This contribution aims to analyze the deterioration behaviour of steel fibre-reinforced high-performance concrete (HPC) in both experiments as well as numerical simulations. For this purpose, flexural tensile tests are carried out on beams with different fibre contents and suitable damage indicators are established to describe and calibrate the damage behaviour numerically using a phase-field model approach. In addition to conventional measurement methods, the tests are equipped with acoustic emission sensors in order to obtain a more precise picture of crack evolution by observing acoustic events. It is shown that, in addition to classical damage indicators, such as stiffness degradation and absorbed energy, various acoustic indicators, such as the acoustic energy of individual crack events, can also provide information about the damage progress. For the efficient numerical analysis of the overall material behaviour of fibre-reinforced HPC, a phenomenological material model is developed. The data obtained in the experiments are used to calibrate and validate the numerical model for the simulation of three-point bending beam tests. To verify the efficiency of the presented numerical model, the numerical results are compared with the experimental data, e.g., load-CMOD curves and the degradation of residual stiffness. © 2022 by the authors. Licensee MDPI, Basel, Switzerland.

We construct two-dimensional, two-phase random heterogeneous microstructures by stochastic simulation using the planar Boolean model, which is a random collection of overlapping grains. The structures obtained are discretized using finite elements. A heterogeneous Neo-Hooke law is assumed for the phases of the microstructure, and tension tests are simulated for ensembles of microstructure samples. We determine effective material parameters, i.e., the effective Lamé moduli λ∗ and μ∗, on the macroscale by fitting a macroscopic material model to the microscopic stress data, using stress averaging over many microstructure samples. The effective parameters λ∗ and μ∗ are considered as functions of the microscale material parameters and the geometric parameters of the Boolean model including the grain shape. We also consider the size of the Representative Volume Element (RVE) given a precision and an ensemble size. We use structured and unstructured meshes and also provide a comparison with the FE2 method. © 2022, The Author(s).

The pseudo-ductile material behavior of fiber reinforced high performance concrete is mainly characterized by the fiber pullout process. Thereby, complex fiber–concrete interactions, i.e. interface debonding, concrete micro cracks, slippage, adhesion and further unknown processes, are commonly investigated in single-fiber pullout tests. The study in this contribution is based on the experimental results of Gebuhr etal., (2019) and compares three different numerical models applied to the fiber pullout test. An accurate and efficient model for fiber pullout behavior forms the basis for the prediction of the overall behavior by means of composite models or multi-scale approaches in subsequent studies. © 2020 Elsevier Ltd

Residual stresses are an important issue as they affect both the manufacturing process as well as the performance of the final parts. Taking the whole process chain of hot forming into account, the integrated heat treatment provided by a defined temperature profile during cooling of the parts offers a great potential for the targeted adjustment of the desired residual stress state. The aim of this work is the investigation of technological reproducibility and stability of residual stresses arising from the thermomechanical forming process. For this purpose, a long-term study of residual stresses on hot-formed components is conducted. In order to develop finite element models for hot forming, a comprehensive thermomechanical material characterisation with special focus on phase transformation effects is performed. The numerical model is validated by means of a comparison between residual stress states determined with X-ray diffraction on experimentally processed components and predicted residual stresses from the simulations. © 2021, The Minerals, Metals & Materials Society.

The resulting shapes in production processes of metal components are strongly influenced by deformation induced residual stresses. Dual-phase steels are commonly used for industrial application of, e.g., forged or deep-drawn structural parts. This is due to their ability to handle high plastic deformations, while retaining desired stiffness for the products. In order to influence the resulting shape as well as component characteristics positively it is important to predict the distribution of phase-specific residual stresses which occur on the microscale of the material. In this contribution a comparative study is presented, where two approaches for the numerical simulation of residual stresses are applied. On the one hand a numerically efficient mean field theory is used to estimate on the grain level the total strain, the plastic strains and the eigenstrains based on macroscopic stress, strain and stiffness data. An alternative ansatz relies on a Taylor approximation for the grain level strains. Both approaches are applied to the corrosion-resistant duplex steel X2CrNiMoN22-5-3 (1.4462), which consists of a ferritic and an austenitic phase with the same volume fraction. Mean field and Taylor approximation strategies are implemented for usage in three dimensional solid finite element analysis and a geometrically exact Euler–Bernoulli beam for the simulation of a four-point-bending test. The predicted residual stresses are compared to experimental data from bending experiments for the phase-specific residual stresses/strains which have been determined by neutron diffraction over the bending height of the specimen. © 2021, The Author(s).

The aim of this work is the adjustment of an advantageous compressive residual stress profile in hot-formed components by intelligent process control with tailored cooling from the forging heat. The feasibility and potential are demonstrated in a hot forming process in which cylindrical specimen with an eccentric hole are formed at 1000 °C and subsequently cooled in water from the forging heat. Previous work shows that tensile residual stresses occur in the specimen formed in this way from the material 1.3505. Using the presented multi-scale FE models, an alternative process variant is analysed in this work, where advantageous compressive residual stresses can be generated instead of tensile residual stresses through tailored cooling from the forming heat in the specimen. The tailored cooling is achieved by partially exposing the specimen to a water-air spray. In this way, the local plastification can be influenced by inhomogeneous strains due to thermal and transformation-induced effects in order to customise the resulting residual stress distribution. The scientific challenge of this work is to generate different residual stresses in the surface of the specimen without changing the geometrical and microstructural properties. It is demonstrated that influencing the residual stresses and even reversing the stress sign is possible using smart process control during cooling. © 2021, The Author(s).

In this contribution, the two-scale analysis of residual stress states in a hot bulk formed part with subsequent cooling in the framework of the FE 2-method is presented. The induction of specific residual stress states in order to improve a component’s properties is an area of current research. In general, residual stresses can be induced inside a component in different ways, e.g., quenching, phase transformation in hot forming processes or dislocation movements. It is widely known that different types of residual stresses can be characterized based on the scale the type acts on. In addition to the macroscopic residual stress analysis, in which residual stresses of first type are considered, this contribution specifically analyzes the microscopic residual stress evolution as a consequence of the cooling of the component. © 2021, The Author(s).

In production engineering, current research focuses on the induction of targeted residual stress states in components in order to improve their properties. Therein, the combination of experiment and simulation plays an important role. In this contribution, a focus is laid on the investigation of hot forming processes with subsequent cooling. A numerical approach is presented to analyze the distribution of residual stresses resulting from cooling of a cylinder with an eccentric hole made of chromium-alloyed steel. The occurring phase transformation, which is evoked by cooling, is considered in order to compute residual stress distributions inside the material. © 2021, The Author(s).

In production engineering, current research focuses on the induction of targeted residual stress states in components in order to improve their properties rather than follow the usual path of minimizing residual stresses to prevent failure. In this contribution, a focus is laid on the investigation of the subsequent cooling process of hot bulk formed parts. Such cooling of a component leads to a microscopic phase transformation, which has to be considered in order to compute residual stresses inside the material. A numerical approach based on a phenomenological macroscopic material model is presented to depict the related stress evolution. © 2021, The Minerals, Metals & Materials Society.

We present a numerical two-scale simulation approach of the Nakajima test for dual-phase steel using the software package FE2TI, a highly scalable implementation of the well known homogenization method FE2. We consider the incorporation of contact constraints using the penalty method as well as the sample sheet geometries and adequate boundary conditions. Additional software features such as a simple load step strategy and prediction of an initial value by linear extrapolation are introduced. The macroscopic material behavior of dual-phase steel strongly depends on its microstructure and has to be incorporated for an accurate solution. For a reasonable computational effort, the concept of statistically similar representative volume elements (SSRVEs) is presented. Furthermore, the highly scalable nonlinear domain decomposition methods NL-FETI-DP and nonlinear BDDC are introduced and weak scaling results are shown. These methods can be used, e.g., for the solution of the microscopic problems. Additionally, some remarks on sparse direct solvers are given, especially to PARDISO. Finally, we come up with a computationally derived Forming Limit Curve (FLC). © The Author(s) 2020.

This study proposes a micromechanical modeling scheme to predict relevant mechanical behavior of DP600 steel for the sheet metal forming process. This study can be divided into two parts which are the prediction of the advanced anisotropic initial yield function by means of microstructure-based simulations and the investigation of microstructure changes during the sheet metal forming process. Firstly, based on the quantitative microstructure characterization of DP600 steel by EBSD analysis, the obtained statistical information of important microstructural features is used to generate a microstructure model with the help of an advanced dynamic microstructure generator (ADMG), which combines a particle simulation method with radical Voronoi tessellation. In the next step, finite element simulations with a non-local crystal plasticity model for the individual grains are conducted. With the help of these simulations, the crystal plasticity parameters are adapted to match the experiments. The resulting parameterized microstructure model of DP600 steel is then applied to various loading conditions to investigate the corresponding mechanical responses. For the second part, macroscopic simulations of the bending process are performed and local deformation fields of the location of interest are captured and imposed as boundary conditions on the microstructure model to study the changes in the microstructural features. © 2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the 23rd International Conference on Material Forming.

This paper aims at extending the existing knowledge regarding the pull-out behavior of single steel fibers embedded in high- and ultra-high-strength concretes with compressive strengths exceeding 100 MPa. Apart from the compressive strength, straight fibers, and fibers with hooked-ends as well as different embedded lengths are considered. The experiments have shown that the bond strength for straight fibers increases with an increasing compressive strength, mechanical anchorage in terms of hooks multiply the load-bearing capacity. For hooked end fibers in ultra-high performance concrete (UHPC) fiber rupture occurred. Special attention was paid to characterize scatter adequately. Especially for hooked-end fibers in UHPC conventional slip-wise averaging does not represent the maximum load well. Therefore, a more precise approach basing on characteristic points is introduced. The experimental results are presented to be used for calibration and validation of numerical models. As an example, an elasto-plastic phase-field model using a Drucker-Prager yield condition is developed, which represents the pull-out behavior of straight fibers satisfyingly. © 2019 fib. International Federation for Structural Concrete

In this contribution, the deterioration behaviour of steel fibre reinforced high performance concrete (HPC) beams in flexural tests is described and analysed. The focus is drawn on quasi-static low-cycle tests in an attempt to derive the progression of stiffness, plastic deformation, and the hysteresis energy of individual load cycles depending on the amount of fibres used per specimen. To achieve this goal, different fibre quantities from plain concrete up to 1.5 vol.-% are looked into. Additionally, a prospect is given on the usability of the described deterioration indicators to predict the concrete’s performance in high-cycle loading processes. Lastly, a short outline for the applicability of the acquired data on a numeric model using a phase-field approach as described in Pise et al. (2018) will be given. © 2019 Taylor & Francis Group, London, UK.

Residual stresses in components are a central issue in almost every manufacturing process, as they influence the performance of the final part. Regarding hot forming processes, there is a great potential for defining a targeted residual stress state, as many adjustment parameters, such as deformation state or temperature profile, are available that influence residual stresses. To ensure appropriate numerical modeling of residual stresses in hot forming processes, comprehensive material characterization and suitable multiscale Finite Element (FE) simulations are required. In this paper, experimental and numerical investigations of thermo-mechanically processed steel alloy 1.3505 (DIN 100Cr6) are presented that serve as a basis for further optimization of numerically modeled residual stresses. For this purpose, cylindrical upsetting tests at high temperature with subsequently cooling of the parts in the media air or water are carried out. Additionally, the process is simulated on the macroscale and compared to the results based on the experimental investigations. Therefore, the experimentally processed specimens are examined regarding the resulting microstructure, distortions, and residual stresses. For the investigation on a smaller scale, a numerical model is set up based on the state-data of the macroscopic simulation and experiments, simulating the transformation of the microstructure using phase-field theory and FE analysis on micro- and meso-scopic level. © 2019 by the author. Licensee MDPI, Basel, Switzerland.

In this contribution, a constitutive framework of elasto-plastic phase-field model of fracture is applied to simulation of pullout test of single steel fiber embedded in high-performance concrete and is compared to experimental results. For the description of the mechanical behavior the Drucker-Prager plasticity model is used. The aim is to examine the pullout behavior of a single steel fiber and its influence on the overall material behavior. Therefore, the mechanical behavior of high-performance concrete is studied in experiments. The predictive capability of the above mentioned model is analyzed in detail by simulating pullout tests and calibrated using experimental data. © 2019 Taylor & Francis Group, London, UK.

The microstructure of dual-phase steels consisting of a ferrite matrix with embedded martensite inclusions is the main contributor to the mechanical properties such as high ultimate tensile strength, high work hardening rate, and good ductility. Due to the composite structure and the wide field of applications of this steel type, a wide interest exists in corresponding virtual computational experiments. For a reliable modeling, the microstructure should be included. For that reason, in this paper we follow a computational strategy based on the definition of a representative volume element (RVE). These RVEs will be constructed by a set of tomographic measurements and mechanical tests. In order to arrive at more efficient numerical schemes, we also construct statistically similar RVEs, which are characterized by a lower complexity compared with the real microstructure but which represent the overall material behavior accurately. In addition to the morphology of the microstructure, the austenite–martensite transformation during the steel production has a relevant influence on the mechanical properties and is considered in this contribution. This transformation induces a volume expansion of the martensite phase. A further effect is determined in nanoindentation test, where it turns out that the hardness in the ferrite phase increases exponentially when approaching the martensitic inclusion. To capture these gradient properties in the computational model, the volumetric expansion is applied to the martensite phase, and the arising equivalent plastic strain distribution in the ferrite phase serves as basis for a locally graded modification of the ferritic yield curve. Good accordance of the model considering the gradient yield behavior in the ferrite phase is observed in the numerical simulations with experimental data. © 2015, Springer-Verlag Berlin Heidelberg.

A metallurgical material description of the flow behavior for finite element (FE) simulations was developed. During hot compression tests, the dynamic microstructure evolution is modeled on the example of high-strength martensitic steel MS-W 1200. Compression tests at 900-1000 °C with a strain rate of 0.1 s-1 on fine-grain and coarse-grain samples were performed. An analysis of the flow behavior identified a strong correlation between the dynamic recrystallization kinetics and the initial microstructure. The regression analysis has been used to determine correction factors of the new model to describe the dynamic recrystallization. A good agreement between FE simulation and measurement shows the validity of the new model. A metallurgical material description of the flow behavior for finite element (FE) simulations is developed. During hot compression tests, the dynamic microstructure evolution is modeled on the example of high-strength martensitic steel MS-W 1200. An analysis of the flow behavior identifies a strong correlation between the dynamic recrystallization kinetics and the initial microstructure. © 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

The magneto-electric (ME) coupling of multiferroic materials is of high interest for a variety of advanced applications like in data storage or sensor technology. Since the ME coupling of single-phase multiferroics is too low for technical applications, the manufacturing of composite structures becomes relevant. These composites generate the effective ME coupling as a strain-induced product property. Several experiments on composite multiferroics showed remarkable ME coefficients that are orders of magnitudes higher than those of single-phase materials. The present paper investigates the arising effective product properties of two-phase ME composites by simulating the coupling behavior using a two-scale finite element (FE2) homogenization approach. By means of this method, microstructures with different volume fractions of the individual phases and associated macroscopic ME coupling coefficients are considered. We investigate the influence of different magnetization states by means of the non-linear dissipative magnetostriction material model originally established in [1]. © 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

In modern engineering, micro-heterogeneous materials are designed to satisfy the needs and challenges in a wide field of technical applications. The effective mechanical behavior of these materials is influenced by the inherent microstructure and therein the interaction and individual behavior of the underlying phases. Computational homogenization approaches, such as the FE2 method have been found to be a suitable tool for the consideration of the influences of the microstructure. However, when real microstructures are considered, high computational costs arise from the complex morphology of the microstructure. Statistically similar RVEs (SSRVEs) can be used as an alternative, which are constructed to possess similar statistical properties as the realmicrostructure but are defined by a lower level of complexity. These SSRVEs are obtained from a minimization of differences of statistical measures and mechanical behavior compared with a real microstructure in a staggered optimization scheme, where the inner optimization ensures statistical similarity and the outer optimization problem controls themechanical comparativity of the SSRVE and the real microstructure. The performance of SSRVEs may vary with the utilized statistical measures and the parameterization of the microstructure of the SSRVE.With regard to an efficient construction of SSRVEs, it is necessary to consider statistical measures which can be computed in reasonable time and which provide sufficient information of the real microstructure.Minkowski functionals are analyzed as possible basis for statistical descriptors of microstructures and compared with other well-known statistical measures to investigate the performance. In order to emphasize the general importance of considering microstructural features by more sophisticated measures than basic ones, i.e. volume fraction, an analysis of upper bounds on the error of statistical measures and mechanical response is presented. © Springer International Publishing Switzerland 2015.

In this paper an extended optimization procedure is proposed for the construction of statistically similar RVEs (SSRVEs) which are defined as artificial microstructures showing a lower complexity than the associated real microstructures. This enables a computationally efficient discretization required for numerical calculations of microscopic boundary value problems and leads therefore to more efficient computational two-scale schemes. The optimization procedure is staggered and consists of an outer and an inner optimization problem. The outer problem treats different types of morphology parameterizations, different sets of statistical measures and different sets of weighting factors needed in the inner problem to minimize differences of mechanical errors that compare the response of the SSRVE with a target (real) microstructure. The inner problem minimizes differences of statistical measures describing the microstructure morphology for fixed parameterization type, statistical measures and weighting factors. The main contribution here is the analysis of new microstructure descriptors based on tensor-valued Minkowski functionals, whose numerical calculation requires less time compared to e.g. lineal-path functions. Thereby, a more efficient inner optimization problem can be realized and thus, an automated solution of the outer optimization problem becomes more practicable. Representative examples demonstrate the performance of the proposed method. It turns out that the evaluation of objective functions formulated in terms of the Minkowski functionals is almost 2000 times faster than functions taking into account lineal-path functions. © 2015 Elsevier Ltd. All rights reserved.

Advanced high strength steels, such as dual-phase steel (DP steel), provide advantages for engineering applications compared to conventional high strength steel. The main constituents of DP steel on the microscopic level are martensitic inclusions embedded in a ferritic matrix. A way to include these heterogeneities on the microscale into the modeling of the material is the FE2- method. Herein, in every integration point of a macroscopic finite element problem a microscopic boundary value problem is attached, which consists of a representative volume element (RVE) often defined as a segment of a real microstructure. From this representation, high computational costs arise due to the complexity of the discretization which can be circumvented by the use of a Statistically Similar RVE (SSRVE), which is governed by similar statistical features as the real target microstructure but shows a lower complexity. For the construction of such SSRVEs, an optimization problem is constructed which consists of a least-square functional taking into account the differences of statistical measures evaluated for the real microstructure and the SSRVE. This functional is minimized to identify the SSRVE for which the similarity in a statistical sense is optimal. The choice of the statistical measures considered in the least-square functional however play an important role. We focus on the construction of SSRVEs based on the volume fraction, lineal-path function and spectral density and check the performance in virtual tests. Here the response of the individual SSRVEs is compared with the real target microstructure. Further higher order measures, some specific Minkowski functionals, are investigated regarding their applicability and efficiency in the optimization process. © 2014 The Authors. Published by Elsevier Ltd.

In this paper a method is presented for the construction of two- and three-dimensional statistically similar representative volume elements (SSRVEs) that may be used in computational two-scale calculations. These SSRVEs are obtained by minimizing a least-square functional defined in terms of deviations of statistical measures describing the microstructure morphology and mechanical macroscopic quantities computed for a random target microstructure and for the SSRVE. It is shown that such SSRVEs serve as lower bounds in a statistical sense with respect to the difference of microstructure morphology. Moreover, an upper bound is defined by the maximum of the least-square functional. A staggered optimization procedure is proposed enabling a more efficient construction of SSRVEs. In an inner optimization problem we ensure that the statistical similarity of the microstructure morphology in the SSRVE compared with a target microstructure is as high as possible. Then, in an outer optimization problem we analyze mechanical stress–strain curves. As an example for the proposed method two- and three-dimensional SSRVEs are constructed for real microstructure data of a dual-phase steel. By comparing their mechanical response with the one of the real microstructure the performance of the method is documented. It turns out that the quality of the SSRVEs improves and converges to some limit value as the microstructure complexity of the SSRVE increases. This converging behavior gives reason to expect an optimal SSRVE at the limit for a chosen type of microstructure parameterization and set of statistical measures. © 2014, Springer-Verlag Berlin Heidelberg.

Purpose - The purpose of this paper is to present a computational framework for the simulation of patient-specific atherosclerotic arterial walls. Such simulations provide information regarding the mechanical stress distribution inside the arterial wall and may therefore enable improved medical indications for or against medical treatment. In detail, the paper aims to provide a framework which takes into account patient-specific geometric models obtained by in vivo measurements, as well as a fast solution strategy, giving realistic numerical results obtained in reasonable time. Design/methodology/approach - A method is proposed for the construction of three-dimensional geometrical models of atherosclerotic arteries based on intravascular ultrasound virtual histology data combined with angiographic X-ray images, which are obtained on a routine basis in the diagnostics and medical treatment of cardiovascular diseases. These models serve as a basis for finite element simulations where a large number of unknowns need to be calculated in reasonable time. Therefore, the finite element tearing and interconnecting-dual primal (FETI-DP) domain decomposition method is applied, to achieve an efficient parallel solution strategy. Findings - It is shown that three-dimensional models of patient-specific atherosclerotic arteries can be constructed from intravascular ultrasound virtual histology data. Furthermore, the application of the FETI-DP domain decomposition method leads to a fast numerical framework. In a numerical example, the importance of three-dimensional models and thereby fast solution algorithms is illustrated by showing that two-dimensional approximations differ significantly from the 3D solution. Originality/value - The decision for or against intravascular medical treatment of atherosclerotic arteries strongly depends on the mechanical situation of the arterial wall. The framework presented in this paper provides computer simulations of stress distributions, which therefore enable improved indications for medical methods of treatment. © Emerald Group Publishing Limited.

For the direct incorporation of micromechanical information into macroscopic boundary value problems, the FE2-method provides a suitable numerical framework. Here, an additional microscopic boundary value problem, based on evaluations of representative volume elements (RVEs), is attached to each Gauss point of the discretized macrostructure. However, for real random heterogeneous microstructures the choice of a "large" RVE with a huge number of inclusions is much too time-consuming for the simulation of complex macroscopic boundary value problems, especially when history-dependent constitutive laws are adapted for the description of individual phases of the mircostructure. Therefore, we propose a method for the construction of statistically similar RVEs (SSRVEs), which have much less complexity but reflect the essential morphological attributes of the microscale. If this procedure is prosperous, we arrive at the conclusion that the SSRVEs can be discretized with significantly less degrees of freedom than the original microstructure. The basic idea for the design of such SSRVEs is to minimize a least-square functional taking into account suitable statistical measures, which characterize the inclusion morphology. It turns out that the combination of the volume fraction and the spectral density seems not to be sufficient. Therefore, a hybrid reconstruction method, which takes into account the lineal-path function additionally, is proposed that yields promising realizations of the SSRVEs. In order to demonstrate the performance of the proposed procedure, we analyze several representative numerical examples. © 2010 Springer-Verlag.

A main problem of direct homogenization methods is the high computational cost, when we have to deal with large random microstructures. This leads to a large number of history variables which needs a large amount of memory, and moreover a high computation time. We focus on random microstructures consisting of a continuous matrix phase with a high number of embedded inclusions. In this contribution a method is presented for the construction of statistically similar representative volume elements (SSRVEs) which are characterized by a much less complexity than usual random RVEs in order to obtain an efficient simulation tool. The basic idea of the underlying procedure is to find a simplified SSRVE, whose selected statistical measures under consideration are as close as possible to the ones of the original microstructure. © 2010 Springer Science+Business Media B.V.

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. The 3D discretization results in a large system of equations; therefore, a parallel algorithm is applied to solve the equilibrium problem. Domain decomposition methods like the Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) method are designed to solve large linear systems of equations, that arise from the discretization of partial differential equations, on parallel computers. Their numerical and parallel scalability, as well as their robustness, also in the incompressible limit, has been shown theoretically and in numerical simulations. We are using a dual-primal FETI method to solve nonlinear, anisotropic elasticity problems for 3D models of arterial walls and present some preliminary numerical results. © 2009 Springer-Verlag.