Partitioned Nonlinear Structural Analysisof Wind Turbines Using BeamDyn∗

Qi Wang†, Michael A. Sprague‡, Jason Jonkman§, and Bonnie Jonkman¶

National Renewable Energy Laboratory, Golden, CO 80401

We review BeamDyn as a module coupled under the FAST modularization framework, which providesloose coupling between multi-physics modules in time-domain simulations. BeamDyn, which is a beam solverbased on Legendre spectral finite elements and geometrically exact beam theory, was created for high-fidelitypredictive simulations of modern, highly flexible wind turbine blades. After reviewing the underlying Beam-Dyn theory and implementation, we describe the FAST coupling algorithm, which was designed for wind tur-bine analysis. Numerical experiments with a simple beam-spring-damper-mass system are used to verify anddemonstrate some of the features of BeamDyn and the loose-coupling algorithm. Realistic wind turbine simu-lations are performed with BeamDyn for the NREL 5-MW reference turbine, and we compare results againstthe established FAST-ElastoDyn blade model. We demonstrate that the 5-MW blade can be accurately simu-lated with only six nodes in the BeamDyn blade model when evaluated with an accurate quadrature schemethat captures all provided sectional properties.

I. Introduction

Recently, researchers at the National Renewable Energy Laboratory (NREL) developed BeamDyn1–4, an open-source nonlinear structural dynamics module for composite wind turbine blade analysis, in the FAST modularizationframework. BeamDyn, which is founded on geometrically exact beam theory (GEBT), was created to accurately sim-ulate the large, nonlinear deflections of modern wind turbine blades that are designed with aero-elastic tailoring andcomplicated composite structures. GEBT was first proposed by Reissner5 and then extended to three-dimensional(3D) beams by Simo6 and Simo and Vu-Quoc7. Readers are referred to Hodges,8 in which comprehensive derivationsand discussions on nonlinear composite-beam theories can be found. In the BeamDyn implementation, the GEBTequations are discrretized spatially with Legendre spectral finite elements (LSFEs), which are p-type high-order el-ements that combine the accuracy of global spectral methods with the geometric modeling flexibility of low-orderh-type elements. More details on BeamDyn can be found in Wang et al.2

FAST is NREL’s flagship multi-physics engineering tool for simulating both land-based and offshore wind turbinesunder realistic operating conditions. The established FAST blade model included in the module ElastoDyn is inca-pable of predictive analysis of highly flexible, composite wind turbine blades. FAST has been reformulated under anew modularized framework that provides a rigorous means by which various mathematical systems are implementedin distinct modules and coupled to other modules. The restructuring of FAST greatly enhanced flexibility and expand-ability to enable further developments of functionality without the need to recode established modules. These modulesare interconnected to solve for the coupled dynamic response of wind turbines.9,10 In previous work, this frameworkwas extended to handle the non-matching spatial grids at interfaces and non-matching temporal meshes that allowmodule solutions to advance with different time increments and different time integrators.11

In this paper, the formulation of the beam theory is first reviewed and then the coupling algorithm of BeamDynwith other modules is presented. Numerical examples are provided to demonstrate the capabilities of BeamDyn andits coupling algorithm when it is running in a coupled-to-FAST mode.

∗The submitted manuscript has been offered by employees of the Alliance for Sustainable Energy, LLC (Alliance), a contractor of the U.S.Government under Contract No. DE-AC36-08GO28308. Accordingly, the U.S. Government and Alliance retain a nonexclusive royalty-free licenseto publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes.†Research Engineer, National Wind Technology Center, AIAA Professional Member. Email: Qi.Wang2@nrel.gov‡Senior Research Scientist, Computational Science Center, AIAA Professional Member.§Senior Engineer, National Wind Technology Center, AIAA Professional Member.¶Senior Scientist, National Wind Technology Center, AIAA Professional Member.

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II. Formulation and Implementation

A. Geometrically Exact Beam Theory

This section briefly reviews the geometrically exact beam theory. Further details on the content of this section canbe found in many other papers12 and textbooks.8,13 Figure 1 shows a beam in its initial undeformed and deformedstates. A reference frame bi is introduced along the beam axis for the undeformed state and a frame Bi is introducedalong each point of the deformed beam axis. The curvilinear coordinate x1 defines the intrinsic parameterization ofthe reference line.

B 1

B 2

B 3

Deformed State

Undeformed State

r

R

R

s

r

u

x1

b 1

b 2

b 3

R ˆ

r ˆ

Figure 1: A beam deformation schematic.

In this paper, matrix notation is used to denote vectorial or vectorial-like quantities. For example, an underlinedenotes a vector u, a bar denotes a unit vector n, and a double underline denotes a tensor ∆. Note that sometimes theunderlines only denote the dimension of the corresponding matrix. The governing equations of motion for geometri-cally exact beam theory (without structural damping) can be written as13

h− F ′ = f (1)

g + ˙uh−M ′ + (x′0 + u′)TF = m (2)

where h and g are the linear and angular momenta resolved in the inertial coordinate system, respectively, F and Mare the beam’s sectional force and moment resultants, respectively, u is the one-dimensional (1D) displacement ofa point on the reference line, x0 is the position vector of a point along the beam’s reference line, and f and m arethe distributed force and moment, respectively, applied to the beam structure. The notation (·)′ indicates a derivativewith respect to beam axis x1 and ˙(·) indicates a derivative with respect to time. The tilde operator (·) defines a skew-symmetric tensor corresponding to the given vector. In the literature, it is also termed as “cross-product matrix.” Forexample, given nT = [n1 n2 n3],

n =

0 −n3 n2n3 0 −n1−n2 n1 0

(3)

The constitutive equations relate the velocities to the momenta and the 1D strain measures to the sectional resultantsas {

hg

}=M

{uω

}(4){

FM

}= C

{εκ

}(5)

whereM and C are the 6×6 sectional mass and stiffness matrices, respectively (note that they are not really tensors); εand κ are the 1D strains and curvatures, respectively; and, ω is the angular velocity vector that is defined by the rotationtensor R as ω = axial(R RT ). The axial vector a associated with a second-order tensor A is denoted a = axial(A)

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and its components are defined as

a = axial(A) =

a1a2a3

=1

2

A32 −A23

A13 −A31

A21 −A12

(6)

The 1D strain measures are defined as {εκ

}=

{x′0 + u′ − (R R

0)ı1

k

}(7)

where k = axial[(RR0)′(RR0)T ] is the sectional curvature vector resolved in the inertial basis and ı1 is the unit vectoralong x1 direction in the inertial basis. Note that these three sets of equations, including equations of motion (1) and(2), constitutive equations (4) and (5), and kinematical equations (7), provide a full mathematical description of beamelasticity problems.

For a displacement-based finite-element implementation, there are six degrees of freedom at each node: threedisplacement components and three rotation components. Here, q denotes the elemental displacement array as qT =[uT pT

], where u is the displacement and p is the rotation-parameter vector. The acceleration array can thus be

defined as aT =[uT ωT

]. For nonlinear finite-element analysis, the discretized forms of displacement, velocity, and

acceleration are written as

q(x1) = N q qT =[uT pT

](8)

v(x1) = N v vT =[uT ωT

](9)

a(x1) = N a aT =[uT ωT

](10)

where N is the shape function matrix and (·) denotes a column matrix of nodal values.

B. Spatial and Temporal Discretization

As described above, the GEBT model is discretized in space with LSFEs14, which are a type of high-order p-type finiteelement (FE). Elements have p + 1 nodes located at the Gauss-Lobatto-Legendre points, where p is the polynomialorder of the Lagrangian-interpolant basis functions. For a given number of nodes, we have shown1,2 that LSFEsfor GEBT-based models can be dramatically more accurate than low-order elements. As a tool in BeamDyn, anLSFE evaluated with an appropriate quadrature scheme provides the option to model a modern, highly flexible turbineblade with a single element. The choice of numerical quadrature is described in the next section. The GEBT-LSFEequations are time-integrated with a second-order-accurate generalized-alpha algorithm that is equipped with user-defined numerical damping (see Wang et al.3 for details).

C. Finite-Element Quadrature

Numerical integration (quadrature) of the finite-element inner products over an element domain is required in the FEformulation. Typically, the quadrature rule employed in a finite-element implementation is Gauss-Legendre (GL), forwhich the number of quadrature points is chosen based on the polynomial order of the underlying FE basis functions.In the case where material properties vary significantly over an element domain, the accuracy of the quadrature isdegraded, which can affect the overall accuracy of the solution. If the number of quadrature points is fixed to the FEbasis-function order, accuracy is increased by either increasing number of elements (h-refinement) or the order of theelements (p-refinement). However, if the quadrature order is chosen for accurate evaluation of FE inner products, thenthe choice in FE resolution can be chosen based on overall solution accuracy.

For wind turbine blade analysis, the material sectional properties are defined discretely at ns stations along thebeam axis. BeamDyn is equipped with two quadrature options: Gauss-Legendre quadrature and trapezoidal-rule (TR)quadrature. For GL quadrature, nq = p + 1, where nq is the number of quadrature points and p is the order of theLSFE (however, the GL quadrature point locations differ from the p+ 1 GLL nodal locations). Material properties arelinearly interpolated from the nearest-neighbor discrete stations. Depending on the nature of the material properties,an increase in the element order p could instigate a dramatically different response, because the quadrature points maycapture different material properties. For TR quadrature, nq = ns + (ns − 1) × (j − 1) = (ns − 1) × j + 1, wherej is a positive integer that is user specified. Trapezoidal-rule quadrature enables a user to model a modern turbineblade defined by many cross-sectional property stations with few node points (i.e., p� ns) while capturing all of theprovided material properties. For example, the widely used NREL 5-MW reference wind turbine15 blade is defined by49 stations along the blade axis. If one were using first-order FEs with a fixed quadrature scheme, at least 48 elementswould be required to accurately capture the material data in the FE inner products. BeamDyn, with the GEBT modeland LSFE p-type discretization, is equipped to model a wind turbine blade with a single element. LSFE discretization

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with TR quadrature is an effective modeling approach when the beam deformation can be described accurately withrelatively few FE nodes, despite the large number of material-property stations. However, for a given element orderand nq � p, solutions will be more expensive than if nq ≈ p because inner products are evaluated at least once pertime step.

D. Module-Coupling Algorithm

The FAST modularization framework9,11,16 was created to loosely couple multi-physics modules for time-domainsimulation. Each module’s internal variables are described by parameters (as constants) and states that are eithercontinuous-in-time, discrete-in-time, constraints, or “other” (e.g., logicals). Modules interact through input-outputrelationships, where output quantities are derived from inputs and states, and where inputs and outputs are defined onspatial meshes, and a predictor-corrector algorithm is employed to improve stability and accuracy of the time-updateof the coupled system. The modularization environment provides utilities for coupling non-matching meshes in spaceand time. A detailed description of the coupling algorithm employed here can be found in Sprague et al.11, which wesummarize as follows for the simplified case where each module is advanced with the same time increment ∆t, andwhere we ignore details regarding the mapping of information between modules with non-matching meshes. Assumethat we know all states, inputs, and outputs at time t. In order to advance the states of all modules from time t tot+ ∆t,

1. Using linear or quadratic extrapolation of known inputs, approximate the inputs at t+ ∆t.

2. Update the states of all modules to t+ ∆t.

3. Solve the global system of input-output equations at t + ∆t. Depending on the relationship between modulesand the module output equations, this system solve can range from a simple transfer of information to a fullnonlinear-system solve.

4. Either accept the states, inputs, and outputs, or apply a correction by repeating step (2) with the inputs calculatedin Step (3), and then repeating Step (3).

It was shown, using simple numerical examples, in Sprague et al.11,16 that employing one or more correction steps canincrease the accuracy of the coupled simulation and can increase stability by permitting the use of larger time incre-ments. However, these increases in accuracy and stability must be weighed against the additional cost of additional“update state” and output calculations for each module.

III. Numerical Examples

A. Example 1: Partitioned Analysis

In our first example, we have an oversimplified representation of a blade connected to a hub, where a BeamDyn beamis “clamped” to the mass in a spring-damper-mass (SDM) system as shown in Figure 2. We use this example to verifyour numerical implementation and examine the accuracy and numerical behavior of the coupling algorithm. Here, therelevant output from BeamDyn is translation reaction force, whereas its inputs are root motion (translational displace-ment, velocity, and acceleration). The inputs and outputs are the same, but swapped, for the SDM system. The timeintegrator in BeamDyn is a second-order implicit generalized-alpha algorithm (see Wang et al.3), and the time integra-tor for the SDM system is fourth-order Adams-Bashforth-Moulton (ABM4), which is a predictor-corrector algorithm.The material properties, coordinate system, and geometric parameters can be found in Figure 2. The cross-sectiondimensions of the beam is 0.1 m × 0.1 m. The natural frequency of the uncoupled mass-spring-damper system is6.28 rad/s and the first five distinct natural frequencies for the uncoupled beam (in a clamped/cantilevered configu-ration) are 0.26, 1.72, 5.78, 22.62, and 24.21 rad/s, respectively, as determined by a refined ANSYSr Mechanicalmodal analysis. The first five distinct natural frequencies of the coupled system, obtained by ANSYS modal analy-sis, are 0.25, 0.85, 1.80, 4.76, and 9.34 rad/s, respectively. For time-domain analysis, the system has quiescent initialconditions with an initial displacement of 0.1 m from the neutral position in the positive Z direction (see Figure 2).

For verification, we analyzed the system in ANSYS using 60 BEAM188 elements (cantilevered to a point masswith spring and damper elements) and the time increment was 10−5 s. The Newmark-β time integrator is adoptedin ANSYS without numerical damping. For the BeamDyn results, the beam was discretized by a single eighth-orderelement, and the time increment was 10−5 s. BeamDyn inner products were evaluated with standard Gauss-Legendrequadrature. Unless otherwise noted, BeamDyn numerical damping was disabled and there was no correction step inthe coupling algorithm. Based on initial conditions corresponding to translating the mass and beam 0.1-m along Zand letting go, the root and tip displacements of the beam calculated with the two models are shown in Figure 3.Excellent agreement is demonstrated between the ANSYS and BeamDyn results. The root horizontal velocity and

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Figure 2: Schematic showing a 0.1 m × 0.1 m beam clamped to a spring-damper-mass system with system propertiesand dimensions.

0 1 2 3 4 5

Time (s)

0.10

0.05

0.00

0.05

0.10

0.15

Uz(m

)

ANSYSBeamDyn

(a) Root Displacement

0 1 2 3 4 5

Time (s)

0.3

0.2

0.1

0.0

0.1

0.2

0.3

Uz(m

)

ANSYSBeamDyn

(b) Tip Displacement

Figure 3: Beam-root and -tip horizontal-displacement histories calculated with ANSYS and BeamDyn for the beam-SDM system (Figure 2). The ANSYS and BeamDyn models were both time integrated with ∆t = 10−5 s. TheANSYS model had 60 BEAM188 elements, and the BeamDyn model had a single eighth-order element.

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0 1 2 3 4 5

Time (s)

0.6

0.4

0.2

0.0

0.2

0.4

0.6

Vz(m/s

)

ANSYSBeamDyn

(a) Root Velocity

0 1 2 3 4 5

Time (s)

4

3

2

1

0

1

2

3

4

Az(m/s

2)

ANSYSBeamDyn

(b) Root Acceleration

Figure 4: Beam-root horizontal velocity and acceleration histories calculated with ANSYS and BeamDyn for thebeam-SDM system (Figure 2). Both models were time integrated with ∆t = 10−5 s. The ANSYS model had 60BEAM188 elements, and the BeamDyn model had a single eighth-order element.

acceleration are also compared between the current model and the ANSYS benchmark solution in Figure 4. Again,excellent agreement is shown.

For the BeamDyn-SDM system, we consider first stability of the coupling algorithm for which the beam is dis-cretized by a single fifth-order LSFE. Figure 5 shows the maximum stable time increments for the BeamDyn-SDMsystem, obtained by numerical experiments against the number of correction iterations in the FAST loose-couplingscheme. Results are shown with and without numerical damping in the BeamDyn module. Increasing the numberof correction iterations increases the allowable time increment for stability for up to three corrections. In movingfrom three to four corrections, however, we see no increase in stability for the numerical-damping-free simulations,and we even see a small decrease in stability with BeamDyn numerical damping. Overall, however, the inclusion ofnumerical damping in BeamDyn increases the allowable time increment for stable solutions. The computational sav-ings offered by a larger stable time increment provided by additional correction steps must be considered against theadditional computational cost of the correction steps. For example, adding a single correction step makes each timestep about twice as expensive compared with no correction. We also note that the maximum stable time increment forthe uncoupled BeamDyn model (without numerical damping) is 5× 10−3 s. Thus, the coupled-system is more stiff.

0 1 2 3 4

Number of Corrections

10-4

10-3

10-2

Maxim

um

tim

e incr

em

ent

(s) ρ∞ =1.0

ρ∞ =0.9

Figure 5: Maximum stable time increments as a function of the number of correction iterations in the predictor-corrector coupling algorithm for the BeamDyn-SDM system. Results are shown with (ρ∞ = 0.9) and without (ρ∞ =1.0) numerical damping in the BeamDyn module.

We consider here the accuracy of the coupled model for the BeamDyn-SDM system by examining solutions cal-culated with various spatial and temporal discretization levels. Accuracy was quantified by root-mean-square (RMS)error, εRMS , which was calculated as

εRMS =

√∑nmax

k=0 [Ukz − Ub(tk)]2∑nmax

k=0 [Ub(tk)]2(11)

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and where Ub(t) is the benchmark solution calculated with a BeamDyn-SDM solution calculated with a 9-node Beam-Dyn model and ∆t = 5 × 10−6 s. Figure 6 shows the RMS error of the horizontal tip displacements as a function ofthe total number of BeamDyn nodes for two different time increments (1 × 10−4 s and 2 × 10−4 s); BeamDyn hadno numerical damping, and there were no correction steps in the loose-coupling algorithm. The results indicate that∆t = 2 × 10−4 s provides time-increment-independent solutions. We see approximately second-order convergencewith spatial refinement (increase in the number of nodes), and results indicate that a fifth-order element (six nodes)provides results of sufficient accuracy. Table 1 shows RMS errors of tip-displacement histories for the BeamDyn-SDMsystem for several combinations of time increment and number of corrections in the loose-coupling algorithm. We seethat, for this system, increasing the number of corrections permits larger time increments while maintaining sufficientaccuracy.

3 4 5 6 7 8

Number of Nodes

10-3

10-2

10-1

100

101

ε RMS(U

z)

O(n2 )

∆t=1E−4s

∆t=2E−4s

Figure 6: Normalized RMS error of horizontal tip displacement histories as a function of total number of nodes forthree different time increments. The dashed line shows ideal second-order convergence. The benchmark solution wascalculated with a 9-node LSFE and ∆t = 5× 10−6 s.

∆t (s) # of corrections εRMS

2 ×10−4 0 3.85 ×10−2

5 ×10−4 1 5.78 ×10−2

1 ×10−3 2 3.85 ×10−2

3 ×10−3 3 3.83 ×10−2

3 ×10−3 4 3.82 ×10−2

Table 1: RMS errors of tip-displacement histories for the BeamDyn-SDM system for several combinations of timeincrement and number of corrections in the loose-coupling algorithm. Results are shown for a BeamDyn model with6 nodes and no numerical damping.

B. Example 2: NREL 5-MW Wind Turbine

The second example is an analysis of the NREL 5-MW reference wind turbine15, which has straight 61.5 m blades.We examine simulation results where the three blades are modeled by BeamDyn or ElastoDyn. The blade structural-dynamics model in the ElastoDyn module of FAST is applicable to straight isotropic blades dominated by bending. Themodel includes two flapwise-bending modes and one edgewise-bending mode, coupled through a structural pre-twist,but neglects axial, shear, and torsional degrees of freedom and mass and elastic offsets from the pitch axis. Severalgeometric and kinematic nonlinearities are accounted for, including radial shortening and centrifugal, Coriolis, andgyroscopic loading.

We examine here the numerical performance of two different BeamDyn quadrature methods (c.f., Section II.C),Gauss-Legendre and trapezoidal, for this realistic-blade analysis. As described above, the sectional properties forthe NREL 5-MW reference turbine blade are defined at 49 stations along its 61.5 m length. First, a cantileveredblade under a uniformly distributed static force of magnitude 104 N/m along the flap direction is analyzed. Figure 7shows the tip displacement in the flap direction as a function of the number of nodes. Monotonic convergence oftip-displacement is shown for the trapezoidal-quadrature results with an increasing number of nodes. The convergencerate of tip displacements for Gauss-Legendre quadrature, however, is non-monotonic. As described in Section II.C, the

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trapezoidal-rule quadrature captures all 49 material-data stations regardless of the number of element nodes, whereasthe particular material data incorporated by Gauss-Legendre quadrature varies with the number of element nodes. Thisis also demonstrated in the calculation of total blade mass as shown in Figure 8. The total blade mass as calculatedwith trapezoidal quadrature is independent of the number of nodes, whereas the mass calculated by Gauss quadraturedepends on the number of nodes in the element, and a large number of nodes is required for an accurate total-masscalculation. We note that a small scaling factor has been applied to the calculation of blade mass so that it is consistentwith ElastoDyn. In all subsequent calculations with BeamDyn, trapezoidal quadrature is employed.

Figure 7: Tip deflections of a cantilevered NREL 5-MW blade under uniformly distributed load as a function ofthe number of nodes in a single-element BeamDyn model where finite-element inner-products were calculated withGauss-Legendre or trapezoidal-rule quadrature.

Figure 8: Total blade mass of an NREL 5-MW reference blade as a function of the number of nodes in a single-element BeamDyn model where finite-element inner-products were calculated with Gauss-Legendre or trapezoidal-rule quadrature.

Next, we studied the time step sizes required for stable simulation of BeamDyn in stand-alone and couple-to-FASTmodes. Figure 9 shows the maximum time step size versus the number of nodes for a BeamDyn model composed ofa single element. In the stand-alone mode, we used FAST as the driver but with all coupling options disabled so thatthe blade rotated at a fixed speed loaded only by gravity. For the coupled-to-FAST case, we conducted an aero-servo-elastic wind turbine analysis under a mean wind speed of 12 m/s2 with turbulence, which is certification test case#26 in the FAST archive. BeamDyn numerical damping was disabled and there were no correction iterations in thecoupling algorithm. We see that the two-way coupling between BeamDyn and FAST requires significantly smallertime increments for stable solutions.

Finally, we studied the performance of BeamDyn in the coupled FAST analysis. Figures 10 and 11 show the tipflap displacement histories under different time and space discretizations. Note that all of the quantities studied hereare defined in the body-attached blade reference coordinate system following the IEC standard, where the X direction

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5 6 7 8 9 10 11 12

Number of Nodes

10-4

10-3

10-2

10-1

Maxim

um

tim

e incr

em

ent

(s) Stand-alone

FAST

Figure 9: Maximum stable time increments vs number of FE nodes for an NREL 5-MW blade, where the blade ismodeled with a single element.

is toward the suction side of the airfoil, the Y direction is toward the trailing edge, and the Z direction is toward theblade tip from root. These results demostrated that, for this system, results that are grid independent (in space andtime) can be obtained with ∆t = 2× 10−3 s and a single fifth-order element for each blade.

Figure 10: Blade tip deflection histories along flap direction obtained using different time increments, where the bladeis modeled with a single fifth-order element.

We compared the results obtained by BeamDyn with those obtained by ElastoDyn for the coupled analysis (asdescribed above). The BeamDyn blades were each modeled with a single fifth-order element, and the FAST-BeamDynsystem was time-integrated with ∆t = 2× 10−3 s as demanded for numerical stability. The FAST-ElastoDyn systemwas time integrated with ∆t = 1.25 × 10−2 s. The tip displacements of blade 1 are shown in Figure 12. Resultsfor BeamDyn are shown with and without off-diagonal terms in the sectional mass matrices (the latter is for moredirect comparison to ElastoDyn). Good agreement can be observed between the ElastoDyn and BeamDyn results. Itis noted that because of the Trapeze effect and elastic stretching considered in BeamDyn, the mean value of the axialdisplacement in Figure 12c by BeamDyn is different from ElastoDyn. Figure 13 and 14 shows the root reaction forcesand moments calculated by BeamDyn and ElastoDyn. Again, good agreement can be found in these quantities. Wenote the spurious spikes in the Mpitch histories, which will be addressed in a future release of BeamDyn. While wesee noticeable differences in the tip displacement histories, it is interesting to note the excellent agreement betweenElastoDyn and BeamDyn results for root reaction forces and moments. This is because the NREL 5-MW bladefeatures are well accommodated by the approximations behind the ElastoDyn model. In particular, ElastoDyn is wellsuited for modeling the NREL 5-MW blade because

• the blade is naturally straight,

• the lowest modes excited by wind are dominated by bending,

• there are no cross-sectional couplings induced by anisotropic composite laminate layups,

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Figure 11: Blade tip deflection histories along flap direction obtained using different BeamDyn refinements, wheremodels were time integrated with ∆t = 2× 10−3 s.

• torsion, extension, and shear effects are mostly negligible (the first torsional mode natural frequency is wellabove rated rotor speed; the blade aspect ratio is high, etc.),

• the deflections are moderate enough (not too large) that they can be accurately captured by the geometric non-linear terms included in ElastoDyn, and

• the mass-center offsets are small and do not cause a large change in response.

The benefits of moving from an efficient model like ElastoDyn to a higher-fidelity model that is more computationallyexpensive like BeamDyn will be best seen for turbine blades that do not satisfy the simplifying features, e.g., thosewith aero-elastically tailored curved blades.17

IV. Conclusion

In this paper, we examined the beam theory and the coupling algorithm for partitioned mechanical-system anal-ysis. The governing equations for BeamDyn were reviewed and the coupling schemes between BeamDyn and othermechanical modules were explained. For wind turbine blade analysis, which features a large number of cross-sectionalstations along the blade axis, we implemented a trapezoidal numerical integration so that all the cross-sectional data,including inertial and stiffness matrices, can be used in the analysis. The users do not need to pick certain stationsfrom a large set of data nor numerically interpolate them, which usually introduces errors. This new spatial integrationmethod and the coupling algorithm have been verified in the numerical examples. The results obtained by trapezoidalquadrature were monotonously converged while those obtained by conventional Gauss integration were randomly con-verged depending on the stations chosen and interpolation method. In the coupled analysis, exponential convergencerate can be seen in the BeamDyn results, which is a prominent feature of the LSFEs. In the full turbine analysis,reasonable agreement between BeamDyn and ElastoDyn results is found. More verification and validation results canbe found in Guntar et al.17.

Acknowledgments

This work was supported by the U.S. Department of Energy under Contract No. DE-AC36-08GO28308 with theNational Renewable Energy Laboratory. Funding for the work was provided by the DOE Office of Energy Efficiencyand Renewable Energy, Wind and Water Power Technologies Office.

References1Wang, Q. and Sprague, M. A., “A Legendre spectral finite element implementation of geometrically exact beam theory,” Proceedings of the

54th Structures, Structural Dynamics, and Materials Conference, Boston, Massachusetts, April 2013.2Wang, Q., Sprague, M. A., Jonkman, J., and Johnson, N., “Nonlinear Legendre spectral finite elements for wind turbine blade dynamics,”

Proceedings of the 32nd ASME Wind Energy Symposium, National Harbor, Maryland, January 2014, also published as NREL tech. report NREL/CP-2C00-60759.

3Wang, Q., Sprague, M. A., Jonkman, J., and Johnson, N., “BeamDyn: A high-fidelity wind turbine blade sovler in the FAST modularframework,” Proceedings of SciTech 2015, Kissimmee, Florida, 2015, also published as NREL tech. report NREL/CP-5000-63165.

4Wang, Q., “BeamDyn,” https://nwtc.nrel.gov/BeamDyn, October 2015, [Online; accessed 11-December-2015].5Reissner, E., “On one-dimensional large-displacement finite-strain beam theory,” Studies in Applied Mathematics LII, 1973, pp. 87–95.6Simo, J. C., “A finite strain beam formulation. The three-dimensional dynamic problem. Part I,” Computer Methods in Applied Mechanics

and Engineering, Vol. 49, 1985, pp. 55–70.

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(a) Flap direction

(b) Edge direction

(c) Axial direction

Figure 12: Comparisons of blade 1 tip displacements between ElastoDyn and BeamDyn results.

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(a) Fflap

(b) Fedge

(c) Faxial

Figure 13: Comparisons of root reaction forces between ElastoDyn and BeamDyn results

7Simo, J. C. and Vu-Quoc, L., “A three-dimensional finite-strain rod model. Part II,” Computer Methods in Applied Mechanics and Engineer-ing, Vol. 58, 1986, pp. 79–116.

8Hodges, D. H., Nonlinear Composite Beam Theory, AIAA, 2006.9Jonkman, J. M., “The new modularization framework for the FAST wind turbine CAE tool,” Proceedings of the 51st AIAA Aerospace

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(a) Medge

(b) Mfalp

(c) Mpitch

Figure 14: Comparisons of root reaction moments between ElastoDyn and BeamDyn results

Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Grapevine, Texas, January 2013.10Jonkman, J. and Jonkman, B., “FAST v8,” https://nwtc.nrel.gov/FAST8, October 2015, [Online; accessed 11-December-2015].11Sprague, M. A., Jonkman, J., and Jonkman, B., “FAST modular framework for wind turbine simulation: new algorithms and numerical

examples,” Proceedings of the 33rd ASME Wind Energy Symposium, Kissimmee, Florida, January 2015.12Yu, W. and Blair, M., “GEBT: A general-purpose nonlinear analysis tool for composite beams,” Composite Structures, Vol. 94, 2012,

pp. 2677–2689.13Bauchau, O. A., Flexible Multibody Dynamics, Springer, 2010.14Ronquist, E. M. and Patera, A. T., “A Legendre Spectral Element Method for the Stefan Problem,” International Journal for Numerical

Methods in Engineering, Vol. 24, 1987, pp. 2273–2299.15Jonkman, J., Butterfield, S., Musial, W., and Scott, G., “Definition of a 5-MW Reference Wind Turbine for Offshore System Development,”

Tech. Rep. NREL/TP-500-38060, National Renewable Energy Laboratory, 2009.

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16Sprague, M., Jonkman, J., and Jonkman, B., “FAST modular wind turbine CAE tool: non matching spatial and temporal meshes,” Proceed-ings of the 32nd ASME Wind Energy Symposium, National Harbor, Maryland, January 2014.

17Guntur, S., Jonkman, J., Schreck, S., Jonkman, B., Q.Wang, Sprague, M. A., Singh, M., Hind, M., and Sievers, R., “FAST v8 Verificationand Validation for a MW-scale Wind Turbine with Aeroelastically Tailored Blades,” Proceedings of the 34th ASME Wind Energy Symposium, SanDiego, California, 2016.

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