A BEM-isogeometric method with application to the wavemaking

Proceedings of the ASME 2011 30th International Conference on Ocean, Offshore and Arctic Engineering OMAE 2011 June 19-24, 2011, Rotterdam, The Netherlands

OMAE2011-49159

A BEM-ISOGEOMETRIC METHOD WITH APPLICATION TO THE WAVEMAKING RESISTANCE PROBLEM OF SHIPS AT CONSTANT SPEED K.A. Belibassakis(1,2), Th.P. Gerostathis(1), K.V. Kostas(1), C.G. Politis(1) P.D. Kaklis(2), A.I. Ginnis(2) and C. Feurer(2) (1)

Dept. of Naval Architecture, Technological Educational Institute of Athens Ag. Spyridonos 12210, Athens, GREECE. Email: [email protected]

(2)

School of Naval Architecture and Marine Engineering, National Technical University of Athens Zografos 15773, Athens, GREECE. Email: [email protected]

ABSTRACT In the present work IsoGeometric Analysis (IGA), initially proposed by Hughes et al (2005), is applied to the solution of the boundary integral equation associated with the NeumannKelvin (NK) problem and the calculation of the wave resistance of ships, following the formulation by Brard (1972) and Baar & Price (1988). As opposed to low-order panel methods, where the body is represented by a large number of quadrilateral panels and the velocity potential is assumed to be piecewise constant (or approximated by low degree polynomials) on each panel, the isogeometric concept is based on exploiting the NURBS basis, which is used for representing exactly the body geometry and adopts the very same basis functions for approximating the singularity distribution (or in general the dependent physical quantities). In order to examine the accuracy of the present method, in a previous paper Belibassakis et al (2009), numerical results obtained in the case of submerged bodies are compared against analytical and benchmark solutions and low-order panel method predictions, illustrating the superior efficiency of the isogeometric approach. In the present paper we extent previous analysis to the case of wavemaking resistance problem of surface piercing bodies. The present approach, although focusing on the linear NK problem which is more appropriate for thin ship hulls, it carries the IGA novelty of integrating CAD systems for shiphull design with computational hydrodynamics solvers. 1. INTRODUCTION The investigation of ship resistance in calm water is a significant problem due to its specific importance in ship powering prediction and optimisation of ship hulls. This problem is complicated since ship resistance is dependent on

both viscous and gravitational effects. For computational purposes, the calculation of viscous and wavemaking resistance are usually considered separately. Wave-making resistance is a very important component, which sometimes could contribute more than 50% of the total resistance of a ship (especially for relatively full hull forms and/or at high speeds). Experience has shown that the wavemaking resistance component is quite sensitive to design parameters and significant reduction can be achieved without affecting cargo capacity. The capability to predict and minimize wave resistance in the early stages of the design is, therefore, very important. During the last 50 years, the interest in numerical methods for calculating ship wave resistance has been constantly growing. Computations are performed using a variety of techniques, ranging from the simple Michell's thin ship theory to fully nonlinear RANS methods; recent advances are presented in the reports by ITTC (see ITTC 2005, 2008 and the references cited therein). The application of three dimensional potential flow theory to the steady ship motion problem results in an essentially non-linear boundary value problem (due to the nonlinear character of the boundary condition at the free surface), from which the unknown velocity potential and the free-surface disturbance must be calculated. Boundary Element Method (BEM) is a widely used approach to solve potential flow problems in marine hydrodynamics and especially the wave resistance and the ideal wave pattern of ships advancing with steady forward speed. There exist two main types of elementary singularities used in the implementation of the method. The first type uses the Kelvin wave source, satisfying the field equation and all the boundary conditions except the bodyboundary condition, as the elementary singularity. The major advantages of such a scheme are the automatic satisfaction of the radiation condition and the definition of the resulting

1

Copyright © 2011 by ASME

Boundary Integral Equation only on the ship hull (see, e.g., Wehausen 1973). The second type uses the simple Rankine source (the fundamental solution of Laplace equation) as the elementary singularity. The method was first presented by Dawson (1977) and since then it has been widely applied as a practical method to predict wave resistance. Many improvements have also been made to account for non-linear effects; see, e.g., Nakos & Sclavounos (1990), Raven (1996), Bertram (2000), Bal (2008). Considerable efforts have been devoted to increase efficiency and accuracy by introducing several variations, as the patch method, the desingularized method, the RAPID method etc. This approach has the advantage of employing a simple elementary singularity, but on the other hand, the resulting boundary integral equation is extended over the ship hull and the free surface leading to increased computational requirements. In the present work, IsoGeometric Analysis (IGA), proposed by Hughes et al (2005, 2008) and so far used in the context of Finite Element Method (see, e.g., Cottrell et al 2007), is applied, apparently for the first time, to the solution of the boundary integral equation associated with the linearised Neumann-Kelvin (NK) problem. In this version of the problem, the non-linear effects stemming from the presence of the unknown free surface are neglected, while the threedimensional character of the fluid flow is fully retained. Independently of its own practical interest in some cases, e.g. slender ships and/or low forward speed, the robust and accurate solution of this linear problem represents a useful first stage before dealing with the complete non-linear problem. For the solution we shall use the Boundary Element Method (BEM) implemented by means of a Kelvin wave source distribution over the wetted part of the hull. The satisfaction of the bodyboundary condition leads to an Integral Equation defined on the wetted part of the hull and its intersection with the free surface, Brard (1972) and Baar & Price (1988). Integral equation formulation of Laplace boundary value problems has been established as one of the standard tools for calculating inviscid, incompressible flow characteristics (velocity and pressure) around 2-D and 3-D bodies and geometrical systems; see, e.g., Hess (1975), Katz-Plotkin (1991). Some of the most important advantages of this approach include the reduction of dimensionality, facilitating calculations around complex geometrical configurations (especially in 3D), consistent handling of conditions at infinity, high convergence rates when the domain boundary and boundary data are (relatively) smooth and easy implementation to optimization (inverse-type) problems. As concerns the numerical solution, although various other techniques are available (such as spectral methods using basis functions of global support on the boundary), Boundary Element Methods (BEM or panel methods) serve today as the main tool, especially in the case of non-linear and time-dependent problems; see e.g. Brebbia et al (1984), Paris & Canas (1997), Brebbia (2002). In the low-order BEM (or panel method) the body surface is usually discretised to a finite number of elements or patches, each carrying a simple distribution of the unknown function; see, e.g., Hess (1975). On the contrary, high-order BEM, characterised by an increased order of approximation both with respect to geometry and the surface singularity distribution, have the property of faster convergence, as element-size vanishes, and yield more accurate results with coarser grid

resolutions (see, e.g., Gennaretti et al, 1998). The latter is found to be quite important especially at places/subregions where the solution presents physical discontinuities and/or singularities which are not well treated by low-order methods. In this direction, high-order panel methods based on B-spline and/or non-uniform rational B-spline (NURBS) representations have recently appeared in the literature, for potential-flow problems. In the sequel, we briefly present some of these works. A 3D method for wave-body interaction through a Rankine boundary element approach, based on Maniar (1995) and satisfying the body-boundary condition directly on the exact computer-aided design (CAD) surfaces, has been applied to multi-body seakeeping design optimization by Peltzer et al (2008). The high-order hydrodynamic boundary element method overlays the NURBS geometry and yields stable wave and motion integration in time (Kring 1995). Kim & Shin (2003), solve the three-dimensional radiation and diffraction problem using a NURBS representation of the body geometry and a B-spline basis for the unknown potential. Datta & Sen (2006, 2007), solve, in the time domain, the three-dimensional ship motions problem with forward speed. The problem is formulated in the time domain using the transient free-surface Green function. The body geometry is represented by either B-spline or NURBS, depending on the hull type, whereas the unknown field variables are represented via B-spline basis functions. Moreover, Kim et al (2007) use a higher-order panel method, based on B-spline representation for both the geometry and the solution, with application to the analysis of steady flow around marine propellers. Also, Gao & Zou (2008) solve the 3-D radiation and diffraction problem with forward speed, using the Rankine source distribution method, in conjunction with a NURBS surface to precisely represent the body geometry, whereas the velocity potential on the body surface is represented by B-splines. As opposed to high-order panel methods outlined above, where the body and the associated physical quantities are expressed via different basis functions, the IsoGeometric (IGA) approach, Hughes et al (2005), Cottrell et al (2009), is based on using the very same basis for representing both the body geometry and the physical quantities, under the constraint that the chosen basis is able to represent accurately the geometry in question. In a recent work by Politis et al (2009), IGA is exploited for the solution of the exterior Neumann problem in 2D, reformulated as boundary integral equation by using single-layer distributions, see, e.g., Kress (1989). In order to examine the accuracy of this method, numerical results obtained for a circular and a free-form contour, represented exactly as NURBS curves, and various types of forcing, are compared against analytical and benchmark solutions. The error is compared against low- and higher-order panel method predictions, illustrating very high rates of algebraic convergence, ranging from O ( N −4 ) to O ( N −10 ) , if mesh refinement, through knot insertion, is combined with degree elevation, N standing for the total number of degrees of freedom. This result, compared with the low rate O ( N −1 ) of the classical panel method, is found to be promising for boundary integral equations of potential flow problems associated with ship and submerged bodies in steady motion. In particular, the application of the IGA to the wavemaking resistance problem of submerged bodies has been examined in a previous paper by 2


tangential vector along the waterline, directed as shown in Fig.1. The integral representation, Eqs. (1),(2), permits us to automatically satisfy the linearised condition on the undisturbed free surface and the conditions at infinity. Using the above, the NK problem is equivalently reformulated as a boundary, integral equation (BIE) defined on the wetted surface S = ∂D ,

μ ( P) 2

− ∫ μ (Q ) S

−

the authors (Belibassakis et al 2009). In the case of spheroids and ellipsoids numerical results obtained by IGA method were compared against the analytical and benchmark solutions without and with free-surface effects (Farell 1973), as well as against low-order panel method predictions, illustrating the convergence aspects of the present approach. In this paper we extent previous analysis to the case of wavemaking resistance problem of surface piercing bodies. The present approach, although focusing on the linear NK problem which is appropriate for thin ship hulls, it has the novelty of bringing together modern integrated CAD systems for ship-hull design with computational hydrodynamics solvers. 2. FORMULATION OF THE PROBLEM Let Oxyz be a right-handed rectangular coordinate system with z- axis directed vertically upwards; see Fig. 1. We consider the flow of a uniform stream with velocity U = (-U, 0, 0) of an ideal fluid with a free surface incident upon a surface piercing body D. Following the formulation by Brard (1972) and Baar & Price (1988) for the linearised Neumann-Kelvin problem, the disturbance potential (φ) due to the steady translation of the ship in infinite water depth is represented by

∫ μ ( Q ) G ( P, Q ) dS ( Q ) + S

+ k −1 ∫ μ ( Q ) G * ( P, Q ) nx ( Q )τ y ( Q ) dw ( Q ) ,

(1)

WL

where S = ∂D denotes the wetted boundary of the body and WL the waterline, μ is the source density on S, and G is the NK- Green’s function: 4π G ( P, Q ) = r −1 − R −1 + G * ( P, Q ) , Q ∈ S , P ∈ IR −3 \ D ,

(2)

where IR −3 = { x, y, z ≤ 0} is used to denote the half space below the undisturbed free surface z=0. In the above equation, →

→

r = QP , R = Q ′P , where Q ′ is the image of Q with respect to the z=0 plane (the undisturbed free surface), and G * ( P, Q ) stands for the regular part of the NK Green’s function, consisting of exponential decaying and wavelike components (for more details see Baar & Price 1988). Moreover, n ( P ) = ( nx ,n y ,nz ) denotes the normal vector

directed outwards the body and

∂n ( P )

dS ( Q )

∂G* ( P, Q ) 1 Q nx ( Q )τ y ( Q ) dw ( Q ) = − g ( P ) , μ ( ) ∫ k WL ∂n ( P ) P, Q ∈ S (3)

Fig. 1. Steadily translated ship.

ϕ ( P) =

∂G ( P, Q )

τ ( P ) = (τ x ,τ y ,τ z )

the

where

g ( P ) = −U ⋅ n ( P )

are

the boundary data, and

k = g / U is the characteristic wavenumber, controlling the wavelength of the transverse ship waves, which is directly connected with the square inverse of the corresponding Froude number F = U / gL , with L denoting the max length of the body. From the solution of the above integral equation, various quantities, such as velocity, pressure distribution and ship wave pattern can be obtained. Specifically, total flow velocity, and pressure are readily obtained by the relations 2

ρ

(U 2 − w2 ) − ρ gz , 2 and the free-surface elevation as follows w = U + ∇ϕ ,

p = p∞ +

η ( x, y ) = (U / g ) ⋅ ϕ x ( x, y; z = 0 ) ,

(4) (5)

where ρ denotes the fluid density and g is the gravitational acceleration. In the case of a fully submerged body the above formulation is modified by dropping the waterline integral in Eqs. (1) and (3). 3. THE ISOGEOMETRIC B.E.M. IGA philosophy is equivalent to approximating the field quantities (dependent variables) of the boundary-value problem in question by the very same basis that is being used for representing accurately the geometry of the involved bodyboundary. In the case of the boundary integral equation (3), for which we are interested in, the dependent variable is the density μ, distributed over the closed surface S. For the latter, we shall presume that it can be accurately represented as a regular parametric NURBS surface. In this connection it should be stressed that, although NURBS is not a requisite IGA ingredient, we adopt it because it represents a thoroughly developed tool and in widespread use in CAD technology. To proceed, let us consider, for x ∈ S , the following NURBS representation of the wetted body n1

n2

x ( t1 , t2 ) = ∑ ∑ di1i2 Ri1i2 , k1k2 ( t1 , t2 ) ,

(6a)

i1 = 0 i2 = 0

wi1i2 Bi1k1 ( t1 ) Bik22 ( t2 )

Ri1i2 , k1k2 ( t1 , t2 ) =

n1

n2

∑ ∑ wl1l2 Blk11 ( t1 ) Blk22 ( t2 )

,

(6b)

l1 = 0 l2 = 0

where

( t1 , t2 )

I1 × I 2 = ⎡⎣t

Start 1

3

are the parameters ranging in the domain End 1

,t

⎤⎦ × ⎡⎣t2Start , t2End ⎤⎦ , wi1i2 are the weights, di1i2 are Copyright © 2011 by ASME

the control points and Ri1i2 , k1k2 ( t1 , t2 ) are rational B-spline basis functions of degree k1 and k2 in the t1 and t2 directions, respectively, defined over knot vectors J1 and J 2 along the two parametric intervals I1 and I2, respectively. For simplicity we concentrate here to body surfaces represented by a single NURBS patch. For example, a NURBS representation for the ellipsoid can be generated by applying a shear transformation to a bi-quadratic NURBS representation of the sphere given in Piegl & Tiller (1997); see Fig. 2. In the case of surface piercing bodies we consider NURBS representations where the isoparametric curve x ( t1 , t2 = t2End ) , t1 ∈ I1 coincides with the undisturbed free-surface plane (z=0). The B-Spline representation of the classical Wigley semi-hull is created by sweeping the parabolic waterline curve on a parabola representing the hull mid section; see Fig. 3. Representations of more complex ship hull geometries and geometrical configurations can be obtained via multipatch NURBS surfaces applying modern CAD tools and techniques.

Fig. 2. NURBS representation of a 5:1:1 prolate spheroid.

In IGA we employ the same representation for the unknown source-sink surface distribution (μ) on the parametric ( t1 , t2 ) space

μ ( t1 , t2 ) =

n1 + A1 n2 + A 2

∑ ∑μ i1 = 0 i2 = 0

i1i2

Ri(1iA21,Ak21)k2 ( t1 , t2 ) =

n+l

∑ μ R( ) ( t , t ) , i

l i ,k

1

2

(7)

Fig. 3. B-spline representation of the half-symmetric part of Wigley parabolic hull.

i =0

where μi1i2 are the (unknown) coefficients associated with the above expansion and A1 , A 2 denote the numbers of additional knots inserted in I1 and I 2 , respectively, in order to increase the approximation power of the expansion. The bold index notation, appearing in the last member of Eq. (7) enables avoiding appearance of multiple indices. In the above way, sequences of nested, finite-dimensional spaces S

k (l)

(

= S k1k2 J1( 1 ) , J 2( A

A2 )

) are produced, to which the BIE (3)

is

projected. Several methods are available for defining a projection and discretising the BIE (3), like Galerkin and collocation methods (see e.g. Kress 1989, Sec.13). In the present work, a collocation scheme (Belibassakis et al 2009) is used to obtain the approximation. Thus, let G Pj = Pj1 j2 = x t1, j1 , t2, j2 denote a set of collocation points on S,

(

)

where t1, j1 , j1 =0,…, n1 + A1 , and t2, j2 , j2 =0,…, n2 + A 2 , lie in I1 and I2, respectively. Next, we define the so-called “induced velocity” factors as follows: u i ( Pj ) =

t1 = t1B t2 = t2B

∫ ∫

(

Rik ( t1 , t2 ) ∇ Pj G (l)

)

α ( t1 , t2 ) dt1dt2 ,

(8)

t1 = t1A t2 = t2A 2

where a = e1 × e 2 is the metric tensor determinant, defined trough the covariant base vectors, e1 = ∂x / ∂t1 , e 2 = ∂x / ∂t2 ,

{ } (l)

(9)

and supp Rik = ⎡⎣t , t ⎤⎦ × ⎡⎣t , t ⎤⎦ . In the case where the A 1

B 1

A 2

B 2

(l)

support of Rik reaches the undisturbed free-surface plane (z=0) the additional term t1 = t1B

∫

t1 = t1A

(

Rik( l ) t1 , t2 = t2End

) (∇

Pj

)

G* nx ( t1 )τ y ( t1 ) e1 ( t1 ) dt1

Fig.4. An enhanced collocation net for the Wigley parabolic hull, obtained after inserting 16x8 knots per parametric interval (dof=781). should be added to the right-hand side of Eq. (8) in order to account for the waterline integral effect. The “induced velocity” factors provide the velocity at any point Pj ∈ IR 3 induced by a source-sink distribution (single layer potential) on

{

}

the surface part E = supp Rik( ) ( t1 , t2 ) with density Rik( l ) ( t1 ,t2 ) . l

Since the function Rik( l ) is Hölder continuous within its support, the integral of Eq. (8) exists and is well defined everywhere except for points Pj ∈ E \ ∂E where it exists as a Cauchy principal value integral (see, e.g., Mikhlin et al 1965). This result breaks down at points Pj lying on the boundary ∂E . To avoid this problem in the present implementation such points are slightly shifted, as explained below. Integration of singular kernels

The Green’s function G ( P,Q ) , Eq. (2), contains both a regular and a singular part, as follows, G ( P,Q ) = Greg ( P,Q ) + Gsin g ( P,Q ) ,

P,Q ∈ S ,

(10a)

where

4


Greg ( P,Q ) = ( − R −1 + G* ( P,Q ) ) ( 4π ) , Gsin g ( P,Q ) = ( 4π r ) . (10b)

points Pj as follows

For the calculation of the regular part of Green’s function (G*), involving exponential decaying and wavelike components, a procedure based on Newman (1987a,b) analysis is followed. However, the series expansion associated with the wavelike component becomes unstable as the points P, Q approach the z=0 plane, leading to numerical instabilities particularly to G* derivatives (see also the discussion by Marr & Jackson 1999). At present this problem is numerically treated by vertically downshifting the whole surface of the ship by a small parameter of the order of Ο(λ/100), where λ=2π/k is the wavelength. Future work is focused on optimizing the calculation of G*, as e.g., described in Marr & Jackson (1999). No such numerical correction is needed in the case of fully submerged bodies.

Πj =

−1

−1

In accordance with Eq.(10a), the integral of Eq. (8) is split to two parts as follows ui ( Pj ) = ui ,reg ( Pj ) + ui ,sin g ( Pj ) .

(11)

The integration of the regular part ( u i ,reg ) is easily obtained by using standard numerical quadrature rules. In the present version, Gauss-Kronrod quadrature formulas are implemented, using points and weights ranging from 21 to 51; see, e.g., Press et al (1992), Shampine (2008). For the evaluation of the singular part ( u i ,sin g ) we distinguish three possible cases as follows: (i) FAR case: The point Pj ∉ E and its distance from E is greater than 2 times its size (e.g. measured by its diagonal in Cartesian space). In this case, the integrand behaves regularly and the same as before 21-51 points Gauss-Kronrod quadrature formulae are applied. (ii) NEAR Case: The point Pj ∉ E and its distance from E is less than 2 times its size. Numerical experience has shown that, if Pj is coming very close to E numerical instabilities could occasionally appear. Current work is directed towards developing an alternative scheme for the evaluation of the integral in this case, based on appropriate transformation techniques, permitting the numerical grid (used for the integration) to become finer as the projection Pj′ of Pj to E is approached; see e.g., Telles (1987), Telles & Oliveira (1994) and Voutsinas & Bergeles (1992). (iii) SELF case: The point Pj ∈ E , lying strictly in the interior of E. In this case the kernel is singular and the integral is considered in the sense of the Cauchy principal value (see, e.g., Mikhlin 1965). For the evaluation of Cauchy principal value, an ε-neighborhood of the singularity at Pj is introduced, cuttingoff of the singularity, and subsequently the limit as ε → 0 is calculated (Mikhlin 1965). An important aspect concerning the implementation of the discrete Cauchy principal value integral is the connection between the cut-off parameter ε and the discretization parameter h (equivalent meshsize for the numerical integration) so that h → 0 , as ε → 0 . The discrete scheme On the basis of the above, we define the projection (Π) of the integral operator to its image values at the collocation

n+l

∑μ i =0

i

(

)

⎡u i ( Pj ) ⋅ n ( Pj ) ⎤ , Pj = Pj j = x t1, j , t2, j ∈ S , (12) 1 2 1 2 ⎣ ⎦

where the unit normal vector is n = e1 × e 2 / a . Using the above approximation, the BIE (3) is finally projected in the S

k (l)

(

= S k1k2 J1( 1 ) , J 2(

∑ μ {R( ) ( t n+l

i

i =0

l ik

1, j1

A

A2 )

) spaces, as follows:

) (

, t2, j2 − 2 u i ( Pj ) ⋅ n ( Pj )

for all

)} = −2 g ( P ) , j

(

)

Pj = Pj1 j2 = x t1, j1 , t2, j2 ∈ S ,

(13)

which constitutes the final linear algebraic system with respect to the unknowns μi = μi1i2 . The above approximation of the BIE (3) is of collocation type and it can be considered as a projection of the continuous integral operator to the finitedimensional space S k ( l ) , which is materialized through an interpolation at the collocation points. In this way, in order to guarantee that this projection remains non singular, a choice is to use as collocation points the ones corresponding to the Greville abscissas; see, e.g., Farin (2001). Enhanced approximations are obtained by knot insertion, that could also be combined with degree elevation (see also Politis et al 2009, Belibassakis et al 2009). As an example, we present in Fig.4 an enhanced collocation net for the Wigley parabolic hull obtained after inserting 16x8 knots per parametric interval comprising of N=781 degrees of freedom (dof). We note here that, in some cases, some Greville points are located on the l boundary of the support of a NURBS basis function Rik( ) ( t1 , t2 ) (as, e.g. in the case of multiple knots) rendering in this way problematic the calculation of the Cauchy principal value integral in (11). To overcome this difficulty, these points are slightly shifted from their original position in such a way so that the symmetry of the arrangement of the collocation points in the parameter domain is maintained, as far as possible. After the numerical solution is obtained, velocities are obtained everywhere using Eqs. (1) and (4a). Also, free-surface elevation is calculated from the x-gradient of the NK potential on z=0 (Eq.5), and pressure distribution is obtained by application of Eq. (4b). Finally, forces and moments are calculated by means of pressure integration on S, combined with the appropriate components of the normal vector and other geometrical quantities. 4. NUMERICAL RESULTS & DISCUSSION

In order to test the accuracy and the convergence of the proposed IGA_BEM method, numerical results for fully submerged and surface piercing bodies, moving steadily in deep water, are presented and discussed. Corresponding results concerning application to flow problems in infinite and semiinfinite domains, including comparison with analytical solution and study of the convergence of the present method can be found in Belibassakis et al (2009). As an example, the velocity distribution along the top meridian on the xz plane of a 3-axial ellipsoid with axes ratio 2:1:0.5, at low submergence d/L=0.16 below a horizontal plane where a homogeneous Dirichlet condition is applied, is shown in Fig.5. 5


knots inserted 2

dof 153

L2error (relative) --

4

325

0.0063

8

861

0.0029

Fig. 5. Velocity distribution along the top meridian on the xz plane of 3-axial ellipsoid (axes ratio 2:1:0.5), at low submergence d/L=0.16.

Fig. 7. Wave resistance coefficient (103Cw) of a prolate spheroid of axes ratio 5:1:1, at low submergence d/L=0.16, vs. the Froude number (F)

Fig. 6. Free-surface elevation generated by a prolate spheroid of axes ratio 5:1:1, at Froude number Fr=0.5, translated at low submergence d/L=0.16.

Fig. 8. Free-surface elevation generated by Wigley parabolic hull, at Froude number Fr=0.316, as calculated by the present method.

In this figure the convergence characteristics of the present method concerning both distributed quantities as well as the decay of integrated L2-error are illustrated as dof increases.

CW =

Submerged spheroid under the free-surface In this section results are presented concerning the solution of the linear NK wave-making problem for a fully submerged body. The case considered here deals with a prolate spheroid with axes a=0.5m, b=c=0.1m, of length L=2a=1m, moving with constant forward speed beneath the free surface. The calculated wave pattern of the above prolate spheroid, at low submergence (d/L=0.16), translated steadily at U=1.57m/s, which corresponds to Froude number F = U / gL =0.5, is shown in Fig. 6. The wave resistance ( RW ) at various speeds is obtained by pressure integration on the surface of the submerged body, and the corresponding wave resistance coefficient is calculated by

RW = S −1 ∫ ∫ CP nx a dt1dt2 , 2 0.5 ρU S u v

(14)

where CP stands for the pressure coefficient Cp =

p − p∞ 2 = 1 − ( w / U ) − 2 gz / U 2 , 2 0.5 ρU

with w = w = U + ∇ϕ

(15)

denoting the total flow velocity and

nx being the x-component of the normal vector. Other important coefficients as the ones concerning sinkage and trim are calculated by similar formulas. In Fig. 7 the wave resistance coefficient CW of the same body at low submergence d/L=0.16 is plotted vs. the Froude number, as calculated by Eq.(14). We observe that the predictions obtained by the present method converge rapidly to the reference solution (Doctors & Beck

6


Fig. 9. Pressure coefficient Cp on the Wigley hull for F = 0.267 as calculated by the present method using dof=781.

Fig. 11. Wave profile for F = 0.316 alongside the Wigley hull. Comparison with experimental data and computations.

Fig. 10. Pressure coefficient Cp on the Wigley hull for F = 0.316 as calculated by the present method using dof=781.

Fig. 12. Wave resistance coefficient (103Cw) parabolic hull vs. the Froude number (F)

1987), shown by using thick line. We also note in Fig.7 that present IGA results, obtained by inserting 2 and 4 knots per parametric interval, corresponding to 135 and 325 dof, respectively, are much more accurate than the ones obtained by the low-order panel method using a grid of 16x20 elements (on the surface of the whole body), which corresponds to 320 dof (shown by using crosses).

circles. We see in this figure that our results agree relatively well with theoretical predictions and measured data over the whole length of the ship with the exception of an area near the stern, where present method leads to overestimation of the wave profile. Finally, in Fig. 12 the wave resistance coefficient CW of the of the Wigley parabolic hull is plotted vs. the Froude number, as calculated by the present IGA method and N= 781 dof, with the aid of Eq. (14). We observe that the predictions obtained by the present method lie within the range of experimental data for this hull and agree well with predictions by other boundary element methods (e.g., Bal 2008). Similarly as before, the present method solution is found to be more accurate than the one obtained by the low-order panel method with the same dof.

Surface piercing body

Finally the present method is applied to the calculation of the wave field and the wave resistance of the standard parabolic Wigley hull, with L/B = 10, L/T = 16, B/T = 1.6. The calculated wave pattern of this hull translated steadily at U=0.9897m/s, which corresponds to Froude number F = U / gL =0.316, is shown in Fig. 8. Colorplots of the pressure distributions on the hull surface for Froude numbers F = 0.267 and 0.316, are presented in Figs. 9, 10, respectively, as obtained by the present method, using N= 781 dof. For the latter Froude number, predictions obtained by the present method concerning the wave profile alongside the Wigley hull are compared against results by other methods and experimental data (Maskew et al, 1997). Present method results are denoted by using red line, while results from the USAERO code are denoted by using solid lines with symbols and experimental data by using open

of

Wigley

CONCLUSIONS

In the present work IsoGeometric Analysis, initially proposed by Hughes et al (2005), is applied to the solution of the boundary integral equation associated with the NeumannKelvin problem and the calculation of the wave resistance of ships. The isogeometric concept is based on exploiting the NURBS basis for the representation the body geometry and adopting the very same basis for approximating the singularity distribution or, in general, the dependent physical quantities. To 7


this respect the present approach, although focusing on the simple NK problem, it has the novelty of bringing together modern integrated CAD systems for ship-hull design with CFD solvers. The enhanced accuracy and efficiency of the present method has been demonstrated, by comparing numerical results obtained in the case of submerged spheroids and ellipsoids against the analytical solution and low-order panel method predictions. Future work is planned towards the detailed analysis of rates of convergence of the present method, its extension to treat more realistic hull surfaces represented by multiple and/or trimmed surfaces, and finally, its application to the optimization of ship hulls with respect to wave resistance. ACKNOWLEDGMENTS Financial support by the FP7 EC-Project EXCITING (SCP8-2007GA-218536) is gratefully acknowledged. Also, the authors would like to thank Dr A.Theodoulides from HRS (also an EXCITING partner) for providing access to a software implementation of the NK source function.

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