guy¹, Michel Bergmann²

¹IFP Energies Nouvelles, 1-4 Avenue du Bois Préau, Rueil-Malmaison, France.

²INRIA Bordeaux, 200 Avenue de la Vieille Tour, Talence, France.

E-mail: caroline.le-guern@ifpen.fr

**Abstract.** Wind turbine developments have led to more powerful offshore rotor systems, creating complex aero-hydro-servo-elastic behaviors that require numerical analysis. A mutliphysics problem can be modelled using several approaches: a "monolithic" method where the systems are considered as a unique block or a "partitioned coupling" where the different domains are solved sequentially. The latter is the retained technique for this work in which different solvers are related to distinct equations that require independent time discretisations. The focus is set on coupling a wind turbine's mechanical solver with an aerodynamic code, the latter being modelled through an inviscid vortex method and the structural aspect using a finite element method. The study aims to present the implementation process and comparison of different time integration coupling schemes within a large wind turbine simulation framework where aeroelastic effects are studied on the IEA 15MW wind turbine. A "Conventional Serial Staggered" (CSS) scheme is used as a reference coupling technique, allowing independent time integration while limiting coupling to a single time step. Alternatively, a resembling "subcycling" architecture is proposed and implemented to remedy cases with different temporal discretisations and reduce computational costs of CSS when using the inviscid vortex method.

## 1. Introduction

Offshore wind turbines are rapidly evolving technologies for which new design challenges are raised. Amongst the latter are their rotor sizes, with blades attaining lengths of more than 100 m for which a double constraint that links mass to structural stiffness has to be taken into account. Designing rigid blades, using a set of materials, results in a significant mass, while pursuing a lighter overall blade entails greater structural flexibility. Hence, accurately capturing the structural dynamics of a wind turbine across various conditions and configurations is indispensable for the design phase. Mutli-physics, and more specifically in this case aeroelastic behaviours, which take into account the interaction between the wind turbine's structure and the surrounding aerodynamics, require numerical modelling. When used for design purposes, the latter need to employ high performance simulation tools, capable of capturing aero-elastic effects with sufficient accuracy without compromising on the overall computational chain's performance. For that matter, two coupling categories can be retained: the "monolithic" method which joins the different physics into a single system solved all-together, or a "partitioned" approach [1] where physics are solved separately, the interaction being modelled through data exchange at a common interface. For industrial practices, monolithic solving is rather unmanageable. It is

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mostly used in academic cases (academic codes often have a limited number of functionalities
compared to those used for industrial purposes) or for specific coupling problems with little to
no flexibility on the full system's equations. On the other hand, partitioned techniques allow
for the re-use of established stand-alone solvers with coupling algorithms ensuring interaction
between codes. If partitioned coupling is often referred to as "loose coupling" in comparison
to "monolithic" methods, it offers a greater flexibility in terms of modelling approaches with a
potential reduction in computational costs at the expense of possible stability and accuracy losses
[2]. In the present work, two partitioned schemes, namely a "Conventional Serial Staggered"
(CSS) as well as a "Subcycling" technique [3] will be considered in a fluid/structure coupling
framework. For that matter, the structural finite element solver DeepLines [4] has been coupled
to several aerodynamic libraries that use various methods. Amongst the latter, the Blade
Element Momentum (BEM) [5] method is one of the most commonly used in wind turbine
aerodynamic simulations, as its computational cost makes it an indisputably well-suited design
tool. Advantages of BEM are nonetheless countered by the theory's hypothesis which constrains,
for instance, loads evaluation in a rotor plane - assumption that might be violated for a
large offshore (floating) wind turbine with highly flexible blades. In the present work, a less
assumption-heavy code using a free wake inviscid vortex method [6], CASTOR [7], is used for
aerodynamic loads computations. While higher fidelity is ensured by the vortex method, its use
in aeroelastic simulations makes overall computational costs higher rendering it impractical as a
viable design tool. This work aims at demonstrating how implementing alternative partitioned
coupling techniques allows for a significant computational time reduction in comparison to CSS
procedures. The first part of this work will describe the different modelling tools that are made
available for aeroelastic computation purposes, with a focus on the free wake vortex method and
its constraints when coupled to a structural solver. The aeroelastic simulation framework on the
large IEA 15MW [8] reference wind turbine will then be presented. In a second part, the focus
will be set on existing coupling methods as well as the implementation process of a partitioned
scheme in our coupling framework. The last part will present the results of the adapted subcycling
scheme compared to the original CSS coupling where significant computational cost reduction
is achieved and where changes in stability and accuracy will be discussed.

## 2. Numerical aeroelastic simulation

### 2.1. Aero-servo-elastic simulation tool

The aeroelastic modelling tool used throughout this work is DeepLines Wind [4]. It allows for multi-physics modelling, in part for offshore wind turbine simulations that gather aero-hydro-servo-elastic fields. In this work, the focus is set on the aero-servo-elasticity side of the problem, in which the wind turbine's structure is operated by a control system, namely the ROSCO controller [9]. The structural aspect of the wind turbine is modelled using a finite element method, where the tower and blades are modelled through 3D non-linear geometrically exact beam elements with large transformations for the latter. The aerodynamic side of the problem can be computed using various codes, amongst which AeroDeeP [10] and CASTOR use respectively a semi-analytical BEM method and a free wake inviscid vortex method. When considering a temporal analysis, a graphical representation of the coupling between components is depicted in fig. 1. At each time step, the structural solver transfers rotor and wind speed data to the controller (step a). In return, the controller gives blade pitch and rotation velocity instructions to the structure (step b). The fluid solver calculates the aerodynamic forces acting on the structure and transfers them to the structural code (step d). The latter thus computes the structural displacements which are then transferred back to the fluid (step c). All the previously mentioned data exchange between aero-servo-elastic components are done through solver communications which occur only once per time step in this general case.
<figure><img src="image_5.png" /><figcaption>Figure 1: Aero-servo-elastic simulation communication scheme.</figcaption></figure>

## 2.2. Aerodynamic code

In the present work, the chosen aerodynamic solver for aeroelastic simulation purposes is CASTOR. This code is a GPU accelerated solver which uses a 3D free-wake inviscid vortex method based on the lifting-line theory [11]. Solving Euler equations, which in this case are written in vorticity form, is achieved using a Lagrangian method, in which Lagrangian markers delimit sets of vortex filaments. The latter are used in order to model the vorticity field behind the rotor as well as the blade's lifting line and can be categorised into so called "trail" and "shed" filaments (for the wake) as well as "bound" filaments (for the blade). Their graphical representation is illustrated in fig. 2 and additional detail on their generation, as well as the overall vortex method algorithm, can be found in [7]. As previously mentioned, vortex method solvers are frequently not considered as well-suited design tools due to their high computational costs. This comes mainly from the coupling of such method to a servo-elastic solver which can impose small time steps in comparison to a stand alone vortex code (typically for the IEA 15MW wind turbine the first blade mode's frequency is around 0.5Hz leading to a structural time step of order $\Delta t = 0.01$s for an accurate resolution while the vortex method's ideal resolution rate is around $5^\circ$ - $10^\circ$, which corresponds to a larger time step). In this case, the structural displacements received by the fluid solver at each time step provoke the generation of an extra row of shed filaments in the wake, for which the new set of Lagrangian markers are interconnected through extra trail filaments. Inductions from wake filaments on bound filaments, as well as inductions from the wake on each Lagrangian marker are then computed and kept in memory in order to estimate aerodynamic forces. This is the most costly part of the vortex algorithm, for which the computational costs are scaled with the squared number of filaments. Thus, when coupled to a servo-elastic solver with restrictive time step, the vortex code will generate a more filament dense wake, hence increasing again the overall computational cost. To remedy this, several approaches can be considered such as filament merging methods, process that was used in [7], or partitioned time integration coupling schemes where different temporal discretisations are permitted. The latter is the retained technique for this work, but both approaches could be complimentary.

<figure><img src="image_6.png" /><figcaption>Figure 2: Schematic representation of a rotating blade and the associated wake filament discretisation.</figcaption></figure>
### 2.3. Study case

The study case is that of the reference IEA 15MW wind turbine subject to a constant wind of velocity $U_{\text{inf}} = 10.6 \, \text{m.s}^{-1}$ with a tip speed ratio of $\lambda = 8.5$. The structural code's time step is driven by the control system which imposes a coupling time step of order $\Delta t = 0.01 \, \text{s}$ (equivalent to a rotation of $0.45^\circ$ per time step). When analysing the initial blade modes of the IEA 15MW turbine, eigenfrequencies are observed around $0.5 \, \text{Hz}$ [8]. This results in the structural characteristic time being of comparable order to the control's time step. On the other hand, the inviscid vortex method is known to have an optimal wake emission at a frequency of $5^\circ$ to $10^\circ$ per time step [12, 13] showing a clear over-resolution of the fluid problem with the fore-mentioned controller. Two different partitioned coupling schemes are studied in this work, namely a CSS algorithm and a subcycling technique which are the focus of the next part of this paper. The effects of the subcycling scheme, which allows for the aerodynamics to be solved at their own rate, will be compared to that of CSS, in terms of computational cost and accuracy.

## 3. Time integration coupling methods for multi-physics problems

### 3.1. Context

The modelling of multi-physics problems can be achieved through "monolithic" or "partitioned coupling" techniques. In the first case, the different physics' governing equations are intertwined which allows for a simultaneous time integration: this method is often referred to as a "strong coupling" [3]. The stability of such a method is counteracted by its high computational costs as well as poor flexibility in terms of code and algorithm reuse. On the other hand, partitioned coupling allows for the separation of time integration of different physics as the governing equations are dissociated. Together with the reduced computational costs of such technique comes other advantages [14]: each physical domain can have its own modelling method (finite element on one side and finite volume on the other), along with distinct physical and time discretisations (with non matching meshes, and explicit/implicit integration schemes) as well as different modelling codes that can further be developed independently from one another. Although partitioned coupling algorithms show many benefits, they are often referred to as "loose coupling schemes". The latter present certain drawbacks, mainly in terms of stability. This can be explained by the communication and data transfer between the codes which introduce a lag at each time step, where an induced disequilibrium between physics can become either cumulative or compensated throughout time. The following part will introduce the different partitioned coupling methods that were considered throughout this work.

### 3.2. Partitioned coupling methods

The conventional serial staggered [15] scheme is one of the most commonly used amongst partitioned methods as it is one of the simplest configurations for sub-domain time integration separation. The different steps of such algorithm are depicted in fig. 3, where the evolution of aerodynamic and structural problems, as considered in this work, are presented. In order to go from time steps $n$ to $n + 1$, four main steps are carried out:

1) structural displacements $u^n$ and velocities $\dot{u}^n$ computed at time $t_n$ are transferred from the structural to the aerodynamic solver. This allows the aerodynamic solver to determine the positions at which the new filaments will be emitted as well as compute the relative velocity which is defined as the sum of the induced, structural and wind velocities, in order to compute aerodynamic forces.

2) time integration of the fluid problem is conducted and aerodynamic forces $F_f^{n+1}$ induced from structural displacements are obtained at time $t_{n+1}$

3) aerodynamic forces are transferred to the structural solver $F_s^{n+1}$
4) time integration of the structural problem is carried out to obtain structural displacements $u^{n+1}$ induced from aerodynamic forces at time $t_{n+1}$

In steps 2 and 4 of the above described coupling scheme, time integration is achieved through an explicit scheme on the aerodynamic's side and by implicit integration for the structure. This can vary from one problem to another, considering which physics is solved first and whether predictions are made on the first transmitted data's state at time $t_{n+1}$. In the present work, and more specifically for the free vortex wake method code coupled to the FEM solver, a Forward Euler (FE) scheme is coupled to a Newmark method for the fluid and structural problems respectively. Implicit Newmark integration schemes are often used in structural codes, while the explicit Forward Euler scheme is easier to manage with filament emissions as it allows the induced velocity to be computed only once per time step. One can see through fig. 3 that a single time step is used both for fluid and structural time integration as well as coupling time step for data transfer. This can become a limitation when two physics have different characteristic times, thus limiting the coupling time step to the smallest one. In such scenario, one of the two physics is "over-resolved", thus increasing computational costs.

<figure><img src="image_7.png" /><figcaption>Figure 3: Conventional Serial Staggered scheme between F a fluid solver and S a structural solver.</figcaption></figure>

The next algorithm focuses on a partitioned scheme that prevents this "over-resolution" problem. The "subcycling" scheme's architecture is depicted in fig. 4 in which the structural problem represents the physics with lowest characteristic time in comparison to the fluid. Subcycling allows for an extra time step to be added to the coupling scheme: $\Delta t_s$ being the structure's time step and $\Delta t_f$ counting as both the fluid integration as well as coupling time steps. The main stages of this algorithm are very similar to that of CSS: extra resolution steps are added to the structure where intermediate stages of structural displacements are computed $u^{n+1,1} \dots u^{n+1,k}$ (stages 3 through 5) while the aerodynamic data is kept constant.

<figure><img src="image_8.png" /><figcaption>Figure 4: Subcycling scheme between F a fluid solver and S a structural solver.</figcaption></figure>

### 3.3. Adapted subcycling algorithm

In fig. 5 the focus is set on the use of an integrated meta model (more detail on this are given in the next section) in the revisited subcycling algorithm. At each coupling time step
that corresponds to the aerodynamic's resolution rate, or $\Delta t_f$, the solver using the vortex method will directly communicate aerodynamic forces to the structural code. In the intermediate stages, which correspond to the structural solver's characteristic time ($\Delta t_s$), the meta model is itself in communication with DeepLines. The meta model estimates the state of the fluid in these intermediate stages by extrapolating computed aerodynamic forces from previous steps. Different types of regression techniques considered and implemented in this work were constant, linear and polynomial extrapolations but more sophisticated methods could also be implemented in this framework (ARIMA [16] models for instance for more complex aerodynamic forces forecasting scenarios such as in turbulent wind conditions).

<figure><img src="image_9.png" /><figcaption>Figure 5: Subcycling scheme with meta model.</figcaption></figure>

### 3.4. *Implementation details*

The implementation process as well as some specifications on the new coupling method will be described next. In order to achieve subcycled coupling between DeepLines and CASTOR, one has to revisit the original coupling implementation that was used for CSS aeroelastic simulations. The latter consists in connecting DeepLines' "façade" to the appropriate APIs (Application Programming Interface) in order to call the wanted dynamic link library (dll) associated to a specific aerodynamic solver. That way, at each time step, using specific API functions, information could be transferred from the structural code to the aerodynamic solver and vice versa.

Figure 6 depicts the revisited coupling architecture to which an additional coupling component denoted "coupler" as well as an extra aerodynamic library "meta model" were added. The "coupler" module now allows for the call of two aerodynamic libraries, namely one that corresponds to a "high fidelity resolution" meaning the use of an actual fluid solver (using the vortex method or other) to compute aerodynamic forces, and a "low fidelity" dll (meta model) that would only generate data relative to aerodynamic forces. Two coupling time steps are now defined, one that corresponds to the structural solver's time step and one relative to the fluid solver's resolution rate. That way, the structural solver can receive information regarding fluid forces at each time step through the meta model without the aerodynamic problem having to always be resolved through an expensive solver.
<figure><img src="image_10.png" /><figcaption>Figure 6: Coupling implementation architecture.</figcaption></figure>

# 4. Results

## 4.1. Wake length convergence

The precision of the aerodynamic forces computation using the free vortex wake method is dependent on the wake's length [13] and thus the number of vortex filaments kept behind the rotor. A convergence analysis of aerodynamic forces as a function of the wake length was performed using the subcycling algorithm with 7 sub-iterations. Owing to the high computational costs of CSS and to very similar results obtained using subcycling schemes (presented in the next section), the convergence test on the subcycled algorithm gives a good approximation of the converged wake length at a minor cost. Results of the latter are shown in fig. 7 where the sectional normal forces converge towards a solution for simulations in which a number of filaments equivalent to 10 to 15 rotations are kept in the wake. For all the following simulations in this paper, a fixed wake length equivalent to 13.5 rotor rotations are kept in the wake.

<figure><img src="image_11.png" /><figcaption>Figure 7: Convergence of sectional aerodynamic forces at radius location $r = 70$ m function of different wake lengths.</figcaption></figure>

## 4.2. Subcycling case

In this first case, a 200 s aeroelastic simulation, coupling DeepLines to CASTOR, was run using various coupling algorithms. The considered reference case uses a CSS scheme, for which the coupling time step is set to $\Delta t_c = 0.01$ s. A subcycling algorithm with a growing number of regression iterations using constant forecasting was implemented for comparison. The obtained
sectional aerodynamic forces between the CSS and the subcycling methods are given in fig. 8. One can see that as sub-iterations are added to the coupling algorithm, a slight difference in amplitude in aerodynamic forces is observed in comparison to the reference case. A small phase offset is also generated with the sub-iterations which could be due to the estimation of induced displacements from the constant forecasted aerodynamic forces received by the structural solver, thus possibly causing a lag between the time evolution of both physics. This is observed until a subcycled structure with 7 iterations, thus allowing the aerodynamics' integration time step to be set at $\Delta t_f = 0.08$s, above which the coupling algorithm diverges. Table 1 shows the total CPU times of the test case using the different coupling algorithms. By adding sub-iterations to the coupling scheme and thus allowing the fluid problem to be solved at a slower rate, total CPU time is drastically reduced with up to more than 250 times faster computing between a 7-iteration subcycled scheme and the CSS method. This is due to the reduction of wake filament emissions from the vortex method and the direct meta model use for aerodynamic data communication.

<figure><img src="image_12.png" /><figcaption>Figure 8: Sectional aerodynamic forces comparison for CSS method and subcycling scheme using constant regression at radius location $r = 70$ m.</figcaption></figure>

<table><caption>Table 1: Total CPU times for CSS and subcycled schemes.</caption><thead><tr><th>Method</th><th>CSS</th><th>1 sub-it</th><th>2 sub-it</th><th>3 sub-it</th><th>4 sub-it</th><th>5 sub-it</th><th>7 sub-it</th></tr></thead><tbody><tr><td>Total CPU time</td><td>10j16h</td><td>35h06min</td><td>11h24min</td><td>5h01min</td><td>2h55min</td><td>1h51min</td><td>57min</td></tr></tbody></table>

Results regarding meta model predictions using linear and second order polynomial regressions (the latter using the three previously solver computed sectional forces) are shown in fig. 9. The tendencies of both forecasting techniques are similar to that of the constant regression behaviour: adding subcycles induces differences in the aerodynamic forces amplitudes while also creating a slight lag compared to CSS. In this case, the maximum iterations for subcycles using linear and polynomial forecasting are 4 and 2 respectively, above which the subcycling algorithm diverges. Tables 2 and 3 gather the relative errors in amplitude and mean height compared to CSS of the different subcycled algorithms. It shows how for all subcycling schemes, the amplitude and mean height of normal sectional aerodynamic forces are slightly underestimated compared
to CSS. One can also notice that for a given number of subcycles, the relative error in mean height is lowest when using a constant regression scheme, while polynomial forecasting generates the highest relative error.

<figure><img src="image_13.png" /><figcaption>Figure 9: Sectional aerodynamic forces comparison for CSS method and subcycling scheme using linear and polynomial regression at radius location $r = 70m$.</figcaption></figure>

<table><caption>Table 2: Relative error of amplitude between times 142s and 152s compared to CSS of sub-iterations methods with constant, linear and polynomial regression.</caption><thead><tr><th>Sub-it number</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th></tr></thead><tbody><tr><td>Constant regr err (%)</td><td>3.8</td><td>3.38</td><td>3.39</td><td>3.2</td><td>2.93</td><td>3.15</td><td>2.26</td></tr><tr><td>Linear regr err (%)</td><td>3.0</td><td>3.48</td><td>2.94</td><td>3.14</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Polynomial regr err (%)</td><td>1.85</td><td>2.17</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr></tbody></table>

<table><caption>Table 3: Relative error of mean height between times 142s and 152s compared to CSS of sub-iterations methods with constant, linear and polynomial regression.</caption><thead><tr><th>Sub-it number</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th></tr></thead><tbody><tr><td>Constant regr err (%)</td><td>0.78</td><td>0.77</td><td>0.74</td><td>0.75</td><td>0.73</td><td>0.74</td><td>0.79</td></tr><tr><td>Linear regr err (%)</td><td>1.09</td><td>1.33</td><td>1.26</td><td>1.30</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Polynomial regr err (%)</td><td>1.72</td><td>1.55</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr></tbody></table>

## 5. Conclusion

In this work, different partitioned coupling algorithms were studied in a fluid/structure interaction framework. The FEM structural code DeepLines was coupled to the inviscid vortex method solver CASTOR where coupling algorithms were analysed on the large IEA 15MW reference wind turbine. In a first part, it was argued how a higher fidelity method was needed in terms of aerodynamic modelling for design purposes especially for larger offshore wind turbine models. It was shown that when coupled to a servo-elastic solver, the free vortex wake
method used with usual coupling methods remains expensive to be considered for industrial use
purposes. This is mainly due to the wake discretisation of the vortex method which is directly
connected to the time step imposed by the coupling algorithm inducing highly dense filament
wakes. Existing partitioned coupling techniques, including CSS and subcycling techniques
were introduced and the latter's implementation and adaptation to the CASTOR-DeepLines
framework was presented. Results showed a very promising computation time gain when making
use of subcycling procedures, with aerodynamic forces amplitudes slightly underestimated in
comparison to the reference CSS solution. The stability of subcycled schemes was shown to
be afflicted by the number of sub-iterations imposed: for constant, linear and polynomial
forecasting a critical number of structural iterations exists above which the coupling process
diverges. Perspectives for this work could include using an adaptive time step where in the
transitional starting up phase a subcycled algorithm with only 1 or 2 sub-iterations would be
used and where more sub-iterations would be allowed in the steady state stage, thus limiting
stability issues while reducing computational costs.

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