Dissertations / Theses on the topic 'Dirichlet boundary condition'
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Yang, Xue. "Neumann problems for second order elliptic operators with singular coefficients." Thesis, University of Manchester, 2012. https://www.research.manchester.ac.uk/portal/en/theses/neumann-problems-for-second-order-elliptic-operators-with-singular-coefficients(2e65b780-df58-4429-89df-6d87777843c8).html.
Full textCheaytou, Rima. "Etude des méthodes de pénalité-projection vectorielle pour les équations de Navier-Stokes avec conditions aux limites ouvertes." Thesis, Aix-Marseille, 2014. http://www.theses.fr/2014AIXM4715.
Full textMotivated by solving the incompressible Navier-Stokes equations with open boundary conditions, this thesis studies the Vector Penalty-Projection method denoted VPP, which is a splitting method in time. We first present a literature review of the projection methods addressing the issue of the velocity-pressure coupling in the incompressible Navier-Stokes system. First, we focus on the case of Dirichlet conditions on the entire boundary. The numerical tests show a second-order convergence in time for both the velocity and the pressure. They also show that the VPP method is fast and cheap in terms of number of iterations at each time step. In addition, we established for the Stokes problem optimal error estimates for the velocity and pressure and the numerical experiments are in perfect agreement with the theoretical results. However, the incompressibility constraint is not exactly equal to zero and it scales as O(varepsilondelta t) where $varepsilon$ is a penalty parameter chosen small enough and delta t is the time step. Moreover, we deal with the natural outflow boundary condition. Three types of outflow boundary conditions are presented and numerically tested for the projection step. We perform quantitative comparisons of the results with those obtained by other methods in the literature. Besides, a theoretical study of the VPP method with outflow boundary conditions is stated and the numerical tests prove to be in good agreement with the theoretical results. In the last chapter, we focus on the numerical study of the VPP scheme with a nonlinear open artificial boundary condition modelling a singular load for the unsteady incompressible Navier-Stokes problem
Choulli, Mourad. "Identifiabilite d'un parametre dans une equation parabolique non lineaire monodimensionnelle." Toulouse 3, 1987. http://www.theses.fr/1987TOU30245.
Full textMatsui, Kazunori. "Asymptotic analysis of an ε-Stokes problem with Dirichlet boundary conditions." Thesis, Karlstads universitet, Institutionen för matematik och datavetenskap (from 2013), 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:kau:diva-71938.
Full textSTINT (DD2017-6936) "Mathematics Bachelor Program for Efficient Computations"
Couture, Chad. "Steady States and Stability of the Bistable Reaction-Diffusion Equation on Bounded Intervals." Thesis, Université d'Ottawa / University of Ottawa, 2018. http://hdl.handle.net/10393/37110.
Full textPERROTTA, Antea. "Differential Formulation coupled to the Dirichlet-to-Neumann operator for scattering problems." Doctoral thesis, Università degli studi di Cassino, 2020. http://hdl.handle.net/11580/75845.
Full textMarco, Alacid Onofre. "Structural Shape Optimization Based On The Use Of Cartesian Grids." Doctoral thesis, Universitat Politècnica de València, 2018. http://hdl.handle.net/10251/86195.
Full textLa competitividad en la industria actual impone la necesidad de generar nuevos y mejores diseños. El tradicional procedimiento de prueba y error, usado a menudo para el diseño de componentes mecánicos, ralentiza el proceso de diseño y produce diseños subóptimos, por lo que se necesitan nuevos enfoques para obtener una ventaja competitiva. Con el desarrollo del Método de los Elementos Finitos (MEF) en el campo de la ingeniería en la década de 1970, la optimización de forma estructural surgió como un área de aplicación prometedora. El entorno industrial cada vez más exigente implica ciclos cada vez más cortos de desarrollo de nuevos productos. Por tanto, la naturaleza iterativa de los procesos de optimización de forma, que supone el análisis de gran cantidad de geometrías (para las se han de usar modelos numéricos de gran tamaño a fin de limitar el efecto de los errores intrínsecamente asociados a las técnicas numéricas), puede incluso disuadir del uso de estas técnicas. Esta Tesis se centra en la formulación de una metodología 3D basada en el Cartesian-grid Finite Element Method (cgFEM) como herramienta para un análisis numérico eficiente y robusto. Esta metodología pertenece a la categoría de técnicas de discretización Immersed Boundary donde el concepto clave es extender el problema de análisis estructural a un dominio de aproximación, que contiene la frontera del dominio físico, cuya discretización (mallado) resulte sencilla. El uso de mallados cartesianos proporciona una plataforma natural para la optimización de forma estructural porque el dominio numérico está separado del modelo físico, que podrá cambiar libremente durante el procedimiento de optimización sin alterar la discretización subyacente. Otro argumento positivo reside en el hecho de que la generación de malla se convierte en una tarea trivial. La discretización del dominio numérico y su manipulación, en coalición con la eficiencia de una estructura jerárquica de datos, pueden ser explotados para ahorrar coste computacional. Sin embargo, estas ventajas pueden ser cuestionadas por varios problemas numéricos. Básicamente, el esfuerzo computacional se ha desplazado. Del uso de costosos algoritmos de mallado nos movemos hacia el uso de, por ejemplo, esquemas de integración numérica elaborados para poder capturar la discrepancia entre la frontera del dominio geométrico y la malla de elementos finitos que lo embebe. Para ello, utilizamos, por un lado, una formulación de estabilización para imponer condiciones de contorno y, por otro lado, hemos desarrollado nuevas técnicas para poder captar la representación exacta de los modelos geométricos. Para completar la implementación de un método de optimización de forma estructural se usa una formulación adjunta para derivar las sensibilidades de diseño requeridas por los algoritmos basados en gradiente. Las derivadas no son sólo variables requeridas para el proceso, sino una poderosa herramienta para poder proyectar información entre diferentes diseños o, incluso, proyectar la información para crear mallas h-adaptadas sin pasar por un proceso completo de refinamiento h-adaptativo. Las mejoras propuestas se reflejan en los ejemplos numéricos presentados en esta Tesis. Estos análisis muestran claramente el comportamiento superior de la tecnología cgFEM en cuanto a precisión numérica y eficiencia computacional. En consecuencia, el enfoque cgFEM se postula como una herramienta adecuada para la optimización de forma.
Actualment, amb la competència existent en la industria, s'imposa la necessitat de generar nous i millors dissenys . El tradicional procediment de prova i error, que amb freqüència es fa servir pel disseny de components mecànics, endarrereix el procés de disseny i produeix dissenys subòptims, pel que es necessiten nous enfocaments per obtindre avantatge competitiu. Amb el desenvolupament del Mètode dels Elements Finits (MEF) en el camp de l'enginyeria en la dècada de 1970, l'optimització de forma estructural va sorgir com un àrea d'aplicació prometedora. No obstant això, a causa de la natura iterativa dels processos d'optimització de forma, la manipulació dels models numèrics en grans quantitats, junt amb l'error de discretització dels mètodes numèrics, pot fins i tot dissuadir de l'ús d'aquestes tècniques (o d'explotar tot el seu potencial), perquè al mateix temps els cicles de desenvolupament de nous productes s'estan acurtant. Esta Tesi se centra en la formulació d'una metodologia 3D basada en el Cartesian-grid Finite Element Method (cgFEM) com a ferramenta per una anàlisi numèrica eficient i sòlida. Esta metodologia pertany a la categoria de tècniques de discretització Immersed Boundary on el concepte clau és expandir el problema d'anàlisi estructural a un domini d'aproximació fàcil de mallar que conté la frontera del domini físic. L'utilització de mallats cartesians proporciona una plataforma natural per l'optimització de forma estructural perquè el domini numèric està separat del model físic, que podria canviar lliurement durant el procediment d'optimització sense alterar la discretització subjacent. A més, un altre argument positiu el trobem en què la generació de malla es converteix en una tasca trivial, ja que la discretització del domini numèric i la seua manipulació, en coalició amb l'eficiència d'una estructura jeràrquica de dades, poden ser explotats per estalviar cost computacional. Tot i això, estos avantatges poden ser qüestionats per diversos problemes numèrics. Bàsicament, l'esforç computacional s'ha desplaçat. De l'ús de costosos algoritmes de mallat ens movem cap a l'ús de, per exemple, esquemes d'integració numèrica elaborats per poder capturar la discrepància entre la frontera del domini geomètric i la malla d'elements finits que ho embeu. Per això, fem ús, d'una banda, d'una formulació d'estabilització per imposar condicions de contorn i, d'un altra, desevolupem noves tècniques per poder captar la representació exacta dels models geomètrics Per completar la implementació d'un mètode d'optimització de forma estructural es fa ús d'una formulació adjunta per derivar les sensibilitats de disseny requerides pels algoritmes basats en gradient. Les derivades no són únicament variables requerides pel procés, sinó una poderosa ferramenta per poder projectar informació entre diferents dissenys o, fins i tot, projectar la informació per crear malles h-adaptades sense passar per un procés complet de refinament h-adaptatiu. Les millores proposades s'evidencien en els exemples numèrics presentats en esta Tesi. Estes anàlisis mostren clarament el comportament superior de la tecnologia cgFEM en tant a precisió numèrica i eficiència computacional. Així, l'enfocament cgFEM es postula com una ferramenta adient per l'optimització de forma.
Marco Alacid, O. (2017). Structural Shape Optimization Based On The Use Of Cartesian Grids [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/86195
TESIS
Coco, Armando. "Finite-Difference Ghost-Cell Multigrid Methods for Elliptic problems with Mixed Boundary Conditions and Discontinuous Coefficients." Doctoral thesis, Università di Catania, 2012. http://hdl.handle.net/10761/1107.
Full textEschke, Andy. "Analytical solution of a linear, elliptic, inhomogeneous partial differential equation with inhomogeneous mixed Dirichlet- and Neumann-type boundary conditions for a special rotationally symmetric problem of linear elasticity." Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2014. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-149965.
Full textJunior, Vanderley Alves Ferreira. "Problemas de valores de contorno envolvendo o operador biharmônico." Universidade de São Paulo, 2013. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-20032013-083331/.
Full textWe study the boundary value problem {\'DELTA POT. 2\' u = f in \'OMEGA\', \'BETA\' u = 0 in \'PARTIAL OMEGA\', in a bounded open \'OMEGA\'\'THIS CONTAINED\' \'R POT. N\' , under different boundary conditions. The questions of existence and positivity of solutions for this problem are addressed with Dirichlet, Navier and Steklov boundary conditions. We deduce natural boundary conditions through the study of a model for a plate with static load. We also study properties of the first eigenvalue of \'DELTA POT. 2\' and the semi-linear problem { \'DELTA POT. 2\' e o problema semilinear {\'DELTA POT. 2\' u = F (u) in \'OMEGA\' u = \'PARTIAL\'u SUP . \'PARTIAL\' v = 0 in \'PARTIUAL\' \'OMEGA\', for non-linearities like F(t) = \'l t l POT. p-1\', p \' DIFFERENT\' t, p > 0. For such problem we study existence and non-existence of solutions and its positivity
Bringmann, Philipp. "Adaptive least-squares finite element method with optimal convergence rates." Doctoral thesis, Humboldt-Universität zu Berlin, 2021. http://dx.doi.org/10.18452/22350.
Full textThe least-squares finite element methods (LSFEMs) base on the minimisation of the least-squares functional consisting of the squared norms of the residuals of first-order systems of partial differential equations. This functional provides a reliable and efficient built-in a posteriori error estimator and allows for adaptive mesh-refinement. The established convergence analysis with rates for adaptive algorithms, as summarised in the axiomatic framework by Carstensen, Feischl, Page, and Praetorius (Comp. Math. Appl., 67(6), 2014), fails for two reasons. First, the least-squares estimator lacks prefactors in terms of the mesh-size, what seemingly prevents a reduction under mesh-refinement. Second, the first-order divergence LSFEMs measure the flux or stress errors in the H(div) norm and, thus, involve a data resolution error of the right-hand side f. These difficulties led to a twofold paradigm shift in the convergence analysis with rates for adaptive LSFEMs in Carstensen and Park (SIAM J. Numer. Anal., 53(1), 2015) for the lowest-order discretisation of the 2D Poisson model problem with homogeneous Dirichlet boundary conditions. Accordingly, some novel explicit residual-based a posteriori error estimator accomplishes the reduction property. Furthermore, a separate marking strategy in the adaptive algorithm ensures the sufficient data resolution. This thesis presents the generalisation of these techniques to three linear model problems, namely, the Poisson problem, the Stokes equations, and the linear elasticity problem. It verifies the axioms of adaptivity with separate marking by Carstensen and Rabus (SIAM J. Numer. Anal., 55(6), 2017) in three spatial dimensions. The analysis covers discretisations with arbitrary polynomial degree and inhomogeneous Dirichlet and Neumann boundary conditions. Numerical experiments confirm the theoretically proven optimal convergence rates of the h-adaptive algorithm.
Chang, Ning-Shan, and 張寧珊. "Error minimized boundary treatment for Poisson problem with Dirichlet boundary condition." Thesis, 2018. http://ndltd.ncl.edu.tw/handle/3ke5mz.
Full text國立中興大學
應用數學系所
106
This thesis uses Legendre pseudospectral penalty formulation to construct the scheme for solving Poisson problem with Dirichlet boundary condition. In this study, the penalty parameters are determined analytically so that the error of numerical solution is minimized. We conducted the program to executive numerical experiments. The results obtain better accuracy, compared to the boundary conditions enforced strongly.
GUAN, WEN-GI, and 管文麒. "Positive radially symmetric solutions for senilinear elliptic partial differential equation on annualar with Dirichlet-Neumann boundary condition." Thesis, 1989. http://ndltd.ncl.edu.tw/handle/46273846456590631863.
Full textMutnuri, Venkata Satyanand. "Spectral Analysis of Wave Motion in Nonlocal Continuum Theories of Elasticity." Thesis, 2021. https://etd.iisc.ac.in/handle/2005/5437.
Full textLiaw, Cai-Pin, and 廖采頻. "The Null-Field Methods and Conservative schemes of Laplace’s Equation for Dirichlet and Mixed Types Boundary Conditions." Thesis, 2011. http://ndltd.ncl.edu.tw/handle/13133549896617901995.
Full text國立中山大學
應用數學系研究所
99
In this thesis, the boundary errors are defined for the NFM to explore the convergence rates, and the condition numbers are derived for simple cases to explore numerical stability. The optimal convergence (or exponential) rates are discovered numerically. This thesis is also devoted to seek better choice of locations for the field nodes of the FS expansions. It is found that the location of field nodes Q does not affect much on convergence rates, but do have influence on stability. Let δ denote the distance of Q to ∂S. The larger δ is chosen, the worse the instability of the NFM occurs. As a result, δ = 0 (i.e., Q ∈ ∂S) is the best for stability. However, when δ > 0, the errors are slightly smaller. Therefore, small δ is a favorable choice for both high accuracy and good stability. This new discovery enhances the proper application of the NFM. However, even for the Dirichlet problem of Laplace’s equation, when the logarithmic capacity (transfinite diameter) C_Γ = 1, the solutions may not exist, or not unique if existing, to cause a singularity of the discrete algebraic equations. The problem with C_Γ = 1 in the BEM is called the degenerate scale problems. The original explicit algebraic equations do not satisfy the conservative law, and may fall into the degenerate scale problem discussed in Chen et al. [15, 14, 16], Christiansen [35] and Tomlinson [42]. An analysis is explored in this thesis for the degenerate scale problem of the NFM. In this thesis, the new conservative schemes are derived, where an equation between two unknown variables must satisfy, so that one of them is removed from the unknowns, to yield the conservative schemes. The conservative schemes always bypasses the degenerate scale problem; but it causes a severe instability. To restore the good stability, the overdetermined system and truncated singular value decomposition (TSVD) are proposed. Moreover, the overdetermined system is more advantageous due to simpler algorithms and the slightly better performance in error and stability. More importantly, such numerical techniques can also be used, to deal with the degenerate scale problems of the original NFM in [15, 14, 16]. For the boundary integral equation (BIE) of the first kind, the trigonometric functions are used in Arnold [3], and error analysis is made for infinite smooth solutions, to derive the exponential convergence rates. In Cheng’s Ph. Dissertation [18], for BIE of the first kind the source nodes are located outside of the solution domain, the linear combination of fundamental solutions are used, error analysis is made only for circular domains. So far it seems to exist no error analysis for the new NFM of Chen, which is one of the goal of this thesis. First, the solution of the NFM is equivalent to that of the Galerkin method involving the trapezoidal rule, and the renovated analysis can be found from the finite element theory. In this thesis, the error boundary are derived for the Dirichlet, the Neumann problems and its mixed types. For certain regularity of the solutions, the optimal convergence rates are derived under certain circumstances. Numerical experiments are carried out, to support the error made.
Péloquin-Tessier, Hélène. "Partitions spectrales optimales pour les problèmes aux valeurs propres de Dirichlet et de Neumann." Thèse, 2014. http://hdl.handle.net/1866/11511.
Full textThere exist many ways to study the spectrum of the Laplace operator. This master thesis focuses on optimal spectral partitions of planar domains. More specifically, when imposing Dirichlet boundary conditions, we try to find partitions that achieve the infimum (over all the partitions of a given number of components) of the maximum of the first eigenvalue of the Laplacian in all the subdomains. This question has been actively studied in recent years by B. Helffer, T. Hoffmann-Ostenhof, S. Terracini and their collaborators, who obtained a number of important analytic and numerical results. In the present thesis we propose a similar problem, but for the Neumann boundary conditions. In this case, we are looking for spectral maximal, rather than minimal, partitions. More precisely, we attempt to find the maximum over all possible $k$-partitions of the minimum of the first non-zero Neumann eigenvalue of each component. This question appears to be more difficult than the one for the Dirichlet conditions, since many properties of Dirichlet eigenvalues, such as domain monotonicity, no longer hold in the Neumann case. Nevertheless, some results are obtained for 2-partitions of symmetric domains, and specific partitions are found analytically for rectangular domains. In addition, some general properties of optimal spectral partitions and open problems are also discussed.
Fokam, Jean-Marcel. "Forced vibrations via Nash-Moser iterations." Thesis, 2006. http://hdl.handle.net/2152/23960.
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Charron, Philippe. "Théorème de Pleijel pour l'oscillateur harmonique quantique." Thèse, 2015. http://hdl.handle.net/1866/13442.
Full textThe aim of this thesis is to explore the geometric properties of eigenfunctions of the isotropic quantum harmonic oscillator. We focus on studying the nodal domains, which are the connected components of the complement of the nodal (i.e. zero) set of an eigenfunction. Assume that the eigenvalues are listed in an increasing order. According to a fundamental theorem due to Courant, an eigenfunction corresponding to the $n$-th eigenvalue has at most $n$ nodal domains. This result has been originally proved for the Dirichlet eigenvalue problem on a bounded Euclidean domain, but it also holds for the eigenfunctions of a quantum harmonic oscillator. Courant's theorem was refined by Pleijel in 1956, who proved a more precise result on the asymptotic behaviour of the number of nodal domains of the Dirichlet eigenfunctions on bounded domains as the eigenvalues tend to infinity. In the thesis we prove a similar result in the case of the isotropic quantum harmonic oscillator. To do so, we use a combination of classical tools from spectral geometry (some of which were used in Pleijel’s original argument) with a number of new ideas, which include applications of techniques from algebraic geometry and the study of unbounded nodal domains.
Poliquin, Guillaume. "Géométrie nodale et valeurs propres de l’opérateur de Laplace et du p-laplacien." Thèse, 2015. http://hdl.handle.net/1866/13721.
Full textThe main topic of the present thesis is spectral geometry. This area of mathematics is concerned with establishing links between the geometry of a Riemannian manifold and its spectrum. The spectrum of a closed Riemannian manifold M equipped with a Riemannian metric g associated with the Laplace-Beltrami operator is a sequence of non-negative numbers tending to infinity. The square root of any number of this sequence represents a frequency of vibration of the manifold. This thesis consists of four articles all related to various aspects of spectral geometry. The first paper, “Superlevel sets and nodal extrema of Laplace eigenfunction”, is presented in Chapter 1. Nodal geometry of various elliptic operators, such as the Laplace-Beltrami operator, is studied. The goal of this paper is to generalize a result due to L. Polterovich and M. Sodin that gives a bound on the distribution of nodal extrema on a Riemann surface for a large class of functions, including eigenfunctions of the Laplace-Beltrami operator. The proof given by L. Polterovich and M. Sodin is only valid for Riemann surfaces. Therefore, I present a different approach to the problem that works for eigenfunctions of the Laplace-Beltrami operator on Riemannian manifolds of arbitrary dimension. The second and the third papers of this thesis are focused on a different elliptic operator, namely the p-Laplacian. This operator has the particularity of being non-linear. The article “Principal frequency of the p-Laplacian and the inradius of Euclidean domains” is presented in Chapter 2. It discusses lower bounds on the first eigenvalue of the Dirichlet eigenvalue problem for the p-Laplace operator in terms of the inner radius of the domain. In particular, I show that if p is greater than the dimension, then it is possible to prove such lower bound without any hypothesis on the topology of the domain. Such bounds have previously been studied by well-known mathematicians, such as W. K. Haymann, E. Lieb, R. Banuelos, and T. Carroll. Their papers are mostly oriented toward the case of the usual Laplace operator. The generalization of such lower bounds for the p-Laplacian is done in my third paper, “Bounds on the Principal Frequency of the p-Laplacian”. It is presented in Chapter 3. My fourth paper, “Wolf-Keller theorem of Neumann Eigenvalues”, is a joint work with Guillaume Roy-Fortin. This paper is concerned with the shape optimization problem in the case of the Laplace operator with Neumann boundary conditions. The main result of our paper is that eigenvalues of the Neumann boundary problem are not always maximized by disks among planar domains of given area. This joint work is presented in Chapter 4.
Exnerová, Vendula. "Bifurkace obyčejných diferenciálních rovnic z bodů Fučíkova spektra." Master's thesis, 2011. http://www.nusl.cz/ntk/nusl-300427.
Full textHeld, Joachim. "Ein Gebietszerlegungsverfahren für parabolische Probleme im Zusammenhang mit Finite-Volumen-Diskretisierung." Doctoral thesis, 2006. http://hdl.handle.net/11858/00-1735-0000-0006-B39E-E.
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