Electromagnetic flow meters are simple, rugged, non-invasive flow measuring instruments, which are extensively employed in many applications. In particular, they are ideally suited for the flow rate measurement of liquid metals, which serve as coolants in fast breeder reactors. In such applications, theoretical evaluation of the sensitivity turns out to be the best possible choice. Invariably, an evaluation of the associated electromagnetic fields forms the first step. However, due to the complexity of the problem, only numerical field computational approach would be feasible. In the pertinent literature, couple of e orts could be found which employ the well-known Galerkin Finite Element Method (GFEM) for the required task. However, GFEM is known to suffer from the numerical stability problem even at moderate flow rates. This problem is quite common in fluid dynamics area and several stabilization schemes have been suggested as a remedial measure. Among such schemes, the Streamline Upwinding Petrov Galerkin (SU/PG) method is a simple and widely employed approach. The same has been adopted in some of the moving conductor literatures for obtaining a stable solution. Nevertheless, in fluid dynamics literature, it has been shown that the SU/PG solution can suffer from distortion/peaking at the boundary. The remedial measures proposed are nonlinear in nature and hence are computationally demanding. Also, even the SU/PG scheme by itself requires significant additional computation for quadratic and higher order elements. Further, the value of stabilization parameter is not accurately known for 2D and 3D problems. The present work is basically an attempt to address the above problem for flow meter and other rectilinearly moving conductor problems. More specifically, but for the requirement of (graded) structured mesh along the flow direction, it basically aims to address a more general class of problems not just limited to the flow meter. Following the classical approach employed in fluid dynamics literature, first the problem is studied in its 1D form. It was observed that a relatively better performance of GFEM over FDM scheme is basically due to the difference in their Right Hand Side (RHS) terms, which represents the applied magnetic field. Taking clue from this, it was envisaged that a better insight to the numerical problem can be obtained by using the control system theory's transfer function approach. An application of FDM or GFEM to the 1D form of the governing equation, leads to flalge-braic equations with space variable in discrete form. Hence, a Z-transform based approach is employed to relate the applied magnetic field to the vector potential of the resulting reaction magnetic field. It is then shown that the presence of a pole at Z = -1 is basically responsible for the oscillations in the numerical solution. It is then proposed that by using the control systems pole-zero cancellation principle, stability can be brought into the numerical solution. This requires suitable modification of RHS terms in the discretised equations and accordingly, two novel schemes have been proposed which works within the framework of GFEM. In author's considered opinion, the use of Z-transform for analysing the stability of the numerical schemes and the idea of employing pole-zero cancellation to bring in stability, are first of its kind. In the first of the proposed schemes, the pole-zero cancellation is achieved by simply restating the input magnetic field in terms its vector potential. Solving the difference equations given by the application of FDM or GFEM to 1D version of the governing equation, it is analytically shown that the proposed scheme is absolutely stable at high flow rates. However, at midrange of flow rates there is a small error, which is analytically quantified. Then the scheme is applied to the original flow meter problem which has only axially varying applied field and the stability is demonstrated for an extensive range of flow rates. Note that the discretisation along the flow direction was restricted in the above exercise to graded regular mesh, which can readily be realised for problems involving rectilinearly moving conductors. In order to cater for more general cases in which the applied field varies in both axial and transverse directions, a second scheme is developed. Here the RHS term representing the input magnetic field is considered in a generic weighted average form. The required weights are evaluated by imposing apart from the need for an essential zero yielding term, the flux preservation and other symmetry conditions. The stability of this scheme is proven analytically for both 1D and 2D version of the problem using respectively, the 1D and 2D Z-transform based approaches. The analytical inferences are adequately validated with numerical exercises. Also, the small error present for the midrange of flow rates is analytically quantified. Then the second scheme is applied to the actual flow meter with a general magnetic field pro le. The proposed scheme is shown to be very stable and accurate even at very high flow rates. As before, the discretisation was restricted to graded regular mesh along the flow direction. By solving for the standard TEAM No. 9 benchmark problem, applicability of the second scheme for other rectilinearly moving conductor problem has been adequately demonstrated. Even though the problems considered in this work readily permits the use of a graded regular mesh along the flow direction, for the sake of completeness, discretisation with arbitrary quadrilateral and triangular mesh is also considered. The performance of the proposed schemes for such cases even though found to deteriorate, is still shown to be considerably better than the GFEM. In summary, this work has successfully proposed two novel, computationally effcient and stable GFEM schemes for the simulation of electromagnetic flow meters and other rectilin early moving conductor problems.

A triangular B-spline (DMS-spline)-based finite element method (TBS-FEM) is proposed along with possible enrichment through discontinuous Galerkin, continuous-discontinuous Galerkin finite element (CDGFE) and stabilization techniques. The developed schemes are also numerically explored, to a limited extent, for weak discretizations of a few second order partial differential equations (PDEs) of interest in solid mechanics. The presently employed functional approximation has both affine invariance and convex hull properties. In contrast to the Lagrangian basis functions used with the conventional finite element method, basis functions derived through n-th order triangular B-splines possess (n ≥ 1) global continuity. This is usually not possible with standard finite element formulations. Thus, though constructed within a mesh-based framework, the basis functions are globally smooth (even across the element boundaries). Since these globally smooth basis functions are used in modeling response, one can expect a reduction in the number of elements in the discretization which in turn reduces number of degrees of freedom and consequently the computational cost. In the present work that aims at laying out the basic foundation of the method, we consider only linear triangular B-splines. The resulting formulation thus provides only a continuous approximation functions for the targeted variables. This leads to a straightforward implementation without a digression into the issue of knot selection, whose resolution is required for implementing the method with higher order triangular B-splines. Since we consider only n = 1, the formulation also makes use of the discontinuous Galerkin method that weakly enforces the continuity of first derivatives through stabilizing terms on the interior boundaries. Stabilization enhances the numerical stability without sacrificing accuracy by suitably changing the weak formulation. Weighted residual terms are added to the variational equation, which involve a mesh-dependent stabilization parameter. The advantage of the resulting scheme over a more traditional mixed approach and least square finite element is that the introduction of additional unknowns and related difficulties can be avoided. For assessing the numerical performance of the method, we consider Navier’s equations of elasticity, especially the case of nearly-incompressible elasticity (i.e. as the limit of volumetric locking approaches). Limited comparisons with results via finite element techniques based on constant-strain triangles help bring out the advantages of the proposed scheme to an extent.

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The paper is concerned with the finite element resolution of layers appearing in singularly perturbed problems. A special anisotropic grid of Shishkin type is constructed for reaction diffusion problems. Estimates of the finite element error in the energy norm are derived for two methods, namely the standard Galerkin method and a stabilized Galerkin method. The estimates are uniformly valid with respect to the (small) diffusion parameter. One ingredient is a pointwise description of derivatives of the continuous solution. A numerical example supports the result. Another key ingredient for the error analysis is a refined estimate for (higher) derivatives of the interpolation error. The assumptions on admissible anisotropic finite elements are formulated in terms of geometrical conditions for triangles and tetrahedra. The application of these estimates is not restricted to the special problem considered in this paper.

This paper presents three new coupling methods for interior penalty discontinuous Galerkin finite element methods and boundary element methods. The new methods allow one to use discontinuous basis functions on the interface between the subdomains represented by the finite element and boundary element methods. This feature is particularly important when discontinuous Galerkin finite element methods are used. Error and stability analysis is presented for some of the methods. Numerical examples suggest that all three methods exhibit very similar convergence properties, consistent with available theoretical results.

The numerical solution of the convection-diffusion-reaction problem is considered in two and three dimensions. A stabilized finite element method of Galerkin/Least squares type accomodates diffusion-dominated as well as convection- and/or reaction- dominated situations. The resolution of boundary layers occuring in the singularly perturbed case is accomplished using anisotropic mesh refinement in boundary layer regions. In this paper, the standard analysis of the stabilized Galerkin method on isotropic meshes is extended to more general meshes with boundary layer refinement. Simplicial Lagrangian elements of arbitrary order are used.

A high-order hybrid discontinuous Galerkin finite element method (DG-FEM) is developed for multi-layered curved panels having large deformation and finite strain. The kinematics of the multi-layered shells is presented at first. The Jacobian matrix and its determinant are also calculated. The weak form of the DG-FEM is next presented. In this case, the discontinuous basis functions can be employed for the displacement basis functions. The implementation details of the nonlinear FEM are next presented. Then, the Consistent Orthogonal Basis Function Space is developed. Given the boundary conditions and structure configurations, there will be a unique basis function space, such that the mass matrix is an accurate diagonal matrix. Moreover, the Consistent Orthogonal Basis Functions are very similar to mode shape functions. Based on the DG-FEM, three dedicated finite elements are developed for the multi-layered pipes, curved stiffeners and multi-layered stiffened hydrofoils. The kinematics of these three structures are presented. The smooth configuration is also obtained, which is very important for the buckling analysis with large deformation and finite strain. Finally, five problems are solved, including sandwich plates, 2-D multi-layered pipes, 3-D multi-layered pipes, stiffened plates and stiffened multi-layered hydrofoils. Material and geometric nonlinearities are both considered. The results are verified by other papers' results or ANSYS. / Master of Science

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Mes travaux de thèse portent sur la construction de conditions aux limites absorbantes (CLAs) pour des problèmes de propagation d'ondes posés dans des milieux limités par des surfaces régulières. Ces conditions sont nouvelles car elles prennent en compte non seulement les ondes proagatives (comme la plupart des CLAs existantes) mais aussi les ondes évanescentes et rampantes. Elles sont donc plus performantes que les conditions existantes. De plus, elles sont facilement implémentables dans un schéma d'éléments finis de type Galerkine Discontinu (DG) et ne modifie pas la condition de stabilité de Courant-Friedrichs-Lewy (CFL). Ces CLAs ont été implémentées dans un code simulant la propagation des ondes acoustiques ainsi que dans un code simulant la propagation des ondes en régime harmonique. Les comparaisons réalisées entre les nouvelles conditions et celles qui sont les plus utilisées dans la littérature montrent que prendre en compte les ondes évanescentes et les ondes rampantes permet de diminuer les réflexions issues de la frontière artificielle et donc de rapprocher la frontière artificielle du bord de l'obstacle. On limite ainsi les coûts de calcul, ce qui est un des avantages de mes travaux. De plus, compte tenu du fait que les nouvelles CLAs sont écrites pour des frontières quelconques, elles permettent de mieux adapter le domaine de calcul à la forme de l'obstacle et permettent ainsi de diminuer encore plus les coûts de calcul numérique. / In my PhD, I have worked on the construction of absorbing boundary conditions (ABCs) designed for wave propagation problems set in domains bounded by regular surfaces. These conditions are new since they take into account not only propagating waves (as most of the existing ABCs) but also evanescent and creeping waves. Therefore, they outperform the existing ABCs. Moreover, they can be easily implemented in a discontinuous Galerkin finite element scheme and they do not change the Courant-Friedrichs-Lewy stability condition. These ABCs have been implemented in two codes that respectively simulate the propagation of acoustic waves and harmonic waves. The comparisons performed between these ABCs and the ABCs mostly used in the litterature show that when we take into account evanescent and creeping waves, we reduce the reflections coming from the artificial boundary. Therefore, thanks to these new ABCs, the artificial boundary can get closer to the obstacle. Consequently, we reduce the computational costs which is one of the advantages of my work. Moreover, since these new ABCs are written for any kind of boundary, we can adapt the shape of the computational domain and thus we can reduce again the computational costs.