No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Necessary conditions of first-order for an optimal boundary control problem for viscous damage processes in 2D
Abstract
Controlling the growth of material damage is an important engineering task with plenty of real world applications. In this paper we approach this topic from the mathematical point of view by investigating an optimal boundary control problem for a damage phase-field model for viscoelastic media. We consider non-homogeneous Neumann data for the displacement field which describe external boundary forces and act as control variable. The underlying hyberbolic-parabolic PDE system for the state variables exhibit highly nonlinear terms which emerge in context with damage processes. The cost functional is of tracking type, and constraints for the control variable are prescribed. Based on recent results from [M. H. Farshbaf-Shaker, C. Heinemann: A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media. Math. Models Methods Appl. Sci. 25 (2015), 2749--2793], where global-in-time well-posedness of strong solutions to the lower level problem and existence of optimal controls of the upper level problem have been established, we show in this contribution differentiability of the control-to-state mapping, well-posedness of the linearization and existence of solutions of the adjoint state system. Due to the highly nonlinear nature of the state system which has by our knowledge not been considered for optimal control problems in the literature, we present a very weak formulation and estimation techniques of the associated adjoint system. For mathematical reasons the analysis is restricted here to the two-dimensional case. We conclude our results with first-order necessary optimality conditions in terms of a variational inequality together with PDEs for the state and adjoint state system.
... To put our work into perspective, let us mention some further approaches related to fracture optimization. In [35,36] control of a viscous damage model was considered in a continuous setting, shape optimization techniques were utilized in [37,38], and an approach where the propagation of a crack was limited through controlling the release of the associated energy was used in [39,40]. An optimal control problem of a two-field damage model, and a nonsmooth (viscous damage) coupled system, was analyzed in [41,42]. ...
We prove local quadratic convergence of the sequential quadratic programming (SQP) method for an optimal control problem of tracking type governed by one time step of the Euler-Lagrange equation of a time discrete regularized fracture or damage energy minimization problem. This lower-level energy minimization problem % describing the fracture process contains a penalization term for violation of the irreversibility condition in the fracture growth process and a viscous regularization term. Conditions on the latter, corresponding to a time step restriction, guarantee strict convexity and thus unique solvability of the Euler Lagrange equations. Nonetheless, these are quasilinear and the control problem is nonconvex. For the convergence proof with localization of the SQP-method, we follow the approach from \cite{Troeltzsch:1999}, utilizing strong regularity of generalized equations and arguments from \cite{HoppeNeitzel:2020} for -localization. 2020 Mathematics Subject Classification 90C55, 49M41, 49M15.
We prove second-order optimality conditions for an optimal control problem of tracking-type for a time-discrete regularized phase-field fracture or damage propagation model. The energy minimization problem describing the fracture process contains a penalization term for violation of the irreversibility condition in the fracture growth process, as well as a viscous regularization corresponding to a time-step restriction in a temporal discretization of the problem. In the control problem, the energy minimization problem is replaced by its Euler–Lagrange equations. While the energy minimization functional is convex due to the viscous approximation, the associated Euler–Lagrange equations are of quasilinear type, making the control problem nonconvex. We prove second-order necessary as well as second-order sufficient optimality conditions without two-norm discrepancy.
A theory of continuum damage mechanics is developed within the framework of the principle of virtual power. Because the damage in a solid results from microscopic movements, we decided to include the power of these microscopic movements in the power of the internal forces. The microscopic velocities are related to the damage rate. The power of the internal forces we choose depends on the damage velocity and its gradient to take into account the interactions. Models issued from this theory are presented. The are coherent from the mechanical and mathematical points of view. The numerical computations show no mesh sensitivity. The describe with good agreement the main experimental properties. Concrete is chosen as an example to illustrate the theory. A model using two damage quantities is also presented. It yields the correct description of the unilateral phenomenon observed in concrete. Finally, an extension of existing models is proposed to descride the fatigue damage behaviour.
The present contribution investigates shape optimisation problems for a class of semilinear elliptic variational inequalities with Neumann boundary conditions. Sensitivity estimates and material derivatives are firstly derived in an abstract operator setting where the operators are defined on polyhedral subsets of reflexive Banach spaces. The results are then refined for variational inequalities arising from minimisation problems for certain convex energy functionals considered over upper obstacle sets in . One particularity is that we allow for dynamic obstacle functions which may arise from another optimisation problems. We prove a strong convergence property for the material derivative and establish state-shape derivatives under regularity assumptions. Finally, as a concrete application from continuum mechanics, we show how the dynamic obstacle case can be used to treat shape optimisation problems for time-discretised brittle damage models for elastic solids. We derive a necessary optimality system for optimal shapes whose state variables approximate desired damage patterns and/or displacement fields.
This contribution deals with a class of models combining isotropic damage with plasticity. It has been inspired by a work by Freddi and Royer-Carfagni [FRC10], including the case where the inelastic part of the strain only evolves in regions where the material is damaged. The evolution both of the damage and of the plastic variable is assumed to be rate-independent. Existence of solutions is established in the abstract energetic framework elaborated by Mielke and coworkers (cf., e.g., [Mie05, Mie11b]).
The quasistatic rate-independent damage combined with linearized plasticity
with hardening at small strains is investigated. The fractional-step time
discretisation is devised with the purpose to obtain a numerically efficient
scheme converging possibly to a physically relevant stress-driven solutions,
which however is to be verified a-posteriori by using a suitable integrated
variant of the maximum-dissipation principle. Gradient theories both for damage
and for plasticity are considered to make the scheme numerically stable with
guaranteed convergence within the class of weak solutions. After finite-element
approximation, this scheme is computationally implemented and illustrative
2-dimensional simulations are performed.
A typical phase field approach for describing phase separation and coarsening phenomena in alloys is the Cahn-Hilliard model. This model has been generalized to the so-called Cahn-Larché system by combining it with elasticity to capture non-neglecting deformation phenomena, which occur during phase separation and coarsening processes in the material. In order to account for damage effects, we extend the existing framework of Cahn-Hilliard and Cahn-Larché systems by incorporating an internal damage variable of local character. This damage variable allows to model the effect that damage of a material point is influenced by its local surrounding. The damage process is described by a unidirectional rate-dependent evolution inclusion for the internal variable. For the introduced Cahn-Larché systems coupled with rate-dependent damage processes, we establish a suitable notion of weak solutions and prove existence of weak solutions.
In 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 {\em 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 . 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 \emph{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.
In this work, we introduce a degenerating PDE system with a time-depending
domain for complete damage processes under time-varying Dirichlet boundary
conditions. The evolution of the system is described by a doubly nonlinear
differential inclusion for the damage process and a quasi-static balance
equation for the displacement field which are strongly nonlinearly coupled.
In our proposed model, the material may completely disintegrate which is
indispensable for a realistic modeling of damage processes in elastic
materials. Complete damage theories lead to several mathematical problems since
for instance coercivity properties of the free energy are lost and, therefore,
several difficulties arise.
For the introduced complete damage model, we propose a classical formulation
and a corresponding suitable weak formulation in an SBV-framework. The main
aim is to prove existence of weak solutions for the introduced degenerating
model. In addition, we show that the classical differential inclusion can be
regained from the notion of weak solutions under certain regularity assumptions
which is a novelty in the theory of complete damage models of this type.
Damage of a linearly-responding material that can completely disintegrate is addressed at small strains. Using time-varying
Dirichlet boundary conditions we set up a rateindependent evolution problem in multidimensional situations. The stored energy
involves the gradient of the damage variable. This variable as well as the stress and energies are shown to be well defined
even under complete damage, in contrast to displacement and strain. Existence of an energetic solution is proved, in particular,
by detailed investigating the Γ-limit of the stored energy and its dependence on boundary conditions. Eventually, the theory
is illustrated on a one-dimensional example.
In this paper, we analyze a PDE system arising in the modeling of phase
transition and damage phenomena in thermoviscoelastic materials. The resulting
evolution equations in the unknowns \theta (absolute temperature), u
(displacement), and \chi (phase/damage parameter) are strongly nonlinearly
coupled. Moreover, the momentum equation for u contains \chi-dependent elliptic
operators, which degenerate at the pure phases (corresponding to the values
\chi=0 and \chi=1), making the whole system degenerate. That is why, we have to
resort to a suitable weak solvability notion for the analysis of the problem:
it consists of the weak formulations of the heat and momentum equation, and,
for the phase/damage parameter \chi, of a generalization of the principle of
virtual powers, partially mutuated from the theory of rate-independent damage
processes. To prove an existence result for this weak formulation, an
approximating problem is introduced, where the elliptic degeneracy of the
displacement equation is ruled out: in the framework of damage models, this
corresponds to allowing for partial damage only. For such an approximate
system, global-in-time existence and well-posedness results are established in
various cases. Then, the passage to the limit to the degenerate system is
performed via suitable variational techniques.
The numerical implementation of the model of brittle fracture developed in Francfort and Marigo (1998. J. Mech. Phys. Solids 46 (8), 1319–1342) is presented. Various computational methods based on variational approximations of the original functional are proposed. They are tested on several antiplanar and planar examples that are beyond the reach of the classical computational tools of fracture mechanics.
This paper is concerned with an optimal control problem governed by a regularized fracture model using a phase-field technique. To avoid the non-differentiability due to the irreversibility constraint on the fracture growth, the phase-field fracture model is relaxed using a penalization approach. Existence of a solution to the penalized fracture model is shown and existence of at least one solution for the regularized optimal control problem is established. Moreover, the linearized fracture model is considered and used to establish first order necessary conditions as well as to discuss QP-approximations to the nonlinear optimization problem. A numerical example suggests that these can be used to obtain a fast convergent algorithm.
In this work we investigate a phase field model for damage processes in two-dimensional viscoelastic media with nonhomogeneous Neumann data describing external boundary forces. In the first part we establish global-in-time existence, uniqueness, a priori estimates and continuous dependence of strong solutions on the data. The main difficulty is caused by the irreversibility of the phase field variable which results in a constrained PDE system. In the last part we consider an optimal control problem where a cost functional penalizes maximal deviations from prescribed damage profiles. The goal is to minimize the cost functional with respect to exterior forces acting on the boundary which play the role of the control variable in the considered model. To this end, we prove existence of minimizers and study a family of ``local'' approximations via adapted cost functionals.
In this paper, we study an optimal control problem for the mixed boundary value problem for an elastic body with quasistatic evolution of an internal damage variable. We suppose that the evolution of microscopic cracks and cavities responsible for the damage is described by a nonlinear parabolic equation. A density of surface traction p acting on a part of boundary of an elastic body Ω is taken as a boundary control. Because the initial boundary value problem of this type can exhibit the Lavrentieff phenomenon and non-uniqueness of weak solutions, we deal with the solvability of this problem in the class of weak variational solutions. Using the convergence concept in variable spaces and following the direct method in calculus of variations, we prove the existence of optimal and approximate solutions to the optimal control problem under rather general assumptions on the quasistatic evolution of damage.
We analyze a rate-independent model for damage evolution in elastic bodies. The central quantities are a stored energy functional and a dissipation functional, which is assumed to be positively homogeneous of degree one. Since the energy is not simultaneously (strictly) convex in the damage variable and the displacements, solutions may have jumps as a function of time. The latter circumstance makes it necessary to recur to suitable notions of weak solution. However, the by-now classical concept of global energetic solution fails to describe accurately the behavior of the system at jumps.
Hence, we consider rate-independent damage models as limits of systems driven by viscous, rate-dependent dissipation. We use a technique for taking the vanishing viscosity limit, which is based on arclength reparametrization. In this way, in the limit we obtain a novel formulation for the rate-independent damage model, which highlights the interplay of viscous and rate-independent effects in the jump regime, and provides a better description of the energetic behavior of the system at jumps.
In the modeling of pressurized fractures using phase-field approaches, the irreversibility of crack growth is enforced through an inequality constraint on the temporal derivative of the phase-field function. In comparison to the classical case in elasticity, the presence of the pressure requires the additional constraint and makes the problem much harder to analyze. After temporal discretization, this induces a minimization problem in each time step over a solution dependent admissible set. To avoid solving the resulting variational inequality corresponding to the first order necessary conditions, a penalization approach is used, commonly, to remove the inequality constraint. It is well-known that for large penalty parameters the algorithm suffers from numerical instabilities in the solution process. Consequently, to avoid such a drawback, we propose an augmented Lagrangian algorithm for the discrete in time and continuous in space phase-field problems. The final set of equations is solved in a decoupled fashion. The proposed method is substantiated with several benchmark and prototype tests in two and three dimensions.
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, withcareful 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.
The paper is concerned with the control of the shape of rigid and elastic inclusions and crack paths in elastic bodies. We provide the corresponding problem formulations and analyze the shape sensitivity of such inclusions and cracks with respect to different perturbations. Inequality type boundary conditions are imposed at the crack faces to provide a mutual nonpenetration between crack faces. Inclusion and crack shapes are considered as control functions and control objectives, respectively. The cost functional, which is based on the Griffith rupture criterion, characterizes the energy release rate and provides the shape sensitivity with respect to a change of the geometry. We prove an existence of optimal solutions.
In the framework of linear fracture theory, the Griffith criterion postulates the growth of any crack if the corresponding so-called energy release rate, defined as the variation of the mechanical energy, reaches a critical value. We consider in this Note the optimal location problem which consists in minimizing this rate by applying to the structure an additional boundary load having a support which is disjoint from the support of the initial load possibly responsible of the growth. We give a sufficient well-posedness condition, introduce a relaxed problem in the general case, and then present a numerical experiment which suggests that the original nonlinear problem is actually well-posed. To cite this article: P. Hild et al., C. R. Mecanique 336 (2008).
Damage of an elastic body undergoing large deformations by a "hard-device" load-ing possibly combined with an impact (modeled by a unilateral frictionless contact) of another, ideally rigid body is formulated as an activated, rate-independent process. The damage is assumed to absorb a specific and prescribed amount of energy. A solution is defined by energetic principles of stability and balance of stored and dissipated energies with the work of external loading, realized here through displacement on a part of the boundary. Rigorous analysis by time discretization is performed.
This paper discusses an existence result for energetic solutions of rate-independent damage processes and the temporal regularity of the solution. We consider a body consisting of a physically nonlinearly elastic material undergoing small deformations and partial damage. The present work is a generalization of [16] concerning the properties of the stored elastic energy density as well as the suitable Sobolev space for the damage variable: While previous work assumes that the damage variable z satisfies z in W1, r(Ω) with r > d for Ω ⊂ ℝd, we can handle the case r > 1 by a new technique for the construction of joint recovery sequences. Moreover, this work generalizes the temporal regularity results to physically nonlinearly elastic materials by analyzing Lipschitz- and Hölder-continuity of solutions with respect to time.
We consider a dissipative model recently proposed by M. Frmond to describe the evolution of damage in elastic materials. The corresponding PDEs system consists of an elliptic equation for the displacements with a degenerating elastic coefficient coupled with a variational dissipative inclusion governing the evolution of damage. We prove a local-in-time existence and uniqueness result for an associated initial and boundary value problem, namely considering the evolution in some subinterval where the damage is not complete. The existence result is obtained by a truncation technique combined with suitable a priori estimates. Finally, we give an analogous local-in-time existence and uniqueness result for the case in which we introduce viscosity into the relation for macroscopic displacements such that the macroscopic equilibrium equation is of parabolic type.
We consider a one-dimensional dynamic model that describes the evolution of damage caused by tension in a viscoelastic material. The process is modeled by a coupled set of two differential inclusions for the elastic displacement and damage fields. We establish the existence of local weak solutions. The existence result is derived from the a priori estimates obtained for a sequence of regularized, truncated, and time-retarded approximations. We also establish the existence of the unique weak solution of a simplified version of the model.
Maß-Und Integrationstheorie. Grundwissen Mathematik
- J Elstrodt
J. Elstrodt. Maß-Und Integrationstheorie. Grundwissen Mathematik. Springer Berlin Heidelberg, 2009.
A rate-independent gradient system in damage coupled with plasticity via structured strains
- E Bonetti
- E Rocca
- R Rossi
- M Thomas
E. Bonetti, E. Rocca, R. Rossi, and M. Thomas. A rate-independent gradient system in damage coupled
with plasticity via structured strains. To appear in: ESAIM Proceedings and Surveys, 2016.