Optimizing Variable-Axial Fiber-Reinforced Composite ...

Research ArticleOptimizing Variable-Axial Fiber-Reinforced CompositeLaminates The Direct Fiber Path Optimization Concept

Lars Bittrich1 Axel Spickenheuer1 Joseacute Humberto S Almeida Jr 1 Sascha Muumlller2

Lothar Kroll2 and Gert Heinrich1

1Mechanics and Composite Materials Department Leibniz-Institut fur Polymerforschung Dresden e V Hohe Str 601069 Dresden Germany2Institut fur Strukturleichtbau Technische Universitat Chemnitz 09107 Chemnitz Germany

Correspondence should be addressed to Jose Humberto S Almeida Jr humbertoipfddde

Received 18 September 2018 Revised 17 December 2018 Accepted 16 January 2019 Published 19 February 2019

Academic Editor Arkadiusz Zak

Copyright copy 2019 Lars Bittrich et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The concept of aligning reinforcing fibers in arbitrary directions offers a new perception of exploiting the anisotropic characteristicof the carbon fiber-reinforced polymer (CFRP) composites Complementary to the design concept of multiaxial composites alaminate reinforced with curvilinear fibers is called variable-axial (also known as variable stiffness and variable angle tow) TheTailored Fiber Placement (TFP) technology is well capable of manufacturing textile preforming with a variable-axial fiber designby using adapted embroidery machines This work introduces a novel concept for simulation and optimization of curvilinearfiber-reinforced composites where the novelty relies on the local optimization of both fiber angle and intrinsic thickness build-upconcomitantly This framework is called Direct Fiber Path Optimization (DFPO) Besides the description of DFPO its capabilitiesare exemplified by optimizing a CFRP open-hole tensile specimen Key results show a clear improvement compared to the currentoften used approach of applying principal stress trajectories for a variable-axial reinforcement pattern

1 Introduction

Recently the demand for energy efficient systems leveragedthe use of CFRP lightweight composites in structural com-ponents These materials are increasingly being employed inaeronautical aerospace and automotive applications Due tothe high cost of carbon fibers their efficient usage becomesessential [1] By employing a variable-axial (VA) fiber designstiffness and strength properties may be improved whencomparing to classical CFRP designs [2] Thereby the termVA means varying the fiber orientation at the ply levelThe desired performance of CFRP composites is achievedby guiding the loads almost exclusively along the fiberorientation and thusminimizing the shear load of thematrixFor a technical realization TFP technology which wasdeveloped at Leibniz-Institut fur Polymerforschung Dresden(Germany) is well suited Basics and some applications ofTFP technology are described in [3 4] The placement ofcarbon fibers is usually carried out by stitching dry rovings

as shown in Figure 1The roving is guided through a rotatableroving pipe onto a base material where a sewing threadapplied in the zig-zag-pattern holds it in place

Several approaches have been developed to optimizeVA composites An extensive overview of curvilinear fiber-reinforced composites was recently performed by Ribeiro etal [5] Under the name variable angle tow steeringWeaver etal [6] improved the postbuckling performance of compositepanels with a VA layout whereas Panesar and Weaver [7]optimized blended bistable laminates suitable for morphingflap applications Duvaut et al [8] implemented a varyingfiber density in order to consider local stress intensity Fora similar purpose the local layer thickness was varied byParnas et al [9] as an additional design parameter Grohand Weaver [10] proposed a minimum-mass design of atypical aircraft wing panel under end-compression Khaniet al [11] developed a mathematical optimization algorithmfor variable stiffness panels using lamination parametersVan Campen et al [12] proposed a methodology to convert

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 8260563 11 pageshttpsdoiorg10115520198260563

2 Mathematical Problems in Engineering

Roving pipe

Moving DOFof the base material

Roving

Needle

Sewing threadBobbin

Figure 1 Basic principle of the TFP process

known lamination parameters distribution for a VA compos-ite laminate into realistic fiber angles with minimum lossof structural performance Cho and Rowlands [13] reducedstress concentrations in an open-hole laminate with a geneticalgorithm

In contrast to optimization procedures principal stresscriterion has been often used for deriving curvilinear fiberpaths eg [14ndash17] Kelly et al [18] Waldmann et al [19] andMalakhov and Polilov [20] designed the curvilinear fiber pathbased on the concept of aligning fibers following the load pathby placing fibers along principal stresses Both approacheswere assumed to be optimization criteria although no math-ematical optimization process was explicitly carried out andonly an a priori design criterion was applied Furthermoregradient based numerical optimization processes [9 21ndash23]and optimization approaches based on evolutionary algo-rithms [24ndash28] to design VA composites have been alsodeveloped

However due to the often chosen approach of varyingangles of single quasi-isotropic finite elements (FE) the num-ber of design variables is considerably high also increasingcomputational costs Therefore most examples have beencomputed by using FEmodels with a limited number of finiteelements Increased design freedom causes increased designproblem complexity For example Conti et al [17] found thatusing fiber angles as design variables inevitably leads to anill-behaved objective function with many local minima

Usually a VA fiber pattern can be implied in a varyingdensity of fibers which causes a nonuniform thicknessof the dry preforms Hence this heterogeneous thicknessbuild-up is extremely complex to be accounted for in theanalysis Given this complexity current approaches neglectthe thickness accumulation and consider only the fiber anglevariation

Thus the major criticism of many state-of-the-art opti-mization approaches is that there is no mathematical opti-mization procedure to begin or to operate without necessaryinformation on the manufacturing process due to the lack ofan appropriatemodeling procedure However the knowledgeof the thickness distribution and the local fiber orientationcorresponding to an arbitrary fiber layout which can beproduced by TFP is essential for the part design processSpickenheuer et al [1 29] and Albers et al [30] made initialattempts to separate the optimization process of a curvilinearfiber-reinforced composite manufactured via TFP from theactual numerical models in order to limit the number ofrequired design variables making them independent of the

applied FEmesh resolutionThus once a sufficiently accuratemodeling of VA fiber layouts is established optimizationtechniques can be applied to the fiber pattern This allowsfully utilizing the high degrees of freedom in the designprocess and the maximization of the anisotropic materialcharacteristics of CFRPs

Given the identified gaps in the current state-of-the-art inproperly modeling VA composites this work presents a noveldesign procedure for illustrating the capability for generatinga VA pattern for an open-hole tensile specimen wherean optimal fiber pattern cannot be easily derived Hencethe novel optimization approach called Direct Fiber PathOptimization (DFPO) for VA composites will be introducedand numerically evaluated on the example of an open-holetensile specimen

2 Finite Element Modeling

21 Model Setup According to the state-of-the-art modelingof composite structures is mostly limited to stacking layerswith a constant thickness and a constant fiber angle withineach layer Models for structural analysis of uniform spiralsand single curved tapes of parallel fibers and constantthickness have been successfully applied additionally by anincrease in modeling efforts [31] In this case an analyticdescription for local preform thickness and the fiber ori-entation is known which can be used to build appropriateFE models for structural simulation However the existingmodel limitations are too strong if one plans to applyoptimization strategies to fully exploit the potential of CFRPmanufactured by TFP and to adapt production requirementsAlthough there are many approaches of employing themathematical description of the optimization algorithm todeduce the numerical model eg the geometry of eachiteration step this work is going to describe the modelingindependent of the optimization and thus as a genericmodulefor any VA structure with a similar placement characteristicstrictly following the manufacturing characteristics of TFPThe objective of the modeling is the elastic description ofthe laminate compliance and the prediction of initial failuresbased on a physically based failure criterion A mesoscaledmodel is used to evaluate the specific properties of the TFPprocess following the recommendations raised by Uhlig et al[32]

To generate continuous layers with TFP the rovings haveto be placed with a slight overlap to avoid gaps betweenthem If placed with constant thickness and fiber orientationparallel laminates can be produced for a small range ofdistances between neighboring rovings The thickness 119905 iscalculated according to the following (see Figure 2)

119905 = 119860119903119900V119889 = 119879119905120588119889120593 (1)

where Arov is the roving cross-section area 119889 the distancebetween neighboring rovings 119879119905 the roving fineness 120593 thefiber volume fraction and 120588 the fiber density

For arbitrary nonparallel roving placement the thicknessevaluation becomes more complex As a starting point for

Mathematical Problems in Engineering 3

t

d

Aro

Figure 2 Schematic depiction of (1)

y

x

( x12 y12 )

( x02 y02 )

( x01 y01 )

( x13 y13 )

( x03 y03 )

( x11 y11 )

Figure 3 Placement pattern as a sequence of straight lines

the analytical description of such a preform the placementpath is used which is the basis for the fiber placement witha TFP machine This path or more generally a sequence ofpaths will be referenced as the design pattern The simplestmathematical description is a sequence of straight lines intwo dimensions Curved placement paths eg containingprimitives such as arcs or splines will be approximated witha sequence of short straight lines within the accuracy ofproduction A straight line is defined by the starting point(1199090119894 1199100119894 ) and end point (1199091119894 1199101119894 ) inside the placement plane Allpoints in between are described by

(119909119894 (119904)119910119894 (119904)) = (11990901198941199100119894 ) + (1199091119894 minus 11990901198941199101119894 minus 1199100119894 ) 119904 (2)

where 119904 is the parameterization variable ranging from zero toone Note that either the total sequence of straight lines can beconnected in the case if there is just one fiber path or at leastsome succeeding lines are not connected which representscompletely separate fiber paths as can be seen in Figure 3

This type of design pattern contains only the informationof the fiber placement paths including the length of rovingsbut no information about the width or cross-section area Aformal extension of the straight path information combinedwith the cross-section area 119860119903119900V is the line thickness distribu-tion 119905119897119894119899119890 line for line segments with length 119889119894

119905119897119894119899119890 (119909 119910)= sum119894

119860119903119900V119889119894 int10

120575 (119909 minus 119909119894 (119904)) times 120575 (119910 minus 119910119894 (119904)) 119889119904 (3)

Here the concept of the Dirac delta distribution was usedto define the density functionThis line thickness distribution

function is only an intermediate step as the total fiber volumeis concentrated along the infinitely thin lines and an infinitelyhigh thickness is obtained on the lines and zero elsewhereHowever the function 119905119897119894119899119890 already fulfills the normalizationcondition By integrating over the total design space or anyarea containing all line segments the total fiber volume 119881119905119900119905 isobtained

119881119905119900119905119886119897 = ∬ 119905119897119894119899119890 (119909 119910) 119889119909119889119910 = sum119894

119860119903119900V119889119894 (4)

For practical purposes this thickness distribution is notvery useful as it lacks the information about the width of atypical roving which is placed by an embroidery machineThis width usually depends on the type of rovings usedeg the number of filaments material density and mostimportantly on a machine parameter the width of the zigzagstitch used to fix the roving on the base material

By convolution with different smoothing functions theinformation about the width of the roving can be added Avery convenient approach is the coarse-graining by convolu-tion with a Gaussian centered at (119909 119910) of width 120590 determinedby the placement width to obtain the Gaussian thicknessdistribution

119905120590 (119909 119910) = 121205871205902 ∬ 119905119897119894119899119890 (1199091015840 1199101015840)sdot exp(minus(119909 minus 1199091015840)2 + (119910 minus 1199101015840)221205902 ) 11988911990910158401198891199101015840 (5)

The coarse-graining is done by integrating the functionin the whole plane of 1199091015840 1199101015840 By using the definition of theline thickness distribution 119905119897119894119899119890 a solution for this convolutioncan be expressed in terms of error functions This makes anumerical implementation very fastThis Gaussian thicknessdistribution represents a Gaussian weighted average of theroving volume density in the area around the point at whichthe thickness needs to be computed For single straightrovings this thickness distribution leads to a Gaussian cross-section area which roughly approximates the real cross-section areas for the TFP process as shown in microsectionsby Uhlig et al [32] Other smoothing functions such as acylindrical average approximate the cross-section of a singleroving to a closer degree However the resulting laminatesexhibit many discontinuities which negatively influence theconvergence of the modeling With the Gaussian thicknessdistribution the laminate boundary needs to be defined by acut-off thickness as the Gaussian is nowhere exactly zero

The main challenge for numerical modeling is to obtainthe geometry and the fiber orientation based on the place-ment pattern of a single layer Successive layers are stackedon top of each other without regard to draping behaviorwhich is fine as long as thickness gradients of the lowerlayers are small enough The description is restricted tolayers of noncrossing rovings or at least to roving placementswhere overlapping rovings cross at small angles such thatan element wise average of fiber orientation is meaningfulNote that for many examples which contain a self-crossing


roving path the layer can be split into smaller layers withnoncrossing rovings In Figure 4 a schematic descriptionof the modeling procedure is shown Based on a two-dimensional (2D) mesh of the planar design space a three-dimensional finite element model is derived using localizedinformation of the Gaussian thickness distribution and theaveraged fiber orientation The thickness is evaluated at eachcorner node and the fiber orientation at the center of eachelement The fiber orientation is well defined for linear linesegments The elemental fiber orientation is averaged by athickness weighted average of all linear line segments whichcontribute to the total thickness at the center point of eachelement

Successive layers can be stacked on top of each otherThe resulting three-dimensional (3D) FE model represents apiece-wise linearization per element of the locally averagedcharacteristics namely thickness and fiber angle Alterna-tively the thickness and fiber angle can be combined at thecenter of the FE into a 2D layered shell element descriptionto obtain a model for the same fiber layout with less com-putational cost The main difference arises from neglectingthe out-of-plane component of the fiber orientation andthickness gradients within an element

Next two numerical examples are considered by usingDFPO For both cases the following parameters are usedfiber volume fraction (120593) of 58 roving fineness (119879119905) of400 tex density (120588) of 176 gcmminus3 and width smoothingparameter (120590) of 1 mm

22 Case 1 Open-Hole Tensile Specimen An open-hole spec-imen under tensile loading is chosen to demonstrate themodeling capabilities and the optimization of VA laminatesby employing the DFPO approachTheir specimen geometryand dimensions are presented in Figure 5(a) In order todirectly evaluate the capabilities of the proposed optimizationframework the specimen comprises two layers achieved bystacking a carbon fiber TFP layer (layer to be optimized) ontop of the base material (plusmn45∘ woven fabric with area weightof 256 1198921198982) Figure 6 shows in detail the two-layer open-hole specimen in study

Based on a 2D meshing of the supporting plane thelocal thickness is evaluated at each node for each laminatelayer along the FE mesh as it is shown in blue color scale(Figure 7(b)) In addition the elemental fiber orientation(Figure 7(a)) is set as the averaged fiber orientation at thecenter of each element In areas where the current consideredfiber pattern places no rovings the thickness computationyields effectively zero However to provide a continuousmesh in this case a very small thickness of 0001 mm is setat the corresponding nodes and the corresponding elementmaterial properties are set to resin properties (blue elementsin Figure 6) The corresponding FE model additionallyincorporates at the bottom of the laminate a layer of constantthickness (024 mm) of base material as Figure 6 depicts

Symmetrical boundary conditions are applied along allaxes of the specimen The load is applied at the top-edge ofthe specimen These details can be seen in Figure 6 Finiteelement simulations are carried out in ANSYS APDL using

xyz

Plane based onthickness analysis

Area for fiberorientation analysis

Area for thickness distribution analysis

Segment of thevariable-axial fiber pattern

Plane of reference nodes

Adapted elementcoordinate system

Figure 4 Schematic description of the modeling structure

quadratic SOLID186 and linear SOLID185 elements (ANSYSlibrary reference)

23 Case 2 Narrow-Middle Tensile Specimen In order toprovide another example for the applicability of the pro-posed DFPO framework a sample under the same loadingconditions has been considered For that a narrow-middlespecimen under tensile loading is analyzed and optimizedDetails on the geometry and dimensions of the narrow-middle tensile specimen are shown in Figure 5(b) In orderto evaluate the capabilities of DFPO similarly to the open-hole specimen the sample consists of two layers attained bystacking a carbon fiber TFP layer (layer to be optimized) ontop of the base material (plusmn 45∘ carbon fiber woven fabric withareal weight of 256 119892119898minus2) The material properties of bothUD carbon fiberepoxy TFP layer and the carbon fiberepoxywoven fabric laminated composites used in the FE modelsand optimizations are presented in Table 1

3 Optimization Process

The optimization problem for the fiber path is describedby the minimization of an objective function in whichcompliance minimization is the objective function whichanalogously stands for stiffness maximization (119878) under vari-ation of each roving placement path Ci Within the contextof the actual optimization two compliances are aimed to beminimized as follows

119878119905119894ff119899119890119904119904 119900119901119905119894119898119894119911119886119905119894119900119899 min119862119894

(maxΩ

119906119910) (6)

119878119905119903119890119899119892119905ℎ 119900119901119905119894119898119894119911119886119905119894119900119899 min119862119894

(maxΩ

119872119868119860) (7)

where minimizing the maximum of the displacement in 119910-direction is the objective function for stiffness optimizationwhereas minimization of the maximum of MIA (modeinteraction parameter) is the objective function for strengthoptimizationThis MIA parameter is related to the physicallybased failure mode concept developed by Cuntze [33] Withthis criterion it is possible to distinguish several failuremodes namely tension and compression induced failuremodes for fiber failure and compression tension and shearinduced inter-fiber-failure modes Cuntzersquos Failure Mode


200

mm

20 mm

80 mm(a)

200

mm

80 mm

40 mm

(b)

Figure 5 Geometry and dimensions for the open-hole (a) and narrow-middle (b) tensile specimens reinforced with UD fibers (referencelayouts for the optimizations)

Table 1 Material properties for both TFP and base material layers

TFP layer unidirectional CFRP1198641(119866119875119886) 1198642 = 1198643(119866119875119886) 11986612 = 11986613 = 11986623(119866119875119886) ]12 = ]13 = ]23132 956 576 0258Base material CFRP woven fabric (plusmn45∘)1198641 = 1198642(119866119875119886) 1198643(119866119875119886) 11986612 = 11986613 = 11986623(119866119875119886) ]12 = ]13 = ]2362 767 417 0033

Top view

F

Optimized fiber layer

Base material

Front viewDetail 1

Detail 1

Figure 6Quarter-symmetrical FEmodel of an open-hole specimenhighlighting both TFP-optimized (red for carbon fiber and blue forresin) and base material (magenta) layers

Concept (FMC) is based on the stress and strengths quanti-ties which means that MIA (failure parameter) is calculatedbased on the stress state of the laminate at each interactionalong the analysis In other words if 119872119868119860 ge 1 then thelaminate fails analogously if 119872119868119860 lt 1 the laminate is safeAdditionally all failure modes can be combined into a singlenumerical value suitable for optimization with the modeinteraction (MIA) quantity Since the whole formulation ofCuntzersquos FMC is very extensive its full description can be seenin [33]

Mathematically the dimensionality of the optimizationproblem of even a single roving path is infinite Howeverdue to limited production accuracy the placement path canbe modeled using a finite set of parameters within someplacement path representation

The optimization flowchart is implemented and pre-sented in Figure 8 The parameterized fiber layout is repre-sented by a finite set of coefficients eg spline control pointsThe 2D fiber path is computed which in turn is analyzedby the 3D modeling tool to generate the finite elementmodel The local thickness and fiber orientation are takeninto account Loads and boundary conditions are applied andthen the model is solved Based on this solution the targetoptimization value (compliance minimization or stiffness


100

50

0

minus50

minus100minus40

[mm]

[mm

]

minus20 0 20 40

(a)

100

80

20 01

02

Thickness[mm]

03

04

05

00

40

60

0

[mm

]0

[mm]10 20 30 40

y

x

(b)

Figure 7 Geometry (black contour) fiber rovings (red lines) (a) and thickness distribution for the open-hole tensile test specimen (b)

Start with initialfiber layout

Optimized fiber layout

Yes

No

Fiber pathparameter

optimization(BOBYQA)

ndash New fiber layout

3D-FE-model restricted todesign space with boundary

conditions and loads

Performance evaluationeg compliance

Change inperformance value small

Figure 8 Optimization procedure for Direct Fiber Path Optimization


0000700006000050000400003000020000100000

SOLID 185SOLID 186

Number of elements

max

(ＯＳ) [

mm

]

104 105

(a)

max

(MIA

)

SOLID 185SOLID 186

Number of elements

040

045

050

055

104 105

(b)

Figure 9 Convergence study for maximum displacement (a) and maximum MIA (b) with respect to the number of elements by using themodeling approach

maximization) is derived The optimization value is the soleinput value for gradient free optimization algorithms suchas BOBYQA (Bound Optimization BY Quadratic Approx-imation) by Powell [34] which can modify the fiber pathparameters within predefined boundaries to achieve a min-imal displacement value As long as no gradients are derivedonly gradient free optimization algorithms can be usedBOBYQA provides a fast converging algorithm for smoothoptimization functions due to its quadratic approximationalso implementing box constraints that can be used to restrictthe fiber pattern towithin reasonable locations Details on theoptimization parameters are given in Section 32 In generalother optimization values such as failure stress can beapplied However the convergence to overall good solutionsis much better for stiffness optimization in comparison tostrength optimization Thus for a strength optimization astiffness optimized layout is used as an initial layout

31 Convergence Study For the use in optimization proce-dures the numerical model must be sufficiently stable andfree of mesh dependence once otherwise numerical fluctu-ations lead to nonconverging behavior in the optimizationalgorithm

For layers that fully cover the design space such thatneighboring rovings overlap ie maximum displacementin 119910-direction max(119906119910) (Figure 9(a)) and maximum MIA(Figure 9(b)) the simulation converges or stabilizes withincreasing the number of elements (119873) as Figure depictsRegarding stiffness optimization (Figure 9(a)) the FE modelcomposed of quadratic elements (SOLID 186) easily con-verges for any element size whereas for the FE modelwith linear elements (SOLID 185) the model converges wellwith a minimum number of 20000 elements On the otherhand for strength optimization (Figure 9(b)) both elementtypes take a while to converge but for a mesh density of200000 elements the FE model converges for both linearand quadratic elements In this way for the stiffness objectivefunction the mesh with 20000 elements has been employedin all further optimization and FE analyses

The convergence is only achieved if the boundaries of therovings overlap the previous and next rovings thus formingcontinuous layers without gaps If the rovings do not fill the

wholemesh this basemesh elements need to be aligned alongthe bounding contour of the fiber layers to allow a realisticmaterial description per element

32 Open-Hole and Narrow-Middle Specimens OptimizationIn this section the parameterization of the fiber layout isdescribed in more detail For both examples only the 0∘layer is optimized However in general multiple layers can beparameterized in a similar way and the collective parametersets are combined to form a single optimization parametervector A basis or an initial fiber layout is chosen and theparameterization describes only modifications of this layoutFor the 0∘ layer of both examples a layout of equidistantstraight and parallel fibers is chosen as an initial layoutDeviations from this layout are restricted to shifts in 119909-direction (see coordinate system in Figure 6) which limitspossible layouts to angles of less than 90∘ between fiberorientation and the load which is parallel to the 119910-directionIn addition closed loops cannot be described with such anapproach The angle limitation is useful especially if multiplelayers are considered where fiber layers are assigned to spe-cific ldquotasksrdquo which should not be exchanged between layersduring the optimization (Closed loops and abruptly endingfibers within the part are also impractical for production withTFP) Similar to Nagendra et al [35] the fiber path is modeledbased on 2D cubic B-splines However only deviations fromthe initial path are described the straight and parallel fiberlayout in this case with the spline functionsThe x-coordinateof the placement path is given by

119909 997888rarr 119891 (119906119896 V) = 119873119909sum119894=1

119873119910sum119895=1

119901119894119895119861119894 (119906119896) 119861119895 (V) + 119886119906119896 (8)

where 119861119894 are spline basis functions and the control points 119901119894119895define the optimization parametersThe linear scaling factors119886 and 119887 determine the total length scale An equidistant setof 119906119896 defines the different rovings next to each other in x-direction and the total set of curves for each roving path alongthe y-direction is given by variation of V

119862119896 (119906119896 V) 997888rarr (119891 (119906119896 V)119887V ) (9)


(a) (b) (c)

Figure 10 Fiber layouts for open-hole specimens reference layout with equidistant and parallel fibers (a) stiffness optimization (DFPO) (b)and principal stress design (c) The TFP layer is placed on top of the base material

For 119901119894119895 = 0 the initial layout with straight fibers 119862119896(119906119896 V)is obtained By fixing119901119894119895 = 0 for 119895 = 1 and 119895 = 2 the boundaryconditions of equidistant rovings in the clamping area with asmooth transition can be fulfilled The demand for smoothrovings also at the symmetry line 119910 = 0 leads to additionalrestrictions of 119901119894119895 = 119901119894119895+1 for 119895 = 119873119910 minus 1 The optimizationparameters for both examples are 16 independent controlpoints (119873119909 = 4 119873119910 = 7) at the beginning and increase upto 112 obtained by node insertion after BOBYQA algorithmconverges for a lower resolution In principle BOBYQAalgorithm converges even for larger number of optimizationparameters of several hundreds of parameters However themanufacturing precision limits meaningful increase of theresolution The optimization is considered to be convergedif the control points do not change by more than 0005 mmbetween successive iterations The initial resolution of 16parameters converges in about 60 iterations and takes about10 min in a typical workstation

4 Results and Discussion

Figure 10 shows the various layouts of the open-hole spec-imen The reference layout with equidistant and parallelfibers is given in Figure 10(a)) the stiffness optimizationresult is in Figure 10(b)) and for comparison the result of aprincipal stress orientation of fibers is given in Figure 10(c))The optimization results provide a different solution whencompared to previously optimized fiber pattern for open-hole tensile specimen as can be seen in [9 14 28] wherethey employed the principal stress criterion Not surprisinglyDFPO achieves better improvement than those ones Thedisturbance of fibers reaches much farther away from the

hole such that globally straighter fibers with overall similarlength are obtained

In addition to the open-hole specimen another exampleis provided to demonstrate the potential of the DFPOframework for another case Then a tensile specimen with anarrow section in themiddle is considered where the ratio ofthe narrow section to the full width is 50Due to the smoothtransition region of the narrowed section the principal stresslayout (Figure 11(c)) works very well in this case and morefiber rovings divert from the straight path (Figure 11(a))The DFPO solution is qualitatively similar to the open-holesolution but with stronger fiber concentration due to thestronger narrowing of the defect (Figure 11(b)) In addition inthis case the effect of the optimization using DFPO is muchmore ldquoglobalrdquo compared to the principal stress layout

Figure 12 presents the stiffness and strength increaseof the optimized fiber layouts relative to the referencedesign (Figure 10(a)) The principal stress oriented layout(Figure 10(c)) yields to a 5 increase in stiffness (Figure 11(a))(20 for the second example) and about 139 increase instrength in terms of Cuntze fiber failure mode interactionmax(MIA) (Figure 12(b)) (237 for the second example)whereas the DFPO-optimized layout (Figure 10(b)) results inabout 9 increase of stiffness (25 for the second example)and 197 increase in strength (275 for the second example)Please note that the boundary conditions of the optimizationswere such that the total number of rovings next to eachother was fixed and thus the volume and mass change fordifferent fiber layouts However the increase in volume of06 for principal stress and 16 for DFPO (59 and 63respectively for the second example) is smaller than the gainin both stiffness and strength


(a) (b) (c)

Figure 11 Fiber layouts for narrow-middle specimens reference layout with equidistant and parallel fibers (a) stiffness optimization (DFPO)(b) and principal stress design (c) The TFP layer is placed on top of the base material

PS oriented DFPO0

5

10

15

20

25

Stiff

ness

incr

ease

[]

(a)PS oriented DFPO

0

50

100

150

200

250

300

Stre

ngth

incr

ease

[]

(b)

Figure 12 Performance increase in stiffness (a) and strength (b) of principal stress oriented fiber layout (PS oriented) and stiffness optimizedlayout (DFPO) compared to reference design containing straight fibers

In contrast to the principal stress design DFPO repre-sents a real optimization procedure and consequently takesglobal andnot just local features of the specimen into accountThe thickness distribution is nonuniform in both cases anda thickness concentration near the defect of the structureis observed In the DFPO case this thickness concentrationextends further from the defect area than in the principalstress layout The fiber length of single rovings is muchmore uniform along each family of specimen for the DFPO-optimized such that the load balance of all rovings under ten-sile load is better Compared to other optimization techniques

where elemental fiber orientations and thickness values areoptimized without correlations induced by endless fibersin DFPO each fiber layout considered in every optimiza-tion iteration is already manufacturable and no subsequentadaptation is necessary Thus these gains obtained by theoptimization can be fully transferred to the application

5 Conclusions

The key objective of this investigation was to present anovel methodology for optimizing the fiber path with a


variable-axial fiber reinforcement design by employing anovel optimization methodology called Direct Fiber PathOptimization (DFPO) The main achievement is the localoptimization of both fiber angle and thickness at eachfinite element along the base mesh in order to reach globaloptimum DFPO demonstrated its capabilities on the opti-mization of both open-hole and narrow-middle examplesunder uniaxial tension For both cases the results show aclear increase in both stiffness and strength compared toa reference design with equidistant straight fiber-reinforcedparallel fibers as well as compared to the principal stressoriented layouts

Data Availability

The data used to support the findings of this study areavailable from the corresponding and first authors (bittrich-larsipfddde) upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to thank K Uhlig for fruitful dis-cussions and E Richter (both from IPF-Dresden) for hissupport with the figures The financial support of DFGgrants HE 446629-1 and KR 171319-1 is also gratefullyacknowledged Jose Humberto S Almeida Jr acknowledgesCAPES and Alexander von Humboldt Foundations for thefinancial support

References

[1] J H S Almeida M L Ribeiro V Tita and S C AmicoldquoStacking sequence optimization in composite tubes underinternal pressure based on genetic algorithm accounting forprogressive damagerdquo Composite Structures vol 178 pp 20ndash262017

[2] A Spickenheuer Zur Fertigungsgerechten Auslegung Von Faser-Kunststoff-Verbundbauteilen Fur Den Extremen Leichtbau Auf-basis Des Variabelaxialen Fadenablageverfahrens Tailored FiberPlacement [PhD Thesis] Technische Universitat DresdenFakultat Maschinenwesen 2014

[3] PMattheij K Gliesche andD Feltin ldquoTailored fiber placement- mechanical properties and applicationsrdquo Journal of ReinforcedPlastics and Composites vol 17 no 9 pp 774ndash786 1998

[4] C Cherif Ed Textile Werkstoffe fur den Leichtbau SpringerBerlin Heidelberg New York 2011

[5] P Ribeiro H Akhavan A Teter and J Warminski ldquoA reviewon the mechanical behaviour of curvilinear fibre compositelaminated panelsrdquo Journal of Composite Materials vol 48 no22 pp 2761ndash2777 2014

[6] P M Weaver K D Potter K Hazra M A R Saverymutha-pulle and M T Hawthorne ldquoBuckling of variable angle towplates From concept to experimentrdquo in Proceedings of the 50thAIAAASMEASCEAHSASC Structures Structural Dynamicsand Materials Conference USA May 2009

[7] A S Panesar and P M Weaver ldquoOptimisation of blendedbistable laminates for a morphing flaprdquo Composite Structuresvol 94 no 10 pp 3092ndash3105 2012

[8] GDuvaut G Terrel F Lene andV E Verijenko ldquoOptimizationof fiber reinforced compositesrdquo Composite Structures vol 48pp 83ndash89 2000

[9] L Parnas S Oral and U Ceyhan ldquoOptimum design ofcomposite structures with curved fiber coursesrdquo CompositesScience and Technology vol 63 no 7 pp 1071ndash1082 2003

[10] R M J Groh and P M Weaver ldquoMass optimization of variableangle tow variable thickness panels with static failure andbuckling constraintsrdquo in 56th AIAAASCEAHSASC StructuresStructural Dynamics andMaterials Conference Kissimmee FlaUSA 2015

[11] A Khani S T Ijsselmuiden M M Abdalla and Z GurdalldquoDesign of variable stiffness panels formaximum strength usinglamination parametersrdquoComposites Part B Engineering vol 42no 3 pp 546ndash552 2011

[12] J M J F Van Campen C Kassapoglou and Z Gurdal ldquoGen-erating realistic laminate fiber angle distributions for optimalvariable stiffness laminatesrdquoComposites Part B Engineering vol43 no 2 pp 354ndash360 2012

[13] H K Cho and R E Rowlands ldquoReducing tensile stress con-centration in perforated hybrid laminate by genetic algorithmrdquoComposites Science and Technology vol 67 no 13 pp 2877ndash2883 2007

[14] M W Tosh and D W Kelly ldquoOn the design manufacture andtesting of trajectorial fibre steering for carbon fibre compositelaminatesrdquoComposites Part A Applied Science andManufactur-ing vol 31 no 10 pp 1047ndash1060 2000

[15] R Rolfes J Tessmer R Degenhardt H Temmen P Burmannand J Juhasz ldquoNew design tools for lightweight structuresBHV Topping and CA Mota Soaresrdquo in Progress in Com-putational Structures Technology Saxe-Coburg PublicationsStirling Scotland 2004

[16] S SetoodehMM Abdalla and Z Gurdal ldquoDesign of variable-stiffness laminates using lamination parametersrdquo CompositesPart B Engineering vol 37 no 4-5 pp 301ndash309 2006

[17] H Moldenhauer ldquoBerechnung variabler faserverlaufe zur opti-mierung von compositestrukturenrdquo Lightweight Design vol 4no 1 pp 51ndash56 2011

[18] DWKelly PHsu andMAsudullah ldquoLoad paths and load flowin finite element analysisrdquo Engineering Computations (SwanseaWales) vol 18 no 1-2 pp 304ndash313 2001

[19] W Waldmann R Heller R Kaye and L Rose ldquoAdvances instructural loadflow visualisation and applications to optimalshapes (dsto-rr-0166)rdquo Technical Report Aeronautical andMaritime Research Laboratory Airframes and Engines Divi-sion Melbourne Australia 1999

[20] A V Malakhov and A N Polilov ldquoDesign of compositestructures reinforced curvilinear fibres using FEMrdquo CompositesPart A Applied Science and Manufacturing vol 87 pp 23ndash282016

[21] Y Katz R T Haftka and E Altus ldquoOptimization of fiberdirections for increasing the failure load of a plate with aholerdquo in Proceedings of the American Society for Composites 4thTechnical Conference Composite Materials Systems pp 62ndash71Blacksburg Virginia 1989

[22] GDuvaut G Terrel F Lene andVVerijenko ldquoOptimization offiber reinforced compositesrdquo Composite Structures vol 48 no1-3 pp 83ndash89 2000


[23] H K Cho and R E Rowlands ldquoOptimizing fiber direction inperforated orthotropic media to reduce stress concentrationrdquoJournal of Composite Materials vol 43 no 10 pp 1177ndash11982009

[24] J Wisniewski ldquoOptimal design of reinforcing fibres in multi-layer composites using genetic algorithmsrdquo Fibres amp Textiles inEastern Europe vol 12 no 3 pp 58ndash63 2004

[25] X LegrandD Kelly A Crosky andDCrepin ldquoOptimisation offibre steering in composite laminates using a genetic algorithmrdquoComposite Structures vol 75 no 1-4 pp 524ndash531 2006

[26] K Dems and J Wisniewski ldquoOptimal fibres arragement incomposite materialrdquo in Proceedings 8th World Congress onStructural and Multidisciplinary Optimization pp 1ndash10 LisboaPortugal 2009

[27] J Turant and K Dems ldquoDesign of fiber reinforced compositedisks using evolutionary algorithmrdquo in Proceedings 8th WorldCongress on Structural and Multidisciplinary OptimizationLisboa Portugal 2009

[28] J Bardy X Legrand and A Crosky ldquoConfiguration of agenetic algorithm used to optimise fibre steering in compositelaminatesrdquo Composite Structures vol 94 no 6 pp 2048ndash20562012

[29] A Spickenheuer M Schulz K Gliesche and G HeinrichldquoUsing tailored fibre placement technology for stress adapteddesign of composite structuresrdquo Plastics Rubber and Compos-ites vol 37 no 5-6 pp 227ndash232 2008

[30] A Albers N Majic and D Troll ldquoModeling approaches for thesimulation of curvilinear fiber-reinforced polymer compositesrdquoin Proceedings NAFEMS Seminar Progress in Simulating Com-posites Wiesbaden Germany 2011

[31] K Uhlig A Spickenheuer L Bittrich and G Heinrich ldquoDevel-opment of a highly stressed bladed rotor made of a CFRPusing the tailored fiber placement technologyrdquo Mechanics ofComposite Materials vol 49 no 2 pp 201ndash210 2013

[32] KUhligM Tosch L Bittrich et al ldquoMeso-scaled finite elementanalysis of fiber reinforced plastics made by Tailored FiberPlacementrdquo Composite Structures vol 143 pp 53ndash62 2016

[33] R G Cuntze ldquoEfficient 3D and 2D failure conditions for UDlaminae and their application within the verification of thelaminate designrdquo Composites Science and Technology vol 66no 7-8 pp 1081ndash1096 2006

[34] M J D Powell The BOBYQA algorithm for bound constrainedoptimization without derivatives Department of Applied Math-ematics andTheoretical Physics NA06 2009

[35] S Nagendra S Kodiyalam J Davis and V ParthasarathyldquoOptimization of tow fiber paths for composite designrdquo TheAmerican Institute of Aeronautics and Astronautics - AIAAJournal vol 95-1275 pp 1031ndash1041 1995

Hindawiwwwhindawicom Volume 2018

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Submit your manuscripts atwwwhindawicom


Roving pipe

Moving DOFof the base material

Roving

Needle

Sewing threadBobbin

Figure 1 Basic principle of the TFP process

known lamination parameters distribution for a VA compos-ite laminate into realistic fiber angles with minimum lossof structural performance Cho and Rowlands [13] reducedstress concentrations in an open-hole laminate with a geneticalgorithm

In contrast to optimization procedures principal stresscriterion has been often used for deriving curvilinear fiberpaths eg [14ndash17] Kelly et al [18] Waldmann et al [19] andMalakhov and Polilov [20] designed the curvilinear fiber pathbased on the concept of aligning fibers following the load pathby placing fibers along principal stresses Both approacheswere assumed to be optimization criteria although no math-ematical optimization process was explicitly carried out andonly an a priori design criterion was applied Furthermoregradient based numerical optimization processes [9 21ndash23]and optimization approaches based on evolutionary algo-rithms [24ndash28] to design VA composites have been alsodeveloped

However due to the often chosen approach of varyingangles of single quasi-isotropic finite elements (FE) the num-ber of design variables is considerably high also increasingcomputational costs Therefore most examples have beencomputed by using FEmodels with a limited number of finiteelements Increased design freedom causes increased designproblem complexity For example Conti et al [17] found thatusing fiber angles as design variables inevitably leads to anill-behaved objective function with many local minima

Usually a VA fiber pattern can be implied in a varyingdensity of fibers which causes a nonuniform thicknessof the dry preforms Hence this heterogeneous thicknessbuild-up is extremely complex to be accounted for in theanalysis Given this complexity current approaches neglectthe thickness accumulation and consider only the fiber anglevariation

Thus the major criticism of many state-of-the-art opti-mization approaches is that there is no mathematical opti-mization procedure to begin or to operate without necessaryinformation on the manufacturing process due to the lack ofan appropriatemodeling procedure However the knowledgeof the thickness distribution and the local fiber orientationcorresponding to an arbitrary fiber layout which can beproduced by TFP is essential for the part design processSpickenheuer et al [1 29] and Albers et al [30] made initialattempts to separate the optimization process of a curvilinearfiber-reinforced composite manufactured via TFP from theactual numerical models in order to limit the number ofrequired design variables making them independent of the

applied FEmesh resolutionThus once a sufficiently accuratemodeling of VA fiber layouts is established optimizationtechniques can be applied to the fiber pattern This allowsfully utilizing the high degrees of freedom in the designprocess and the maximization of the anisotropic materialcharacteristics of CFRPs

Given the identified gaps in the current state-of-the-art inproperly modeling VA composites this work presents a noveldesign procedure for illustrating the capability for generatinga VA pattern for an open-hole tensile specimen wherean optimal fiber pattern cannot be easily derived Hencethe novel optimization approach called Direct Fiber PathOptimization (DFPO) for VA composites will be introducedand numerically evaluated on the example of an open-holetensile specimen

2 Finite Element Modeling

21 Model Setup According to the state-of-the-art modelingof composite structures is mostly limited to stacking layerswith a constant thickness and a constant fiber angle withineach layer Models for structural analysis of uniform spiralsand single curved tapes of parallel fibers and constantthickness have been successfully applied additionally by anincrease in modeling efforts [31] In this case an analyticdescription for local preform thickness and the fiber ori-entation is known which can be used to build appropriateFE models for structural simulation However the existingmodel limitations are too strong if one plans to applyoptimization strategies to fully exploit the potential of CFRPmanufactured by TFP and to adapt production requirementsAlthough there are many approaches of employing themathematical description of the optimization algorithm todeduce the numerical model eg the geometry of eachiteration step this work is going to describe the modelingindependent of the optimization and thus as a genericmodulefor any VA structure with a similar placement characteristicstrictly following the manufacturing characteristics of TFPThe objective of the modeling is the elastic description ofthe laminate compliance and the prediction of initial failuresbased on a physically based failure criterion A mesoscaledmodel is used to evaluate the specific properties of the TFPprocess following the recommendations raised by Uhlig et al[32]

To generate continuous layers with TFP the rovings haveto be placed with a slight overlap to avoid gaps betweenthem If placed with constant thickness and fiber orientationparallel laminates can be produced for a small range ofdistances between neighboring rovings The thickness 119905 iscalculated according to the following (see Figure 2)

119905 = 119860119903119900V119889 = 119879119905120588119889120593 (1)

where Arov is the roving cross-section area 119889 the distancebetween neighboring rovings 119879119905 the roving fineness 120593 thefiber volume fraction and 120588 the fiber density

For arbitrary nonparallel roving placement the thicknessevaluation becomes more complex As a starting point for


t

d

Aro


y

x

( x12 y12 )

( x02 y02 )

( x01 y01 )

( x13 y13 )

( x03 y03 )

( x11 y11 )



(119909119894 (119904)119910119894 (119904)) = (11990901198941199100119894 ) + (1199091119894 minus 11990901198941199101119894 minus 1199100119894 ) 119904 (2)



119905119897119894119899119890 (119909 119910)= sum119894

119860119903119900V119889119894 int10




119881119905119900119905119886119897 = ∬ 119905119897119894119899119890 (119909 119910) 119889119909119889119910 = sum119894

119860119903119900V119889119894 (4)













xyz












119878119905119894ff119899119890119904119904 119900119901119905119894119898119894119911119886119905119894119900119899 min119862119894

(maxΩ

119906119910) (6)

119878119905119903119890119899119892119905ℎ 119900119901119905119894119898119894119911119886119905119894119900119899 min119862119894

(maxΩ

119872119868119860) (7)



200

mm

20 mm

80 mm(a)

200

mm

80 mm

40 mm

(b)




Top view

F


Base material

Front viewDetail 1

Detail 1






100

50

0

minus50

minus100minus40

[mm]

[mm

]

minus20 0 20 40

(a)

100

80

20 01

02

Thickness[mm]

03

04

05

00

40

60

0

[mm

]0

[mm]10 20 30 40

y

x

(b)




Yes

No

Fiber pathparameter









0000700006000050000400003000020000100000

SOLID 185SOLID 186

Number of elements

max

(ＯＳ) [

mm

]

104 105

(a)

max

(MIA

)

SOLID 185SOLID 186

Number of elements

040

045

050

055

104 105

(b)








119909 997888rarr 119891 (119906119896 V) = 119873119909sum119894=1

119873119910sum119895=1

119901119894119895119861119894 (119906119896) 119861119895 (V) + 119886119906119896 (8)


119862119896 (119906119896 V) 997888rarr (119891 (119906119896 V)119887V ) (9)


(a) (b) (c)









(a) (b) (c)


PS oriented DFPO0

5

10

15

20

25

Stiff

ness

incr

ease

[]

(a)PS oriented DFPO

0

50

100

150

200

250

300

Stre

ngth

incr

ease

[]

(b)




5 Conclusions




Data Availability




Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018









t

d

Aro


y

x

( x12 y12 )

( x02 y02 )

( x01 y01 )

( x13 y13 )

( x03 y03 )

( x11 y11 )



(119909119894 (119904)119910119894 (119904)) = (11990901198941199100119894 ) + (1199091119894 minus 11990901198941199101119894 minus 1199100119894 ) 119904 (2)



119905119897119894119899119890 (119909 119910)= sum119894

119860119903119900V119889119894 int10




119881119905119900119905119886119897 = ∬ 119905119897119894119899119890 (119909 119910) 119889119909119889119910 = sum119894

119860119903119900V119889119894 (4)













xyz












119878119905119894ff119899119890119904119904 119900119901119905119894119898119894119911119886119905119894119900119899 min119862119894

(maxΩ

119906119910) (6)

119878119905119903119890119899119892119905ℎ 119900119901119905119894119898119894119911119886119905119894119900119899 min119862119894

(maxΩ

119872119868119860) (7)



200

mm

20 mm

80 mm(a)

200

mm

80 mm

40 mm

(b)




Top view

F


Base material

Front viewDetail 1

Detail 1






100

50

0

minus50

minus100minus40

[mm]

[mm

]

minus20 0 20 40

(a)

100

80

20 01

02

Thickness[mm]

03

04

05

00

40

60

0

[mm

]0

[mm]10 20 30 40

y

x

(b)




Yes

No

Fiber pathparameter









0000700006000050000400003000020000100000

SOLID 185SOLID 186

Number of elements

max

(ＯＳ) [

mm

]

104 105

(a)

max

(MIA

)

SOLID 185SOLID 186

Number of elements

040

045

050

055

104 105

(b)








119909 997888rarr 119891 (119906119896 V) = 119873119909sum119894=1

119873119910sum119895=1

119901119894119895119861119894 (119906119896) 119861119895 (V) + 119886119906119896 (8)


119862119896 (119906119896 V) 997888rarr (119891 (119906119896 V)119887V ) (9)


(a) (b) (c)









(a) (b) (c)


PS oriented DFPO0

5

10

15

20

25

Stiff

ness

incr

ease

[]

(a)PS oriented DFPO

0

50

100

150

200

250

300

Stre

ngth

incr

ease

[]

(b)




5 Conclusions




Data Availability




Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018















xyz












119878119905119894ff119899119890119904119904 119900119901119905119894119898119894119911119886119905119894119900119899 min119862119894

(maxΩ

119906119910) (6)

119878119905119903119890119899119892119905ℎ 119900119901119905119894119898119894119911119886119905119894119900119899 min119862119894

(maxΩ

119872119868119860) (7)



200

mm

20 mm

80 mm(a)

200

mm

80 mm

40 mm

(b)




Top view

F


Base material

Front viewDetail 1

Detail 1






100

50

0

minus50

minus100minus40

[mm]

[mm

]

minus20 0 20 40

(a)

100

80

20 01

02

Thickness[mm]

03

04

05

00

40

60

0

[mm

]0

[mm]10 20 30 40

y

x

(b)




Yes

No

Fiber pathparameter









0000700006000050000400003000020000100000

SOLID 185SOLID 186

Number of elements

max

(ＯＳ) [

mm

]

104 105

(a)

max

(MIA

)

SOLID 185SOLID 186

Number of elements

040

045

050

055

104 105

(b)








119909 997888rarr 119891 (119906119896 V) = 119873119909sum119894=1

119873119910sum119895=1

119901119894119895119861119894 (119906119896) 119861119895 (V) + 119886119906119896 (8)


119862119896 (119906119896 V) 997888rarr (119891 (119906119896 V)119887V ) (9)


(a) (b) (c)









(a) (b) (c)


PS oriented DFPO0

5

10

15

20

25

Stiff

ness

incr

ease

[]

(a)PS oriented DFPO

0

50

100

150

200

250

300

Stre

ngth

incr

ease

[]

(b)




5 Conclusions




Data Availability




Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018









200

mm

20 mm

80 mm(a)

200

mm

80 mm

40 mm

(b)




Top view

F


Base material

Front viewDetail 1

Detail 1






100

50

0

minus50

minus100minus40

[mm]

[mm

]

minus20 0 20 40

(a)

100

80

20 01

02

Thickness[mm]

03

04

05

00

40

60

0

[mm

]0

[mm]10 20 30 40

y

x

(b)




Yes

No

Fiber pathparameter









0000700006000050000400003000020000100000

SOLID 185SOLID 186

Number of elements

max

(ＯＳ) [

mm

]

104 105

(a)

max

(MIA

)

SOLID 185SOLID 186

Number of elements

040

045

050

055

104 105

(b)








119909 997888rarr 119891 (119906119896 V) = 119873119909sum119894=1

119873119910sum119895=1

119901119894119895119861119894 (119906119896) 119861119895 (V) + 119886119906119896 (8)


119862119896 (119906119896 V) 997888rarr (119891 (119906119896 V)119887V ) (9)


(a) (b) (c)









(a) (b) (c)


PS oriented DFPO0

5

10

15

20

25

Stiff

ness

incr

ease

[]

(a)PS oriented DFPO

0

50

100

150

200

250

300

Stre

ngth

incr

ease

[]

(b)




5 Conclusions




Data Availability




Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018









100

50

0

minus50

minus100minus40

[mm]

[mm

]

minus20 0 20 40

(a)

100

80

20 01

02

Thickness[mm]

03

04

05

00

40

60

0

[mm

]0

[mm]10 20 30 40

y

x

(b)




Yes

No

Fiber pathparameter









0000700006000050000400003000020000100000

SOLID 185SOLID 186

Number of elements

max

(ＯＳ) [

mm

]

104 105

(a)

max

(MIA

)

SOLID 185SOLID 186

Number of elements

040

045

050

055

104 105

(b)








119909 997888rarr 119891 (119906119896 V) = 119873119909sum119894=1

119873119910sum119895=1

119901119894119895119861119894 (119906119896) 119861119895 (V) + 119886119906119896 (8)


119862119896 (119906119896 V) 997888rarr (119891 (119906119896 V)119887V ) (9)


(a) (b) (c)









(a) (b) (c)


PS oriented DFPO0

5

10

15

20

25

Stiff

ness

incr

ease

[]

(a)PS oriented DFPO

0

50

100

150

200

250

300

Stre

ngth

incr

ease

[]

(b)




5 Conclusions




Data Availability




Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018









0000700006000050000400003000020000100000

SOLID 185SOLID 186

Number of elements

max

(ＯＳ) [

mm

]

104 105

(a)

max

(MIA

)

SOLID 185SOLID 186

Number of elements

040

045

050

055

104 105

(b)








119909 997888rarr 119891 (119906119896 V) = 119873119909sum119894=1

119873119910sum119895=1

119901119894119895119861119894 (119906119896) 119861119895 (V) + 119886119906119896 (8)


119862119896 (119906119896 V) 997888rarr (119891 (119906119896 V)119887V ) (9)


(a) (b) (c)









(a) (b) (c)


PS oriented DFPO0

5

10

15

20

25

Stiff

ness

incr

ease

[]

(a)PS oriented DFPO

0

50

100

150

200

250

300

Stre

ngth

incr

ease

[]

(b)




5 Conclusions




Data Availability




Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018









(a) (b) (c)









(a) (b) (c)


PS oriented DFPO0

5

10

15

20

25

Stiff

ness

incr

ease

[]

(a)PS oriented DFPO

0

50

100

150

200

250

300

Stre

ngth

incr

ease

[]

(b)




5 Conclusions




Data Availability




Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018









(a) (b) (c)


PS oriented DFPO0

5

10

15

20

25

Stiff

ness

incr

ease

[]

(a)PS oriented DFPO

0

50

100

150

200

250

300

Stre

ngth

incr

ease

[]

(b)




5 Conclusions




Data Availability




Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018










Data Availability




Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018





























Journal of












Journal of







Volume 2018






Volume 2018








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