35K86 Nonlinear parabolic unilateral problems and nonlinear parabolic variational inequalities [See also 35R35, 49J40]
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- English (4)
Keywords
- C-stationarity (1)
- Cahn-Hilliard reaction system (1)
- Cahn-Hilliard system (1)
- Yosida regularization (1)
- arclenght reparameterization (1)
- distributied optimal control (1)
- double-obstacle potential (1)
- doubly non-linear differential inclusion (1)
- existence (1)
- mathematical programming with equilibrium constraints (MPEC) (1)
Application Area
- C (4)
In this paper, we introduce and study analytically a vectorial Cahn-Hilliard reaction model coupled with rate-dependent damage processes. The recently proposed Cahn-Hilliard reaction model can e.g. be used to describe the behavior of electrodes of lithium-ion batteries as it includes both the intercalation reactions at the surfaces and the separation into different phases. The coupling with the damage process allows considering simultaneously the evolution of a damage field, a second important physical effect occurring during the charging or discharging of lithium-ion batteries.
Mathematically, this is realized by a Cahn-Larché system with a non-linear Newton boundary condition for the chemical potential and a doubly non-linear differential inclusion for the damage evolution. We show that this system possesses an underlying generalized gradient structure which incorporates the non-linear Newton boundary condition. Using this gradient structure and techniques from the field of convex analysis we are able to prove constructively the existence of weak solutions of the
coupled PDE system.
We consider a non-isothermal multi-phase field model.
We subsequently discretize implicitly in time and with
linear finite elements. The arising algebraic problem is
formulated in two variables where one is the multi-phase
field, and the other contains the inverse temperature field.
We solve this saddle point problem numerically by a
non-smooth Schur-Newton approach using truncated
non-smooth Newton multigrid methods. An application in
grain growth as occurring in liquid phase crystallization
of silicon is considered.
This paper focuses on
rate-independent damage in elastic bodies. Since the driving energy is nonconvex,
solutions may have jumps as a function of time, and in this situation it is known that the classical concept
of energetic solutions for rate-independent systems
may fail to accurately describe the
behavior of the system at jumps.
Therefore, we resort to the (by now well-established) vanishing viscosity approach to rate-independent modeling
and approximate the model by its viscous regularization.
In fact, the analysis of the latter PDE system presents
remarkable difficulties, due to its highly nonlinear character.
We tackle it by combining a variational approach to a class of abstract doubly nonlinear evolution equations, with
careful regularity estimates tailored to this specific system relying on a q-Laplacian type gradient regularization of the damage variable.
Hence, for the viscous problem we conclude the existence of weak solutions satisfying a
suitable energy-dissipation inequality that is the starting point for the vanishing viscosity analysis.
The latter leads to the notion of (weak) parameterized
solution to our rate-independent system,
which encompasses the influence of viscosity in the description of the jump regime.
In this paper we study the distributed optimal control for the Cahn-Hilliard system. A general class of free energy potentials is allowed which, in particular, includes the double-obstacle potential. The latter potential yields an optimal control problem of a parabolic variational inequality which is of fourth-order in space. We show the existence of optimal controls to approximating problems where the potential is replaced by a mollified version of its Moreau-Yosida approximation. Corresponding first order optimality conditions for the mollified problems are given. For this purpose a new result on the continuous Fr\'echet differentiability of superposition operators with values in Sobolev spaces is established. Besides the convergence of optimal controls of the mollified problems to an optimal control of the original problem, we also derive first order optimality conditions for the original problem by a limit process. The newly derived stationarity system corresponds to a function space version of C-stationarity.