Higher Order BEM

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    ELSEVIER

    Applied Ocean Research 17 1995) 11-11

    0 1995 Elsevier Science Limited

    0141-1187(95)00007-0

    Printed in Great Britain. All rights reserved

    0141-I 187/95/ 09.50

    New higher-order boundary element methods for

    wave diffraction/radiation

    B. Teng* R. Eatock Taylor-t

    Depart ment of Engineeri ng Science, Uni versit y of Oxf ord, O xford OX I 3PJ, U

    (Received 2 February 1995; accepted 19 April 1995)

    This paper describes some higher-order boundary element methods and presents a

    novel integral equation for the calculation of the wave diffraction and radiation

    problem. A higher-order element discretisation of the resulting integral equation

    is used. An examination of the convergence and CPU time is carried out, and the

    results demonstrate the advantages of the new method.

    INTRODUCTION

    The analysis of wave diffraction and radiation by a body

    at or near a free surface requires the solution of a

    boundary value problem based on Laplaces equation.

    Because the problem involves an unbounded domain,

    the integral equation method brings certain advantages

    to its solution. Based on this method, the constant panel

    representation introduced by Hess and Smith is

    widely used, for example by Faltinsen and Michelsen,

    Garrison3 Inglis and Price,4 and Korsmeyer et al5

    In this, the submerged body surface alone is discretised

    by a set of quadrilateral or triangular flat panels.

    Pulsating sources or sinks are placed at the centres of

    all panels, the strengths of these sources being constant

    in each panel. For a body with a curved surface,

    however, such a representation allows leaks between

    panels, with the result that large numbers of panels

    are required to achieve computations with sulhcient

    accuracy.

    Recently, the use of higher-order element methods for

    this problem has been investigated, for example by Liu

    et CZI.,~*~ atock Taylor and Chat? and Eatock Taylor

    and Teng. Higher-order element methods are believed,

    in general, to give more accurate results than the

    constant panel method for the same computational

    effort, although there appear to be no direct compar-

    isons for the water wave problem. For complex

    structures and analysis of the non-linear diffraction

    *Present address: Department of Civil Engineering, Dalian

    University of Technology, Dalian 116024, Peoples Republic

    of China.

    In higher-order element methods, special attention

    has to be paid to the specification of solid angles at

    nodes on the body surface, and to some singular

    integrals when field points are very close to a source

    point. The integration of the singular values where the

    slope of the surface is discontinuous only exists in the

    Cauchy principal value (CPV) sense, so special techni-

    ques have to be used. Most researchers use indirect

    methods to avoid computing them directly. A method of

    cancelling the solid angle and CPV integrals was devised

    by Eatock Taylor and Chau. It combines the integral

    equation outside the body with another inside the body,

    obtained using the same pulsating source Green func-

    tion as outside. In this paper, we have derived another

    novel integral equation, by using a simple Green

    function inside the body, which satisfies a rigid lid

    condition on the free surface, and a weakly rigid

    condition on the sea bed. This equation appears to be an

    improvement over that of Eatock Taylor and Chau.*

    tTo whom all correspondence should be addressed.

    In the following, we describe the new formulation,

    problem by Stokes expansion techniques, accuracy is a

    vital factor in a successful calculation, together with

    minimum computer storage and CPU time. In a higher-

    order boundary element approach to this analysis,

    the body surface is discretized by a set of curved

    elements, and the velocity potentials at nodes on the

    element sides and corners become the unknowns. The

    velocity potential and its derivatives inside an element

    are expressed in terms of the corresponding nodal values

    and shape functions. Thus higher-order element

    methods are convenient for the calculation of wave

    run-up and second-order forces on structures, where the

    potential at the water surface and its spatial derivatives

    are needed.

    71

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    72

    B. Teng R. Eatock Taylor

    and give results from some numerical experiments.

    These were carried out to investigate the efficiency of the

    method, and to study convergence in relation to meshes

    both on the body surface and the inner sea bed. The

    latter mesh is a feature of the new method, but is only

    required for structures reaching to the seabed.

    2 THE BASIC INTEGRAL EQUATION

    In order to illustrate the various approaches, we

    consider the first-order wave diffraction and radiation

    problem. It should be noted, however, that the novel

    formulation introduced here has also been applied to

    the calculation of second-order diffraction problems,

    where it is particularly effective. We assume here that

    the first-order incident waves have frequency w, and

    the time factor of the complex potentials is taken as

    e iwf.We use the oscillating source G(x, x0) as Greens

    function, which satisfies the linear free-surface bound-

    ary condition, the radiation condition at infinity and

    the impermeable condition on the horizontal seabed at

    depth d. Here x and x0 are the field and source points,

    respectively. Use of Greens identity leads to the

    following Fredholm integral equation over the body

    surface Ss

    C(XO)~(XO)Js.G(;;o)b(x)ds

    =-

    G(x, x01 x)ds (1)

    SB

    Here 4(x) is the unknown scattering potential associated

    with a normal velocity V(x) prescribed on Ss. The

    positive direction of the normal to the body surface is

    defined as being out of the fluid. C(Q) has the value

    unity if x0 lies strictly inside the fluid region, and zero if

    x0 is outside the fluid. When x0 is on the body surface,

    47rC is the solid angle over which the fluid is viewed

    from x0 (2?rC at the water line where the body surface

    intersects the free surface).

    In the higher-order boundary element method, the

    body surface can be discretized by Na isoparametric

    elements. After introducing shape functions hk([, r]) in

    each element, we can write the velocity potential and its

    derivatives within an element in terms of nodal values in

    the form

    (2)

    where K is the number of nodes in the element, 4k are

    the nodal potentials, and (

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    New higher-o rder boundary element methods

    73

    diagonal terms of the left-hand side matrix [A].) To

    deal with these difficulties, a number of alternative

    approaches have been developed and applied to the

    wave diffraction problem.

    Eatock Taylor and Chau applied Noblesses contin-

    uous integral equation

    to the higher-order boundary

    element method, and obtained an indirect formulation,

    For later comparison we call this method A. By

    applying Greens theorem inside the body, they obtained

    the following integral equation

    (1- c>+o>

    Is,

    G;;

    ) (x0)ds

    =U

    J

    ,

    G(X>%MXo)ds

    where Sw is the inner water plane, and v = w*/g. Eatock

    Taylor and Chau then combined eqn (5) with eqn (1)

    to obtain the new integral equation

    +

    JJ

    x, x0>

    &

    @(x0) 4(x))ds

    SB

    =--

    J J

    W, o>x>s 6)

    SB

    In this equation, the coefficient C has been eliminated

    and the singular parts in the double-layer integration are

    cancelled. Instead, attention has to be paid to the

    calculation of an integral on the inner water plane.

    For floating bodies, the following alternative integral

    equation has been developed by Wu and Eatock

    Taylor12

    The auxiliary Green function Go, which corresponds to a

    rigid free surface condition, is defined by

    Go(x,

    x0) =

    -&-

    where

    Y =

    [R2+

    (2 -

    zo)2]12,

    l =

    [R+

    (2 +

    zo)2]12,

    and

    R =

    [(x - xo)2 (y - yo)2]12

    (8)

    in Cartesian axes with the z-axis measured vertically

    upwards from the mean free surface. This equation

    avoids the integral on the inner water plane, and the

    singular kernel in the integral on the body surface is still

    cancelled.

    In the case of a body extending to the seabed, one can

    use a Green function Gt, which corresponds to an

    infinite sum of images of the foregoing sources and their

    reflections about the horizontal seabed

    where

    r2,, = [R* + (z - z. - 2nd) ]

    1/2

    r3n = [R2+ (z + zo + 2nd)]j2

    r4 = [R2 + (z - z. + 2nd) ]

    l/2

    r5,, = [R2 (z + z. - 2nd)2] 12

    (11)

    We can then obtain another integral equation as

    follows

    This equation avoids the integration on the inner water

    plane, but the calculation of G1 is time consuming. Since

    eqns (12) and (7) have the same form, we designate the

    solution based on these equations as method B.

    We can avoid the drawbacks in the above methods, by

    using the simple Green function

    G2 = -f

    7r

    f,;+ +

    1

    >

    (13)

    in the integral equation