Interior Modes Research Articles

Comparison of substructuring methods with interface reduction

For complex dynamical systems, substructuring techniques can be applied to simulate the large system more efficiently by handling it as a collection of smaller components. In order to couple these components, the description of the interface between components is very important. Classic substructuring techniques, such as the Craig-Bampton method, reduce the interior modes of the system, but don’t reduce the interface description. This can cause unnecessary degrees of freedom kept in the final assembled model. While there have been a few substructuring methods published that do include some amount of interface reduction, they are much more rare and even more rarely applied in practice. Therefore, the goal of this work is to survey these techniques and compare them with a new, line-based orthogonal polynomial interface reduction technique. These methods will be applied to a set of example problems in order to determine their validity and convergence compared to the full model.

Black hole interior Petz map reconstruction and Papadodimas-Raju proposal

We study the reconstruction of the bulk operators in AdS/CFT when the geometry contains a black hole. The black hole exterior can be mapped to the CFT via a very simple Petz map which coincides with the HKLL map reconstruction of the black hole exterior. For the interior modes of the bulk theory, using the definition of the Petz recovery channel in modular theory, we can find the mapping from the black hole interior to the dual boundary theory. In the case of the evaporating black hole, it is expected that the interior modes map to some operators that have support only on the bath system, the cavity that absorbs the Hawking radiation. The most important observation that we have here is that in the case that we have a typical black hole microstate in the bulk, the CFT dual of the interior modes that we can find using the Petz recovery channel are exactly the operators that so-called “mirror operator “ in the Papadodimas-Raju proposal. Therefore, we can interpret Papadodimas-Raju proposal as an example of the Petz map reconstruction. It may help us answer some open questions about their procedure.

An improvement of generalized Bloch mode synthesis method-based model order reduction technique for band-structure computation of periodic structures

In this paper, an improvement of generalized Bloch mode synthesis (GBMS) method-based model order reduction technique is presented for accelerating the band-structure computation of periodic structures. The improvement is implemented through employing firstly the algebraic substructuring, which enables the mass and stiffness matrices of a single unit cell partitioned into a set of small substructures automatically in the algebraic perspective without considering the physical domain. Then these substructures are reduced via the enhanced Craig-Bampton (ECB) method. The boundary sets are divided into interior boundary and exterior boundary components. The reduced interior modes and interior boundary sets are regarded as the new interior sets relative to the exterior boundary component, and its size can be further reduced via the second level ECB method. Local-level characteristic constraint (L-CC) mode reduction technique is employed for exterior boundary reduction for further reducing the computational cost. The coupling residual modes of the two-level ECB processes are compensated after imposing the Bloch boundary conditions. Two numerical examples are presented to demonstrate the potential benefits of the proposed GBMS-based model-order reduction technique in accelerating band-structure computation of periodic structures. The results show that the proposed method can effectively improve the computational efficiency and have good computational convergence compared with the generalized Bloch mode synthesis with algebraic condensation method.

Evaluation of Bonding Interior Layer Failure Strength Modes in the Composite Pavement

A tack coat is a minimal coating of asphalt cement, cut-back asphalt, or asphalt emulsion to an existing pavement surface between layers to guarantee proper bonding between the two layers and longitudinal and transverse Joints. Numerous researchers have assessed interlayer adhesion employing failure-mode behavior tests, such as pull-off, direct shear, and torsion testing. This study aims to quantify the best tensile resistance obtained using PG (76-10) modified asphalt cement with polymer 4.5%, Sikadur®-31 CF (S.E) usage at elevated temperatures between +25 °C and +45 °C, an epoxy-resin-based adhesive, and repair mortar with specific fillers. And Nitomortar TC2000 epoxy (F.E) resin-based sealing compound from Fosrok Company. All are applied on concrete surfaces at a rate of 0.5 kg/m2 except for Nitomortar, which depends on layer thickness ranges between (1-2.5) mm instead of the application rate. The Proceq DYNA Z16 pull–off tester is used to measure the tensile strength at a rate of 2lb/s. It is found that the average tensile strength of the tack coat materials is (0.607, 1.481, 2.622) MPa, respectively. It concluded that (F.E) epoxy has the maximum tensile strength. Also, adding polymer to asphalt increased adhesiveness. Failure strength modes of interior bonding varied between cohesive failure adhesive, adhesive failure, and cohesive failure in the concrete substrate.

A modal-based Partition of Unity Finite Element Method for elastic wave propagation problems in layered media

The time-harmonic propagation of elastic waves in layered media is simulated numerically by means of a modal-based Partition of Unity Finite Element Method (PUFEM). Instead of using the standard plane waves or the Bessel solutions of the Helmholtz equation to design the discretization basis, the proposed modal-based PUFEM explicitly uses the tensor-product expressions of the eigenmodes (the so-called Love and interior modes) of a spectral elastic transverse problem, which fulfil the coupling conditions among layers. This modal-based PUFEM approach does not introduce quadrature errors since the coefficients of the discrete matrices are computed in closed-form. A preliminary analysis of the high condition number suffered by the proposed method is also analyzed in terms of the mesh size and the number of eigenmodes involved in the discretization. The numerical methodology is validated through a number of test scenarios, where the reliability of the proposed PUFEM method is discussed by considering different modal basis and source terms. Finally, some indicators are introduced to select a convenient discrete PUFEM basis taking into account the observability of cracks located on a coupling boundary between two adjacent layers.

Classical Love number for quantum black holes

We present a method for comparing the classical and quantum calculations of the electric quadrupolar Love number $k_2$ and show that our previous derivation of the quantum Love number of a quantum blackhole matches exactly the classical calculation of $k_2$ when quantum expectation values are replaced by the corresponding classical quantities, as dictated by the Bohr correspondence principle. The standard derivation of $k_2$ for classical relativistic stars relies on fixing boundary conditions on the surface of the star for the Einstein equations in the presence of an external perturbing field. An alternative method for calculating $k_2$ uses properties of the spectrum of the non-relativistic fluid modes of the star. We adopt this alternative method and use it to derive an effective description of the interior modes in terms of a collection of driven harmonic oscillators characterized by different frequencies and amplitudes. We compare these two classical methods and find that most of the interior information can be integrated out, reducing the problem of calculating $k_2$ to fixing a single boundary condition for the perturbed Einstein equations on the surface of the deformed star. We then determine this single boundary condition in terms of the spectrum of the object and proceed to identify the relationship between classical quantities and quantum expectation values and to verify the agreement between the results of the effective classical calculation and the quantum calculation.

Interplay of Surface and Interior Modes in Exciton-Phonon Coupling at the Nanoscale.

Exciton-phonon coupling (EXPC) plays a key role in the optoelectronic properties of semiconductor nanocrystals (NCs), but a microscopic picture of EXPC is still lacking, particularly regarding the magnitude and scaling with NC size, the dependence on phonon frequency, and the role of the NC surface. The computational complexity associated with accurately describing excitons and phonons has limited previous theoretical studies of EXPC to small NCs, noninteracting electron-hole models, and/or a small number of phonon modes. Here, we develop an atomistic approach for describing EXPC in NCs of experimentally relevant sizes. We validate our approach by calculating the reorganization energies, a measure of EXPC, for CdSe and CdSe-CdS core-shell NCs, finding good agreement with experimental measurements. We demonstrate that exciton formation distorts the NC lattice primarily along the coordinates of low-frequency acoustic modes. Modes at the NC surface play a significant role in smaller NCs while interior modes dominate for larger systems.

Gravitational SL(2, \u211d) algebra on the light cone

In a region with a boundary, the gravitational phase space consists of radiative modes in the interior and edge modes at the boundary. Such edge modes are necessary to explain how the region couples to its environment. In this paper, we characterise the edge modes and radiative modes on a null surface for the tetradic Palatini-Holst action. Our starting point is the definition of the action and its boundary terms. We choose the least restrictive boundary conditions possible. The fixed boundary data consists of the radiative modes alone (two degrees of freedom per point). All other boundary fields are dynamical. We introduce the covariant phase space and explain how the Holst term alters the boundary symmetries. To infer the Poisson brackets among Dirac observables, we define an auxiliary phase space, where the SL(2, ℝ) symmetries of the boundary fields are manifest. We identify the gauge generators and second-class constraints that remove the auxiliary variables. All gauge generators are at most quadratic in the fundamental SL(2, ℝ) variables on phase space. We compute the Dirac bracket and identify the Dirac observables on the light cone. Finally, we discuss various truncations to quantise the system in an effective way.

𝑯(curl 2)-Conforming Spectral Element Method for Quad-Curl Problems

Abstract In this paper, we propose an H ⁢ ( curl 2 ) \boldsymbol{H}(\mathbf{curl}^{2}) -conforming spectral elements to solve the quad-curl problem on cubic meshes in three dimensions. Starting with generalized vectorial Jacobi polynomials, we first construct the basis functions of the H ⁢ ( curl 2 ) \boldsymbol{H}(\mathbf{curl}^{2}) -conforming spectral elements using the contravariant transform together with the affine mapping from the reference cube onto each physical element. Falling into four categories, interior modes, face modes, edge modes, and vertex modes, these H ⁢ ( curl 2 ) \boldsymbol{H}(\mathbf{curl}^{2}) -conforming basis functions are constructed in an arbitrarily high degree with a hierarchical structure. Next, H ⁢ ( curl 2 ) \boldsymbol{H}(\mathbf{curl}^{2}) -conforming spectral element approximation schemes are established to solve the boundary value problem as well as the eigenvalue problem of quad-curl equations. Numerical experiments demonstrate the effectiveness and efficiency of the ℎ-version and the 𝑝-version of our H ⁢ ( curl 2 ) \boldsymbol{H}(\mathbf{curl}^{2}) -conforming spectral element method.

Interior quasinormal modes and strong cosmic censorship

Recent work has argued that the strong cosmic censorship (SCC) conjecture is violated by near-extremal Reissner-Nordstr\"om de Sitter (RNdS) black holes but respected by Kerr-de Sitter black holes. It has also been shown that the conjecture is violated by near-extremal BTZ black holes. The latter result relies on a coincidence between "exterior" and "interior" quasinormal frequencies. If this coincidence were to occur also for RNdS or Kerr-dS then it would significantly modify the conclusions of earlier work. In this paper, it is demonstrated that this coincidence does not occur for RNdS or Kerr-dS and so the conclusions of the earlier work remain valid.

Observer-dependent black hole interior from operator collision

We present concrete construction of interior operators for a black hole which is perturbed by an infalling observer. The construction is independent from the initial states of the black hole while dependent only on the quantum state of the infalling observer. The construction has a natural interpretation from the perspective of the boundary operator's growth, resulting from the collision between operators accounting for the infalling and outgoing modes. The interior partner modes are created once the infalling observer measures the outgoing mode, suggesting that the black hole interior is observer dependent. Implications of our results on various conceptual puzzles, including the firewall puzzle, the Marolf-Wall puzzle, and the information problem, are discussed. Explicit counterarguments against the $\mathrm{ER}=\mathrm{EPR}$ approach by Maldacena and Susskind are also presented.

An infalling observer and behind the horizon cutoff

Using Papadodimas and Raju construction of operators describing the interior of a black hole, we present a general relation between partition functions of operators describing inside and outside the black hole horizon. In particular for an eternal black hole the partition function of the interior modes may be given in terms those partition functions associated with the modes of left and right exteriors. By making use of this relation we observe that setting a finite UV cutoff will enforce us to have a cutoff behind the horizon whose value is fixed by the UV cutoff. The resultant cutoff is in agreement with what obtained in the context of holographic complexity.

Reduced-Order Modeling of Bladed Disks Considering Small Mistuning of the Disk Sectors

A model order reduction method based on the component mode synthesis for mistuned bladed disks is introduced, with one component for the disk and one component for each blade. The interface between the components at the blade roots is reduced using the wave-based substructuring (WBS) method, which employs tuned system modes. These system modes are calculated first, and used subsequently during the reduction of the individual components, which eliminates the need to build a partially reduced intermediate model with dense matrices. For the disk, a cyclic Craig–Bampton (CB) reduction is applied. The deviations of the stiffness and mass matrices of individual disk sectors are then projected into the cyclic basis of interior and interface modes of the disk substructure. Thereby, it is possible to model small disk mistuning in addition to large mistuning of the blades.

Transition from Molecular Vibrations to Phonons in Atomically Precise Cadmium Selenide Quantum Dots.

We use micro-Raman spectroscopy to measure the vibrational structure of the atomically precise cadmium selenide quantum dots Cd35Se20X30L30, Cd56Se35X42L42, and Cd84Se56X56L56. These quantum dots have benzoate (X) and n-butylamine (L) ligands and tetrahedral (Td) shape with edges that range from 1.7 to 2.6 nm in length. Investigating this previously unexplored size regime allows us to identify the transition from molecular vibrations to bulk phonons in cadmium selenide quantum dots for the first time. Room-temperature Raman spectra have broad CdSe peaks at 175 and 200 cm-1. Density functional theory calculations assign these peaks to molecular surface and interior vibrational modes, respectively, and show that the interior, surface, and ligand atom motion is strongly coupled. The interior peak intensity increases relative to the surface peak as the cluster size increases due to the relative increase in the polarizability of interior modes with quantum dot size. The Raman spectra do not change with temperature for molecular Cd35Se20X30L30, while the interior peak narrows and shifts to higher energy as temperature decreases for Cd84Se56X56L56, a spectral evolution typical of a phonon. This result shows that the single bulk unit cell contained within Cd84Se56X56L56 is sufficient to apply a phonon confinement model, and that Cd56Se35X42L42, with its 2.1 nm edge length, marks the boundary between molecular vibrations and phonons.

Mode selection for reduced order modeling of mechanical systems excited at resonance

Welding, food cutting, atomizing, cleaning and deagglomerating are just a few common uses for ultrasonic resonators, which are carefully designed to operate excited in resonance. Finite element analysis is nearly always adopted for predicting and improving resonator performances. Large number of small elements are usually needed to guarantee accuracy. As a consequence, models have typically very large dimensions, and hence considerable computational and ill conditioning problems arise. Model reduction techniques can be extremely useful to keep model dimensions to a minimum. In this paper a new ranking method, called Interior Mode Ranking (IMR), is introduced for the selection of the interior normal modes in the Craig Bampton reduction technique, which is one of the most popular model reduction methods, often available in commercial finite element software packages. The IMR method allows ranking the interior modes analytically by comparing the contributions provided by the interior modes of the subsystem with constrained boundary conditions to the dynamics of interest of the complete system (with actual boundary conditions). The method is general and can be applied to any resonator in the reduction at the system level. Here it is employed to obtain an accurate reduced-order model of an ultrasonic welding bar horn. The results achieved by the method are compared with those yielded by other ranking techniques. The comparison shows that the IMR method outperforms the other ranking techniques and leads to accurate representations of the excited modes.

Geometry of the infalling causal patch

The firewall paradox states that an observer falling into an old black hole must see a violation of unitarity, locality, or the equivalence principle. Motivated by this remarkable conflict, we analyze the causal structure of black hole spacetimes in order to determine whether all the necessary ingredients for the paradox fit within a single observer's causal patch. We particularly focus on the question of whether the interior partner modes of the outgoing Hawking quanta can, in principle, be measured by an infalling observer. Since the relevant modes are spread over the entire sphere, we answer a simple geometrical question: can any observer see an entire sphere behind the horizon? We find that for all static black holes in 3+1 and higher dimensions, with any value of the cosmological constant, no single observer can see both the early Hawking radiation and the interior modes. We present a detailed description of the causal patch geometry of the Schwarzschild black hole in 3+1 dimensions, where an infalling observer comes closest to being able to measure the relevant modes.

Energy-Based Optimal Ranking of the Interior Modes for Reduced-Order Models under Periodic Excitation

This paper introduces a novel method for ranking and selecting the interior modes to be retained in the Craig-Bampton model reduction, in the case of linear vibrating systems under periodic excitation. The aim of the method is to provide an effective ranking of such modes and hence an optimal sequence according to which the interior modes should be progressively included to achieve a desired accuracy of the reduced-order model at the frequencies of interest, while keeping model dimensions to a minimum. An energy-based ranking (EBR) method is proposed, which exploits analytical coefficients to evaluate the contribution of each interior mode to the forced response of the full-order system. The application of the method to two representative systems is discussed: an ultrasonic horn and a vibratory feeder. The results show that the EBR method provides a very effective ranking of the most important interior modes and that it outperforms other state-of-the-art benchmark techniques.

Violations of the Equivalence Principle by a Nonlocally Reconstructed Vacuum at the Black Hole Horizon

If information escapes from an evaporating black hole, then field modes just outside the horizon must be thermally entangled with distant Hawking radiation. But for an infalling observer to find empty space at the horizon, the same modes would have to be entangled with the black hole interior. Thus, unitarity appears to require a "firewall" at the horizon. Identifying the interior with the distant radiation promises to resolve the entanglement conflict and restore the vacuum. But the map must adjust for any interactions, or else the firewall will reappear if the Hawking radiation scatters off the cosmic microwave background. Such a map produces a "frozen vacuum," a phenomenon that is arguably worse than a firewall. An infalling observer is unable to excite the vacuum near the horizon. This allows the horizon to be locally detected and so violates the equivalence principle.

A Surface-Aware Projection Basis for Quasigeostrophic Flow

Abstract Recent studies indicate that altimetric observations of the ocean’s mesoscale eddy field reflect the combined influence of surface buoyancy and interior potential vorticity anomalies. The former have a surface-trapped structure, while the latter are often well represented by the barotropic and first baroclinic modes. To assess the relative importance of each contribution to the signal, it is useful to project the observed field onto a set of modes that separates their influence in a natural way. However, the surface-trapped dynamics are not well represented by standard baroclinic modes; moreover, they are dependent on horizontal scale. Here the authors derive a modal decomposition that results from the simultaneous diagonalization of the energy and a generalization of potential enstrophy that includes contributions from the surface buoyancy fields. This approach yields a family of orthonormal bases that depend on two parameters; the standard baroclinic modes are recovered in a limiting case, while other choices provide modes that represent surface and interior dynamics in an efficient way. For constant stratification, these modes consist of symmetric and antisymmetric exponential modes that capture the surface dynamics and a series of oscillating modes that represent the interior dynamics. Motivated by the ocean, where shears are concentrated near the upper surface, the authors consider the special case of a quiescent lower surface. In this case, the interior modes are independent of wavenumber, and there is a single exponential surface mode that replaces the barotropic mode. The use and effectiveness of these modes is demonstrated by projecting the energy in a set of simulations of baroclinic turbulence.

From Isolated Metaatoms to Photonic Metamaterials: Evolution of the Plasmonic Near-Field

Metamaterials are artificial media which can provide optical properties not available from natural materials. These properties often result from the resonant excitation of plasmonic modes in the metallic building blocks ("metaatoms") of the metamaterial. Electromagnetic interactions between the metaatoms significantly modify the resonances of the individual metaatoms and influence the optical properties of the whole metamaterial. To better understand these interactions, we study in this Letter the evolution of the plasmonic near-field in the course of the transition from an isolated metaatom, in our case a split-ring resonator (SRR), to a photonic metamaterial via electron energy-loss spectroscopy. For small SRR ensembles, we observe the formation of discrete optical bright and dark modes due to coupling of the metaatoms. Large SRR arrays reveal a quasi-continuum of modes in the interior and distinct edge modes at the boundaries of the array. Our experimental results are in excellent agreement with numerical calculations.