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.
In this work, the parallel scalability of the finite element program for the sim- ulation of fracture in elastic-plastic solids is analyzed. The in house finite element program, from here on referred to as PLEANv1.0, is written as a user element and a user material model for the research code Finite Element Analysis Program (FEAP) developed by Prof. R. L. Taylor from the University of California Berke- ley. After introducing the fundamentals of the background theory of the finite element method, the phasefieldmethod of fracture and the underlying constitu- tive relations of elastic-plastic solids, the scalability analysis is performed by solv- ing a mechanical problem with 3D unit cube geometry and uniaxial tensile load. The scalability and performance of the PLEANv1.0 is obtained for computations performed over a high performance computing (HPC) cluster using 576 CPUs. A comparison of the scalability and performance of PLEANv1.0 with the research code FEAP for different solution options is also discussed. The promising scala- bility of PLEANv1.0 allows solution of very large problems in a computationally efficient manner by using the HPC cluster.
the interface directly. Instead, it treats the interface as a diffuse boundary of finite, but not infinitely small thickness, and derives the model equations from an energy functional describing the total energy of the examined sys- tem. Thus, it is no longer necessary to work out the precise parameters for the sharp interface formulation for numerical simulation. However, it is possible to derive the sharp interface formulation from the phase-field model equations via an asymptotic analysis by expanding and matching the model equations both in the bulk and near the interface (in the latter case via a coordinate transformation in which the width of the interface is approaching zero) - which is a useful tool for making quantitative statements about the kinetics of solidification phenomena [24, 74]. Still, as the dynamics of phase transitions are usually highly nonlinear, numerical simulations are usually required for deeper understanding. In general, the phase-fieldmethod has proved to be a powerful tool for studying front evolution problems under multiphysical influences [23, 28], oftentimes accompanied by the formation of several distinct phases [28, 62] or grains of multiple orientations [3, 35] and interactions of different governing mechanisms over several time and length scales [17, 26].
Then we investigated the coupled problem of diffusion-limited solid-state transforma- tions in the presence of lattice strain. Apart from phasefield simulations in a chan- nel geometry, these transformations were also studied in free space using steady state Green’s function methods. In comparison to the accurate and computationally very effi- cient Green’s function technique, the fully dynamic phasefieldmethod had the advantage of higher flexibility. In particular, the phasefield simulations cover also the transient states and are, for instance, easily applied to the more complicated scenario of collective growth of several thermally and elastically interacting seeds. However, restricting to steady states in relatively wide channels the two methods exhibit a comparable behavior. In studying single and bicrystal growth for a number of different structural transformations, we found that the arising elastic effects have a strong influence on the pattern selection. The differ- ent structural transitions lead to a very rich behavior, and we obtained a surprisingly large variety of patterns already for the relatively simple model considered here. Interestingly, some of the selected steady state velocities were found to be much higher than the corre- sponding velocities from classical dendritic growth selected by the anisotropy of surface tension.
Cam-Lai Nguyen 1 , Abdel Hassan Sweidan 1 , Yousef Heider 1,∗ , and Bernd Markert 1
1 Institute of General Mechanics, RWTH Aachen University. Eilfschornsteinstr. 18, 52062, Aachen, Germany
In this work, the problem of brittle fracture in a fluid-saturated porous material is extended by considering the non-isothermal states of the sample. The temperature field will affect the problem in two aspects: 1) Temperature-dependent material param- eters, such as elasticity modulus (E) and critical energy release rate (Gc). 2) Thermal expansion due to thermo-mechanical volume coupling. In hydraulic fracturing, we further study the effect of the temperature difference between the injected fluid and the surrounding porous media ambient on the crack behavior. The modeling of the porous media domain is based on the macroscopic theory of porous media (TPM), whereas the phase-fieldmethod (PFM) is applied to approximate the sharp crack edges by diffusive ones. In the numerical implementation, the coupled system of partial differential equations will be solved using the FEM in order to simulate the heat transition in the crack and non-crack regions.
In this contribution, we firstly provided a brief overview of the phase-field model for crack propagation in isotropic and anisotropic materials, respectively. Next, we presented the phase-field model for anisotropic fracture which can distinguish the loading under tension and compression. The fully coupled monolithic solution scheme within the finite element framework was formulated. Comprehensive parameter studies for the proposed phase-field model have been done and the results have been analyzed. Simulations of the anisotropic fracture within the lower value of the crack orientation θ are also considered, though the resulting crack path is no longer in alignment with the predefined crack orientation, and the opposite direction with fluctuating crack paths can be observed in the results. Furthermore, it is necessary to account for such phenomena that the widely used anisotropic materials (e.g. woods) consist of the lower value of the crack orientation in the structure and it can play an important role in material design processes. This topics will be addressed elsewhere. Representative numerical examples of the crack propagation in solar-grade polycrystalline silicon are carried out which can validate its capability of modelling of inter- and transgranular fracture process. Last but not least, the damage and failure analysis of solar-grade polycrystalline silicon using phase-fieldmethod will also be compared with experimental results in future.
Understanding the mechanisms of dynamical mass generation and confinement are the cen- tral objectives of research related to the QCD phase diagram. While dynamical mass gener- ation is tied to the breaking of chiral symmetry, already the exact definition of confinement is controversial. Respectively, there are several scenarios in discussion how to describe con- finement from a fundamental point of view. One of these is motivated by a mutual order parameter, the Polyakov loop. Its expectation value is sensitive to center symmetry breaking and likewise indicates deconfinement . Additionally, Gattringer  found that the Polyakov loop can be related to the spectrum of the Dirac operator, if the boundary con- ditions are generalized. This relation is important since it can be seen in analogy to the Banks–Casher relation  that connects the chiral condensate to the infrared eigenvalues of the Dirac operator. Thus, the eigenvalues of the Dirac operator could serve as the connect- ing degrees of freedom between confinement and chiral symmetry breaking. In the course of research for a better understanding of the contribution of the separate eigenvalues [166–169], yet another observable connected to the Polyakov loop was discovered, the dressed Polyakov loop [170, 171]. This observable can be pictured as a collection of loops winding once around the temporal direction of the spacetime manifold. While the winding number is fixed by projection, in principle all possible detours in spatial directions contribute 11 . In the limit
In Section 2.4, we will give a precise derivation of the main model studied in this thesis, after we have introduced some necessary notation and definitions at the beginning of Chapter 2. Nevertheless, we give a first description of the mathematical setting of the main model for introductory purpose. The model involves three variables: the mechanical displacement u, the electric field φ and the polarization P . The former two variables are given by an elliptic piezo-system, with coefficients which are functions in variable P (see (2.14) below). Thus roughly speaking, once P is given, the variable (u, φ) can be uniquely determined by certain elliptic existence theory. Hence it suffices to find a solution P which fulfills the evolutionary law
of the nanowire surfaces with the wide-band gap materials effectively suppresses surface states, substantially improves optical quality of nanowires , and also provides a routine for fabrication of novel radial heterostructures. To summarise, a large amount of contributions and reviews on semiconductor nanowires reveals an upsurge of interest in their fundamental properties as well as possibilities for technological applications [10–14]. As a result, various nanowire-based functional building blocks for electronic and photonic devices have been already demonstrated. For instance, light-emitting diodes , solar cells , transistors , or room-temperature lasers . The rapid development of the field of nanowires may also result in various successful technological spin-offs, e.g. CrayoNano , and have an effect on some important aspects of our life such as an effective and sustainable energy conversion. In this work, of particular interest is the study of nanoscale phenomena and properties of GaAs semiconductor nanowires, which arise from their unique crystalline structure. Whereas bulk and thin film GaAs have a zinc blende equilibrium crystal structure, nanowires made of GaAs may also exhibit regions with a wurtzite type crystal phase. Since the symmetry of the crystal structure defines the major material properties of semiconductors, the wurtzite phase is expected to have a band gap, band offsets and effective masses that are different from those of the zinc blende phase of the same material. For GaAs nanowires, there is an ongoing debate about the exact values of the band gap of wurtzite GaAs and band offsets between zinc blende and wurtzite GaAs [19–25]. In addition, nanowires typically exhibit a mixture of zinc blende and wurtzite phases along the growth direction. Due to the different band gap energies of these phases, alternation of zinc blende and wurtzite segments produces crystal-phase heterostructures. Optical and electronic properties of such nanowires depend on the lateral extend and order of crystal-phase nanodomains, which have a typical length scale from one to a few tens of nanometers. Therefore, nanowires are considered as important model structures for fundamental studying optical and electronic properties of wurtzite type GaAs and zinc blende – wurtzite crystal-phase heterostructures. The experimental challenge of this study is to probe and resolve optical and electronic properties of GaAs nanowires with a mixture of zinc blende and wurtzite crystal phases alternating at the nanometer length scale.
In the present contribution a non-variational phase-field approach is presented which allows for the description of the damage evolution in the open-cell foam materials. The topological property characterising the microstructure such as the edge-connectivity is used as the basis for the development of the proposed approach. The numerical results show that the connectivity-based variable drives the evolution of the damage phasefield. The presented approach provides advantages in the description of the microstructural changes resulting from the damage process in open-cell structures. The possibility of incorporating the directional data of the microstructure by means of the orientation distribution function approximated with fabric tensors is examined. The performance of the proposed formulation and the effects of the model parameters on the overall behaviour of the model are shown by means of the respective numerical examples and parameter studies.
The behavior of electric charged particles in a classical electromagnetic field has been extensively studied in the literature. However, the behavior of color charged particles (quarks and gluons) in a classical chromofield has been less studied. The gluon case is particularly interesting because there is no analogue in the electromagnetic theory, since photons do not interact with the classical electromagnetic field. One important issue is to derive the equation of motion for the gluon in the presence of chromofield and to define and calculate the classical current produced by the gluon. As we know it is easy to define and understand the color current for quarks in the classical chromofield. But strictly defining the color current for gluons classically is not an easy task, mainly due to its quantum and bosonic nature. We plan to address some aspects of these issues in this paper.
In the earlier researches of FPI, its applications are only in quasi-static measurements, e.g. strain and thermal expansion measurements. For the conventional FPI, the direction cannot be determined instantly. Therefore, an additional aid for directional discrimination is necessary. For this reason, the conventional FPI which is using the fringe counting method (FCM) would not be able to determine the movement direction of the object [1, 2]. Figure 1 and 2 are the structure and interferometric signal of FCM FPI. This disadvantage is also one of the reasons that conventional FPI is hard to be utilized in dynamic length measurement.
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.
The audibility of constant phase shifts can be regarded as special issue of the audibility of phase distortion and group delay distortion, cf. [13–18], often evaluated with allpass filters. From these works it is known, that audibility is strongly dependent of the signal’s waveform and spectrum and the amount of the group delay in the critical bands. Generally, sensitivity for phase/group delay distortions decreases with increasing frequency. For low frequency content a different pitch and for high frequency content ringing and different lateralization is reported for group delay distortions. The polarity of highly transient signals plays a role for the audi- bility. It was often shown, that training on phase/group delay distorted audio content increases the sensitivity to detect them. To the authors’ knowledge to date, the perceptual impact of the constant phase shift has not been studied yet. It is of great interest whether the existence or absence of such a phase shift is audible, and in the special context of sound field synthesis, if this affects the authenticity of the synthesized sound fields. The paper discusses the signal processing fundamentals of discrete-time constant phase shift in Sec. II. In Sec. III a listening test is presented for selected audio content and phase shifts to initially evaluate the audibility of constant phase shifts. Sec. IV concludes the paper.
The development in laser science was one of the driving forces for strong-field physics. One of the most prominent processes here is the recollision mechanism, whereas the (semiclassical) simple man’s model (SMM) delivers an intuitive under- standing . In the first step an atom or molecule is ionized in the electric field of a laser pulse. Afterwards, the electron is driven away from the parent ion (step two) and recollides with it in the final third step. Besides the generation of an attosecond pulse, which is a result of the recombination between electron and ion, another important process can be observed - the emission of high-energy electrons due to elastic (re-)scattering. Similar to the photons in HHG, the electrons contain information about the interacting particles, allowing insights into the temporal dynamics of the sample as well as into its structure.
where B and B are closed, linear, self-adjoint, positive operators on a Hilbert space H . The ﬁrst positive result was obtained by Faˇsangov´a and Pr¨uss in [FP99, FP01], where the authors develop a method which combines techniques from nonlinear Volterra equations in ﬁnite dimensions (cf. [GLS90]) and harmonic analysis of vector-valued functions (cf. [Chi98]). The main problem of this approach is that in order to establish convergence to an equilibrium one has to assume that the set of stationary points of (0.0.3) is discrete, a condition that is not easy to verify and not fulﬁlled in general. Recently, Chill and Faˇsangov´a [CF05], using ideas from Dafermos [Daf70] and [AF01], were able to prove that under suitable conditions on the kernel a any global bounded solution u of (0.0.3) converges to a steady state, provided that the functional E satisﬁes the ojasiewicz-Simon inequality near some ϑ ∈ ω(u) . Note that the latter allows to avoid additional assumptions on the set of equilibria.
Since differences in the turbulent kinetic energy were evident, the ensemble and spatial- averaged turbulent kinetic energy were calculated to investigate the general progression more thoroughly. Figure 6.40 shows the turbulent kinetic energy and the standard deviation of the turbulent kinetic energy for the reference case and the four injection pressures at N = 1,500 rpm. It can be clearly seen that the ensemble and spatial- averaged turbulent kinetic energy is twice as high for the reference case compared to the flow field with injection during the early intake stroke. One explanation could be an increase in inertia since the flow field has to move the fuel droplets. The higher inertia acts as a low-pass filter that dampens the small scale fluctuations that cause an increase in turbulent kinetic energy during the intake stroke. However, the general progression of the turbulent kinetic energy for the reference and the injection cases is similar as a decrease in the turbulent kinetic energy is evident for increasing crank angles. Furthermore, the increase in the turbulent kinetic energy at late compression is evident for all cases. The injection cases reach values in the range of the reference case. Similar to the vorticity distribution, a higher injection pressure leads to a decreased turbulent kinetic energy. However, the deviation is within the standard deviation of the measurements and therefore not statistically significant. In conclusion, the injection suppresses small scale fluctuations since the general turbulent kinetic energy drops to values well below the reference value during the early intake stroke. However, the increase near tdc combustion, which is also reported in Borée  and Bücker et al. ), ensures a good mixture formation since the tumble decays into small scale vortical structures, which can be seen by a general increase in the turbulent kinetic energy. Influence of higher RPM