Modal Transformation Research Articles

A stable model order reduction scheme for an acoustic finite element model with perfectly matched layers

Perfectly matched layers (PMLs) are often used in an exterior acoustic finite element model to simulate the Sommerfeld radiation condition. However, the time-domain model is often too large to efficiently handle the transient simulation due to the frequency-dependent stretching function and the number of elements needed per wavelength. Krylov subspace-based model order reduction can alleviate this problem by only matching the important moments of the original model. However, stability-preservation is not guaranteed, which is the key to perform time-domain simulation. This work proposes a stability-preserving model order reduction scheme for the time-domain PMLs. By assessing the stability conditions of time-domain PMLs, a two-step strategy is applied. Firstly, a one-sided split basis is used to preserve the form of Lyapunov function of the original model such that a stable intermediate reduced order model (ROM) results. Secondly, an unsplit basis is applied on the modal transformation of this intermediate ROM to reduce the size further. The proposed method is verified by numerical simulations.

Cross-Modal Face Super-Resolution Based on Quasi-Siamese Domain Transfer Fusion Network

In this paper, we propose a Cross-Modal Face Super-Resolution (CMFSR) method to construct high-resolution (HR) facial images from low-resolution (LR) cross-modal facial images captured respectively by disjoint visible light (VIS) and near-infrared (NIR) cameras. Due to the coupling of modality transformation and information fusion, CMFSR is more difficult to obtain HR reconstructed results compared with traditional super-resolution. To solve this problem, a Quasi-Siamese Domain Transfer Fusion Network (QSDTFN) for CMFSR is proposed in this paper, whose two branches transfer two LR face modality to HR face modality by domain transfer respectively. Different from two completely independent branches in the traditional pseudo-siamese network, only the HR-to-LR face transfer processes of the two branches in our quasi-siamese network are independent, while the LR-to-HR face transfer processes are coupled. This coupled module called the Adaptive Weighted Domain Transfer Fusion Module (AWDTFM) disentangles the modality and identity information in the two LR faces, thus achieving modality transformation and identity information fusion simultaneously. In order to strengthen the optimization on the process of CMFSR, this method further introduces the backward QSDTFN to form a higher-level bidirectional structure with the forward QSDTFN, and specifically designs two types of losses: intra-network loss and inter-network loss, to constrain the modality and identity consistencies within one QSDTFN and between two QSDTFNs respectively. The experimental results on the challenging LR cross-modal face datasets demonstrate that the proposed method performs favorably against the state-of-the-art methods.

Model order reduction of time-domain acoustic finite element simulations with perfectly matched layers

This paper presents a stability-preserving model reduction method for an acoustic finite element model with perfectly matched layers (PMLs). PMLs are often introduced into an unbounded domain to simulate the Sommerfeld radiation condition. These layers act as anisotropic damping materials to absorb the scattered field, of which the material properties are frequency- and coordinate-dependent. The corresponding time-domain model size is often very large due to this frequency-dependent property and the number of elements needed per wavelength. Therefore, to enable efficient transient simulations, this paper proposes a two-step method to generate stable reduced order models (ROMs) of such systems. Firstly, the modified and stable version of PMLs is projected by a one-sided split basis, which gives a stable intermediate ROM. Secondly, the intermediate ROM is modified to satisfy the stability-preserving condition by applying the modal transformation. Applying any one-sided model order reduction method on this modified model leads to a stable and small ROM. This two-step method is further extended to account for the locally-conformal PML model by reformulating it in curvilinear coordinates, which works for arbitrary convex truncated domains. The proposed method is successfully verified by several numerical simulations.

A cross-modal high-resolution image generation approach based on cloud-terminal collaboration for low-altitude intelligent network

The advancement of digitization and automation in Low Altitude Intelligent Networking (LAIN) is constrained by limited computational resources and the absence of a dedicated modal transformation mechanism, affecting the performance of latency-sensitive missions. This study addresses these challenges by proposing a Downscaling Reconstruction Multi-scale Locally Focused Generative Adversarial Network (DR-MFGAN) with Federated Learning (FL). This integration employs wavelet transform downscaling and zero-shot residual learning techniques to create noise-suppressed image pairs, ultimately facilitating high-quality image reconstruction. The core network structure is composed of multidimensional residual blocks and generative confrontation network, and feature extraction is further enhanced through cross channel attention mechanism. Finally, distributed training based on Federated Learning ensures the training effectiveness of nodes with small data volumes.Experimental results demonstrate significant improvements: an 18.18% reduction in Mean Squared Error (MSE), a 33.52% increase in Peak Signal to Noise Ratio (PSNR), and a 39.54% improvement in Learned Perceptual Image Patch Similarity (LPIPS). The edge terminal can provide high-resolution imagery with limited data, achieving precise cross-modal transformations. This approach enhances LAIN capabilities, addressing computational and transformation challenges to support critical latency-sensitive missions.

Quantitative study on the flow control of a 1:2 rectangular cylinder based on the leading-edge suction

The present study employed the leading-edge suction flow control on a 1:2 rectangular cylinder to control its unsteady aerodynamics. To study the suppression on aerodynamic forces, the surface pressures were measured at the Reynolds number of 1.28 × 104. A dimensionless suction momentum coefficient (Cμ) was defined to quantify the suction flow rate. The detailed flow visualization was conducted on the wake flows as well as the leading-edge separated flows to reveal the control mechanism. The proper orthogonal decomposition (POD) analysis was conducted on the wake flows to study the modal transformations. Results show that the leading-edge suction flow control could significantly affect the flow separation-reattachment characteristics and the present control processing could be divided into five stages with increasing Cμ. The lift fluctuation could reach a maximum reduction of 66.4%. Meanwhile, the flow separation at the leading edge was gradually suppressed and eliminated with increasing Cμ. The wake flow was compressed in the transverse region and suppressed in the turbulent kinetic energy (TKE) as well as the Reynolds shear stress (RSS), which should be responsible for the suppression on the lift fluctuation.

Defect recognition of photovoltaic panels based on adaptive dimensional feature aggregation convolutional neural network

Abstract Under the policy framework of achieving carbon neutrality goals and addressing climate change, the importance of photovoltaic power generation will become more prominent, therefore the detection of defects in photovoltaic panels will become increasingly important. In order to accurately detect defects in photovoltaic panels, this paper proposes an adaptive dimensional feature aggregation algorithm that combines Deformable Convolutional (DCN) and C3 convolutional layers to form C3_ DCN module enhances the generalization ability of convolutional kernels; By incorporating the Cross Modal Transformer Attention Mechanism (CoTAttention) and Context Enhancement Module (CAM) at the output end, the model has the ability to adaptively allocate computing resources and enhance the ability to detect small targets; Using Focal EIoU as the loss function reduces the slow convergence speed and inaccurate results of known box regression methods. The experimental results show that for the PVEL-AD industrial detection dataset, the improved model has an average accuracy of 91.8%, which is 1.5% higher than the original model.

Vibration and stability of axially moving beam on multiple intermediate elastic supports with transfer matrix method

This study delves into the vibration characteristics and stability of an axially moving beam with multiple intermediate elastic supports. Initially, the motion equation of the beam is developed, setting the stage for an in-depth investigation of its free vibration characteristics through the transfer matrix method. This analysis covers the beam’s natural frequencies and critical velocity when moving at a constant velocity. Additionally, the dynamic behavior of the beam is investigated when it is subjected to constant axial movement and a concurrent traveling load. The multi-scale method is used to analyze the instability regions of the axially moving beam on multiple intermediate elastic supports as disturbance of the axial movement approaches subharmonic and sum-type combined resonances. The investigation further considers how alterations in the stiffness and configuration of the elastic supports influence the dynamic behavior and stability of the beam. Unique dynamic characteristics arise when the axially moving beam is supported by multiple elastic elements, contrasting with those observed in beams without such supports. An increase in support stiffness enhances the critical velocity at which the beam axially moves, potentially causing modal transformations of the structure prior to reaching this critical velocity, and diminishing the instability regions associated with sum-type combined resonances under disturbance velocities.

Active vibration isolation on electromechanical coupling suspension system of a quarter-vehicle via modal space controller design

The active vibration isolation of a quarter-vehicle suspension system using an electromagnetic damper was achieved by designing the feedback gain of a linear quadratic regulator (LQR) and H-∞ controllers in modal coordinates. The efficacy of vibration isolation was analyzed in the frequency domain, while the performance of vibration suppression and passengers’ comfort quality were evaluated in the time domain. The research results demonstrated that by employing modal coordinate transformation in state space, the designated system’s resonance peaks can be directly suppressed in the frequency domain. The findings also indicate that the active control in modal coordinates can achieve a higher vibration suppression percentage per unit control effort in comparison to traditional active control in physical space. In other words, the modal controller is more efficient in vibration suppression than the controller designed in physical space.

Transferable non-invasive modal fusion-transformer (NIMFT) for end-to-end hand gesture recognition.

Objective.Recent studies have shown that integrating inertial measurement unit (IMU) signals with surface electromyographic (sEMG) can greatly improve hand gesture recognition (HGR) performance in applications such as prosthetic control and rehabilitation training. However, current deep learning models for multimodal HGR encounter difficulties in invasive modal fusion, complex feature extraction from heterogeneous signals, and limited inter-subject model generalization. To address these challenges, this study aims to develop an end-to-end and inter-subject transferable model that utilizes non-invasively fused sEMG and acceleration (ACC) data.Approach.The proposed non-invasive modal fusion-transformer (NIMFT) model utilizes 1D-convolutional neural networks-based patch embedding for local information extraction and employs a multi-head cross-attention (MCA) mechanism to non-invasively integrate sEMG and ACC signals, stabilizing the variability induced by sEMG. The proposed architecture undergoes detailed ablation studies after hyperparameter tuning. Transfer learning is employed by fine-tuning a pre-trained model on new subject and a comparative analysis is performed between the fine-tuning and subject-specific model. Additionally, the performance of NIMFT is compared to state-of-the-art fusion models.Main results.The NIMFT model achieved recognition accuracies of 93.91%, 91.02%, and 95.56% on the three action sets in the Ninapro DB2 dataset. The proposed embedding method and MCA outperformed the traditional invasive modal fusion transformer by 2.01% (embedding) and 1.23% (fusion), respectively. In comparison to subject-specific models, the fine-tuning model exhibited the highest average accuracy improvement of 2.26%, achieving a final accuracy of 96.13%. Moreover, the NIMFT model demonstrated superiority in terms of accuracy, recall, precision, and F1-score compared to the latest modal fusion models with similar model scale.Significance.The NIMFT is a novel end-to-end HGR model, utilizes a non-invasive MCA mechanism to integrate long-range intermodal information effectively. Compared to recent modal fusion models, it demonstrates superior performance in inter-subject experiments and offers higher training efficiency and accuracy levels through transfer learning than subject-specific approaches.

Closed-Form Solutions to Continuous-Time Algebraic Riccati Equation for Second-Order Systems

Abstract Two closed-form methods to solve the continuous-time algebraic Riccati equation (CARE) for second-order systems in terms of the mass, damping, and stiffness matrices are presented. One method utilizes the modal transformation of mass and stiffness matrices, and the other does not require this transformation. Hundreds of high-dimensional second-order systems are used to show that these methods achieve similar or better accuracy compared to the state-of-the-art, while significantly reducing the computation time. Furthermore, advantages of these methods are illustrated in vibration control problems.

Construction and evolutionary pattern of the coupling relationship network of regional sustainable development in China

Climate change poses a serious threat to the survival and development of humankind. Accordingly, sustainable socioeconomic transformation has become a major global concern. This study quantitatively assesses the process and speed of regional-level sustainable development in China, as well as the degree of coupling and the degree of coupling coordination among Sustainable Development Goals (SDGs). To this end, based on regional development and data monitoring in China, we construct a regional sustainable development assessment system containing 111 indicators, which combines existing sustainable development assessment indicators with the indicators of the 2030 Agenda for Sustainable Development. On that basis, using network topology construction and modal transformation, we propose a framework for analyzing the coupling relationships and evolutionary patterns of regional sustainable development. The study found that the score for SDGs at the national level increased from 50.80 in 2010 to 60.86 in 2020, showing a good development trend and the differences between provinces and regions was gradually reduced. The development speed of regional sustainable transformation shows a change from decreasing to increasing. In recent years, there is a significant change in the acceleration of regional sustainable development in China, with the vast majority of provinces having positive acceleration and only three provinces having negative acceleration. China's regional sustainable development shows that mutual coordination and promotion among provinces was higher in the eastern area than in the central region, which was higher than in the western area. The sustainable processes of each region and the coordination of sustainable development steadily increased toward a high level of coupling, all achieving coordinated development. The degree of coupling coordination and the SDG scores in each region were positively correlated with economic development, reflecting the role of economic development in the promotion of sustainability. However, this positive correlation showed a tendency of strengthening and then weakening. The correlation of the evolution patterns in the eastern–central, eastern–western, and central–western regions over time presented a trend of first weakening, then strengthening, and then weakening. Although the process of promoting sustainable development in western and central regions over time is weak synergistic consistency with that in eastern regions, overall, China's regional sustainable development presents a development model of multi-province linkage and multi-goal coordination.

DDViT: Advancing lithology identification on FMI image logs through a dual modal transformer model with less information drop

Lithology is an essential topic in oil and gas reservoir studies. Lithological observation lays the foundation for assessing oil and gas prospects and guides future exploration and development. Currently, the prevailing approach in lithological observation heavily depends on the manual analysis of core samples. However, such an approach is highly subjective and time-consuming. With the development of deep learning, some automated deep learning-based methods have been proposed for lithology interpretation from logging curves. However, the Fullbore Formation MicroImager (FMI) image logging, while widely used in the oil field exploration and development process, is occasionally deployed and utilized for lithology identification. In this work, we proposed a Dual-modal Drop-less-information Vision Transformer (DDViT), an FMI image lithology identification model based on transformer architecture. DDViT uses a dual-modal architecture to identify lithology using two different image modalities, namely dynamic FMI images (FMI_DYN) and static FMI images (FMI_STAT). These modalities reflect the local and overall characteristics, respectively. Furthermore, DDViT uses a less-information dropping module to drop the blank band information inherent in the FMI images to make our model more rational and stable. DDViT achieved a 90.81% lithology identification accuracy on Fengxi Well A of the western Qaidam Basin, providing a new approach to lithology identification and demonstrating the great potential of deep learning in geological images.

An Electromagnetic Load Identification Method Based on the Polynomial Structure Selection Technique

Electromagnetic loads can effectively monitor motor health and improve motor design. Considering the weak correlation of the modal shape and Chebyshev orthogonal polynomial in the space-time independent electromagnetic load identification method, a proposed method combining the polynomial structure selection technique together with limited measured displacement responses is presented, in which an error reduction ratio is used to pick out the significant mode shape matrix and the Chebyshev orthogonal polynomial. The time-history function of the electromagnetic load is reconstructed by combining the significant mode shape matrix and the identified concentrated load through modal transformation, and the corresponding spatial distribution function is fitted by the significant Chebyshev orthogonal polynomial. Eventually, a comparative numerical study considering the selection of significant components and measurement noise is carried out to prove the effectiveness of the presented method.

Incongruity-aware multimodal physiology signals fusion for emotion recognition

Various physiological signals can reflect the human’s emotional states objectively. How to take advantage of the common as well as complementary properties of different physiological signals in representing the emotional states is an interesting problem. Although various models have been constructed to fuse multimodal physiological signals for emotion recognition, the possible incongruity existing among different physiological signals in representing the emotional states and the redundancy resulted from the fusion, which may affect the performance of the fusion schemes seriously, were seldom considered. To this end, a fusion model, which can eliminate the incongruity among different physiological signals and reduce the information redundancy to some extent, is proposed. First, one physiological signal is chosen as the primary modality due to its prominent performance in emotion recognition, and the remaining physiological signals are viewed as the auxiliary modalities. Secondly, the Cross Modal Transformer (CMT) is adopted to optimize the features of the auxiliary modalities by eliminating the incongruity among them, and then Low Rank Fusion (LRF) is performed to eliminate information redundancy caused by fusion. Thirdly, the modified CMT (MCMT) is constructed to enhance the feature of the primary modality by that of each optimized auxiliary modality feature. Fourthly, Self-Attention Transformer (SAT) is performed on the concatenation result of all the enhanced primary modality features to take full advantage of the common as well complementary properties among them in representing the emotional states. Finally, the enhanced primary modality feature and the optimized auxiliary features are fused by concatenation for emotion recognition. Extensive experimental results on DEAP and WESAD datasets demonstrate that (i) The incongruity does exist among different physiological signals, and the CMT-based auxiliary modality feature optimization strategy can eliminate the incongruity prominently; (ii) The emotion prediction accuracy of the primary modality can be enhanced by the auxiliary modality; (iii) All the key modules in the proposed model, CMT, LRF, and MCMT, contribute to the performance enhancement of the proposed model; iv) The proposed model outperforms State-Of-The-Art (SOTA) models in emotion recognition task.

Towards optimal [formula omitted]-variational autoencoders combined with transformers for reduced-order modelling of turbulent flows

Variational autoencoders (VAEs) have shown promising potential as artificial neural networks (NN) for developing reduced-order models (ROMs) in the context of turbulent flows. In this study, we propose a method that combines β-VAEs for modal decomposition and transformer neural networks for temporal-dynamics prediction in the latent space to develop ROMs. We apply our method to an existing database of a turbulent flow around a wall-mounted square cylinder obtained by direct numerical simulation (DNS). A parametric study is performed to investigate the effects of the hyperparameters of the proposed β-VAEs and determine the optimal values. For the first time, we incorporate the consideration of the complexity of architecture into our studies, providing new insights into hyperparameter selection for β-VAEs, which remains a challenging problem for optimising model performance. Results regarding the influence of the different hyperparameters and guidelines to design these architectures are reported. Our optimal model achieves a reconstruction accuracy of 97.18% of the entire dataset using only ten modes. Subsequently, we employ the transformer models to identify latent-space temporal dynamics learned by the optimal β-VAE model and build ROMs to predict instantaneous fields. The resulting model achieves promising accuracy in temporal-dynamics predictions and yields energy reconstruction levels of 96.5% and 83% for a field 25 and 50 steps into the future, respectively, showcasing the potential of the transformer in predicting the temporal dynamics. Overall, the proposed method has potential applications in advanced flow control and fundamental studies of complex turbulent flows.

Using the Exact Equivalent π-Circuit Model for Representing Three-Phase Transmission Lines Directly in the Time Domain

This paper presents a novel three-phase transmission line model for electromagnetic transient simulations that are executed directly within the time domain. This model relies on distributed and frequency-dependent parameters, as well as employs modal transformation for its implementation. The single-phase model of the exact equivalent π-circuit is utilized for each propagation mode. This model combines discrete components, such as resistors, inductors, and capacitors, to accurately emulate the transmission line behavior via linear circuit elements. This model can be seamlessly integrated into various electrical circuit simulation software, thus allowing easy utilization and incorporating time-varying elements to analyze transmission lines. The JMarti model, which comes by default in the Alternative Transient Program, and the numerical Laplace transform method implemented in MATLAB were utilized to validate the proposed solution across various scenarios. An advantage of this model is its independence from the prior calculation of travel time and its exemption from convolutions, inverse Laplace transforms, and Fourier transforms, thus streamlining the simulation process.

Graph Sampling-Based Multi-Stream Enhancement Network for Visible-Infrared Person Re-Identification.

With the increasing demand for person re-identification (Re-ID) tasks, the need for all-day retrieval has become an inevitable trend. Nevertheless, single-modal Re-ID is no longer sufficient to meet this requirement, making Multi-Modal Data crucial in Re-ID. Consequently, a Visible-Infrared Person Re-Identification (VI Re-ID) task is proposed, which aims to match pairs of person images from the visible and infrared modalities. The significant modality discrepancy between the modalities poses a major challenge. Existing VI Re-ID methods focus on cross-modal feature learning and modal transformation to alleviate the discrepancy but overlook the impact of person contour information. Contours exhibit modality invariance, which is vital for learning effective identity representations and cross-modal matching. In addition, due to the low intra-modal diversity in the visible modality, it is difficult to distinguish the boundaries between some hard samples. To address these issues, we propose the Graph Sampling-based Multi-stream Enhancement Network (GSMEN). Firstly, the Contour Expansion Module (CEM) incorporates the contour information of a person into the original samples, further reducing the modality discrepancy and leading to improved matching stability between image pairs of different modalities. Additionally, to better distinguish cross-modal hard sample pairs during the training process, an innovative Cross-modality Graph Sampler (CGS) is designed for sample selection before training. The CGS calculates the feature distance between samples from different modalities and groups similar samples into the same batch during the training process, effectively exploring the boundary relationships between hard classes in the cross-modal setting. Some experiments conducted on the SYSU-MM01 and RegDB datasets demonstrate the superiority of our proposed method. Specifically, in the VIS→IR task, the experimental results on the RegDB dataset achieve 93.69% for Rank-1 and 92.56% for mAP.

Fault location of untransposed double-circuit transmission lines based on an improved Karrenbauer matrix and the QPSO algorithm

Some double-circuit transmission lines are untransposed, which results in complex coupling relations between the parameters of the transmission lines. If the traditional modal transformation matrix is directly used to decouple the parameters, it can lead to large errors in the decoupled modal parameter, errors which will be amplified in the fault location equation. Consequently, it makes the fault location results of the untransposed double-circuit transmission lines less accurate. Therefore, a new modal transformation method is needed to decouple the parameter matrix of untransposed double-circuit transmission lines and realize the fault location according to the decoupled modal parameter. By improving the basis of the Karrenbauer matrix, a modal transformation matrix suitable for decoupling parameters of untransposed double-circuit transmission lines is obtained. To address the difficulties in solving the fault location equation of untransposed double-circuit transmission lines, a new fault location method based on an improved Karrenbauer matrix and the quantum-behaved particle swarm optimization (QPSO) algorithm is proposed. Firstly, the line parameter matrix is decomposed into identical and inverse sequence components using the identical-inverse sequence component transformation. The Karrenbauer matrix is then transformed to obtain the improved Karrenbauer matrix for untransposed double-circuit transmission lines and applied to identical and inverse sequence components to solve the decoupled modal parameter. Secondly, based on the principle that voltage magnitudes at both ends are equal, the fault location equation is expressed using sequence components at each end, and the QPSO algorithm is introduced to solve the equation. Finally, the feasibility and accuracy of the proposed method are verified by PSCAD simulation. The simulation results fully demonstrate that the innovative improvement on the basis of the traditional modal transformation matrix in this paper can realize the modal transformation of the complex coupling parameters of the untransposed double-circuit transmission lines. It causes almost no errors in the decoupling process. The QPSO algorithm can also solve the fault location equation more accurately. The new fault location method can realize the accurate fault location of untransposed double-circuit transmission lines.

An active piezoelectric damping vibration control system for the sting used in wind tunnel

In wind tunnel tests, the cantilever sting supporting system often suffers from low-frequency and large-amplitude resonance due to its inherent low structural damping characteristic, resulting in the degradation of data quality and structural safety. To improve wind tunnel testing safety and data accuracy, this paper is dedicated to establish an active vibration control system using piezoelectric stack actuators. A novel methodology of vibration monitoring based on modal transformation, which uses measured strain and a Strain-to-Displacement Transformation (SDT) matrix to reconstruct dynamic displacement field, is proposed herein. Meanwhile, strain sensor positions are optimized by an improved Particle Swarm Optimization (PSO) algorithm to reduce systematic estimation errors of this method. Furthermore, a Back-Propagated Neutral Network (BPNN) is established to implement a self-adaptive control strategy. A series of verification tests are performed to demonstrate the validity of the proposed system. Experimental results indicate that the relative Root Mean Square Error (RMSE) between estimated vibration displacement and measured vibration displacement is less than 3%, and a vibration attenuation of over 14 dB/Hz is achieved in ground tests, proving the superiority of this intelligent active vibration suppression system.

Load-independent multi-objective sensor placement method for localization and reconstruction of external excitations under interval uncertainties

The direct measurement of external excitations under operation conditions remains challenging in many engineering applications. This study proposes a method of optimal sensor placement for the best localization and reconstruction of external excitations when considering multi-source uncertainties. The excitation inversion is achieved in the framework of minimizing the discrepancy between the forward and inversed modal loads. To accurately obtain the modal force in the time-domain excitation reconstruction algorithm, three combined indicators of the interested modal matrix are defined, which integrate measures of effectiveness, stability, orthogonality, and robustness of inverse modal transformation. The optimal sensor placement is determined by multi-objective particle swarm optimization and Pareto optimal solution. In addition, to alleviate the computational burden of loading localization, a surrogate model based on an interval kriging interpolation model and sequential sampling strategies is established for the forward modal forces. And a time-dependent interval error of modal forces inspired by the theory of second-order narrow bounds in system reliability is developed to guide the inversion process. Notably, the developed sensor selection and excitation inversion methods are independent of the type, location, and time evolution of external excitations. Eventually, two numerical examples and an experimental case are discussed to demonstrate the validity and feasibility of the developed approach.