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Author: Lasarzik, Robert ×

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    Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models
    (Boston, Mass. : De Gruyter, 2022) Eiter, Thomas; Hopf, Katharina; Lasarzik, Robert
    We study a model for a fluid showing viscoelastic and viscoplastic behavior, which describes the flow in terms of the fluid velocity and a symmetric deviatoric stress tensor. This stress tensor is transported via the Zaremba-Jaumann rate, and it is subject to two dissipation processes: one induced by a nonsmooth convex potential and one by stress diffusion. We show short-time existence of strong solutions as well as their uniqueness in a class of Leray-Hopf-type weak solutions satisfying the tensorial component in the sense of an evolutionary variational inequality. The global-in-time existence of such generalized solutions has been established in a previous work. We further study the limit when stress diffusion vanishes. In this case, the above notion of generalized solutions is no longer suitable, and we introduce the concept of energy-variational solutions, which is based on an inequality for the relative energy. We derive general properties of energy-variational solutions and show their existence by passing to the nondiffusive limit in the relative energy inequality satisfied by generalized solutions for nonzero stress diffusion.
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    Weak-strong uniqueness for the general Ericksen-Leslie system in three dimensions
    (Springfield, Mo. : American Institute of Mathematical Sciences, 2018) Emmrich, Etienne; Lasarzik, Robert
    We study the Ericksen-Leslie system equipped with a quadratic free energy functional. The norm restriction of the director is incorporated by a standard relaxation technique using a double-well potential. We use the relative energy concept, often applied in the context of compressible Euler- or related systems of fluid dynamics, to prove weak-strong uniqueness of solutions. A main novelty, not only in the context of the Ericksen-Leslie model, is that the relative energy inequality is proved for a system with a nonconvex energy.
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    Modelling and simulation of flame cutting for steel plates with solid phases and melting
    (Berlin ; Heidelberg : Springer, 2020) Arenas, Manuel J.; Hömberg, Dietmar; Lasarzik, Robert; Mikkonen, Pertti; Petzold, Thomas
    The goal of this work is to describe in detail a quasi-stationary state model which can be used to deeply understand the distribution of the heat in a steel plate and the changes in the solid phases of the steel and into liquid phase during the flame cutting process. We use a 3D-model similar to previous works from Thiébaud (J. Mater. Process. Technol. 214(2):304–310, 2014) and expand it to consider phases changes, in particular, austenite formation and melting of material. Experimental data is used to validate the model and study its capabilities. Parameters defining the shape of the volumetric heat source and the power density are calibrated to achieve good agreement with temperature measurements. Similarities and differences with other models from literature are discussed. © 2020, The Author(s).
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    Topology optimization subject to additive manufacturing constraints
    (Berlin ; Heidelberg : Springer, 2021) Ebeling-Rump, Moritz; Hömberg, Dietmar; Lasarzik, Robert; Petzold, Thomas
    In topology optimization the goal is to find the ideal material distribution in a domain subject to external forces. The structure is optimal if it has the highest possible stiffness. A volume constraint ensures filigree structures, which are regulated via a Ginzburg–Landau term. During 3D printing overhangs lead to instabilities. As a remedy an additive manufacturing constraint is added to the cost functional. First order optimality conditions are derived using a formal Lagrangian approach. With an Allen-Cahn interface propagation the optimization problem is solved iteratively. At a low computational cost the additive manufacturing constraint brings about support structures, which can be fine tuned according to demands and increase stability during the printing process.
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    Existence of weak solutions to a dynamic model for smectic-A liquid crystals under undulations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Emmrich, Etienne; Lasarzik, Robert
    A nonlinear model due to Soddemann et al. [37] and Stewart [38] describing incompressible smectic-A liquid crystals under flow is studied. In comparison to previously considered models, this particular model takes into account possible undulations of the layers away from equilibrium, which has been observed in experiments. The emerging decoupling of the director and the layer normal is incorporated by an additional evolution equation for the director. Global existence of weak solutions to this model is proved via a Galerkin approximation with eigenfunctions of the associated linear differential operators in the three-dimensional case.
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    Measure-valued solutions to the Ericksen-Leslie model equipped with the Oseen-Frank energy
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Lasarzik, Robert
    In this article, we prove the existence of measure-valued solutions to the EricksenLeslie system equipped with the OseenFrank energy. We introduce the concept of generalized gradient Young measures. Via a Galerkin approximation, we show the existence of weak solutions to a regularized system and attain measure-valued solutions for vanishing regularization. Additionally, it is shown that the measure-valued solution fulfills an energy inequality.
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    On the existence of generalized solutions to a spatio-temporal predator-prey system
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Hömberg, Dietmar; Lasarzik, Robert; Plato, Luisa
    In this paper we consider a pair of coupled non-linear partial differential equations describing the interaction of a predator-prey pair. We introduce a concept of generalized solutions and show the existence of such solutions in all space dimension with the aid of a regularizing term, that is motivated by overcrowding phenomena. Additionally, we prove the weak-strong uniqueness of these generalized solutions and the existence of strong solutions at least locally-in-time for space dimension two and three.
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    Weak-strong uniqueness for measure-valued solutions to the Ericksen-Leslie model equipped with the Oseen-Frank free energy
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Lasarzik, Robert
    We analyze the EricksenLeslie system equipped with the OseenFrank energy in three space dimensions. Recently, the author introduced the concept of measure-valued solutions to this system and showed the global existence of these generalized solutions. In this paper, we show that suitable measure-valued solutions, which fulfill an associated energy inequality, enjoy the weak-strong uniqueness property, i. e. the measure-valued solution agrees with a strong solution if the latter exists. The weak-strong uniqueness is shown by a relative energy inequality for the associated nonconvex energy functional.
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    Approximation and optimal control of dissipative solutions to the Ericksen-Leslie system
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Lasarzik, Robert
    We analyze the EricksenLeslie system equipped with the OseenFrank energy in three space dimensions. Recently, the author introduced the concept of dissipative solutions. These solutions show several advantages in comparison to the earlier introduced measure-valued solutions. In this article, we argue that dissipative solutions can be numerically approximated by a relative simple scheme, which fulfills the norm-restriction on the director in every step. We introduce a semi-discrete scheme and derive an approximated version of the relative-energy inequality for solutions of this scheme. Passing to the limit in the semi-discretization, we attain dissipative solutions. Additionally, we introduce an optimal control scheme, show the existence of an optimal control and a possible approximation strategy. We prove that the cost functional is lower semi-continuous with respect to the convergence of this approximation and argue that an optimal control is attained in the case that there exists a solution admitting additional regularity.
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    Dissipative solution to the Ericksen-Leslie system equipped with the Oseen-Frank energy
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Lasarzik, Robert
    We analyze the EricksenLeslie system equipped with the OseenFrank energy in three space dimensions. The new concept of dissipative solutions is introduced. Recently, the author introduced the concept of measure-valued solutions to the considered system and showed global existence as well as weak-strong uniqueness of these generalized solutions. In this paper, we show that the expectation of the measure valued solution is a dissipative solution. The concept of a dissipative solution itself relies on an inequality instead of an equality, but is described by functions instead of parametrized measures. These solutions exist globally and fulfill the weak-strong uniqueness property. Additionally, we generalize the relative energy inequality to solutions fulfilling different nonhomogeneous Dirichlet boundary conditions and incorporate the influence of a temporarily constant electromagnetic field. Relying on this generalized energy inequality, we investigate the long-time behavior and show that all solutions converge for the large time limit to a certain steady state.