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The extended finite element method (XFEM) offers an elegant tool to model material discontinuities and cracks within a regular mesh, so that the element edges do not necessarily coincide with the discontinuities. This allows the modeling of propagating cracks without the requirement to adapt the mesh incrementally. Using a regular mesh offers the advantage, that simple refinement strategies based on the quadtree data structure can be used to refine the mesh in regions, that require a high mesh density. An additional benefit of the XFEM is, that the transmission of cohesive forces through a crack can be modeled in a straightforward way without introducing additional interface elements. Finally different criteria for the determination of the crack propagation angle are investigated and applied to numerical tests of cracked concrete specimens, which are compared with experimental results.

Major problems of applying selective sensitivity to system identification are requirement of precise knowledge about the system parameters and realization of the required system of forces. This work presents a procedure which is able to deriving selectively sensitive excitation by iterative experiments. The first step is to determine the selectively sensitive displacement and selectively sensitive force patterns. These values are obtained by introducing the prior information of system parameters into an optimization which minimizes the sensitivities of the structure response with respect to the unselected parameters while keeping the sensitivities with respect to the selected parameters as a constant. In a second step the force pattern is used to derive dynamic loads on the tested structure and measurements are carried out. An automatic control ensures the required excitation forces. In a third step, measured outputs are employed to update the prior information. The strategy is to minimize the difference between a predicted displacement response, formulated as function of the unknown parameters and the measured displacements, and the selectively sensitive displacement calculated in the first step. With the updated values of the parameters a re-analysis of selective sensitivity is performed and the experiment is repeated until the displacement response of the model and the actual structure are conformed. As an illustration a simply supported beam made of steel, vibrated by harmonic excitation is investigated, thereby demonstrating that the adaptive excitation can be obtained efficiently.

The modeling of crack propagation in plain and reinforced concrete structures is still a field for many researchers. If a macroscopic description of the cohesive cracking process of concrete is applied, generally the Fictitious Crack Model is utilized, where a force transmission over micro cracks is assumed. In the most applications of this concept the cohesive model represents the relation between the normal crack opening and the normal stress, which is mostly defined as an exponential softening function, independently from the shear stresses in tangential direction. The cohesive forces are then calculated only from the normal stresses. By Carol et al. 1997 an improved model was developed using a coupled relation between the normal and shear damage based on an elasto-plastic constitutive formulation. This model is based on a hyperbolic yield surface depending on the normal and the shear stresses and on the tensile and shear strength. This model also represents the effect of shear traction induced crack opening. Due to the elasto-plastic formulation, where the inelastic crack opening is represented by plastic strains, this model is limited for applications with monotonic loading. In order to enable the application for cases with un- and reloading the existing model is extended in this study using a combined plastic-damage formulation, which enables the modeling of crack opening and crack closure. Furthermore the corresponding algorithmic implementation using a return mapping approach is presented and the model is verified by means of several numerical examples. Finally an investigation concerning the identification of the model parameters by means of neural networks is presented. In this analysis an inverse approximation of the model parameters is performed by using a given set of points of the load displacement curves as input values and the model parameters as output terms. It will be shown, that the elasto-plastic model parameters could be identified well with this approach, but require a huge number of simulations.

The Element-free Galerkin Method has become a very popular tool for the simulation of mechanical problems with moving boundaries. The internally applied Moving Least Squares approximation uses in general Gaussian or cubic weighting functions and has compact support. Due to the approximative character of this method the obtained shape functions do not fulfill the interpolation condition, which causes additional numerical effort for the imposition of the essential boundary conditions. The application of a singular weighting function, which leads to singular coefficient matrices at the nodes, can solve this problem, but requires a very careful placement of the integration points. Special procedures for the handling of such singular matrices were proposed in literature, which require additional numerical effort. In this paper a non-singular weighting function is presented, which leads to an exact fulfillment of the interpolation condition. This weighting function leads to regular values of the weights and the coefficient matrices in the whole interpolation domain even at the nodes. Furthermore this function gives much more stable results for varying size of the influence radius and for strongly distorted nodal arrangements than classical weighting function types. Nevertheless, for practical applications the results are similar as these obtained with the regularized weighting type presented by the authors in previous publications. Finally a new concept will be presented, which enables an efficient analysis of systems with strongly varying node density. In this concept the nodal influence domains are adapted depending on the nodal configuration by interpolating the influence radius for each direction from the distances to the natural neighbor nodes. This approach requires a Voronoi diagram of the domain, which is available in this study since Delaunay triangles are used as integration background cells. In the numerical examples it will be shown, that this method leads to a more uniform and reduced number of influencing nodes for systems with varying node density than the classical circular influence domains, which means that the small additional numerical effort for interpolating the influence radius leads to remarkable reduction of the total numerical cost in a linear analysis while obtaining similar results. For nonlinear calculations this advantage would be even more significant.

In engineering science the modeling and numerical analysis of complex systems and relations plays an important role. In order to realize such an investigation, for example a stochastic analysis, in a reasonable computational time, approximation procedure have been developed. A very famous approach is the response surface method, where the relation between input and output quantities is represented for example by global polynomials or local interpolation schemes as Moving Least Squares (MLS). In recent years artificial neural networks (ANN) have been applied as well for such purposes. Recently an adaptive response surface approach for reliability analyses was proposed, which is very efficient concerning the number of expensive limit state function evaluations. Due to the applied simplex interpolation the procedure is limited to small dimensions. In this paper this approach is extended for larger dimensions using combined ANN and MLS response surfaces for evaluating the adaptation criterion with only one set of joined limit state points. As adaptation criterion a combination by using the maximum difference in the conditional probabilities of failure and the maximum difference in the approximated radii is applied. Compared to response surfaces on directional samples or to plain directional sampling the failure probability can be estimated with a much smaller number of limit state points.

The design and application of high performance materials demands extensive knowledge of the materials damage behavior, which significantly depends on the meso- and microstructural complexity. Numerical simulations of crack growth on multiple length scales are promising tools to understand the damage phenomena in complex materials. In polycrystalline materials it has been observed that the grain boundary decohesion is one important mechanism that leads to micro crack initiation. Following this observation the paper presents a polycrystal mesoscale model consisting of grains with orthotropic material behavior and cohesive interfaces along grain boundaries, which is able to reproduce the crack initiation and propagation along grain boundaries in polycrystalline materials. With respect to the importance of modeling the geometry of the grain structure an advanced Voronoi algorithm is proposed to generate realistic polycrystalline material structures based on measured grain size distribution. The polycrystal model is applied to investigate the crack initiation and propagation in statically loaded representative volume elements of aluminum on the mesoscale without the necessity of initial damage definition. Future research work is planned to include the mesoscale model into a multiscale model for the damage analysis in polycrystalline materials.

A fast solver method called the multigrid preconditioned conjugate gradient method is proposed for the mechanical analysis of heterogeneous materials on the mesoscale. Even small samples of a heterogeneous material such as concrete show a complex geometry of different phases. These materials can be modelled by projection onto a uniform, orthogonal grid of elements. As one major problem the possible resolution of the concrete specimen is generally restricted due to (a) computation times and even more critical (b) memory demand. Iterative solvers can be based on a local element-based formulation while orthogonal grids consist of geometrical identical elements. The element-based formulation is short and transparent, and therefore efficient in implementation. A variation of the material properties in elements or integration points is possible. The multigrid method is a fast iterative solver method, where ideally the computational effort only increases linear with problem size. This is an optimal property which is almost reached in the implementation presented here. In fact no other method is known which scales better than linear. Therefore the multigrid method gains in importance the larger the problem becomes. But for heterogeneous models with very large ratios of Young's moduli the multigrid method considerably slows down by a constant factor. Such large ratios occur in certain heterogeneous solids, as well as in the damage analysis of solids. As solution to this problem the multigrid preconditioned conjugate gradient method is proposed. A benchmark highlights the multigrid preconditioned conjugate gradient method as the method of choice for very large ratio's of Young's modulus. A proposed modified multigrid cycle shows good results, in the application as stand-alone solver or as preconditioner.

The present paper is part of a comprehensive approach of grid-based modelling. This approach includes geometrical modelling by pixel or voxel models, advanced multiphase B-spline finite elements of variable order and fast iterative solver methods based on the multigrid method. So far, we have only presented these grid-based methods in connection with linear elastic analysis of heterogeneous materials. Damage simulation demands further considerations. The direct stress solution of standard bilinear finite elements is severly defective, especially along material interfaces. Besides achieving objective constitutive modelling, various nonlocal formulations are applied to improve the stress solution. Such a corrective data processing can either refer to input data in terms of Young's modulus or to the attained finite element stress solution, as well as to a combination of both. A damage-controlled sequentially linear analysis is applied in connection with an isotropic damage law. Essentially by a high resolution of the heterogeneous solid, local isotropic damage on the material subscale allows to simulate complex damage topologies such as cracks. Therefore anisotropic degradation of a material sample can be simulated. Based on an effectively secantial global stiffness the analysis is numerically stable. The iteration step size is controlled for an adequate simulation of the damage path. This requires many steps, but in the iterative solution process each new step starts with the solution of the prior step. Therefore this method is quite effective. The present paper provides an introduction of the proposed concept for a stable simulation of damage in heterogeneous solids.

Advanced finite elements are proposed for the mechanical analysis of heterogeneous materials. The approximation quality of these finite elements can be controlled by a variable order of B-spline shape functions. An element-based formulation is developed such that the finite element problem can iteratively be solved without storing a global stiffness matrix. This memory saving allows for an essential increase of problem size. The heterogeneous material is modelled by projection onto a uniform, orthogonal grid of elements. Conventional, strictly grid-based finite element models show severe oscillating defects in the stress solutions at material interfaces. This problem is cured by the extension to multiphase finite elements. This concept enables to define a heterogeneous material distribution within the finite element. This is possible by a variable number of integration points to each of which individual material properties can be assigned. Based on an interpolation of material properties at nodes and further smooth interpolation within the finite elements, a continuous material function is established. With both, continuous B-spline shape function and continuous material function, also the stress solution will be continuous in the domain. The inaccuracy implied by the continuous material field is by far less defective than the prior oscillating behaviour of stresses. One- and two-dimensional numerical examples are presented.

In this paper an adaptive heterogeneous multiscale model, which couples two substructures with different length scales into one numerical model is introduced for the simulation of damage in concrete. In the presented approach the initiation, propagation and coalescence of microcracks is simulated using a mesoscale model, which explicitly represents the heterogeneous material structure of concrete. The mesoscale model is restricted to the damaged parts of the structure, whereas the undamaged regions are simulated on the macroscale. As a result an adaptive enlargement of the mesoscale model during the simulation is necessary. In the first part of the paper the generation of the heterogeneous mesoscopic structure of concrete, the finite element discretization of the mesoscale model, the applied isotropic damage model and the cohesive zone model are briefly introduced. Furthermore the mesoscale simulation of a uniaxial tension test of a concrete prism is presented and own obtained numerical results are compared to experimental results. The second part is focused on the adaptive heterogeneous multiscale approach. Indicators for the model adaptation and for the coupling between the different numerical models will be introduced. The transfer from the macroscale to the mesoscale and the adaptive enlargement of the mesoscale substructure will be presented in detail. A nonlinear simulation of a realistic structure using an adaptive heterogeneous multiscale model is presented at the end of the paper to show the applicability of the proposed approach to large-scale structures.