Modal Expansion Research Articles

Robustness and reliability of state-space, frame-based modeling for thermoacoustics

The Galerkin modal expansion is a well-known method used to develop reduced order models for thermoacoustics. A known issue is the appearance of Gibbs-type oscillations on velocity fluctuations at the interface between subdomains and at boundary conditions. Recent work of Laurent et al. (2019) [20] and Laurent et al. (2021) [23] have shown that it is possible to overcome this issue by using an over-completed frame, instead of a Galerkin modal basis. However, the low-order modeling based on this frame modal expansion may generate spurious modes. In this paper, the origin of these non-physical modes is identified and a method is proposed to automatically remove them from the outcome. By preventing any interaction between the physical and non-physical components, the proposed methodology drastically improves the robustness and reliability of the frame modal expansion modeling for thermoacoustics.

Computation of emitter-plasmon interactions using an axis-symmetric model for off-axis dipoles

The axis-symmetric modeling technique is based on expanding vector fields in cylindrical harmonics and computing the response on a two-dimensional cross-section separately for each azimuthal harmonic, significantly reducing computational costs. However, it has limitations when dealing with dipoles placed away from the symmetry axis due to challenges in the expansion of angular modes. To address this, we propose a reformulated axis-symmetric model based on the Fourier expansion of the delta function distribution concerning the azimuthal variable. This model is validated using standard Mie theory for off-axis dipoles and applied to study multiple-emitter-plasmon interactions. The emission properties of a non-cooperative ensemble near a plasmonic nanoparticle are observed to scale with the number of emitters considered, N. Notably, a Dicke effect-like superradiance (N2-dependence) is observed when a spatially disordered ensemble of dipoles oscillates collectively inside a plasmonic dimer gap. This kind of high-level cooperative quantum phenomenon is of high interest in fields such as quantum optics and light-harvesting.

Sloshing in a tank with elastic side walls and a membrane cover

The coupled problem of liquid sloshing in a tank with a membrane cover and elastic side walls is investigated. The velocity potential theory is used for the flow and the linear elastic theory for the cover and side walls. For the former, the vertical mode expansion is used, in which all the roots of the dispersion relationship are first found. The expansion automatically satisfies the governing equation, the tank bottom, and the tank cover conditions. The deflections of the side walls are expanded into a cosine series together with a four term polynomial, which is vital for the procedure to succeed. These expansions are then matched through the dynamic and kinematic equations of the plates, and the problem is completed by imposing the edge conditions of the plate and membrane. Through the combined matrix equation, the natural frequencies of the system are obtained. In addition, the nature of the dispersion relationship of the membrane is analyzed. Explicit solutions are obtained for some special cases, and the link with the free surface sloshing is established. Extensive numerical results are provided, and their physics is analyzed.

A novel shape sensing approach based on the coupling of Modal Virtual Sensor Expansion and iFEM: Numerical and experimental assessment on composite stiffened structures

Shape sensing, i.e. the reconstruction of the displacement field of a structure from discrete strain measurements, is becoming crucial for the development of a modern Structural Health Monitoring framework. Nevertheless, an obstacle to the affirmation of shape sensing as an efficient monitoring system for existing structures is represented by its requirement for a significant amount of sensors. Two shape sensing methods have proven to exhibit complementary characteristics in terms of accuracy and required sensors that make them suitable for different applications, the inverse Finite Element Method (iFEM) and the Modal Method (MM). In this work, the formulations of these two methods are coupled to obtain an accurate shape sensing approach that only requires a few strain sensors. In the proposed procedure, the MM is used to virtually expand the strains coming from a reduced number of strain measurement locations. The expanded set of strains is then used to perform the shape sensing with the iFEM. The proposed approach is numerically and experimentally tested on the displacement reconstruction of composite stiffened structures. The results of these analyses show that the formulation is able to strongly reduce the number of required sensors for the iFEM and achieve an extremely accurate displacement reconstruction.

Dynamic force identification considering modeling errors using modal expansion method and relevant vector regression algorithm

Accurate identification and estimation of external forces on launch vehicles, aircraft, and other in-service engineering structures plays a vital role in structural design and health monitoring. Considering structural modeling errors, this paper develops a novel force identification method based on modal expansion and relevant vector regression (RVR). Initially, the incomplete and noisy experimental modal data are used to modify the stiffness and mass matrices of the finite element (FE) model, reducing the impact of modeling errors on force identification accuracy. Subsequently, to account for noise disturbances and potential information about unknown initial conditions in the measured acceleration responses, the external forces and the unknown initial conditions are respectively expanded using trigonometric functions and the mode shapes based on the function fitting technique. On this basis, the force identification equation is established for the updated model. The RVR algorithm is integrated into modal expansion and force identification. Firstly, the mode shapes of the FE model are utilized to expand the incomplete and noisy experimental mode shapes to their full degree of freedom, significantly enhancing the accuracy of the model updating. Secondly, by combining the measured acceleration responses, the base coefficients can be solved sparsely, effectively identifying the dynamic forces even when unknown initial conditions are present.

A hidden 2d CFT for self-dual Yang-Mills on the celestial sphere

Self-dual Yang-Mills theory admits an underlying infinite dimensional symmetry algebra, which has been obtained from mode expansion of Mellin transformed 4d scattering amplitudes and separately, Koszul duality on twistor space. In this paper, we propose to derive an explicit 2d realization of the algebra by performing a particular gauge transformation on the twistor action for self-dual Yang-Mills. The gauge parameter used in the transformation generates pure gauge connections corresponding to large gauge transformations on 4d Minkowski space, which localises part of the twistor action to a ℂℙ1 on ℂ* reduction of twistor space. Under a projection, it can be mapped to the celestial sphere at the light-cone cut of the origin on I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{I} $$\\end{document}. Geometrically, this is the common boundary celestial sphere shared by Euclidean AdS3 or Lorentzian dS3 slices of Minkowski space. We comment on the geometric meaning of the derivation from the perspective of minitwistor spaces of the 3d slices embedded in 4d Minkowski space. Using the action functional of this 2d CFT, we compute its stress-energy tensor and central charge. By a further marginal deformation, we calculate correlation functions of current algebra generators purely from the 2d side which reproduce full 4d tree-level form factors.

Damage detection method for square steel tube based on CS-NME algorithm via ultrasonic guided waves

The utilization of ultrasonic guided wave (UGW) technology has garnered considerable attention in non-destructive testing field, particularly for steel components. However, for square steel tubes with noncircular cross sections, this method encounters severer frequency dispersion, which hampers the accuracy of damage identification. To address this issue, the approach utilizing the normal mode expansion (NME) method combined with cuckoo search (CS) algorithm to develop a layout strategy for ultrasonic transducer position and length is proposed, which enables single mode excitation of UGW while reducing the interference from redundant clutter signals, thereby enhancing the accuracy of damage detection. The efficacy of the proposed approach is evaluated via numerical simulations and laboratory experiments. The results show that this method can significantly reduce the interference of other modes by optimizing the location and length of ultrasonic transducers, and successfully detect the hole and crack damage on the square steel tube.

Beyond quantum annealing: optimal control solutions to maxcut problems

Quantum Annealing (QA) relies on mixing two Hamiltonian terms, a simple driver and a complex problem Hamiltonian, in a linear combination. The time-dependent schedule for this mixing is often taken to be linear in time: improving on this linear choice is known to be essential and has proven to be difficult. Here, we present different techniques for improving on the linear-schedule QA along two directions, conceptually distinct but leading to similar outcomes: 1) the first approach consists of constructing a Trotter-digitized QA (dQA) with schedules parameterized in terms of Fourier modes or Chebyshev polynomials, inspired by the Chopped Random Basis algorithm for optimal control in continuous time; 2) the second approach is technically a Quantum Approximate Optimization Algorithm (QAOA), whose solutions are found iteratively using linear interpolation or expansion in Fourier modes. Both approaches emphasize finding smooth optimal schedule parameters, ultimately leading to hybrid quantum–classical variational algorithms of the alternating Hamiltonian Ansatz type. We apply these techniques to MaxCut problems on weighted 3-regular graphs with N = 14 sites, focusing on hard instances that exhibit a small spectral gap, for which a standard linear-schedule QA performs poorly. We characterize the physics behind the optimal protocols for both the dQA and QAOA approaches, discovering shortcuts to adiabaticity-like dynamics. Furthermore, we study the transferability of such smooth solutions among hard instances of MaxCut at different circuit depths. Finally, we show that the smoothness pattern of these protocols obtained in a digital setting enables us to adapt them to continuous-time evolution, contrarily to generic non-smooth solutions. This procedure results in an optimized QA schedule that is implementable on analog devices.

Computation Theory of Large-Scale Partially Coherent Imaging by the Modified Modal Expansion Method

The numerical calculation of partially coherent imaging involves a fourfold integral, which is numerically complex and impracticable to be calculated directly. The use of coherent-mode decomposition (CMD) can make this problem more manageable but finding the coherent-modes for complicated partial coherent fields (without already-known coherent-mode expansion) are rather computationally intensive. In this letter, a modified modal expansion method is proposed, which significantly reduces the requirement of computational resources. The propagation of partial coherence in imaging systems with extremely large sampling number could be handled by an ordinary computer. A comparison between the new method and the traditional method in terms of memory resource requirements and computational time consumption is also detailed in this article. We will also show that this method could deal with the anisoplanatic imaging cases while maintaining the same computational efficiency.

Dynamic Model Reduction for Viscously Damped Structures with Statically Determinate Interfaces

A novel model reduction method for viscously damped structures with statically determinate interfaces, such as spacecraft flexible appendages, is proposed. The paper presents a derivation of the complete complex modal expansion of the interface dynamic stiffness of these structures. Based on the identity relation for all complex modes, which is obtained during the derivation, it is found that the interface acceleration impedance can be expressed as a rational fraction with high accuracy using only low-order complex modes. Using this rational fraction as an approximation model, numerical results of the frequency response can be fitted. The fitted interface acceleration impedance can be applied to real-time control as a reduced model in the form of a transfer function. Furthermore, it can be transformed into the form of system matrices by introducing auxiliary variables, which then participate in the dynamic analysis of the assembly. The reduction process circumvents complex modal analysis and necessitates only the results of frequency responses. Thanks to the powerful ability of conventional finite element software to perform frequency response analysis, this reduction method can be used for large-scale complex models in actual engineering applications.

Validation of a model-based dual-band modal decomposition and expansion approach for fatigue monitoring of offshore wind turbine foundations

The fatigue limit state strongly drives the lifetime of offshore wind turbine support structures. Despite steady advancements in their design, the accumulation of fatigue damage over the lifetime of the structure remains uncertain; one way to improve certainty is to directly monitor structural loads at fatigue-critical locations, e.g., near or below the mudline. However, directly monitoring the loads at these locations is often not possible, since these locations do not allow for easy installation or retrofitting of sensors. To this end, model-based virtual sensing techniques can be employed to estimate the structural response at unmeasured locations based on vibration response data and a finite element model which can accurately describe the modal and operational deflection shapes of a specific turbine [1,2,3]. For this contribution, a modal decomposition and expansion (MDE) algorithm [4] will be used to predict strain time series at fatigue-critical locations, based on acceleration and strain data obtained from the tower of an operational OWT. Emphasis will be put on sparse data situations in which a realistic amount of sensor data is used as input. Fibre-optical strain gauges close to and below the mudline level (Fig 1) will be used for direct validation of the extrapolation capabilities. The required finite element model is built using an in-house developed and verified integrated FE model class [5,6], in combination with a unique database which contains all available geotechnical and structural data of offshore wind turbines (owimetadatabase) [7,8]. This combination allows us to generate fleet-wide, turbine-specific, finite elements models for several wind farms. The results will be presented in terms of time series (e.g., Fig. 2) and fatigue damage-related metrics. Preliminary results show that strain time series estimated with the MDE method show good agreement with the validation data.

Equivalent Circuit Modeling of a Novel Reconfigurable Metasurface With Independent Control of Amplitude and Phase Based on Floquet Modal Expansion

Equivalent Circuit Modeling of a Novel Reconfigurable Metasurface With Independent Control of Amplitude and Phase Based on Floquet Modal Expansion

Insight into local defect resonance and its in-situ measurement based on flexible nano-composite sensors

The local defect resonance has been proven to be effective for damage detection, and the methods based on LDR for damage evaluation have demonstrated appealing capabilities of selective assessment and applicability to complex structures. To explain the generation of LDR, models based on vibration theories have been developed in which the defect boundaries are assumed to be fixed. Nevertheless, existing models can only give an accurate prediction of frequency for fundamental LDR mode owing to the deviation of their assumptions from the reality. In addition, the measurement of LDR is currently limited to noncontact technologies, hampering its application to in-situ and online monitoring of inspected structures. To overcome these deficiencies, an analytical model is proposed in this investigation to gain an insight into the generation of LDR from the perspective of wave reflection at defect boundaries and wave superposition. In this model, the interaction of guided waves with defect is scrutinized using a normal mode expansion method to obtain the phase shift of the reflected waves at the defect boundaries. On this basis, the formation of standing waves is analyzed to interpret the occurrence of the resonance of guided waves, whereby the relation of the LDR frequencies and the defect parameters can be obtained. On top of this analytical investigation, the sensing of the LDR using a novel nano-composite sensor is attempted. Flexible and lightweight, these sensors can form a dense sensing network, with which the LDR response in the inspected region can be perceived. Using the LDR-based method and nano-composite sensors, this approach can give a damage characterization in an in-situ and online manner for complex structures. Experimental validations of the proposed approach is performed in which delaminations in composite structures are identified and assessed using the LDR response perceived with the nano-composite sensors.

Vibration control of thin shallow paraboloidal shells with light-activated shape memory polymers

With the applications of thin-walled paraboloidal shell structure in aviation, aerospace and communication systems, smart-structure based active vibration control becomes essential to improve the performance of thin-walled parabolic structures. Light-activated shape memory polymer (LaSMP) is a novel actuator material that exhibits dynamic Young's modulus and strain properties under ultraviolet (UV) light exposures, which can be used for non-contact vibration control. This study focuses on investigating the vibration control performance of a shallow parabolic thin shell, with simply supported boundary conditions, with laminated LaSMP patches. Based on the LaSMP strain model, the vibration control effect of LaSMP patch on the thin shell is discussed. A finite element model of the simply supported parabolic thin shell is developed to obtain the natural frequencies. The modal expansion method is employed to investigate the vibration control of the shallow parabolic thin shell laminated with LaSMPs, and independent modal responses are calculated. According to the different control mode, the effects of LaSMP actuators on vibration control of shallow paraboloid shell are studied in three cases with varying actuation position and coverage area of LaSMP actuators. The results indicate that LaSMP can control the vibration of the thin shell by reducing the vibration amplitude and the final vibration control effect is determined by the mode shape, the initial LaSMP strain, and the location and area covered by LaSMP actuators. By establishing the model between the LaSMP actuator force and the attenuations in modal amplitude, this study provides an analytical tool for the application of LaSMPs in vibration control of flexible structures.

Numerical study on guided-wave reflection and transmission at water pipe joint using hybrid finite element method

AbstractThis study investigates guided-wave reflection and transmission at a water pipe joint. The system comprises a linearly elastic pipe filled with water with a joint that is modeled as a discontinuity of the solid region. Wave reflection and transmission are solved using the finite element method (FEM) with radiation conditions for reflected and transmitted guided waves into infinite waveguides. For the radiation conditions, the reflected and transmitted waves are expressed by modal expansion using the semi-analytical finite-element (SAFE) dispersion analysis method. This study extends the hybrid SAFE-FEM to the coupled fluid–solid axisymmetric problem. Numerical results demonstrate that the hybrid SAFE-FEM provides sufficiently accurate solutions. The propagation modes, similar to the modes in a solid pipe, are strongly or perfectly reflected by the joint. However, the modes are transmitted through the joint with little scattering after they converge to the modes in a water bar. The crossing of dispersion curves with those for modes in a solid pipe causes mode conversion and induces scattering attenuation.

Unified model modal expansion method for full field reconstruction and inversion under mechanical and thermal loading

In this paper, a novel method termed UMM-T&S (Unified Model Modal reconstruction and inversion method for Temperature field and Structural field under limited sensors) is proposed. This method aims to enhance high-precision structural performance inversion and reconstruction under mechanical and thermal loading conditions, particularly in the context of structural health monitoring within the aviation and aerospace sectors. The approach begins by deriving the “temperature modal” solution of the heat conduction equation, drawing upon the modal expansion techniques used in multi-degree-of-freedom systems. This is followed by constructing a parameter mapping relationship within the unified model, enabling the solution of both the structural field (encompassing strain and displacement) and the temperature field through a single structural model modal expansion. Furthermore, the paper details a numerical validation using two different temperature fields and a thermal loading experimental verification on a stiffened plate under five distinct working conditions. The proposed UMM-T&S method yields lower errors (ranging between 10%-40%) and more stable performance across various complex working conditions, in comparison to existing methods. The analysis also delves into the sources of errors and their cumulative impacts.In conclusion, the innovative UMM-T&S method offers a swift, precise, and robust solution for the intricate task of reconstructing and inverting structural performance under combined mechanical and thermal stresses. The method's efficacy and potential are substantiated through experimental validation and comparative analysis against an established methodology, underscoring its capability to effectively tackle the challenges faced in structural monitoring under such demanding conditions.

Second-order water wave properties over Roseau’s three-parameter bottom topography

Summary In his book Asymptotic Wave Theory (North-Holland 1976), Roseau develops the only known exact solution for linear wave propagation over a shoal. Remarkably, almost half a century later, the only properties of that solution which appear to have been used to validate more wide-ranging models are the limiting values of wave height in the far field. The primary aim of this work is to remedy this situation by providing a robust computational setting where the solution may be fully evaluated randomly both on the surface and the interior, speedily and to a high degree of accuracy. Such ‘benchmark’ solutions will facilitate a much more rigorous approach to validation of more general alternative approximation models. One such has some of its numerical results here examined and subsequently largely vindicated by the present calculations. Computation of the absolutely convergent solution integrals is supported by cubic spline approximation of an integrand component which satisfies a certain difference equation and the work is extended to cover bottom profiles with overhang including the extreme case where the overhang degenerates to a semi-infinite flat plate. The further development of solutions at second-order reveals details of mean Eulerian current and the attendant Stokes drift enabling mass transport computations; all illustrated through the entire depth with a range of contouring and arrows plots. The force field that drives a second harmonic component is calculated and shows excellent agreement with its equivalent in the more general modal expansion model applied to a non-identical but somewhat similar profile. This further validates the modelling approach and application described in that work.

Finite element model updating and response prediction of a frame structure based on optimal sensor placement

This paper proposes an approach for finite element (FE) model updating and response prediction of frame structures based on optimal sensor placement (OSP), which integrates sensor placement optimization, mode expansion, and model updating techniques. Firstly, sensor optimization layout and modal testing analysis are conducted on a bolted laboratory frame structure. The covariance-driven stochastic subspace identification (SSI-COV) method identifies the real-world structure’s natural frequencies, modal damping, and mode shapes. Secondly, the complete mode shapes are expanded using the measured incomplete modal data from the limited number of sensors. Thirdly, a multi-objective function based on frequency and mode shapes is established to adjust the parameters of the FE model. This ensures that the updated model accurately represents the dynamic properties of the actual structure within a specific frequency range. Finally, the Rayleigh damping of the frame structure is estimated, and the damping matrix is assembled to enhance the accuracy of dynamic response prediction in the updated model. By comparing the response prediction results of the updated FE model with and without considering the updated damping effects to the measurement data of the real-world structure, it is demonstrated that the proposed method considering updated damping effects can more effectively predict the structural response.

Study of a cubic cavity resonator for gravitational waves detection in the microwave frequency range

The direct detection of gravitational waves (GWs) of frequencies above MHz has recently received considerable attention. In this work, we present a precise study of the reach of a cubic cavity resonator to GWs in the microwave range, using for the first time tools allowing to perform realistic simulations. Concretely, the boundary integral—resonant mode expansion (BI-RME) 3D method, which allows us to obtain not only the detected power but also the detected voltage (magnitude and phase), is used here. After analyzing three cubic cavities for different frequencies and working simultaneously with three different degenerate modes at each cavity, we conclude that the sensitivity of the experiment is strongly dependent on the polarization and incidence angle of the GW. The presented experiment can reach sensitivities up to 1×10−19 at 100 MHz, 2×10−20 at 1 GHz, and 6×10−19 at 10 GHz for optimal angles and polarizations, and where in all cases we assumed an integration time of Δt=1 ms. These results provide a strong case for further developing the use of cavities to detect GWs. Moreover, the possibility of analyzing the detected voltage (magnitude and phase) opens a new interferometric detection scheme based on the combination of the detected signals from multiple cavities. Published by the American Physical Society 2024

A phase field approach to the fracture simulation of quasi-brittle structures with initial state

To solve the simulation problem of crack propagation in solid structures with initial stress, this paper proposes a phase field approach that considers the initial state of the quasi-brittle structure. By using the initial equilibrium iteration method, the estimated initial state of the structure, including initial stress, is transformed into internal stress loads of the structure. Thus, the problem of non-convergence caused by inaccurate initial state estimation of the structure in engineering calculations was solved, and an improved subproblem staggered iteration method was adopted to simulate crack propagation of the structure under external loads. The universality and computational accuracy of this method for different phase field models were verified by using pre made notched thin plates for the expansion of mode I and II cracks in different initial states. Among them, the calculation error using the PF_CZM model is within 1%, and the calculation error of the AT2 model increases with the initial stress, but is less than 7%. At the same time, the crack growth problem of cracked structures under different uniform initial stresses is accurately simulated, which reflects the applicability of this method to actual engineering. In addition, the study also verified the universal applicability of the adaptive refinement strategy proposed based on the AT2 model for other phase field models.