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Load Case Characterisation for Gradient Based Optimisation with anAircraft Global Finite Element Model
Dharmasaroja, A., Armstrong, C., Murphy, A., & Robinson, T. T. (2014). Load Case Characterisation forGradient Based Optimisation with an Aircraft Global Finite Element Model. Paper presented at 4th AircraftStructural Design Conference , Belfast, United Kingdom.
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Download date:09. Aug. 2019
https://pure.qub.ac.uk/portal/en/publications/load-case-characterisation-for-gradient-based-optimisation-with-an-aircraft-global-finite-element-model(1521ec66-a43d-4d26-b457-06a58841730c).html
Load case characterisation for gradient based optimisation with an aircraftglobal finite element model
Atipong Dharmasaroja, Cecil G Armstrong, Adrian A Murphy, Trevor T RobinsonQueen’s University Belfast, BT9 5AH, UK
Abstract
Due to the dynamically-varying nature of aircraft load-ing scenarios, a large volume of global load cases aregenerated and required to be analysed in the Global Fi-nite Element Model. One method to reduce the numberof load cases is to use Singular Value Decomposition(SVD) to derive a smaller set of characteristic distri-butions which represents all the global load cases. Theanalysis result for this set of characteristic loads can besuperimposed to create the internal load distributionsfor all the original load cases, with acceptable accuracy.In the structural optimisation process the load distribu-tions change as the local components are optimised, soit is useful to calculate the sensitivities of local compo-nent design variables to the local load distribution. Thispaper proposes a variant of the sensitivity calculationprocess, which is appropriate for large scale gradientbased optimisation. By using the SVD of the set of loadcases, the number of sensitivity calculations, can be sig-nificantly reduced.
Nomenclature
AbbreviationDOF Degrees Of FreedomGFEM Global Finite Element ModelLC Load caseRF Reserve FactorSE Structural ElementSMT Shear, Moment and TorqueSV D Singular Value Decomposition
Roman Symbolsc constant valuel characteristic load vectorn number of calculations, components, etc.u left eigenvectorv right eigenvector
x design variableA full load case matrixC local stiffness matrixD displacement matrixF external force matrixK global stiffness matrixL characteristic load matrixM linear relationship matrixN internal load matrixS scaling matrixU left eigenvector matrixV right eigenvector matrix
Greek Symbolsδ approximation errorε relative approximation errorσ singular valuesΣ singular values matrix
Subscripts and Superscriptsk reduced rank of the matrixr full rank of the matrixT transpose of the matrix
1. Introduction
During the preliminary design stage, an aircraftglobal finite element model (GFEM) is optimised to ob-tain the minimum structural weight. The optimisationis done by changing the design parameters whilst sat-isfying all the design constraints, which are usually interms of reserve factors. In a large airframe model theremay be over 105 degrees of freedom, and thousands ofdesign parameters are involved during the optimisationprocess.
The number of design parameters has become evenlarger since composite materials have become a majormaterial for airframe design. These have additional de-sign parameters, such as ply thickness and angle, which
can be optimised simultaneously with the geometric de-sign parameters [1]. As a result, the size of the ac-tual problem being optimised in the industry these dayscould be very large.
One of the difficulties is that large number of loadcases needed to be evaluated to ensure that the designmeets the target performance in different operationalscenarios. Changing one of these design parameters canalter the load paths through the entire structure, so it isvital to optimise all the design parameters at the sametime.
In the current aircraft load and structural designprocess, load cases are normally provided by the loadsdepartment in terms of an array of data. The array sizecould be very large due to several factors. Firstly, thestructural motion of an aircraft and the external pertur-bations are dynamic, which means varying over a spec-ified time domain. Secondly, the conditions themselvesthat the aircraft has to encounter vary enormously [2].Load cases are generated at every specified time in-stance to replicate the dynamic motion at those oper-ating conditions, so there are great deal of information.However, some of them might be repeated or not evensignificant.
One of the load case reduction techniques was em-ployed in the optimisation process [3]. Load cases werefiltered using the max-min envelope approach, whichselects only the extreme loads near the periphery of theenvelope. A significant number of load cases was elim-inated. As the down-selection is entirely based on theload values and not on the actual structural response ofthe loads themselves, some of the neglected load casesthat lie well inside the envelope may cause the structuralfailure.
The recent research at Queen’s University Belfast[4], [5] showed that the singular value decomposition(SVD) can create a small set of characteristic loads thatrepresents the behaviour of the entire load case set. Thecharacteristic loads can be linearly combined to con-struct the matrix of the original load cases effectively.This paper will show how such an approach can be im-plemented into the optimisation process and contributesto significant computational saving.
2. Singular Value Decomposition
SVD is an elegant way to factorise a general rect-angular matrix into a product of simpler matrices. Sup-pose A is a real m by n matrix (m > n) rank r (r ≤ n).The SVD of A is generally expressed as
A =UΣV T (1)
The products obtained from the SVD comprise of
two orthogonal matrices U and V and a diagonal matrixΣ.
The Σ matrix is a diagonal matrix in which the di-agonal entries are the singular values σ arranged in or-der of magnitude. The matrices U and V , sometimescalled left and right eigenvector matrices, are formu-lated from a set of orthonormal vectors i.e. u1, ...,umand vT1 , ...,v
Tn . The u and v are in Rn and Rm, which are
the column and row space of the A matrix respectively.Equation (1) can be alternatively written in the form ofvectors asA =
[ u1 · · ·ur ur+1 · · ·um ]
σ1 . . . 0σr
0 0
vᵀ1...
vᵀrvᵀr+1
...vᵀn
(2)
All the vectors are linearly independent, so theyform the orthonormal bases of the row and columnspaces. Therefore, the SVD essentially classifies thebases of the row and column of the original matrix intotwo separate matrices and prioritises their magnitudesby the singular value matrix.
Additionally, there are only r singular values ac-cording to the rank of the original matrix. The singularvalues from σr+1, ...,σm are all zero. As a result, thevectors ur+1, ...,um and vTr+1, ...,v
Tn do not contribute
any meaning to the matrix A. Only r components arerequired from each matrix to construct the original Amatrix.
2.1. Reduced-rank approximation
One of the most important applications of theSVD is the reduced-rank approximation. As previouslyshown in equation (2), the SVD requires only r compo-nents to construct the original matrix A and fully pre-serves all of its information.
Since the singular values in diagonal of Σ are ar-ranged in decreasing order, the last few singular valuescould be very small. Let the SVD of A beA =
[ u1 · · ·uk uk+1 · · ·ur ]
σ1
. . . 0σk
σk+1
0. . .
σr
vᵀ1...
vᵀkvᵀk+1
...vᵀr
(3)
If σk >> σk+1, the approximation of A is given by
A≈ Ak =UkΣkV Tk (4)
such that Ak is an approximation of the original matrixA obtained from a reduced dimension of SVD matri-ces: Uk,Σk and V Tk . Equation (4) is usually called thereduced-rank approximation or the rank-k approxima-tion. This idea was initially given by Eckart and Young[6] and is very useful when there are only few signifi-cant singular values in the matrix A. The matrix A canbe approximated from a few singular values along witha smaller set of Uk and Vk.
2.2. Error Quantification
Error arising from the reduced-rank approximationis unavoidable; however, it could be restricted by us-ing a sufficient number of singular values. A methodused to quantify the approximating error can be derivedbased on the Frobenius norm, which is the square-rootof the sum of the squares of all elements in the matrix.
Using the fact that the Frobenius norm is invariantunder orthogonal transformation [7], the matrices thatconsists of a set of orthogonal vectors i.e. U,V or Uk,V Tkare all invariant under the Fronenius norm. Therefore,the Frobenius norm of A can be described by
‖A‖F = ‖UΣV T‖F = ‖Σ‖F =√
σ21 + · · ·+σ2r (5)
Equation (5) implies that the Frobenius norm of Acan be determined by the singular values alone. Sim-ilarly, the error from the reduced-rank approximationonly depends on the ignored singular values. Denoter− k as the partitioning starting from kth rank to rthrank; the approximation error is neatly defined by
‖ε‖F = ‖Σr−k‖F =√
σ2k+1 + · · ·+σ2r (6)
Finally, the relative error between the reduced-rankapproximation and the original matrix can be fully de-scribed by
δ =‖ε‖F‖A‖F
=‖Σr−k‖F‖Σ‖F
=
√σ2k+1 + · · ·+σ2rσ21 + · · ·+σ2r
(7)
3. Characteristic loads
In the situation where an array of load cases pro-vided for the analysis is very large, the SVD can be veryuseful. The main reason is that the loading patterns de-composed by the SVD are classified into the most fun-damental forms. The rank-k approximation can elimi-nate unimportant information from the original data. Ifthe array is arranged appropriately, the SVD can greatlyreduce the redundancy of the data in the array.
3.1. Formulation of the characteristic loads
Let an array of external loads (forces or moments)in the dth degree of freedom (DOF) be
ad =
f11 f12 · · · f1 jf12 f22 · · · f2 j...
.... . .
...fi1 fi2 · · · fi j
(8)such that i is the number of load cases, and j is the num-ber of nodes where the loads are applied. For a modelwith k DOFs, a matrix A consisting of k DOF loads canbe written as
A =
a1 a2 · · · ak (9)
For example, a wing model which has 100 loadcases with 3DOF (Shear, Moment and Torque) appliedat 20 stations along the wing span will have 100 rowsand 60 columns. The number of rows in A is the same asthat in a. The number of its columns is expanded by thenumber of DOF. Hence, the total size of A is determinedfrom
size(A) = m×n = i× ( j× k) (10)
Performing the SVD on A will obtain 3 matrices:U , S and V T . The basis vectors of the row and columnof the matrix are categorised into the U and V respec-tively. Each of them is amplified or shrunk by the cor-responding singular values (σ ).
In order to derive a set of data that represents thecharacteristics of the loads, simply combine Σ and V Tby
L = ΣV T (11)
This can be presented in terms of vectors as
L =
`̀̀1...`̀̀r
= σ1v
T1
...σrvTr
(12)Equation (12) shows that all the vectors `̀̀1, ..., `̀̀r
representing all the bases of the load space are arrangedin order of magnitude. The matrix L is called the Char-acteristic load matrix as it contains all the basis vectorsthat characterise the loading behaviours (vT ) and theircorresponding magnitudes (σ ).
Considering the SVD of A in equation (1), it canalternatively be written in terms of characteristic loadsas
A =UL (13)
Applying the approximation theorem in equation(4) equation (13) can be expressed by k number of `̀̀vectors by
A≈ Ak =UkLk (14)
such that
Lk =
`̀̀1...`̀̀k
= σ1v
T1
...σrvTk
(15)Although at this stage, the definition of character-
istic load appears to be just regrouping matrices into anewly defined variable, the usefulness of the conceptwill become more apparent during the analysis phase.
3.2. Balancing the magnitudes of the loadingmatrix
Preprocessing a matrix by balancing the relativemagnitude of its components improves the result of aneigen value problem if it is poorly scaled [8]. Thisapproach is indeed applicable for computation of theSVD.
Consider the loading matrix A given in equation(9). The loadings are a combination between forcesand moments, their values and units are hence compar-atively different. Additionally, the loading at each sta-tion can be relatively different i.e. shear forces near thewing root are generally higher than those near the tip.The key idea here is attempting to make the values ineach column balanced relative to the others before per-forming the SVD. Consequently, the reference value,which should be obtained from every entry in each ofthe column vectors in A, is required.
Recall that the Euclidean norm of a vector in Rp isthe total length of vector in p dimensional space. There-fore, it can effectively be used as the reference value.Introduce a diagonal matrix S where its diagonal entryis the inverse norm of the corresponding column vectorof A as
S =
‖a1‖−1
‖a2‖−1. . .
‖an‖−1
(16)such that an is a column vector in A. Consequently,a balanced matrix A′ of any given loading matrix A iscomputed by
A′ = AS (17)
Notice that each column vector in A is multipliedby the corresponding diagonal entry in S. As a result, allthe column vectors in A′ are balanced by their relative
magnitude. Performing the SVD on this matrix resultsin
A′ =U ′Σ′V ′T (18)
To convert the loading matrix back to its originalscale, the matrix A′ in equation (17) can simply be mul-tiplied by the inverse of S as
A = A′S−1 (19)
Substitute equation (18) into (19) gives an alterna-tive SVD formulation, which can be written in terms ofcharacteristic loads by
A =U ′Σ′V ′T S−1 =U ′L′S−1 (20)
Equation (20) clearly indicates that all the infor-mation in the matrix A is fully preserved. Referring toequations 14, the SVD of A can be expressed by usingrank k approximation by
A≈ Ak =U ′kΣ′kV ′Tk S−1 =U ′kL′kS−1 (21)
Figure 1 and 2 illustrate the magnitudes (absolute)of the loads from the unbalanced and balanced matrix.
020
4060
80100
120140
160180
0
5
10
15
20
25
300
1
2
3
4
5
6
7
8
9x 105
LoadsLC
Mag
nitu
des
Figure 1: Original magnitudes of SMT loadings
020
4060
80100
120140
160180
0
5
10
15
20
25
300
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
LoadsLC
Magnitudes
Figure 2: Balanced magnitudes of SMT loadings
4. Analysis of characteristic loads
In the aircraft structural design process, a GFEMis evaluated at each external load case to attain internal
load paths going into each sub-structure. The quantityof external load cases therefore directly indicates thenumber of analyses. When the analysis is performedlinearly, incorporating the SVD into the process is pos-sible and could significantly reduce the computationaleffort.
4.1. Linear analysis of characteristic loads
Suppose an output matrix of some linear responsessuch as internal loads (N) is requested from a GFEManalysis. N is typically related to the external force ma-trix (F) by a single matrix (M) that contains only mate-rial and geometric properties of the structure as
N = MF (22)
To integrate the SVD into the finite element analy-sis, the A matrix of loads must be used instead of F . Theformat of F in finite element analysis is slightly differ-ent from the SVD format (A) shown in equation (10).Its rows and columns are generally expressed by
size(F) = n×m = (k× j)× i (23)
Notice that the size of the matrix F is essentiallyAT . The only required operation is repositioning therows in AT from ( j×k) to (k× j), which can be achievedby utilising a permutation matrix (P). This is done toensure that rearranging the matrix will not cause anyincompatibility in the multiplication process.
The permutation matrix is a square matrix whoserows and columns contain a single element with thevalue “1” and “0” elsewhere and normally used to re-shape the current matrix into a desired pattern [9]. Therelationship between the F and AT can be defined as
F = PAT (24)
Both of A and F formats are fixed and governed bythe same set of variables. Therefore, the pattern associ-ating between the two matrices can be found and easilybe programmed. Once the perturbation matrix is con-structed, the formats of the A and F are interchangeable.Equation (22) is rewritten as
N = MPAT (25)
4.2. Superposition of characteristic loads
Since A can be decomposed by the SVD into char-acteristic load L and the corresponding U matrices,equation (25) can be replaced by
N = MPLTUT (26)
Px
Ny
Ny
Nx
Nx
Nxy
Nxy
Figure 3: The in-plane loadings governed in a structural element
This equation is denoted as the superposition ofcharacteristic loads, which is applicable for any fi-nite element applications where the models and the re-quested outputs are linearly related.
The explanation of this equation is that LT is re-arranged to the new format and then analysed in finiteelement analysis. The result is the product of MPLT .The multiplication between MPLT and UT produces thesimilar result with analysing AT in equation (25).
Nevertheless, equation (26) still requires the anal-ysis result of at least r characteristic loads, which doesnot exploit the full advantage of the SVD. Instead, thereduced rank approximation in equation (20) can be em-ployed in order to further reduce the computational ef-fort. This can be written as
N ≈ Nk = MPLTk UTk (27)
Equation (27) shows that all internal load results(NT ) analysed from the full load cases in A can be ap-proximated from the analysis of only k load cases mul-tiplied with the corresponding matrix product.
4.3. Error from the superposition of the k char-acteristic loads
In theory as long as all the characteristic loadsare included, the superposition should give exactly thesame outcome as analysing the full load cases. How-ever, the error will be unavoidable when a fewer numberof characteristic loads is used.
The ideal scenario is being able to evaluate the ap-proximation error without running the full analysis. Re-call that N is the internal load matrix due to the directanalysis of A. The error occuring from the superposi-tion of k characteristic loads to determine Nk is givenby
ε = ‖N−Nk‖F (28)
Substituting equation (26) and (27) into (28) gives
ε = ‖MPLTUT −MPLTk UTk ‖F (29)
Since U and Uk are orthogonal invariant, then
ε = ‖MP(LT −LTk )‖F = ‖MPLTr−k‖F (30)
As a result, the relative error is equal to
δ =‖ε‖F‖A‖F
=‖MPLTr−k‖F‖MPLT‖F
(31)
Equation (31) involves two characteristic loadsterms: L, Lk and some constant matrices: M, P.From two matrix properties namely submultiplicative(‖xy‖F ≤ ‖x‖F‖y‖F ) [10] and orthogonal invariant, thisequation can be rewritten as
δ ≤‖MP‖F‖LTr−k‖F‖MP‖F‖LT‖F
=‖Σr−k‖F‖Σ‖F
≤
√σ2k+1 + · · ·+σ2rσ21 + · · ·+σ2r
(32)
The approximation error at the internal load level isessentially defined by the magnitude of singular values.This equation implies that it should not be greater thanand possess a similar trend with the error at the externalload level in equation (7).
5. SVD based sensitivity analysis
Among the numerical procedures available today,gradient based optimisation is one of the most powerfultechniques capable of handling a large-scale structuraloptimisation problem involving multiple design param-eters, load cases and constraints. This paper presents atechnique that could be suitable for efficient computingin numerical optimisation applications, especially in alarge-scale gradient based optimisation.
5.1. Sensitivity of the reserve factor constraintswith respect to the design variables
In gradient based optimisation process, sensitivitiesof the performance or the constraints, which define themost efficient searching directions toward the optimum,are required by the optimiser. The sensitivities are thechanges in performance or constraint functions with re-spect to the changes in design parameter. Generally theconstraints functions are defined in the form of reservefactors (RF) e.g. RF > 1, in order to ensure that thestructure will be able to withstand the loads.
The RF for a given structure is a function of bothdesign parameters (x) and the internal loads (N); the full
sensitivity equation must be expressed by the chain ruleof differentiation as
dRF(x,N)dx
=∂RF(x,N)
∂x+
∂RF(x,N)∂N
· dN(x)dx
(33)
Typically, calculating the sensitivities is one of themost expensive parts of the gradient based optimisationprocess because of a multitude of constraints, which aresubjected to many loads and design parameters. Figure4 shows an example of some typical design variables ina stringer-panel structural element.
stringer pitch
stringer height
stringer thickness
panel thickness
Figure 4: Typical design variables in a stringer-panel structure
The calculation process is done separately by twomain steps. The first step involves determining the twopartial derivatives: ∂RF(x,N)/∂x and ∂RF(x,N)/∂N.These two terms are generally calculated via the fi-nite differencing, which is computing the difference be-tween the perturbed reserve factor and the existing re-serve factor over a small change in design variables orinternal forces. Most of the reserve factor calculationsare non-linear.
The last part of the chain, the sensitivity of inter-nal forces with respect to design variables dN(x)/dx, isperformed in a separate step. This process is of inter-est because, unlike the reserve factors, all the internalforces are static responses, which can be described lin-early and easily be computed in standard finite elementprograms. Therefore, it is possible to linearly combinedthe analysis results of a reduced set of characteristicloads to approximate the sensitivity due to the full setof original load cases.
5.2. Sensitivity of the internal loads with re-spect to the design variables
Fundamentally in finite element analysis, the sen-sitivity or the gradient of the static responses such as in-ternal load, stress and strain energy is computed basedon the displacement responses due to the applied exter-nal loads [11]. Since the internal load (N) is typically afunction of displacements (D) and design variables (x),the gradient computation can be described by the calcu-lus chain rule as
dNdx
=∂N∂x
+∂N∂D· ∂D
∂x(34)
Three partial derivative terms are required to com-pute the full sensitivity. The internal load appearing inthe equation (34) is generally associated with an ele-ment stiffness matrix (C) and the displacement matrixas
N =CD (35)
The formulation of C depends on the type of thestructure being modelled. Since the explicit expressionof internal load as a function of design variable is usu-ally available, the terms ∂N/∂x and ∂N/∂D can nor-mally be obtained by directly differentiating equation(35) as follows:
∂N∂x
=∂C∂x
D (36)
∂N∂D
=C (37)
Calculating a partial derivative of the displacementwith respect to the design variable ∂D/∂x is slightlymore complicated since there is no explicit equation forthe displacement to differentiate.
The standard practice to obtain this term in finiteelement analysis is to differentiate the static equilibriumequation that describes the displacement of a structurewhen subjected to the applied external force (F) in thestatic equilibrium state by a global stiffness matrix (K)as
F = KD (38)
Differentiating and rearranging this equation yields
KdDdx
=∂F∂x− ∂K
∂xD (39)
The terms on the right hand side of equation (39)are referred as Pseudo-Loads. Usually, the applied ex-ternal forces are independent of the design variables, sothe term ∂K/∂x can be eliminated. Subsequently, re-arrange this equation to obtain only ∂D/∂x on the lefthand side as
∂D∂x
=−K−1 ∂K∂x
D (40)
Equation (39) can be solved directly to obtain theterm ∂D/∂x. Since the stiffness matrix (K) and thedisplacement matrix (D) are available from running thefirst finite element analysis, the change of the stiffnessmatrix with respect to the change in the design variable(∂K/∂x) can be computed, either by means of directdifferentiation or via finite differencing as implementedin Nastran [12].
Substituting equation (36), (37) and (40) into equa-tion (34), the sensitivity of the internal loads (N) withrespect to the design variable (x) can be expressed in
terms of displacement (D) and stiffness matrices (C, K)as
dNdx
=
(∂C∂x−CK−1 ∂K
∂x
)D (41)
5.3. Sensitivity of the internal loads with re-spect to the external applied forces
In order to validly apply the superposition of thecharacteristic loads, the linear relationship between thesensitivity of the internal loads (dN/dx) to the appliedexternal loads should firstly be be determined.
This task is relatively straightforward since the re-lationship between the sensitivity of the internal loads(N) as a function of displacement (D) has been estab-lished in equation (41). D is simply the displacement re-sult, which can expliciltly obtained from equation (38)as K−1F . Substituting K−1F into equation (41) yieldsthe equation of the sensitivity in terms of external forcesas
dNdx
=
(∂C∂x−CK−1 ∂K
∂x
)K−1F (42)
Thus, the sensitivity of any internal load can be ex-pressed by
dNdx
= MF (43)
Equation (43) implies that the sensitivity and theexternal force matrix are linearly related by the M ma-trix. As it is already in the same format as equation (22),the superposition of k characteristic loads described inequation (27) is possible and can be given by
dNdx
= MPLTk UTk (44)
Equation (44) describes the finite element processthat analyses k characteristic loads (L) which has beenrearranged by the permutation matrix (P) and subse-quently multiplied by the coefficient matrix (U).
6. Cost analysis
There are many factors influencing the cost of thesensitivity analysis, but only three major terms are men-tioned here: number of calculations, analysis time andthe size of transferred data. Being able to quantify theseterms allows the benefit of the load case reduction to beclearly understood.
6.1. Number of calculations
In linear calculation, the size of an output array nor-mally implies the number of calculations needed to be
Figure 5: Example of the sensitivity matrix on a GFEM structure for a singleload case
done. Figure 5 shows the typical array of a GFEM struc-ture. The size of the sensitivity matrix as similar to thisfigure can roughly be estimated by
ntotal = (nN×nSE)×nLC×nx (45)
such that nN , nSE , nLC and nx represents the number ofinternal loads, structural elements, load cases and de-sign variables respectively.
The number of internal forces in a structural assem-bly depends on the type of structural element. For ageneral wing GFEM involving panel and stringer struc-tural elements, there are 3 in-plane loads on each paneland 1 axial load on the stringer.
6.2. Analysis time
The analysis time is a very crucial factor in theoptimisation process because the process may be re-peated for many iterations before the optimal designis achieved. Reducing the sensitivity calculation timein each optimisation cycle contributes to the significantamount of time saving of the entire design process. Notonly the design operation can be controlled within thelimited time frame, but this also encourages the design-ers to experiment with more data i.e. load cases anddesign parameters.
Due to the linear relationship between time andnumber of calculations, the rate of increase of the anal-ysis time can be determined from just a single finite el-ement run of any number of load cases. This can beexpressed by
slope =timenLC
nLC(46)
6.3. Data size
In the aerospace industry, engineering programsare usually operated in the central computers via an in-ternal network. Significantly large files may potentiallycause some problems as follows:
• Hard disk capacity: Since the files have to bestored in the computer’s hard disks, the full discspace prevents any new jobs to be invoked, espe-cially when a company operates its software on themain computer servers.
• Memory: Printing and reading a very large filesize consume considerable amount of memory andsometimes cause the system to crash.
• Network transfer: Transferring a set of large filesthrough the system network could sometimes takea very long time to complete. It also consumes thenetwork bandwidth and may limit the access speedof other users.
The size of files generated by a finite element pro-gram could be very large depending on which types offiles are required. The files that contain finite elementproperties and relevant equations are normally constantregardless of the number of load cases. However, thesize of output files depends greatly on the size of theoutput matrix, which is proportional to the number ofload cases in static sensitivity analysis.
Similar to the analysis time, the data size is in-creased linearly as the number of load cases rises. Theslope of the graph can be expressed as
slope =sizenLC
nLC(47)
7. Results & discussion
The experiment was conducted on the actual load-ing data on a wing and a typical wing GFEM so that theimpact of the characteristic load reduction on the sensi-tivity analysis process can be clearly investigated.
The loading data was provided by the load depart-ment in Airbus (UK) as an array of load cases that weregenerated from a typical real aircraft database contain-ing a mix of ground and flight load cases. It was similarto the one that would be used in the actual design pro-cess. The data was already desensitised by using a toolmapping the loads from a specific geometry to a simplebox-beam representation.
7.1. Load case characterisation result
The size of the initial loading array is [100×162].The SVD was performed on this matrix to extract dif-ferent sets of characteristic loads Lk such that k =1, ...,100. It was computed using the predefined SVDfunction in MATLAB, which is extremely fast and effi-cient. For this specific scenario where there are less than30×104 entries in a single matrix, the computational ef-fort is considered insignificant. The characteristic loadswere then linearly combined with the corresponding Ukmatrix to approximate the A matrix. The SVD was alsocarried out for the balanced case.
0 20 40 60 80 1000
10
20
30
40
50
LC
Erro
r (%
)
UnbalancedBalanced
Figure 6: Original magnitudes of SMT loadings
Figure 6 demonstrates that the approximation er-ror reduces when the number of characteristic loads in-creases. Error drops to almost zero at around 20 char-acteristic loads for the original data and 25 for the bal-anced load matrix.
7.2. Balancing
The result suggests that the balancing increases theapproximation error calculated in the Frobenius norm.This happens probably because the Frobenius normmeasures the magnitude of the overall matrix, and themagnitude of some loading values i.e. moments is toodominant in the unbalanced case. Since the SVD tendsto captures higher magnitudes, only the moment valueswere well captured.
However in the analysis stage, other values such asshear forces are also important. More detailed inves-tigation, for example error of each loading componentat the individual station, is required before the decisionbetween the original and balancing approaches can bemade.
Further study focusing on the effect of balancinghas been conducted. Figure 7 shows that the relative er-rors at each wing station, which were calculated from
the sum of the relative error at each column of the ma-trix divided by the number of columns. The balancingimproves the load cases reconstruction quality in gen-eral.
1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
60
70
80
90
100
LC
Erro
r (%
)
UnbalancedBalanced
Figure 7: Average error per station of the 10 lowest magnitude cases
A plot of the shear force in the chord-wise direc-tion of the wing around the wing root stations is alsoshown in figure 8. It can clearly be seen that balancingimproves the mapping quality.
0 2 4 6 8 10−1000
−900
−800
−700
−600
−500
WingUstation
X−ax
isUsh
earUf
orce
U(N)
ActualUnbalancedBalanced
Figure 8: The reconstruction results of the x-axis shear forces around the wingroot stations of a low magnitude load case
The future work, the error in other norms (e.g. in-finity norm, 2-norm) can be investigated to minimisedthe error in the quantities of interest.
7.3. Analysis of characteristic loads results
Different sets of characteristic load cases, rangedfrom 1 to 100, were carried over to analyse in the wingGFEM. The 100 actual load cases were also analysedto be used as the based reference in the error calcula-tion. Only the results from the balancing method willbe shown here.
From several input files generated, the main filethat is of concern is the one that contains the loadinginformation. The characteristic loadings were firstly
transformed into the GFEM geometry using the 6DOFshear moment diagram and then inserted into the inputfile. The analysis was performed in the MD Nastran2011, which has been installed on a Windows 7 Enter-prise 64bit machine with Intel Xeon 3.10 GHz proces-sor and 16 GB memory.
Sensitivity results were superimposed from a set ofcharacteristic load cases. Figure 9 shows the approxi-mation error of the analysis and the external loads. Thesuperposition error is eliminated when the number ofcharacteristic loads is around 25. The trend of the re-sults complies with equation (32), which suggests thatthe superposition should possess a similar trend with theerror at the external loads level. The maximum errors in
0 20 40 60 80 1000
10
20
30
40
50
LC
Erro
r (%
)
External loadSensitivity
Figure 9: The comparison between the errors from the external and the analysislevel
each design variable column are also less than 5% rela-tive its norm. Hence, 25 characteristic loads should besufficient to approximate the sensitivity values.
7.4. Cost analysis
Three types of information were studied includingdata size, analysis time against the error from each char-acteristic load run against the approximation error. Theanalysis time and the size of output files were obtainedfrom equation (46) and (47) respectively. The differ-ences between the actual sensitivity matrix and the ap-proximated were normalised by the Frobenius norm andcompared with the Frobenius norm of the actual matrixto obtain the approximation errors. These 3 types of in-formation were plotted together as shown in figure 10.
The relationship between the approximation errorand computation effort is inversely proportional. Theerror approaches zero when increasing the number ofcharacteristic loads to around 25 whereas the file sizeand the analysis time continue to increase. Instead ofrunning the full 100 load cases, the analysis of only 25characteristic loads would provide computational sav-ings of around 3 times.
0 10 20 30 40 50 60 70 80 90 1000
5
10
15
20
Erro
rb(%
)
0
50
100
150
200
TIm
eb(s
)
LC
ErrorTimeFilebsize
10 20 30 40 50 60 70 80 90 100 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
File
bSiz
eb(g
b)
0
50
100
150
200
TIm
eb(s
)
LC
ErrorTimeFilebsize
Figure 10: Approximation error, analysis time and file size at different numbersof characteristic loads of a single structural element
It is important to remark that only a structural ele-ment consisting of 1 stringer and 2 panels with around60 design variables was run in this experiment. Thereare approximately, according to equation (47), around7×60 = 420 entries in the sensitivity matrix generated.
In contrast, a typical industrial wing GFEM modelis an assembly of approximately 500 structural ele-ments. The number of rows and columns of the ma-trix will increase by a factor of 5002. Hence for a fullwing GFEM, the size of the sensitivity matrix would bearound 420× 5002 = 10.5× 106 per single load case.Eliminating 1 load case significantly reduces the com-putational effort.
Table 1 demonstrates the potential cost reductionthat can be achieved when using 25 characteristic loadsinstead of 100 load cases in the sensitivity calculationof a full wing GFEM.
Table 1: Comparison of computational effort between the full and characteristicload cases analysis (estimated for a full wing GFEM)
Analysis type Time (hr) Size (TB) Error(%)100 full LCs 1194 114 -
25 characteristic LCs 298 28 0.22
8. Conclusion
This paper presents a new approach to characteriseaircraft loading and its application in the gradient basedoptimisation process. A set of smaller important load-ings called characteristic loads can be identified fromthe full loadings by performing SVD. The linear rela-tionship between the sensitivity matrix of the internalloads with respect to the design variables and the exter-nal loading matrix in a finite element model indicatesthat it is possible to analysed the characteristic loads andthen use linear combination of the reduced rank SVDapproximation to create the full set of sensitivity results.
Using the Frobenius norm as the error indicatorprovides a good and efficient way of decision makingof how many characteristic loads are required to createwell approximated results. Detailed error investigationmay be required, especially in an application where theaccuracy is crucial. More characteristic loads can be in-cluded to improve the accuracy but will also increasethe analysis time. The balancing method, which is anapproach to adjusting the magnitude of the loading ma-trix before performing the SVD, will also improve theaccuracy.
In the experiment, different sets of characteristicload cases were extracted from 100 load cases. The re-sult suggests that only a quarter of the full load casesis sufficient to create the original results. The set ofcharacteristic loads were then analysed in the wingGFEM to create the approximated sensitivity results.By using only a quarter of the original load cases inthe analysis, significant computational reduction can beachieved. For a typical GFEM wing, it effectively elim-inates around 10.5×106 calculations for each load caseremoved. The study also concludes that the analysistime and the file size were reduced by 75%.
9. Acknowledgement
The authors would like to thank Airbus for thesponsorship of this project. We’d also like to give spe-cial thanks to Nicolae Lucian Iorga and James Barronof Airbus for technical assistance and information pro-vided.
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