Academic literature on the topic 'Stochastic time domain spectral element method'

A novel method to solve exterior Helmholtz problems in the case of multipole excitation and random input data is developed. The infinite element method is applied to compute the sound pressure field in the exterior fluid domain. The consideration of random input data leads to a stochastic infinite element formulation. The generalized polynomial chaos expansion of the random data results in the spectral stochastic infinite element method. As a solution technique, the non-intrusive collocation method is chosen. The performance of the spectral stochastic infinite element method is demonstrated for a time-harmonic problem and an eigenfrequency study.

Abstract The stochastic method of simulating ground motions from finite faults is validated against strong-motion data from the M 6.7 1994 Northridge, California, earthquake. The finite-fault plane is subdivided into elements, each element is assigned a stochastic ω2 spectrum, and the delayed contributions from all subfaults are summed in the time domain. Simulated horizontal acceleration time histories and Fourier spectra at 28 rock sites are compared with observations. We first perform simulations using the slip distribution on the causative fault derived from strong-motion, teleseismic, GPS, and leveling data (Wald et al., 1996). We then test the performance of the method using quasi-random distributions of slip and alternative hypocenter locations; this is important because the rupture initiation point and slip distribution are in general not known for future earthquakes. The model bias is calculated as the ratio of the simulated to the observed spectrum in the frequency band of 0.1 to 12.5 Hz, averaged over a suite of rock sites. The mean bias is within the 95% confidence limits of unity, showing that the model provides an accurate prediction of the spectral content of ground motions on average. The maximum excursion of the model bias from the unity value, when averaged over all 28 rock stations, is a factor of approximately 1.6; at most frequencies, it is below a factor of 1.4. Interestingly, the spectral bias and the standard deviation of the stochastic simulations do not depend on whether the fault slip distribution and hypocenter location are based on data or are randomly generated. This suggests that these parameters do not affect the accuracy of predicting the average characteristics of ground motion, or they may have their predominant effect in the frequency range below about 0.1 Hz (below the range of this study). The implication is that deterministic slip models are not necessary to produce reasonably accurate simulations of the spectral content of strong ground motions. This is fortunate, because such models are not available for forecasting motions from future earthquakes. However, the directivity effects controlled by the hypocenter location are important in determining peak ground acceleration at individual sites. Although the method is unbiased when averaged over all rock sites, the simulations at individual sites can have significant errors (generally a factor of 2 to 3), which are also frequency dependent. Factors such as local geology, site topography, or basin-propagation effects can profoundly affect the recordings at individual stations. To generate more accurate site-specific predictions, empirical responses at each site could be established.

In order to study the random vibration performance of trains running on continuous beam bridge with vertical track irregularity, a time-domain framework of random analysis on train-bridge coupling system is established. The vertical rail irregularity is regarded as a random process. A multibody mass-spring-damper model is employed to represent a moving railway car and the bridge system is simulated by finite elements. By introducing the pseudo excitation algorithm into the train-bridge interaction dynamic system, expressions of the mean value, standard deviation, and power spectral density of the nonstationary random dynamic responses of bridge and vehicles are derived. Monte-Carlo simulations are implemented to validate the presented method. A comprehensive analysis of the train-bridge coupling system with vertical track irregularity is conducted focusing on the effect of the randomness of the vertical rail irregularity on the dynamic behavior of the running train and the three-span continuous concrete bridge. Moreover, stochastic characteristics of the indicator for assessing the safety and the riding quality of the railway cars running on continuous beam bridge are carried out, which may be a useful reference in the dynamic design of the bridge.

This thesis considers the development and analysis of a hybrid spectral element method for the solution of two-dimensional wave scattering problems in the time domain. The components, namely a quadrilateral formulation of the diagonal mass matrix spectral element method and a triangular formulation of the spectral discontinuous Galerkin finite element method, are introduced and tested separately before being coupled to form the final hybrid procedure. Subsequently, a simple circular scattering problem is analysed to validate the computational model and various methods of curved boundary representation are tested to assess their impact on solution accuracy. Finally, a range of two-dimensional wave scattering problems are modelled, showing the computational efficiency of the higher order approximation in comparison with low order linear models.

Aerospace structures are different from other structures due to the difference in the required properties for an aerospace application. The primary motivation of this kind of structure lies in its lightweight nature, need for enough strength to withstand different loading conditions, durability, high performance, etc. The choice of materials for the aerospace structures is mainly metals, alloys, ceramics, and composites. The use of composites is increasing rapidly for aerospace structures due to its lightweight nature, high specific strength, tailoring properties to meet the design needs, fatigue and corrosion resistance, etc. The structural analysis and modelling of different aerospace structures to obtain a particular level of performance depend on a deep understanding of the material properties, and structural and geometric characteristics. So, to achieve a reliable structure, variability among the material and its effect on analysis and design should be addressed. In this thesis, we explore the modelling and analysis of uncertainty and its effect on different structures which have aerospace applications. In particular laminated composites, sandwich composites and periodic structures are considered. Computational methods which need less CPU time are also explored to facilitate efficient uncertainty quantification. In search of the effect of uncertainty on the failure curves of laminated composites, we have considered a random variable based modelling approach to obtain the most conservative Tsai- Wu failure envelopes considering material as well as ply angle uncertainty. The uncertainty analysis is performed using Monte Carlo simulation (MCS). Meso-scale elastic constants of the carbon/epoxy composite material are considered as uncertain with 5% coefficient of variation (COV). The innermost points of the stochastic realizations of failure curves form the most conservative modified Tsai-Wu failure envelopes, and these curves are used as constraint functions to perform the weight optimization problem using particle swarm optimization algorithm. Later, AS4/8552 carbon epoxy laminated composite with material and ply angle uncertainties is considered to analyze the effect of uncertainties on the failure strength properties, and their probability distributions are also obtained. For uncertainty influenced design, three ranges of failure curves (μ − 3σ, μ, μ + 3σ) (μ = mean, σ = standard deviation) are chosen as constraint functions to observe the effect of uncertainty on the design of composite structures. Also, a new heuristic directional bat algorithm (dBA) is explored for the constrained minimum weight composite laminate design optimization problem. We also focus on the modelling aspects of spatial variability of material and geometric parameters for sandwich composite beam structures. The elastic and geometric properties of the sandwich beam are considered as a non-Gaussian random field, and MCS along with the computationally efficient time-domain spectral element method (TSEM) is proposed considering a higher order sandwich panel theory which can capture the core compatibility. Further, a stochastic time domain spectral element method (STSEM) is proposed for both Timoshenko and sandwich beams. Discretization of the non-Gaussian random field is performed using expansion optimal linear estimation and optimal linear estimation. The analysis shows the effect of different variability and their impact on the various response characteristics of beam structures. The computational efficiency of STSEM is shown through the Timoshenko beam problem. Results show that STSEM lessens the CPU time for computation. The effect of uncertainty is quantified considering static, free vibration and dynamic analysis. Aerospace structures are subject to transient loading which consists of high-frequency content and noise from undesired vibration. The use of periodic structures helps to control or reduce the undesired vibration. In this thesis, the effect of material uncertainty on the stop band characteristics of periodic bar and beam structures is also investigated. Periodicity is common in aerospace structures (e.g., fuselage structure). The basic building blocks of any structure are bar, beam and plate structures which also influence the design of the whole structure. The periodic analysis of these structures is essential to investigate the dispersion characteristics. The periodic analysis is carried out for 1-D bar and beam considering uncertainty in the material properties. A TSEM based transfer matrix formulation and wave finite element method is proposed to carry out the analysis. The analysis shows a considerable saving of CPU time and also quantifies the individual effects of random parameters on the stop bands of a periodic bar and Timoshenko beam. Parametric analysis is performed and insight is given for design aspects regarding the stop bands in the frequency range of interest.

In this study we propose a fast hybrid spectral-element time-domain (SETD) / finite-element time-domain (FETD) method for transient analysis of multiscale electromagnetic problems, where electrically fine structures with details much smaller than a typical wavelength and electrically coarse structures comparable to or larger than a typical wavelength coexist.

Simulations of multiscale electromagnetic problems, such as electromagnetic interference (EMI), electromagnetic compatibility (EMC), and electronic packaging, can be very challenging for conventional numerical methods. In terms of spatial discretization, conventional methods use a single mesh for the whole structure, thus a high discretization density required to capture the geometric characteristics of electrically fine structures will inevitably lead to a large number of wasted unknowns in the electrically coarse parts. This issue will become especially severe for orthogonal grids used by the popular finite-difference time-domain (FDTD) method. In terms of temporal integration, dense meshes in electrically fine domains will make the time step size extremely small for numerical methods with explicit time-stepping schemes. Implicit schemes can surpass stability criterion limited by the Courant-Friedrichs-Levy (CFL) condition. However, due to the large system matrices generated by conventional methods, it is almost impossible to employ implicit schemes to the whole structure for time-stepping.

To address these challenges, we propose an efficient hybrid SETD/FETD method for transient electromagnetic simulations by taking advantages of the strengths of these two methods while avoiding their weaknesses in multiscale problems. More specifically, a multiscale structure is divided into several subdomains based on the electrical size of each part, and a hybrid spectral-element / finite-element scheme is proposed for spatial discretization. The hexahedron-based spectral elements with higher interpolation degrees are efficient in modeling electrically coarse structures, and the tetrahedron-based finite elements with lower interpolation degrees are flexible in discretizing electrically fine structures with complex shapes. A non-spurious finite element method (FEM) as well as a non-spurious spectral element method (SEM) is proposed to make the hybrid SEM/FEM discretization work. For time integration we employ hybrid implicit / explicit (IMEX) time-stepping schemes, where explicit schemes are used for electrically coarse subdomains discretized by coarse spectral element meshes, and implicit schemes are used to overcome the CFL limit for electrically fine subdomains discretized by dense finite element meshes. Numerical examples show that the proposed hybrid SETD/FETD method is free of spurious modes, is flexible in discretizing sophisticated structure, and is more efficient than conventional methods for multiscale electromagnetic simulations.

Dissertation

Ultrasonic guided wave-based detection of structural defects or damages is one of the promising methods for structural health monitoring (SHM). A significant proportion of structural elements in an aircraft structure and other civil and marine structures are thin structures and those components can be monitored using ultrasonic guided wave. Piezoelectric transducers have been extensively applied to ultrasonic wave based non-destructive inspection (NDI). SHM is considered an extension of NDI adding a new dimension to the way future structures will be designed. In recent years, research on the behavior of guided wave in structures, especially targeted for aerospace application, received enormous significance. Guided wave interaction with structural features and defects/damages creates complicated wave field patterns. Experimental observation is limited to the availability of sophisticated instrumentation and tomography techniques to visualize the wave pattern even on the surface of a structure and simulation is the only way various details regarding internal patterns of the waves in a complex geometry can be analyzed. Analytical approach to deal with the wave propagation in a structure is limited to simple geometries and simple forms of damage. Guided wave in a simple geometry like beam/plate is modeled using analytical as well as semi-analytical methods. But when it comes to complicated problems like structures with actual damage/fracture or geometrical complexity which includes curvature and components with joints etc., one requires efficient computational schemes to simulate the behavior. Contents x There are various numerical tools developed in last few decades, such as finite difference, finite element, boundary element methods etc. In high frequency guided wave propagation problems, higher-order modes participate. Moreover, to detect a small damage, a high-frequency wave is needed. To deal with such kind of problems incorporating standard polynomial incorporation based finite element schemes need very fine mesh, which makes the computations of such problem enormously expensive, sometimes prohibitive. Time domain spectral element (TSFE) method is an efficient numerical method that can capture higher order field and therefore can deal with wave propagation problems efficiently and with better accuracy. TSFE uses higher order highly convergent interpolation functions based on the Chebyshev or Lobatto nodal distributions. TSFE based on the Lobatto nodal distribution and Legendre-Lobatto quadrature rule makes the mass matrix diagonal, which reduces the computer memory requirement to a great extent. As the number of nodes increases, accuracy increases exponentially and hence the term spectral finite element. The present thesis incorporates this idea and formulates a TSFE scheme to simulate guided wave propagation in laminated composite materials with damages such as material uncertainty/degradation, micro-cracks, and delaminations. Various benchmark problems are solved to validate the simulated results and establish superior convergence properties. Because of anisotropy of composite laminate and direction dependent properties of its constituent, composite laminate has various damage modes including matrix crack, fiber break, delamination etc. Among those damage modes, in this thesis, a special focus is given on delamination detection problem as it grows in the interfacial plane under repeated loading and reduces load-carrying capability to a great extent. Those damage modes are internal hence invisible. In wave propagation based detection methods, delamination can be identified and localized from the wave scattering from it. But it is of great interest to quantify the damage in terms of various parameters such as delamination length as well as the thickness and position of the laminate. Delamination scatters an incident wave and the strength of the reflection depends on the frequency/wavelength, length and thickness position of the delamination for a given structure. Simulation results show that the near-field effect of the damaged region provides crucial information about the scattering and reflected wave characteristics. Delamination in a laminate divides the damaged region into sub-laminates which are thinner compared to the base laminate. Each sub-laminate behaves like a separate waveguide. The vibration of the sub-laminates during the propagation of the wave through the damaged region Contents xi is of great interest. The energy of scattered waves and dissipation/conversion of energy due to the damage depends on the resonance characteristics of the sub-laminates. Therefore, the resonance phenomenon is correlated to the damage quantification problem with the help of simulation. Detection of damage near the structural boundary is one of the most challenging tasks as the scattering from the damage is overlapped by the strong reflection from the boundary. Effect of incident wave frequency/wavelength on the delamination near a structural boundary is studied. Damaged region behaves like the material degradation and in some frequency range the energy is trapped inside the damaged region and slow dissipation/conversion of the trapped energy into other forms of vibration creates a significant difference in the reflected wave and the simulated results help to identify the presence of damage. Impact-induced damage in composite has a great influence on the integrity and life of a composite structure. In most cases, initially, it develops material degradation in terms of matrix cracks at micro-scale. Although in composite structural design, micro-scale matrix cracks are not considered, however, as these micro-cracks coalesce, it gives initiate delamination. Under a severe dynamic impact loading, such small size delamination can grow and can lead to catastrophic failure of the structure. Ultrasonic wave propagation in composite with matrix crack is one of the major subjects of study which can predict the delamination onset in the composite. In the present thesis, wave scattering due to matrix cracks is studied and behavior of wave reflection from matrix cracking zone is investigated for various damage severity, which is expressed in terms of matrix crack density. Moreover, the matrix cracks along with delamination initiated from the zone, which is a kind of mixed-mode damage zone, appears more commonly in a composite structure than an ideal single-mode damage like a sharp crack or delamination. A mixed-mode damage complicates the modeling problem to be dealt with considering complex nature of near-field scattering of the incident wave. In the present thesis, this aspect is studied in details and damage severity effects are correlated to the scattered wave packet properties. Guided wave has a special characteristic that it is guided by the material media geometrically even when the structure is curved. This advantage can be exploited in developing damage detection scheme, for example, by bringing the scattered wave field located behind the curvature in a structural component without direct access to the surface where inspection cannot be carried out using local methods. Another important aspect of Contents xii guided wave is that propagation through a curved region not only produces reflection, it also generates mode converted waves which appear in both the reflected and transmitted waves. In the present thesis, wave transmissibility and signal loss at various frequencies and the effect of the radius of curvature is studied in detail. The simulation results provide a new insight regarding the wave mode and frequency for inspection for a given curved structure. Delamination near the curved region and T-joint is modeled and simulation shows correlation where the wave scattering due to delamination is possible to discriminate from that due to curved junctions. In composite, material uncertainty is inherent because of limited control over the fabrication processes, which in turn affects the proportion of the constituent materials or the fiber orientation. Significant material property variation can take place due to the variability in fiber volume fraction or the distortion in the stacking angle. The location of a damage and various other parameters are directly influenced by the material property variations. Therefore, the deterministic study is not sufficient to deal with these problems. In the present thesis, a Monte-Carlo method based simulation of wave scattering is carried out. The study primarily focuses on the problem of quantification of uncertainty in various damage detection parameters such as wave scattering coefficient and variation in the time of flight of the scattered packet, wave velocity etc. Detailed analysis is carried out regarding how the simulation based inspection method can be developed that gives better insight on the probabilistic distribution of the detection parameters of interests.

Inverse problems are very challenging as these problems involve, finding the cause by analyzing the effects. In structural dynamics problems, the effects are normally measured in terms of dynamic responses in structures. These responses which are used to find the cause generally have partial data, embedded with measurement noise, and are truncated. Due to these problems, inverse problems are generally ill-posed in most cases as against forward problems. In this dissertation, we solve five different types of inverse problems involving high-frequency transient loads. All these problems are solved using the time-domain spectral element method (TSFEM) along with experimental or numerically simulated responses. The dissertation starts with the formulation of the forward problem, which is obtaining the responses from known input forces. The general formulation of TSFEM of composite Timoshenko beam is derived. The isotropic beam formulation is shown as a special case in this formulation. Five different inverse problems solved in the thesis are: 1. Force identification problem: A new algorithm is developed using a 1-D waveguide, involving an eight noded spectral finite element. The force identification is carried out, using a few measured responses on the structure, and using TSFEM we reconstruct the input force. This is followed by a portal frame example to demonstrate the wave reflection complexities. New procedures are developed to use various types of response data like displacement, velocity, acceleration, and strain to identify the force. 2. Material identification problem: A new procedure making use of the developed TSFEM, few responses, and nonlinear least square techniques are used to determine the material properties. Also, we show the case, in which we derive the material properties without force input consideration. 3. Crack location detection problem: A new procedure is developed using TSFEM and mechanics of crack. Three methods are described, in which the first method uses only responses and wave speeds to determine the location of the crack. In the second method, force reconstruction using the measured responses is carried out and this, in turn, is used to determine the location of the crack. The third method uses the residues of the actual force and the reconstructed forces using the healthy beam matrices and cracked beam responses. A new procedure to identify the crack location using a general force input pulse having many frequency components is also developed. 4. Material defect identification: Material defects like voids or density changes are identified using TSFEM. Location and magnitude of defect are identified using response computation and using the method of residues. 5. Porous location and identification in a composite material: TSFEM is used to construct a porous element and this is used along with a healthy beam to generate the responses. A force reconstruction algorithm is used to identify the location of the porous element. The Force residue method to identify the location of the defect is also demonstrated. Further, we make use of the material identification algorithm with a few modifications to evaluate all the parameters for the porous element.

A micro-macro simulation algorithm for the calculation of polymeric flow is developed and implemented. The algorithm couples standard finite element techniques to compute velocity and pressure fields with stochastic simulation techniques to compute polymer stress from simulated polymer dynamics. The polymer stress is computed using a microscopic-based rheological model which combines aspects of network and reptation theory with aspects of continuum mechanics. The model dynamics include two Gaussian stochastic processes each of which is destroyed and regenerated according to a survival time randomly generated from the material’s relaxation spectrum. The Eulerian form of the evolution equations for the polymer configurations are spatially discretized using the discontinuous Galerkin method. The algorithm is tested on benchmark contraction domains for a polyisobutylene (PIB) solution. In particular, the flow in the abrupt die entry domain is simulated and the simulation results are compared with experimental data. The results exhibit the correct qualitative behavior of the polymer and agree well with the experimental data.