In this thesis, least-squares based finite element models (LSFEM) for the Poisson equation and Navier-Stokes equation are presented. The least-squares method is simple, general and reliable. Least-squares formulations offer several computational and theoretical advantages. The resulting coefficient matrix is symmetric and positive-definite. Using these formulations, the choice of approximating space is not subject to any compatibility condition. The Poisson equation is cast as a set of first order equations involving gradient of the primary variable as auxiliary variables for the mixed least-square finite element model. Equal order C0 continuous approximation functions is used for primary and auxiliary variables. Least-squares principle was directly applied to develop another model which requires C1continous approximation functions for the primary variable. Each developed model is compared with the conventional model to verify its performance. Penalty based least-squares formulation was implemented to develop a finite element for the Navier Stokes equations. The continuity equation is treated as a constraint on the velocity field and the constraint is enforced using the penalty method. Velocity gradients are introduced as auxiliary variables to get the first order equivalent system. Both the primary and auxiliary variables are interpolated using equal order C0 continuous, p-version approximation functions. Numerical examples are presented to demonstrate the convergence characteristics and accuracy of the method.

Sudarytas Puasono lygties sprendimo per „rutuliukų“ potencialus algoritmas. Šiuo metodu Puasono lygties sprendimo uždavinys suvedamas į tiesinių algebrinių lygčių sistemos sprendimą. Sudaryta ir išbandyta matematiniu paketu MATHCAD to sprendimo programa. Palyginti gauti sprendiniai su tais, kurie gaunami analiziškai, įvertintas gautų sprendinių tikslumas. Šį sprendimo būdą galima panaudoti realiems fizikiniams potencialams paskaičiuoti, turint galvoje realų potencialą su kuriuo realūs krūviai. / It consists of Poisson equation solution in the "ball" potential algorithm. In this method the Poisson equation, the decision problem are reduced to linear algebraic equations system solution. Created and tested a mathematical package MATHCAD program for that decision. Compared to solutions with those obtained analytically, estimated to obtain accurate solutions. This solution can be used to calculate the real physical potentials, given the real potential of the real workloads.

Thesis (PhD)--University of South Australia, 2001.

The present document is the final master thesis report written by Marc PILORGET,student at SUPAERO (home institution) and KTH (Royal Institute of Technology,Exchange University). This six months internship was done at DASSAULT AVIATION(Airframe engineering department) based in Saint-Cloud, France. It spanned from the 5thof July to the 23rd of December. The thesis work aims at developing an SPH (SmoothParticle Hydrodynamics) calculation method for ditching and implementing it in the finiteelement software ELFINI® developed by DASSAULT. Ditching corresponds to a phasewhen the aeroplane is touching the water. The problematic of ditching has always beenan area of interest for DASSAULT and the whole aeronautical industry. So far, only testsand simple analytical calculations have been performed. Most of the work was carried bythe NACA (National Advisory Committee for Aeronautics) in the late 70's. However in thepast decade, a new method for fluid-structure coupling problems has been developed. Itis called SPH. The basic principle is the following: the domain is represented by means ofparticles and each particle of fluid is treated separately and submitted to the Navier-Stokes equations. The particle is influenced by the neighbouring particles with a weightfunction depending on the distance between the two particles. Particles are also placed atthe interface solid-fluid: they are called limit particles. The final purpose of this SPHmethod is to access to the structural response of an aircraft when ditching. The crucialinterest of such a method compared to methods used so far is the absence of mesh. Theanalysis of large deformation problems by the finite element method may require thecontinuous remeshing of the domain to avoid the breakdown of the calculation due toexcessive mesh distortion. When considering ditching or other large deformationsproblems, the mesh generation is a far more time-consuming task than the constructionand solution of a discrete set of equations. For DASSAULT-AVIATION, the long termobjective is to get a numerical tool able to model ditching. The SPH method is used tosolve the equations for the fluid and is coupled with a finite element method for thestructure. So far, the compressible solver for 2D geometries has been implemented.Tests are going to be performed to ensure the program’s robustness. Then theincompressible solver for 2D geometries will be studied both theoretically andnumerically.

This thesis describes a project in which algorithms are developed for the rapid and accurate solution of Poisson's equation in the presence of a dielectric boundary and multiple point charges. These algorithms are then used to perform Brownian dynamics simulations on realistic models of biological ion channels. An iterative method of solution, in which the dielectric boundary is tiled with variable sized surface charge sectors, provides the flexibility to deal with arbitrarily shaped boundaries, but is too slow to perform Brownian dynamics. An analytical solution is derived, which is faster and more accurate, but only works for a toroidal boundary. Finally, a method is developed of pre-calculating solutions to Poisson's equation and storing them in tables. The solution for a particular configuration of ions in the channel can then be assembled by interpolation from the tables and application of the principle of superposition. This algorithm combines the flexibility of the iterative method with greater speed even than the analytical method, and is fast enough that channel conductance can be predicted. The results of simulations for a model single-ion channel, based on the acetylcholine receptor channel, show that the narrow pore through the low dielectric strength medium of the protein creates an energy barrier which restricts the permeation of ions. They further show that this barrier can be removed by dipoles in the neck of the channel, but that the barrier is not removed by shielding by counter-ions. The results of simulations for a model multi-ion channel, based on a bacterial potassium channel, show that the model channel has conductance characteristics similar to those of real potassium channels. Ions appear to move through the model multi-ion channel via rapid transitions between a series of semi-stable states. This observation suggests a possible physical basis for the reaction rate theory of channel conductance, and opens up an avenue for future research.

This paper is concerned with mesh refinement techniques for treating elliptic boundary value problems in domains with re- entrant edges and corners, and focuses on numerical experiments. After a section about the model problem and discretization strategies, their realization in the experimental code FEMPS3D is described. For two representative examples the numerically determined error norms are recorded, and various mesh refinement strategies are compared.

In this thesis we will investigate how the SBP-SAT finite difference method behave with and without an interface. As model problem, we consider the heat equation with piecewise constant coefficients. The thesis is split in two main parts. In the first part we look at the heat equation in one-dimension, and in the second part we expand the problem to a two-dimensional domain. We show how the SAT-parameters are chosen such that the scheme is dual consistent and stable. Then, we perform numerical experiments, now looking at the static case. In the one-dimensional case we see that the second order SBP-SAT method with an interface converge with an order of two, while the second order SBP-SAT method without an interface converge with an order of one.

This paper is concerned with mesh refinement techniques for treating elliptic boundary value problems in domains with re- entrant edges and corners, and focuses on numerical experiments. After a section about the model problem and discretization strategies, their realization in the experimental code FEMPS3D is described. For two representative examples the numerically determined error norms are recorded, and various mesh refinement strategies are compared.