Finite Structures Research Articles

Thermoelectric Joule heating in crack problems in nano-sized structures

Crack problems in nano-sized elastic structures under the Joule heating are analyzed by the finite element method (FEM). The variational principle is applied for derivation of coupled electrothermomechanical governing equations. The temperature field is described by classical Fourier′s law and it is not influenced by mechanical field. Transient heat conduction is considered since it is the slowest process among all other coupled fields here. Quasi-static conditions are considered for mechanical and electrical fields. Due to higher derivatives in governing equations the conventional FEM cannot be applied. Therefore, the mixed FEM with the C0 continuous interpolation for displacements and strains is developed. The influence of the material characteristic length on variation of crack opening displacements in finite nano-sized structures under an electric current load is investigated.

Low frequency band gap and wave propagation mechanism of resonant hammer circular structure

The two circular structures with local resonance resonant hammers proposed in this paper have the comprehensive performance of small size, easy fabrication and low frequency vibration and sound damping. In the study, Bloch’s theorem is used to calculate the energy band curves of the two structures and to analyze the vibration forms of the structures at the edge of the band gap. At the same time, the integrated analysis means of phase constant surface, group velocity and phase velocity are innovatively used to explore the real propagation path of waves and provide experience for topology optimization. The position of resonant hammers, the mirror symmetry structure and the size of resonant hammers are adjusted to optimize the band gap for both structures. Finally, the vibration transmission characteristics of the finite period structure are calculated to verify the band gap and evaluate the vibration isolation performance. The results show that the new structure has low-frequency noise reduction performance, and the band gap frequency can be further reduced by topology optimization. This study presents an innovative approach for achieving low-frequency noise reduction, analysis of wave propagation paths and lightweight design.

Low frequency band gap and vibration suppression mechanism of innovative multiphase metamaterials

In order to achieve low frequency vibration and noise reduction, researchers have found and explored many types of metamaterial structures. Based on this, an innovative multiphase metamaterial structure composed of hard materials and soft materials is proposed in this paper. According to Bloch’s theorem, the band gap characteristics of the proposed structure are calculated using the finite element method, and the formation mechanism of the band gap is analyzed. In addition, the effects of the stiffness and density of the two materials on the band gap are studied in detail. Then, the phase velocity, group velocity and wave propagation direction are calculated according to a certain frequency of the specific dispersion curve, and the important information of wave propagation in the structure is obtained. Finally, according to the transmission characteristics of vibration in a finite periodic structure, it is verified that the existence of band gap can inhibit the vibration transmission, and it is further clarified that the proposed structure has a good performance in the design of low-frequency band gaps with large amplitude.

Looking towards the future: patient-specific computational modeling to optimize outcomes for transcatheter mitral valve repair.

Severe mitral valve regurgitation (MR) is a heart valve disease that progresses to end-stage congestive heart failure and death if left untreated. Surgical repair or replacement of the mitral valve (MV) remains the gold standard for treatment of severe MR, with repair techniques aiming to restore the native geometry of the MV. However, patients with extensive co-morbidities may be ineligible for surgical intervention. With the emergence of transcatheter MV repair (TMVR) treatment paradigms for MR will evolve. The longer-term outcomes of TMVR and its effectiveness compared to surgical repair remain unknown given the differing patient eligibility for either treatment at this time. Advances in computational modeling will elucidate answers to these questions, employing techniques such as finite element method and fluid structure interactions. Use of clinical imaging will permit patient-specific MV models to be created with high accuracy and replicate MV pathophysiology. It is anticipated that TMVR technology will gradually expand to treat lower-risk patient groups, thus pre-procedural computational modeling will play a crucial role guiding clinicians towards the optimal intervention. Additionally, concerted efforts to create MV models will establish atlases of pathologies and biomechanics profiles which could delineate which patient populations would best benefit from specific surgical vs. TMVR options. In this review, we describe recent literature on MV computational modeling, its relevance to MV repair techniques, and future directions for translational application of computational modeling for treatment of MR.

Optical properties of chiral three-dimensional photonic crystals

We perform a theoretical and numerical study of the optical properties of both direct and inverse three-dimensional (3D) chiral woodpile structures and a corresponding chiral Bragg stack, also known as a Reusch pile. We compute transmission spectra in the helical direction for finite crystals and photonic band structures, where we ensure the effective index of all structures to be the same. We find that both 3D structures show dual-band circular dichroism, where light with a particular circular polarization state---either left- or right-handed---reveals a broad stop band with major attenuation, whereas the other polarization state is transmitted nearly unimpeded. We observe such gaps in different frequency ranges with alternating handedness. The appearance of the circular-polarized gaps is successfully interpreted with a physical model wherein the circular polarization either corotates or counterrotates with the chiral structure, thereby effectively leading to a spatially dependent or a constant refractive index, and thus to the presence or absence of a gap. We find that the presence or absence of circularly polarized gaps is tuned by the transverse periods perpendicular to the chiral axis in both the 3D direct and inverse chiral woodpile crystal structures. The tunability of circular dichroism may find applications as on-chip sensors for circularly polarized light and in the on-chip detection of chiral molecules.

실시간 상태-공간 식별 기반 제어기를 이용한 유한 국부공진 메타물질 평판의 적응형 진동 저감

To improve the internal sound quality of an electric vehicle, the noises (tire and road noises), vibration, and discomfort should be reduced effectively. Compared to the conventional damping materials, the locally resonant metamaterial is advantageous in terms of reducing the vibration in the low frequency band and reducing the weight through a stop band, where vibration is not transmitted. However, the locally resonant metamaterial installed in a finite structure fails to form an ideal stop band. Additionally, there is a problem of occurrence vibration mode at an arbitrary frequency. Accordingly, there is a need for an active vibration control method capable of learning and actively reducing the vibration modes occurring at arbitrary frequencies within the stop band in real-time. In this study, a real-time state-space identification-based active vibration control method was proposed to reduce the vibration modes occurring at arbitrary frequencies within the stationary band of local resonance metamaterials. The active vibration control method proposed in this study, combined with the existing passive vibration reduction method, can be utilized to effectively reduce the vibration for random disturbance of a fluctuating system.

Gradient V-Shaped and N-Shaped Seismic Metamaterials.

Seismic metamaterials provide an innovative alternative in earthquake engineering by reducing the hazards from seismic waves without modifying the existing structures. Although many seismic metamaterials have been proposed, a design for a broad bandgap at low frequencies is still in demand. In this study, two novel seismic metamaterials, V- and N-shaped designs, are proposed. We found that by adding a line to the letter V, turning the V-shaped design into an N-shaped design, the bandgap can be broadened. Both the V- and N-shaped designs are arranged in a gradient pattern to combine the bandgaps from metamaterials with different heights. Using only concrete as the base material for the design makes the proposed seismic metamaterial cost effective. Finite element transient analysis and band structures are in good agreement, validating the accuracy of the numerical simulations. Surface waves are effectively attenuated over a broad range of low frequencies using the gradient V- and N-shaped seismic metamaterials.

Hermitian and non-Hermitian Weyl physics in synthetic three-dimensional piezoelectric phononic beams

Recently, the evolution of the Weyl point (WP) caused by the introduction of nonhermiticity into Weyl semimetals has aroused great research interest. We consider elastic flexural wave propagation in a phononic beam containing piezoelectric materials and introduce nonhermiticity through active regulation of external circuits. Considering a synthetic parameter space constituted by the one-dimensional Bloch wave vector and two geometrical parameters, we demonstrate that a double WP (DWP) arises at the band crossing. Then we study its evolution from the hermitic to nonhermitic situation under the effect of the active piezoelectric materials. We find that the DWP in the hermitic case evolves into a Weyl degenerate line and a Weyl hollow ring as concerns the real and imaginary parts of the Weyl frequencies, respectively. The formation mechanisms of the DWPs, lines, and rings are explained through the Hamiltonian of the system. Further, we observe the changes of the DWP and degenerate line in the transmission spectra of finite structures. Finally, we discuss the synthetic Fermi arc interface states through the analysis of the reflected phase vortices. In this paper, we provide insights into the high-dimensional Hermitian and non-Hermitian physics in elastic wave systems using synthetic dimensions.

Study on Band Gap Characteristics of Phononic Crystal Double-Layer Beam Structure Attached by Multilayer Cylinder

Intended for the vibration and noise control problems faced by many engineering fields, a fresh variety of phononic crystal beam structure was constructed by attaching a one-dimensional periodic multilayer cylinder to a double-layer beam structure. Utilizing the finite element method and the Bloch theorem, the vibration modes of the band structure, the critical point of the band gap and the associated finite structure’s vibration transmission is estimated, and then the band gap characteristics of the structure are comprehensively studied. The results show that reasonable parameter design can achieve vibration and noise control in a certain frequency range. Based on the modal analysis, the mechanism of band gap opening is revealed. By comparing the single-layer beam and double-layer beam with the same parameters, the advantages of the double-layer beam in vibration reduction and noise reduction are shown. The study’s findings offer a fresh concept for ship engineering disciplines including vibration and noise reduction technology.

Design of novel two-dimensional single-phase chiral phononic crystal assembly structures and study of bandgap mechanism

In this paper, novel two-dimensional single-phase three-ligament, four-ligament and six-ligament chiral structure models are proposed. The advantage is that it is easy to manufacture with few components, has good sound insulation and vibration damping performance, and is more practical. The dispersion information of the structure is calculated by combining Bloch's theorem and the finite element method. By varying the geometric parameters of the new structural single-cell and simulating the vibration transmission of the finite period structure to check the vibration suppression capability of the optimized structure. Three combined structures were constructed to compare their energy band properties with the base structure. The wave propagation characteristics within the combined structure are also analyzed by comprehensive means. It is shown that the optimized structure has a significantly lower bandgap down to below 1 kHz and a significantly wider bandgap width up to 3.6 kHz. The combined structure can open up a wider omnidirectional bandgap for multi-frequency and broadband vibration attenuation. The flexibility of the structure combination gives it a large potential for use, providing a new solution for broadband and multi-frequency vibration and noise reduction, while plane wave analysis provides theoretical support for structural optimization.

Fixed points on band structures of non-Hermitian models: Extended states in the bandgap and ideal superluminal tunneling

Space-time reflection symmetry (PT symmetry) in non-Hermitian electronic models has drawn much attention over the past decade mainly because it guarantees that the band structures calculated under open boundary conditions be the same as those calculated under periodic boundary conditions. PT symmetry in electromagnetic (EM) models, which are usually borrowed from electronic models, has also been of immense interest, mainly because it leads to ``exceptional'' parameter values below which non-Hermitian operators have real eigenvalues, although PT symmetry is not the sole symmetry that allows such exceptional parameter values. In this work, we examine one-dimensional PT-symmetric non-Hermitian EM models to introduce novel concepts and phenomena. We introduce the band-structure concept of ``fixed points,'' which leads to ``bidirectional'' reflection zeros in the corresponding finite structures, contrary to a common belief about the EM structures with PT symmetry (and without parity inversion and time-reversal symmetries). Some of the fixed points manifest themselves as, what we name, ``extended states in the bandgap'' on the band structure while some other fixed points are the ``turning points'' of the band structure. The extended states in the bandgap are in fact the dual of the well-known ``bound states in the continuum,'' while the turning points allow us to observe ``ideal'' superluminal tunneling in the corresponding finite structures. By ``ideal'' superluminal tunneling, we mean the case where not only the transmission coefficient has an almost uniform phase over a broad bandwidth, but also the magnitudes of the transmission and reflection coefficients are almost equal to unity and zero, respectively, over the bandwidth.

Algebraic Theories of AUNU Permutation Using Group Action with Non- Prime Order

Permutation groups have played important roles in method of enumerating combinatorial objects in recent times. However, the study of the applicability AUNU permutation groups using group action is challenging. In this paper, a new method of constructing group action using the subsequences of (123)- avoiding of AUNU Permutation patterns is provided. The method used the generating function of AUNU groups in the construction to obtain some geometric structures of orbits resembling to lattice. The research further established some algebraic properties of group action as well as orbit and stabilizer in permutation groups which are relevant to the study of finite group structures. The paper also providedsome motivations for the study of group action in an abstract domain.

Dual Ramsey properties for classes of algebras

Almost any reasonable class of finite relational structures has the Ramsey property or a precompact Ramsey expansion. In contrast to that, the list of classes of finite algebras with the precompact Ramsey expansion is surprisingly short. In this paper we show that any nontrivial variety (that is, equationally defined class of algebras) enjoys various dual Ramsey properties. We develop a completely new set of strategies that rely on the fact that left adjoints preserve the dual Ramsey property, and then treat classes of algebras as Eilenberg–Moore categories for a monad. We show that finite algebras in any nontrivial variety have finite dual small Ramsey degrees, and that every finite algebra has finite dual big Ramsey degree in the free algebra on countably many free generators. As usual, these come as consequences of ordered versions of the statements.

Tailored Topological Edge Waves via Chiral Hierarchical Metamaterials

Precise manipulation of the direction and re-direction of vibrational wave energy is a key demand in wave physics and engineering. We consider the paradigm of a finite frame-like structure and the requirement to channel energy away from critical regions, leaving them vibration-free, and redirect energy along edges towards energy concentrators for damping or energy harvesting. We design an exemplar frame metamaterial, combining two distinct areas of wave physics. Firstly, topological edge states taking an unconventional tetrachiral lattice. We control these highly localised protected edge states leveraging a hierarchy of scales through the addition of micro-resonators that impose tuneable symmetry breaking and reconfigurable mass. This allows us to achieve precise positional control in the macro-scale frame lattice, thereby opening opportunities for robust signal transport and vibration control. Experiments, theory, simulation are all utilised to provide a comprehensive analysis and interpretation of the physics.

Sandwich Plate Structure Periodically Attached by S-Shaped Oscillators for Low Frequency Ship Vibration Isolation.

Locally resonant phononic crystals are a kind of artificial periodic composite material/structure with an elastic wave band gap that show attractive application potential in low-frequency vibration control. For low-frequency vibration control problems of ship power systems, this paper proposes a phononic crystal board structure, and based on the Bloch theorem of periodic structure, it uses a finite element method to calculate the band structure and the displacement fields corresponding to the characteristic mode and vibration transmission curve of the corresponding finite periodic sandwich plate structure, and the band gap characteristics are studied. The mechanism of band gap formation is mainly attributed to the mode coupling of the phononic crystal plate structure. Numerical results show that the sandwich plate structure has a double periodicity, so it has a multi-stage elastic wave band gap, which can fully inhibit the transmission of flexural waves and isolate the low-frequency flexural vibration. The experimental measurements of flexural vibration transmission spectra were conducted to validate the accuracy and reliability of the numerical calculation method. On this basis, the potential application of the proposed vibration isolation method in a marine power system is discussed. A vibration isolation platform mounted on a steel plate is studied by numerical simulation, which can isolate low-frequency vibration to protect electronic equipment and precision instruments.

Strong Bending Distortion of a Supercoronene Graphene Model upon Adsorption of Lanthanide Atoms

Numerous applications of graphene involve quasi-infinite sheets, as well as finite structures with edges, pores, graphene quantum dots, etc. In theoretical studies of adsorption of diverse chemical species, including single atoms, molecules, cations, and anions, graphene usually behaves as a very rigid planar structure. However, we found that when adsorbing lanthanide atoms, finite size structures, represented by the widely used supercoronene model, can undergo considerable distortion, and the degree of distortion depends on the number of unpaired electrons, reaching a maximum for Gd (eight unpaired electrons). Lanthanides closely approach the supercoronene surface and increase the interaction energy. Extrapolating to real-world systems, one can expect the existence and magnitude of lanthanide-induced distortion to depend on the size of graphene structures. Quasi-infinite or very large graphene sheets are too rigid to undergo such bending, but it becomes tangible for graphene quantum dots and for atom adsorption closer to graphene edges.

A note on k-cyclic modal pseudocomplemented De Morgan algebras

Symmetric and k-cyclic structure of modal pseudocomplemented De Morgan algebras was introduced previously. In this paper, we first present the construction of epimorphisms between finite symmetric (or 2-cyclic) modal pseudocomplemented De Morgan algebras. Furthermore, we compute the cardinality of the set of all epimorphism between finite structures. Secondly, we present the construction of finite free algebras on the variety of k-cyclic modal pseudocomplemented De Morgan algebras and display how our computations are in fact generalizations to others in the literature. Our work is strongly based on the properties of epimorphisms and automorphisms and the fact that the variety is finitely generated.

Tutorial on vibration and sound

Over the years I have written many tutorials on vibroacoustics. This tutorial summarizes much of the material in those articles, starting with structural flexural vibrations and how they radiate sound. I explain structural modes of vibration, how they sum into structural mobilities, and how to use infinite plate theory to estimate the means of those mobilities. I also show that as structural damping increases, the mobilities of finite structures converge to that of an infinite structure. Next, you will see how structural flexural waves radiate sound into adjacent acoustic spaces. The radiated sound power may be normalized to a nondimensional radiation efficiency. Next we will examine fluid loading effects on structures, including loading from exterior and internal acoustic regions. I will show how exterior heavy fluid loading (from water) significantly mass loads and radiation damps structural vibrations. I will also show how internal air cavities can stiffen low order structural modes, and in some cases cause coupled oscillator behavior when structural and cavity mode resonance frequencies coincide. Finally, I will explore the fundamentals of sound transmission loss through structural barriers. My complete tutorial series on vibroacoustics is available at hambricacoustics.com.

Tree-Depth and the Formula Complexity of Subgraph Isomorphism

For a fixed “pattern” graph , the colored -subgraph isomorphism problem (denoted by ) asks, given an -vertex graph and a coloring , whether contains a properly colored copy of . The complexity of this problem is tied to parameterized versions of and , among other questions. An overarching goal is to understand the complexity of , under different computational models, in terms of natural invariants of the pattern graph . In this paper, we establish a close relationship between the formula complexity of and an invariant known as tree-depth (denoted by). is known to be solvable by monotone formulas of size . Our main result is an lower bound for formulas that are monotone or have sublogarithmic depth. This complements a lower bound of Li, Razborov, and Rossman [SIAM J. Comput., 46 (2017), pp. 936–971] relating tree-width and circuit size. As a corollary, it implies a stronger homomorphism preservation theorem for first-order logic on finite structures [B. Rossman, An improved homomorphism preservation theorem from lower bounds in circuit complexity, in 8th Innovations in Theoretical Computer Science Conference, LIPIcs. Leibniz Int. Proc. Inform. 67, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, Germany, 2017, 27]. The technical core of this result is an lower bound in the special case where is a complete binary tree of height , which we establish using the pathset framework introduced in B. Rossman [SIAM J. Comput., 47 (2018), pp. 1986–2028]. (The lower bound for general patterns follows via a recent excluded-minor characterization of tree-depth [W. Czerwiński, W. Nadara, and M. Pilipczuk, SIAM J. Discrete Math., 35 (2021), pp. 934–947; K. Kawarabayashi and B. Rossman, A polynomial excluded-minor approximation of treedepth, in Proceedings of the 2018 Annual ACM-SIAM Symposium on Discrete Algorithms, 2018, pp. 234–246]. Additional results of this paper extend the pathset framework and improve upon both the best known upper and lower bounds on the average-case formula size of when is a path.

Resonance mode analysis of finite plate strip with acoustic black holes: the gap between bandgap and attenuation band

Acoustic Black Hole (ABH) lattice structures show promise for achieving broadband bandgap in lightweight design. Existing ABH lattice research usually assumes that the bandgaps of infinite periodic structure are roughly the same as the attenuation band with strong energy attenuation in the finite counterpart. This work is concerned with comparison of the real attenuation bandwidth of finite periodic ABH structures and the bandgap of the infinite counterpart. The plate strips consisting of different numbers of periodic ABH elements are considered. It is observed that the bandgap-predicted attenuation band is split into two narrow attenuation bands, so that the periodic ABHs fails to ensure a broadband and continuous attenuation in the finite scenarios. Results show that there are resonance modes of finite plate strips with ABHs falling into the bandgap due to the boundary reflection and would result in high transmission peaks within the bandgap-predicted attenuation band. This unexpected phenomenon suggests the gap between the attenuation bandwidth of finite periodic structure and the bandgap of the corresponding infinite one. Analysis shows that the resonance mode of finite plate strips, which reduces the bandgap-predicted attenuation bandwidth, can be tuned by geometrically disturbing the periodicity and symmetry of the finite periodic configuration.