Buckling Optimization of Laminated Hybrid Composite Shell Structures Using Discrete Material Optimization

Transcription

1 6 th World Congress on Structural and Multidisciplinary Optimization Rio de Janeiro, 3 May - 3 June 25, Brazil Buckling Optimization of Laminated Hybrid Composite Shell Structures Using Discrete Material Optimization Erik Lund 1, Lennart Kühlmeier 2 and Jan Stegmann 1 1 Institute of Mechanical Engineering, Aalborg University, Pontoppidanstraede 11, DK-922 Aalborg East, Denmark 2 Vestas Wind Systems A/S, Smed Sørensens Vej 5, 695 Ringkøbing, Denmark el@imeaaudk, lenku@vestascom, js@imeaaudk 1 Abstract The design problem of maximizing the buckling load factor of laminated hybrid composite shell structures is investigated using the so-called Discrete Material Optimization (DMO) approach The design optimization method is based on ideas from multi-phase topology optimization where the material stiffness is computed as a weighted sum of candidate materials, thus making it possible to solve discrete optimization problems using gradient based techniques and mathematical programming The potential of the DMO method to solve the combinatorial problem of proper choice of material, stacking sequence and fiber orientation simultaneously is illustrated for two benchmark plate examples, and ongoing work on buckling optimization of a wind turbine blade test section is outlined 2 Keywords: Laminate design optimization, composite structures, discrete material optimization, topology optimization 3 Introduction The size of wind turbines has increased dramatically in the last decade, and today a standard wind turbine blade has a length between 4 and 6 meters These high performance, hybrid material structures are lightweight and may exhibit maximum tip displacements of up to about 25% of the length before they fail due to local buckling on the compressive side of the blade Thus, in order to improve the structural performance the design objective is to increase the buckling load factor, taking weight considerations into account This paper deals with this design problem for wind turbine blades specifically, but the methodology can be applied to any laminated hybrid composite shell structure The outer shape of a wind turbine blade is determined by aerodynamic considerations and therefore in general not subject to change These structures consist of stiff fiber reinforced polymers such as Glass or Carbon Fiber Reinforced Polymers (GFRP/CFRP) together with foam and different types of wood stacked in a number of layers and bonded together by a resin The design problem is then to determine the best stacking sequence by proper choice of material and fiber orientation of each FRP layer in order to obtain the desired structural performance For complicated geometries like wind turbine blades this is a very challenging design problem that calls for use of sophisticated structural optimization tools, and it is the aim of this paper to describe the development of the so-called Discrete Material Optimization (DMO) approach for design optimization using buckling behavior as criteria function The DMO method is based on ideas from multi-phase topology optimization (see Sigmund and co-workers [1, 2]) where the material stiffness (or density) is computed as a weighted sum of candidate materials In this way the discrete problem of choosing the best material (with the right orientation) is converted to a continuous formulation where the design variables are the scaling factors (or weight functions) on each candidate material The method has been successfully applied for maximum stiffness design and eigenfrequency design using as many as 12 candidate materials at each point and several hundred thousands of design variables in total, see [3], [4] and [5], and it is being extended to buckling criteria in this work The application of the DMO method to local strain criteria can be found in these proceedings in the paper [6] 4 Linear Buckling Analysis of Laminated Composite Shell Structures The finite element method is used for determining the maximum buckling load factor of the laminated composite structure A linear buckling analysis is used, ie, the structure is assumed to be perfect with no geometrical imperfections and the buckling load found will be an upper limit for the real value The laminated composite is typically composed of multiple materials and multiple layers, and the laminated shell structures may, in general, be curved or doubly-curved The materials used in this work may be fiber reinforced polymers oriented at a given angle θ k for layer k or it may be softer isotropic core materials All materials are assumed to behave linearly elastic and the structural behavior of the laminate is described using an equivalent single layer theory where the layers are assumed to be perfectly bonded together and thus, displacements and strains will be continuous across the thickness Such theories are known to be sufficiently accurate for modelling of the structural stiffness but may not be adequate for describing interlaminar effects such as delamination 1

2 The shell elements used are derived using the degenerated solid approach and are thus based on first order shear deformation theory For computational efficiency the element used for the design optimization in this work is a four node isoparametric shell element with full integration where the problem of shear locking is avoided by using the method of assumed natural strains for the transverse shear interpolation, see [7], but several other quadratic and cubic shell finite elements have also been implemented The element stiffness matrix, K e, is obtained as the sum of layer stiffnesses over the total number of layers, N l K e = N l k=1 (B l k) T C l kb l kdv (1) In Eq (3) the matrix B l k is the strain-displacement matrix of the k th layer and C l k is the constitutive matrix for layer k The global stiffness matrix K is obtained by summation over all elements, N e, and the static equilibrium equation for the structure may be written as KD = F (2) Here D is the global displacement vector and F the global load vector Based on the displacement field the element layer stresses σ l k can be computed, whereby the stress stiffening effects due to the mechanical loading can be evaluated by computing the element initial stress stiffness matrix K e σ (also termed the geometric stiffness matrix) K e σ = N l k=1 (G l k) T S l kg l kdv (3) Here G l k is an element layer matrix that contains shape functions and their derivatives and S l k a matrix containing initial layer stresses σ l k By summation over all elements the global stress stiffness matrix K σ is obtained, and the linearized buckling problem can be established as (K + λ jk σ)φ j =, j = 1, 2, (4) where the eigenvalues are assumed ordered by magnitude, such that λ 1 is the lowest eigenvalue, ie, the buckling load factor, and Φ 1 is the corresponding eigenvector 5 Design Sensitivity Analysis and Optimization of Linear Buckling Problems The objective of the design problem considered is to maximize the lowest buckling load factor of the laminated composite structure using gradient based techniques, and thus the buckling load factor sensitivities should be computed in an efficient way 51 Design Sensitivity Analysis of Simple Eigenvalues The design variables are termed x i, i = 1,, I, and the direct approach to obtain the eigenvalue sensitivity in case of a distinct, ie simple eigenvalue λ j, is to differentiate Eq (4) with respect to a design variable x i, premultiply by Φ T j and make use of Eq (4), then the following expression is obtained for the eigenvalue sensitivity in case of a simple, ie distinct, eigenvalue λ j, see, eg, [8] and [9] dλ j dx i = Φ T j dk dk σ + λ j Φj (5) dx i dx i where it has been assumed that the eigenvectors have been K σ-orthonormalized, such that Φ T j K σφ j = 1 For the DMO parametrization proposed in this work where the geometry is fixed and only the material is changed, the stiffness matrix derivative dk/dx i only involves the derivative of the element layer constitute matrix C l k with respect to x i In the next section the DMO interpolation scheme used for determining the constitute matrix is described, and the derivative of this interpolation is computed analytically The stress stiffness matrix is an implicit function of the displacement field, ie K σ = K σ(d(x),x), which must be taken into account as dk σ dx i = Kσ x i + Kσ D dd dx i (6) Thus, the displacement sensitivities dd/dx i must be computed which is done efficiently using the direct differentiation approach, ie, the static equilibrium equation, Eq (2), is differentiated with respect to a design variable x i K dd = F K D (7) dx i x i x i where the load sensitivity F/ x i is zero for the DMO design variables used, unless volume forces are considered The displacement sensitivities are computed analytically by solving Eq (7) for each design variable x i, reusing 2

3 the stiffness matrix K from the analysis, but the stress stiffness matrix sensitivities dk σ/dx i in Eq (6) are computed by central difference approximations at the element level This approach is mainly used in order to facility easy implementation of sensitivity analysis for all different types of laminate design variables and no inaccuracy problems have been observed for the central difference approximation of element stress stiffness matrix sensitivities 52 Design Sensitivity Analysis of Multiple Eigenvalues So far multiple eigenvalues and corresponding eigenvectors have not been mentioned In this case the eigenvectors are not unique, which complicates the sensitivity analysis and optimization due to the non-differentiability of the eigenvalues In such situations the sensitivity analysis described in [1] is used together with the optimization algorithm developed in [11] The details are omitted here for brevity 53 The Mathematical Programming Problem In case of only simple, distinct eigenvalues, the optimization problem of maximizing the buckling load factor λ 1 is reformulated using a bound formulation, see [12], as Objective : max x, β β Subject to : λ j β, j = 1,, N λ M M x i x i x i, i = 1,, I (8) where M is the upper limit on the mass M of the structure By introducing the bound parameter β the lowest N λ eigenvalues are considered when solving the minimax problem of maximizing the buckling load factor The derivative of the mass constraint in Eq (8) is also computed analytically, and having obtained all the necessary design sensitivities, the mathematical programming problem is solved using the Method of Moving Asymptotes by Svanberg [13] The closed loop of analysis, design sensitivity analysis and optimization is repeated until convergence in terms of no change of the design variables is reached or until the maximum number of design iterations have been performed 6 The Discrete Material Optimization Approach The design parametrization method applied in this work is denoted Discrete Material Optimization (DMO), that can be used for efficient design of general laminated composite shell structures, see [3, 4, 5] The approach developed is to formulate the optimization problem using a parametrization that allows us to do efficient gradient based optimization on real-life problems while reducing the risk of obtaining a local optimum solution To this end we will use the mixed materials strategy suggested by Sigmund and co-workers [1, 2] for multi-phase topology optimization, where the total material stiffness is computed as a weighted sum of candidate materials In the present context this means that the stiffness of each layer of the laminated composite will be computed from a weighted sum of a finite number of candidate constitutive matrices, each representing a given lay-up of the layer Consequently, the design variables are no longer the fiber angles or layer thicknesses but the scaling factors (or weight functions) on each constitutive matrix in the weighted sum For example, we could choose a stiff orthotropic material oriented at three angles θ 1 =, θ 2 = 45 and θ 3 = 9 and a soft isotropic material, thereby obtaining a problem having four design variables per layer The objective of the optimization is then to drive the influence of all but one of these constitutive matrices to zero for each ply by driving all but one weight function to zero As such, the methodology is very similar to that used in topology optimization This is further emphasized by the fact that penalization is used on the design variables to make intermediate values un-economical At the beginning of the optimization, the constitutive matrices used in the analysis thus consist of contributions from several candidate materials, but at the end of the design optimization, the parametrization for the weight functions has to fulfill the demand, that one distinct candidate material has been chosen 61 Parametrization for Single Layered Laminate Structures As in topology optimization the parametrization of the DMO formulation is invoked at the finite element level The element constitutive matrix, C e, for a single layered laminate structure may in general be expressed as a sum over the element number of candidate material configurations, n e : C e = n e i=1 w ic i = w 1C 1 + w 2C w n ec n e, w i 1 (9) where each candidate material is characterized by a constitutive matrix C i The weight functions w i must all have values between and 1 in order to be physically allowable Furthermore, in case of solving buckling 3

4 problems or having a mass constraint as in the optimization problems studied here, it is necessary that the sum of the weight functions is 1, ie, ne i=1 wi = 1 If this demand is not fulfilled, physically meaningless results will be obtained Several new parametrization schemes have been developed, see details in [3, 4], and we give here a short outline of the most effective implementation The DMO parametrization schemes are also described in the paper [6] in these proceedings We apply for each element a number of design variables x e i, i = 1,, n e, and write w i = ŵ i = (x e i) p ŵ i, i = 1,, n e ne k=1 ŵk e n j=1; j i 1 (xe j) p (1) To push the design variables x e i towards and 1 the SIMP method known from topology optimization has been adopted by introducing the power, p, as a penalization of intermediate values of x e i The power p is typically set to 2 in the beginning of the optimization process and then increased by 1 for every 1 design iterations until p is 6 Moreover, the term (1 x e j) j i is introduced such that an increase in x e i results in a decrease of all other weight functions Finally, the weights have been normalized in order to satisfy the constraint that the sum of the weight functions is 1 Note that the expression in Eq (1) means that complicated additional constraints on the design variables x e i are avoided and only simple box constraints have to be dealt with It should be noted that the normalization introduced in Eq (1) makes the interpolation less effective in driving the weights to /1 since the normalization alters the effect of the penalization Consequently, the normalized weighting scheme converges slower to a design where a distinct material have been selected than if the weight functions ŵ i are used 62 Parametrization for Multi Layered Laminate Structures The only difference between single and multi layered laminate structures is that the interpolation given above has to be used for all layers, ie, the layer constitutive matrix C l k for layer k is computed as C l k = where n l is the number of candidate materials for the layer n l i=1 w ic i (11) 63 Patch Design Variables The design variables x i may be associated with each finite element of the model or the number of design variables may be reduced by introducing patches, covering larger areas of the structure This is a valid approach for practical design problems since laminates are typically made using fiber mats covering larger areas 7 Benchmark Examples In this section two benchmark examples illustrate the use of the DMO approach for buckling design of laminated plates 71 Single Layered Simply Supported Plate The first example is an academic benchmark example of determining the fiber angles for buckling design of a simply supported plate subjected to inplane loading The plate has dimension 5 5 m and consists of one layer with thickness 1 m The plate is assumed to be made of unidirectional glass/epoxy, ie an orthotropic material with E x = Pa, E y = E z = Pa, ν = 3, G xy = G xz = Pa, G yz = Pa, and ρ = 191 kg/m 3 The distributed compression load is 1 N/m, and the first buckling mode of the plate can be seen on Figure 1 The plate is modelled using 4 four node shell elements, and first the buckling load factor is maximized using continuous fiber angles as design variables The optimization is performed using initial fiber angles of and 9, respectively, and the results can be seen in Figure 2 The fiber angle optimization illustrates the potential problem of using fiber angles directly as design variables due to many local minima that exists The design obtained when all fiber angles initially are oriented at 9 is seen to be rather different from the case when all angles initially are Next a DMO optimization is executed where the orthotropic material may be oriented at, ±45, and 9 This yields a total of 16 design variables, and the discrete fiber angles obtained can be seen on Figure 3 From Figure 3 it is seen that in 85% of the elements a distinct choice of candidate angle is obtained In sheardominated areas of the plate the unidirectional material is not suited, thus the optimizer cannot find a distinct fiber angle 4

5 Fixed in x and z Distributed compression load Fixed in z y z x Fixed in y and z Fixed in z Figure 1: Out-of-plane buckling mode for the compression loaded simply supported plate when all fiber angles are oriented at wrt the x-axis Initial fiber angles: Initial fiber angles: 9 Figure 2: Continuous fiber angle optimization of single layered plate simply supported at all edges Left: the buckling load factor is increased from 237 to 315 if the initial fiber angles are Right: the buckling load factor is increased from 198 to 285 if the initial fiber angles are 9 If the number of DMO variables is increased, such that the candidate angles are, ±15, ±3, ±45, ±6, ±75 and 9, the design seen on Figure 4 is obtained In Figure 4 the chosen fiber angles are plotted, and as before there is no distinct choice of angle in 15% of the area The buckling load factor is again increased to 315 as in the case of using continuous fiber angles oriented at initially There is seen to be good agreement between the fiber angles obtained using the two different optimization approaches 5

6 All chosen candidate angles Converged fiber angles Figure 3: DMO fiber angle optimization of single layered plate simply supported at all edges The orthotropic material may be oriented at, ±45, and 9 The buckling load factor is increased from 232 to 315 Left: all chosen candidate angles are shown In 85% of the elements a distinct choice of candidate angle is obtained Right: the fiber angles that have converged Figure 4: DMO fiber angle optimization of single layered plate simply supported at all edges The orthotropic material may be oriented at, ±15, ±3, ±45, ±6, ±75 and 9 The buckling load factor is increased from 232 to 315 In real-life structures the fiber angles are typically constant over larger areas, and this fact reduces the number of DMO design variables when solving realistic laminated design problems However, the element-wise fiber angle optimizations shown in this section are useful from a benchmark point of view in order to investigate the properties of the optimization approach Layer Simply Supported Plate The next example illustrates the potential of the DMO method for solving the combinatorial problem of proper choice of material, stacking sequence and fiber orientation simultaneously for maximum buckling load design of the simply supported plate just presented 6

7 The plate is divided into 16 layers of equal thickness but now half of the domain should be filled with soft material The other candidate material is an isotropic polymer foam material and has E x = Pa, ν = 35, and ρ = 8kg/m 3 The upper and lower layers have 4 DMO design variables per element associated with the orthotropic material oriented at, ±45, and 9, respectively, and the remaining 14 interior layers have 5 DMO variables, allowing the optimizer also to choose the soft isotropic material The mass constraint is set such that 1/2 of the total volume should be filled with soft material The topology and fiber orientations of the orthotropic material for all 16 layers can be seen on Figure 5 Layer 1) Layer 2) Layer 3) Layer 4) Layer 5) Layer 6) Layer 7) Layer 8) Layer 9) Layer 1) Layer 11) Layer 12) Layer 13) Layer 14) Layer 15) Layer 16) Figure 5: Optimized material directions (fiber angles) for maximum buckling load factor design of 16-layer simply supported plate when 5 DMO variables per element are used (soft material together with orthotropic material oriented at, ±45, and 9 ) White means that the isotropic soft material has been selected The 16-layer plate example clearly illustrates the potential of the DMO approach for doing simultaneous material selection, material orientation and stacking sequence design This is a key feature of the DMO approach developed The optimization method may not obtain full convergence everywhere in the domain, but from an engineering point of view, the method is capable of providing much insight into the best design of the problem 7

8 8 Preliminary Results: Optimization of Wind Turbine Blade Test Section The authors are currently investigating the use of the DMO approach for design of wind turbine blades, and some preliminary results are given in this section The blades are normally the most expensive single component of a wind turbine, and pushing the material utilization to the limit is a necessity for light and cost effective blades As a consequence of the minimum material design strategy the structures are becoming thin-walled and buckling problems must be addressed Today s blades are high performance, hybrid material structures with lengths of up to 6 meters with many very challenging design problems involved For the design study a generic airfoil section has been used In the present case a NACA airfoil section with a cord width of 117 m has been chosen As with most structural designs for modern wind turbine blades the 9 m section is a hybrid structure Here both carbon and glass fibres as well as birch and balsa wood together with foam materials may be used Production and test facilities for such a 9 m airfoil test section have been developed, see [14], and it is the purpose of the design study to obtain an optimized design that will be tested destructively in a four-point bending configuration in order to verify the buckling performance The cross section of the test section together with the model used can be seen on Figure 6 Leeward Leading edge panel Spar cap Trailing edge panel Windward Web Fixed all dof Stiff material Patch no Patch 115 Fixed layup y etc Anti-symmetry enforced at symmetry plane x-z x z Distributed compression load Figure 6: The NACA airfoil section studied together with the model of the test section The design domain consists of the 15 patches in the spar cap Anti-symmetric boundary conditions are assumed The objective of the optimization is to maximize the buckling load of the structure Typically there is a design constraint on the global bending stiffness of the blade since fatigue issues often drive the design Hence the maximum strain of the material is a constraining factor Today s carbon fibre materials have excellent fatigue properties and thus the material is strained still harder to gain the maximum utilization of the expensive carbon fibre Thus the structure should be buckling stable at the highest possible strain Because of the constraint on 8

9 the global bending stiffness of the blade the stiffest material will always be oriented in the axial direction of the blade The design problem and the analysis model used are defined on Figure 6 In the 4-point bending test facility developed a 2 m test section in the middle of the blade will have a uniform moment distribution, and in the analysis model this 2 m section is extended with areas consisting of stiff material in order to simulate the boundary conditions in the test The airfoil section is very close at being symmetric, and for simplicity this is assumed Thus antisymmetric boundary conditions are applied as illustrated on Figure 6 The leading and trailing edge panels are sandwich structures made of ±45 GFRP as face sheets and foam material as core, and in this preliminary study they cannot be changed In a similar way the webs made of ±45 GFRP are fixed in the design problem studied The design domain is restricted to the spar cap which is divided into 15 patches as shown on Figure 6 Each patch has 1 layers, and the two outer layers at top and bottom are reserved for GFRP oriented at ±15, ±3, and ±45 whereas the 6 inner layers may consist of stiff unidirectional CFRP material together with foam material The mass constraint is set such that 2/3 of the 6 inner layers should consist of foam material The linear buckling analysis yields a lowest buckling mode as shown on figure 7, and the buckling load factor obtained is in good agreement with the collapse load found using a geometrically nonlinear analysis Figure 7: The lowest buckling mode for the initial design of the 9 m test section About 8 four node shell elements are used for the analysis The linear buckling problem yields a number of lowest eigenvalues very closely spaced, and these are taken into account in the optimization problem when the bound formulation is used, see Eq (8) However, the analysis revealed a problem with the analysis code used since the corner stiffness where the webs meet the spar cap, ie, where three shell elements are connected, seems to be incorrect This is a problem associated with the shell formulation currently used, and this problem is currently being corrected Thus, the optimization result shown may not be optimal as the corner stiffness has a great influence on the local buckling modes Figure 8: Preliminary results of optimized layup of spar cap ±3 and ±45 refers to the orientation of GFRP in the two outer layers, 9 means CFRP and foam material Please note that inaccuracies of the analysis model have been observed and thus the results mainly serve as illustration of the ongoing work In part of the domain the DMO approach did not yield a distinct choice of material, thus the layup is partly based on postprocessing of intermediate values of the weight functions in Eq (1) The layup obtained is shown on Figure 8 The preliminary results show that the unidirectional CFRP material 9

10 should be placed at the top as expected, but CFRP material is also distributed through the thickness at patch 4 and 12 where the webs join the spar cap As the local buckling modes are very dependent of the stiffness of these connections this material distribution solution has to be investigated carefully with more accurate models which is ongoing work It was expected that the rest of the CFRP material would be positioned at the bottom of the inner layers For the two outer layers the optimal fiber angle varies between ±3 and ±45 at both the top and bottom The same orientation will be used for the whole domain when manufacturing the test section, and thus the next step will be to enforce the same orientation for these layers for all 15 patches 9 Summary In this paper the Discrete Material Optimization approach has been applied with the purpose of maximizing the buckling load factor of laminated hybrid composite shell structures, and several examples have documented the potential of the parametrization method for multi material optimization The parametrization used may not converge to a distinct /1 design everywhere in the domain but still the method provides much insight into the optimal solution to the material distribution problem 1 References [1] O Sigmund and S Torquato Design of materials with extreme thermal expansion using a three-phase topology optimization method Journal of the Mechanics and Physics of Solids, 45: , 1997 [2] LV Gibiansky and O Sigmund Multiphase composites with extremal bulk modulus Journal of the Mechanics and Physics of Solids, 48: , 2 [3] J Stegmann Analysis and optimization of laminated composite shell structures PhD Thesis, Institute of Mechanical Engineering, Aalborg University, Denmark, 24 Special report no 54, available at wwwimeaaudk/ js [4] J Stegmann and E Lund Discrete material optimization of general composite shell structures Int Journal for Numerical Methods in Engineering, 62(14):29 227, 25 [5] E Lund and J Stegmann On structural optimization of composite shell structures using a discrete constitutive parameterization Wind Energy, 8(1):19 124, 25 [6] J Stegmann and E Lund Discrete material optimization of laminated composite shell structures using local strain criteria In Proc 6th World Congresses of Structural and Multidisciplinary Optimization, Rio de Janeiro, Brazil, 3 May - 3 June, 25 9 pages [7] EN Dvorkin and KJ Bathe A continuum mechanics based four-node shell element for general nonlinear analysis Engineering Computations, 1:77 88, 1984 [8] R Courant and D Hilbert Methods of mathematical physics, volume 1 Interscience Publishers, New York, 1953 [9] WH Wittrick Rates of change of eigenvalues, with reference to buckling and vibration problems Journal of the Royal Aeronautical Society, 66:59 591, 1962 [1] AP Seyranian, E Lund, and N Olhoff Multiple eigenvalues in structural optimization problems Structural Optimization, 8:27 227, 1994 [11] E Lund Finite element based design sensitivity analysis and optimization PhD Thesis, Institute of Mechanical Engineering, Aalborg University, Denmark, 1994 Special report no 23, available at wwwimeaaudk/ el [12] MP Bendsøe, N Olhoff, and JE Taylor A variational formulation for multicriteria structural optimization Journal of Structural Mechanics, 11: , 1983 [13] K Svanberg The method of moving asymptotes - a new method for structural optimization Numerical Methods in Engineering, 24: , 1987 [14] L Kühlmeier, OT Thomsen, and E Lund Large scale buckling experiment and validation of predictive capabilities In Proc ICCM15 - Fifteenth International Conference on Composite Materials, Durban, South Africa, June 27 - July 1, 25 1