A FULLY INTEGRATED SAIL-RIG ANALYSIS METHODvm2330.sgvps.net/~syrftest/images/library/20160514165849.pdf · A FULLY INTEGRATED SAIL-RIG ANALYSIS METHOD ... geometric data make the

The Second International Conference on Innovation in High Performance Sailing Yachts, Lorient, France

© 2010: Royal Institution of Naval Architects

A FULLY INTEGRATED SAIL-RIG ANALYSIS METHOD S Malpede, and F Nasato, SMAR Azure Ltd, UK SUMMARY This paper presents a method to analyse the rigs of sailing yachts in a timely, cost-effective and versatile manner. Mast and rig design cover an important role in maximising sailing yacht performance. The design of optimal rigs should consider realistic sail loads; conversely the design of an optimal sail-plan should take into account particular rig configuration and structural behaviour. The developed method, by virtually simulating the structural behaviour of rigs under aero elastic sails loads in up-wind sailing conditions facilitates optimal rig design. The method consists of two main components: (1) DESIGN, where rig and sails geometry and materials are defined; (2) ANALYSIS, where the rig is analysed under the calculated sails loads in specified trim and sailing conditions. Design, meshing and analysis phases are fully integrated in a single software platform, which allows the comparison of alternative solution. The sail - rig analysis is non-linear and can be applied to fibre-membrane sails. NOMENCLATURE AWA Apparent Wind Angle (deg) AWS Apparent Wind Speed (kn) CP pressure coefficient CDrive drive force coefficient CSide side force coefficient CE centre of effort (m) � efficiency EUWS equivalent uniform wind speed (kn) KE Element Stiffness Matrix p pressure (N m-2) p∞ asymptotic flow pressure (N m-2) Q∞ asymptotic dynamic pressure (N m-2) TWS True Wind Speed (kn) TGWS top gradient wind speed (kn) V flow velocity at a control point (msec-1) V∞ asymptotic flow velocity (msec-1) VMG Velocity Made Good (kn) 1. INTRODUCTION This paper presents a method to optimize the design of modern sailing yacht rigs. The optimization is performed via a fully integrated design and analysis method that predicts the sail-rig aeroelastic behaviour in specific sailing and trimming conditions. Mast and rig design cover an important role in maximising the performance of sailing yachts. Weight, shape and materials are the main aspects of the mast design. The design of optimal rigs should take into account realistic sail loads, conversely the design of optimal sail-plan shall consider for particular rig configuration and structural behaviour (mast bend, forestay sag). The challenge is to facilitate both. The developed method, by virtually simulating the structural behaviour of various rig configurations under the aeroelastic sail loads in up-wind sailing conditions allows the design of optimal rigs. The method performs the aeroelastic analysis of sails through the interaction of the structural nonlinear finite element model with a

vortex lattice aerodynamic model [9]. The final sail loads –evaluated in defined sailing conditions- and the trim loads are applied to the various rig components. The rig structural analysis is non-linear and takes into account the different structural properties of the rig elements. The method is fully integrated in a single software platform and it is also valid in case of fibre-membrane sails. The main features are: (i) the geometrical definition of sail shapes and rig components (ii) the creation of a fluid-dynamic model and finite element model of sailplan, (iii) the iterative aero-elastic solution for the sailplan, (iv) the rig finite element model and structural analysis. The structural analysis of rigs can be solved with a variety of commercial available finite element analysis (FEA) software. Those software applications require user-defined sail-loads. The existing mast design procedures are based almost entirely on empirical evaluation of the loads due to the sails and know-how. When it is crucial to optimise the sail-rig performance, designers often use RANS code to calculate the sail loads. Indeed, not only are these procedures very expensive in terms of time and human and technical resources needed, but also it can be very laborious to transfer the sail loads on the rig finite element model (FEM) avoiding numerical flaw. At present a fully integrated rig-sail analysis system does not exist. The paper presents a fully integrated, versatile and cost-effective sail-rig design system. The emphasis is placed on providing a practical application of rig design, which will allow further improvements over current design methods. 2. SAIL – RIG ANALYSIS This section describes the method implemented to calculate mast bend, luff sag and the rig elements tensions under defined tuning loads and aeroelastic sail loads.



The sail-rig analysis method consists of five phases, as shown in figure 1. In phase 1, the designer creates an initial geometry of the rig, by defining mast, spreaders, boom, bowsprit and stays. For each rig element, it is possible to set position, sections and material properties. In phase 2, the designer generates the sail shapes and assigns the material properties. Hence, in phase 3, the wind loads acting on the sails in specific sailing and trimming conditions are calculated. Subsequently, in phase 4, the sail structural analysis evaluates the flying sail-shape and stress distribution. In case the calculated flying sail shapes differ from the designed sail shapes, it is necessary to carry out the aeroelastic analysis that estimates the flying sailplan shape in equilibrium with the wind and trim loads acting on the sails. At this point, the sail loads can be transferred on the rig. Finally –in phase 5- the rig structural analysis allows calculating stay loads and sag, the tensions on the verticals and diagonals, the spars deformed shape and compression load under the specified tuning loads and the calculated sail loads. Section 3 illustrates the sail aeroelastic analysis. Section 4 describes the rig structural analysis. Section 5 presents how the method can be applied to optimise a rig of a TP52. 3. SAIL AEROELASTIC ANALYSIS

METHOD The implemented sail ‘static aeroelastic analysis’ code [9] automatically performs the following actions: 1. Generate a panel model, by placing the nodal and

control points on the designed sail-shape; 2. Perform the aerodynamic analysis in defined sailing

and trim conditions to calculate the wind loads; 3. Apply the wind loads, evaluated in 2., on the current

sails finite element model; 4. Calculate the deformed sail-shape and stress

distribution with the structural analysis; 5. Carry the aeroelastic analysis out by repeating the

actions 1) to 4), considering the deformed sail-shape evaluated in 4, until convergence.

Convergence is achieved when the flying sail shape evaluated with the aerolastic analysis is the sail-shape in static equilibrium with the known external loads in the particular sailing and trimming conditions set by the designer. It also provides the wind loads and their distribution on the flying sail-shape and the stress distribution. Section 3.1 describes the aerodynamic analysis method. Section 3.2 describes the sail structural analysis method. Details of validation tests are also included.

3.1 AERODYNAMIC ANALYSIS The evaluation of the aerodynamic loads on the sailplan is carried out using a modified version of the Vortex Lattice Method (M.V.L.M.), which represents the most widely used technique for solving the inviscid sail aerodynamic problem. In 1968, Milgram, [10], started to apply this method to sail aerodynamic analysis. After his basic development, the method has been developed in several ways and is demonstrated to be an effective tool for sail analysis. Low costs in terms of computational and human resources make Vortex Lattice methods popular and efficient method with respect to RANS techniques and wind tunnel tests, which are mostly used at the final stage of the design. Phase 1: RIG DESIGN Geometry definition of: • Mast • Spars • Collar • Forestay /Backstay/ Stays • Bowsprit For each element it is possible to set geometry and material properties

Phase 2: SAIL DESIGN

Geometry definition of • Mainsail • Genoa • Spinnaker/Gennaker • Gaff

For each sail, it is possible to design the fiber layout and set material properties

Phase 3: SAIL AERODYNAMIC ANALYSIS Set:

• Sailing conditions: AWA & AWS for each heel angle and Righting Moments

• Sheeting angles The resulting pressure force will be used to calculate the flying sail-shape

Phase 4: SAIL STUCTURAL ANALYSIS Set: • Trim loads (e.g. sheets) • Constraints

The resulting sail structural loads will be transferred on the rig.



Phase 5: RIG STRUCTURAL ANALYSIS For specified tuning loads and calculated sail loads, the following are calculated: • Forestay/Removable stay Load & Sag • Verticals & Diagonals tension on both

sides • Mast Bend and Compression • Collar Forces

Figure 1: Sail-Rig Analysis Method The implemented M.V.L.M. converts a sail surface into a vortex sheet, which means that the surface has zero thickness and the jump in velocity across it is equal to the local strength of the vortex sheet. The local vorticity is evaluated by imposing the condition that there is no flow through the surface of the sail and, hence, the velocity field is tangent to the surface. Furthermore, to obtain a unique solution, it is assumed that the flow separates from the sail surface at the trailing edge. For an extensive description of the method please refer to Malpede-Baraldi [9]. Essential assumptions are: i. Flow-field is irrotational everywhere ii. No flow separation on the sail surface. Whereas the condition (i) can be considered satisfied for high Reynolds number, as sails are thin lifting surface working in reasonable high Reynolds number in a low-speed flow, the assumption (ii) is more problematic as sails are cambered lifting surfaces working at high attack angle and a wrong trim can generate even leading edge separation. Thus, the method accuracy decreases for high angles of attack. The M.V.L.M. calculates the jump of the tangential flow velocity over the sail. Thus, the aerodynamic loads are calculated via the Bernoulli principle for calculating the pressure coefficient distribution: Cp = (p-p∞ )/ Q∞ = 1-(V/ V∞ )2 (1) Where Cp, p and V are respectively: pressure coefficient, static pressure and the velocity at a control point, while p∞, Q∞ and V∞ are the static pressure, dynamic pressure and velocity of the asymptotic flow. 3.1 (a) Validation Using Wind Tunnel Testing The scarcity of experimental data for loads on yacht sails makes the validation for sails quite problematic. The authors have carried out many tests comparing the results obtained with published case studies, but the paucity of geometric data make the reproduction of the geometry very approximate. Thus, in this paper is presented the comparison of the aerodynamic analysis results [9] calculated with the implemented method and evaluated in a wind tunnel at the University of Sydney for sets of sails of a scaled 18ft skiff model.

The two sailplans are composed of the same jib and two different styles of mainsail: a square-top and a pin-head (or elliptical) mainsail. The wind tunnel tests were carried out by Evan Walker (Aerospace Engineering – University of Sydney) [16]. The main phases of the study were the following: 1. SMAR Azure designed the sails and produced the file of panels to be manufactured. 2. Evan Walker produced the sails, carried out the wind tunnel tests at the University of Sydney and measured the flying sail shape from photos taken during the tests. 3. Then, SMAR Azure reproduced the flying sail shape and calculated the aerodynamic forces in the same conditions used in the wind tunnel tests and compared the results. The wind tunnel tests conditions are shown on table 1. Figure 2 and 3 show the wind tunnel model, the M.V.L.M. mesh and the calculated pressure distribution.

Wind Angle 21 deg Top Gradient Wind Speed 13.4* knots & 15.9* knots Heeling Angle 0 deg Jib Sheeting Angle 12 deg Mainsail Sheeting Angle 0.5 deg

Table 1: wind tunnel test conditions (*Wind velocities measured at 1 meter high)

Figure 2: square top main case - from left, wind tunnel model, M.V.L.M mesh and pressure force

Figure 3: elliptical main case - from left, wind tunnel model, M.V.L.M mesh and pressure force It is important to note that the wind tunnel tests were carried out in an apparent wind speed that varies with the height, to simulate a realistic wind gradient. Table 2 and 3 show the results of the validation study.



Wind tunnel tests reveal very little difference between the two sailplans [16]. This validation study has concluded that, on the basis of the heeling moment as a control parameter to individuate the equivalent uniform wind speed (EUWS), the method presented: 1. Slightly underestimates the height of the centre of effort by 15% at most. This difference is due to the fact that the aerodynamic analysis carried out in the implemented M.V.L.M considers the uniform wind flow and therefore the sails develop higher pressure forces in the lower part with respect the forces developed by the sails in the gradient wind flow in the wind tunnel. 2. Slightly overestimates the total aerodynamic force in a range not higher than 20%. This discrepancy can be explained by the fact that the centre of effort calculated with the M.V.L.M. is lower (see point 1) and a bigger force is needed to achieve the same heeling moment to calculate the EUWS with a smaller arm. 3. Estimates a total force direction in a range not higher than 5 degrees aft. This discrepancy is believed to be due to inaccuracies in the geometrical data measurement taken from the pictures: in particular the entry angle and twist. This variation in the direction of the totals force explains the differences in the thrust (underestimated) and side force (overestimated). Overall the results of the validation exercise are considered satisfactory.

TGWS [kn]

EUWS [kn]

Total Force [N]

Angle [deg

Thrust [N]

Side [N]

CE Height

[m] 15.9* 11.6 74 3.1 11.2 0.50

12.4 12.1 79 2.4 11.8 0.47 +4% +5 -23% +6% -5%

13.4* 8.6 74 2.3 8.3 0.51 11.6 9.6 77 2.2 9.4 0.46 +12% -4% +13% -10%

Table 2: square top mainsail: M.V.L.M results vs Wind tunnel Results (grey rows) *flow velocity at the mast head

Table 3: pin top mainsail: M.V.L.M results vs Wind tunnel Results (grey rows) *flow velocity at the mast head

3.2 SAILS STRUCTURAL ANALYSIS The structural analysis consist of computing the deformation and stress distribution of the rig-sail system by using a nonlinear finite element method where geometric nonlinearities are taken into account and material properties are linear [15]. The approach is well suited for the problem involved where large displacements are encountered and the material is expected to work within the linear region of the stress-strain function; no yielding is modelled. Since a nonlinear problem is solved, Raphson’s method is used to find the deformed equilibrium state given a load perturbation. A linear relation between strain and displacement is assumed: this approximation is certainly verified when displacements are small within each load iteration by a suitable choice of the load perturbation (load step). [9] Two finite element types are considered: a membrane element modelling the sail and a beam element modelling battens. For a detailed numerical approach to the membrane element modelling and wrinkling and validation method refer to Malpede Baraldi [9] A unified approach for the modelling of the sail, rig and battens was selected (see section 4) 4. RIG STRUCTURAL ANALYSIS A nonlinear analysis is required especially for battens where large displacements are expected and it allows the determination of member buckling. The element stiffness matrix [KE] is based on beam-column theory in order to account for stress stiffening and softening in the presence of axial load. Beam-column equations with bending about two axes and torsion are employed to generate the element stiffness and solved with the finite difference method in order to generate the various stiffness contributions of each finite element. In this fashion, the effect of member displacement upon bending moment is included and the bending stiffness becomes a function of the axial load. The rotational degrees of freedom of the beam elements are introduced in the linear system discretizing the sail and battens, while the stiffness relative to the linear displacement degrees of freedom is simply added to the existing membrane nodes. This approach implies a one to one interface between the membrane mesh and the beam mesh. The approximation does not take into account the sliding contact between sail and inner nodes of the batten. Beam column [7] finite elements are used to simulate spars (masts, spreaders and boom). Running rigging

TGWS [kn]

EUWS [kn]

Total Force [N]

Angle [deg

Thrust [N]

Side [N]

CE Height

[m] 15.9* 13.8 74 3.7 13.3 0.54

14.4 15.5 78 3.2 15.2 0.47 +13% -14% +15% -13%

13.4* 11.0 77 2.5 10.7 0.56 13.4 13.2 78 2.8 12.9 0.47 +20% +11% +20% -15%



elements (shrouds, stays, vang and mainsheet) are treated as cable finite elements [7]. The cable element is not capable either of carrying bending loads or compressive loads. The stress is assumed to be uniform over the entire element. To compute the forestay sag the forestay itself the finite element model is composed by a sequence of beam column elements, as cable finite elements would have determined instability. To avoid the compression on the cable finite elements a “compression check” is performed for each iteration, allowing cable to be slack without transferring any reaction to the rig. Cables can be tuned by a virtual turnbuckle, specifying the strain of every finite element. Also the mast jack can be tuned, pushing the mast upwards. Weight is considered as external force applied to the rig, computed by the properties of the different rig elements and taking in account the appropriate heeling angle. The results of the simulation are: • the compression load on every beam column element

(mast, spreaders and boom) • the tensile load on cables and the breaking load ratio • the maximum displacement and the position on the

mast where it occurs (from the undeformed configuration)

• the longitudinal and lateral mast bend curve and forestay sag curve and their maximum values

• the reaction on the hull at the mast step and at the stays and shrouds chainplates

Several validation of the software has been performed. Nonlinear cantilever beam test cases have been found in literature, from Ansys and Patran simulations [2], [12], [13], [8]. The authors have replicated the geometry and load cases in the presented method and have compared the resultant deflection at the tip. The cantilever beam has been loaded with a concentrated force and, separately, with a concentrated moment. Large deformations are involved and nonlinear analysis has revealed to be determinant. Furthermore, results have been validated against commercial available software for a simple fractional rig with no spreaders and with spreaders. Comparisons between the published results and those calculated with the presented method are very positive, showing difference of less than a percentage point. 5. OPTIMAL RIG DESIGN- CASE STUDY At the present time, even for cruising yachts, it is very important to maximize the sailing performance. Maximising the contribution of the rig to sailing performance means (i) minimizing the weight of the overall structure and (ii) optimizing the section shape. Thus, optimal mast design is ruled by a realistic estimation of the maximum operating loading conditions.

Rig loads will define the spars and the stays dimensions and which material to use. This section describes how the method described in this paper can be used to optimise a TP52 mast and rig design, by showing how the sail loads are evaluated and applied to the rig. Following the analysis procedure described in section 2 and figure 1, after having developed the geometry of the rig, mainsail and jib –shown in figure 4-, the sails panel models were created to carry out the aerodynamic analysis in the sailing and trimming condition that maximise the VMG in light wind.

Figure 4: TP52 sailplan – trimmer and masthead view Table 4 shows the input data and results of the aerodynamic analysis. Figure 5 shows the pressure distribution on the sail. The aerodynamic loads are transferred on the finite element model of the jib and mainsail to perform the structural analysis via the implemented finite element analysis (FEA), as described in section 3.2 Table 5 shows the structural analysis settings and results. On the left of figure 6, the sailplan flying shape is shown. The Von Mises stress distribution is shown in figure 7. As the maximum displacement evaluated on both sails is approximately 0.8% of the relative luff length, it was decided to verify the flying sail shape and final loads by carrying out the aeroelastic analysis. Table 6 shows the results after the aeroelastic analysis. Comparing the results of the aeroelastic analysis with the one obtained from the aerodynamic (phase 2) and structural analysis (phase 3), it is interesting to note that



the calculated flying sailplan shape is slightly more efficient.

TWS 8 kn Cp Jib 1.66 VMG AWA 22° Cp Main 1.37

VMG AWS 14 kn CDrive 0.51-Jib 0.28-Main

Heeling angle 10 deg CSide 1.58-Jib 1.31 -Main

Sheeting angle

7° jib 0° main

Heeling Moment 77KNm

Table 4: TP52-Aerodynamic analysis input and results

Figure 5: TP52- pressure distribution and wake shape FEA settings Results Jib Main

Sailcloth GPL21 Maximum stress 38.7MPa 44.6MPa

Battens Carbon Maximum displacement 0.15m 0.16m

Halyard load

6,6KN Jib 10,6KN Main Clew load 6220N 9050N

Table 5: TP52- structural analysis settings and results In fact, both sails develop a lower drive and side force than the one calculated in the first aerodynamic analysis, but whereas the drive force is only 3% lower, the side force is almost 8% lower. These results are due to the fact that the calculated flying sail-shapes are slightly flatter than the designed one and the leech is more open than the designed one. Thus the efficiency (�=Cdrive/Cside) measured as the ratio between the drive and side coefficients are: �(design shape)=0.27 �(flying shape)=0.28 (2) Pressure and stress distribution are very similar to the ones calculated in the aerodynamic analysis and structural analysis phases (see fig 5 and 7).

Figure 6: flying sails shape calculated with the structural analysis (left) and with the aeroelastic analysis (right). Flying sail shapes (red) are characterised by a slightly more open leech and flatter shape than the initial one (blue). Aero Performance Structural Analysis Results

Jib

Cp 1.55 Max disp 0.20

CDrive 0.48 Max Stress 38.5MPa

CSide 1.47 Position Max Stress HEAD

Main

Cp 1.27 Max disp 0.20

CDrive 0.27 Max Stress 42.2MPa

CSide 1.21 Position Max Stress HEAD

Table 6: Aeroelastic Analysis Results The sail loads evaluated with the aeroelastic analysis are then transferred onto the rig. Prior to start the rig analysis, it is possible to set up: - Length of each beam/cable element - Thickness of spars and stays. In alternative, it is

possible to input the area and inertia moment for each mast panel, boom, spreader and crane.

- Tuning Loads, including mast jack The weight of each element is also taken into account. The loads applied are: - Weight loads, which are automatically calculated - Sail Loads, calculated with the aeroelastic analysis - Tuning Loads (backstay = 90mm / forestay= 30mm



Figure 8 shows the analysis results in case the spars are built with light aluminium and all stays in Nitronic. The mast bend is plotted with a high zoom factor to show the deformation. Table 6 summarises the results for the same load conditions, when the mast tube thickness and stays tube section vary. The purpose of the case study presented in this paper is not to exhaust the debate about optimal mast design or to provide standard dimension for an optimal mast and rig model. Moreover, this study enlightens the need for accurate evaluation of the sails loads when performance is the key.

Figure 7: Von Mises stress distribution (red =high stress) Spar thickness 8mm Stays thickness 10mm Mast Bend L 1.18% Position 57 Mast Bend Side 0.02% Position 75 Luff Sag L 0.27 Position 51 Luff Sag Side 0.44 Position 58 Spar thickness 6mm Stays thickness 8mm Mast Bend L 1.17% Position 57 Mast Bend Side 0.02% Position 81 Luff Sag L 0.27% Position 51 Luff Sag Side 0.44 Position 58 Table 7: optimization of rig weight

Figure 8: mast bend and result graphical user interface. 6. CONCLUSIONS The development of a fully integrated sail-rig analysis has been presented. The emphasis is placed on providing a practical application of rig design, which will allow further improvements over earlier methods. The development of the integrated sail-rig analysis and design system has met the objectives within the mission statement, more specifically: The algorithms written for the various design, mesh

and analysis phases have been developed to produce an efficient, flexible and user-friendly analysis method, which can be used on a personal computer.

The analytic method constitutes an enhancement to the current rig and sail design approach.

The use of this system is time-effective and versatile: it calculates the rig loads under sail loads and tuning loads and evaluates alternative solutions.

Considering the overall practicality of the system, current results are promising. The case study has shown three important results: 1. The utmost need for a versatile and integrated sail-

rig design tool that allows a first evaluation of the optimal rig dimensions and material properties.

2. The importance of a quick evaluation of mast bend, luff sag for optimal sail plan design and yacht design.

Further developments will concentrate on adding the aerodynamic influence of the rig on the sails and non-stationary effects. Current diversifications include (i) the development of a complete input customised for the Velocity Prediction Program (V.P.P.) for use in the performance prediction for the entire sailboat and (ii) the development of a method to carry our the structural analysis of the sail and rig under the pressure loads resulting from typical RANS analysis method.



7. ACKNOWLEDGEMENTS Special thanks to the technical team of SMAR-Azure Ltd, Dr. Donald W. MacVicar and Mr. Stephen Jordan, for their continuous and pro-active contribution to the development of the method and graphics presented in this paper. 8. REFERENCES 1. ANDERSON, J.D. jr., ‘Fundamentals of Aerodynamics’, 2nd ed., McGraw-Hill International Edition, 1991 2. ANSYS, ‘Intermediate Tutorials, NonLinear Analysis of a Cantilever Beam’, http://2doworld.com/ansys-and-etabs/ansys-intermediate-tutorials-nonlinear-analysis-a-cantilever-beam.html, 2008 3. FUKASAWA, T., KATORI, M., ‘Numerical approach to the aeroelastic responses of three-dimensional flexible sails’, The 11th Chesapeake Sailing Yacht Symposium, SNAME, Annapolis USA, January, 1993 4. FIDDES, S.P., GAYDON, J.H., ‘A new vortex lattice method for calculating the flow past yacht’, Journal of Wind Engineering and Industrial Aerodynamics, 1996 5. GARRET, R., ‘The symmetry of Sailing’, Sheridan House, NY., 1996 6. JACKSONS, P. S., ‘The analysis of the three-dimensional sails’, Proceeding of the 10th Canadian Congress of Applied Mechanics, 1985 7. LEVY, R., SPILLERS, W.R., ‘Analysis of Geometrically Nonlinear Structures’, Springer, 1993 8. MARCHAJ, C.A., ‘AERO-HIDRODYNAMICS OF SAILING’, 2nd edn., Adlard Coles Nautical, London, 1994 9. MALPEDE, S, BARALDI, A, ‘A fully integrated method for optimising fiber-membrane’, Proceedings of the 3rd High Performance Yacht Design Conference, 2008 10. MILGRAM, J.H., ‘The analytical design of yacht sails’, Annual Meeting of the Society of Naval Architects and Marine Engineers, 1968 11. NASATO, F, WALKER, E, ‘Validation of AzureProject Aerodynamic Analysis using Wind Tunnel Testing’, Internal Report, 2010 12. MSC Patran tutorials www.mscsoftware.com/support/online_ex/Patran/Pat322/Lec01_Lesson01_L_NL_An_Cant_Beam.pdf 13. ‘Plastic bending of a clamped beam’, http://www.eng.utah.edu/~me7540/ANSYS_Example_BeamBend.pdf, 2006. 14. SCOTT, J. A., HU, Y., ‘Experiences of sparse direct symmetric solvers’, ACM Trans. Math. Softw., 2007 15. TABARROK, B. & QIN Z., ‘Non linear analysis of tension structures’, Computers & Structures, 1992 16. WALKER, E., AULD, D., ‘An aerodynamic analyis of square topped mainsail’, Thesis University of Sydney, 2009

17. ZIENKIEWICZ, O.C., ‘The finite element method in Engineering science’, McGraw Hill, London, 1971 18. ZLATEV, Z., WANASNIEW, S J., SCHAUMBURG, K., ‘Solution of Large and Sparse Systems of Linear Algebraic Equations’, Lecture Notes in Computer Science, Springer, 1981 9. AUTHORS BIOGRAPHY Dr. Sabrina Malpede is the co-founder and Managing Director of SMAR Azure Ltd. For the past eleven years, she has been involved in sail-design, at first for her doctorate and then within SMAR Azure for the development of products and services required by the Industry. She is a graduate with honours in Aeronautical Engineering at the University of Naples (Italy) and has a Ph.D. in Sail Design from the University of Glasgow (UK). Francesco Nasato holds the current position of Research and Development Manager at SMAR-Azure Ltd. He is responsible for the development of the development of the aerodynamic and structural analysis methods of the SMAR-Azure Ltd technology. He graduated in Aeronautical Engineering at the University of Padova and has a M.Sc in Yacht Design at University of Venice (Italy).

Documents

A FULLY INTEGRATED SAIL-RIG ANALYSIS METHODvm2330.sgvps.net/~syrftest/images/library/20160514165849.pdf · A FULLY INTEGRATED SAIL-RIG ANALYSIS METHOD ... geometric data make the