Arenas, Manuel J.

Lasarzik, Robert

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Relative energy approach withapplication to optimal control offlame cutting of steel platesWIAS-Days 2019 Manuel J. ArenasRobert LasarzikResearch Group 4Nonlinear Optimization andInverse ProblemsHead:Prof. Dr. Dietmar Hömberg4The relative energy approach is a natural tool to compare different solutions to systems of cou-pled nonlinear evolution equations. It may be used to quantify the error of a simplifying modelingassumption, which is often crucial to perform reasonable numerical simulations. Such numeri-cal simulations proved to be an important tool, e.g., to predict the outcome of heat treatmentsof steel parts. Flame cutting as such a part of the production-process in steel manufacturingneeds to be designed in an optimal way to gain a work piece with the desired properties.Relative energy inequalityTwo solutions u, ũ are compared in terms of the relative energy E and relative dissipation WUse the convex structure of the energy, i.e., for a convex function η : Rd→[0,∞)it holds E (u|ũ) = η(u)−η(ũ)−η ′(ũ)(u− ũ)≥ 0Flexible tool to deduce several properties of associated solutionsE (u|ũ)(t)+∫ t0W (u|ũ)ds≤ E (u|ũ)(0)+∫ t0K (ũ)E (u|ũ)ds+∫ t0A (ũ)(u− ũ)dsrelative dissipation W , measure of regularity K , associated operator AWeak-strong uniqueness (see [1], [3])Continuous dependence on data(see [1], [3])A posteori estimates for approximations(see [2])Singular limitsLong-time behavior (see [1])Dissipative solutions (see [1])Optimal control (see [2])GRONWALL’s inequality allows to derive estimatesApplicable to very general solution concepts like entropic or dissipative solutionsApplication to induction hardening processε∂tθ +(v ·∇)θ −∇ · (κ(θ)∇θ) = σ(θ)|∂tA|2 , n ·∇θ = 0σ(θ)(∂tA−∇φ)+∇×(1µ∇×A)= 0 , n×A = 0Existence of weak entropic solutions for this model fulfilling an relative energy inequalityproviding weak-strong uniqueness of solutions (see [3])Relative energy:E (θ ,A|θ̃ ,Ã) =∫Ωε(θ − θ̃ − θ̃(logθ − log θ̃))dx+12µ∥∥∇×A−∇×Ã∥∥2L2Approximate system: Simplification of Maxwell’s potential formulation in the frequency domainand quasi stationary approximation of the energy balance (entropy production rate)(v ·∇)θ̃ −∇ · (κ(θ̃)∇θ̃) = ω∫ ·+ω/2·−ω/2|Ã|2 ds , n ·∇θ̃ = 0σ(θ̃)(iωÃ−∇φ)+∇×(1µ∇×Ã)= 0 , n× Ã = 0Derive error estimate depending on frequency ω under the condition that tω2 ≤CεE (θ ,A|θ̃ ,Ã)(t)+∫ t0W (θ ,A|θ̃ ,Ã)e∫ ts K (ũ)dτ ds≤ E (θ ,A|θ̃ ,Ã)(0)e∫ t0 K (ũ)ds+ c((ω2+1) log(ω2+1)+ω2)The error is small for high frequencies, i.e., ω→0 and possible small constants εQuasi-stationary model suitablefor optimal controlAverage of JOULE heat preventsundesired oscillationsSufficiently high frequenciesguarantee a suitableapproximationInduction preheating for flame cuttingFlame cutting causes a heat affected zone (HAZ)around the cutting edgeCold cracks in the HAZ can appear due to tensilestress, hydrogen content and microstructurechangesUltra-high strength steel thick plates require apreheating stage to avoid cracksLocal induction preheating is a novel and efficientmethod to be developedNumerical simulation of flame cuttingQuasi stationary 3D model for the temperature (θ ) andthe molten steel phase (z)FEM is used within pdelib2 to solve the systemThe full convection phase equation requires astabilization method like SUPG(v ·∇)z = f (z,θ)Numerical simulation of induction preheatingSkin effect hinders achieving uniform 300◦Cin thickness along the cutting line with onlyone coil on topHeat conduction spreads the temperaturefrom top to bottomOptimal controlρCp (θ)(v ·∇)θ −∇ · (κ (θ)∇θ) = QF−ρL · f (θ ,z)+u(v ·∇)z = f (z,θ)The Joule heat is substituted by the control term u that is parametrized representation that canbe associated with a current frequency and intensityThe heat equation is coupled with a set of quasi stationary rate laws for the solid-solid andsolid-liquid transformations of steel during flame cuttingThe functional J determines the difference between the temperature field and a desired oneand the temperature gradients in the direction rJ(θ ,u) =12∫Ω(θ −θd(x))2 dx+α12∫Ω|∇θ ·r|2 dx+α22∫Ωu2 dxSummary and referencesRelative energy inequalities have been applied to:ERICKSEN–LESLIE model describing liquid crystals (highly nonlinear coupled system, whichevolution is driven by an nonconvex energy functional on a nonoriantable manifold)Induction Hardening (multi-field, multi-scale system)[1] R. Lasarzik. Dissipative solution to the Ericksen–Leslie system equipped with the Oseen–Frank energy.Z. Angew. Math. Phys., 70:8, 2019[2] R. Lasarzik. Approximation and optimal control of dissipative solutions to the Ericksen–Leslie system.WIAS-Preprint 2535, 2018.[3] D. Hömberg, R. Lasarzik, T. Petzold. Weak entropy solutions to a model in induction hardening, existence and weak-strong uniquenessin preparation[4] M.J. Arenas, D. Hömberg, T. Petzold, P. Mikkonen. Modelling and Simulation of Flame Cutting and Metallurgy for Steel Platesin preparationOutlookRelativeenergiesRelative energy as a general toolEstimate under what condition a simplification is reasonable,compare different approximationsModel hierarchies (adapt model to special situation)FlamecuttingPhenomenological definition of the heat source for flame cuttingNumerics and analysis of optimal control problemGoal: find suitable distance between induction coil and torchCorrelate optimal dissipated power to current intensity and frequencyStudy solutions from optimal control for different values for velocityContactManuel J. ArenasRobert LasarzikE robert.lasarzik@wias-berlin.de WIASMohrenstr. 3910117 BerlinFunding Cooperation

Relative energy approach with application to optimal control of flame cutting of steel plates

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