Finite element methods for the Stokes system based on a Zienkiewicz type N-simplex

Hermite interpolation is increasingly showing to be a powerful numerical solution tool, as applied to different kinds of second order boundary value problems. In this work we present two Hermite finite element methods to solve viscous incompressible flows problems, in both two- and three-dimension space. In the two-dimensional case we use the Zienkiewicz triangle to represent the velocity field, and in the three-dimensional case an extension of this element to tetrahedra, still called a Zienkiewicz element. Taking as a model the Stokes system, the pressure is approximated with continuous functions, either piecewise linear or piecewise quadratic, according to the version of the Zienkiewicz element in use, that is, with either incomplete or complete cubics. The methods employ both the standard Galerkin or the Petrov–Galerkin formulation first proposed in Hughes et al. (1986) [18], based on the addition of a balance of force term. A priori error analyses point to optimal convergence rates for the PG approach, and for the Galerkin formulation too, at least in some particular cases. From the point of view of both accuracy and the global number of degrees of freedom, the new methods are shown to have a favorable cost-benefit ratio, as compared to velocity Lagrange finite elements of the same order, especially if the Galerkin approach is employed.

Modelos sin malla en simulación numérica de estructuras aeroespaciales

La presente Tesis Doctoral aborda la introducción de la Partición de Unidad de Bernstein en la forma débil de Galerkin para la resolución de problemas de condiciones de contorno en el ámbito del análisis estructural. La familia de funciones base de Bernstein conforma un sistema generador del espacio de funciones polinómicas que permite construir aproximaciones numéricas para las que no se requiere la existencia de malla: las funciones de forma, de soporte global, dependen únicamente del orden de aproximación elegido y de la parametrización o mapping del dominio, estando las posiciones nodales implícitamente definidas. El desarrollo de la formulación está precedido por una revisión bibliográfica que, con su punto de partida en el Método de Elementos Finitos, recorre las principales técnicas de resolución sin malla de Ecuaciones Diferenciales en Derivadas Parciales, incluyendo los conocidos como Métodos Meshless y los métodos espectrales. En este contexto, en la Tesis se somete la aproximación Bernstein-Galerkin a validación en tests uni y bidimensionales clásicos de la Mecánica Estructural. Se estudian aspectos de la implementación tales como la consistencia, la capacidad de reproducción, la naturaleza no interpolante en la frontera, el planteamiento con refinamiento h-p o el acoplamiento con otras aproximaciones numéricas. Un bloque importante de la investigación se dedica al análisis de estrategias de optimización computacional, especialmente en lo referente a la reducción del tiempo de máquina asociado a la generación y operación con matrices llenas. Finalmente, se realiza aplicación a dos casos de referencia de estructuras aeronáuticas, el análisis de esfuerzos en un angular de material anisotrópico y la evaluación de factores de intensidad de esfuerzos de la Mecánica de Fractura mediante un modelo con Partición de Unidad de Bernstein acoplada a una malla de elementos finitos. ABSTRACT This Doctoral Thesis deals with the introduction of Bernstein Partition of Unity into Galerkin weak form to solve boundary value problems in the field of structural analysis. The family of Bernstein basis functions constitutes a spanning set of the space of polynomial functions that allows the construction of numerical approximations that do not require the presence of a mesh: the shape functions, which are globally-supported, are determined only by the selected approximation order and the parametrization or mapping of the domain, being the nodal positions implicitly defined. The exposition of the formulation is preceded by a revision of bibliography which begins with the review of the Finite Element Method and covers the main techniques to solve Partial Differential Equations without the use of mesh, including the so-called Meshless Methods and the spectral methods. In this context, in the Thesis the Bernstein-Galerkin approximation is subjected to validation in one- and two-dimensional classic benchmarks of Structural Mechanics. Implementation aspects such as consistency, reproduction capability, non-interpolating nature at boundaries, h-p refinement strategy or coupling with other numerical approximations are studied. An important part of the investigation focuses on the analysis and optimization of computational efficiency, mainly regarding the reduction of the CPU cost associated with the generation and handling of full matrices. Finally, application to two reference cases of aeronautic structures is performed: the stress analysis in an anisotropic angle part and the evaluation of stress intensity factors of Fracture Mechanics by means of a coupled Bernstein Partition of Unity - finite element mesh model.

Application of the homogenization techniques to the determination of the effective flexure constants of plate structural systems

A Mindlin plate with periodically distributed ribs patterns is analyzed by using homogenization techniques based on asymptotic expansion methods. The stiffness matrix of the homogenized plate is found to be dependent on the geometrical characteristics of the periodical cell, i.e. its skewness, plan shape, thickness variation etc. and on the plate material elastic constants. The computation of this plate stiffness matrix is carried out by averaging over the cell domain some solutions of different periodical boundary value problems. These boundary value problems are defined in variational form by linear first order differential operators on the cell domain and the boundary conditions of the variational equation correspond to a periodic structural problem. The elements of the stiffness matrix of homogenized plate are obtained by linear combinations of the averaged solution functions of the above mentioned boundary value problems. Finally, an illustrative example of application of this homogenization technique to hollowed plates and plate structures with ribs patterns regularly arranged over its area is shown. The possibility of using in the profesional practice the present procedure to the actual analysis of floors of typical buildings is also emphasized.

Rapid solution of finite element equations on locally refined grids by multi-level methods /

"UILU-ENG 80 1712."

An iterative method for solving nonsymmetric linear systems with dynamic estimation of parameters /

Vita.

Numerical studies of the Stone algorithm and comparisons with Alternating Direction Implicit methods /

Thesis (M.S.)--University of Illinois, 1970.

A note on bifurcation from an interval

We study the global bifurcation of nonlinear Sturm-Liouville problems of the form -(pu')' + qu = lambda a(x)f(u), b(0)u(0) - c(0)u' (0) = 0, b(1)u(1) + c(1)u'(1) = 0 which are not linearizable in any neighborhood of the origin. (c) 2005 Published by Elsevier Ltd.

Dynamic simulation of chemical plant

A CSSL- type modular FORTRAN package, called ACES, has been developed to assist in the simulation of the dynamic behaviour of chemical plant. ACES can be harnessed, for instance, to simulate the transients in startups or after a throughput change. ACES has benefited from two existing simulators. The structure was adapted from ICL SLAM and most plant models originate in DYFLO. The latter employs sequential modularisation which is not always applicable to chemical engineering problems. A novel device of twice- round execution enables ACES to achieve general simultaneous modularisation. During the FIRST ROUND, STATE-VARIABLES are retrieved from the integrator and local calculations performed. During the SECOND ROUND, fresh derivatives are estimated and stored for simultaneous integration. ACES further includes a version of DIFSUB, a variable-step integrator capable of handling stiff differential systems. ACES is highly formalised . It does not use pseudo steady- state approximations and excludes inconsistent and arbitrary features of DYFLO. Built- in debug traps make ACES robust. ACES shows generality, flexibility, versatility and portability, and is very convenient to use. It undertakes substantial housekeeping behind the scenes and thus minimises the detailed involvement of the user. ACES provides a working set of defaults for simulation to proceed as far as possible. Built- in interfaces allow for reactions and user supplied algorithms to be incorporated . New plant models can be easily appended. Boundary- value problems and optimisation may be tackled using the RERUN feature. ACES is file oriented; a STATE can be saved in a readable form and reactivated later. Thus piecewise simulation is possible. ACES has been illustrated and verified to a large extent using some literature-based examples. Actual plant tests are desirable however to complete the verification of the library. Interaction and graphics are recommended for future work.

On the numerical solution of a Cauchy problem in an elastostatic half-plane with a bounded inclusion

We propose an iterative procedure for the inverse problem of determining the displacement vector on the boundary of a bounded planar inclusion given the displacement and stress fields on an infinite (planar) line-segment. At each iteration step mixed boundary value problems in an elastostatic half-plane containing the bounded inclusion are solved. For efficient numerical implementation of the procedure these mixed problems are reduced to integral equations over the bounded inclusion. Well-posedness and numerical solution of these boundary integral equations are presented, and a proof of convergence of the procedure for the inverse problem to the original solution is given. Numerical investigations are presented both for the direct and inverse problems, and these results show in particular that the displacement vector on the boundary of the inclusion can be found in an accurate and stable way with small computational cost.

A variational method and approximations of a Cauchy problem for elliptic equations

A Cauchy problem for general elliptic second-order linear partial differential equations in which the Dirichlet data in H½(?1 ? ?3) is assumed available on a larger part of the boundary ? of the bounded domain O than the boundary portion ?1 on which the Neumann data is prescribed, is investigated using a conjugate gradient method. We obtain an approximation to the solution of the Cauchy problem by minimizing a certain discrete functional and interpolating using the finite diference or boundary element method. The minimization involves solving equations obtained by discretising mixed boundary value problems for the same operator and its adjoint. It is proved that the solution of the discretised optimization problem converges to the continuous one, as the mesh size tends to zero. Numerical results are presented and discussed.

Determining the temperature from Cauchy data in corner domains

We consider the problem of reconstruction of the temperature from knowledge of the temperature and heat flux on a part of the boundary of a bounded planar domain containing corner points. An iterative method is proposed involving the solution of mixed boundary value problems for the heat equation (with time-dependent conductivity). These mixed problems are shown to be well-posed in a weighted Sobolev space.

A procedure for the reconstruction of a stochastic stationary temperature field

An iterative procedure is proposed for the reconstruction of a temperature field from a linear stationary heat equation with stochastic coefficients, and stochastic Cauchy data given on a part of the boundary of a bounded domain. In each step, a series of mixed well-posed boundary-value problems are solved for the stochastic heat operator and its adjoint. Well-posedness of these problems is shown to hold and convergence in the mean of the procedure is proved. A discretized version of this procedure, based on a Monte Carlo Galerkin finite-element method, suitable for numerical implementation is discussed. It is demonstrated that the solution to the discretized problem converges to the continuous as the mesh size tends to zero.

Quasi-separation of the biharmonic partial differential equation

In this paper, we consider analytical and numerical solutions to the Dirichlet boundary-value problem for the biharmonic partial differential equation on a disc of finite radius in the plane. The physical interpretation of these solutions is that of the harmonic oscillations of a thin, clamped plate. For the linear, fourth-order, biharmonic partial differential equation in the plane, it is well known that the solution method of separation in polar coordinates is not possible, in general. However, in this paper, for circular domains in the plane, it is shown that a method, here called quasi-separation of variables, does lead to solutions of the partial differential equation. These solutions are products of solutions of two ordinary linear differential equations: a fourth-order radial equation and a second-order angular differential equation. To be expected, without complete separation of the polar variables, there is some restriction on the range of these solutions in comparison with the corresponding separated solutions of the second-order harmonic differential equation in the plane. Notwithstanding these restrictions, the quasi-separation method leads to solutions of the Dirichlet boundary-value problem on a disc with centre at the origin, with boundary conditions determined by the solution and its inward drawn normal taking the value 0 on the edge of the disc. One significant feature for these biharmonic boundary-value problems, in general, follows from the form of the biharmonic differential expression when represented in polar coordinates. In this form, the differential expression has a singularity at the origin, in the radial variable. This singularity translates to a singularity at the origin of the fourth-order radial separated equation; this singularity necessitates the application of a third boundary condition in order to determine a self-adjoint solution to the Dirichlet boundary-value problem. The penultimate section of the paper reports on numerical solutions to the Dirichlet boundary-value problem; these results are also presented graphically. Two specific cases are studied in detail and numerical values of the eigenvalues are compared with the results obtained in earlier studies.

An iterative procedure for solving a Cauchy problem for second order elliptic equations

An iterative method for reconstruction of solutions to second order elliptic equations by Cauchy data given on a part of the boundary, is presented. At each iteration step, a series of mixed well-posed boundary value problems are solved for the elliptic operator and its adjoint. The convergence proof of this method in a weighted L2 space is included. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A procedure for the temperature reconstruction in corner domains from Cauchy data

An iterative procedure is proposed for the reconstruction of a stationary temperature field from Cauchy data given on a part of the boundary of a bounded plane domain where the boundary is smooth except for a finite number of corner points. In each step, a series of mixed well-posed boundary value problems are solved for the heat operator and its adjoint. Convergence is proved in a weighted L2-space. Numerical results are included which show that the procedure gives accurate and stable approximations in relatively few iterations.