multi phase field method

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Phase-field modeling of multi-domain evolution in ferromagnetic shape memory alloys and of polycrystalline thin film growth

Phase-field modeling of multi-domain evolution in ferromagnetic shape memory alloys and of polycrystalline thin film growth

The Landau-Lifshitz-Gilbert Eq. (4.8), described in Chap. 4, is the widely accepted equation for the time-evolution of the magnetic moments in fer- romagnetic materials. There are some major issues that must be dealt with when a solution method is numerically implemented. The magnetic moments locally precede around the axis of a so called effective magnetic field, and in the equilibrium state the directions of the effective magnetic field and the local moments locally coincide. The demagnetization field, arising from the interdependent interaction of the magnetic moments, is a non-negligible addend of this effective field. The long-range character of the demagnetization field makes it hard to compute. This chapter deals with two major topics: The first section discusses problems with the nu- merical integration of Eq. (4.8) and summarizes an unconditional stable explicit one-step Euler integration scheme that will be used to compute the updates as proposed by Lewis and Nigam [112, 12]. The second sec- tion discusses the difficulties in conjunction with the calculation of the demagnetization field. Two solution schemes will be presented to calcu- late the demagnetization field efficiently for different boundary conditions, both will rely on FFT techniques: One solution method assumes a finitely extended specimen, the other an in all three spatial dimensions periodic RVE, cut out of a surrounding specimen.
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Phase-field simulations of multi-component solidification and coarsening based on thermodynamic datasets

Phase-field simulations of multi-component solidification and coarsening based on thermodynamic datasets

The work presented in this PhD thesis was carried out as part of the “Center of Computational Materials Science and Engineering (CCMSE)”, which was a joint research project of different universities in the German state of Baden-Württemberg. The aim of this project was to make progress in the interdisciplinary field of computer-aided material science and one of the investigated subjects was the formation of microstructures in casting processes. Because aluminum-silicon is an industrially relevant non-ferrous alloy system with excellent casting properties [3, 4, 5], it was chosen as an exemplary object of investigation. The particular purpose of the present PhD thesis is to simulate the solidification of this alloy by utilizing thermodynamic data provided by the CALPHAD method [6]. As part of the same project, the phase-field model based on the grand potential formulation of Choudhury and Nestler [7] was developed and implemented simultaneously to my doctoral studies. The core of this model are thermodynamic functions and the key to ensure their quantitativeness is the utilization of accurate data. For this reason, a large part of the present thesis is about the coupling with thermodynamic databases to provide the specific input parameters needed for the grand potential model. This thesis is intended to give an overview about possible coupling approaches and to analyze the different strategies theoretically and at the example of real systems. On the basis of the discussed coupling framework, the solidification of Al-Si under different conditions is investigated, proving the capability of the new phase-field model to cope with real alloy systems. As a validation of the model and its implementation, the simulation studies are designed for the comparison with analytical solutions, such as the well established theories of Mullins and Sekerka [8] or Lipton, Glicksman and Kurz [9]. As a further application, the discussed simulation framework is applied to study the diffusion controlled process of Ostwald ripening in a solid iron-copper alloy.
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Phase field modeling of ferroelectrics with
point defects

Phase field modeling of ferroelectrics with point defects

Bassiouny et al. first considered the multi-axial loading case. They introduced the Helmholtz free energy and used the yield surface in plasticity and work hardening as a source of reference. The theory described the initial polarization and the hysteresis loop under cyclic loading, but did not present the butterfly-loop [28, 29]. Based on the motion of domain walls, Huber et al. developed a micromechanical constitutive law, which is similar to crystal sliding [30]. Chen et al. develpoed a model which introduced volume fraction of domains as an internal variable, considering the interaction between crystals by a mean field method and obtained the behaviour of polycrystalline ferroelectrics [31]. Lu et al. conducted a study based on micromechanics and established a criterion taking into account the difference between the 90 ◦ switching and the 180 ◦ switching by a thermodynamic approach [32]. Shaikh et al. proposed a domain switching crite- rion for a generalized electromechanical loading based on an estimation of the existing domain switching criteria for ferroelectrics [33]. Kamlah and Tsakmakis constructed a phenomenolog- ical model of ferroelectricity for general loading histories [34]. Kamlah et al. also presented a complete phennomenological theory. They introduced several nonlinear functions to describe the switching behaviour and defined the domain switching driving forces for different loading cases [35]. McMeeking et al. presented a phenomenological theory. They defined the domain switching criterion similar to the yield surface in plasticity theory [36, 37]. Based on domain- switching mechanisms, Zhang et al. proposed a new domain-switching criterion [38].
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Free-boundary problem of crack dynamics: phase field modeling

Free-boundary problem of crack dynamics: phase field modeling

ior of such multi-cracked materials. Two analytical approximation schemes, the effective medium theory and the differential homogenization method, are frequently used in the literature to predict the elastic behavior. Since the behavior of cracks depends on the local elastic fields, which in turn depend sensitively on the geometric crack distribution, it is a priori unclear how important interaction effects are and how the analytical ap- proaches include them. Since our finite-difference based phase-field code could faithfully reproduce the behavior of a single crack, we compared simulations of static inclusions with the predictions we obtained with the effective medium theory and the differential homogenization method. As an additional test, we performed finite element simulations, which were always in good agreement with the results obtained by the phase-field calcu- lations. This was done for several two- and three-dimensional systems. We could show that it is favorable for the analytic treatment to convert plane strain problems into a two-dimensional representation wherever possible, since the resulting equations are much simpler to treat analytically. In all cases, we found that the effective medium theory underestimates the stiffness of the system, while the differential homogenization overesti- mates it. For a two-dimensional sheet containing circular holes, our numerical results were in good agreement with results found in the literature (Fig. 6.3). For the corresponding system in three dimensions, we found an initial increase of the effective Young’s modulus for auxetic materials under plane strain conditions (Fig. 6.5). For systems that contain cracks, our proposed approach simplifies the analytical calculations tremendously. The two-dimensional treatment predicts a simple exponential decay of the elastic modulus as a function of the crack density and exhibits the inability of the differential homogenization method to predict percolation. The simple conversion of the results for two dimensions into the three-dimensional representation prove to be equivalent to previous results from the literature, where the stiffness increase for materials with negative Poisson ratios was also seen (Fig. 6.7).
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Field-temperature phase diagram and entropy landscape of CeAuSb2

Field-temperature phase diagram and entropy landscape of CeAuSb2

P 4/nmm structure, moderate electrical anisotropy, and strong magnetic anisotropy [ 1 ]. Although it has not been widely stud- ied, it shows strong phenomenological similarities with other cerium-based compounds that have received intense interest. A major theme of study of these systems is to understand and tune the balance between Kondo and Ruderman-Kittel- Kasuya-Yosida (RKKY) interaction: RKKY interaction cou- ples localized spins and favors a magnetically ordered ground state, while strong Kondo interaction quenches local spins and yields a heavy Fermi liquid. Data presented in this paper, and comparison with other compounds, suggest that CeAuSb 2 is on the border, with the effects of both Kondo and RKKY inter- actions apparent in its bulk properties, but neither dominating. A field-temperature phase diagram, for field along the easy axis (the c axis), of CeAuSb 2 is shown in Fig. 1 . The indicated phase boundaries are from this work; however, many of its basic features were published in Refs. [ 2 , 3 ]. At zero field, there is a N´eel transition at T N = 6.6 K. As the field is increased, there are two first-order metamagnetic transitions, at 2.8 and 5.6 T; at 1.5 K the magnitudes of the metamagnetic jumps
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Multi-method decision support framework for supply network design

Multi-method decision support framework for supply network design

In the case study undertaken in this research, data analytics and green field anal- ysis were used to determine the cost items. Green field analysis is a GIS/center of gravity based approach (Russel and Taylor, 2010), which seeks to determine the geographic coordinates for a potential new facility. Computations are based on minimum transportation cost (calculated as “distance” * “Product Amount”) (AnyLogistix, 2018) in consideration of aggregated demand for each customer and product, customer locations (direct distance between customers and DCs/Ware- houses), and service distance (or number of facilities to locate) (AnyLogistix, 2018). To perform green field analysis, 2 years of operational information on historical demand (by location, amount, and time distribution), product flows, and costs were required. A template in MS Excel was shared with the company to facili- tate data provision. The company provided the historical data referring to the biennium 2016 and 2017.
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Phase-Field Modelling of Crack Propagation in Anisotropic Polycrystalline Materials

Phase-Field Modelling of Crack Propagation in Anisotropic Polycrystalline Materials

Polycrystalline materials are widely used in engineering and material science applications, e.g. automobile, aerospace or renewable energy. The macroscopic defects are generally strongly influenced by the fracture be- havior of the polycrystalline materials at meso- and microscopic level. In this paper, the proposed phase-field model for anisotropic fracture, which accounts for the preferential cleavage directions within each randomly ori- ented crystal, as well as an anisotropic material behavior with cubic symmetries, has been used to simulate the complex crack pattern in solar-grade polycrystalline silicon. Furthermore, the proposed phase-field model allows to distinguish the loading under tension and compression. The finite element implementation of the model has been realized by using a monolithic solution scheme. Three representative numerical examples are carried out, i.e. anisotropic crack propagation (i) in a sole material, (ii) in a bi-material with different crack orientation and (iii) in multi-grains with randomly distributed anisotropy. It is demonstrated that the proposed phase-field model is capable of characterizing fracture propagation in anisotropic solids under static loading.
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Peritectic phase transformation in the Fe-Mn and Fe-C system utilizing simulations with phase-field method

Peritectic phase transformation in the Fe-Mn and Fe-C system utilizing simulations with phase-field method

Table 2 shows the parameters used in the simulations under directional solidification conditions for the Fe–Mn system. Different simulations were performed with different values of ␦–␥ interfacial tension in the range presented in this table in order to check: (1) how the ␦–␥ interfacial ten- sion affects the ␥-phase thickness during the steady-state growth on ␦-phase (during the peritectic reaction) in the sim- ulation; and, (2) how the simulation results can be compared with the analytical theory presented briefly in Section 3 . Thus, the ␥-phase thicknesses were measured during the steady- state growth on the ␦-phase at a distance of 3.1 ␮m from the triple-point position. The position of the measurements was defined arbitrarily but considering the interferences of mea- surements excessively close to the triple-point or extremely far way of this point (the influence of the peritectic transfor- mation – the growth of ␥-phase into the liquid and ␦-phase). In addition to that, the liquid concentration next to the ␥–L interface was determined in the simulations and these deter- mined values were assumed suitable for being utilized as x L/ B in the analytical model. The other composition terms of the analytical model were determined in accordance with the phase diagram data considering x L/ B . Another assumption is that the speed of the ␥-phase growth is equal to the pulling speed of the unidirectional solidification on the steady-state. Thus, the analytical values and the simulation results were obtained for directional solidification with thermal gradient equal to 100 K/cm and pulling speed equal to 5.0 and 10.0 ␮m/s. The numerical domain for this investigation was set equal to 60 ␮m × 32 ␮m.
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Phase-Field Modeling of Relaxor Ferroelectrics and Related Composites

Phase-Field Modeling of Relaxor Ferroelectrics and Related Composites

Although extensive efforts have been made to explore the physical origin of relaxors, there are limited works on the discussion of the electromechanical couplings in relaxors, which are more important in applications. Phase-field model has been proved to be an efficient tool to study the evolution of domain structure in ferroelectrics, [ 13 , 14 , 15 , 16 ] and the order parameter can be fully coupled with electrical and mechanical quantities. Recently, the phase-field methods have also been employed in the field of relaxors, see Refs. [ 17 , 18 ]. Rather than generic re- laxor models, these works regards the relaxor features are the results of either point defects or localized nanoscale polar volumes inside a different phase. Moreover, the piezoresponse cannot be directly simulated from these models. Therefore, a generic fully coupled phase-field relaxor model which can reproduce and predict relaxor characteristics found in experiments is required. There are many important issues which could be investigated with the phase-field relaxor model. Here more attention is paid to the following two topics. (1) The role of relaxors in the relaxor/ferroelectric composite structure. Experimentally, the relaxor/ferroelectric composites show excellent large-signal piezoresponse. Understanding the role of relaxors in such compos- ites is important for the designing of relaxor-based piezoelectric devices. (2) The role of relaxors in the core-shell structure. The core-shell structures have been found in some relaxor systems, for instance, 0.75Bi 1/2 Na 1/2 TiO 3 -0.25SrTiO 3 (BNT-25ST). The phase-field relaxor model will help in understanding the coupling effect at the core-shell interface and the abnormal macro- scopic electromechanical behaviors in such core-shell structured relaxors. In the following, these two questions are elaborated in detail.
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Linear-quadratic mean-field-type games: A direct method

Linear-quadratic mean-field-type games: A direct method

Received: 4 January 2018; Accepted: 31 January 2018; Published: 12 February 2018 Abstract: In this work, a multi-person mean-field-type game is formulated and solved that is described by a linear jump-diffusion system of mean-field type and a quadratic cost functional involving the second moments, the square of the expected value of the state, and the control actions of all decision-makers. We propose a direct method to solve the game, team, and bargaining problems. This solution approach does not require solving the Bellman–Kolmogorov equations or backward–forward stochastic differential equations of Pontryagin’s type. The proposed method can be easily implemented by beginners and engineers who are new to the emerging field of mean-field-type game theory. The optimal strategies for decision-makers are shown to be in a state-and-mean-field feedback form. The optimal strategies are given explicitly as a sum of the well-known linear state-feedback strategy for the associated deterministic linear–quadratic game problem and a mean-field feedback term. The equilibrium cost of the decision-makers are explicitly derived using a simple direct method. Moreover, the equilibrium cost is a weighted sum of the initial variance and an integral of a weighted variance of the diffusion and the jump process. Finally, the method is used to compute global optimum strategies as well as saddle point strategies and Nash bargaining solution in state-and-mean-field feedback form.
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Regional gravity field recovery using the point mass method

Regional gravity field recovery using the point mass method

In practice, there are usually two strategies to determine the magnitudes and positions of the point mass RBFs. The first is that a set of point mass RBFs with an initial guess of their magnitudes and positions are given; then these parameters are updated iteratively. In this case, the crucial issue is how to choose the number of used point mass RBFs and their initial positions properly. This is a hard task, and no explicit rules can be given. Although, the larger the number of used point mass RBFs is, the better fit of the data can be obtained, it nevertheless raises the risk of over-parameterization as well as numerical instabilities. Therefore, this strategy will not be considered in this thesis. In the second strategy, the point mass RBFs are searched for one by one with a simultaneous determination of the magnitude and position for each RBF within an iterative nonlinear approach (e.g., Barthelmes, 1986; Lehmann, 1993; Claessens et al., 2001). Compared to the first strategy, the advantages of the second strategy are that: (1) the number, magnitudes, and positions of the RBFs can be determined automatically based on the data distribution and accuracy as well as signal variations, making the resulting point mass model adaptive to the data; (2) the solution of a medium- or large-scale nonlinear equation system is avoided (e.g., thousands or ten thousands of model parameters), and instead, a series of small-scale nonlinear equation systems has to be solved (e.g., tens or hundreds of model parameters). The issue of computational complexity and numerical instabilities for a small-scale non- linear equation system is insignificant, and hence the regularization term in the objective function given by Eq. (3.61) is not taken into consideration in the numerical computations in this thesis. How- ever, the cost of the automatic selection of the point mass RBFs for the modeling is that several model factors should be defined properly by the user, which will be discussed later. Often, when the search and optimization of the point mass RBFs are finished, the resulting magnitudes and positions for all searched RBFs can be regarded as the final solution. To improve the solution, we propose to do a further adjustment of the magnitudes for all selected point mass RBFs, while keeping the obtained positions fixed. This leads to the two-step point mass method.
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A Field-Matching Method for Sound Field Synthesis for Large Scale Sound Reinforcement Systems

A Field-Matching Method for Sound Field Synthesis for Large Scale Sound Reinforcement Systems

Keywords: Sound reinforcement, field-matching, sound field synthesis. 1. INTRODUCTION There are several popular methods to reproduce a sound field of a combination of virtual sources through an array of loudspeakers: WaveFieldSynthesis (1,2), Higher Order Ambisonics (HOA) (3), Vector Based Amplitude Panning (3) to name just a few. While being well established, each of them poses restrictions on the array geometry, loudspeakers characteristics or the audience area. Loudspeakers are often assumed to be monopoles or dipoles and have to be located right next to each other. This works well for small rooms but not so well for larger areas. Systems like HOA only work for a “sweet spot” and not for an audience area. Modifications of WFS have been developed to account for loudspeaker directivity, however, loudspeakers are required to be identical and located on a straight line or in a circle (4,5).
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Phase Calibration Of Multi-Baseline Pol-InSAR Data Stacks

Phase Calibration Of Multi-Baseline Pol-InSAR Data Stacks

As regards the radiometric fidelity, the analysis of the pixels F and G is noteworthy. Unfortunately, in absence of a ground truth, it is not possible to state exactly which is the inaccurate profile (i.e. the one still affected by miscalibrations), unless a further analysis is carried out. However, a possible reference can be the BF profile, as it is less prone to miscalibration residuals. In Fig. 4.8 both the BF and ABF profiles are reported after calibration with the ME and the CS method. Figure refers to the cell F, where a main volume contribution is present at 15m height and a less powerful echo from the ground scatterer is located at 5m. The BF profiles are almost superimposed (at least in a height interval close to the mainlobe) and the power relation between the two maxima is almost the same for both the ME and CS profiles. Since the BF is radiometrically linear, this power relation can be assumed as reference. Considering the ABF filtering, the CD-ME profile exhibits a power relation between the peaks in the order of the one in the corresponding BF profile, with a stronger contribution from the volume and a weaker one from the surface. The profile appears sharp. On the other hand, the shape of the CS profile is very different, and the two peaks are almost identically powerful. In this case the radiometric fidelity is not preserved, and the explanation of the change in the profile behavior has to be found in a residual contribution of phase errors which are well compensated after the ME-based calibration. In order to better investigate this features and to have a clear validation of the last conclusions, a further simulated analysis has been performed. In particular, an error free profile with two scatterers has been simulated in such a way that the BF spectrum of the simulated profile matches with the one retrieved from real data [see Fig. 4.9(a)]. Then, the ABF profile has been calculated and compared with the one obtained after CD-ME calibration. A good agreement between the two profiles has been found [see Fig. 4.9(b)]. Since the simulated one has been obtained in absence of errors, this means that the proposed method has been able to estimate the calibration phases better than the CS-based.
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A Method of Calculating Bound States in a Unified Field Model 

A Method of Calculating Bound States in a Unified Field Model 

In a model in which the usual elementary particles (leptons, quarks, photons, weak bosons, gluons, and so on) are bound states of truly elementary fermions we present a method for the [r]

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Signal interpolation method for quadrature phase-shifted Fabry-Perot interferometer

Signal interpolation method for quadrature phase-shifted Fabry-Perot interferometer

Quadrature phase-shifted Fabry-Perot interferometer is a kind of FPI which has the orthogonal signal pattern. Orthogonal signal is the most common way for the signal processing in industry application. Almost every signal processing of high speed commercial displacement measurement devices are based on this signal pattern. For this reason, QFPI have highly potential of high speed positioning in nanometer scale.

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Framing supply chain visibility through a multi-field approach

Framing supply chain visibility through a multi-field approach

In this paper, we seek to build a framework of the SCV concept. To extend the study beyond the boundaries of the SCM approach, we studied the no- tion of visibility in general. A multidisciplinary overview of the concept of visibility was performed and complemented by an exploratory study in an industrial company in order to gather field experiences concerning in- stances of a lack of visibility. At each step of the research, we sought to an- swer the following questions: why is it important to have visibility? (RQ1); what do we want to have visibility of? (RQ2); what factors, conditions and/or tools could facilitate or hinder visibility? (RQ3).
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Framing supply chain visibility through a multi-field approach

Framing supply chain visibility through a multi-field approach

In this paper, we seek to build a framework of the SCV concept. To extend the study beyond the boundaries of the SCM approach, we studied the no- tion of visibility in general. A multidisciplinary overview of the concept of visibility was performed and complemented by an exploratory study in an industrial company in order to gather field experiences concerning in- stances of a lack of visibility. At each step of the research, we sought to an- swer the following questions: why is it important to have visibility? (RQ1); what do we want to have visibility of? (RQ2); what factors, conditions and/or tools could facilitate or hinder visibility? (RQ3).
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Color current induced by gluons in background field method of QCD

Color current induced by gluons in background field method of QCD

terms [1]. This formalism needs to be extended by incorporating classical chromofield. The evolution of partons can be obtained by solving the Schwinger-Dyson equation defined on a closed-time-path. This is formidable task because it involves a non-local non-linear integrodifferential equation and because of its quantum nature. There are two scales in the system: the quantum (microscopic) scale and the statistical-kinetic (macroscopic) one. When the statistical-kinetic scale is much larger than the quantum one, the Schwinger-Dyson equation may be recasted into a much simpler form of the kinetic Boltzmann equation by a gradient expansion. The Boltzmann equation describes evolution of the particle distribution function in the phase space of momentum-coordinate, and can be solved numerically for practical purpose. For a non-equilibrium system of quarks and gluons in a chromofield, the distribution function also depends on the classical color charge of the parton, since the color is exchanged between the chromofield and partons and among partons themselves. In this case, the Boltzmann equation also describes the evolution of the parton distribution function in the color space [2,3]. While the Boltzmann equation describes the kinetic and color evolution of the hard parton system, soft partons are normally treated as a coherent classical field whose evolution may be described by a equation which is similar to Yang- Mills equation. Therefore, one should study the transport problem for hard partons in the presence of a classical background field. For example, in high energy heavy-ion collisions, minijets (which are hard partons) are initially produced and then propagate in a classical chromofield created by the soft partons [4–6]. In this situation it is necessary to derive the equation of the gluon and the classical chromofield to study the formation and equilibration of quark-gluon plasma. Hence we see that the issues we want to address in this paper have practical implication in ultra-relativistic heavy ion collisions.
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Image Space Tensor Field Visualization Using a LIC-like Method

Image Space Tensor Field Visualization Using a LIC-like Method

allowing it to dynamically create geometry on the GPU. How these shaders may be useful for solving the mentioned problem has to be investigated. Due to its image space nature, our approach is not able to achieve the ex- actly same visual results during advection as the LIC algorithm would achieve. This could cause small areas of the composited surface being blurry under certain awkward perspectives. This may be solved using a more sophisticated integra- tion method during advection in combination with textures, able to store real un- clamped floats with their full precision, which is not the case for standard textures. Another feature, needing more investigation, is the variation of the spot den- sity and spot size for the input noise texture mapped to the geometry. By doing this, it will be possible to let certain metrics, like the eigenvalue field, further in- fluence the input texture, resulting in a more sophisticated visual representation of the tensor field’s physical properties.
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Quantum phase transitions in magnetic systems: application of coupled cluster method

Quantum phase transitions in magnetic systems: application of coupled cluster method

Quantum phase transitions between semiclassical magnetically ordered phases and magnetically disordered quantum phases which are driven by frustration attract much interest, see, e.g., Ref. [2]. In particular, frustration may lead to the breakdown of semi- classical N´ eel LRO in 2D quantum antiferromagnets. There are a variety of models which are known to exhibit the so-called frustration effects. A typical model is the Shastry-Sutherland antiferromagnet introduced in the eighties [74], which has special arrangement of frustrating next-nearest-neighbor J 2 bonds on the square lattice, cf. Fig. 4.2. We note that for bonds of equal strength, i.e., J 1 = J 2 , the Shastry-Sutherland model is equivalent to a Heisenberg model on one of the eleven uniform Archimedean lattices [1]. Although the initial motivation to study this special frustrated square- lattice antiferromagnet is related to the existence of a simple singlet-product eigenstate (which becomes the ground state (GS) for strong frustration), the renewed interest in the last years was stimulated by the discovering of the new quantum phase in SrCu(BO 3 ) 2 [14,75] which can be understood in terms of the Shastry-Sutherland model (see Fig. 4.1). Although the GS of this model in the limit of small frustration J 2 and large J 2 is well understood, the GS phase at moderate J 2 is still a matter of discussion.
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