CASD: Computer Aided Ship Design Panagiotis Kaklis joint work with - - PowerPoint PPT Presentation

CASD: Computer Aided Ship Design

Panagiotis Kaklis

joint work with

A.-A.I. Ginnis, K.V. Kostas & C. Feurer

National Technical University of Athens (NTUA) School of Naval Architecture and Marine Engineering (Sname) Ship Design Laboratory (SDL)

October 5, 2010

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 1 / 26

preamble

a bit of history

in Farin’s ”History of Curves and Surfaces in CAGD” it is stated: The

earliest recorded use of curves in a manufacturing environment seems to go back to early AD Roman times, for the purpose of shipbuilding. A ship’s ribs were produced based on templates which could be reused many

• times. Thus a vessel’s basic geometry could

be stored and did not have to be recreated every time. These techniques were perfected by the Venetians from the 13th to the 16th century.

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 2 / 26

preamble

a bit of history

The form of the ribs was defined in terms of tangent continuous circular arcs, NURBS in modern parlance. Ship hull was obtained by varying the ribs’ shapes along the keel, an early manifestation of today’s tensor product surface definitions. No drawings existed to define a ship hull; these became popular in England in the 1600s.

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 3 / 26

preamble

a bit of history

The classical spline, a wooden beam which is used to draw smooth curves, was probably invented then. The earliest available mention of a spline seems to be from 1752 by Monceau. This shipbuilding connection, described by Horst Nowacki (2000), was the earliest use of constructive geometry to define free-form shapes.

More modern developments linking marine and CAGD techniques may be found in, e.g., Theilheimer and Starkweather (1961), Berger et al (1966), Mehlum and Sorenson (1971) and Rogers and Satterfield (1980).

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 4 / 26

preamble

a special issue on CASD (Elsevier, JCAD, vol. 42, 2010)

CASD: Some Recent Results and Steps ahead in Theory, Methodology and Practice dedicated to Prof. H. Nowacki

• G. Holbach (TUB, DE), X. Ye (Zhejiang Univ., CN) and PK (NTUA, GR)

1 Five Decades of Computer-Aided Ship Design 2 Mesh-based Morphing Method for Rapid Hull Form Generation 3 Hull-form Optimization in Calm and Rough Water 4 An Integrated Method for Predicting Resistance of Low-cB Ships 5 Springback Adjustment for Multi-point Forming of Thick Plates in

Shipbuilding

6 Application of a New Multi-agent Hybrid Co-evolution Based Particle

Swarm Optimization Methodology in Ship Design

7 Holistic Ship design Optimization 8 VELOS: a VR Platform for Ship-evacuation Analysis

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 5 / 26

preamble

a special issue on CASD (Elsevier, JCAD, vol. 42, 2010)

CASD: Some Recent Results and Steps ahead in Theory, Methodology and Practice dedicated to Prof. H. Nowacki

• G. Holbach (TUB, DE), X. Ye (Zhejiang Univ., CN) and PK (NTUA, GR)

1 Five Decades of Computer-Aided Ship Design 2 Mesh-based Morphing Method for Rapid Hull Form Generation 3 Hull-form Optimization in Calm and Rough Water 4 An Integrated Method for Predicting Resistance of Low-cB Ships 5 Springback Adjustment for Multi-point Forming of Thick Plates in

Shipbuilding

6 Application of a New Multi-agent Hybrid Co-evolution Based Particle

Swarm Optimization Methodology in Ship Design

7 Holistic Ship design Optimization 8 VELOS: a VR Platform for Ship-evacuation Analysis

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 5 / 26

preamble

a special issue on CASD (Elsevier, JCAD, vol. 42, 2010)

CASD: Some Recent Results and Steps ahead in Theory, Methodology and Practice dedicated to Prof. H. Nowacki

• G. Holbach (TUB, DE), X. Ye (Zhejiang Univ., CN) and PK (NTUA, GR)

1 Five Decades of Computer-Aided Ship Design 2 Mesh-based Morphing Method for Rapid Hull Form Generation 3 Hull-form Optimization in Calm and Rough Water 4 An Integrated Method for Predicting Resistance of Low-cB Ships 5 Springback Adjustment for Multi-point Forming of Thick Plates in

Shipbuilding

6 Application of a New Multi-agent Hybrid Co-evolution Based Particle

Swarm Optimization Methodology in Ship Design

7 Holistic Ship design Optimization 8 VELOS: a VR Platform for Ship-evacuation Analysis

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 5 / 26

preamble

a special issue on CASD (Elsevier, JCAD, vol. 42, 2010)

CASD: Some Recent Results and Steps ahead in Theory, Methodology and Practice dedicated to Prof. H. Nowacki

• G. Holbach (TUB, DE), X. Ye (Zhejiang Univ., CN) and PK (NTUA, GR)

1 Five Decades of Computer-Aided Ship Design 2 Mesh-based Morphing Method for Rapid Hull Form Generation 3 Hull-form Optimization in Calm and Rough Water 4 An Integrated Method for Predicting Resistance of Low-cB Ships 5 Springback Adjustment for Multi-point Forming of Thick Plates in

Shipbuilding

6 Application of a New Multi-agent Hybrid Co-evolution Based Particle

Swarm Optimization Methodology in Ship Design

7 Holistic Ship design Optimization 8 VELOS: a VR Platform for Ship-evacuation Analysis

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 5 / 26

preamble

a special issue on CASD (Elsevier, JCAD, vol. 42, 2010)

CASD: Some Recent Results and Steps ahead in Theory, Methodology and Practice dedicated to Prof. H. Nowacki

• G. Holbach (TUB, DE), X. Ye (Zhejiang Univ., CN) and PK (NTUA, GR)

1 Five Decades of Computer-Aided Ship Design 2 Mesh-based Morphing Method for Rapid Hull Form Generation 3 Hull-form Optimization in Calm and Rough Water 4 An Integrated Method for Predicting Resistance of Low-cB Ships 5 Springback Adjustment for Multi-point Forming of Thick Plates in

Shipbuilding

6 Application of a New Multi-agent Hybrid Co-evolution Based Particle

Swarm Optimization Methodology in Ship Design

7 Holistic Ship design Optimization 8 VELOS: a VR Platform for Ship-evacuation Analysis

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 5 / 26

preamble

a special issue on CASD (Elsevier, JCAD, vol. 42, 2010)

CASD: Some Recent Results and Steps ahead in Theory, Methodology and Practice dedicated to Prof. H. Nowacki

• G. Holbach (TUB, DE), X. Ye (Zhejiang Univ., CN) and PK (NTUA, GR)

1 Five Decades of Computer-Aided Ship Design 2 Mesh-based Morphing Method for Rapid Hull Form Generation 3 Hull-form Optimization in Calm and Rough Water 4 An Integrated Method for Predicting Resistance of Low-cB Ships 5 Springback Adjustment for Multi-point Forming of Thick Plates in

Shipbuilding

6 Application of a New Multi-agent Hybrid Co-evolution Based Particle

Swarm Optimization Methodology in Ship Design

7 Holistic Ship design Optimization 8 VELOS: a VR Platform for Ship-evacuation Analysis

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 5 / 26

preamble

a special issue on CASD (Elsevier, JCAD, vol. 42, 2010)

CASD: Some Recent Results and Steps ahead in Theory, Methodology and Practice dedicated to Prof. H. Nowacki

• G. Holbach (TUB, DE), X. Ye (Zhejiang Univ., CN) and PK (NTUA, GR)

1 Five Decades of Computer-Aided Ship Design 2 Mesh-based Morphing Method for Rapid Hull Form Generation 3 Hull-form Optimization in Calm and Rough Water 4 An Integrated Method for Predicting Resistance of Low-cB Ships 5 Springback Adjustment for Multi-point Forming of Thick Plates in

Shipbuilding

6 Application of a New Multi-agent Hybrid Co-evolution Based Particle

Swarm Optimization Methodology in Ship Design

7 Holistic Ship design Optimization 8 VELOS: a VR Platform for Ship-evacuation Analysis

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 5 / 26

CASD as a MOP

CASD as a Multiobjective Optimization Problem (MOP)

Given a set of geometric, engineering and operational constraints find an optimal, versus a set of costs, ship hull. There is no unique optimal solution but rather a set of efficient solutions, also known as Pareto-optimal solutions, characterized by the fact that their cost cannot be improved in one dimension without being worsened in another. The set of all Pareto solutions, the Pareto front, represents the problem trade-offs. Sampling this set in a representative manner is a very useful aid in decision making.

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 6 / 26

CASD as a MOP

a MOP example in CASD

2 costs: an energy- and an environment-oriented cost

(Papanikolaou(2010)):

1 Rt = Rf + Rw where Rf

denotes the frictional and RW the wave resistance

2 W: the average wave height

along a longitudinal wave cut at a certain distance from the vessel’s center line.

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 7 / 26

CASD as a MOP

basic elements of MOP in CASD

design parameters:

• ship’s main dimensions (length, beam, side depth, draft) or ratios

between them, possibly extended to include the entire ship’s hull form→ SHIP PARAMETRIC MODEL

• the arrangement of spaces
• (main) structural elements
• networking elements (piping, electrical, etc)

costs (criteria, merit functions, goals):

• economic (profit of the initial investment)
• performance (in calm water and in seaways)
• safety
• ships’s strength (including fatigue)

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 8 / 26

CASD as a MOP

basic elements of MOP in CASD

constraints:

• owners’s specifications related to:
• cargo capacity (deadweight, payload)
• service speed, range
• financial data (profit expectations, interest rates),
• market conditions (demand and supply data)
• costs for major materials (steel, fuel), etc.
• the above list may be extended with criteria characterized by uncertainty

with respect to their actual values and being assessed by probabilistic models

• inequalities/equalities resulting from regulatory frameworks pertaining

to safety

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 9 / 26

an OP in CASD

• ptimizing a parent container ship versus wave resistance

Given a parent container ship, modify appropriately its geometry so that the cost functional 3

i=1 wiR(vi) is

minimized, with R(v) denoting the wave resistance of a ship advancing with constant forward speed v.

speed vi (knots) 20 25 27 weight wi 0.35 0.50 0.15

Container ships represent commercially interesting ship hulls while at the same time are slender enough to ensure that the Neumann-Kelvin model will provide a satisfactory approximation of the ship wave resistance. slenderness ⇐ ⇒ B

L = T L = o(1),

L : length, B : beam, T : draft

• (NTUA/SDL - CAGD/CAD/VR Group)

SAGA, Fall School 2010 October 5, 2010 10 / 26

an OP in CASD

• ptimizing a parent container ship versus wave resistance

Given a parent container ship, modify appropriately its geometry so that the cost functional 3

i=1 wiR(vi) is

minimized, with R(v) denoting the wave resistance of a ship advancing with constant forward speed v.

speed vi (knots) 20 25 27 weight wi 0.35 0.50 0.15

Container ships represent commercially interesting ship hulls while at the same time are slender enough to ensure that the Neumann-Kelvin model will provide a satisfactory approximation of the ship wave resistance. slenderness ⇐ ⇒ B

L = T L = o(1),

L : length, B : beam, T : draft

• (NTUA/SDL - CAGD/CAD/VR Group)

SAGA, Fall School 2010 October 5, 2010 10 / 26

an OP in CASD

Neumann-Kelvin model

basic hypotheses:

• ideal (⇔ incompressible, inviscid,

irrotational) liquid → potential theory for the flow field

• linearized free-surface condition on the

plane z = 0

• exact body-boundary condition
• radiation condition:
• utgoing waves decaying as

O(R−1/2), R =

• (x2 + y2)
• Boundary Integral Equation (BIE) over the wetted hull

surface

• hybrid method: coupling the Finite Element Method

(FEM) in a finite volume around the ship with a multipole expansion method in the exterior of the finite volume

− → U : ship velocity ϕ: disturbance potential w = − → U + ∇ϕ: total velocity p = p∞ + ρ

2 (U2 − w2) − ρgz:

pressure η = (U/g)ϕx(x, y, z = 0): free-surface elevation

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 11 / 26

an OP in CASD

Neumann-Kelvin model

basic hypotheses:

• ideal (⇔ incompressible, inviscid,

irrotational) liquid → potential theory for the flow field

• linearized free-surface condition on the

plane z = 0

• exact body-boundary condition
• radiation condition:
• utgoing waves decaying as

O(R−1/2), R =

• (x2 + y2)
• Boundary Integral Equation (BIE) over the wetted hull

surface

• hybrid method: coupling the Finite Element Method

(FEM) in a finite volume around the ship with a multipole expansion method in the exterior of the finite volume

− → U : ship velocity ϕ: disturbance potential w = − → U + ∇ϕ: total velocity p = p∞ + ρ

2 (U2 − w2) − ρgz:

pressure η = (U/g)ϕx(x, y, z = 0): free-surface elevation

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 11 / 26

an OP in CASD

Neumann-Kelvin model

basic hypotheses:

• ideal (⇔ incompressible, inviscid,

irrotational) liquid → potential theory for the flow field

• linearized free-surface condition on the

plane z = 0

• exact body-boundary condition
• radiation condition:
• utgoing waves decaying as

O(R−1/2), R =

• (x2 + y2)
• Boundary Integral Equation (BIE) over the wetted hull

surface

• hybrid method: coupling the Finite Element Method

(FEM) in a finite volume around the ship with a multipole expansion method in the exterior of the finite volume

− → U : ship velocity ϕ: disturbance potential w = − → U + ∇ϕ: total velocity p = p∞ + ρ

2 (U2 − w2) − ρgz:

pressure η = (U/g)ϕx(x, y, z = 0): free-surface elevation

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 11 / 26

an OP in CASD

Neumann-Kelvin model

basic hypotheses:

• ideal (⇔ incompressible, inviscid,

irrotational) liquid → potential theory for the flow field

• linearized free-surface condition on the

plane z = 0

• exact body-boundary condition
• radiation condition:
• utgoing waves decaying as

O(R−1/2), R =

• (x2 + y2)
• Boundary Integral Equation (BIE) over the wetted hull

surface

• hybrid method: coupling the Finite Element Method

(FEM) in a finite volume around the ship with a multipole expansion method in the exterior of the finite volume

− → U : ship velocity ϕ: disturbance potential w = − → U + ∇ϕ: total velocity p = p∞ + ρ

2 (U2 − w2) − ρgz:

pressure η = (U/g)ϕx(x, y, z = 0): free-surface elevation

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 11 / 26

ship-hull PM

ship-hull Parametric Model (PM)

“Given a parent container ship, modify appropriately its geometry”

• the availability of a SHIP PARAMETRIC MODEL, that is able to:
• reconstruct the parent container hull
• construct a rich set of container hulls that:

1 fulfill a set of constraints inherited from the parent hull 2 remain shape-similar to the parent hull.

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 12 / 26

ship-hull PM

constraints inherited from the parent hull

global parameters constraint

• verall length

±5% draft ±10% displacement ±1% length of parallel midbody ±10% start of the Flat Of Bottom (FOB) ±10%

• displacement: The weight of water of the

displaced volume of the ship, which equals the weight of the ship and cargo.

Parallel Midbody FOS Fore FOS Aft. FOB Aft. FOB Fore

M0

M1 M2 M3 M4 (NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 13 / 26

ship-hull PM

constraints inherited from the parent hull

local (bow, bulb, stern) parameters constraint bulb length +30% transom behind AP ±10% stern root height ±10% stern root start ±10%

FP AP Baseline WL

S_TransomSlopeAngle S_TransomBehind S_PropellerClearance S_RootHeight

S_ShaftHeight S_ShaftExit

S_RootStart

S_ShaftRadius

B_BulbRiseLength B_BulbRise B_BulbLength B_BulbTopPosition B_BulbCenter B_ForwardOverhang B_BulbRadius

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 14 / 26

constructing a PM for container ships

constructing a parametric model for container ships

Design parameters can be classified into 4 groups:

1 The global group ← ship’s principal dimensions [effect: global] 2 The 2nd category ← parameters involved in the generation of the

midship part [effect: global, as the midship part is both the initial and supporting entity in the construction of the hull parametric model]

3 3rd and 4th categories ← parameters involved in the generation of

the bow and stern parts of the ship [effect: local]

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 15 / 26

constructing a PM for container ships

constructing the Control Curve Network (CCN)

Design parameters in conjunction with shape-preserving interpolation techniques provide CCN(0) ← the initial version of Control Curve Network. CCN(0) comprises ship’s profile, FOS & FOB boundary curves, midship section, sections capturing shape transition in bow and stern areas.

• n shape-preserving interpolation techniques → the lecture by Prof. Menelaos

Karavelas on Thursday.

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 16 / 26

constructing a PM for container ships

constructing the Control Curve Network (CCN)

Design parameters in conjunction with shape-preserving interpolation techniques provide CCN(0) ← the initial version of Control Curve Network. CCN(0) comprises ship’s profile, FOS & FOB boundary curves, midship section, sections capturing shape transition in bow and stern areas.

• n shape-preserving interpolation techniques → the lecture by Prof. Menelaos

Karavelas on Thursday.

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 16 / 26

constructing a PM for container ships

enhancing CCN(0) with tangent information

CCN(0) is used in conjunction with constraints apparently induced from ship’s geometry, e.g.,

• symmetry with respect to the ship’s center plane,
• planarity of FOB/FOS area,

for producing cross-tangent ribbons over its elements. CCN(0) {unit cross-tangent-vector ribbons} = CCN(0,t) ← enhanced initial version of CCN

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 17 / 26

constructing a PM for container ships

enhancing CCN(0,t) with auxiliary curves

Using information inherited from CCN(0,t) and a set of internal parameters, CCN(0) is enriched with a set of auxiliary curves in order to:

• use them as shape stabilizers in the transitional areas of bow and

stern, where shape changes rapidly,

• eventually have a network of principally quadrilateral topology

CCN(0) {auxiliary curves} = CCN(1) ← semifinal version of CCN as with CCN(0), CCN(1) can be enhanced along its elements with unit cross-tangent-vector ribbons → CCN(1,t) ←final version of CCN.

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 18 / 26

constructing a PM for container ships

examples of auxiliary curves:

• guide curves c W1,c W2 and

c W3 secure and control the saddle shape the waterline-entry area.

• guide curve c BowG1 separates the

bulbous part from the bow area and passes through the inflection point

• f c S2.

c_S1 c_W

c_W1 c_W3 c_W2

c_BowProfile

2 4 3

c _ B O W S E P c_S1 c_S2 c_W

c_W1 c_W3 c_W2

c_BowProfile c_BowG1 c_Deck Midship Section c_FOB_bow c_FOS_bow

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 19 / 26

constructing a PM for container ships

container-ship hull generation

The sought-for G1−continuous surface is then obtained by solving a series

• f Hermite-type local problems of the form: construct a patch that

interpolates a closed boundary along with its tangent-plane distribution, defined by CCN(1,t).

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 20 / 26

a PM implementation

implementing the proposed PM for container ships

parent ship KRISO Container Ship (KCS) is part of the CFD Workshop benchmark suit of problems; see at the home page of the CFD Workshop in Tokyo 2005.

KCS has been proposed by the Norwegian Classification Society DNV (Det Norske Veritas), NTUA’s partner in the FP7 project EXCITING: EXACT GEOMETRY SIMULATION FOR OPTIMIZED DESIGN OF VEHICLES AND VESSELS (2008-2011), coordinated by Professor Bert J¨ uttler (JKU).

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 21 / 26

a PM implementation

basic features of PM implementation

• parametric-modeling environment: CATIA
• generation process controlled by 29 design (exposed) and 46 internal

parameters

• the final Control Curve Network CCN(1,t) consists of quintic NURBS

curves

• the generated semi-hull is at least G1−continuous and comprises 34

surface patches that are principally tensor-product NURBS surfaces of degree 5x5

• generation of a hull instance for a given parametric set takes less than

5 sec on a modern pc

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 22 / 26

a PM implementation

performance issues

is PM able to reproduce the parent hull KCS satisfactorily? KCS (above) and PM instance (below)

integral properties KCS PM instance ∇(m3) 79, 727.00 79, 719.00 LCB (m) 107.49 107.49 Cb (m) 0.677 0.677

• ∇: volume displacement
• LCB: Longitudinal Center of Buoyancy
• block coefficient Cb=∇/LBT

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 23 / 26

a PM implementation

performance issues

can PM guarantee the integrity of the outcome? is PM able to produce a rich set of container hulls that remain similar with the parent hull KCS?

hull instances with significantly varying length between perpendiculars: Lpp = 146m (above), Lpp = 470m (below)

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 24 / 26

a PM implementation

performance issues

hull instances with varying the design parameter Bulblength: B BulbLength=8.8m (above), B BulbLength=6.5 (below)

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 25 / 26

a PM implementation

issues of interest

1 investigate/modify/enrich the methodology for solving the

Hermite-type local problems versus the:

• self-intersection risk
• need to handle shape constraints

2 measure shape similarity between PM instances 3 introduce a notion of measurable ”richness” of PM-outcome sets 4 develop an optimization-oriented LOD approach for PM instances 5 investigate the compatibility/efficiency of the geometry

representation, provided by PM, versus the method used for solving the NK problem ← isogeometric analysis

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 26 / 26

references

[1] G. Farin, A History of Curves and Surfaces in CAGD. [2] H.L. Duhamel du Monceau. El´ ements de l’Architecture Navale ou Trait´ e Pratique de la Construction des Vaissaux. 1752. Paris. [3] H. Nowacki. Splines in shipbuilding. Proc. 21st Duisburg colloquium on marine technology, May 2000. [4] F. Theilheimer and W. Starkweather. The fairing of ship lines on a high speed computer. Numerical Tables Aids Computation, vol. 15,

• pp. 338-355, 1961.

[5] S. Berger, A. Webster, R. Tapia, and D. Atkins. Mathematical ship

• lofting. J Ship Research, vol. 10, pp. 203-222, 1966.

[6] E. Mehlum and P. Sorenson. Example of an existing system in the shipbuilding industry: the AUTOKON system. Proc. Roy. Soc. London, Series A, vol. 321, pp. 219-233, 1971.

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 26 / 26

references

[7] D. Rogers and S. Satterfield. B-spline surfaces for ship hull design. Computer Graphics (Proc. Siggraph 80), vol. 14, no. 3, pp. 211-217, 1980. [8] A. Papanikolaou. Holistic ship design optimization. Computer-Aided Design (JCAD), vol. 42, pp. 1028-1044, 2010.

(NTUA/SDL - CAGD/CAD/VR Group) SAGA, Fall School 2010 October 5, 2010 26 / 26