Dissertations / Theses on the topic 'High–Order Spectral Methods'
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1
Kannan, Ravishekar. "High order spectral volume and spectral difference methods on unstructured grids." [Ames, Iowa : Iowa State University], 2008.
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Vanharen, Julien. "High-order numerical methods for unsteady flows around complex geometries." Phd thesis, Toulouse, INPT, 2017. http://oatao.univ-toulouse.fr/17967/1/vanharen.pdf.
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This work deals with high-order numerical methods for unsteady flows around complex geometries. In order to cope with the low-order industrial Finite Volume Method, the proposed technique consists in computing on structured and unstructured zones with their associated schemes: this is called a hybrid approach. Structured and unstructured meshes are then coupled by a nonconforming grid interface. The latter is analyzed in details with special focus on unsteady flows. It is shown that a dedicated treatment at the interface avoids the reflection of spurious waves. Moreover, this hybrid approach is validated on several academic test cases for both convective and diffusive fluxes. The extension of this hybrid approach to high-order schemes is limited by the efficiency of unstructured high-order schemes in terms of computational time. This is why a new approach is explored: The Spectral Difference Method. A new framework is especially developed to perform the spectral analysis of Spectral Discontinuous Methods. The Spectral Difference Method seems to be a viable alternative in terms of computational time and number of points per wavelength needed for a given application to capture the flow physics.
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Hao, Zhaopeng. "High-order numerical methods for integral fractional Laplacian: algorithm and analysis." Digital WPI, 2020. https://digitalcommons.wpi.edu/etd-dissertations/612.
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The fractional Laplacian is a promising mathematical tool due to its ability to capture the anomalous diffusion and model the complex physical phenomenon with long-range interaction, such as fractional quantum mechanics, image processing, jump process, etc. One of the important applications of fractional Laplacian is a turbulence intermittency model of fractional Navier-Stokes equation which is derived from Boltzmann's theory. However, the efficient computation of this model on bounded domains is challenging as highly accurate and efficient numerical methods are not yet available. The bottleneck for efficient computation lies in the low accuracy and high computational cost of discretizing the fractional Laplacian operator. Although many state-of-the-art numerical methods have been proposed and some progress has been made for the existing numerical methods to achieve quasi-optimal complexity, some issues are still fully unresolved: i) Due to nonlocal nature of the fractional Laplacian, the implementation of the algorithm is still complicated and the computational cost for preparation of algorithms is still high, e.g., as pointed out by Acosta et al \cite{AcostaBB17} 'Over 99\% of the CPU time is devoted to assembly routine' for finite element method; ii) Due to the intrinsic singularity of the fractional Laplacian, the convergence orders in the literature are still unsatisfactory for many applications including turbulence intermittency simulations. To reduce the complexity and computational cost, we consider two numerical methods, finite difference and spectral method with quasi-linear complexity, which are summarized as follows. We develop spectral Galerkin methods to accurately solve the fractional advection-diffusion-reaction equations and apply the method to fractional Navier-Stokes equations. In spectral methods on a ball, the evaluation of fractional Laplacian operator can be straightforward thanks to the pseudo-eigen relation. For general smooth computational domains, we propose the use of spectral methods enriched by singular functions which characterize the inherent boundary singularity of the fractional Laplacian. We develop a simple and easy-to-implement fractional centered difference approximation to the fractional Laplacian on a uniform mesh using generating functions. The weights or coefficients of the fractional centered formula can be readily computed using the fast Fourier transform. Together with singularity subtraction, we propose high-order finite difference methods without any graded mesh. With the use of the presented results, it may be possible to solve fractional Navier-Stokes equations, fractional quantum Schrodinger equations, and stochastic fractional equations with high accuracy. All numerical simulations will be accompanied by stability and convergence analysis.
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Junior, Carlos Breviglieri. "High-order unstructured spectral finite volume method for aerodynamic applications." Instituto Tecnológico de Aeronáutica, 2010. http://www.bd.bibl.ita.br/tde_busca/arquivo.php?codArquivo=1133.
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An implicit finite volume algorithm is developed for higher-order unstructured computation of inviscid compressible flows. The Spectral Finite Volume method is used to achieve high-order spatial discretization of the domain, coupled with a matrix-free LU-SGS algorithm to solve the linear systems arising from implicit time marching of the governing equations, avoiding the explicit storage of the flux Jacobian matrices. A new limiter formulation for the high-order terms of the reconstruction polynomial is introduced. The issue of mesh refinement in accuracy measurements for unstructured meshes is investigated. A straightforward methodology is applied for accuracy assessment of the higher-order unstructured approach based on entropy levels and direct solution error calculation. The accuracy, fast convergence and robustness of the proposed higher-order unstructured solver for different speed regimes are demonstrated via several known test cases from the literature for the 2nd-, 3rd- and 4th-order discretizations. The possibility of reducing the computational cost required for a given level of accuracy using high-order discretization is demonstrated. The main features of the present methodology include the reconstruction algorithm that yields 2nd-, 3rd- and 4th-order spatially accurate schemes, an implicit time march algorithm, high-order domain boundaries representation and a hierarchical moment limiter to treat flow solution discontinuities.
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Lundquist, Tomas. "High order summation-by-parts methods in time and space." Doctoral thesis, Linköpings universitet, Beräkningsmatematik, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-126172.
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This thesis develops the methodology for solving initial boundary value problems with the use of summation-by-parts discretizations. The combination of high orders of accuracy and a systematic approach to construct provably stable boundary and interface procedures makes this methodology especially suitable for scientific computations with high demands on efficiency and robustness. Most classes of high order methods can be applied in a way that satisfies a summation-by-parts rule. These include, but are not limited to, finite difference, spectral and nodal discontinuous Galerkin methods. In the first part of this thesis, the summation-by-parts methodology is extended to the time domain, enabling fully discrete formulations with superior stability properties. The resulting time discretization technique is closely related to fully implicit Runge-Kutta methods, and may alternatively be formulated as either a global method or as a family of multi-stage methods. Both first and second order derivatives in time are considered. In the latter case also including mixed initial and boundary conditions (i.e. conditions involving derivatives in both space and time). The second part of the thesis deals with summation-by-parts discretizations on multi-block and hybrid meshes. A new formulation of general multi-block couplings in several dimensions is presented and analyzed. It collects all multi-block, multi-element and hybrid summation-by-parts schemes into a single compact framework. The new framework includes a generalized description of non-conforming interfaces based on so called summation-by-parts preserving interpolation operators, for which a new theoretical accuracy result is presented.
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Vadsola, Mayank. "High-Order Spectral Element Method Simulation of Flow Past a 30P30N Three-Element High Lift Wing." Thesis, Université d'Ottawa / University of Ottawa, 2020. http://hdl.handle.net/10393/40964.
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The purpose of a multi-element high lift device is to increase lift dramatically
while controlling the stall limit. The fluid flow over a multi-element high lift device
has been explored widely both experimentally and numerically at high Reynolds
numbers (O(10^6 )). The numerical simulations use turbulence models and hence
details of the flow are not yet available. Low Reynolds number (O(10^4 )) flows
over high lift devices have not been explored until recently. These lower Reynolds
number flows have applications in the development of small aerial vehicles. The
present work discusses both two-dimensional and three-dimensional direct numer-
ical simulations of fluid flow over a 30P30N three-element high lift system using a
high-order spectral element method code, Nek5000, that solves the incompressible
Navier-Stokes equations. The intricate geometry of the multi-element device poses
a challenge for the high-order spectral element method. We study the complex
flow physics in the slat cove region and the wake/shear layer interaction over a
30P30N three-element high lift device. The targeted cases are at Reynolds num-
bers based on stowed chord lengths (Rec ) of 8.32 × 10^3 , 1.27 × 10^4 , and 1.83 × 10^4 at angle of attack of 4. A critical interval for Rec has previously been found
between 1.27 × 10^4 and 1.38 × 10^4 in experiments. This divides the flow into
two types: when Rec is below the critical interval, no roll-up is observed in the
slat cove and Görtler vortices dominate the slat wake; however when the Rec is
above the critical interval, a roll-up is observed in the slat cove and co-existence
of streamwise and spanwise vortices is confirmed in the slat wake. We confirm
the presence of the critical interval from the simulations performed at three values of Rec . Lift and drag analysis is provided along with pressure coefficient plots
for each element of the multi-element airfoil. Different vortical structures are also
identified in the transition of flow from two dimensions to three dimensions. The
relevant validation is performed with the available experimental data.
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Harris, Robert Evan. "An adaptive quadrature-free implementation of the high-order spectral volume method on unstructured grids." [Ames, Iowa : Iowa State University], 2008.
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Thomas, Gregory Robert. "A combined high-order spectral and boundary integral equation method for modelling wave interactions with submerged bodies." Thesis, Monterey, California. Naval Postgraduate School, 1996. http://hdl.handle.net/10945/8098.
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Thomas, Gregory Robert. "A combined high-order spectral and boundary integral equation method for modelling wave interactions with submerged bodies." Thesis, Massachusetts Institute of Technology, 1997. http://hdl.handle.net/1721.1/17432.
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Barnes, Caleb J. "An Implicit High-Order Spectral Difference Method for the Compressible Navier-Stokes Equations Using Adaptive Polynomial Refinement." Wright State University / OhioLINK, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=wright1315591802.
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Huismann, Immo. "Computational fluid dynamics on wildly heterogeneous systems." TUDPress, 2018. https://tud.qucosa.de/id/qucosa%3A74002.
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In the last decade, high-order methods have gained increased attention. These combine the convergence properties of spectral methods with the geometrical flexibility of low-order methods. However, the time step is restrictive, necessitating the implicit treatment of diffusion terms in addition to the pressure. Therefore, efficient solution of elliptic equations is of central importance for fast flow solvers. As the operators scale with O(p · N), where N is the number of degrees of freedom and p the polynomial degree, the runtime of the best available multigrid algorithms scales with O(p · N) as well. This super-linear scaling limits the applicability of high-order methods to mid-range polynomial orders and constitutes a major road block on the way to faster flow solvers.
This work reduces the super-linear scaling of elliptic solvers to a linear one. First, the static condensation method improves the condition of the system, then the associated operator is cast into matrix-free tensor-product form and factorized to linear complexity. The low increase in the condition and the linear runtime of the operator lead to linearly scaling solvers when increasing the polynomial degree, albeit with low robustness against the number of elements. A p-multigrid with overlapping Schwarz smoothers regains the robustness, but requires inverse operators on the subdomains and in the condensed case these are neither linearly scaling nor matrix-free. Embedding the condensed system into the full one leads to a matrix-free operator and factorization thereof to a linearly scaling inverse. In combination with the previously gained operator a multigrid method with a constant runtime per degree of freedom results, regardless of whether the polynomial degree or the number of elements is increased.
Computing on heterogeneous hardware is investigated as a means to attain a higher performance and future-proof the algorithms. A two-level parallelization extends the traditional hybrid programming model by using a coarse-grain layer implementing domain decomposition and a fine-grain parallelization which is hardware-specific. Thereafter, load balancing is investigated on a preconditioned conjugate gradient solver and functional performance models adapted to account for the communication barriers in the algorithm. With the new model, runtime prediction and measurement fit closely with an error margin near 5 %.
The devised methods are combined into a flow solver which attains the same throughput when computing with p = 16 as with p = 8, preserving the linear scaling. Furthermore, the multigrid method reduces the cost of implicit treatment of the pressure to the one for explicit treatment of the convection terms. Lastly, benchmarks confirm that the solver outperforms established high-order codes.
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Singh, Pranav. "High accuracy computational methods for the semiclassical Schrödinger equation." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/274913.
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The computation of Schrödinger equations in the semiclassical regime presents several enduring challenges due to the presence of the small semiclassical parameter. Standard approaches for solving these equations commence with spatial discretisation followed by exponentiation of the discretised Hamiltonian via exponential splittings. In this thesis we follow an alternative strategy${-}$we develop a new technique, called the symmetric Zassenhaus splitting procedure, which involves directly splitting the exponential of the undiscretised Hamiltonian. This technique allows us to design methods that are highly efficient in the semiclassical regime. Our analysis takes place in the Lie algebra generated by multiplicative operators and polynomials of the differential operator. This Lie algebra is completely characterised by Jordan polynomials in the differential operator, which constitute naturally symmetrised differential operators. Combined with the $\mathbb{Z}_2$-graded structure of this Lie algebra, the symmetry results in skew-Hermiticity of the exponents for Zassenhaus-style splittings, resulting in unitary evolution and numerical stability. The properties of commutator simplification and height reduction in these Lie algebras result in a highly effective form of $\textit{asymptotic splitting:} $exponential splittings where consecutive terms are scaled by increasing powers of the small semiclassical parameter. This leads to high accuracy methods whose costs grow quadratically with higher orders of accuracy. Time-dependent potentials are tackled by developing commutator-free Magnus expansions in our Lie algebra, which are subsequently split using the Zassenhaus algorithm. We present two approaches for developing arbitrarily high-order Magnus--Zassenhaus schemes${-}$one where the integrals are discretised using Gauss--Legendre quadrature at the outset and another where integrals are preserved throughout. These schemes feature high accuracy, allow large time steps, and the quadratic growth of their costs is found to be superior to traditional approaches such as Magnus--Lanczos methods and Yoshida splittings based on traditional Magnus expansions that feature nested commutators of matrices. An analysis of these operatorial splittings and expansions is carried out by characterising the highly oscillatory behaviour of the solution.
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Vazquez, Thais Godoy. "Funções de interpolação e regras de integração tensorizaveis para o metodo de elementos finitos de alta ordem." [s.n.], 2008. http://repositorio.unicamp.br/jspui/handle/REPOSIP/263500.
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Orientador: Marco Lucio BittencourtTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica
Made available in DSpace on 2018-08-10T12:57:32Z (GMT). No. of bitstreams: 1 Vazquez_ThaisGodoy_D.pdf: 11719751 bytes, checksum: c6d385d6a6414705c9f468358b8d3bea (MD5) Previous issue date: 2008
Resumo: Este trabalho tem por objetivo principal o desenvolvimento de funções de interpolaçao e regras de integraçao tensorizaveis para o Metodo dos Elementos Finitos (MEF) de alta ordem hp, considerando os sistemas de referencias locais dos elementos. Para isso, primeiramente, determinam-se ponderaçoes especficas para as bases de funçoes de triangulos e tetraedros, formada pelo produto tensorial de polinomios de Jacobi, de forma a se obter melhor esparsidade e condicionamento das matrizes de massa e rigidez dos elementos. Alem disso, procuram-se novas funçoes de base para tornar as matrizes de massa e rigidez mais esparsas possiveis. Em seguida, escolhe-se os pontos de integraçao que otimizam o custo do calculo dos coeficientes das matrizes de massa e rigidez usando as regras de quadratura de Gauss-Jacobi, Gauss-Radau-Jacobi e Gauss-Lobatto-Jacobi. Por fim, mostra-se a construçao de uma base unidimensional nodal que permite obter uma matriz de rigidez praticamente diagonal para problemas de Poisson unidimensionais. Discute-se ainda extensoes para elementos bi e tridimensionais
Abstract: The main purpose of this work is the development of tensor-based interpolation functions and integration rules for the hp High-order Finite Element Method (FEM), considering the local reference systems of the elements. We first determine specific weights for the shape functions of triangles and tetrahedra, constructed by the tensorial product of Jacobi polynomials, aiming to obtain better sparsity and numerical conditioning for the mass and stiffness matrices of the elements. Moreover, new shape functions are proposed to obtain more sparse mass and stiffness matrices. After that, integration points are chosen that optimize the cost for the calculation of the coefficients of the mass and stiffness matrices using the rules of quadrature of Gauss-Jacobi, Gauss-Radau-Jacobi and Gauss-Lobatto-Jacobi. Finally, we construct an one-dimensional nodal shape function that obtains an almost diagonal stiffness matrix for the 1D Poisson problem. Extensions to two and three-dimensional elements are discussed.
Doutorado
Mecanica dos Sólidos e Projeto Mecanico
Doutor em Engenharia Mecânica
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Bassili, Niclas, and Douglas Eriksson. "An evaluation of deterministic prediction of ocean waves using pressure data to assist a Wave Energy Converter." Thesis, KTH, Skolan för industriell teknik och management (ITM), 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-279600.
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Currently, existing devices for extracting electrical power from ocean waves all suffer from problems with low efficiency due to a lack of information about the incoming waves in advance. The complex dynamic nonlinear characteristics of the ocean make the prediction of these incoming waves a great challenge. This paper aims to investigate a deterministic short-term wave-by-wave prediction system, that can accurately predict the wave height and timing of the incoming waves, based on measurements from a submerged pressure sensor. The complete prediction process contains three steps, namely reconstruction, assimilation, and prediction. A nonlinear Weakly Dispersive Reconstruction method (WDM) is firstly employed to accurately calculate the surface elevation from the measured pressures. Afterwards, a variational assimilation method is used to convert the time series surface elevation to a spatial wavefield, to obtain initial conditions for the prediction. Lastly, a High Order Spectral Method (HOSM) deterministically predicts the evolution of the 2D irregular wavefield based on the acquired initial conditions. To verify the performance of this proposed prediction system, tests were conducted with data from irregular sea states with varying wave parameters, generated by simulations as well as model experiments in the controlled environment of a wave tank. The results show that the surface elevation can be predicted within 5% from the reference, for a future period of about 10 seconds for wavefields commonly experienced by a wave energy converter. Based on the results, a prediction is possible, but the accuracy heavily depends on the current sea state and the chosen prediction distance.The results have been compared against similar tests made using radar data and proven to be a viable alternative for certain sea states. In conclusion, pressure measurements, as a mean to sample an ocean wavefield, is shown to be a good option when combined with nonlinear reconstruction and prediction methods to assist the power harvesting capabilities of a wave energy converter.Nuvarande enheter för att extrahera elektrisk energi från havsvågor lider av stora problem med låg effektivitet på grund av brist på information om de inkommande vågorna. Det komplexa ickelinjära dynamiska beteendet hos havsvågor gör förutsägelsen av dem till en stor utmaning. Det här arbetet syftar till att undersöka ett deterministiskt kortsiktigt system för att förutspå våg för våg, som noggrant kan förutspå våghöjd och tidpunkt för de inkommande vågorna, baserat på mätdata från en dränkbar trycksensor. Den kompletta förutsägelseprocessen innehåller tre steg, rekonstruktion, assimilering och förutsägelse. En ickelinjär weakly dispersive reconstruction method används först för att med hög noggrannhet beräkna ythöjningen från det uppmätta trycket. Därefter, används en variational assimilation method för att konvertera en tidsserie av ythöjningen till ett rumsligt vågfält, för att erhålla initialvillkor för förutsägelsen. Slutligen används en High Order Spectral Method för att deterministiskt förutspå utvecklingen av det tvådimensionella oregelbundna vågfältet baserat på de förvärvade initialvillkoren. För att verifiera prestandan av det föreslagna förutsägelsesystemet, så genomfördes tester med data från olika oregelbundna havstillstånd med varierande parametrar, genererade av simuleringar, såväl som modellexperiment utförda i en kontrollerad miljö i form av en vågtank. Resultaten från testerna visar att ythöjningen förutspås inom 5% från referensen, för en period på 10 sekunder framåt i tiden, för vågor som ett vågkraftverk vanligtvis utsätts för. Baserat på resultatet, så är det möjligt att förutspå inkommande vågor, men noggrannheten beror kraftigt på det aktuella havstillståndet och det valda avståndet för förutsägelsen. Resultaten har jämförts mot liknande tester gjorda med radardata och visat sig vara ett genomförbart alternativ för vissa havstillstånd. Sammanfattningsvis visas det att tryckmätningar, som ett medel för att mäta ett havsvågfält, är ett bra alternativ när de kombineras med ickelinjära rekonstruktions- och förutsägelsemetoder för att hjälpa till att öka ett vågkraftverks energigenerering.
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Akeab, Imad. "High frequency scattering and spectral methods." Doctoral thesis, Linnéuniversitetet, Institutionen för fysik och elektroteknik (IFE), 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-57871.
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This thesis consists of five parts. The first part is an introduction with references to some recent work on 2D electromagnetic scattering problems at high frequencies. It also presents the basic integral equation types for impenetrable objects and the standard elements of the method of moments. An overview of frequency modulated radar at low frequencies is followed by summaries of the papers. Paper I presents an accurate implementation of the method of moments for a perfectly conducting cylinder. A scaling for the rapid variation of the solution improves accuracy. At high frequencies, the method of moments leads to a large dense system of equations. Sparsity in this system is obtained by the modification of the path in the integral equation. The modified path reduces the accuracy in the deep shadow. In paper II, a hybrid method is used to handle the standing waves that are prominent in the shadow for the cylindrical TE case. The shadow region is treated separately, in a hybrid scheme based on a priori knowledge about the solution. An accurate method to combine solutions in this hybrid scheme is presented. In paper III, the surface current in the shadow zone of a convex or a concave scatterer is approximated by extracting the dominant waves. An accurate technique based on the symmetric discrete Fourier transform is used to extract the complex wavenumbers and amplitudes for those waves. The dominant waves constitute a concise form of scaling that is used to improve the performance of the method of moments. The effect of surface curvature on the dominant waves has been investigated in this work. In paper IV, frequency modulated continuous wave radar (FMCW) at low frequency is studied as a way to locate targets that are normally not detected by conventional radar. Three separate platforms with isotropic antennas are used for this purpose. The trilateration method is a way to locate the targets accurately by means of spectral techniques. The problem of ghost targets has been studied for monostatic and multistatic radar. In the case of confluent echoes in the spectra, potentially missing echoes are reinserted in order to locate all targets. The Capon method is used to obtain high resolution spectra and thus reduce the confluence problem. The need for bandwidth is also reduced.
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Postell, Floyd Vince. "High order finite difference methods." Diss., Georgia Institute of Technology, 1990. http://hdl.handle.net/1853/28876.
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Rossi, Lorenzo. "Functional renormalization group: higher order flows and pseudo-spectral methods." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2019. http://amslaurea.unibo.it/18120/.
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Diamo una rapida introduzione alle idee su cui si basa il Gruppo di Rinormalizzazione Funzionale e deriviamo l’equazione di flsso esatta per l’azione effettiva media nella formulazione proposta da Wetterich e Morris. Lavorando in tale contesto, discutiamo i più comuni troncamenti LPA e LPA’ e li applichiamo a una teoria con un campo scalare reale in D dimensioni. Discutiamo la struttura dei puntifissi della teoria in dimensioni arbitrarie tramite un metodo numerico di shooting e concentrandoci sul caso D = 3, che corrisponde alla classe di universalità del modello di Ising, studiamo le soluzioni di punto fissi e gli esponenti critici tramite espansioni polinomiali. Come risultato originale deriviamo una nuova equazione di flusso esatta al secondo ordine nel tempo RG e consideriamo analoghi schemi di approssimazione LPA e LPA’, applicandoli ad una teoria scalare in D = 3. Per risolvere le nuove e più complesse equazioni diventa necessaria una tecnica numerica più potente, pertanto i metodi psuedo-spettrali vengono introdotti e applicati con successo al problema. Infine, il confronto tra i risultati trovati con i diversi schemi di approssimazione permette di mettere in luce l’affdabilità e i limiti dei diversi troncamenti e di identificare promettenti ulteriori sviluppi.
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Strauss, Michael. "Spectral pollution and higher order projection methods for operator pencils." Thesis, King's College London (University of London), 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.497989.
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Kudo, Jun S. M. Massachusetts Institute of Technology. "Robust adaptive high-order RANS methods." Thesis, Massachusetts Institute of Technology, 2014. http://hdl.handle.net/1721.1/95563.
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Thesis: S.M., Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2014.Cataloged from PDF version of thesis.
Includes bibliographical references (pages 89-94).
The ability to achieve accurate predictions of turbulent flow over arbitrarily complex geometries proves critical in the advancement of aerospace design. However, quantitatively accurate results from modern Computational Fluid Dynamics (CFD) tools are often accompanied by intractably high computational expenses and are significantly hindered by the lack of automation. In particular, the generation of a suitable mesh for a given flow problem often requires significant amounts of human input. This process however encounters difficulties for turbulent flows which exhibit a wide range of length scales that must be spatially resolved for an accurate solution. Higher-order adaptive methods are attractive candidates for addressing these deficiencies by promising accurate solutions at a reduced cost in a highly automated fashion. However, these methods in general are still not robust enough for industrial applications and significant advances must be made before the true realization of robust automated three-dimensional turbulent CFD. This thesis presents steps towards this realization of a robust high-order adaptive Reynolds-Averaged Navier-Stokes (RANS) method for the analysis of turbulent flows. Specifically, a discontinuous Galerkin (DG) discretization of the RANS equations and an output-based error estimation with an associated mesh adaptation algorithm is demonstrated. To improve the robustness associated with the RANS discretization, modifications to the negative continuation of the Spalart-Allmaras turbulence model are reviewed and numerically demonstrated on a test case. An existing metric-based adaptation framework is adopted and modified to improve the procedure's global convergence behavior. The resulting discretization and modified adaptation procedure is then applied to two-dimensional and three-dimensional turbulent flows to demonstrate the overall capability of the method.
by Jun Kudo.
S.M.
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Bissessur, Prithiraj. "Unsteady aerodynamics using high-order methods." Thesis, University of Southampton, 2007. https://eprints.soton.ac.uk/49924/.
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Unsteady flows occur in many applications of engineering interest. One category of unsteady flows occur as self-sustaining oscillatory fluid motion, such as the flow over rectangular cavities. There has been a significant amount of research performed on this topic over the years, both experimentally and numerically. The unsteady flow over rectangular cavities is the case study in this research. In this work, a generic numerical solver is developed and written to predict the near-field aerodynamics of unsteady fluid motions at low Mach numbers. High order numerical schemes are employed to this effect. The Detached Eddy Simulation (DES) method is considered for the turbulence modelling part. At the start of this project, the combination of high order Computational Aeroacoustics (CAA) numerical schemes, non-reflecting boundary conditions and DES constituted a state of the art approach to the simulation of unsteady compressible flow phenomena at low Mach numbers. In the numerical study of 2D cavities, a number of cases with different length-to depth (L/D) ratios were considered. Under the same flow conditions, the relation of the L/D to the radiated sound in the farfield is sought. It is found that the nature of the flow interaction with the downstream corner, which changes with L/D, dictates the directivity and amplitude of the sound field observed at a far distance from the source. To gain more insight into the topology of 3D cavity flows, an experimental study using non-intrusive measurement techniques is outlined. This explains the work performed on 3D cavities with different spanwise dimensions. A detailed flow visualisation of the meanflow patterns in various measurements planes describes the presence of strong 3D features. In particular, the symmetrical flow behaviours at relatively large width-to-depth (W/D) ratios of 3 and 2 are highlighted. This provides the justification to employ a symmetry condition in the 3D DES study. Therefore, the final case study is based on the numerical simulation of a 3D cavity geometry where only half of the cavity is simulated. The observations from the 2D simulations and the experimental work provided a basis of the expectations of this test case. Again, a correlation between the near-field aerodynamics and the farfield sound is sought. The 3D cavity showed (as in the 2D cases) a preferred directivity in the farfield.
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Villardi, de Montlaur Adeline de. "High-order discontinuous Galerkin methods for incompressible flows." Doctoral thesis, Universitat Politècnica de Catalunya, 2009. http://hdl.handle.net/10803/5928.
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Aquesta tesi doctoral proposa formulacions de Galerkin discontinu (DG) d'alt ordre per fluxos viscosos incompressibles. Es desenvolupa un nou mètode de DG amb penalti interior (IPM-DG), que condueix a una forma feble simètrica i coerciva pel terme de difusió, i que permet assolir una aproximació espacial d'alt ordre. Aquest mètode s'aplica per resoldre les equacions de Stokes i Navier-Stokes. L'espai d'aproximació de la velocitat es descompon dins de cada element en una part solenoidal i una altra irrotacional, de manera que es pot dividir la forma dèbil IPM-DG en dos problemes desacoblats. El primer permet el càlcul de les velocitats i de les pressions híbrides, mentre que el segon calcula les pressions en l'interior dels elements. Aquest desacoblament permet una reducció important del número de graus de llibertat tant per velocitat com per pressió. S'introdueix també un paràmetre extra de penalti resultant en una formulació DG alternativa per calcular les velocitats solenoidales, on les pressions no apareixen. Les pressions es poden calcular com un post-procés de la solució de les velocitats. Es contemplen altres formulacions DG, com per exemple el mètode Compact Discontinuous Galerkin, i es comparen al mètode IPM-DG.
Es proposen mètodes implícits de Runge-Kutta d'alt ordre per problemes transitoris incompressibles, permetent obtenir esquemes incondicionalment estables i amb alt ordre de precisió temporal. Les equacions de Navier-Stokes incompressibles transitòries s'interpreten com un sistema de Equacions Algebraiques Diferencials, és a dir, un sistema d'equacions diferencials ordinàries corresponent a la equació de conservació del moment, més les restriccions algebraiques corresponent a la condició d'incompressibilitat.
Mitjançant exemples numèrics es mostra l'aplicabilitat de les metodologies proposades i es comparen la seva eficiència i precisió.
This PhD thesis proposes divergence-free Discontinuous Galerkin formulations providing high orders of accuracy for incompressible viscous flows.
A new Interior Penalty Discontinuous Galerkin (IPM-DG) formulation is developed, leading to a symmetric and coercive bilinear weak form for the diffusion term, and achieving high-order spatial approximations. It is applied to the solution of the Stokes and Navier-Stokes equations. The velocity approximation space is decomposed in every element into a solenoidal part and an irrotational part. This allows to split the IPM weak form in two uncoupled problems. The first one solves for velocity and hybrid pressure, and the second one allows the evaluation of pressures in the interior of the elements. This results in an important reduction of the total number of degrees of freedom for both velocity and pressure.
The introduction of an extra penalty parameter leads to an alternative DG formulation for the computation of solenoidal velocities with no presence of pressure terms. Pressure can then be computed as a post-process of the velocity solution. Other DG formulations, such as the Compact Discontinuous Galerkin method, are contemplated and compared to IPM-DG.
High-order Implicit Runge-Kutta methods are then proposed to solve transient incompressible problems, allowing to obtain unconditionally stable schemes with high orders of accuracy in time. For this purpose, the unsteady incompressible Navier-Stokes equations are interpreted as a system of Differential Algebraic Equations, that is, a system of ordinary differential equations corresponding to the conservation of momentum equation, plus algebraic constraints corresponding to the incompressibility condition.
Numerical examples demonstrate the applicability of the proposed methodologies and compare their efficiency and accuracy.
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Svärd, Magnus. "Stable High-Order Finite Difference Methods for Aerodynamics." Doctoral thesis, Uppsala universitet, Avdelningen för teknisk databehandling, 2004. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-4621.
Full textAbstract:
In this thesis, the numerical solution of time-dependent partial differential equations (PDE) is studied. In particular high-order finite difference methods on Summation-by-parts (SBP) form are analysed and applied to model problems as well as the PDEs governing aerodynamics. The SBP property together with an implementation of boundary conditions called SAT (Simultaneous Approximation Term), yields stability by energy estimates. The first derivative SBP operators were originally derived for Cartesian grids. Since aerodynamic computations are the ultimate goal, the scheme must also be stable on curvilinear grids. We prove that stability on curvilinear grids is only achieved for a subclass of the SBP operators. Furthermore, aerodynamics often requires addition of artificial dissipation and we derive an SBP version. With the SBP-SAT technique it is possible to split the computational domain into a multi-block structure which simplifies grid generation and more complex geometries can be resolved. To resolve extremely complex geometries an unstructured discretisation method must be used. Hence, we have studied a finite volume approximation of the Laplacian. It can be shown to be on SBP form and a new boundary treatment is derived. Based on the Laplacian scheme, we also derive an SBP artificial dissipation for finite volume schemes. We derive a new set of boundary conditions that leads to an energy estimate for the linearised three-dimensional Navier-Stokes equations. The new boundary conditions will be used to construct a stable SBP-SAT discretisation. To obtain an energy estimate for the discrete equation, it is necessary to discretise all the second derivatives by using the first derivative approximation twice. According to previous theory that would imply a degradation of formal accuracy but we present a proof that this is not the case.
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Svärd, Magnus. "Stable high-order finite difference methods for aerodynamics /." Uppsala : Acta Universitatis Upsaliensis : Univ.-bibl. [distributör], 2004. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-4621.
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Matar, Samir A. "Numerical methods for high-order boundary-value problems." Thesis, Brunel University, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.293058.
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Iqbal, Kashif H. "Comparison of high-order methods on unstructured grids." Thesis, Cranfield University, 2013. http://dspace.lib.cranfield.ac.uk/handle/1826/8274.
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A high-order Discontinuous Galerkin (DG) method is formulated and implemented on the Cranfield University’s 3D unstructured Finite Volume Method (FVM) code (UCNS3D), for both linear and non-linear hyperbolic conservation laws and for test-cases which exhibit both smooth and discontinuous solutions. As both DG and FVM are developed on the same solver platform, this enables the use of any procedures which are common to both the methods, thus, ensuring the closest possible compari-son.
The initial part of the thesis details the basic concepts and derivation of the discon-tinuous Galerkin method in the 1D space for the advection equation, which is then extended to the 3D space for a hyperbolic system.
Prior to comparing the FVM and DG methods, the DG method implementation is verified. The verification is a combination of a theoretical and numerical approach which endeavours to minimize any potential programming errors.
Following the verification of the DG method, the FVM and DG methods are compared for numerous flows: the linear advection equation and Euler equations, sufficiently smooth testcases, and testcases which require a limiter to suppress Gibb’s oscillations.
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Velechovsky, Jan. "High-order numerical methods for laser plasma modeling." Thesis, Bordeaux, 2015. http://www.theses.fr/2015BORD0098/document.
Full textAbstract:
Cette thèse présente le développement d’une méthode ALE pour la modélisation del’interaction laser–plasma. La particularité de cette méthode est l’utilisation d’une étape de projectiond’ordre élevé. Cette étape de projection consiste en une interpolation conservative des quantitésconservatives du maillage Lagrangien sur un maillage régularisé. Afin d’éviter les oscillationsnumériques non-physiques, les flux numériques d’ordre élevé sont combinés avec des fluxnumériques d’ordre moins élevé. Ces flux numériques sont obtenu en considérant les quantitésconservatives constantes par morceaux. Cette méthode pour la discrétisation cellule–centrée consisteà préserver les maximums locaux pour la densité, la vitesse et l’énergie interne. Aspects particuliersde la méthode sont appliquées pour la projection la quantité de mouvement pour la discrétisation’staggered’. Nous l’utilisons ici dans le cadre de la projection sous la forme de la méthode FluxCorrection Remapping (FCR). Dans cette thèse le volet applicatif concerne la modélisation del’interaction d’un laser énergétique avec de plasma et des matériaux microstructures. Un intérêtparticulier est porté à la modélisation de l’absorption du laser par une mousse de faible densité.L’absorption se fait à deux échelles spatiales simultanément. Ce modèle d’absorption laser à deuxéchelles est mis en oeuvre dans le code PALE hydrodynamique. Les simulations numériques de lavitesse de pénétration du laser dans une mousse à faible densité sont en bon accord avec lesdonnées expérimentalesThis thesis presents the overview and the original contributions to a high–orderArbitrary Lagrangian–Eulerian (ALE) method applicable for the laser–generated plasma modeling withthe focus to a remapping step of the ALE method. The remap is the conservative interpolation of theconservative quantities from a low–quality Lagrangian grid onto a better, smoothed one. To avoidnon–physical numerical oscillations, the high–order numerical fluxes of the reconstruction arecombined with the low–order (first–order) numerical fluxes produced by a standard donor remappingmethod. The proposed method for a cell–centered discretization preserves bounds for the density,velocity and specific internal energy by its construction. Particular symmetry–preserving aspects of themethod are applied for a staggered momentum remap. The application part of the thesis is devoted tothe laser radiation absorption modeling in plasmas and microstructures materials with the particularinterest in the laser absorption in low–density foams. The absorption is modeled on two spatial scalessimultaneously. This two–scale laser absorption model is implemented in the hydrodynamic codePALE. The numerical simulations of the velocity of laser penetration in a low–density foam are in agood agreement with the experimental data
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Kress, Wendy. "High Order Finite Difference Methods in Space and Time." Doctoral thesis, Uppsala : Acta Universitatis Upsaliensis : Univ.-bibl. [distributör], 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-3559.
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Tsoutsanis, Panagiotis. "Very high-order methods for 3D arbitrary unstructured grids." Thesis, Cranfield University, 2009. http://hdl.handle.net/1826/4511.
Full textAbstract:
Understanding the motion of fluids is crucial for the development and analysis of new designs
and processes in science and engineering. Unstructured meshes are used in this context
since they allow the analysis of the behaviour of complicated geometries and configurations
that characterise the designs of engineering structures today. The existing numerical methods
developed for unstructured meshes suffer from poor computational efficiency, and their applicability
is not universal for any type of unstructured meshes. High-resolution high-order
accurate numerical methods are required for obtaining a reasonable guarantee of physically
meaningful results and to be able to accurately resolve complicated flow phenomena that
occur in a number of processes, such as resolving turbulent flows, for direct numerical simulation
of Navier-Stokes equations, acoustics etc.
The aim of this research project is to establish and implement universal, high-resolution, very
high-order, non-oscillatory finite-volume methods for 3D unstructured meshes. A new class
of linear and WENO schemes of very high-order of accuracy (5
th
) has been developed. The
key element of this approach is a high-order reconstruction process that can be applied to any
type of meshes. The linear schemes which are suited for problems with smooth solutions,
employ a single reconstruction polynomial obtained from a close spatial proximity. In the
WENO schemes the reconstruction polynomials, arising from different topological regions,
are non-linearly combined to provide high-order of accuracy and shock capturing features.
The performance of the developed schemes in terms of accuracy, non-oscillatory behaviour
and flexibility to handle any type of 3D unstructured meshes has been assessed in a series of
test problems. The linear and WENO schemes presented achieve very high-order of accuracy
(5
th
). This is the first class of WENO schemes in the finite volume context that possess highorder
of accuracy and robust non-oscillatory behaviour for any type of unstructured meshes.
The schemes have been employed in a newly developed 3D unstructured solver (UCNS3D).
UCNS3D utilises unstructured grids consisted of tetrahedrals, pyramids, prisms and hexahedral
elements and has been parallelised using the MPI framework. The high parallel efficiency
achieved enables the large scale computations required for the analysis of new designs and
processes in science and engineering.
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Bowen, Matthew K. "High-order finite difference methods for partial differential equations." Thesis, Loughborough University, 2005. https://dspace.lboro.ac.uk/2134/13492.
Full textAbstract:
General n-point formulae for difference operators and their errors are derived in terms of elementary symmetric functions. These are used to derive high-order, compact and parallelisable finite difference schemes for the decay-advection-diffusion and linear damped Korteweg-de Vnes equations. Stability calculations are presented and the speed and accuracy of the schemes is compared to that of other finite difference methods in common use. Appendices contain useful tables of difference operators and errors and present a stability proof for quadratic inequalities. For completeness, the appendices conclude with the standard Thomas method for solving tri-diagonal systems.
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Mazzieri, Ilario. "Non-conforming high order methods for the elastodynamics equation." Nice, 2012. http://www.theses.fr/2012NICE4014.
Full textAbstract:
Dans cette thèse, on présente une nouvelle approche de discrétization qui combine les méthodes aux éléments spectraux Discontinuous Galerkin (DGSE) and Mortar (MSE) avec des méthodes convenables de discretization en temps pour la simulation de la propagation des ondes élastiques dans les milieux hétérogènes. Pour surmonter les limites des approches existantes on applique le paradigme non conforme au niveau des sous domaines. On montre que les formulations obtenues sont stables, ont des propriétés d’approximation optimales, et souffrent d’erreurs de dispersion et dissipation négligeables. Les méthodes DGSE et MSE sont finalement utilisées pour résoudre de vrais problèmes sismiques dans des domaines tridimensionnelsIn this thesis, we present a new discretization approach to combine the Discontinuous Galerkin Spectral Element (DGSE) and the Mortar Spectral Element (MSE) methods with suitable time advancing schemes for the simulation of the elastic wave propagation in heterogeneous media. To overcome the limitations of the existing approaches we apply the non-conforming paradigm only at the subdomain level. We show that the resulting formulations are stable, enjoy optimal approximation properties, and suffer from low dispersion and dissipation errors. Applications of the DGSE and MSE methods to simulate realistic seismic wave propagation problems in three dimensions are also considered
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Grundvig, Dane Scott. "High Order Numerical Methods for Problems in Wave Scattering." BYU ScholarsArchive, 2020. https://scholarsarchive.byu.edu/etd/8617.
Full textAbstract:
Arbitrary high order numerical methods for time-harmonic acoustic scattering problems originally defined on unbounded domains are constructed. This is done by coupling recently developed high order local absorbing boundary conditions (ABCs) with finite difference methods for the Helmholtz equation. These ABCs are based on exact representations of the outgoing waves by means of farfield expansions. The finite difference methods, which are constructed from a deferred-correction (DC) technique, approximate the Helmholtz equation and the ABCs to any desired order. As a result, high order numerical methods with an overall order of convergence equal to the order of the DC schemes are obtained. A detailed construction of these DC finite difference schemes is presented. Details and results from an extension to heterogeneous media are also included. Additionally, a rigorous proof of the consistency of the DC schemes with the Helmholtz equation and the ABCs in polar coordinates is also given. The results of several numerical experiments corroborate the high order convergence of the proposed method. A novel local high order ABC for elastic waves based on farfield expansions is constructed and preliminary results applying it to elastic scattering problems are presented.
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Riera, Pau. "Conservative high order collocation methods for nonlinear Schrödinger equations." Thesis, Stockholms universitet, Fysikum, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-194703.
Full textAbstract:
In this thesis, we investigate the numerical solution of time-dependent nonlinear Schrödinger equations (more specifically, the Gross-Pitaevskii equation) that appear in the modeling of Bose-Einstein condensates. Since the model is known to conserve important physical invariants, such as mass and energy of the condensate, our goal is to study the importance of reproducing the conservation on the discrete level. The reliability of conservative, compared to non-conservative, methods shall be studied through high order collocation methods for the time discretization and finite element-based space discretizations. In particular, this includes symplectic discontinuous Galerkin time-stepping methods, as well as Continuous Petrov-Galerkin methods. The methods shall be tested for a problem with a known analytical solution, namely two interacting solitons in 1D. This problem is a suitable choice due to its high sensitivity to oscillations of the energy and difficulty to approximate for large time scales.
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Vila, Pérez Jordi. "Low and high-order hybridised methods for compressible flows." Doctoral thesis, Universitat Politècnica de Catalunya, 2021. http://hdl.handle.net/10803/671889.
Full textAbstract:
The aerospace community is challenged as of today for being able to manage accurate overnight computational fluid dynamics (CFD) simulations of compressible flow problems. Well-established CFD solvers based on second-order finite volume (FV) methods provide accurate approximations of steady-state turbulent flows but are incapable to produce reliable predictions of the full flight envelope. Alternatively, promising high-order discretisations, claimed to permit feasible highfidelity simulations of unsteady turbulent flows, are still subject to strong limitations in robustness and efficiency, placing their level of maturity far away from industrial requirements. In consequence, the CFD paradigm is immersed at this point into the crossroads outlined by the inherent limitations of low-order methods and the yet immature state of high-order discretisations. Accordingly, this thesis develops a twofold strategy for the high-fidelity simulation of compressible flows introducing two methodologies, at the low and high-order levels, respectively, based on hybridised formulations.
First, a new finite volume paradigm, the face-centred finite volume (FCFV) method, is proposed for the formulation of steady-state compressible flows. The present methodology describes a hybrid mixed FV formulation that, following a hybridisation process, defines the unknowns of the problem at the face barycentres.
The problem variables, i.e. the conservative quantities and the stress tensor and heat flux in the viscous case, are retrieved with optimal first-order accuracy inside each cell by means of an inexpensive postprocessing step without need of reconstruction of the gradients. Hence, the FCFV solver preserves the accuracy of the approximation even in presence of highly stretched or distorted cells, providing a solver insensitive to mesh quality. In addition, the FCFV method is a monotonicity-preserving scheme, leading to non-oscillatory approximations of sharp gradients without resorting to shock capturing or limiting techniques. Finally, the method is robust in the incompressible limit and is capable of computing accurate solutions for flows at low Mach number without the need of introducing specific pressure correction strategies.
In parallel, the high-order hybridisable discontinuous Galerkin (HDG) method is reviewed in the context of compressible flows, presenting an original unified framework for the derivation of Riemann solvers in hybridised formulations. The framework includes, for the first time in an HDG context, the HLL and HLLEM Riemann solvers as well as the traditional Lax-Friedrichs and Roe solvers. The positivity preserving properties of HLL-type Riemann solvers are displayed, demonstrating their superiority with respect to Roe in supersonic cases. In addition, HLLEM specifically outstands in the approximation of boundary layers because of its shear preservation, which confers it an increased accuracy with respect to HLL and Lax-Friedrichs.
An extensive set of numerical benchmarks of practical interest is introduced along this study in order to validate both the low and high-order approaches.
Different examples of compressible flows in a great variety of regimes, from inviscid to viscous laminar flows, from subsonic to supersonic speeds, are presented to verify the accuracy properties of each of the proposed methodologies and the performance of the introduced Riemann solvers.La comunitat aeroespacial té el repte a dia d’avui de poder tractar amb precisió simulacions de mecànica de fluids computacional (CFD) de problemes de flux compressible en càlcul nocturn. Programes convencionals de simulació CFD basats en mètodes de volums finits (VF) de segon ordre ofereixen aproximacions precises de fluxos turbulents estacionaris però són incapaços de produir prediccions fidels de l’entorn de vol complet. Alternativament, les discretitzations prometedores d’alt ordre, de les quals s’espera que permetin simulacions accessibles d’alta fidelitat per a fluxos turbulents transitoris, encara estan subjectes a fortes limitacions en eficiència i robustesa, delimitant-ne el nivell de maduresa encara lluny de requeriments industrials. En conseqüència, el paradigma del CFD es troba immers ara mateix en la cruïlla delimitada per les limitacions inherents dels mètodes de baix ordre i l’estat encara immadur de les discretitzacions d’alt ordre. D’acord amb això, aquesta tesi desenvolupa una estratègia doble per a la simulació d’alta fidelitat de flux compressible introduint dues metodologies, als nivells de baix i alt ordre, respectivament, basades en formulacions híbrides. Primer, es proposa un nou paradigma de VF, el mètode de volums finits centrats en les cares (FCFV), per a la formulació de fluxos compressible estacionaris. Aquesta metodologia descriu una formulació mixta híbrida de VF que, seguint un procés d’hibridització, defineix les incògnites del problema als baricentres de les cares. Les variables del problema -quantitats conservatives i tensor de tensions i flux de calor en el cas viscós- són obtingudes amb precisió òptima de primer ordre dins de cada element mitjançant una etapa de postprocessat de cost reduït sense la necessitat de reconstrucció dels gradients. Amb això, el mètode FCFV preserva la qualitat de l’aproximació fins i tot en presència d’elements amb un alt estretament o distorsió, donant lloc a un mètode insensible a la qualitat de la malla. A més a més, el mètode de FCFV és un esquema preservador de monotonia, donant lloc a aproximacions no oscil·latòries de forts gradients sense necessitat d’utilitzar mètodes de captura de xocs o limitadors. Finalment, el mètode és robust en el límit incompressible i és capaç de calcular amb precisió solucions de fluxos amb nombre de Mach baix sense haver d’introduir estratègies específiques de correcció de pressió. En paral·lel, es presenta una revisió del mètode híbrid de Galerkin discontinu (HDG) d’alt ordre en el context de flux compressible, presentant un marc unificat per a la derivació de fluxos numèrics del problema de Riemann en formulacions híbrides. El marc inclou per primera vegada en un entorn HDG, els fluxos numèrics d’HLL i HLLEM, així com els tradicionals de Lax-Friedrichs i Roe. Es mostren les propietats de preservació de positivitat dels fluxos de tipus HLL, que demostren la seva superioritat respecte els de Roe en casos supersònics. Addicionalment, el mètode d’HLLEM destaca especialment en l’aproximació de capes límit com a resultat de la seva preservació d’esforços tallants, la qual li confereix una precisió afegida respecte les d’HLL i Lax-Friedrichs. Al llarg de l’estudi s’introdueix una llista extensa d’exemples numèrics de referència d’interès pràctic per tal de validar les propostes en baix i alt ordre. Es presenten diferents exemples de flux compressible en una gran varietat de règims, des de flux invíscid fins a flux laminar viscós, des de velocitats subsòniques fins a supersòniques, per tal de verificar la precisió de les metodologies proposades i el rendiment dels fluxos numèrics introduïts
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Hyde, Edward McKay Bruno Oscar P. "Fast, high-order methods for scattering by inhomogeneous media /." Diss., Pasadena, Calif. : California Institute of Technology, 2003. http://resolver.caltech.edu/CaltechETD:etd-08142002-182101.
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Lin, Yuan. "High-order finite difference methods for solving heat equations /." Available to subscribers only, 2008. http://proquest.umi.com/pqdweb?did=1559848541&sid=1&Fmt=2&clientId=1509&RQT=309&VName=PQD.
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Thesis (Ph. D.)--Southern Illinois University Carbondale, 2008."Department of Mathematics." Keywords: High-order finite difference, Heat equations Includes bibliographical references (p. 64-68). Also available online.
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Zachariadis, Zacharias Ioannis. "High resolution and high order methods for RANS modelling and aerodynamic optimization." Thesis, Cranfield University, 2008. http://hdl.handle.net/1826/3806.
Full textAbstract:
With the optimisation of fixed aerodynamic shapes reaching its limits,
the active flow control concept increasingly attracts attention of both
academia and industry. Adaptive wing technology, and shape morphing
airfoils in particular, represents a promising way forward. The
aerodynamic performance of the morphing profiles is an important
issue affecting the overall aerodynamic performance of an adaptive
wing.
A new concept of active flow, the Active Camber concept has been
investigated. The actuator is integrated into the aerofoil and aerofoil
morphing is realized via camber deformation. In order to identify
the most aerodynamically efficient designs, an optimisation study has
been performed using high resolution methods in conjunction with a
two equation eddy viscosity model.
Several different types of previously proposed compressible filters, including
monotone upstream-centered schemes for conservation laws
(MUSCL) and weighted essential non-oscillatory (WENO) filters, are
incorporated and investigated in the present research. The newly
developed CFD solver is validated and the effect that high resolution
methods have on turbulent flow simulations is highlighted. The outermost
goal is the development of a robust high resolution CFD method
that will efficiently and accurately simulate various phenomena, such
as shock/boundary layer interaction, flow separation and turbulence
and thus provide the numerical framework for analysis and aerodynamic
aerofoil design.
With respect to the latter a multi-objective integrated design system
(MOBID) has been developed that incorporates the CFD solver and
a state-of-the-art heuristic optimisation algorithm, along with an efficient
parametrization technique and a fast and robust method of
propagating geometric displacements. The methodologies in the MOBID
system resulted in the identification of the design vectors that
revealed aerodynamic performance gains over the datum aerofoil design.
The Pareto front provided a clear picture of the achievable
trade-offs between the competing objectives.
Furthermore, the implementation of different numerical schemes led to
significant differences in the optimised airfoil shape, thus highlighting
the need for high-resolution methods in aerodynamic optimisation.
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Wang, Tengyao. "Spectral methods and computational trade-offs in high-dimensional statistical inference." Thesis, University of Cambridge, 2016. https://www.repository.cam.ac.uk/handle/1810/260825.
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Spectral methods have become increasingly popular in designing fast algorithms for modern highdimensional datasets. This thesis looks at several problems in which spectral methods play a central role. In some cases, we also show that such procedures have essentially the best performance among all randomised polynomial time algorithms by exhibiting statistical and computational trade-offs in those problems. In the first chapter, we prove a useful variant of the well-known Davis{Kahan theorem, which is a spectral perturbation result that allows us to bound of the distance between population eigenspaces and their sample versions. We then propose a semi-definite programming algorithm for the sparse principal component analysis (PCA) problem, and analyse its theoretical performance using the perturbation bounds we derived earlier. It turns out that the parameter regime in which our estimator is consistent is strictly smaller than the consistency regime of a minimax optimal (yet computationally intractable) estimator. We show through reduction from a well-known hard problem in computational complexity theory that the difference in consistency regimes is unavoidable for any randomised polynomial time estimator, hence revealing subtle statistical and computational trade-offs in this problem. Such computational trade-offs also exist in the problem of restricted isometry certification. Certifiers for restricted isometry properties can be used to construct design matrices for sparse linear regression problems. Similar to the sparse PCA problem, we show that there is also an intrinsic gap between the class of matrices certifiable using unrestricted algorithms and using polynomial time algorithms. Finally, we consider the problem of high-dimensional changepoint estimation, where we estimate the time of change in the mean of a high-dimensional time series with piecewise constant mean structure. Motivated by real world applications, we assume that changes only occur in a sparse subset of all coordinates. We apply a variant of the semi-definite programming algorithm in sparse PCA to aggregate the signals across different coordinates in a near optimal way so as to estimate the changepoint location as accurately as possible. Our statistical procedure shows superior performance compared to existing methods in this problem.
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Chiang, Weng Cheng Venus. "High-order finite difference methods for solving convection diffusion equations." Thesis, University of Macau, 2008. http://umaclib3.umac.mo/record=b1807119.
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Weggler, Lucy Verfasser], and Sergej [Akademischer Betreuer] [Rjasanow. "High order boundary element methods / Lucy Weggler. Betreuer: Sergej Rjasanow." Saarbrücken : Saarländische Universitäts- und Landesbibliothek, 2012. http://d-nb.info/1051586801/34.
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Rothauge, Kai. "The discrete adjoint method for high-order time-stepping methods." Thesis, University of British Columbia, 2016. http://hdl.handle.net/2429/60285.
Full textAbstract:
This thesis examines the derivation and implementation of the discrete adjoint method for several time-stepping methods. Our results are important for gradient-based numerical optimization in the context of large-scale model calibration problems that are constrained by nonlinear time-dependent PDEs. To this end, we discuss finding the gradient and the action of the Hessian of the data misfit function with respect to three sets of parameters: model parameters, source parameters and the initial condition. We also discuss the closely related topic of computing the action of the sensitivity matrix on a vector, which is required when performing a sensitivity analysis. The gradient and Hessian of the data misfit function with respect to these parameters requires the derivatives of the misfit with respect to the simulated data, and we give the procedures for computing these derivatives for several data misfit functions that are of use in seismic imaging and elsewhere. The methods we consider can be divided into two categories, linear multistep (LM) methods and Runge-Kutta (RK) methods, and several variants of these are discussed. Regular LM and RK methods can be used for ODE systems arising from the semi-discretization of general nonlinear time-dependent PDEs, whereas implicit-explicit and staggered variants can be applied when the PDE has a more specialized form. Exponential time-differencing RK methods are also discussed. The implementation of the associated adjoint time-stepping methods is discussed in detail. Our motivation is the application of the discrete adjoint method to high-order time-stepping methods, but the approach taken here does not exclude lower-order methods. All of the algorithms have been implemented in MATLAB using an object-oriented design and are written with extensibility in mind. For exponential RK methods it is illustrated numerically that the adjoint methods have the same order of accuracy as their corresponding forward methods, and for linear PDEs we give a simple proof that this must always be the case. The applicability of some of the methods developed here to pattern formation problems is demonstrated using the Swift-Hohenberg model.Science, Faculty of
Mathematics, Department of
Graduate
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Tam, Anita W. "High-order spatial discretization methods for the shallow water equations." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2001. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp04/NQ58942.pdf.
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Mattsson, Ken. "Summation-by-Parts Operators for High Order Finite Difference Methods." Doctoral thesis, Uppsala : Acta Universitatis Upsaliensis : Univ.-bibl. [distributör], 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-3434.
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Marais, Neilen. "Efficient high-order time domain finite element methods in electromagnetics." Thesis, Stellenbosch : University of Stellenbosch, 2009. http://hdl.handle.net/10019.1/1499.
Full textAbstract:
Thesis (DEng (Electrical and Electronic Engineering))--University of Stellenbosch, 2009.The Finite Element Method (FEM) as applied to Computational Electromagnetics (CEM), can beused to solve a large class of Electromagnetics problems with high accuracy and good computational efficiency. For solving wide-band problems time domain solutions are often preferred; while time domain FEM methods are feasible, the Finite Difference Time Domain (FDTD) method is more commonly applied. The FDTD is popular both for its efficiency and its simplicity. The efficiency of the FDTD stems from the fact that it is both explicit (i.e. no matrices need to be solved) and second order accurate in both time and space. The FDTD has limitations when dealing with certain geometrical shapes and when electrically large structures are analysed. The former limitation is caused by stair-casing in the geometrical modelling, the latter by accumulated dispersion error throughout the mesh. The FEM can be seen as a general mathematical framework describing families of concrete numerical method implementations; in fact the FDTD can be described as a particular FETD (Finite Element Time Domain) method. To date the most commonly described FETD CEM methods make use of unstructured, conforming meshes and implicit time stepping schemes. Such meshes deal well with complex geometries while implicit time stepping is required for practical numerical stability. Compared to the FDTD, these methods have the advantages of computational efficiency when dealing with complex geometries and the conceptually straight forward extension to higher orders of accuracy. On the downside, they are much more complicated to implement and less computationally efficient when dealing with regular geometries. The FDTD and implicit FETD have been combined in an implicit/explicit hybrid. By using the implicit FETD in regions of complex geometry and the FDTD elsewhere the advantages of both are combined. However, previous work only addressed mixed first order (i.e. second order accurate) methods. For electrically large problems or when very accurate solutions are required, higher order methods are attractive. In this thesis a novel higher order implicit/explicit FETD method of arbitrary order in space is presented. A higher order explicit FETD method is implemented using Gauss-Lobatto lumping on regular Cartesian hexahedra with central differencing in time applied to a coupled Maxwell’s equation FEM formulation. This can be seen as a spatially higher order generalisation of the FDTD. A convolution-free perfectly matched layer (PML) method is adapted from the FDTD literature to provide mesh termination. A curl conforming hybrid mesh allowing the interconnection of arbitrary order tetrahedra and hexahedra without using intermediate pyramidal or prismatic elements is presented. An unconditionally stable implicit FETD method is implemented using Newmark-Beta time integration and the standard curl-curl FEM formulation. The implicit/explicit hybrid is constructed on the hybrid hexahedral/tetrahedral mesh using the equivalence between the coupled Maxwell’s formulation with central differences and the Newmark-Beta method with Beta = 0 and the element-wise implicitness method. The accuracy and efficiency of this hybrid is numerically demonstrated using several test-problems.
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Baranda, Inok Antonio Filipe. "Investigation of high-order, high-resolution methods for axisymmetric turbulent jet usin ILEs." Thesis, Cranfield University, 2011. http://dspace.lib.cranfield.ac.uk/handle/1826/7317.
Full textAbstract:
This Philosophiae Doctor thesis presents the motivation, objectives and reasoning behind
the undertaken project. This research, study the capability of compressible Implicit Large
Eddy Simulation (ILES) in predicting free shear layer flows, under different free stream
regimes (Static and Co-flow jets).
Unsteady flows or jet flows are non-uniform in structure, temperature, pressure and
velocity. Turbulent mixing is of particular importance for the developing of this class of
flows. As a shear layer is formed immediately downstream of the jet exhaust, an early linear
instability involving exponential growth of small perturbations is introduced at the jet
discharge. Beyond this development stage, in the non-linear Kelvin-Helmholtz instability
region large scale vortex rings roll up, and their dynamics of formation and merging become
the defining feature of the transitional shear flow into fully developed regime. This
class of flows is particularly relevant to numerical predictions, as the extreme nature of
the flow in question is considered as a benchmark; however, experimental data should be
selected carefully as some results are controversial. To qualify the behaviour of unsteady
flows, some important criteria have been selected for the analysis of the flow quantities
at different regions of the flow field (average velocities, Reynolds stresses and dissipation
rates). A good estimation of high-order statistics (Standard Deviation, Skewness and
Kurtosis) correspond to mathematical steadiness and convergence of results. From the
physical point of view, similarity analysis between jet’s wake sections reveals physical
steadiness in results. Spectral analysis of the different regions of the flow field could be
used as a sign that the energy cascade is correctly predicted or efficiently enough since this
is where the smallest scales are usually present and which in effect require to be modelled
by the different numerical schemes. Cont/d.
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Baranda, Inok Antonio Filipe. "Investigation of high-order, high-resolution methods for axisymmetric turbulent jet using ILEs." Thesis, Cranfield University, 2011. http://dspace.lib.cranfield.ac.uk/handle/1826/7317.
Full textAbstract:
This Philosophiae Doctor thesis presents the motivation, objectives and reasoning behind the undertaken project. This research, study the capability of compressible Implicit Large Eddy Simulation (ILES) in predicting free shear layer flows, under different free stream regimes (Static and Co-flow jets). Unsteady flows or jet flows are non-uniform in structure, temperature, pressure and velocity. Turbulent mixing is of particular importance for the developing of this class of flows. As a shear layer is formed immediately downstream of the jet exhaust, an early linear instability involving exponential growth of small perturbations is introduced at the jet discharge. Beyond this development stage, in the non-linear Kelvin-Helmholtz instability region large scale vortex rings roll up, and their dynamics of formation and merging become the defining feature of the transitional shear flow into fully developed regime. This class of flows is particularly relevant to numerical predictions, as the extreme nature of the flow in question is considered as a benchmark; however, experimental data should be selected carefully as some results are controversial. To qualify the behaviour of unsteady flows, some important criteria have been selected for the analysis of the flow quantities at different regions of the flow field (average velocities, Reynolds stresses and dissipation rates). A good estimation of high-order statistics (Standard Deviation, Skewness and Kurtosis) correspond to mathematical steadiness and convergence of results. From the physical point of view, similarity analysis between jet’s wake sections reveals physical steadiness in results. Spectral analysis of the different regions of the flow field could be used as a sign that the energy cascade is correctly predicted or efficiently enough since this is where the smallest scales are usually present and which in effect require to be modelled by the different numerical schemes. The flow solver has been reviewed and improved. The former, a revised version of the reconstruction numerical schemes (WENO 5th and WENO 9th orders) has been performed and tested, the correspondent results have been compared against analytical data; the latter, correction of the method to compute the Jacobian of the transformation (singularity correction), by changing from the standard algebraic to geometric method, and augmented with transparent boundary condition, giving mathematical and physical meaning to the obtained results. The flow solver improvements and review have been verified and validated through simulations of a compressible Convergent-Divergent Nozzle (CDN), and the standard and a modified version of the Shock tube test cases, where the results are gained with minimal modelling effort. The study of numerical errors associated with the simulations of turbulent flows, for unsteady explicit time step predictions, have been performed and a new formula proposed. Ten different computational methods have been employed in the framework of ILES and computations have been performed for a jet flow configuration for which experimental data and DNS are available. It can be seen that a numerical error bar can be defined that takes into account the errors arising from the different numerical building blocks of the simulation method. The effects of different grids, Riemann solvers and numerical reconstruction schemes have been considered, however, the approach can be extended to take into account the effects of the initial and boundary conditions as well as subgrid scale modelling, if applicable. From the physical analysis several observations were established, revealing that differences in terms of jet’s core size are not an important parameter in terms of quantification and qualification of predictions, in other words, data should be reduced to the jet’s inertial reference system. Moreover, the comparative study has been performed to identify the differences between Riemann solvers (CBS and HLLC), Low Mach number Limiting/ Corrections (LMC), numerical reconstruction schemes (MUSCL and WENO) and spatial order of accuracy (2nd-order LMC, 5th-order LMC and 9th-order schemes) in combination with the most efficient cost/resolution discretization level (Medium mesh). The comparisons between results reveals for the Static and Co-Flow jets that the CBS MUSCL 5th-order LMC and the HLLC MUSCL 5th-order LMC as the most accurate schemes in predicting this class of flows, accordingly. Furthermore, the selected numerical methods show to be in accordance with the empirical (Static) and experimental (Co-flow) results in terms of resonance frequency and/or Strouhal number; also, the expected behaviour in terms of spectral energy decay rate throughout the jet’s central line is observed. To conclude the study of the Static jet case, a possible explanation for the jet’s buoyancy effect is presented.
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Ekelschot, Dirk. "Mesh adaptation strategies for compressible flows using a high-order spectral/hp element discretisation." Thesis, Imperial College London, 2016. http://hdl.handle.net/10044/1/43340.
Full textAbstract:
An accurate calculation of aerodynamic force coe cients for a given geometry is of fundamental importance for aircraft design. High-order spectral/hp element methods, which use a discontinuous Galerkin discretisation of the compressible Navier-Stokes equations, are now increasingly being used to improve the accuracy of flow simulations and thus the force coe cients. To reduce error in the calculated force coe cients whilst keeping computational cost minimal, I propose a p-adaptation method where the degree of the approximating polynomial is locally increased in the regions of the flow where low resolution is identified using a goal-based error estimator. We initially calculate a steady-state solution to the governing equations using a low polynomial order and use a goal-based error indicator to identify parts of the computational domain that require improved solution accuracy and increase the approximation order there. We demonstrate the cost-effectiveness of our method across a range of polynomial orders by considering a number of examples in two- and three-dimensions and in subsonic and transonic flow regimes. Reductions in both the number of degrees of freedom required to resolve the force coe cients to a given error, as well as the computational cost, are both observed in using the p-adaptive technique. In addition to the adjoint-based p-adaptation strategy, I propose a mesh deformation strategy that relies on a thermo-elastic formulation. The thermal-elastic formulation is initially used to control mesh validity. Two mesh quality indicators are proposed and used to illustrate that by heating up (expanding) or cooling down (contracting) the appropriate elements, an improved robustness of the classical mesh deformation strategy is obtained. The idea is extended to perform shock wave r-adaptation (adaptation through redistribution) for high Mach number flows. The mesh deformation strategy keeps the mesh topology unchanged, contracts the elements that cover the shock wave, keeps the number of elements constant and the computation as e cient as the unrefined case. The suitability of r-adaptation for shock waves is illustrated using internal and external compressible flow problems.
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Sherer, Scott Eric. "Investigation of high-order and optimized interpolation methods with implementation in a high-order overset grid fluid dynamics solver /." The Ohio State University, 2002. http://rave.ohiolink.edu/etdc/view?acc_num=osu1486462702465327.
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Stoyanov, Miroslav. "Reduced Order Methods for Large Scale Riccati Equations." Diss., Virginia Tech, 2009. http://hdl.handle.net/10919/27832.
Full textAbstract:
Solving the linear quadratic regulator (LQR) problem for partial differential equa-
tions (PDEs) leads to many computational challenges. The primary challenge comes
from the fact that discretization methods for PDEs typically lead to very large sys-
tems of differential or differential algebraic equations. These systems are used to form
algebraic Riccati equations involving high rank matrices. Although we restrict our
attention to control problems with small numbers of control inputs, we allow for po-
tentially high order control outputs. Problems with this structure appear in a number
of practical applications yet no suitable algorithm exists. We propose and analyze so-
lution strategies based on applying model order reduction methods to Chandrasekhar
equations, Lyapunov/Sylvester equations, or combinations of these equations. Our nu-
merical examples illustrate improvements in computational time up to several orders of
magnitude over standard tools (when these tools can be used). We also present exam-
ples that cannot be solved using existing methods. These cases are motivated by flow
control problems that are solved by computing feedback controllers for the linearized
system.Ph. D.
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Gargallo, Peiró Abel. "Validation and generation of curved meshes for high-order unstructured methods." Doctoral thesis, Universitat Politècnica de Catalunya, 2014. http://hdl.handle.net/10803/275977.
Full textAbstract:
In this thesis, a new framework to validate and generate curved high-order meshes for complex models is proposed. The main application of the proposed framework is to generate curved meshes that are suitable for finite element analysis with unstructured high-order methods. Note that the lack of a robust and automatic curved mesh generator is one of the main issues that has hampered the adoption of high-order methods in industry. Specifically, without curved high-order meshes composed by valid elements and that match the domain boundary, the convergence rates and accuracy of high-order methods cannot be realized. The main motivation of this work is to propose a framework to address this issue.
First, we propose a definition of distortion (quality) measure for curved meshes of any polynomial degree. The presented measures allow validating if a high-order mesh is suitable to perform finite element analysis with an unstructured high-order method. In particular, given a high-order element, the measures assign zero quality if the element is invalid, and one if the element corresponds to the selected ideal configuration (desired shape and nodal distribution). Moreover, we prove that if the quality of an element is not zero, the region where the determinant of the Jacobian is not positive has measure zero. We present several examples to illustrate that the proposed measures can be used to validate high-order isotropic and boundary layer meshes.
Second, we develop a smoothing and untangling procedure to improve the quality for curved high-order meshes. Specifically, we propose a global non-linear least squares minimization of the defined distortion measures. The distortion is regularized to allow untangling invalid meshes, and it ensures that if the initial configuration is valid, it never becomes invalid. Moreover, the optimization procedure preserves, whenever is possible, some geometrical features of the linear mesh such as the shape, stretching, straight-sided edges, and element size. We demonstrate through examples that the implementation of the optimization problem is robust and capable of handling situations in which the mesh before optimization contains a large number of invalid elements. We consider cases with polynomial approximations up to degree ten, large deformations of the curved boundaries, concave boundaries, and highly stretched boundary layer elements.
Third, we extend the definition of distortion and quality measures to curved high-order meshes with the nodes on parameterized surfaces. Using this definition, we also propose a smoothing and untangling procedure for meshes on CAD surfaces. This procedure is posed in terms of the parametric coordinates of the mesh nodes to enforce that the nodes are on the CAD geometry. In addition, we prove that the procedure is independent of the surface parameterization. Thus, it can optimize meshes on CAD surfaces defined by low-quality parameterizations.
Finally, we propose a new mesh generation procedure by means of an a posteriori approach. The approach consists of modifying an initial linear mesh by first, introducing high-order nodes, second, displacing the boundary nodes to ensure that they are on the CAD surface, and third, smoothing and untangling the resulting mesh to produce a valid curved high-order mesh. To conclude, we include several examples to demonstrate that the generated meshes are suitable to perform finite element analysis with unstructured high-order methods.
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Duru, Kenneth. "Perfectly Matched Layers and High Order Difference Methods for Wave Equations." Doctoral thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-173009.
Full textAbstract:
The perfectly matched layer (PML) is a novel technique to simulate the absorption of waves in unbounded domains. The underlying equations are often a system of second order hyperbolic partial differential equations. In the numerical treatment, second order systems are often rewritten and solved as first order systems. There are several benefits with solving the equations in second order formulation, though. However, while the theory and numerical methods for first order hyperbolic systems are well developed, numerical techniques to solve second order hyperbolic systems are less complete. We construct a strongly well-posed PML for second order systems in two space dimensions, focusing on the equations of linear elasto-dynamics. In the continuous setting, the stability of both first order and second order formulations are linearly equivalent. We have found that if the so-called geometric stability condition is violated, approximating the first order PML with standard central differences leads to a high frequency instability at most resolutions. In the second order setting growth occurs only if growing modes are well resolved. We determine the number of grid points that can be used in the PML to ensure a discretely stable PML, for several anisotropic elastic materials. We study the stability of the PML for problems where physical boundaries are important. First, we consider the PML in a waveguide governed by the scalar wave equation. To ensure the accuracy and the stability of the discrete PML, we derived a set of equivalent boundary conditions. Second, we consider the PML for second order symmetric hyperbolic systems on a half-plane. For a class of stable boundary conditions, we derive transformed boundary conditions and prove the stability of the corresponding half-plane problem. Third, we extend the stability analysis to rectangular elastic waveguides, and demonstrate the stability of the discrete PML. Building on high order summation-by-parts operators, we derive high order accurate and strictly stable finite difference approximations for second order time-dependent hyperbolic systems on bounded domains. Natural and mixed boundary conditions are imposed weakly using the simultaneous approximation term method. Dirichlet boundary conditions are imposed strongly by injection. By constructing continuous strict energy estimates and analogous discrete strict energy estimates, we prove strict stability.