We present a mathematical model to describe laser beam welding based
on the heat equation. Since the material coeff cients depend on the temperature, this leads to a quasi-linear parabolic partial differential equation. It is our goal to prevent solidif cation cracking. We address this problem by means of optimal control. It is the intensity prof le of the laser beam which acts as the control function. The main challenge is the formulation of a suitable objective function. In particular, high velocities of the solidif cation interface need to be properly penalized in order to deal with and avoid cracking phenomena

Bergmann, Jean Pierre

Bielenin, Martin

Herzog, Roland A.

Hildebrand, Jörg

Riedel, Ilka

Schricker, Klaus

Trunk, Carsten

Worthmann, Karl

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     Technische Universität Ilmenau Institut für Mathematik Preprint No. M 17/09 Prevention of solidification cracking during pulsed laser beam welding  Jean Pierre Bergmann, Martin Bielenin, Roland A. Herzog, Jörg Hildebrand, Ilka Riedel, Klaus Schricker, Carsten Trunk and Karl Worthmann  August 2017  Impressum: Hrsg.:  Leiter des Instituts für Mathematik Weimarer Straße 25 98693 Ilmenau Tel.:  +49 3677 69-3621 Fax:  +49 3677 69-3270 http://www.tu-ilmenau.de/math/ URN: urn:nbn:de:gbv:ilm1-2017200422 Prevention of solidifcation cracking during pulsed laser beam weldingJean Pierre Bergmann, Martin Bielenin, Roland A. Herzog,Jörg Hildebrand, Ilka Riedel, Klaus Schricker,Carsten Trunk, and Karl WorthmannAbstractWe present a mathematical model to describe laser beam welding basedon the heat equation. Since the material coeff cients depend on the temper-ature, this leads to a quasi-linear parabolic partial differential equation. Itis our goal to prevent solidif cation cracking. We address this problem bymeans of optimal control. It is the intensity prof le of the laser beam whichacts as the control function. The main challenge is the formulation of a suit-able objective function. In particular, high velocities of the solidif cation in-terface need to be properly penalized in order to deal with and avoid crackingphenomena.Keywords: laser beam, welding, heat equation, quasi-linear parabolic equation,solidif cation cracking, optimal control1 IntroductionHot cracking is one of the major challenges in laser welding of aluminum alloys ofthe 2XXX, 5XXX and 6XXX series. In comparison to continuous wave laser weld-ing processes, pulsed laser welding has a higher tendency to generate hot cracksbecause the pulsed mode leads to higher cooling rates and thus shorter solidif ca-tion times and higher strain rates [4].A conventional pulsed laser welding beam is simply on or off, i.e., it is de-scribed by a simple rectangular shape of laser beam power vs. time. By contrast,temporal pulse shaping involves varying the laser beam power within the pulseduration [5]. In [5] pulse shaping is investigated and the laser pulses comprisea cooling time in which the laser power is gradually ramped down, see also [6].1Several studies showed that this method reduced hot cracking in pulsed laser weld-ing by lowering the solidif cation rate of the weld pool [7]. Due to the additionalcooling phase of the pulse shape, the welding speed decreases, which reduces theeff ciency of the welding process [1]. It is our goal to design the laser pulse in sucha way that the welding seam solidif es in a crack-free manner while high weld-ing speeds are maintained. To this end we derive, proceeding differently than [8],a mathematical model to determine an optimal pulse form as the solution of anoptimal control problem.2 System Dynamics and ConstraintsLet Ω ⊂ R3 be an open cuboid whose surface ∂Ω is subdivided in the three disjointareas A, B, and ∂Ω\(A ∪ B) as in the f gure. The temperature distribution in thecuboid Ω is described by the heat equationc(T (x, t)) ρ(T (x, t))∂T (x, t)∂t= div (λ(T (x, t))∇T (x, t)) , for x ∈ Ω, t ≥ 0with (temperature dependent) heat capacity c def ned by c(T ) = 715+51 ln(T ) inJkg Kas well as material density ρ in kgm3and thermal conductivity λ in WmKgivenbyρ(T ) =2753 − 0.2T T0 ≤ T < 585◦C,4250 − 2.8T 585 ◦C ≤ T < 650 ◦C,2513 − 0.1T 650 ◦C ≤ T,andλ(T ) =176 + 0.4T T0 ≤ T < 585◦C,10−5 · (1000 − 1.4T ) 585 ◦C ≤ T < 650 ◦C,110 650 ◦C ≤ T.ΩBAThe initial condition is T (x, 0) = T0 = 25 ◦C for all x ∈ Ω. On A ∪ B we have2Robin boundary condition,λ(T (x, t))∂T (x, t)∂n={P (t) on A,k (T 40 − T (x, t)4) + h(T0 − T (x, t)) on B,with laser intensity P in Wm2(to be determined) and coeff cients k = 2.26 · 10−9in Wm2 K4and h = 5 in Wm2 K. On ∂Ω \ (A ∪ B) the Dirichlet boundary conditionT (x, t) = T0 is used.3 Interface VelocityIn contrast to a two-phase Stefan problem (cf., e.g., [3]), we have a temperaturecorridor during which the material solidif es from a liquid to a solid state. Thistransition allows us to eschew the classical Stefan condition in the description ofthe partial differential equation. Instead of an isothermal line as considered in theclassical Stefan problem [2] we consider the boundary surfaceC(t) := {x ∈ A ∪B | 585 ◦ ≤ T (x, t) ≤ 650 ◦}def ned by the temperature corridor of interest. For a given typical laser intensityP : [0, tf ] → [0, pmax], the material is heated up before it starts to cool downafter the melting process. There exists a time instant at which C(t) performs itstransition from the empty set at t = 0 to a two dimensional manifold (which tem-porarily contains a hole) before it contracts to a single point and, f nally, vanishes.Solidif cation cracks typically occur during the cooling phase due to the tempera-ture gradient becoming too large within the surface C(t). We therefore considerthe interface velocity for each isothermal line within the temperature corridor ofinterest,ddtT (x(t), t) = Tx(x(t), t) · x˙(t) + Tt(x(t), t) = 0 (3.1)with x(t) ∈ C(t). We are only interested in the component of the derivative x˙(t)which is perpendicular to the isothermal line. This results inx˙(t) = α(x(t), t)Tx(x(t), t)⊤where α is a scalar function. Hence, x˙(t) is parallel to the spatial derivative ofthe temperature distribution Tx(x(t), t). Plugging this equation into Equation (3.1)yieldsα(x(t), t) · ‖Tx(x(t), t)‖2 + Tt(x(t), t) = 0.3Since we are interested in the absolute value, we def ne v(x(t), t) := ‖x˙(t)‖, whichleads tov(x(t), t) =|Tt(x(t), t)|‖Tx(x(t), t)‖. (3.2)4 Objective FunctionalIn addition to high velocities of thermal isolines near the melt temperature, exces-sive control efforts should be penalized. Moreover, two additional terms are addedto the objective, which essentially ensure that the surface is heated up suff ciently sothat a melting process is triggered and, secondly, take care that the cooling processis over at the time tf . Hence the objective function consists of several summands.Since we are indeed intested in the isoline velocities only during the coolingphase, we propose to use the uni-directional penalty term∫A∪B∫ tf0max{−Tt(x, t)‖Tx(x, t)‖− vmax, 0}2· χ(T (x, t)) dt dx (4.1)in our cost functional. The indicator function χ is def ned asχ(T ) :={1 if 585 ◦ ≤ T ≤ 650 ◦,0 otherwise.The second term in the objective function isβP2∫ tf0P (t)2 dtwith weighting factor βP > 0, which penalizes the control effort. An alternative tothe L2-norm would be to penalize the absolute value, i.e.βP2∫ tf0|P (t)|dt.Since the process is supposed to be completed at time tf we add the followingterms to the objective functional which ensure that both the melting and the coolingprocess have to occur during the time interval [0, tf ]:βf2∫Ωmax {T (x, tf )− 585◦, 0}2 dx+βx02(‖T (x0, ·)‖Lp(0,tf ) − 650◦)2(4.2)with weighting coeff cients βf and βx0 and p≫ 2. The point x0 has to be suitablychosen such that a temperature above the threshold of 650 ◦ at that point impliesthat the melting process is suff ciently triggered. In addition, (4.2) implies thata leisurely preheating of the material is excluded from being an optimal controlstrategy.45 Conclusions and Future WorkA mathematical model considering temperature dependent material properties ispresented with the intention of determining an optimal temporal laser pulse shapeas the solution of an optimal control problem in order to prevent solidif cationcracking during pulsed laser beam welding. Further investigations will deal withthe experimental validation of the mathematical model and the analysis and numer-ical solution of the proposed optimal control problem.References[1] J. P. Bergmann, Schlussbericht des AIF:IGF-Vorhaben IGF 16.260 N, 2014.[2] M.K. Bernauer and R. Herzog, SIAM Journal on Scientif c Computing 33(1),342-363 (2011).[3] S. Gupta, The Classical Stefan Problem. Basic Concepts, Modelling and Anal-ysis (North-Holland, Amsterdam, 2003).[4] S. Katayama, Solidif cation phenomena of weld metals, Solidif cation crack-ing mechanism and cracking susceptibility (3rd report) Welding International15(8), 627-636 (2001).[5] C.G. Tseng and W. F. Savage, The effect of arc oscillation, Welding Journal50(11), 777-786 (1971).[6] J. Zhang, D.C. Weckmann, and Y. Zhou, Effects of temporal pulse shapingon cracking susceptibility of 6061-T6 Aluminum Nd:YAG laser welds, WeldingJournal 87(1), 18-30 (2008).[7] A. Matsunawa, S. Katayama, and Y. Fujita, Laser welding of aluminum al-loys – Defect formation and their suppression methods Proc. 7th Conf. Jointsin Aluminium, (Woodhead Publishing Ltd., Cambridge, 1999), 65-76.[8] V. Petzet, H. J. Pesch, A. Prikhodovsky, and V. Ploshikhin, Different optimiza-tion models for crack-free laser welding. Proc. Appl. Math. Mech. 5(1), 755-756 (2005).5Contact informationJean Pierre BergmannDepartment of Mechanical Engineering, Technische Universität Ilmenau,Postfach 100565, D-98684 Ilmenau, Germanyjeanpierre.bergmann@tu-ilmenau.deMartin BieleninDepartment of Mechanical Engineering, Technische Universität Ilmenau,Postfach 100565, D-98684 Ilmenau, Germanymartin.bielenin@tu-ilmenau.deRoland A. HerzogFaculty of Mathematics, Technische Universität Chemnitz,D-09107 Chemnitz, Germanyroland.herzog@mathematik.tu-chemnitz.deJörg HildebrandDepartment of Mechanical Engineering, Technische Universität Ilmenau,Postfach 100565, D-98684 Ilmenau, Germanyjoerg.hildebrand@tu-ilmenau.deIlka RiedelFaculty of Mathematics, Technische Universität Chemnitz,D-09107 Chemnitz, Germanyilka.riedel@mathematik.tu-chemnitz.deKlaus SchrickerDepartment of Mechanical Engineering, Technische Universität Ilmenau,Postfach 100565, D-98684 Ilmenau, Germanyklaus.schricker@tu-ilmenau.deCarsten TrunkInstitute for Mathematics, Technische Universität Ilmenau,Postfach 100565, D-98684 Ilmenau, Germanycarsten.trunk@tu-ilmenau.deKarl WorthmannInstitute for Mathematics, Technische Universität Ilmenau,Postfach 100565, D-98684 Ilmenau, Germany6karl.worthmann@tu-ilmenau.de7

Prevention of solidification cracking during pulsed laser beam welding

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Prevention of solidification cracking during pulsed laser beam welding

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