The discrete points at the front, rear and side keyhole wall computed by (A.12) can be used to construct a level set function whose zero level set represents the whole keyhole geometry. Let xfi = (xfi,0, zi), xri = (xri,0, zi) and xsi1 = (xsi, yis, zi) = (xHSi , yis, zi) resp. xsi2 = (xsi,−ysi, zi) = (xHSi ,−yis, zi),i= 1, . . . , M, be the coordinates of the points at the front, rear and side keyhole wall. For every depth layer zi, we approximate the keyhole shape by two half-ellipses using the front and rear keyhole wall points as semi-major axes and the side keyhole wall points as semi-minor axes, see FigureA.4.
x y
(xHS,0)
(xr,0) = (ˆxr−xHS,0) (xf,0) = (ˆxf −xHS,0) (xs, ys) = (xHS, ys)
(xs,−ys) = (xHS,−ys)
Figure A.4 Keyhole wall approximated at a fixed depth and with a heat source location (xHS,0) in both coordinate systems.
By doing this, the half-ellipse connectingxfi,xsi1 and xsi2 is given by y2 = (yis)2−(ysi)2
(xfi)2(x−xsi)2. (A.15) and the half-ellipse connectingxri,xsi1 and xsi2 is given by
y2 = (yis)2−(ysi)2
(xri)2(x−xsi)2. (A.16) Based on this equations, we introduce the level set functions
φ1|zi(x) =φ(x, y, zi)
=
⎧
⎪⎪
⎨
⎪⎪
⎩
…
(y2−(ysi)2−(yis)2
(xfi)2(x−xsi)2), forx≥xHS
…
(y2−(ysi)2−(y(xsir)2
i)2(x−xsi)2), forx < xHS
, (A.17)
which are signed distance functions whose zero level sets Γ1|zi represents the (2D) keyhole wall for each depth layer zi, i= 1, . . . , M in thex-y plane. In the vertical direction, the different ellipses (or equivalently the level set functions) are linearly interpolated between the z-layers
to obtain a continuous function φ1 whose zero level set Γ1 defines the 3D keyhole geometry.
Thereby, φ1 has to be extended for z < zM in an arbitrarily but continuous way. For problem (5.88), the keyhole geometry is considered as constant and moves along the welding line at the given welding speed V⃗L. Hence, our sharp interface between keyhole and molten area is given by
Γ1(t) ={x∈Ω : (x−(t−t0)V⃗L)∈Γ1(t0), t∈[t0, tf]}. (A.18) Remark A.2. While in a first step, the discrete keyhole Γh is computed using a very small step size ∆z for a high precision approximation of the keyhole shape and depth, the level set functionφ1 and the corresponding zero level set Γ1 are constructed using only a set of keyhole wall coordinates which contains only about 10% of the previously computed points including the first and the last one. This does not represent any remarkable reduction in precision, as the interfaces are in the end linearly interpolated during the XFEM enrichment of elements.
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