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陳立華 and Lap-wah Samson Chan. "The application of boundary collocation method to fracture problems." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1994. http://hub.hku.hk/bib/B31211197.
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Kapolka, Tyler J. (Tyler Joseph). "A partial state collocation method for covariance optimal control." Thesis, Massachusetts Institute of Technology, 2018. http://hdl.handle.net/1721.1/119908.
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Thesis: S.M., Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, 2018.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Cataloged student-submitted from PDF version of thesis.
Includes bibliographical references (pages 121-123).
An overview is presented for two methods of incorporating the covariance in an optimal control problem. Including the covariance in the optimal control problem can be beneficial in the field of navigation where it is desirable to design trajectories which either minimize navigation error or maximize observability for instrument calibration. The full state collocation method uses Legendre Gauss Radau collocation to discretize the deterministic states and controls as well as the unique elements of the covariance matrix. The problem is then transcribed to a nonlinear progamming problem (NLP) and is solved with an NLP solver. This method, however, results in problems with many constraints and variables, which is computationally expensive. The partial state collocation method, the main focus of this thesis, collocates the deterministic states and controls but uses a shooting method to incorporate the covariance matrix. The problem is then transcribed to a nonlinear programming problem, which has fewer constraints and variables than the full state collocation method. Both of these methods are demonstrated by solving for the trajectory that minimizes the final position uncertainty for a spacecraft reentering Earth's atmosphere. The problem is tested with different sized covariance matrices, which shows how the time it takes to solve the problem increases as the covariance matrix increases in size. The partial state collocation method is generally faster and converges in fewer NLP iterations than the full state collocation method. As the covariance matrix increases in size, the time it takes to solve the problem increases at a smaller rate for the partial state collocation method.
by Tyler J. Kapolka.
S.M.
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Kattelans, Thorsten. "The least squares spectral collocation method for incompressible flows." Berlin Köster, 2009. http://d-nb.info/997987812/04.
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Chan, Lap-wah Samson. "The application of boundary collocation method to fracture problems /." [Hong Kong] : University of Hong Kong, 1994. http://sunzi.lib.hku.hk/hkuto/record.jsp?B1379372X.
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Tian, Haitao. "Taylor meshless method for thin plates." Thesis, Paris, ENSAM, 2019. http://www.theses.fr/2019ENAM0036.
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Une nouvelle classe de méthodes sans maillage - Taylor Meshless Method (TMM) - a été introduite. Elle repose sur une solution explicite des équations aux dérivées partielles dans le domaine à l’aide des extensions de séries de Taylor. Parce que la PDE est résolue analytiquement dans le domaine, on est réduit à un problème de frontière discret dont la taille est plutôt petite.L’efficacité de TMM a été vérifiée en résolvant certaines équations aux dérivées partielles. Dans les cas étudiés, TMM est robuste et efficace. Pour les problèmes linéaires 2D, un domaine suffit pour résoudre les problèmes de valeurs limites.TMM est utilisé pour résoudre les problèmes de plaques de Kirchhoff. Les techniques de TMM permettent de réduire considérablement le degré de liberté, de manière à augmenter le degré de polynôme à un niveau très élevé. Les plaques sandwich laminées sont étudiées à l'aide de TMM. Différents cas sont considérés pour tester l'efficacité et le rendement de la méthode. L'erreur montre une convergence exponentielle avec l'augmentation du degré de polynômes.TMM est combiné à la méthode asymptotique-numérique (ANM) pour résoudre les problèmes de déviation importante des plaques minces. Les équations non linéaires sont développées sous la forme de séries de puissances, ce qui conduit le problème à une série d'équations linéaires. La longueur de l'étape est déterminée automatiquement par une technique de suivi de chemin fiable. La précision et l'efficacité de ANM-TMM sont vérifiées à l'aide de ces exemples et la méthode peut facilement être étendue à d'autres problèmes non linéaires.Basé sur le travail des problèmes de flexion, le flambement des plaques minces est étudié. Cette approche tire pleinement parti de la technique de suivi de chemin. Ainsi, le processus de flambement peut être illustré de manière beaucoup plus précise que celle d'autres méthodes. La performance de l'approche est examinée par une série de problèmes de flambement de référence.Les problèmes de plissement de la membrane sont étudiés. Différentes tensions et imperfections sont imposées pour tester leur influence sur les motifs de rides finaux. Les résultats montrent que TMM peut réaliser des simulations convergentes avec de très petites imperfections et des charges de tension comparées aux méthodes par éléments finis. L’approche de l’analyse de la membrane ridée par la TMM est bien établieA new class of meshless method – Taylor Meshless Method (TMM) - has been introduced that relies on an explicit solution of the Partial Differential Equations inside the domain with the help of Taylor series expansions. Because the PDE is solved analytically in the domain, one is reduced to a discrete boundary problem whose size is rather small.The effectiveness and efficiency of TMM have been verified by solving some partial differential equations. In the cases that have been studied, TMM is robust and effective. For 2D linear problems, one domain is sufficient to solve boundary value problems.TMM is used to solve Kirchhoff plate problems. Techniques in TMM help to reduce the degree of freedoms significantly so that one can increase the degree of the polynomials to a very high level. Laminated sandwich plates are studied by using TMM. Different cases are considered to test the effectiveness and efficiency of the method. The error shows exponential convergence with the increase of degree of polynomials.TMM is combined with asymptotic-numerical method (ANM) to solve large deflection problems of thin plates. The nonlinear equations are expanded in the form of power series, which leads the problem to a series of linear equations. The step length is determined automatically by a reliable path following technique. The accuracy and efficiency of ANM-TMM is verified through these examples and the method can be easily extended to other nonlinear problems.Based on the work of bending problems, the buckling of thin plates are studied. This approach fully takes the advantage of the path following technique. Thus the buckling process can be illustrated much more accurate than that by other methods. The performance of the approach is investigated by a series of benchmark buckling problems.The membrane wrinkling problems are studied. Different tension loads and imperfections are imposed to test their influence on final wrinkle patterns. The results show that TMM can accomplish convergent simulations with very small imperfections and tension loads in comparison with finite element methods. The approach of wrinkled membrane analysis by TMM has been well established
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Tarang, Mare. "Stability of the spline collocation method for Volterra integro-differential equations." Online version, 2004. http://dspace.utlib.ee/dspace/bitstream/10062/793/5/Tarang.pdf.
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Green, Jennifer Neal. "Modeling spider webs as multilinked structures using a Chebyshev pseudospectral collocation method." Thesis, Virginia Tech, 2018. http://hdl.handle.net/10919/83564.
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Spiders weave intricate webs for catching prey, providing shelter and setting mating rituals; arguably the most notable of these creations is the orb web. In this thesis we model the essential vibrations of orb webs by first considering a spider web as a multilinked structure of elastic strings. We then solve the associated eigenvalue problem using a Chebyshev pseudospectral collocation method to discretize the system. This thesis first examines the vibrations of webs with uniform material properties throughout, then investigates the effects of using biologically realistic material parameters for silks within a single web. Understanding how spiders detect and react to the vibrations produced by their webs is of interest for both biological and engineering applications.Master of Science
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Hernando, Quintanilla Francisco. "Pseudospectral collocation method for viscoelastic guided wave problems in generally anisotropic media." Thesis, Imperial College London, 2016. http://hdl.handle.net/10044/1/34915.
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In Non-Destructive Evaluation (NDE) applications guided waves are attractive to perform rapid inspections of long lengths and large areas. However, they are complicated, therefore it is important to have as much information and understanding about their physical properties as possible in order to design the most efficient and robust inspection process as well as to draw the correct conclusions from the measurement results. The main piece of information to gain insight into the guided wave's properties is dispersion curves which, for isotropic structures such as plates and cylinders, have been available for many years. There are many robust algorithms which are currently used to compute them: finite element simulations, partial wave based root finding routines (PWRF) and semi-analytical finite element simulations (SAFE). These methodologies have been generalized and also used to study and compute dispersion curves of more complicated anisotropic materials though the range of tractable cases was limited. Although robust, all these approaches present several challenges, mostly computational, such as missing modes (PWRF), the so called "large-fd" problem (PWRF), artificially increased stiffness (FE, SAFE) or improvement of dispersion curve tracing routines (FE, PWRF, SAFE). In addition, when studying complicated anisotropic materials with a low degree of symmetry or unusual axes configurations where propagation does not take place along any of the principal axes, PWRF routines are frequently unreliable and one must resort to specific SAFE simulations which also present their own challenges and, depending on the SAFE scheme used, can yield spurious modes which need to be carefully filtered. Recently, Pseudospectral Methods (Galerkin and Collocation schemes), were introduced in the field of elastic guided waves, providing a powerful, yet strikingly and conceptually simple alternative to the above algorithms by successfully finding the dispersion curves in isotropic structures and in some simple anisotropic problems. However, a systematic and general approach for accurately and robustly computing dispersion curves of guided waves in anisotropic media, up to the most general case of triclinic symmetry, has not yet been developed. The goal of the work presented in this thesis is to develop such a tool by means of the Pseudospectral Collocation Method (SCM) and to take advantage of its particular features to make it as robust as possible. Firstly, a PSCM scheme is developed for computing dispersion curves of guided waves in anisotropic elastic media by finding all the frequencies for a given value of the real wavenumber. The results are validated with the existing literature as well as with the results provided by the software DISPERSE developed in the NDT group at Imperial College London. Many of the most remarkable features of the PSCM (spectral accuracy, speed, and its failure to miss modes for instance) are already observed in this simple, yet important, class of problems in elastic media. Secondly, guided waves in viscoelastic anisotropic media are studied. In this case, modes present attenuation due to material damping which is reflected in the wavenumber being complex. In order to handle complex wavenumbers the PSCM schemes developed for elastic materials are appropriately extended by means of the Companion Matrix Method. It will be seen that, apart from lowly attenuated propagating modes, all the other highly attenuated modes are found, yielding the full three-dimensional spectrum of the problem under consideration. Moreover, when the PSCM schemes for viscoelastic media are used to compute the dispersion curves of guided waves in an elastic medium, all the remaining, imaginary as well as complex, roots of the elastic problem which were not computed by the simpler PSCM elastic schemes are found, providing the full three-dimensional picture of the dispersion curves. These PSCM schemes, as any other of the aforementioned approaches, only find pairs (\omega,k). If dispersion curves are to be plotted, those pairs must be linked correctly in order to plot the desired dispersion curves, which is non-trivial when crossings amongst modes occur. Motivated by this, an investigation of the parity and coupling properties of guided wave solutions is carried out in detail for all crystal classes. This investigation provides a robust alternative to conventional tracing routines and avoids the problem of mode crossings by exploiting the parity and coupling properties of the solutions. Finally, the most complicated problems involving embedded structures are investigated by including a Perfectly Matched Layer (PML) in the previously developed PSCM schemes for viscoelastic media. The dispersion curves for leaky and trapped modes in an isotropic elastic plate and in a similar cylinder immersed in an infinite ideal fluid are found, showing very good agreement with the results given by PWRF routines in a large range of frequencies. Last, but not least, an illustration of a two-dimensional PSCM scheme is presented to study a vibrating membrane. The results are compared with the available analytical solution showing again excellent agreement.
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Chandra, Sekhar D. "Stochastic engineering simulations using sparse grid collocation method and Kriging based approaches." Thesis, University of Southampton, 2017. https://eprints.soton.ac.uk/418266/.
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The estimation of probabilistic moments is central to robust design process. Typically one would like to estimate the mean and variance of some performance critical metric such as stress, life, etc., of a component in any engineering system, aiming a robustly optimized design that is less sensitive to the input variations/uncertainties. For complex aerospace engineering systems such as aero-engine, a single numerical simulation of any component can often take a substantial amount of time and few samples can be afforded at which the deterministic simulations can be carried out. Considering the variations in the parameters and performing a large number of simulations on such problems is unrealistic and necessitate the improvement of existing UQ approaches. In this study, we present the significance of probabilistic moment estimation approaches for uncertainty quantification and its importance in robust design optimization studies. The background for few popular approaches is provided, where emphasis is put on sparse grid collocation method, adaptive sparse grid collocation approach and Kriging based Bayesian approaches. A non-intrusive multi-point adaptive strategy using sparse grid based collocation design and Kriging based approaches is proposed to reduce the problems arising in high dimensional probabilistic moment estimation studies. The comparison of multi-point adaptive approach with other existing approaches for probabilistic moment estimation in terms of efficiency and accuracy is provided. Further on, the effectiveness of the proposed approach is demonstrated for few mathematical test functions and stochastic structural problems with varying dimensionality and strong interaction among the random variables.
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Posireddy, Sudhir Reddy. "Optimization of Meshfree Method with Distance Fields using Localized Solution Structure and Radial Basis Function Collocation Method." FIU Digital Commons, 2009. http://digitalcommons.fiu.edu/etd/279.
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To avoid the meshing problems with traditional engineering analysis methods many meshfree methods have been developed, and one of the most powerful methods proposed is meshfree method with distance fields. This research work aims at reducing the computational cost of meshfree method with distance fields. The main idea of the proposed approach is to apply a solution structure operator only to those basis functions whose supports are in the vicinity of the boundary and leave other basis functions unaffected. Unfortunately, straightforward implementation of this approach leads to elevated errors in partial derivatives of the solution. To overcome this drawback I propose to modify distance fields in such a way that they behave as a Euclidean distance in the region near the boundary and have smooth transition to a constant value within some distance away from the boundary. The uniqueness of the proposed method over other approaches is its adaptability to all kinds of boundary conditions. This narrow band technique improves the computational cost of meshfree method with distance fields with reasonable impact on accuracy. Another technique proposed in this work is to glue the global solution structures of meshfree method with distance fields with radial basis functions (RBF) and collocation technique. RBF with collocation method itself is proved to give good accurate results with less computational cost [12]. So using RBF-collocation technique with meshfree method with distance fields demands for more accuracy even with less computational cost. Later, these techniques are applied to solve heat transfer problems and the results are compared with global solution techniques to show that the proposed methods are close to global approach and computationally very effective.
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Multani, Sahib Singh. "Pseudospectral Collocation Method Based Energy Management Scheme for a Parallel P2 Hybrid Electric Vehicle." The Ohio State University, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=osu1587653689067271.
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Regele, Jonathan D. "Numerical modeling of acoustic timescale detonation initiation using the Adaptive Wavelet-Collocation Method." Connect to online resource, 2008. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3303847.
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Netuzhylov, Hennadiy. "A space-time meshfree collocation method for coupled problems on irregularly-shaped domains." Braunschweig : CSE, Computational Sciences in Engineering, Techn. Univ, 2009. http://d-nb.info/994492537/34.
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Darbeheshti, Neda. "Modification of the least-squares collocation method for non-stationary gravity field modelling." Thesis, Curtin University, 2009. http://hdl.handle.net/20.500.11937/2228.
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Geodesy deals with the accurate analysis of spatial and temporal variations in the geometry and physics of the Earth at local and global scales. In geodesy, least-squares collocation (LSC) is a bridge between the physical and statistical understanding of different functionals of the gravitational field of the Earth. This thesis specifically focuses on the [incorrect] implicit LSC assumptions of isotropy and homogeneity that create limitations on the application of LSC in non-stationary gravity field modeling. In particular, the work seeks to derive expressions for local and global analytical covariance functions that account for the anisotropy and heterogeneity of the Earth's gravity field.Standard LSC assumes 2D stationarity and 3D isotropy, and relies on a covariance function to account for spatial dependence in the observed data. However, the assumption that the spatial dependence is constant throughout the region of interest may sometimes be violated. Assuming a stationary covariance structure can result in over-smoothing, e.g., of the gravity field in mountains and under-smoothing in great plains. The kernel convolution method from spatial statistics is introduced for non-stationary covariance structures, and its advantage in dealing with non-stationarity in geodetic data is demonstrated.Tests of the new non-stationary solutions were performed over the Darling Fault, Western Australia, where the anomalous gravity field is anisotropic and non-stationary. Stationary and non-stationary covariance functions are compared in 2D LSC to the empirical example of gravity anomaly interpolation. The results with non-stationary covariance functions are better than standard LSC in terms of formal errors and cross-validation. Both non-stationarity of mean and covariance are considered in planar geoid determination by LSC to test how differently non-stationarity of mean and covariance affects the LSC result compared with GPS-levelling points in this area. Non-stationarity of the mean was not very considerable in this case, but non-stationary covariances were very effective when optimising the gravimetric quasigeoid to agree with the geometric quasigeoid.In addition, the importance of the choice of the parameters of the non-stationary covariance functions within a Bayesian framework and the improvement of the new method for different functionals on the globe are pointed out.
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Darbeheshti, Neda. "Modification of the least-squares collocation method for non-stationary gravity field modelling." Curtin University of Technology, Department of Spatial Sciences, 2009. http://espace.library.curtin.edu.au:80/R/?func=dbin-jump-full&object_id=120241.
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Geodesy deals with the accurate analysis of spatial and temporal variations in the geometry and physics of the Earth at local and global scales. In geodesy, least-squares collocation (LSC) is a bridge between the physical and statistical understanding of different functionals of the gravitational field of the Earth. This thesis specifically focuses on the [incorrect] implicit LSC assumptions of isotropy and homogeneity that create limitations on the application of LSC in non-stationary gravity field modeling. In particular, the work seeks to derive expressions for local and global analytical covariance functions that account for the anisotropy and heterogeneity of the Earth's gravity field.Standard LSC assumes 2D stationarity and 3D isotropy, and relies on a covariance function to account for spatial dependence in the observed data. However, the assumption that the spatial dependence is constant throughout the region of interest may sometimes be violated. Assuming a stationary covariance structure can result in over-smoothing, e.g., of the gravity field in mountains and under-smoothing in great plains. The kernel convolution method from spatial statistics is introduced for non-stationary covariance structures, and its advantage in dealing with non-stationarity in geodetic data is demonstrated.
Tests of the new non-stationary solutions were performed over the Darling Fault, Western Australia, where the anomalous gravity field is anisotropic and non-stationary. Stationary and non-stationary covariance functions are compared in 2D LSC to the empirical example of gravity anomaly interpolation. The results with non-stationary covariance functions are better than standard LSC in terms of formal errors and cross-validation. Both non-stationarity of mean and covariance are considered in planar geoid determination by LSC to test how differently non-stationarity of mean and covariance affects the LSC result compared with GPS-levelling points in this area. Non-stationarity of the mean was not very considerable in this case, but non-stationary covariances were very effective when optimising the gravimetric quasigeoid to agree with the geometric quasigeoid.
In addition, the importance of the choice of the parameters of the non-stationary covariance functions within a Bayesian framework and the improvement of the new method for different functionals on the globe are pointed out.
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Booth, Andrew S. "Collocation methods for a class of second order initial value problems with oscillatory solutions." Thesis, Durham University, 1993. http://etheses.dur.ac.uk/5664/.
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We derive and analyse two families of multistep collocation methods for periodic initial-value problems of the form y" = f(x, y); y((^x)o) = yo, y(^1)(xo) = zo involving ordinary differential equations of second order in which the first derivative does not appear explicitly. A survey of recent results and proposed numerical methods is given in chapter 2. Chapter 3 is devoted to the analysis of a family of implicit Chebyshev methods proposed by Panovsky k Richardson. We show that for each non-negative integer r, there are two methods of order 2r from this family which possess non-vanishing intervals of periodicity. The equivalence of these methods with one-step collocation methods is also established, and these methods are shown to be neither P-stable nor symplectic. In chapters 4 and 5, two families of multistep collocation methods are derived, and their order and stability properties are investigated. A detailed analysis of the two-step symmetric methods from each class is also given. The multistep Runge-Kutta-Nystrom methods of chapter 4 are found to be difficult to analyse, and the specific examples considered are found to perform poorly in the areas of both accuracy and stability. By contrast, the two-step symmetric hybrid methods of chapter 5 are shown to have excellent stability properties, in particular we show that all two-step 27V-point methods of this type possess non-vanishing intervals of periodicity, and we give conditions under which these methods are almost P-stable. P-stable and efficient methods from this family are obtained and demonstrated in numerical experiments. A simple, cheap and effective error estimator for these methods is also given.
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Junghanns, P., and U. Weber. "Local theory of a collocation method for Cauchy singular integral equations on an interval." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199801203.
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We consider a collocation method for Cauchy singular integral equations on the interval
based on weighted Chebyshev polynomials , where the coefficients of the operator are
piecewise continuous. Stability conditions are derived using Banach algebra methods,
and numerical results are given.
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Chen, Hongniao, and 陈红鸟. "Incremental displacement collocation method for the determination of fracture properties of quasi-brittle materials." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2013. http://hub.hku.hk/bib/B49799447.
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This thesis presents experimental and numerical investigations on the fracture properties of quasi-brittle materials, including mortar, concrete and graphite. Fracture toughness in terms of the critical stress intensity factor K_IC and fracture energy G_F of the materials were determined through three-point bending tests on centre-notched beams. Furthermore, full-field displacement of the beams subjected to bend was obtained using Electronic Speckle Pattern Interferometry (ESPI) technique.
In order to verify the accuracy of the displacement data measured using the ESPI technique and to obtain reliable deformation information, the displacement and strain errors induced by the rigid-body motions of the specimen were quantified. This study found that the displacement errors were negligible whereas the strain errors were notable and must be eliminated. The influence of different rigid-body motions was analyzed. It was found that the out-of-plane movement of the specimen was critical and affected considerably the accuracy of strain data. Thus the experimental setup was improved accordingly to eliminate the influence of critical rigid-body motions.
Quasi-brittle materials have a finite stress region near the crack tip, known as the fracture process zone (FPZ). The materials exhibit nonlinear fracture behaviour in the FPZ. The cohesive crack model (CCM) is widely used to characterize the nonlinear fracture behaviour of quasi-brittle materials. According to the CCM, all the nonlinear behaviours in the FPZ can be represented by a cohesive crack, and the crack propagation is controlled by the relationship between the cohesive stress and crack opening, namely, the tension softening curve (TSC). Thus an accurate estimation of the TSC is essential.
In order to determine the TSC of quasi-brittle materials, an incremental displacement collocation method (IDCM) was originally developed in this study. In the IDCM, the deformation data measured by the ESPI sensor was analyzed to obtain the crack opening displacement (COD) of the notched specimens. The experimental COD profiles together with the CCM were then integrated into a finite element model to simulate the nonlinear fracture response of the specimen. By minimizing the difference between the computed and measured displacements at selected collocation points, the cohesive stress corresponding to certain crack opening was determined. The entire TSC was constructed in a step-by-step manner following the loading steps.
The IDCM was first applied to estimate the TSC of mortar. By using the estimated TSC, the displacements of the specimen under certain loading levels were computed. By comparing the computed displacements with the experimental data, the reliability of the IDCM and the accuracy of the estimated TSC were verified.
The application of the IDCM was further extended to the determination of the TSCs of concrete and graphite. The parameters used to define the shape of the TSC of the materials were determined using regression analysis. The applicability of those parameters was verified by comparing the TSCs estimated in the present study with those derived by other researchers. Recommendations were put forward to choose appropriate tensile properties of quasi-brittle materials in the numerical simulations. Furthermore, by using the ESPI technique, fracture phenomena of quasi-brittle materials were observed and reported. Such records can greatly enhance the understanding of crack initiation, growth and arrest in quasi-brittle materials, and lead to improvements to the existing fracture models.published_or_final_version
Civil Engineering
Doctoral
Doctor of Philosophy
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Srinivas, Neeraj. "Collocation Method and Model Predictive Control for Accurate Landing of a Mars EDL vehicle." Thesis, Virginia Tech, 2021. http://hdl.handle.net/10919/102736.
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This thesis aims at investigating numerical methods through which the accuracy in landing of a Mars entry-descent-landing (EDL) vehicle can be improved. The methods investigated include the collocation method and model predictive control (MPC). The primary control variable utilized in this study is the bank angle of the spacecraft, which is the angle between the lift vector and the vertical direction. Modulating this vector affects the equations of system of equations and the seven state variables, namely altitude, velocity, latitude, longitude, flight path angle, heading angle and total time taken. An optimizer is implemented which utilizes the collocation method, through which the optimal bank angle is found at every discretized state along the trajectory which are equally separated through a definite timestep, which is a function of the end time state. A 3-sigma wind disturbance model is introduced to the system, as a function of the altitude, which introduces uncertainties to the system, resulting in a final state deviating from the targeted location. The trajectory is split into two parts, for better control of the vehicle during the end stages of flight. The MPC aims at reducing the end state deviation, through the implementation of a predictor-corrector algorithm that propagates the trajectory for a certain number of timesteps, followed by running the optimizer from the current disturbed state to the desired target location. At the end of this analysis, a new set of optimal bank angle are found, which account for the wind disturbances and navigates the EDL vehicle to the desired location.M.S.
Landing on Mars has always been a process of following a set of predetermined instructions by the spacecraft, in order to reach a calculated landing target. This work aims to take the first steps towards autonomy in maneuvering the spacecraft, and finding a method by which the vehicle navigates itself towards the target. This work determines the optimal control scheme a Mars reentry vehicle must have through the atmosphere to reach the target location, and employs method through which the uncertainty in the final landing location is mitigated. A model predictive controller is employed which corrects the disturbed trajectory of the vehicle at certain timesteps, through which the previously calculated optimal control is changed so as to account for the disturbances. The control is achieved by means of changing the bank angle of the spacecraft, which in turn affects the lift and drag experienced by the vehicle. Through this work, a method has been demonstrated which reduces the uncertainty in final landing location, even with wind disturbances present.
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Magoon, Jason. "Application of the B-spline collocation method to a geometrically non-linear beam problem /." Online version of thesis, 2010. http://hdl.handle.net/1850/11587.
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Amireghbali, Aydin. "Elastic Analysis Of A Circumferential Crack In An Isotropic Curved Beam Using Modified Mapping-collocation Method." Master's thesis, METU, 2013. http://etd.lib.metu.edu.tr/upload/12615675/index.pdf.
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The modified mapping-collocation (MMC) method is applied to analyze a circumferential
crack in an isotropic curved beam. Based on the method a MATLAB code was developed to
obtain the stress field. Incorporating the stress correlation technique, the opening and sliding
fracture mode stress intensity factors (SIF)s of the crack for the beam under pure bending
moment load case are calculated.
The MMC method aims to solve two-dimensional problems of linear elastic fracture mechanics
(LEFM) by combining the power of analytic tools of complex analysis (Muskhelishvili
formulation, conformal mapping, and extension arguments) with simplicity of applying the
boundary collocation method as a numerical solution approach.
Qualitatively, a good agreement between the computed stress contours and the fringe shapes
obtained from the photoelastic experiment on a plexiglass specimen is observed. Quantitatively,
the results are compared with that of ANSYS finite element analysis software. The
effect of crack size, crack position and beam thickness variation on SIF values and mode
mixity is investigated.
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Netuzhylov, Hennadiy [Verfasser]. "A space-time meshfree collocation method for coupled problems on irregularly-shaped domains / von Hennadiy Netuzhylov." Braunschweig : CSE, Computational Sciences in Engineering, Techn. Univ, 2009. http://d-nb.info/994492537/34.
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Zendegan, Saeid. "3D trajectory optimization of an acrobatic air race with direct collocation method and quaternion equations of motion." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2019. http://amslaurea.unibo.it/18025/.
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The main purpose of this thesis is to develop a software to optimize the 3D air race trajectory. Air race competitions such as Red Bull are becoming very popular around the world, but behind this eye catching competitions there are always engineers supporting every team to help their pilots to make the fastest lap time in order to win the competition.
Nowadays, by the help of advanced computing systems and proficient optimization algorithm, engineers are able to model the trajectory of these air races and optimize them offline. These optimized trajectories are handed to the pilots before the competition, in order to help them train themselves in a simulator and then with a real aircraft.
In this thesis the equations of motion (EOM) of point-mass model is used for optimization. The first approach was with the EOM based on Euler angles. As we know from flight dynamics, with the Euler angles we will have a singularity in our equations when the aircraft is experiencing the ±π/2 pitch angle. In order to avoid the singularity in the Euler case, we switch to EOM based on Quaternions. Although the quaternions are used for the optimization of the trajectory, we will also represent the results for Euler based EOM.
The software used to develop the code for this optimization is MATLAB 2016b. Algorithm is proposed and implemented by the author. Build-in optimization function (FMINCON) is used to optimize the trajectory based on the proposed algorithm.
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Li, Haojun [Verfasser], and A. [Akademischer Betreuer] Rieder. "Numerical simulation of a micro-ring resonator with adaptive wavelet collocation method / Haojun Li. Betreuer: A. Rieder." Karlsruhe : KIT-Bibliothek, 2011. http://d-nb.info/101481782X/34.
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Cerezo, Graciela M. "Solution Representation and Indentification for Singular neutral Functional Differential Equations." Diss., Virginia Tech, 1996. http://hdl.handle.net/10919/30365.
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The solutions for a class of Neutral Functional Di erential Equations (NFDE) with weakly singular kernels are studied. Using singular expansion techniques, a representation of the solution of the NFDE is obtained by studing an associated Volterra Integral Equation. We study the Collocation Method as a projection method for the approximation of solutions for Volterra Integral Equations. Particulary, the possibility of achieving higher order ap- proximations is discussed. Special attention is given to the choice of the projection space and its relation to the smoothness of the approximated solution. Finally, we study the identification problem for a parameter appearing in the weakly singular operator of the NFDE.Ph. D.
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Slemp, Wesley Campbell Hop. "Integrated Sinc Method for Composite and Hybrid Structures." Diss., Virginia Tech, 2010. http://hdl.handle.net/10919/77111.
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Composite materials and hybrid materials such as fiber-metal laminates, and functionally graded materials are increasingly common in application in aerospace structures. However, adhesive bonding of dissimilar materials makes these materials susceptible to delamination. The use of integrated Sinc methods for predicting interlaminar failure in laminated composites and hybrid material systems was examined. Because the Sinc methods first approximate the highest-order derivative in the governing equation, the in-plane derivatives of in-plane strain needed to obtain interlaminar stresses by integration of the equilibrium equations of 3D elasticity are known without post-processing. Interlaminar stresses obtained with the Sinc method based on Interpolation of Highest derivative were compared for the first-order and third-order shear deformable theories, the refined zigzag beam theory and the higher-order shear and normal deformable beam theory. The results indicate that the interlaminar stresses by the zigzag theory compare well with those obtained by a 3D finite element analysis, while the traditional equivalent single layer theories perform well for some laminates.
The philosophy of the Sinc method based on Interpolation of Highest Derivative was extended to create a novel weak form based approach called the Integrated Local Petrov-Galerkin Sinc Method. The Integrated Local Petrov-Galerkin Sinc Method is easily utilized for boundary-value problem on non-rectangular domains as demonstrated for analysis of elastic and elastic-plastic plane-stress panels with elliptical notches. The numerical results showed excellent accuracy compared to similar results obtained with the finite element method.
The Integrated Local Petrov-Galerkin Sinc Method was used to analyze interlaminar debonding of composite and fiber-metal laminated beams. A double-cantilever beam and a fixed-ratio mixed mode beam were analyzed using the Integrated Local Petrov-Galerkin Sinc Method and the results were shown to correlate well with those by the finite element method. An adaptive Sinc point distribution technique was implemented for the delamination analysis which significantly improved the methods accuracy for the present problem. Delamination of a GLARE, plane-strain specimen was also analyzed using the Integrated Local Petrov-Galerkin Sinc Method. The results correlate well with 2D, plane-strain analysis by the finite element method, including interlaminar stresses obtained by through-the-thickness integration of the equilibrium equations of 3D elasticity.Ph. D.
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Kilic, Bahattin. "Peridynamic Theory for Progressive Failure Prediction in Homogeneous and Heterogeneous Materials." Diss., The University of Arizona, 2008. http://hdl.handle.net/10150/193658.
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The classical continuum theory is not capable of predicting failure without an external crack growth criteria and treats the interface having zero thickness. Alternatively, a nonlocal continuum theory referred to as peridynamic theory eliminates these shortcomings by utilizing formulation that uses displacements, rather than derivatives of displacements, and including material failure in its constitutive relations through the response functions. This study presents a new response function as part of the peridynamic theory to include thermal loading. Furthermore, an efficient numerical algorithm is presented for solution of peridynamic equations. Solution method relies on the discretization of peridynamic equations at collocation points resulting in a set of ordinary differential equations with respect to time. These differential equations are then integrated using explicit methods. In order to improve numerical efficiency of the computations, spatial partitioning is introduced through uniform grids as arrays of linked lists. Furthermore, the domain of interest is divided into subunits each of which is assigned to a specific processor to utilize parallel processing using OpenMP. In order to obtain the static solutions, the adaptive dynamic relaxation method is developed for the solution of peridynamic equations. Furthermore, an approach to couple peridynamic theory and finite element analysis is introduced to take advantage of their salient features. The regions in which failure is expected are modeled using peridynamics while the remaining regions are modeled utilizing finite element method. Finally, the present solution method is utilized for damage prediction of many problems subjected to mechanical, thermal and buckling loads.
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Mesogitis, Tassos. "Stochastic simulation of the cure of advanced composites." Thesis, Cranfield University, 2015. http://dspace.lib.cranfield.ac.uk/handle/1826/9216.
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This study focuses on the development of a stochastic simulation methodology to study the effects of cure kinetics uncertainty, in plane fibre misalignment and boundary conditions uncertainty on the cure process of composite materials. Differential Scanning Calorimetry was used to characterise cure kinetics variability of a commercial epoxy resin used in aerospace applications. It was found that cure kinetics uncertainty is associated with variations in the initial degree of cure, activation energy and reaction order. Image analysis was employed to characterise in plane fibre misalignment in a carbon fibre ±45º non-crimp fabric. The experimental results showed that variability in tow orientation was significant with a standard deviation of about 1.2º. A set of experiments using an infusion set-up was carried out to quantify boundary conditions uncertainty related to tool temperature, ambient temperature and surface heat transfer coefficient using thermocouples (tool/ambient temperature) and heat flux sensors (surface heat transfer coefficient). It was concluded that boundary conditions uncertainty can show considerable short term and long term variability. Conventional Monte Carlo and Probabilistic Collocation Method were integrated with a thermo-mechanical cure simulation model in order to investigate the effect of cure kinetics, fibre misalignment and boundary conditions variability on process outcome. The cure model was developed and implemented using a finite element model incorporating appropriate material sub-models of cure kinetics, specific heat capacity, thermal conductivity, moduli, thermal expansion and cure shrinkage. The effect of cure kinetics uncertainty on the temperature overshoot of a thick carbon fibre epoxy flat panel was investigated using the two stochastic simulation schemes. The stochastic simulation results showed that variability in cure kinetics can introduce a significant scatter in temperature overshoot, presenting a coefficient of variation of about 30%. Furthermore, it was shown that the collocation method can offer an efficient solution with significantly lower computational cost compared to Monte Carlo at comparable accuracy. Stochastic simulation of the cure of an angle shaped carbon fibre-epoxy component within the Monte Carlo scheme showed that fibre misalignment can cause considerable variability in the process outcome. The coefficient of variation of maximum residual stress can reach up to approximately 2% (standard deviation of 1 MPa) whilst qualitative and quantitative variations in final distortion of the cured part occur with the standard deviation in twist and corner angle reaching values of 0.4 º and 0.05º respectively. Simulation of the cure of a thin carbon fibre-epoxy panel within the Monte Carlo scheme indicated that surface heat transfer and tool temperature variability dominate variability in cure time, resulting in a coefficient of variation of about 22%. In addition to Monte Carlo, the effect of surface heat transfer coefficient and tool temperature variations on cure time was addressed using the collocation method. It was found that probabilistic collocation is capable of capturing variability propagation with good accuracy while offering tremendous benefits in terms of computational costs.
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Jannet, Basile. "Influence de la non-stationnarité du milieu de propagation sur le processus de Retournement Temporel (RT)." Thesis, Clermont-Ferrand 2, 2014. http://www.theses.fr/2014CLF22436/document.
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Cette thèse a pour objectif la quantification de l’impact d’incertitudes affectant le processus de Retournement Temporel (RT). Ces aléas, de natures diverses, peuvent avoir une forte influence s’ils se produisent entre les deux étapes du RT. Dans cette optique la méthode de Collocation Stochastique (CS) est utilisée. Les très bons résultats en termes d’efficacité et de précision observés lors de précédentes études en Compatibilité ÉlectroMagnétique (CEM) se confirment ici, pour des problématiques de RT. Cependant, lorsque la dimension du problème à traiter augmente (nombre de variables aléatoires important), la méthode de CS atteint ses limites en termes d’efficacité. Une étude a donc été menée sur les méthodes d’Analyse de Sensibilité (AS) qui permettent de déterminer les parts d’influence respectives des entrées d’un modèle. Parmi les différentes techniques quantitatives et qualitatives, la méthode de Morris et un calcul des indices de Sobol totaux ont été retenus. Ces derniers apportent des résultats qualitatifs à moindre frais, car seule une séparation des variables prépondérantes est recherchée. C’est pourquoi une méthodologie combinant des techniques d’AS avec la méthode de CS a été développée. En réduisant le modèle aux seules variables prédominantes grâce à une première étude faisant intervenir les méthodes d’AS, la CS peut ensuite retrouver toute son efficacité avec une grande précision. Ce processus global a été validé face à la méthode de Monte Carlo sur différentes problématiques mettant en jeu le RT soumis à des aléas de natures variéesThe aim of this thesis is to measure and quantify the impacts of uncertainties in the Time Reversal (TR) process. These random variations, coming from diverse sources, can have a huge influence if they happen between the TR steps. On this perspective, the Stochastique Collocation (SC) method is used. Very good results in terms of effectiveness and accuracy had been noticed in previous studies in ElectroMagnetic Compatibility (EMC). The conclusions are still excellent here on TR problems. Although, when the problem dimension rises (high number of Random Variables (RV)), the SC method reaches its limits and the efficiency decreases. Therefore a study on Sensitivity Analysis (SA) techniques has been carried out. Indeed, these methods emphasize the respective influences of the random variables of a model. Among the various quantitative or qualitative SA techniques the Morris method and the Sobol total sensivity indices have been adopted. Since only a split of the inputs (point out of the predominant RV) is expected, they bring results at a lesser cost. That is why a novel method is built, combining SA techniques and the SC method. In a first step, the model is reduced with SA techniques. Then, the shortened model in which only the prevailing inputs remain, allows the SC method to show once again its efficiency with a high accuracy. This global process has been validated facing Monte Carlo results on several analytical and numerical TR cases subjet to random variations
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Zhang, Jingwei. "Numerical Methods for the Chemical Master Equation." Diss., Virginia Tech, 2009. http://hdl.handle.net/10919/30018.
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The chemical master equation, formulated on the Markov assumption of underlying chemical kinetics, offers an accurate stochastic description of general chemical reaction systems on the mesoscopic scale. The chemical master equation is especially useful when formulating mathematical models of gene regulatory networks and protein-protein interaction networks, where the numbers of molecules of most species are around tens or hundreds. However, solving the master equation directly suffers from the so called "curse of dimensionality" issue. This thesis first tries to study the numerical properties of the master equation using existing numerical methods and parallel machines. Next, approximation algorithms, namely the adaptive aggregation method and the radial basis function collocation method, are proposed as new paths to resolve the "curse of dimensionality". Several numerical results are presented to illustrate the promises and potential problems of these new algorithms. Comparisons with other numerical methods like Monte Carlo methods are also included. Development and analysis of the linear Shepard algorithm and its variants, all of which could be used for high dimensional scattered data interpolation problems, are also included here, as a candidate to help solve the master equation by building surrogate models in high dimensions.Ph. D.
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Preece, Robin. "A probabilistic approach to improving the stability of meshed power networks with embedded HVDC lines." Thesis, University of Manchester, 2013. https://www.research.manchester.ac.uk/portal/en/theses/a-probabilistic-approach-to-improving-the-stability-of-meshed-power-networks-with-embedded-hvdc-lines(b7e4843f-52b4-4ccf-88ad-48f9195b7270).html.
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This thesis investigates the effects of High Voltage Direct Current (HVDC) lines andmulti-terminal grids on power system small-disturbance stability in the presence ofoperational uncertainties. The main outcome of this research is the comprehensiveprobabilistic assessment of the stability improvements that can be achieved through theuse of supplementary damping control applied to HVDC systems.Power systems are increasingly operated closer to stability boundaries in order toimprove their efficiency and economic value whilst a growing number of conventionalcontrolled power plants are being replaced by stochastic renewable generation sources.The resulting uncertainty in conditions can increase the risk of operational stabilityconcerns and should be thoroughly evaluated. There is also a growing necessity toexplore the potential improvements and challenges created by the introduction of newequipment, such as HVDC systems. In recent years, HVDC systems have become moreeconomically competitive and increasingly flexible, resulting in a proliferation ofprojects. Although primarily installed for power transmission purposes, their flexibilityand controllability can provide further benefits, such as the damping of persistentoscillations in the interconnected networks.This work contributes to a number of areas of power systems research, specificallysurrounding the effects of HVDC systems on the small-disturbance stability oftransmission networks. The application and comprehensive assessment of a Wide AreaMeasurement System (WAMS) based damping controller with various HVDC systemsis completed. The studies performed on a variety of HVDC technology types andconfigurations – as well as differing AC test networks – demonstrate the potential forHVDC-based Power Oscillation Damping (POD). These studies include examinationsof previously unexplored topics such as the effects of available modulation capacity andthe use of voltage source converter multi-terminal HVDC grids for POD. Followingthese investigations, a methodology to probabilistically test the robustness of HVDC based damping controllers is developed. This methodology makes use of classificationtechniques to identify possible mitigation options for power system operators whenperformance is sub-optimal. To reduce the high computational burden associated withthis methodology, the Probabilistic Collocation Method (PCM) is developed in order toefficiently identify the statistical distributions of critical system modes in the presenceof uncertainties. Methods of uncertain parameter reduction based on eigenvaluesensitivity are developed and demonstrated to ensure accurate results when the PCM isused with large test systems. Finally, the concepts and techniques introduced within thethesis are combined to probabilistically design a WAMS-based POD controller morerobust to operational uncertainties. The use of the PCM during the probabilistic designresults in rapid and robust synthesis of HVDC-based POD controllers.
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Hung, Chan-Wei, and 洪承緯. "Reproducing Kernel Collocation Method for Nonlinear Iterative Analysis." Thesis, 2015. http://ndltd.ncl.edu.tw/handle/jdj244.
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碩士國立交通大學
土木工程系所
103
In the nonlinear related research of the strong form collocation methods, this is the first work using the reproducing kernel collocation method (RKCM) to solve the semilinear elliptic partial differential equations. As for the iteration schemes, we adopt both the quasi-Newton iteration method and Newton iteration method to solve three examples with the following types of solutions: a trigonometric function, an exponential function, and a trigonometric function combined with a polynomial. Based on our numerical results, the two iteration methods show similar convergence behavior. The Newton iteration method converges faster and is more stable than the quasi-Newton iteration method. But the quasi-Newton iteration method requires less CPU time in each iterative step. Therefore, as the number of collocation points increases, the quasi-Newton iteration method will save more time.
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Kuzmina, M., and S. V. Petrasova. "Method for Automatic Collocation Extraction from Ukrainian Corpora." Thesis, 2018. http://repository.kpi.kharkov.ua/handle/KhPI-Press/46380.
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The article deals with the methods for automatic collocation extraction from Ukrainian corpora. The task of collocation extraction is considered in terms of a corpus-oriented approach [1], based on statistical measures. The term «collocation» is defined as a non-random combination of two words that go together regularly.
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Huang, Yi-Jui, and 黃奕叡. "Buckling Analysis of Nanotubeby Using Spline Collocation Method." Thesis, 2011. http://ndltd.ncl.edu.tw/handle/12376161915704379046.
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碩士國立臺灣大學
土木工程學研究所
99
In this article, I use spline function inferred from Forward Difference Method as a starting point, and it is coordinated with collocation to develop a numerical analyses method, called SCM(Spline Collocation Method).Then, using any order spline function solved early and make a complete B spline value table by calculating repeatedly and it will also be advantageous to our use. In the same time, using MSCM(Modified Spline Collocation Method) inferred from SCM to solve some eigenvalue problems about buckling of nanotube and analysis its every model buckling load and convergence. make a study of the accuracy and astringency by comparing the numerical analyses solutions with exact solutions. And consider a nanotube under axis load and transvers load to solve displacement of middle point, rotation of end point, shear of middle point and draft a deformation diagram and internal force diagram , substitute different boundary condition to solve the numerical analyses solutions. The purpose of this article is used for proving that the advantages of SCM is excellent and it is a numerical analyses method which has accuracy ,convenience and applications. Therefore, SCM is worthy to research in structural analyses in depth.
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Lin, Hung-Sheng, and 林宏昇. "Radial Basis Function Collocation Method for Nonlinear Schrödinger Equations." Thesis, 2012. http://ndltd.ncl.edu.tw/handle/24088146396373938777.
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碩士國立中興大學
應用數學系所
100
In this thesis, we study a radial basis function collocation method (RBFCM) for solving time-independent nonlinear Schrödinger equations (NLSE). A RBFCM with continuation algorithm is presented to trace the solution curve of Gross-Pitaevskii equation (GPE) with an angular momentum rotation term that describes a two dimensional rotating Bose-Einstein condensate (BEC) below a critical temperature. The numerical results show that the RBFCM is highly efficient for solving eigenvalue problems due to its exponential error convergence rate.
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Hsu, Chuan-Yan, and 徐傳硯. "Modified collocation Trefftz method for solving Laplace equations with nonlinear boundary conditions Modified collocation Trefftz method for solving Laplace equations with nonlinear boundary conditions." Thesis, 2012. http://ndltd.ncl.edu.tw/handle/63734423940940754400.
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碩士國立臺灣海洋大學
河海工程學系
100
In this thesis, the combination of modified collocation Trefftz method (MCTM) and exponentially convergent scalar homotopy algorithm (ECSHA) is proposed to analyze Laplace problems with non-linear boundary conditions. These types of physical problems include Signorini problem, heat conduction problem with material nonlinearity and cathode protection problem. MCTM is one kind of boundary-type meshless methods and the numerical solution can be expressed by linear combination of the T-complete functions of the Laplace operator. MCTM is free from mesh and integral-free for spatial discretization. Hence, MCTM can efficiently analyze problem by using few computer resources. On the other hand, MCTM is easy operation and highly accurate by few collocation points. The spatial discretization of MCTM will result in a system of non-linear algebraic equations (NAEs). We used ECSHA to solve NAEs formed by MCTM. ECSHA is derived by concepts of fictitious time and scalar homotopy function and can solve over-determined system or under-determined system. Besides, ECSHA is insensitive to initial guess and can avoid calculating the inverse of Jacobian matrix. Finally, ECSHA is exponentially convergent. In this thesis, we will verify the efficiency and accuracy of the combination of MCTM and ECSHA by some numerical examples. In addition, numerical experiments on testing different parameters are used to verify the stability of the proposed scheme.
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Li, Fang-wen, and 李芳雯. "Radial Basis Collocation Method for Singularly Perturbed Partial Differential Equations." Thesis, 2004. http://ndltd.ncl.edu.tw/handle/06442167851802632152.
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碩士國立中山大學
應用數學系研究所
92
In this thesis, we integrate the particular solutions of singularly perturbed partial differential equations into radial basis collocation method to solve two kinds of boundary layer problem.
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chang, Chih kai, and 張智凱. "The Perturbation and Stability Study of Reproducing Kernel Collocation Method." Thesis, 2010. http://ndltd.ncl.edu.tw/handle/69152660281668592052.
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碩士東海大學
數學系
98
Solving partial differential equations with strong form collocation and nonlocal approximation functions such as orthogonal polynomials and radial basis functions exhibits exponential convergence rate; however, it yields a full matrix and suffers from ill conditioning. In this work, the local approximation functions, reproducing kernel functions, are used as basis functions. This approach offers algebra convergence rate, but the method is stable like the finite element. We provide the perturbation and stability analysis of this approach, and the estimation of condition number of the discrete equation is derived. Condition number is used to measure the solution errors resulting from rounding errors, and it plays a critical role in numerical stability. In addition, the new formulas of condition number, called effective condition numbers, are given. Both matrix and right hand side vector of a linear system are taken into consideration in the estimation of condition number, they offer a better measure of conditioning than traditional condition numbers. Numerical results are also presented to validate the mathematical analysis.
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Lai, chiu kai, and 賴秋凱. "The Convergence and Complexity Study of Reproducing Kernel Collocation Method." Thesis, 2009. http://ndltd.ncl.edu.tw/handle/01408042518529474273.
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碩士東海大學
數學系
97
Abstract The development of meshfree methods can be traced back from two branches, one based on weak Galerkin form and the other based on strong collocation form. In weak form, the local polynomial functions are used as the admissible functions yields algebraic convergence rate. The method is stable like finite element method. However, the need of quadrature rules in the domain integration has caused high computational cost to this class of methods. The strong collocation method based on radial basis function exhibits exponential convergence rate. The method is overshadowed by fully dense and ill-conditioned discrete equation due to non-local functions are used. This confines the application of this method to a small scale problems. In this work, we discuss a reproducing kernel collocation method (RKCM), where the reproducing kernel shape functions with compact support are used as approximation functions. We concentrate on the convergence analysis and the computational complexity for RKCM. The method avoids the domain integration, and leads to well-conditioned discrete equations. An important result extracted from the analysis is that the degree of basis in the reproducing kernel approximation has to be greater than one for convergence. Some numerical experiments provided to validate the results of error analysis.
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Chen, Sheng–Yuan, and 陳昇元. "Analysis of Plates Using SCM with a New Collocation Method." Thesis, 2008. http://ndltd.ncl.edu.tw/handle/28812978619219574434.
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陳駿逸. "On Spectral Collocation Method Applied in Solving Partial Differential Equation." Thesis, 2004. http://ndltd.ncl.edu.tw/handle/88313077179952084918.
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Chen, Yi-Wun, and 陳怡雯. "Study on Solving Seepage Problems Using the Trefftz Collocation Method." Thesis, 2016. http://ndltd.ncl.edu.tw/handle/69890192920989085855.
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碩士國立臺灣海洋大學
河海工程學系
104
This study presents the the numerical solutions for seepage flow in layered soil using the Trefftz collocation method. The Trefftz collocation method is a meshless numerical method with very high accuracy for solving boundary value problems where approximate solutions are expressed as a linear combination of functions automatically satisfy governing equations. To deal with complicated problems for multiply connected domain, the generalized multiple source point boundary collocation Trefftz method which allows many source points in the Trefftz formulation was adopted. In addition, the domain decomposition method which decomposes the problem domain into several simply connected subdomains and to use the Trefftz method in each one was also adopted to solve the seepage flow in layered soil. The validity of of the proposed method is established by conducting several numerical examples in a simply connected domain and a doubly connected domain. Application examples were also carried out using the proposed numerical model. Furthermore, the seepage problems of a vertically layered earth dam with the phreatic surface were also studied. The results revealed that the proposed method can not only obtain numerical solutions for seepage flow in layered soil but also can achieve very high accuracy result in numerical solutions to that of the conventional numerical method.
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Chang, Yung-Chieh, and 張永潔. "Chebyshev Collocation Method for Shallow Water Models with Domain Decomposition." Thesis, 2009. http://ndltd.ncl.edu.tw/handle/68917764508089849119.
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碩士國立交通大學
應用數學系所
97
The spectral methods seek the numerical solutions by a set of known polynomials. The main advantage of using spectral methods for solving atmospheric problems is the high efficiency and conservations of important quadratic quantities such as kinetic energy and enstrophy. Namely, we can get very high accuracy through the exponential convergence. The conservation of the quadratic quantities are important to model the turbulence under strong rotation and stratification. In this paper, we introduce the domain decomposition method to speed up the Chebyshev collocation method. The domain decomposition is to divide the domain into many sub-domains to run the computation in parallel and to exchange the information through the sub-domain boundaries during the time integration. We implement the domain decomposition Chebyshev collocation method with overlapping the sub-domains in one grid spacing interval for 1-D tests such as advection, diffusion and inviscid Burgers equations. We show the exponential convergence property and error characteristics in these tests. In a more realistic atmospheric modeling, we study the spectral method with 2-D shallow water equations. The domain decomposition results compared favorably with that of the single domain calculations. Thus, Chebyshev domain decomposition method may be an efficient alternative method for the atmospheric/oceanic limited area modeling.
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Chen, Chien-Chang, and 陳建彰. "A Chebyshev collocation method for the solution of nonlinear integral equations." Thesis, 2007. http://ndltd.ncl.edu.tw/handle/7cryvk.
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碩士大同大學
應用數學學系(所)
95
A Chebyshev collocation method has been proposed in order to solve nonlinear integral equations. This method is to obtain solution as truncated Chebyshev series and transforms the integral equation to a system of nonlinear equations with unknown Chebyshev coefficients. Some examples are given to illustrate the method. The numerical results using Maple and Matlab are presented and discussed.
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Wu, Chao-Min, and 吳兆民. "The application of Spline Collocation Method in free vibration of beam." Thesis, 2006. http://ndltd.ncl.edu.tw/handle/93723247174609482062.
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Xu, Han-Fu, and 許涵富. "The Analysis of The Immobilization of Enzyme by Orthogonal Collocation Method." Thesis, 1996. http://ndltd.ncl.edu.tw/handle/64442992241225125579.
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碩士國立成功大學
化學工程學系
84
A restricted-diffusion model of immobilized enzyme in porous support has beenestablished in this work. The model is solved by orthogonal method and Runge-Kutta method.The distribution of immobilized enzyme in the pores,and the para-meter of the immobilization process have been investigated by using this model. Our conception is to use glutaraldehyde as the crosslinking agent between supports and urease,and make their binding stable.The exxperiments were operated at 25℃,and used 0.1M phosphate solution buffered at pH6.5.The ureaseconcentration was measured by UV-VIS spectrophotometer. A510 and A400 resins were used as the supports of urease. Because of the differences in resin pore diameter,the amount of adsorbed urease and the effective urea hydrolysis rate were different. Our experiment shows that the amount of urease adsorbed were29.3mg/ml and 12.6mg/ml for A510 and A400 respectively.The amount of urease adsorbed per mililiter of A510 resin is about 2.3 times that of A400 resin.The best activities of these immobilized urease occur at pH6.5 and the rate ofurea hydrolysis is about 31.4 and 20.3 μmol NH3/min mg urease for A510 resinand A400 resin respectively. The parameters in restricted-diffusion model were obtained by curve fitting the experimental data. The system parameter for using A510 resin as support are N=0.018,ψ=0.0059,α=0.0022,φ=60,and kim=1.3e-1 1/sec g ,while system parameter that use A400 resin as support are N=0.036,ψ=0.0065,α=0.292,φ=60and kim=1.8e-3 1/sec g . In accordance with the effect of the whole immobilized process and the hydrolysis rate,we conclude that A510 resin is a better support for enzyme immobilization than A400 resin.
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Hu, Shin Yum, and 胡馨云. "Spectral Collocation Method on solving the linear stability of axisymmetric jet." Thesis, 1995. http://ndltd.ncl.edu.tw/handle/07541619141893067838.
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Huang, Hsu-Hui, and 黃旭輝. "Study of Spline Collocation Method and its Application on Engineering Problems." Thesis, 2009. http://ndltd.ncl.edu.tw/handle/15672539424780408945.
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博士國立臺灣大學
土木工程學研究所
97
In this thesis, we study the spline collocation method (SCM), radial spline collocation method (RSCM) and spline collocation element method (SCEM) for solving engineering problems: beam, beam-column, frame, and plate problem. The popularity of the collocation method is in part due to their conceptual simplicity, wide applicability, and ease of implementation. In comparison to finite element difference methods, the CM provides approximations to the solution and its spatial derivatives at mesh point of the domain of problems. The obvious advantage of collocation method over Galerkin methods is that the calculation of the coefficients in the system of algebraic equations determining the approximate solution is very fast since no integrals need to be evaluated or approximated. Moreover, numerical experiments illustrate that the collocation method provide high order accuracy and super-convergence feature for a wide range of physical and engineering problems.
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Lee, Cheng-Feng, and 李政峰. "High precision computations of multiquadric collocation method for partial differential equations." Thesis, 2006. http://ndltd.ncl.edu.tw/handle/31002340468879174999.
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碩士國立中山大學
應用數學系研究所
94
Multiquadric collocation method is highly efficient for solving partial differential equations due to its exponential error convergence rate. More amazingly, there are two ways to reduce the error: the traditional way of refining the grid, and the unexpected way of simply increasing the value of shape constant $c$ contained in the multiquadric basis function, $sqrt{r^2 + c^2}$. The latter is accomplished without increasing computational cost. It has been speculated that in a numerical solution without roundoff error, infinite accuracy can be achieved by letting $c ightarrow infty$. The ability to obtain infinitely accurate solution is limited only by the roundoff error induced instability of matrix solution with large condition number. Using the arbitrary precision computation capability of {it Mathematica}, this paper tests the above conjecture. A sharper error estimate than previously obtained is presented in this paper. A formula for a finite, optimal $c$ value that minimizes the solution error for a given grid size is obtained. Using residual errors, constants in error estimate and optimal $c$ formula can be obtained. These results are supported by numerical examples.
50
Lin, Chih-Hsun, and 林志勳. "Analysis of Laminated Anisotropic plates and Shells by Chebyshev Collocation Method." Thesis, 2003. http://ndltd.ncl.edu.tw/handle/63943709453361872067.
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博士國立中山大學
機械與機電工程學系研究所
91
The purpose of this work is to solve governing differential equations of laminated anisotropic plates and shells by using the Chebyshev collocation method. This method yields these results those can not be accomplished easily by both Navier’s and Levy’s methods in the case of any kind of stacking sequence in composite laminates with the variety of boundary conditions subjected to any type of loading. The Chebyshev polynomials have the characteristics of orthogonality and fast convergence. They and Gauss-Lobatto collocation points can be utilized to approximate the solution of these problems in this paper. Meanwhile, these results obtained by the method are presented as some mathematical functions that they are more applicable than some sets of data obtained by other methods. On the other hand, by simply mathematical transformation, it is easy to modify the range of Chebyshev polynomials from the interval [-1,1] into any intervals. In general, the research on laminated anisotropic plates is almost focused on the case of rectangular plate. It is difficult to handle the laminated anisotropic plate problems with the non-rectangular borders by traditional methods. However, through the merits of Chebyshev polynomials, such problems can be overcome as stated in this paper. Finally, some cases in the chapter of examples are illustrated to highlight the displacements, stress resultants and moment resultants of our proposed work. The preciseness is also found in comparison with numerical results by using finite element method incorporated with the software of NASTRAN.