Regularity of minimizing maps with values in S2\mathbb{S}^2 and some numerical simulations

International audienc

Prediction of the ultimate behaviour of tubular joints in offshore jacket structures using nonlinear finite element methods

PhD ThesisTubular joints are of great importance in offshore jacket structures. This thesis examines the ultimate state behaviour of tubular joints in offshore structures. In particular, the validity of a nonlinear finite element method was investigated and it was subsequently used to determine the ultimate load behaviour of a range of tubular joints. A geometrically nonlinear, eight node isoparan-letric shell finite element program is developed which allows six degrees of freedom per node. The material laws in the model include elastic and elastoplastic multilaver solution with integration across the thickness. Strain hardening elfects can be included. The nonlinear solution strategies are based on the Newton-Raphson Method. The load is applied in increments where for each step, equilibrium iterations are carried out to establish equilibrium, subject to a given error criterion. To cross the limit point and to select load increments, iterative solution strategies such as the arc length and autoniatic load increments method are adopted. To analyse tubular joints, a simple inesh generator has been developed. Struc- Cural symmetry is exploited to reduce the number of elements. The hibular joijil. is divided into a few regions and by means of a blending function. each region is discret, ised into a joints have been analysed using this finite element method. The numerical results have been compared with experimental tests undertak- en by the Wimpey Offshore Laboratory using large scale specimens. Finally, the applicabiliy of the nonlinear finite element developed here is briefly discussed all potential areas of research in the ultimate behaviour of tubular joints are proposed.British Counci

Numerical simulation of incompressible fluid flow by the spectral element method

Tato diplomová práce prezentuje metodu spektrálních prvků. Tato metoda je použita k řešení stacionárního 2-D laminárního proudění Newtonovské nestlačitelné tekutiny. Proudění je popsáno stacionarní Navier-Stokesovou rovnicí. Dohromady s okrajovou pod- mínkou tvoří Navier-Stokesův problém. Na slabou formulaci této úlohy je aplikována metoda spektrálních prvků. Touto discretizací se získá soustava nelineárních rovnic. K obrdžení lineární soustavy je použita Newtonova iterační metoda. Podorobný algorit- mus tvoří jádro Navier-Stokeseva solveru, který je naprogramován v Matlabu. Na závěr jsou pomocí tohoto solveru řešeny dva příklady: proudění v kavitě a obtékání válce. Přík- lady jsou řešeny pro různé Reynoldsovy čísla. První od 1 do 1000 a druhý od 1 do 100.The thesis presents the spectral element method and its application to a steady 2-D laminar flow of an incompressible Newtonian fluid. Main features of this method are presented in the thesis. The flow is governed by the steady Navier-Stokes equation. Together with boundary data they form the steady Navier-Stokes problem. Its weak form is a starting point for the method. A space discretization is applied and it results into a nonlinear system of equations. Due to this, the nonlinearity has to be treated. To obtain a linear system of equations is the Newton iteration method used. This algorithm forms the kernel of a Navier-Stokes solver that is implemented in Matlab. Finally, there are presented two examples: the lid driven cavity flow and the flow over a cylinder. The first one is solved for Reynolds numbers from 1 to 1000 and the second one for Reynolds numbers from 1 to 100.

Quadratic programming as an extension of linear programming

In the past two decades Mathematical Programming has come to occupy a place of importance in Economic Studies and in Operations Research. Roughly speaking, Mathematical Programming is the analysis of problems of the type: "Find the maximum of a function, when the variables are subject to inequality and equality constraints". The term "Linear Programming" corresponds to the case where, the function to be maximized (the so called objective function) and the equality and inequality constraints are linear. The term "Non-Linear Programming" should then become self-defined. With the introduction of Dantzig's Simplex Method, Linear Programming has become an everyday technique. The same, we regret to say, is not true for Nonlinear Programming because this subject is broader and much more difficult to unify than that of Linear Programming. In fact at present there does exist any unifying theory for Nonlinear Programming. However, we feel that research on this field is gathering tremendous momentum and that in the not too distant future Nonlinear Programming will become both a practical and fundamental tool in many spheres of Science. One of the subject matters of Nonlinear Programming is what we came to call "Quadratic Programming". This name is restricted to the specific problem of maximizing or minimizing a quadratic objective function f(X) = CX + X'DX, where CX is a linear form and X'DX a quadratic form, subject to linear constraints. Historically, Quadratic Programming was the first venture into the theory of Nonlinear Programming. More specifically it is the purpose of this thesis to: (i) Present a unified and simple treatment of the Theory of Concave (Convex) Quadratic Programming (in no way will mathematical rigour be sacrificed for simplicity). (ii) Present a collection of "Simplicial Methods" for solving quadratic programming problems, which are but extensions of the Simplex Method ( for Linear Programming, whose "accuracy" and "convergence" make them completely self-sufficient for the solution of any type of concave (convex) quadratic programming problems

Solution of a few nonlinear problems in aerodynamics by the finite elements and functional least squares methods

The numerical simulation of the transonic flows of idealized fluids and of incompressible viscous fluids, by the nonlinear least squares methods is presented. The nonlinear equations, the boundary conditions, and the various constraints controlling the two types of flow are described. The standard iterative methods for solving a quasi elliptical nonlinear equation with partial derivatives are reviewed with emphasis placed on two examples: the fixed point method applied to the Gelder functional in the case of compressible subsonic flows and the Newton method used in the technique of decomposition of the lifting potential. The new abstract least squares method is discussed. It consists of substituting the nonlinear equation by a problem of minimization in a H to the minus 1 type Sobolev functional space

A parallel finite element algorithm for 3D incompressible flow in velocity-vorticity form

In the last decade, developments and advancement in computer technology, especially the availability of the massively parallel machine, have escalated the numerical treatment of complex fluid flow problems to a new height. Numerical simulation of incompressible viscous fluid flow, often associated with practical industrial and environmental situations, is receiving intense scrutiny to perform in the promising distributed parallel computing environment. On the other hand, the field of computational fluid dynamics continues to explore and exploit unified and versatile formulations, in contention with the notorious divergence-free velocity field constraint, for incompressible Navier-Stokes equations that encompass fluid flow in two- and threedimensions. The velocity-vorticity formulation for the incompressible Navier-Stokes equations is chosen with the full extent to resolve these issues. In the present dissertation, a new finite element implementation for two- and three-dimensional incompressible fluid flow is developed in the velocity-vorticity form. Pressure is eliminated analytically by taking the curl of the momentum equations, and vorticity is introduced as the active variable. The formulation consists of the three derived vorticity transport equations in conjunction with three velocity Poisson equations. Satisfaction of the continuity constraint is cast onto the specific treatment of the kinematic vorticity boundary condition for the no slip wall. A divergence-free solution is guaranteed with equal order finite element interpolation functions for all state variables