Browsing by Author "Rocca, Elisabetta"
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- ItemAdditive manufacturing graded-material design based on phase-field and topology optimization(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Carraturo, Massimo; Rocca, Elisabetta; Bonetti, Elena; Hömberg, Dietmar; Reali, Alessandro; Auricchio, FerdinandoIn the present work we introduce a novel graded-material design for additive manufacturing based on phase-field and topology optimization. The main novelty of this work comes from the introduction of an additional phase-field variable in the classical single-material phase-field topology optimization algorithm. This new variable is used to grade the material properties in a continuous fashion. Two different numerical examples are discussed, in both of them we perform sensitivity studies to asses the effects of different model parameters onto the resulting structure. From the presented results we can observe that the proposed algorithm adds additional freedom in the design, exploiting the higher flexibility coming from additive manufacturing technology.
- ItemAnalysis and simulation of multifrequency induction hardening(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Hömberg, Dietmar; Petzold, Thomas; Rocca, ElisabettaWe study a model for induction hardening of steel. The related differential system consists of a time domain vector potential formulation of the Maxwells equations coupled with an internal energy balance and an ODE for the volume fraction of austenite, the high temperature phase in steel. We first solve the initial boundary value problem associated by means of a Schauder fixed point argument coupled with suitable a-priori estimates and regularity results. Moreover, we prove a stability estimate entailing, in particular, uniqueness of solutions for our Cauchy problem. We conclude with some finite element simulations for the coupled system.
- ItemAnalysis of a diffuse interface model of multispecies tumor growth(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Dai, Mimi; Feireisl, Eduard; Rocca, Elisabetta; Schimperna, GiulioWe consider a diffuse interface model for tumor growth recently proposed in [3]. In this new approach sharp interfaces are replaced by narrow transition layers arising due to adhesive forces among the cell species. Hence, a continuum thermodynamically consistent model is introduced. The resulting PDE system couples four different types of equations: a Cahn-Hilliard type equation for the tumor cells (which include proliferating and dead cells), a Darcy law for the tissue velocity field, whose divergence may be different from 0 and depend on the other variables, a transport equation for the proliferating (viable) tumor cells, and a quasi-static reaction diffusion equation for the nutrient concentration. We establish existence of weak solutions for the PDE system coupled with suitable initial and boundary conditions. In particular, the proliferation function at the boundary is supposed to be nonnegative on the set where the velocity u satisfies u ypsilon 0, where ypsilon is the outer normal to the boundary of the domain. We also study a singular limit as the diffuse interface coefficient tends to zero.
- ItemAnalysis of a tumor model as a multicomponent deformable porous medium(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Krejčí, Pavel; Rocca, Elisabetta; Sprekels, JürgenWe propose a diffuse interface model to describe tumor as a multicomponent deformable porous medium. We include mechanical effects in the model by coupling the mass balance equations for the tumor species and the nutrient dynamics to a mechanical equilibrium equation with phase-dependent elasticity coefficients. The resulting PDE system couples two Cahn--Hilliard type equations for the tumor phase and the healthy phase with a PDE linking the evolution of the interstitial fluid to the pressure of the system, a reaction-diffusion type equation for the nutrient proportion, and a quasistatic momentum balance. We prove here that the corresponding initial-boundary value problem has a solution in appropriate function spaces.
- ItemAsymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modelling tumor growth(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Colli, Pierluigi; Gilardi, Gianni; Rocca, Elisabetta; Sprekels, JürgenThis paper is concerned with a phase field system of Cahn-Hilliard type that is related to a tumor growth model and consists of three equations in gianni terms of the variables order parameter, chemical potential and nutrient concentration. This system has been investigated in the recent papers citeCGH and citeCGRS gianni from the viewpoint of well-posedness, long time bhv and asymptotic convergence as two positive viscosity coefficients tend to zero at the same time. Here, we continue the analysis performed in citeCGRS by showing two independent sets of results as just one of the coefficents tends to zero, the other remaining fixed. We prove convergence results, uniqueness of solutions to the two resulting limit problems, and suitable error estimates
- ItemChallenges in Optimal Control of Nonlinear PDE-Systems(Zürich : EMS Publ. House, 2018) Kunisch, Karl; Leugering, Günter; Rocca, ElisabettaThe workshop focussed on various aspects of optimal control problems for systems of nonlinear partial differential equations. In particular, discussions around keynote presentations in the areas of optimal control of nonlinear/non-smooth systems, optimal control of systems involving nonlocal operators, shape and topology optimization, feedback control and stabilization, sparse control, and associated numerical analysis as well as design and analysis of solution algorithms were promoted. Moreover, also aspects of control of fluid structure interaction problems as well as problems arising in the optimal control of quantum systems were considered.
- ItemChallenges in Optimization with Complex PDE-Systems (hybrid meeting)(Zürich : EMS Publ. House, 2021) Kunisch, Karl; Leugering, Günter; Rocca, ElisabettaThe workshop concentrated on various aspects of optimization problems with systems of nonlinear partial differential equations (PDEs) or variational inequalities (VIs) as constraints. In particular, discussions around several keynote presentations in the areas of optimal control of nonlinear or non-smooth systems, optimization problems with functional and discrete or switching variables leading to mixed integer nonlinear PDE constrained optimization, shape and topology optimization, feedback control and stabilization, multi-criteria problems and multiple optimization problems with equilibrium constraints as well as versions of these problems under uncertainty or stochastic influences, and the respectively associated numerical analysis as well as design and analysis of solution algorithms were promoted. Moreover, aspects of optimal control of data-driven PDE constraints (e.g. related to machine learning) were addressed.
- ItemDamage processes in thermoviscoelastic materials with damage-dependent thermal expansion coefficients(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Heinemann, Christian; Rocca, ElisabettaIn this paper we prove existence of global in time weak solutions for a highly nonlinear PDE system arising in the context of damage phenomena in thermoviscoelastic materials. The main novelty of the present contribution with respect to the ones already present in the literature consists in the possibility of taking into account a damage-dependent thermal expansion coefficient. This term implies the presence of nonlinear couplings in the PDE system, which make the analysis more challenging.
- ItemA diffuse interface model for two-phase incompressible flows with nonlocal interactions and nonconstant mobility(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Frigeri, Sergio; Grasselli, Maurizio; Rocca, ElisabettaWe consider a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids with matched constant densities. This model consists of the Navier-Stokes system coupled with a convective nonlocal Cahn-Hilliard equation with non-constant mobility. We first prove the existence of a global weak solution in the case of non-degenerate mobilities and regular potentials of polynomial growth. Then we extend the result to degenerate mobilities and singular (e.g. logarithmic) potentials. In the latter case we also establish the existence of the global attractor in dimension two. Using a similar technique, we show that there is a global attractor for the convective nonlocal Cahn-Hilliard equation with degenerate mobility and singular potential in dimension three.
- ItemEntropic solutions to a thermodynamically consistent PDE system for phase transitions and damage(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Rocca, Elisabetta; Rossi, RiccardaIn this paper we analyze a PDE system modelling (non-isothermal) phase transitions and damage phenomena in thermoviscoelastic materials. The model is thermodynamically consistent: in particular, no small perturbation assumption is adopted, which results in the presence of quadratic terms on the right-hand side of the temperature equation, only estimated in L1. The whole system has a highly nonlinear character. We address the existence of a weak notion of solution, referred to as entropic, where the temperature equation is formulated with the aid of an entropy inequality, and of a total energy inequality. This solvability concept reflects the basic principles of thermomechanics as well as the thermodynamical consistency of the model. It allows us to obtain global-in-time existence theorems without imposing any restriction on the size of the initial data. We prove our results by passing to the limit in a time discretization scheme, carefully tailored to the nonlinear features of the PDE system (with its entropic formulation), and of the a priori estimates performed on it. Our time-discrete analysis could be useful towards the numerical study of this model.
- ItemExistence of solutions to a two-dimensional model for nonisothermal two-phase flows of incompressible fluids(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Eleuteri, Michela; Rocca, Elisabetta; Schimperna, GiulioWe consider a thermodynamically consistent diffuse interface model describing two-phase flows of incompressible fluids in a non-isothermal setting. The model was recently introduced in [12] where existence of weak solutions was proved in three space dimensions. Here, we aim at studying the properties of solutions in the two-dimensional case. In particular, we can show existence of global in time solutions satisfying a stronger formulation of the model with respect to the one considered in [12]. Moreover, we can admit slightly more general conditions on some material coefficients of the system.
- ItemGeneralized gradient flow structure of internal energy driven phase field systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Bonetti, Elena; Rocca, ElisabettaIn this paper we introduce a general abstract formulation of a variational thermomechanical model, by means of a unified derivation via a generalization of the principle of virtual powers for all the variables of the system, including the thermal one. In particular, choosing as thermal variable the entropy of the system, and as driving functional the internal energy, we get a gradient flow structure (in a suitable abstract setting) for the whole nonlinear PDE system. We prove a global in time existence of (weak) solutions result for the Cauchy problem associated to the abstract PDE system as well as uniqueness in case of suitable smoothness assumptions on the functionals.
- ItemGlobal strong solutions of the full Navier-Stokes and Q-tensor system for nematic liquid crystal flows in 2D: Existence and long-time behavior(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Cavaterra, Cecilia; Rocca, Elisabetta; Wu, Hao; Xu, XiangWe consider a full Navier-Stokes and Q-tensor system for incompressible liquid crystal flows of nematic type. In the two dimensional periodic case, we prove the existence and uniqueness of global strong solutions that are uniformly bounded in time. This result is obtained without any smallness assumption on the physical parameter xi that measures the ratio between tumbling and aligning effects of a shear flow exerting over the liquid crystal directors. Moreover, we show the uniqueness of asymptotic limit for each global strong solution as time goes to infinity and provide an uniform estimate on the convergence rate.
- ItemA model for resistance welding including phase transitions and Joule heating(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Hömberg, Dietmar; Rocca, ElisabettaIn this paper we introduce a new model for solid-liquid phase transitions triggered by Joule heating as they arise in the case of resistance welding of metal parts. The main novelties of the paper are the coupling of the thermistor problem with a phase field model and the consideration of phase dependent physical parameters through a mixture ansatz. The PDE system resulting from our modelling approach couples a strongly nonlinear heat equation, a non-smooth equation for the the phase parameter (standing for the local proportion of one of the two phases) with quasistatic electric charge conservation law. We prove existence of weak solutions in the 3D case, while the regularity result and the uniqueness of solution is stated only in the 2D case. Indeed, uniqueness for the three dimensional system is still an open problem.
- ItemNonisothermal nematic liquid crystal flows with the Ball-Majumdar free energy(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Feireisl, Eduard; Rocca, Elisabetta; Schimperna, Giulio; Zarnescu, ArghirIn this paper we prove the existence of global in time weak solutions for an evolutionary PDE system modelling nonisothermal Landau-de Gennes nematic liquid crystal (LC) flows in three dimensions of space. In our model, the incompressible Navier-Stokes system for the macroscopic velocity u is coupled to a nonlinear convective parabolic equation describing the evolution of the Q-tensor Q, namely a tensor-valued variable representing the normalized second order moments of the probability distribution function of the LC molecules. The effects of the (absolute) temperature theta are prescribed in the form of an energy balance identity complemented with a global entropy production inequality. Compared to previous contributions, we can consider here the physically realistic singular configuration potential f introduced by Ball and Majumdar. This potential gives rise to severe mathematical difficulties since it introduces, in the Q-tensor equation, a term which is at the same time singular in Q and degenerate in theta. To treat it a careful analysis of the properties of f, particularly of its blow-up rate, is carried out.
- ItemA nonlocal phase-field model with nonconstant specific heat(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Krejčí, Pavel; Rocca, Elisabetta; Sprekels, JürgenWe prove the existence, uniqueness, thermodynamic consistency, global boundedness from both above and below, and continuous data dependence for a strong solution to an integrodifferential model for nonisothermal phase transitions under nonhomogeneous mixed boundary conditions. The specific heat is allowed to depend on the order parameter, and the convex component of the free energy may or may not be singular.
- ItemA nonlocal quasilinear multi-phase system with nonconstant specific heat and heat conductivity(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Colli, Pierluigi; Krejˇcí, Pavel; Rocca, Elisabetta; Sprekels, JürgenIn this paper, we prove the existence and global boundedness from above for a solution to an integrodifferential model for nonisothermal multi-phase transitions under nonhomogeneous third type boundary conditions. The system couples a quasilinear internal energy balance ruling the evolution of the absolute temperature with a vectorial integro-differential inclusion governing the (vectorial) phase-parameter dynamics. The specific heat and the heat conductivity $k$ are allowed to depend both on the order parameter $chi$ and on the absolute temperature $theta$ of the system, and the convex component of the free energy may or may not be singular. Uniqueness and continuous data dependence are also proved under additional assumptions.
- ItemOn a diffuse interface model of tumor growth(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Frigeri, Sergio; Grasselli, Maurizio; Rocca, ElisabettaWe consider a diffuse interface model of tumor growth proposed by A. Hawkins-Daruud et al. This model consists of the Cahn-Hilliard equation for the tumor cell fraction φ nonlinearly coupled with a reaction-diffusion equation for ψ which represents the nutrient-rich extracellular water volume fraction. The coupling is expressed through a suitable proliferation functionp(φ) multiplied by the differences of the chemical potentials for φ and ψ. The system is equipped with no-flux boundary conditions which entails the conservation of the total mass, that is, the spatial average of φ+ψ. Here we prove the existence of a weak solution to the associated Cauchy problem, provided that the potential F and p satisfy sufficiently general conditions. Then we show that the weak solution is unique and continuously depends on the initial data, provided that p satisfies slightly stronger growth restrictions. Also, we demonstrate the existence of a strong solution and that any weak solution regularizes in finite time. Finally, we prove the existence of the global attractor in a phase space characterized by an a priori bounded energy.
- ItemOn a non-isothermal diffuse interface model for two-phase flows of incompressible fluids(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Eleuteri, Michaela; Rocca, Elisabetta; Schimperna, GiulioWe introduce a diffuse interface model describing the evolution of a mixture of two different viscous incompressible fluids of equal density. The main novelty of the present contribution consists in the fact that the effects of temperature on the flow are taken into account. In the mathematical model, the evolution of the velocity u is ruled by the Navier-Stokes system with temperaturedependent viscosity, while the order parameter Phi representing the concentration of one of the components of the fluid is assumed to satisfy a convective Cahn-Hilliard equation. The effects of the temperature are prescribed by a suitable form of the heat equation. However, due to quadratic forcing terms, this equation is replaced, in the weak formulation, by an equality representing energy conservation complemented with a differential inequality describing production of entropy. The main advantage of introducing this notion of solution is that, while the thermodynamical consistency is preserved, at the same time the energy-entropy formulation is more tractable mathematically. Indeed, global-in-time existence for the initial-boundary value problem associated to the weak formulation of the model is proved by deriving suitable a-priori estimates and showing weak sequential stability of families of approximating solutions.
- ItemOn a nonlocal Cahn-Hilliard equation with a reaction term(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Melchionna, Stefano; Rocca, ElisabettaWe prove existence, uniqueness, regularity and separation properties for a nonlocal Cahn- Hilliard equation with a reaction term. We deal here with the case of logarithmic potential and degenerate mobility as well an uniformly lipschitz in u reaction term g(x, t, u).