Finite volume effects in chiral perturbation theory with twisted boundary conditions

We study the effects of a finite cubic volume with twisted boundary conditions on pseudoscalar mesons. We apply Chiral Perturbation Theory in the p-regime and introduce the twist by means of a constant vector field. The corrections of masses, decay constants, pseudoscalar coupling constants and form factors are calculated at next-to-leading order. We detail the derivations and compare with results available in the literature. In some case there is disagreement due to a different treatment of new extra terms generated from the breaking of the cubic invariance. We advocate to treat such terms as renormalization terms of the twisting angles and reabsorb them in the on-shell conditions. We confirm that the corrections of masses, decay constants, pseudoscalar coupling constants are related by means of chiral Ward identities. Furthermore, we show that the matrix elements of the scalar (resp. vector) form factor satisfies the Feynman–Hellman Theorem (resp. the Ward–Takahashi identity). To show the Ward–Takahashi identity we construct an effective field theory for charged pions which is invariant under electromagnetic gauge transformations and which reproduces the results obtained with Chiral Perturbation Theory at a vanishing momentum transfer. This generalizes considerations previously published for periodic boundary conditions to twisted boundary conditions. Another method to estimate the corrections in finite volume are asymptotic formulae. Asymptotic formulae were introduced by Lüscher and relate the corrections of a given physical quantity to an integral of a specific amplitude, evaluated in infinite volume. Here, we revise the original derivation of Lüscher and generalize it to finite volume with twisted boundary conditions. In some cases, the derivation involves complications due to extra terms generated from the breaking of the cubic invariance. We isolate such terms and treat them as renormalization terms just as done before. In that way, we derive asymptotic formulae for masses, decay constants, pseudoscalar coupling constants and scalar form factors. At the same time, we derive also asymptotic formulae for renormalization terms. We apply all these formulae in combination with Chiral Perturbation Theory and estimate the corrections beyond next-to-leading order. We show that asymptotic formulae for masses, decay constants, pseudoscalar coupling constants are related by means of chiral Ward identities. A similar relation connects in an independent way asymptotic formulae for renormalization terms. We check these relations for charged pions through a direct calculation. To conclude, a numerical analysis quantifies the importance of finite volume corrections at next-to-leading order and beyond. We perform a generic Analysis and illustrate two possible applications to real simulations.

Efficient adaptive methods based on adjoint equations and truncation error estimation for unstructured grids

El propósito de esta tesis es la implementación de métodos eficientes de adaptación de mallas basados en ecuaciones adjuntas en el marco de discretizaciones de volúmenes finitos para mallas no estructuradas. La metodología basada en ecuaciones adjuntas optimiza la malla refinándola adecuadamente con el objetivo de mejorar la precisión de cálculo de un funcional de salida dado. El funcional suele ser una magnitud escalar de interés ingenieril obtenida por post-proceso de la solución, como por ejemplo, la resistencia o la sustentación aerodinámica. Usualmente, el método de adaptación adjunta está basado en una estimación a posteriori del error del funcional de salida mediante un promediado del residuo numérico con las variables adjuntas, “Dual Weighted Residual method” (DWR). Estas variables se obtienen de la solución del problema adjunto para el funcional seleccionado. El procedimiento habitual para introducir este método en códigos basados en discretizaciones de volúmenes finitos involucra la utilización de una malla auxiliar embebida obtenida por refinamiento uniforme de la malla inicial. El uso de esta malla implica un aumento significativo de los recursos computacionales (por ejemplo, en casos 3D el aumento de memoria requerida respecto a la que necesita el problema fluido inicial puede llegar a ser de un orden de magnitud). En esta tesis se propone un método alternativo basado en reformular la estimación del error del funcional en una malla auxiliar más basta y utilizar una técnica de estimación del error de truncación, denominada _ -estimation, para estimar los residuos que intervienen en el método DWR. Utilizando esta estimación del error se diseña un algoritmo de adaptación de mallas que conserva los ingredientes básicos de la adaptación adjunta estándar pero con un coste computacional asociado sensiblemente menor. La metodología de adaptación adjunta estándar y la propuesta en la tesis han sido introducidas en un código de volúmenes finitos utilizado habitualmente en la industria aeronáutica Europea. Se ha investigado la influencia de distintos parámetros numéricos que intervienen en el algoritmo. Finalmente, el método propuesto se compara con otras metodologías de adaptación de mallas y su eficiencia computacional se demuestra en una serie de casos representativos de interés aeronáutico. ABSTRACT The purpose of this thesis is the implementation of efficient grid adaptation methods based on the adjoint equations within the framework of finite volume methods (FVM) for unstructured grid solvers. The adjoint-based methodology aims at adapting grids to improve the accuracy of a functional output of interest, as for example, the aerodynamic drag or lift. The adjoint methodology is based on the a posteriori functional error estimation using the adjoint/dual-weighted residual method (DWR). In this method the error in a functional output can be directly related to local residual errors of the primal solution through the adjoint variables. These variables are obtained by solving the corresponding adjoint problem for the chosen functional. The common approach to introduce the DWR method within the FVM framework involves the use of an auxiliary embedded grid. The storage of this mesh demands high computational resources, i.e. over one order of magnitude increase in memory relative to the initial problem for 3D cases. In this thesis, an alternative methodology for adapting the grid is proposed. Specifically, the DWR approach for error estimation is re-formulated on a coarser mesh level using the _ -estimation method to approximate the truncation error. Then, an output-based adaptive algorithm is designed in such way that the basic ingredients of the standard adjoint method are retained but the computational cost is significantly reduced. The standard and the new proposed adjoint-based adaptive methodologies have been incorporated into a flow solver commonly used in the EU aeronautical industry. The influence of different numerical settings has been investigated. The proposed method has been compared against different grid adaptation approaches and the computational efficiency of the new method has been demonstrated on some representative aeronautical test cases.

Finite volume modelling of an icefield with subglacial lake

Momentum, mass and energy balance laws provide the tools for the study of the evolution of an icefield covering a subglacial lake. The ice is described as a non-Newtonian fluid with a power-law constitutive relationship with temperature- and stress-dependent viscosity (Glen?s law) [1]. The phase transition mechanisms at the air/ice and ice/water interfaces yield moving boundary formulations, and lake hydrodynamics requires equation reduction for treating the turbulence.

Analysis of underground openings in rock by finite element methods : final report /

"This research was supported by the Advanced Research Projects Agency of the Department of Defense and was monitored by the Bureau of Mines ..."

A study of finite-difference methods as applied to hypersonic viscous shock-layer equations

"Contract no. Nonr 2653(00)."

On Finite Element Methods for 2nd order (semi–) periodic Eigenvalue Problems

We deal with a class of elliptic eigenvalue problems (EVPs) on a rectangle Ω ⊂ R^2 , with periodic or semi–periodic boundary conditions (BCs) on ∂Ω. First, for both types of EVPs, we pass to a proper variational formulation which is shown to fit into the general framework of abstract EVPs for symmetric, bounded, strongly coercive bilinear forms in Hilbert spaces, see, e.g., [13, §6.2]. Next, we consider finite element methods (FEMs) without and with numerical quadrature. The aim of the paper is to show that well–known error estimates, established for the finite element approximation of elliptic EVPs with classical BCs, hold for the present types of EVPs too. Some attention is also paid to the computational aspects of the resulting algebraic EVP. Finally, the analysis is illustrated by two non-trivial numerical examples, the exact eigenpairs of which can be determined.

Prediction of cold shuts using an unstructured mesh finite volume code

Abstract not available

Finite volume modelling of shrinkage porosity formation in metal casting

Abstract not available

A finite-volume computational mechanics framework for multi-physics coupled fluid-stress problems

Abstract not available

A finite volume approach to dynamic fluid structure interaction

Abstract not available

An augmented Lagrangian contact algorithm employing a vertex-based finite volume method

Abstract not available

Multiphysics modelling of materials processing using finite volume unstructured mesh procedures

Abstract not available