tion on the hull (body surface) is that no water flows through the hull. The usual approach in boundary element methods discretises the hull into a number of elements (panels). The boundary condition is then exactly enforced at one point, the collocation point, located approximately at the panel center.

In the ‘patch' method, on the other hand, the total flow through each surface element (patch), and not just at its center, is made to vanish. Using sources distributed over plane or curved panels would lead to complicated integrations; therefore in the patch method simple point sources are used. They are located within the body near to the patch centres. The distance between patch centre and source point may be chosen as the minimum of the following lengths:

Square root of patch area;

1/3 of the local body breadth;

1/2 the radius of longitudinal curvature;

1/2 the radius of transverse curvature.

The results are not sensitive to this distance; in many applications simply 1/10 of the patch length is used.

In the panel method, velocity and pressure can be determined on the hull directly only at the panel centres; at other points, interpolation has to be used. Pressure forces are, typically, determined by multiplying the pressure at the panel centre with the panel area. The patch method aims just to improve this force formula. In the patch method, potential and velocity are determined at the patch corners instead of at the patch centre, i.e. at a reasonable distance from all point sources. The potential at the patch corners allows a better approximation of the *average* velocity within the patch than the value at the panel centre, and combining the potential and the velocity at the patch corners allows to determine an accurate *average* of the pressure within the patch.

For a body in uniform flow to negative *x* direction, the potential is

(1)

*U* is the speed of the uniform flow, *σ* the source strength, *G* the potential of a Rankine point source:

*G*=ln *r* in 2D, and *G*=–1/*r* in 3D, (2)

where is the distance between field point and source point

Let *M*_{i} be the outflow through a patch induced by a point source of unit strength. Then the zero-flow condition for a patch is

(3)

Here is the outward normal on the hull, index *x* (and later *z*) designates the respective component of and *An*_{x} is the projection of the patch area on a plane *x*=constant (with appropriate sign); for the 2d case of Fig.1: *An*_{x}=*y*_{A}*–y*_{B}.

The outflow due to the unit source potential In *r* into all directions is 2*π*. The outflow *M* due to the unit source in passing through the patch in Fig.1 is thus equal to the angle *γ* under which the patch is seen from —both for a straight and a curved patch. If and are the vectors from *S* to *A* and *B* respectively, *γ* is easily determined from the vector and scalar products of and :

(4)

From the value of the potential at the end points *A* and *B,* the average modulus of the velocity is found as |_{A}*–*_{B}|/*l,* where *l* ist the length of the patch. The direction of is parallel to the contour. The velocity at the end points, designated here as is found as .

The pressure force on a straight patch is

(5)

where *v,* the modulus of is not constant. To evaluate this expression, *v* is approximated by the second-order polynomial giving the known values *v*_{A}, *v*_{B} and :

*v=v*_{A}+(6*–*4*v*_{A}–2*v*_{B})*t*+3*(v*_{A}+*v*_{B}–2*)t*^{2} (6)

*t* is the tangential coordinate directed from *A* to *B*. From this expression follows the integral in (5):

(7)