Homotopy Research Articles

Structure design of variable camber wing trailing edge based on multi-block rotating mechanism

The variable camber trailing edge enables the aircraft to maintain the best aerodynamic performance throughout the flight envelope, so as to achieve the ultimate goal of reducing aircraft fuel consumption and air pollutant emissions. It is one of the important characteristics and development direction of the new generation civil aircraft. In order to solve the contradiction between the high load-bearing and the large deformation of the variable camber trailing edge of large aircraft, a structure scheme for variable camber trailing edge based on multi-block rotation was proposed. A parametric optimization method was established for smooth and continuous deformation of multi-block rotating mechanism. The multi-block rotating structure and driving system of variable camber trailing edge were designed, and the deformation function of the variable camber trailing edge demonstrator was preliminarily verified by ground test. The result shows that variable camber wing trailing edge designed by using the present optimization method of multi-block rotating mechanism can realize smooth and continuous morphing. The actual deformation range of the demonstration is 3.9 degrees up to 12.5 degrees down, and the error with the design target is 16.7%. It provides a design reference for solving the engineering application problem of variable camber structure for large aircraft.

Wilson lines and their Laurent positivity

For a marked surface Sigma and a semisimple algebraic group G of adjoint type, we study the Wilson line morphism g_{[c]}:{mathcal {P} }_{G,Sigma } rightarrow G associated with the homotopy class of an arc c connecting boundary intervals of Sigma , which is the comparison element of pinnings via parallel-transport. The matrix coefficients of the Wilson lines give a generating set of the function algebra mathcal {O}({mathcal {P} }_{G,Sigma }) when Sigma has no punctures. The Wilson lines have the multiplicative nature with respect to the gluing morphisms introduced by Goncharov–Shen [18], hence can be decomposed into triangular pieces with respect to a given ideal triangulation of Sigma . We show that the matrix coefficients c_{f,v}^V(g_{[c]}) give Laurent polynomials with positive integral coefficients in the Goncharov–Shen coordinate system associated with any decorated triangulation of Sigma , for suitable f and v.

Research on the breakage and fractal characteristics of calcareous sand

Calcareous sand is widely used in coastal engineering and island construction, which has the characteristic of easy crushing. A series of ring shear and one-dimensional compression creep tests were carried out for the crushing evolution under the influence of large deformation and time length effects. The mechanical characteristics and deformation properties of the single-graded specimens were analyzed, focusing on the crushing characteristics, and fractal properties under the action of different shear displacements, stress levels, initial gradations and creeping motion effects. The results show that the fractal dimension is positively correlated with crushing, the increase of shear displacement, stress level, and initial mean particle size has a facilitating effect on crushing. Under large deformation conditions, this promoting effect gradually tends to slow down, and the characteristic of limit fractal exists. The creep behavior reflects the continuous deformation characteristics of calcareous sand, and the specimen as a whole is still in the stage of crushing development, and the effect of time duration on the fractal is limited.

First-principles study of oxygen segregation and its effect on the embrittlement of molybdenum symmetrical tilt grain boundaries

Elemental segregation can pronouncedly affect the cohesion of grain boundary (GB), which shows tremendous potential in altering the thermal stability and mechanical properties of metals. In this work, we systematically investigate the segregation behavior of oxygen (O) atoms at a series of GBs in molybdenum (Mo), using first principles calculations, to reveal the effect of O segregation on the embrittlement of Mo GBs. It is found that O atoms energetically prefer to segregate at sites in both the GB core and the vicinity of the GB plane. The strengthening energy is strongly associated with the GB structure, and it linearly decreases with the continuous deformation of the GB. O segregation can lead to an increase in the volume of polyhedron sites and a decrease in the charge density among adjacent Mo atoms across the GB plane. The weakened Mo-Mo bonds are believed to disrupt the GB cohesion, thereby increasing the intergranular cleavage tendency of Mo. Interestingly, O segregation does not always trigger GB embrittlement. The Σ5(310)[001] symmetric tilt grain boundary (STGB) with an O atom doped at the cap trigonal prism (CTP) site possesses a strength as strong as the clean GB. O segregation promotes the generation of Mo-O bonds across the GB plane, which counteracts the adverse effect of weakened Mo-Mo bonds. This work deepens the understanding of the possible effect of O segregation on GB cohesion of Mo from the atomic and electronic scales, providing guidance for enhancing the mechanical properties of Mo via GB structure optimization.

Microstructural characterization and mechanical behavior of Mg-AZ31B/Al 6063 bimetallic sheets produced by combining continuous shear deformation with direct extrusion

To enhance the bonding effect and mechanical properties of hot-extruded Mg/Al bimetallic sheets, a novel bimetallic hot extrusion process called direct extrusion and continuous shear deformation (DECS) was proposed based on the shear deformation process. An extrusion die was designed and manufactured, and the influence of extrusion temperature on the microstructure and mechanical properties of the Mg/Al bimetallic sheet was investigated. Results showed that at an extrusion temperature of 330 ℃, defects and cracks appeared at the bonding interface, while favorable metallurgical bonding between Mg/Al bimetallic sheet was achieved at 370 ℃ and 410 ℃. The Mg layer exhibited a nearly complete dynamically recrystallized (DRXed) structure, with the average grain size increasing with higher extrusion temperature. A strong {0001} basal texture was observed in the Mg layer at 330 ℃ and 410 ℃. At 370 ℃, the combined action of shear strain between the bonding interfaces triggered the activation of multiple slip systems in the Mg layer, reducing the basal texture strength. The Al layer exhibited preferred orientations at various extrusion temperatures, particularly the multi-grain < 001 > and < 101 > orientations. The maximum shear strength of the formed sheet was determined to be 18.1 ± 3.7 MPa, with a maximum elongation of 38.4 %.

Two homotopy characterizations of G-ANR spaces for proper actions of Lie groups

For a Lie group G we consider the class G-M of all proper (in the sense of R. Palais) G-spaces that are metrizable by a G-invariant metric. It is proved that if each point x∈X of a proper G-space X admits a Gx-invariant neighbourhood U which is a Gx-ANE then X is a G-ANE, where Gx stands for the stabilizer of x. We give two equivariant homotopy characterizations of proper G-ANR spaces in the class G-M. One of them asserts that a G-space Y∈G-M is a G-ANR iff Y is locally G-contractible and every metrizable closed G-pair (X,A) with X∈G-M has the G-equivariant homotopy extension property with respect to Y (for short, property G-HEP). Another result establishes that a stronger homotopy property than G-HEP also characterizes G-ANR spaces in the class G-M.

FIRST-ORDER HOMOTOPICAL LOGIC

Abstract We introduce a homotopy-theoretic interpretation of intuitionistic first-order logic based on ideas from Homotopy Type Theory. We provide a categorical formulation of this interpretation using the framework of Grothendieck fibrations. We then use this formulation to prove the central property of this interpretation, namely homotopy invariance. To do this, we use the result from [8] that any Grothendieck fibration of the kind being considered can automatically be upgraded to a two-dimensional fibration, after which the invariance property is reduced to an abstract theorem concerning pseudonatural transformations of morphisms into two-dimensional fibrations.

The Free Euler Rigid Body Revisited

We review from a different perspective the approach and solution to the torque-free Euler equations, also called the free asymmetric top equations. We aim to simplify and broaden the study of the asymmetric free rigid body. This is an old but important integrable problem that has two first integrals: the energy and the angular momentum. We reduce this problem by eliminating the time as the independent variable in the three autonomous Euler equations written in cylindrical dimensionless variables, which allows a geometric study of the solution as a function of the cylindrical angle variable ψ, by means of continuous deformations dependent on the two independent parameters κ and e0. The parameter space is divided into six disjoint regions, whose boundaries are the separatices and degenerated cases. The solutions are given in terms of trigonometric functions of the independent cylindric angle ψ.

Universality of the homotopy interleaving distance

As a step towards establishing homotopy-theoretic foundations for topological data analysis (TDA), we introduce and study homotopy interleavings between filtered topological spaces. These are homotopy-invariant analogues of interleavings, objects commonly used in TDA to articulate stability and inference theorems. Intuitively, whereas a strict interleaving between filtered spaces X X and Y Y certifies that X X and Y Y are approximately isomorphic, a homotopy interleaving between X X and Y Y certifies that X X and Y Y are approximately weakly equivalent. The main results of this paper are that homotopy interleavings induce an extended pseudometric d H I d_{HI} on filtered spaces, and that this is the universal pseudometric satisfying natural stability and homotopy invariance axioms. To motivate these axioms, we also observe that d H I d_{HI} (or more generally, any pseudometric satisfying these two axioms and an additional “homology bounding” axiom) can be used to formulate lifts of several fundamental TDA theorems from the algebraic (homological) level to the level of filtered spaces. Finally, we consider the problem of establishing a persistent Whitehead theorem in terms of homotopy interleavings. We provide a counterexample to a naive formulation of the result.

On classifying simplicial modules

In this paper, we prove that the classifying functors W ¯ and Diag ° N are simplicially homotopy equivalent. Using this equivalency, we define classifying simplicial modules and give an alternative definition of (co)homology of crossed modules of algebras.

The development of cataclastic bands in high-porosity sandstones: Insights from ring shear experiments

Investigating the deformation mechanism of cataclastic bands in high-porosity sandstone is crucial for understanding the juxtaposition sealing ability of sandstones in a bed sequence. However, deformation bands that are developed in the field and rock cores do not reflect the evolutionary history of the host rock; therefore, the continuous deformation of high-porosity sandstone cataclastic bands cannot be observed. This paper analyzes the deformation mechanism that affects a cataclastic band when high-porosity sandstone faults form. Based on the latest independently developed high-pressure/low-speed ring shear experimental apparatus, the formation and evolutionary process of cataclastic bands in high-porosity sandstone were studied through the similarities among artificial cores. During the experiments, the effective normal stresses and fault displacement were used as single variables. After the experiments, plunger samples were drilled in the lateral direction, and thin sections were prepared to observe and analyze the thickness and particle characteristics of the deformation bands, thereby characterizing the deformation process of the damage zone. The experimental results reveal that high-porosity sandstone undergoes different intensities of cataclasis during the shearing process, and the evolutionary characteristics can be divided into four primary stages. The particles in the sandstone are successively subjected to rotation, rolling, cracking, cataclasis, and other changes after forced deformation. In terms of the macrostructure, a deformation band can be divided into two layers: an inner zone and an outer zone. From the host rock and the outer zone to the inner zone, the orientational arrangement of the particles along the shear direction becomes increasingly clear, and a greater displacement indicates a stronger orientational arrangement effect of the particles. In terms of the microstructures, the particle sizes in the deformation band are 2–3 orders of magnitude smaller than those in the host rock and exhibit greater roundness. With increasing shear displacement, the thickness of the cataclastic band first increases and then remains constant. With increasing effective normal stress, the thickness first decreases slightly, then increases and finally decreases slightly. Therefore, stress and displacement are important factors controlling cataclasis. The influence of displacement on cataclasis is clear at the initial stage, and stress determines the maximum intensity of cataclasis. Additionally, the increase in displacement will be favorable for roundness when the host rock is poorly sorted. These experimental results provide a theoretical basis for future studies on the effects of cataclastic bands and fluid flow.

Detecting electrochemical changes in a nickel–zinc battery by operando x‐ray computed‐tomography analysis during charge–discharge tests

AbstractThis study developed a method to detect and analyze deterioration in sections of a battery during charge–discharge tests in real time. This method is based on a time‐series analysis using x‐ray computed tomography, three‐dimensional reconstruction of the battery volume, and unsupervised machine learning. The developed method detects not only electrochemical changes in a battery through the conventional voltage‐capacity diagram but also physical changes such as the deterioration of the parts of a battery that cannot be found via human inspection directly from the sliced images of the three‐dimensional reconstructed volumes. Furthermore, the characteristics of these changes inside a battery can be captured through precise analysis using persistent homology, a mathematical machinery, at degrees 0 and 1. This demonstrates that our method can capture both continuous and discrete structural changes (e.g., a continuous deformation of active materials and compounds that are precipitated randomly in the electrolytes) within a battery. As a by‐product, the start of the venting system implemented near the anode of a battery can be detected using the method from a specific cycle during the charge–discharge tests.

The Cold Angular Rolling Process of Copper Sheets: Unraveling Plastic Deformation Behavior and Unveiling Microstructural Transformations

The cold angular rolling process (CARP) is being developed as a continuous severe plastic deformation technique, which can process metal sheets without any length limitations at room temperature. CARP contains cold rolling and equal‐channel angular process components. The sheet thickness is kept consistent before and after CARP, allowing multiple passes of the sheet. The desired microstructure and mechanical properties can be achieved in the processed metallic sheets. The current study is aimed to evaluate the capability of CARP by processing copper sheets with different sheet widths for repetitive passes. The CARP‐treated sheets are examined by lab‐scale X‐ray and high‐energy synchrotron X‐ray diffraction to investigate the evolution in dislocation density, texture, and strain anisotropy, and by tensile testing to identify the bulk mechanical properties. The digital image correlation method is applied to tensile testing so that strain localization within the sample gauge is visualized and deformation behavior is evaluated after yielding till postnecking by estimating the hardening exponent and strain hardening rate of the CARP‐treated sheet. Comparing the reported continuous and multiple‐step processes on Cu and its alloys, the present study confirms that the CARP is potentially a useful sheet process for strengthening ductile metals.

Vietoris thickenings and complexes are weakly homotopy equivalent

Vietoris thickenings and complexes are weakly homotopy equivalent

Optimal parameterizing manifolds for anticipating tipping points and higher-order critical transitions.

A general, variational approach to derive low-order reduced models from possibly non-autonomous systems is presented. The approach is based on the concept of optimal parameterizing manifold (OPM) that substitutes more classical notions of invariant or slow manifolds when the breakdown of "slaving" occurs, i.e., when the unresolved variables cannot be expressed as an exact functional of the resolved ones anymore. The OPM provides, within a given class of parameterizations of the unresolved variables, the manifold that averages out optimally these variables as conditioned on the resolved ones. The class of parameterizations retained here is that of continuous deformations of parameterizations rigorously valid near the onset of instability. These deformations are produced through the integration of auxiliary backward-forward systems built from the model's equations and lead to analytic formulas for parameterizations. In this modus operandi, the backward integration time is the key parameter to select per scale/variable to parameterize in order to derive the relevant parameterizations which are doomed to be no longer exact away from instability onset due to the breakdown of slaving typically encountered, e.g., for chaotic regimes. The selection criterion is then made through data-informed minimization of a least-square parameterization defect. It is thus shown through optimization of the backward integration time per scale/variable to parameterize, that skilled OPM reduced systems can be derived for predicting with accuracy higher-order critical transitions or catastrophic tipping phenomena, while training our parameterization formulas for regimes prior to these transitions takes place.

Wiener–Hopf factorization approach to a bulk-boundary correspondence and stability conditions for topological zero-energy modes

Both the physics and applications of fermionic symmetry-protected topological phases rely heavily on a principle known as bulk-boundary correspondence, which predicts the emergence of protected boundary-localized energy excitations (boundary states) if the bulk is topologically non-trivial. Current theoretical approaches formulate a bulk-boundary correspondence as an equality between a bulk and a boundary topological invariant, where the latter is a property of boundary states. However, such an equality does not offer insight into the stability or the sensitivity of the boundary states to external perturbations. To solve this problem, we adopt a technique known as the Wiener–Hopf factorization of matrix functions. Using this technique, we first provide an elementary proof of the equality of the bulk and the boundary invariants for one-dimensional systems with arbitrary boundary conditions in all Altland–Zirnbauer symmetry classes. This equality also applies to quasi-one-dimensional systems (e.g., junctions) formed by bulks belonging to the same symmetry class. We then show that only topologically non-trivial Hamiltonians can host stable zero-energy edge modes, where stability refers to continuous deformation of zero-energy excitations with external perturbations that preserve the symmetries of the class. By leveraging the Wiener–Hopf factorization, we establish bounds on the sensitivity of such stable zero-energy modes to external perturbations. Our results show that the Wiener–Hopf factorization is a natural tool to investigate bulk-boundary correspondence in quasi-one-dimensional fermionic symmetry-protected topological phases. Our results on the stability and sensitivity of zero-energy modes are especially valuable for applications, including Majorana-based topological quantum computing.

A Shear-Sliding Rigid-Flexible Coupled Skin Variable-Sweep Wing Design and Heat-Fluid-Structure Multifield Coupling Analysis

The variable-sweep wing is very attractive for cross-speed domain aircraft. The shear-sliding rigid-flexible coupled skin variable-sweep wing and its associated mechanism are designed and optimized. The variable-sweep wing has smooth continuous deformation and rigid-flexible coupling features. The calculation model of the sliding skin patch segmentation strategy is established, and the aerodynamic characteristics of the two-dimensional airfoil before and after the deformation of the wing skin are analyzed. According to the deformation characteristics of sliding skin, the configuration of the associated mechanism is determined, and the kinematic characteristics of the reference points of each skin are calculated. The kinematic simulation verifies the force of the mechanism model at the joint of skin surfaces during the deformation process. Considering the aerodynamic heat at supersonic speed, the heat transfer, heat distribution, and structural thermal modes between the flow field and the skin are calculated based on the finite element method. The dynamic characteristics of the swept wing with different flight speeds and different morphologies are analyzed. The natural frequencies are found to be reduced by about 30% to 50% compared to cold models at supersonic speeds. Based on the results of the thermal fluid-solid coupling calculation, the skeleton structure of the swept wing is optimized, and the skeleton structure with 25% mass reduction and better performance is obtained.

Strain‐Induced Toroidal Polar States in Wrinkled Ferroelectric Polymer by Phase‐Field Simulations

AbstractWrinkled ferroelectric polymer films with out‐of‐plane deformation exhibit exotic strain fields that differ from those of conventional epitaxial film systems, offering a new path toward manipulating electrical and mechanical behaviors in ferroelectrics. However, the direct observation of the domain structure evolution in ferroelectric polymer films during continuous deformation is challenging with current experimental approaches. In this study, the strain‐induced evolution of toroidal domain structures in organic poly(vinylidene fluoride‐ran‐trifluoroethylene) (P(VDF‐TrFE)) films with concentric toroidal wrinkle patterns is comprehensively investigated using phase‐field simulations. The results reveal morphology‐ and thickness‐dependent toroidal polar topology in wrinkled P(VDF‐TrFE) films. External strain loading can modulate this ferroelectric domain, which exhibits robust coupling behavior with a mixed strain field of periodic tensile and compressive strains. It is demonstrated that toroidal wrinkled ferroelectric films provide an effective and unique flexible platform for continuously controllable strain engineering in ferroelectrics and a theoretical framework with potential applications in flexible electronic devices.

What is the Seiberg–Witten map exactly?

We give a conceptual treatment of the Seiberg–Witten map as a quasi-isomorphism of differential graded algebras. The corresponding algebras have a very simple form, leading to explicit recurrence formulas for the quasi-isomorphism. Unlike most previous papers, our recurrence relations are nonperturbative in the parameter of non-commutativity. Using the language of quasi-isomorphisms, we give a homotopy classification of ambiguities in Seiberg–Witten maps. Possible generalizations to the Wess–Zumino complexes and some other algebras are briefly discussed.

Atomic understanding of elastic-plastic deformation and crack evolution for single crystal AlN during nanoscratch

Single crystal aluminum nitride (AlN) as a new generation of ultra-wide band gap semiconductor materials is the preferred substrate material for high power, high RF devices, but single crystal AlN is hard and difficult to process. To obtain high-quality substrate requires comprehensive understanding of the material deformation and removal mechanism. In this paper, the molecular dynamics simulation method is used to scratch the AlN on the (0001) plane. Set up multiple sets of scratch depths to study the elastic-plastic deformation, material removal mechanism and crack evolution of single crystal AlN. The simulation results show that with the increase of scratch depth, the single crystal AlN workpiece undergoes elastic deformation to continuous plastic deformation and then to brittle fracture. The deformation mode in the elastic stage is phase-change, from wurtzite-structure to graphite-like (GL) phase. The plastic deformation of single crystal AlN is achieved by dislocations moving on the basal and pyramidal planes, and horizontally extended stacking faults. There is almost no subsurface damage below the stacking faults. When the scratch depth exceeds 6 nm, the material will undergo brittle fracture, and the fracture surface is the cleavage surface {10-10} of single crystal AlN.