Optimal <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-Control in Coefficients for Dirichlet Elliptic Problems: <Emphasis Type="Italic">W</Emphasis>-Optimal Solutions

In this paper, we study a Dirichlet optimal control problem associated with a linear elliptic equation the coefficients of which we take as controls in the class of integrable functions. The coefficients may degenerate and, therefore, the problems may exhibit the so-called Lavrentieff phenomenon and non-uniqueness of weak solutions. We consider the solvability of this problem in the class of W-variational solutions. Using a concept of variational convergence of constrained minimization problems in variable spaces, we prove the existence of W-solutions to the optimal control problem and provide the way for their approximation. We emphasize that control problems of this type are important in material and topology optimization as well as in damage or life-cycle optimization.     

Numerical analysis of dynamic thermoviscoelastic contact with damage of a rod

We describe and analyse a model for a problem of thermoviscoelasticdynamic contact which allows for the general evolution of thematerial damage. The effects on the mechanical properties ofthe material due to crack expansion are described by a damagefield, which measures the decrease in the load-bearing capacityof the material. The damage process is assumed to be reversibleand the microcracks which develop as a result of tension orcompression may grow or disappear. The geometric setting isthat of a 1D rod which may contact a deformable obstacle. Thecontact is modelled by the normal compliance condition and thestress–strain constitutive equation is of Kelvin–Voigttype. The model consists of a coupled system of energy–elasticityequations together with a non-linear parabolic inclusion forthe damage field. The existence of a local weak solution isestablished using penalization, a finite element algorithm forthe solution is constructed and analysed and the results ofnumerical simulations based on this algorithm are presented.The simulations illustrate how the size of the normal compliancecoefficients, the damage rate coefficients and the applied forceaffect the character of the evolution of the damage. In particular,cycles of bonding and debonding can occur.     

A Comparison of Algorithmic Approaches to Damage-Plasticity Modeling in the Context of Gradient-Enhancement

A gradient-enhanced damage-plasticity formulation is proposed, which prevents the loss of well-posedness of the governing field equations in the post-critical damage regime. The non-locality of the formulation then manifests itself in terms of a free energy contribution that penalizes the occurrence of damage gradients. A second penalty term is introduced to force the global damage field to coincide with the internal damage state variable at the Gauss point level. An enforcement of Karush-Kuhn-Tucker (KKT) conditions on the global level can thus be avoided and classical local damage models may directly be incorporated and equipped with gradient enhancement. An important emphasis of this research is to investigate the efficiency and robustness of different algorithmic schemes to locally enforce the KKT conditions in the multi-surface damage-plasticity setting. Response simulations for a representative inhomogeneous boundary value problem are studied to assess the effectiveness of the gradient enhancement regarding stability and mesh objectivity. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal and approximate boundary controls of an elastic body with quasistatic evolution of damage

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. Copyright © 2014 John Wiley & Sons, Ltd.     

Sustainable growth and environmental catastrophes

In the standard AK growth model we introduce the threat of an ecological catastrophe and study the consequences for the economic variables in the long-run. We extend the basic framework by considering two environmental externalities: the first one is local and gives account of the marginal damage from emissions flow; the second one is aggregate, or global, and relates to the extreme damage which may happen if the accumulated stock of pollutants is on the threshold of a worldwide catastrophe. In this context dominated by market failures, we focus on the socially optimal solution and the search of conditions for sustainability. We identify the efficient balanced growth path, which may even show a singularity with trajectories truncating and changing course. In short, with respect to the economy’s long-run performance, we study how environment matters in different ways and at different stages, giving conditions to prevent the catastrophe and preserve (a lower) sustained growth. This paper may also be read as a clear and rigorous example of an optimal control problem involving a pure state-space constraint.     

Minimizing the Mean Damage for Parallel-Series Systems with Two Failure Modes

We address a non-convex optimization problem involving the minimization of the difference between two symmetric functions which are log convex. Specifically we are interested in minimizing the mean system damage of a parallel-series system composed of a maximal number of identical units, each capable of failing in one of two ways.We characterize the solution of the problem, and develop an algorithm to solve large scale versions of it.Numerical results are presented. It is shown that an optimal configuration with n available components may not use all n components.     

A relaxation-based approach to damage modeling

Common material models that take into account softening effects due to damage have the problem of ill-posed boundary value problems if no regularization is applied. This condition leads to a non-unique solution for the resulting algebraic system and a strong mesh dependence of the numerical results. A possible solution approach to prevent this problem is to apply regularization techniques that take into account the non-local behavior of the damage [1]. For this purpose a field function is often used to couple the local damage parameter to a non-local level, in which differences between the local and non-local parameter as well as the gradient of the non-local parameter can be penalized. In contrast, we present a novel approach [2] to regularization that no longer needs a non-local level but directly provides mesh-independent results. Due to the new variational approach we are also able to improve the calculation times and convergence behavior. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Common Regularization for Three Reservoir Optimal Control Problems

Four optimal control problems of reservoir release are investigated. The first problem is to minimize the peak release in order to prevent flood and to reduce the flood height. The second problem is to maximize the lowest release in order to ensure irrigation, water supply, shipping and environment downstream. The third problem is to minimize the flooding duration in order to reduce damage to goods, possessions, plants, levees, etc. It is shown that these three problems may possess infinitely many different optimal solutions, but they all have a common optimal solution, which is the unique optimal solution of the fourth problem. Since this unique optimal solution depends continuously on the input data, the fourth problem is well-posed and it can be considered as a common regularization of the three ill-posed problems.     

A new adaptive approach of the Metropolis-Hastings algorithm applied to structural damage identification using time domain data

In the present work, the formulation and solution of the inverse problem of structural damage identification is presented based on the Bayesian inference, a powerful approach that has been widely used for the formulation of inverse problems in a statistical framework. The structural damage is continuously described by a cohesion field, which is spatially discretized by the finite element method, and the solution of the inverse problem of damage identification, from the Bayesian point of view, is the posterior probability densities of the nodal cohesion parameters. In this approach, prior information about the parameters of interest and the quantification of the uncertainties related to the magnitudes measured can be used to estimate the sought parameters. Markov Chain Monte Carlo (MCMC) method, implemented via the Metropolis-Hastings (MH) algorithm, is commonly used to sample such densities. However, the conventional MH algorithm may present some difficulties, for instance, in high dimensional problems or when the parameters of interest are highly correlated or the posterior probability density is very peaked. In order to overcome these difficulties, a new adaptive MH algorithm (P-AMH) is proposed in the present work. Numerical results related to an inverse problem of damage identification in a simply supported Euler-Bernoulli beam are presented. Synthetic experimental time domain data, obtained with different damage scenarios, and noise levels, were addressed with the aim at assessing the proposed damage identification approach. An adaptive MH algorithm (H-AMH) and the conventional MH algorithm, already consolidated in the literature, were also considered for comparison purposes. The numerical results show that both adaptive algorithms outperformed the conventional MH. Besides, the P-AMH provided Markov chains with faster convergence and better mixing than the ones provided by the H-AMH.     

Parametric shape optimization of biaxial tensile specimen

Common cruciform specimen for biaxial tensile testing of sheet moulding compound, take damage and finally fail in uniaxially loaded areas. When using these specimen, an observation of damage initialization and failure in biaxially loaded areas is, therefore, not possible. In this paper, a parametric shape optimization is described to find a more suitable specimen shape. The parametrization of the specimen is presented. Objective functions are introduced to measure the appropriateness of specimen. A weighted summation transfers the constraint multiobjective optimization problem into a constraint scalar-valued problem. Findings of experiments suggest that a specimen shape with straight, non-tapering arms and slits along the arms is reasonable. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational Phase Field Modeling of Fracture Caused by Fluid Transport in Poroelastic Solids

The numerical modeling of failure mechanisms due to fracture based on sharp crack discontinuities is extremely demanding and suffers in situations with complex crack topologies. This drawback can be overcome by recently developed diffusive crack modeling concepts, which are based on the introduction of a crack phase field. Such an approach is conceptually in line with gradient-extended continuum damage models which include internal length scales. In this paper, we extend our recently outlined mechanical framework [1–3] towards the phase field modeling of fracture in the coupled problem of fluid transport in deforming porous media. Here, extremely complex crack patterns may occur due to drying or hydraulic induced fracture, the so called fracking. We develop new variational potentials for Biot-type fluid transport in porous media at finite deformations coupled with phase field fracture. It is shown, that this complex coupled multi-field problem is related to an intrinsic mixed variational principle for the evolution problem. This principle determines the rates of deformation, fracture phase field and fluid content along with the fluid potential. We develop a robust computational implementation of the coupled problem based on the potentials mentioned above and demonstrate its performance by the numerical simulation of complex fracture patterns. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Micromorphic Homogenisation of a Porous Medium: Application to Size Effects and Quasi-Brittle Damage

The classical theory of continuum mechanics requires that distances over which gradients occur are much larger than the length scales of the microstructure. This requirement is violated if deformation localizes due to material softening which is why the predictions of classical damage models feature certain unrealistic aspects. In the present contribution, a micromorphic model for quasi-brittle damage, derived through homogenisation, is employed in FEM simulations to overcome this problem. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A regularization approach for damage models based on a displacement gradient

Common material models that take into account softening effects due to damage encounter the problem of ill-posed boundary value problems if no regularization is applied. This condition leads to a non-unique solution for the resulting algebraic system and a strong mesh dependence of the numerical results. A possible solution approach to prevent this problem is to apply regularization techniques that take into account the non-local behavior of the damage [1]. For this purpose a field function is used to couple the local damage parameter to a non-local level, in which differences between the local and non-local parameter as well as the gradient of the non-local parameter can be penalized. In contrast, we present a novel approach to regularization in which no field function is needed [2]. Hereto, the regularization is carried out by means of the divergence of the displacements and no additional quantity is necessary since the displacements are already defined on a non-local level. The idea is that with an increasing value of the damage the element's volume will increase as well. This is a result of the softening due to the occurring damage. The increasing volume can be measured by the divergence of the displacements which can be penalized by an additional energy part. The lack of any field function and the regularization by the use of the divergence of the displacements entails several numerical advantages: the computational effort is considerably reduced and the convergence behavior is improved as well. Naturally, the numerical results are mesh independent due to the regularization. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical analysis of an elastic contact problem with damage

In this work, we consider a mathematical model for the quasistatic contact problem between an elastic body and a deformable obstacle, including the effect of the damage of the material, within the framework of the small deformation theory. The numerical analysis of the variational problem is provided using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Finally, a two-dimensional numerical problem is presented to show the performance of the method. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A numerical technique for structural damage detection

A computationally attractive method is proposed in this study to provide an insight to the location and extent of structural damage. The proposed method makes use of the matrix disassembly technique and approaches the damage location and extent problem in a decoupled fashion. First, a scheme is developed to determine the damage location by calculating a damage localization vector, which is derived from the modal residual force criteria. With location determined, the corresponding damage extent can be easily obtained only by the processes of division. The algorithm is applied to a numerical example and its efficiency is demonstrated through damage simulations.     

One-dimensional dynamic thermoviscoelastic contact with damage

The problem of thermoviscoelastic dynamic contact between a rod and a rigid obstacle, when the material damage is taken into account, is modeled and analyzed. The contact is modeled by the normal compliance condition and the stress-strain constitutive equation is of Kelvin-Voigt type. The damage, which describes the reduction of the load carrying capacity of the rod, evolves because of the opening of microcracks as a result of tension or compression. When the damage reaches a critical value at a point on the rod the material cannot carry any load and the system breaks down. Mathematically, this is expressed by the quenching of the solution. The existence of a local weak solution is established using penalization and a priori estimates.     

A SimpleModel for Anisotropic Damage with Applications to Soft Tissues

Arteries are reinforced by helically arranged collagen fibers and posses orthotropic elastic properties. In this paper a polyconvex anisotropic energy is used in order to guarantee the existence of minimizers for the purely elastic boundary value problem. Anisotropic discontinuous damage effects, which are induced by decreasing stiffness of particular fibers, are observed in a certain range of overexpansion. A simple thermodynamical consistent anisotropic damage model is constructed, basing on the assumption that damage mainly takes place in the fiber directions as a result of breakage of collagen cross‐links.Finally a cycled overexpansion of a test material from an artery is analyzed in order to show the main characteristics of the proposed model. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

New damage evolution model of rock material

As a kind of natural engineering material with original defects, there are distinctly nonlinear and anisotropic mechanical behaviors for rock materials. Nevertheless, the rock damage mechanics can solve this problem well. However, for the complexity of mechanical property of rock material, the mature and applicable model to describe the rock failure process and the method to determine the maximum damage value have not been established very well. To solve this problem, one new damage evolution model for rock material has been proposed. In this model, the least energy consumption principle proposed to describe the fracture process of materials is used. Using the experimental data of granite sample under uniaxial compression and the results of numerical tests under uniaxial tension and uniaxial compression, this model is verified. Moreover, the results of the new model have been compared with the results of the tests (numerical test and real test) and the traditional damage model. The comparison shows that the new model has the higher accuracy and better reflects for the fracture process of the granite sample. Moreover, the released damage energies of the new model and Mazars model are different, and the released damage energy of the new model is slightly less than that of the Mazars model.     

Estudio probabilista del daño por choque según la metodología PEER. Aplicación a un paso superior sísmicamente aislado

Bridges with deck supported either on sliding or elastomeric bearings are common in mid-seismicity regions. Pounding between deck and abutments is linked with their main seismic vulnerabilities, which can be assessed by estimating the probability of a given state of damage being reached in a given time period. This paper presents a state of the art methodology used to solve that problem, as well as its application to the assessment of the vulnerability of an overpass placed in Granada area (South-east of Spain), belonging to the previously mentioned typology.The Pacific Earthquake Engineering Research Center methodology will be adapted and applied. Their main steps will be briefly presented, although the identification and characterization of damage likely to occur will be described in more detail. The model of the structure, and specially pounding modeling, will also be detailed.Results are interesting since this bridge can be considered as a representative of a widespread class. The adapted methodology may be applied to similar structures. Also, probability values obtained may serve as benchmarks.     

Large-scale simulation of arterial walls: mechanical modeling

Biological soft tissues appearing in arterial walls are characterized by a nearly incompressible, anisotropic hyperelastic material behavior in the physiological range of deformations. For the representation of such materials we apply a polyconvex strain energy function in order to ensure the existence of minimizers and in order to satisfy the Legendre-Hadamard condition automatically. When arteries are overstretched, discontinuous damage is observed. For the modeling of this effect we apply a damage model, which basically assumes that the damage occurs mainly in fiber direction. For the numerical simulation we consider an atherosclerotic artery and apply a high internal pressure which is comparable to the pressure applied during a balloon-angioplasty. The 3D-discretization results in a large system of equations, therefore, a parallel algorithm using FETI-DP is applied to solve the boundary value problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)