Finite Distortion Research Articles

Origin of the monoclinic distortion and its impact on the electronic properties in KO2

We use the density functional theory and lattice dynamics calculations to investigate the properties of potassium superoxide ${\mathrm{KO}}_{2}$ in which spin, orbital, and lattice degrees of freedom are interrelated and determine the low-temperature phase. After calculating phonon dispersion relations in the high-temperature tetragonal $I4/mmm$ structure, we identify a soft phonon mode leading to the monoclinic $C2/c$ symmetry and optimize the crystal geometry resulting from this mode. Thus we reveal a displacive character of the structural transition with the group-subgroup relation between the tetragonal and monoclinic phases. We compare the electronic structure of ${\mathrm{KO}}_{2}$ with antiferromagnetic spin order in the tetragonal and monoclinic phases. We emphasize that realistic treatment of the electronic structure requires including the local Coulomb interaction $U$ in the valence orbitals of the ${\mathrm{O}}_{2}^{\ensuremath{-}}$ ions. The presence of the ``Hubbard'' $U$ leads to the gap opening at the Fermi energy in the tetragonal structure without orbital order but with weak spin-orbit interaction. We remark that the gap opening in the tetragonal phase could also be obtained when the orbital order is initiated in the calculations with a realistic value of $U$. Finally, we show that the local Coulomb interactions and the finite lattice distortion, which together lead to the orbital order via the Jahn-Teller effect, are responsible for the enhanced insulating gap in the monoclinic structure.

Global Invertibility for Orientation-Preserving Sobolev Maps via Invertibility on or Near the Boundary

By a result of John Ball (1981), a locally orientation preserving Sobolev map is almost everywhere globally invertible whenever its boundary values admit a homeomorphic extension. As shown here for any dimension, the conclusions of Ball's theorem and related results can be reached while completely avoiding the problem of homeomorphic extension. For suitable domains, it is enough to know that the trace is invertible on the boundary or can be uniformly approximated by such maps. An application in Nonlinear Elasticity is the existence of homeomorphic minimizers with finite distortion whose boundary values are not fixed. As a tool in the proofs, strictly orientation-preserving maps and their global invertibility properties are studied from a purely topological point of view.

A dynamic phase field model with no attenuation of wave speed for rapid fracture instability in hyperelastic materials

Numerical experiments on phase field modeling of the fracture in pre-strained hyperelastic materials reveal that the classical mechanical-based dynamic phase field model is ineffective in the framework of non-linear deformation. The aspiration to gain insight into rapid fracture instability motivated us to develop a novel dynamic phase field model characterized by wave velocity invariance, enabling crack propagation at a velocity approach to the asymptotic limit. Given that the numerical treatment of rapid fractures involves extremely high spatiotemporal resolution, robust explicit dynamics and a tried-and-tested multi-level hybrid adaptive mesh algorithm are invoked. More crucially, an original adaptive distorted mesh removal scheme (ADMR) was developed to cope with the intractable finite element mesh distortion problem in large deformation fractures. The detailed numerical implementation for entire procedures is outlined, and its reliability is verified by two quasi-static fracture benchmarks. Utilizing the proposed model and innovative algorithms, the arresting ultrahigh-speed crack oscillation and tip-splitting instabilities captured in the fracture experiments of brittle gels were successfully reproduced. The critical crack velocity at the onset of the instability is also identified, close to the experimental measurements.

Sobolev mappings and moduli inequalities on Carnot groups

We study the mappings that satisfy moduli inequalities on Carnot groups. We prove that the homeomorphisms satisfying the moduli inequalities ($Q$-homeomor\-phisms) with a locally integrable function $Q$ are Sobolev mappings. On this base in the frameworks of the weak inverse mapping theorem, we prove that, on the Carnot groups $\mathbb G,$ the mappings inverse to Sobolev homeomorphisms of finite distortion of the class $W^1_{\nu,\loc}(\Omega;\Omega')$ belong to the Sobolev class $W^1_{1,\loc}(\Omega';\Omega)$.

Detecting quadrupole: a hidden source of magnetic anisotropy for Manganese alloys

Mn-based alloys exhibit unique properties in the spintronics materials possessing perpendicular magnetic anisotropy (PMA) beyond the Fe and Co-based alloys. It is desired to figure out the quantum physics of PMA inherent to Mn-based alloys, which have never been reported. Here, the origin of PMA in ferrimagnetic Mn3− δ Ga ordered alloys is investigated to resolve antiparallel-coupled Mn sites using x-ray magnetic circular and linear dichroism (XMCD/XMLD) and a first-principles calculation. We found that the contribution of orbital magnetic moments in PMA is small from XMCD and that the finite quadrupole-like orbital distortion through spin-flipped electron hopping is dominant from XMLD and theoretical calculations. These findings suggest that the spin-flipped orbital quadrupole formations originate from the PMA in Mn3− δ Ga and bring the paradigm shift in the researches of PMA materials using x-ray magnetic spectroscopies.

Mappings with finite length distortion and Riemann surfaces

Mappings with finite length distortion and Riemann surfaces

Mappings with Finite Length Distortion and Prime Ends on Riemann Surfaces

Abstract The present paper is a continuation of our research that was devoted to the theory of the boundary behavior of mappings in the Sobolev classes (mappings with generalized derivatives) on Riemann surfaces. Here we develop the theory of the boundary behavior of the mappings in the class of FLD (mappings with finite length distortion) first introduced for the Euclidean spaces in the article of Martio–Ryazanov–Srebro–Yakubov at 2004 and then included in the known book of these authors at 2009 on the modern mapping theory. As was shown in the recent papers of Kovtonyuk–Petkov–Ryazanov at 2017, such mappings, generally speaking, are not mappings in the Sobolev classes, because their first partial derivatives can be not locally integrable. At the same time, this class is a natural generalization of the well-known significant classes of isometries and quasiisometries. We prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extensions to the boundary of the mappings with finite length distortion between domains on Riemann surfaces by Caratheodory prime ends. The criterion for the continuous extension of the inverse mapping to the boundary is turned out to be the very simple condition on the integrability of the dilatations in the first power. The criteria for the continuous extension of the direct mappings to the boundary have a much more refined nature. One of such criteria is the existence of a majorant for the dilatation in the class of functions with finite mean oscillation, i.e., having a finite mean deviation from its mean value over infinitesimal disks centered at boundary points. As consequences, the corresponding criteria for a homeomorphic extension of mappings with finite length distortion to the closures of domains by Caratheodory prime ends are obtained.

Multi-Scale Structural Analysis on Rubber Seal for Battery Pack

<div class="section abstract"><div class="htmlview paragraph">A rubber sealing for a water-cooled battery pack plays a significant role to prevent water immersion into the inside of the pack. The appropriate design including the adjacent parts achieves a weight reduction of the battery pack by reducing the battery tray thickness and the quantity of bolts used in the whole battery pack.</div><div class="htmlview paragraph">Generally, finite element analysis (FEA) is effective for the design optimization before proto-typing. However, the application to the sealing for a battery pack requires a large scale analysis, including the complicated contacts and large deformation of the rubber sealing, and results in unpractically long computation time and frequent computation errors due to the finite element distortion.</div><div class="htmlview paragraph">A multi-scale structural analysis and the process on the rubber sealing for the battery pack has been developed to solve the above issues. This approach consists of 3 steps, which are single-unit, entire-scale and detailed structural analysis.</div><div class="htmlview paragraph">The cross-section of rubber sealing was simplified as rectangular shape by modifying the mechanical property of the sealing to meet the reaction force characteristic with the original one through the Step 1. Using this simplified model, the entire-scale analysis including the whole battery pack was carried out to extract the area from the entire seal line at which the seal pressure decreases to less than the requirement or high stress of the battery tray and the cover occurs. Then, the detailed structural analysis at the extracted area was carried out for quantitative evaluation.</div><div class="htmlview paragraph">The developed structural analysis and its work flow can contribute to rubber sealing design optimization by shortening the computation time and reducing the computation errors. In this paper, the developed simulation methodology including the analysis flow are specifically presented. The simulation conditions and results for a battery pack of Battery Electric Vehicle (BEV) are shown as the case study.</div></div>

Mappings with finite length distortion and prime ends on Riemann surfaces

The present paper is a continuation of our research that was devoted to the theory of the boundary behavior of mappings in the Sobolev classes (mappings with generalized derivatives) on Riemann surfaces. Here we develop the theory of the boundary behavior of the mappings in the class of FLD (mappings with finite length distortion) first introduced for the Euclidean spaces in the article of Martio-Ryazanov-Srebro-Yakubov at 2004 and then included in the known book of these authors at 2009 on the modern mapping theory. As was shown in the recent papers of Kovtonyuk-Petkov-Ryazanov at 2017, such mappings, generally speaking, are not mappings in the Sobolev classes, because their first partial derivatives can be not locally integrable. At the same time, this class is a natural generalization of the well-known significant classes of isometries and quasiisometries. We prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extensions to the boundary of the mappings with finite length distortion between domains on Riemann surfaces by Caratheodory prime ends. The criterion for the continuous extension of the inverse mapping to the boundary is turned out to be the very simple condition on the integrability of the dilatations in the first power. The criteria for the continuous extension of the direct mappings to the boundary have a much more refined nature. One of such criteria is the existence of a majorant for the dilatation in the class of functions with finite mean oscillation, i.e., having a finite mean deviation from its mean value over infinitesimal disks centered at boundary points. As consequences, the corresponding criteria for a homeomorphic extension of mappings with finite length distortion to the closures of domains by Caratheodory prime ends are obtained.

On mappings with finite length distortion on Riemann surfaces

On mappings with finite length distortion on Riemann surfaces

On boundary extension of one class of mappings in terms of prime ends

Here we consider the classes of mappings of metric spaces that distort the modulus of families of paths similarly to Poletsky inequality. For domains, which are not locally connected at the boundaries, we obtain results on the boundary extension of the indicated mappings. We also investigate the local and global behaviorof mappings in the context of the equicontinuity of their families. The main statements of the article are proved under the condition that the majorant responsible for the distortion of the modulus of the families of paths has a finite mean oscillation at the corresponding points. The results are applicable to well-known classes of conformal and quasiconformal mappings as well as mappings with a finite distortion.

Creating and Flattening Cusp Singularities by Deformations of Bi-conformal Energy

Mappingsofbi-conformalenergyformthewidestclass of homeomorphisms that one can hope to build a viable extension of Geometric Function Theory with connections to mathematical models of Nonlinear Elasticity. Such mappings are exactly the ones with finite conformal energy and integrable inner distortion. It is in this way, that our studies extend the applications of quasiconformal homeomorphisms to the degenerate elliptic systems of PDEs. The present paper searches a bi-conformal variant of the Riemann Mapping Theorem, focusing on domains with exemplary singular boundaries that are not quasiballs. We establish the sharp description of boundary singularities that can be created and flattened by mappings of bi-conformal energy.

Optimal Extensions of Conformal Mappings from the Unit Disk to Cardioid-Type Domains

The conformal mapping f(z)=(z+1)^2 from {mathbb {D}} onto the standard cardioid has a homeomorphic extension of finite distortion to entire {mathbb {R}}^2 . We study the optimal regularity of such extensions, in terms of the integrability degree of the distortion and of the derivatives, and these for the inverse. We generalize all outcomes to the case of conformal mappings from {mathbb {D}} onto cardioid-type domains.

On mappings with finite length distortion on Riemann surfaces

The present paper is a natural continuation of our previous paper (2017) on the boundary behavior of mappings in the Sobolev classes on Riemann surfaces, where the reader will be able to find the corresponding historic comments and a discussion of many definitions and relevant results. The given paper was devoted to the theory of the boundary behavior of mappings with finite distortion by Iwaniec on Riemannian surfaces first introduced for the plane in the paper of Iwaniec T. and Sverak V. (1993) On mappings with integrable dilatation and then extended to the spatial case in the monograph of Iwaniec T. and Martin G. (2001) devoted to Geometric function theory and non-linear analysis. At the present paper, it is developed the theory of the boundary behavior of the so--called mappings with finite length distortion first introduced in the paper of Martio O., Ryazanov V., Srebro U. and Yakubov~E. (2004) in the spatial case, see also Chapter 8 in their monograph (2009) on Moduli in modern mapping theory. As it was shown in the paper of Kovtonyuk D., Petkov I. and Ryazanov V. (2017) On the boundary behavior of mappings with finite distortion in the plane, such mappings, generally speaking, are not mappings with finite distortion by Iwaniec because their first partial derivatives can be not locally integrable. At the same time, this class is a generalization of the known class of mappings with bounded distortion by Martio--Vaisala from their paper (1988). Moreover, this class contains as a subclass the so-called finitely bi-Lipschitz mappings introduced for the spatial case in the paper of Kovtonyuk D. and Ryazanov V. (2011) On the boundary behavior of generalized quasi-isometries, that in turn are a natural generalization of the well-known classes of bi-Lipschitz mappings as well as isometries and quasi-isometries. In the research of the local and boundary behavior of mappings with finite length distortion in the spatial case, the key fact was that they satisfy some modulus inequalities which was a motivation for the consideration more wide classes of mappings, in particular, the Q-homeomorphisms (2005) and the mappings with finite area distortion (2008). Hence it is natural that under the research of mappings with finite length distortion on Riemann surfaces we start from establishing the corresponding modulus inequalities that are the main tool for us. On this basis, we prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extension to the boundary of the mappings with finite length distortion between domains on arbitrary Riemann surfaces.

On behavior of inverse homeomorphisms in terms of prime ends

As is known, even conformal mappings of plane simply connected domains do not, generally speaking, have continuous boundary extension in the Euclidean sense. One of the minimum requirements necessary for such an extension is the local connectedness of the definition domain of the corresponding map on its boundary. Of course, quite a lot of simply connected domains do not have this property. For example, the unit disk with a cut along the positive part of the real axis is not locally connected at the boundary. In the same way, the mapped domain must also satisfy certain conditions necessary for the continuous extension of a mapping. The situation changes significantly if we are not talking about the Euclidean boundary behavior of mappings, but about extension in terms of the so-called prime ends. In this case, the domain of definition of mappings should be only regular, that is, this domain should be the image of a domain with a locally quasiconformal boundary under some quasiconformal mapping. A similar requirement also applies to the mapped domain. In this article, we study the equicontinuous families of maps at inner and boundary points in the case where the prime ends of the domain serve as boundary points. Relatively speaking, the paper consists of two parts, one of which contains a number of auxiliary statements, and the second, the final part of the work, contains the formulation of the main theorem and its proof. We consider a class of homeomorphisms of Euclidean space, inverse of which distort moduli of families of paths by the Poletsky type inequality. Note that these classes include most well-known mappings, such as conformal mappings, quasiconformal mappings, mappings with finite length and area distortion, and so on. It should be noted that under conformal mappings, distortion of the modulus of families of paths does not occur, therefore, when passing to inverse mappings, we remain in the class under study. A similar situation is in the case of quasiconformal mappings, since, as is known, the inverse mapping to a quasiconformal is also quasiconformal. In more general situations, the studied configurations can turn out to be much more complicated, in particular, the transition to inverse mappings can significantly change their properties (this is confirmed by specific examples of mappings, which are rather easy to construct in this case). This article is actually devoted to the study of this particular case, that is, when we are dealing with a certain family of homeomorphisms with an unbounded characteristic, in addition, mappings inverse to them are studied. In more detail, we consider mappings whose inverse satisfy the upper distortion estimate of the modulus of families of paths with integrable majorant. In the article, we proved that the families of the indicated mappings are equicontinuous both at the inner and boundary points of the domain, provided that the majorant responsible for the distortion of the modulus of the families of paths is integrable, besides that, the definition and mapped domains are regular, and the boundary points are prime ends of the definition domain. The results obtained in the paper are applicable to well-known classes of mappings, such as mappings with bounded and finite distortion, as well as to the Sobolev and Orlicz-Sobolev classes.

Injectivity almost everywhere and mappings with finite distortion in nonlinear elasticity

We show that a sufficient condition for the weak limit of a sequence of $$W^1_q$$-homeomorphisms with finite distortion to be almost everywhere injective for $$q \ge n-1$$, can be stated by means of composition operators. Applying this result, we study nonlinear elasticity problems with respect to these new classes of mappings. Furthermore, we impose loose growth conditions on the stored-energy function for the class of $$W^1_n$$-homeomorphisms with finite distortion and integrable inner as well as outer distortion coefficients.

Research on Image Compression Coding Technology

Absrtact. Image compression technology is the basis of all kinds of media compression and transmission, and its compression effect is directly related to the compression effect of media. In this paper, four commonly used image compression coding methods, namely, undistorted Huffman coding, run-length coding, finite distortion predictive coding and transform coding, are used to compress still images, and the efficiency of each coding method is analyzed. The experimental results show that the two lossless compression coding methods have remarkable compression effect, but the compression efficiency is low. Although the compression ratio of lossy compression coding method is high, the decompressed image has certain distortion.

On the Lower Estimate of the Distortion of Distance for One Class of Mappings

We study the behavior of one class of mappings with finite distortion in the vicinity of the origin. Under certain conditions imposed on the characteristic of quasiconformality, we establish a lower estimate for the distortion of distance under mappings of the indicated kind.

Microbubbles in therapeutic ultrasound

Microbubbles have been developed as ultrasound contrast agents as an emulsion of stabilized gas bubbles that can be injected systemically and circulate throughout the body. Contrast between vascular structures and other tissues is due to the high echogenicity of the bubbles and the significantly different nonlinear response of these bubbles compared to nonlinear signals due to finite amplitude distortion of the ultrasound field. Microbubbles can also be introduced as superheated perfluorocarbon droplets that when activated by ultrasound, vaporize to form gas bubbles in a process termed acoustic droplet vaporization (ADV). The droplets have a range of sizes depending on their purpose in diagnostic and therapeutic application. For therapy, such droplets can be used as vascular agents, can be extravascular such as for delivery in tumor tissue or even formulated into implantable constructs used in tissue engineering and used for spatial/temporal control of factors affecting incorporation of the construct. In each case, formulation can include encapsulate drugs for local release. When bubble are present in an ultrasound field, microbubbles can also affect tissue heating and can undergo stable and/or inertial cavitation resulting in physical effects to tissue ranging from petechial hemorrhage sites to complete cellular disruption. The latter is termed Histotripsy and has developed into an alternative treatment strategy to ultrasound therapies based on the heating of tissue. This presentation will discuss how microbubbles are being used in medical ultrasound and the advances being made in both diagnosis and therapy.

On local behavior of one class of inverse mappings

As is known, the local behavior of maps is one of the most important problems of analysis. This, in particular, relates to the study of mappings with bounded and finite distortion, which have been actively studied recently. As for this work, here we solve the problem of the behavior of maps, the inverse of which satisfies the Poletsky inequality. The main result is the statement about the equicontinuity of the indicated mappings inside the domain in the case when the majorant corresponding to the distortion of the module under the mapping is integrable in the original domain. It should be emphasized that the proof of this result is largely geometric, at the same time, it uses only the conditions of boundedness of the direct and mapped domains and does not involve any requirements on their boundaries. The study of families of mappings inverse to a given class may turn out to be trivial if we are talking about quasiconformal mappings. In the latter case, we do not go beyond the limits of the class under study in the transition to inverse maps. Nevertheless, when studying mappings with unbounded characteristic, this question is quite substantial, as simple examples of the corresponding classes show. The idea of the proof of the main result is based on the fact that the inner points of an arbitrary domain are weakly flat. The last statement can be called the Väisälä lemma, which was established in his monograph and related to families of curves joining two continua between the plates of a spherical condenser. The proof is also based on the fact that the module of families of curves joining two converging continua in a good domain must tend to infinity. In this case, the neighborhood of some inner point of the mapped domain serves as ''good'' region, in which we check the equicontinuity of the inverse family of maps. The results of this article are applicable to many other classes of mappings such as mappings with a finite distortion in the sense of Iwaniec, Sobolev classes on the plane and in space, and so on.