Three-dimensional Axisymmetric Domains Research Articles

Torus-like solutions for the Landau-de Gennes model. Part II: Topology of [formula omitted]-equivariant minimizers

We study energy minimization of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in a three-dimensional axisymmetric domain and in a restricted class of S1-equivariant (i.e., axially symmetric) configurations. We assume smooth and nonvanishing S1-equivariant (e.g. homeotropic) Dirichlet boundary condition and a physically relevant norm constraint (the Lyuksyutov constraint) in the interior. Relying on our previous results in the nonsymmetric setting [16], we prove partial regularity of minimizers away from a possible finite set of interior singularities located on the symmetry axis. For a suitable class of domains and boundary data, we show that for smooth minimizers (torus solutions) the level sets of the signed biaxiality are generically finite unions of tori of revolution. Concerning nonsmooth minimizers (split solutions), we characterize their asymptotic behavior around any singular point in terms of explicit S1-equivariant harmonic maps into S4, whence the generic level sets of the signed biaxiality contains invariant topological spheres. Finally, in the model case of a nematic droplet, we provide existence of torus solutions, at least when the boundary data are suitable uniaxial deformations of the radial anchoring, and existence of split solutions for boundary data which are suitable linearly full harmonic spheres.

The Fourier-finite-element method for Poisson’s equation in three-dimensional axisymmetric domains with edges: Computing the edge flux intensity functions

AbstractIn [Nkemzi and Jung, 2013] explicit extraction formulas for the computation of the edge flux intensity functions for the Laplacian at axisymmetric edges are presented. The present paper proposes a new adaptation for the Fourier-finite-element method for efficient numerical treatment of boundary value problems for the Poisson equation in axisymmetric domains Ω̂ ⊂ ℝ3 with edges. The novelty of the method is the use of the explicit extraction formulas for the edge flux intensity functions to define a postprocessing procedure of the finite element solutions of the reduced boundary value problems on the two-dimensional meridian of Ω̂. A priori error estimates show that the postprocessing finite element strategy exhibits optimal rate of convergence on regular meshes. Numerical experiments that validate the theoretical results are presented.

A posteriori error estimates of finite element method for the time-dependent Darcy problem in an axisymmetric domain

We consider the time dependent Darcy problem in a three-dimensional axisymmetric domain and, by writing the Fourier expansion of its solution with respect to the angular variable, we observe that each Fourier coefficient satisfies a system of equations on the meridian domain. We propose a discretization of these equations in the case of general solution. This discretization relies on a backward Euler’s scheme for the time variable and finite elements for the space variables. We prove a posteriori error estimates that allow for an efficient adaptivity strategy both for the time steps and the meshes. Computations for an example with a known solution are presented which support the a posteriori error estimate.

APPLICATION OF COLLOCATION BEM FOR AXISYMMETRIC TRANSMISSION PROBLEMS IN ELECTRO- AND MAGNETOSTATICS

This paper considers the numerical solution of boundary integral equations for an exterior transmission problem in a three-dimensional axisymmetric domain. The resulting potential problem is formulated in a meridian plane as the second kind integral equation for a boundary potential and the first kind integral equation for a boundary flux. The numerical method is an axisymmetric collocation with equal order approximations of the boundary unknowns on a polygonal boundary. The complete elliptic integrals of the kernels are approximated by polynomials. An asymptotic kernels behavior is analyzed for accurate numerical evaluation of integrals. A piecewise-constant midpoint collocation and a piecewise-linear nodal collocation on a circular arc and on its polygonal interpolation are used for test computations on uniform meshes. We analyze empirically the influence of the polygonal boundary interpolation to the accuracy and the convergence of the presented method. We have found that the polygonal boundary interpolation does not change the convergence behavior on the smooth boundary for the piecewise-constant and the piecewise-linear collocation.

Kansa-RBF Algorithms for Elliptic Problems in Axisymmetric Domains

We employ a Kansa-radial basis function method for the numerical solution of elliptic boundary value problems in three-dimensional axisymmetric domains. We consider problems governed by the Poisson equation, the inhomogeneous biharmonic equation, and the inhomogeneous Cauchy--Navier equations. Appropriate discretizations lead to linear systems, the coefficient matrices of which possess block circulant structures. These systems are then solved efficiently by means of matrix decomposition algorithms and fast Fourier transforms. Methods for choosing appropriate values of the shape parameter are also proposed. The effectiveness of the proposed algorithms is demonstrated by considering several numerical examples.

Mortar spectral method in axisymmetric domains

We consider the Laplace equation posed in a three-dimensional axisymmetric domain. We reduce the original problem by a Fourier expansion in the angular variable to a countable family of two-dimensional problems. We decompose the meridian domain, assumed polygonal, in a finite number of rectangles and we discretize by a spectral method. Then we describe the main features of the mortar method and use the algorithm Strang Fix to improve the accuracy of our discretization.

The Fourier-Nitsche-mortaring for elliptic problems with reentrant edges

The Fourier method is combined with the Nitsche-finite-element method (as a mortar method) and applied to the Dirichlet problem of the Poisson equation in three-dimensional axisymmetric domains with reentrant edges generating singularities. The approximating Fourier method yields a splitting of the 3D problem into a set of 2D problems on the meridian plane of the given domain. For solving the 2D problems bearing corner singularities, the Nitsche-finite-element method with non-matching meshes and mesh grading near reentrant corners is applied. Using the explicit representation of some singularity function of non-tensor product type, the rate of convergence of the Fourier-Nitsche-mortaring is estimated in some H1-like norm as well as in the L2-norm for weak regularity of the solution. Finally, some numerical results are presented.

A TRUNCATED FOURIER/FINITE ELEMENT DISCRETIZATION OF THE STOKES EQUATIONS IN AN AXISYMMETRIC DOMAIN

We consider the Stokes problem in a three-dimensional axisymmetric domain and, by writing the Fourier expansion of its solution with respect to the angular variable, we observe that each Fourier coefficient satisfies a system of equations on the meridian domain. We propose a discretization of this problem which combines Fourier truncation and finite element methods applied to each of the two-dimensional systems. We give the detailed a priori and a posteriori analyses of the discretization and present some numerical experiments which are in good agreement with the analysis.

An Efficient Discretization of the Navier–Stokes Equations in an Axisymmetric Domain. Part 1: The Discrete Problem and its Numerical Analysis

Any solution of the Navier---Stokes equations in a three-dimensional axisymmetric domain admits a Fourier expansion with respect to the angular variable, and it can be noted that each Fourier coefficient satisfies a variational problem on the meridian domain, all problems being coupled due to the nonlinear convection term. We propose a discretization of these equations which combines Fourier truncation and finite element methods applied to each two-dimensional system. We perform the a priori and a posteriori analysis of this discretization.

On the solution of Maxwell's equations in axisymmetric domains with edges

In this paper we present the basic mathematical tools for treating boundary value problems for the Maxwell equations in three-dimensional axisymmetric domains with reentrant edges using the Fourier-finite-element method. We consider both the classical and the regularized time-harmonic Maxwell equations subject to perfect conductor boundary conditions. The partial Fourier decomposition reduces the three-dimensional boundary value problem into an infinite sequence of two-dimensional boundary value problems in the plane meridian domain of the axsiymmetric domain. Here, suitable weighted Sobolev spaces that characterize the solutions of the reduced problems are given, and their trace properties on the rotational axis are proved. In these spaces, it is proved that the reduced problems are well posed, and the asymptotic behavior of the solutions near reentrant comers of the meridian domain is explicitly described by suitable singularity functions. Finally, a finite number of the two-dimensional problems is considered and treated using H 1 -conforming finite elements. An approximation of the solution of the three-dimensional problem is obtained by Fourier synthesis. For domains with reentrant edges, the singular field method is employed to compensate the singular behavior of the solutions of the reduced problems. Emphases are given to convergence analysis of the combined approximations in H 1 under different regularity assumptions on the solution.

Singularities in elasticity and their treatment with Fourier series

AbstractThis paper analyses the regularity of the weak solution of the linear elasticity system in three‐dimensional axisymmetric domains with reentrant edges under prescribed traction on the boundary by means of Fourier series. Using partial Fourier analysis with respect to one space direction (rotational angle), the three‐dimensional boundary value problem (BVP) is decomposed into a sequence of decoupled two‐dimensional boundary value problems on the meridian of the axisymmetric domain. The splitting of the 2D solutions near corners of the meridian domain into regular and singular parts provides coefficients from which the 3D edge singularity functions (generalized stress intensity factors) are derived. Two types of singularity functions are presented, namely, a tensorial type, which needs more smoothness assumption on the right hand side and a non‐tensorial type, which does not demand any further smoothness assumptions.

An h–p adaptive spectral element method for Stokes flow

An h–p adaptive spectral element method for three-dimensional axisymmetric domains featuring an element residual a posteriori error indicator, equidistribution of the error amongst the elements in the initial mesh and simultaneous h and p refinements is presented. The mortar element method is used to efficiently handle the nonconforming decompositions that arise in the adaptive process. Results obtained by applying the adaptive strategy to Stokes flow for different orders of regularity of the righthand side function are compared with the results obtained using a uniform p refinement. The results obtained show the advantages of using the proposed strategy.

Partial Fourier approximation of the Lamé equations in axisymmetric domains

In this paper, we study the partial Fourier method for treating the Lame equations in three-dimensional axisymmetric domains subjected to non-axisymmetric loads. We consider the mixed boundary value problem of the linear theory of elasticity with the displacement u, the body force f ϵ (L2)3 and homogeneous Dirichlet and Neumann boundary conditions. The partial Fourier decomposition reduces, without any error, the three-dimensional boundary value problem to an infinite sequence of two-dimensional boundary value problems, whose solutions un (n = 0, 1, 2,…) are the Fourier coefficients of u. This process of dimension reduction is described, and appropriate function spaces are given to characterize the reduced problems in two dimensions. The trace properties of these spaces on the rotational axis and some properties of the Fourier coefficients un are proved, which are important for further numerical treatment, e.g. by the finite-element method. Moreover, generalized completeness relations are described for the variational equation, the stresses and the strains. The properties of the resulting system of two-dimensional problems are characterized. Particularly, a priori estimates of the Fourier coefficients un and of the error of the partial Fourier approximation are given. Copyright © 1999 John Wiley & Sons, Ltd.

Fourier-Finite-element Approximation of Elliptic Interface Problems in Axisymmetric Domains

The paper deals with the Fourier-finite-element method (FFEM), which combines the approximate Fourier method with the finite-element method, and its application to Poisson-like equations -pΔ 3 u = f in three-dimensional axisymmetric domains Ω. Here, p is a piecewise constant coefficient having a jump at some axisymmetric interface. Special emphasis is given to estimates of the Fourier-finite-element error in the Sobolev space H 1 (Ω), if the interface is smooth or if it meets the boundary of Ω at some edge. In general, the solution u contains a singularity at the interface, which is described by a tensor product representation and treated numerically by appropriate mesh grading in the meridian plane of Ω. The rate of convergence of the combined approximation in H 1 (Ω) is proved to be O(h + N -1 ) (h, N : the parameters of the finite-element- and Fourier-approximation, with h →0, N →∞). The theoretical results are confirmed by numerical experiments.

Fourier–Finite‐element Approximation of Elliptic Interface Problems in Axisymmetric Domains

The paper deals with the Fourier-finite-element method (FFEM), which combines the approximate Fourier method with the finite-element method, and its application to Poisson-like equations −p̂Δ3û = f̂ in three-dimensional axisymmetric domains Ωˆ. Here, pˆ is a piecewise constant coefficient having a jump at some axisymmetric interface. Special emphasis is given to estimates of the Fourier-finite-element error in the Sobolev space H1(Ωˆ), if the interface is smooth or if it meets the boundary of Ωˆ at some edge. In general, the solution û contains a singularity at the interface, which is described by a tensor product representation and treated numerically by appropriate mesh grading in the meridian plane of Ωˆ. The rate of convergence of the combined approximation in H1(Ωˆ) is proved to be 𝒪(h+N−1) (h, N: the parameters of the finite-element- and Fourier-approximation, with h→0, N→∞). The theoretical results are confirmed by numerical experiments.