## Implementation of prescribed temperature model

The welding path is assumed as any valid geometric curve in space on which a local coordinate system moves. From the local coordinate system, a tangential vector and principal normal can be computed at any point on the curve.

Typically, a point is calculated according to the velocity and the time as the leading weld pool travels on the path. Such a point can serve as the centre of the leading weld pool. The locations of other weld pools on the path are specified relative to the centre of the leading pool.

At the centre of each weld pool, an outward normal is calculated. This can be computed as the average of the outward normal of all free surfaces of the elements containing the centre.

For each weld pool, unit vectors for the outward normal and tangential direction define a local coordinate system. The distribution function in a weld pool has a standard form, e. g., a double ellipsoid. The local coordinate system consists of (x1, y1 ,z') in this coordinate system. The transformation matrix from global to local coordinates is:

 1 _ J і і л у z 1 У = y y y 1 i 1 N . N N I
 [Я] =
 (3-8)

To identify the small subset of nodes in the vicinity of the weld pool from the total set of nodes in the mesh, which usually are enormous, a KDTree provides an 0(log N) algorithm.

 (3-9)

A transformation from global to local coordinates can be done on the set of nodes selected by the KDTree.

 Xі x-x0 У = [H] У-Уо 1 z _z~z0_

where x, у and z are global coordinates and xn, yo and z0 define the

origin of a local coordinate system (x, yl, zl).

For any distribution function, including a double ellipsoid, all nodes can then be tested to determine if they are in the domain of the

distribution function. For example, a node is in the front part of an ellipsoid if and only if:

 x >0 (x1)2 , (У)2 , (z1)2

' (3-Ю)

and

 <1.0

(3-11)

 (Xі )2 | (У)2 | (z1)2
 (3-12)

If this is true, the node is prescribed with a temperature: T = T0 expl A

a,

where To is the temperature at the origin, which has the maximum temperature, and A is a constant which has to be evaluated so that T=Tm, the melting point temperature at the boundary of FZ and HAZ.

If:

(x1)2 j (У)2 , (z1)2

 (3-13) (3-14) (3-15)

= 1.0

b2

and

Tm - Tq Є A = In (Tm/T0) Substituting A into equation (3-12);

 T = Tn

ґт

M

T

J

The nodes inside any distribution function can be identified and assigned temperatures in this way.