On the Singular Neumann Problem in Linear Elasticity

The Neumann problem of linear elasticity is singular with a kernel formed by the rigid motions of the body. There are several tricks that are commonly used to obtain a non-singular linear system. However, they often cause reduced accuracy or lead to poor convergence of the iterative solvers. In this paper, different well-posed formulations of the problem are studied through discretization by the finite element method, and preconditioning strategies based on operator preconditioning are discussed. For each formulation we derive preconditioners that are independent of the discretization parameter. Preconditioners that are robust with respect to the first Lam\'e constant are constructed for the pure displacement formulations, while a preconditioner that is robust in both Lam\'e constants is constructed for the mixed formulation. It is shown that, for convergence in the first Sobolev norm, it is crucial to respect the orthogonality constraint derived from the continuous problem. Based on this observation a modification to the conjugate gradient method is proposed that achieves optimal error convergence of the computed solution

On the evaluation of quasi-periodic Green functions and wave-scattering at and around Rayleigh-Wood anomalies

This article presents full-spectrum, well-conditioned, Green-function methodologies for evaluation of scattering by general periodic structures, which remains applicable on a set of challenging singular configurations, usually called Rayleigh-Wood (RW) anomalies (at which the quasi-periodic Green function ceases to exist), where most existing quasi-periodic solvers break down. After reviewing a variety of existing fast-converging numerical procedures commonly used to compute the classical quasi-periodic Green-function, the present work explores the difficulties they present around RW-anomalies and introduces the concept of hybrid “spatial/spectral” representations. Such expressions allow both the modification of existing methods to obtain convergence at RW-anomalies as well as the application of a slight generalization of the Woodbury-Sherman-Morrison formulae together with a limiting procedure to bypass the singularities. (Although, for definiteness, the overall approach is applied to the scalar (acoustic) wave-scattering problem in the frequency domain, the approach can be extended in a straightforward manner to the harmonic Maxwell's and elasticity equations.) Ultimately, this thorough study of RW-anomalies yields fast and highly-accurate solvers, which are demonstrated with a variety of simulations of wave-scattering phenomena by arrays of particles, crossed impenetrable and penetrable diffraction gratings and other related structures. In particular, the methods developed in this article can be used to “upgrade” classical approaches, resulting in algorithms that are applicable throughout the spectrum, and it provides new methods for cases where previous approaches are either costly or fail altogether. In particular, it is suggested that the proposed shifted Green function approach may provide the only viable alternative for treatment of three-dimensional high-frequency configurations with either one or two directions of periodicity. A variety of computational examples are presented which demonstrate the flexibility of the overall approach

On non-coercive mixed problems for parameter-dependent elliptic operators

We consider a (generally, non-coercive) mixed boundary value problem in a bounded domain $D$ of ${\mathbb R}^n$ for a second order parameter-dependent elliptic differential operator $A (x,\partial, \lambda)$ with complex-valued essentially bounded measured coefficients and complex parameter $\lambda$. The differential operator is assumed to be of divergent form in $D$, the boundary operator $B (x,\partial)$ is of Robin type with possible pseudo-differential components on $\partial D$. The boundary of $D$ is assumed to be a Lipschitz surface. Under these assumptions the pair $(A (x,\partial, \lambda),B)$ induces a holomorphic family of Fredholm operators $L(\lambda): H^+(D) \to H^- (D)$ in suitable Hilbert spaces $H^+(D)$ , $H^- (D)$ of Sobolev type. If the argument of the complex-valued multiplier of the parame\-ter in $A (x,\partial, \lambda)$ is continuous and the coefficients related to second order derivatives of the operator are smooth then we prove that the operators $L(\lambda)$ are conti\-nu\-ously invertible for all $\lambda$ with sufficiently large modulus $|\lambda|$ on each ray on the complex plane $\mathbb C$ where the differential operator $A (x,\partial, \lambda)$ is parameter-dependent elliptic. We also describe reasonable conditions for the system of root functions related to the family $L (\lambda)$ to be (doubly) complete in the spaces $H^+(D)$, $H^- (D)$ and the Lebesgue space $L^2 (D)$

Automatic 3D modeling by combining SBFEM and transfinite element shape functions

The scaled boundary finite element method (SBFEM) has recently been employed as an efficient means to model three-dimensional structures, in particular when the geometry is provided as a voxel-based image. To this end, an octree decomposition of the computational domain is deployed and each cubic cell is treated as an SBFEM subdomain. The surfaces of each subdomain are discretized in the finite element sense. We improve on this idea by combining the semi-analytical concept of the SBFEM with certain transition elements on the subdomains' surfaces. Thus, we avoid the triangulation of surfaces employed in previous works and consequently reduce the number of surface elements and degrees of freedom. In addition, these discretizations allow coupling elements of arbitrary order such that local p-refinement can be achieved straightforwardly

A new transform approach to biharmonic boundary value problems in circular domains with applications to Stokes flows

In this thesis, we present a new transform approach for solving biharmonic boundary value problems in two-dimensional polygonal and circular domains. Our approach provides a unified general approach to finding quasi-analytical solutions to a wide range of problems in Stokes flows and plane elasticity. We have chosen to analyze various Stokes flow problems in different geometries which have been solved using other techniques and present our transform approach to solve them. Our approach adapts mathematical ideas underlying the Unified transform method, also known as the Fokas method, due to Fokas and collaborators in recent years. We first consider Stokes flow problems in polygonal domains whose boundaries consist of straight line edges. We show how to solve problems in the half-plane subject to different boundary conditions along the real axis and we are able to retrieve analytical results found using other techniques. Next, we present our transform approach to solve for a flow past a periodic array of semi-infinite plates and for a periodic array of point singularities in a channel, followed by a brief discussion on how to systematically solve problems in more complex channel geometries. Next, we show how to solve problems in circular domains whose boundaries consist of a combination of straight line and circular edges. We analyze the problems of a flow past a semicircular ridge in the half-plane, a translating and rotating cylinder above a wall and a translating and rotating cylinder in a channel.Open Acces

Scaled Boundary Finite Element Method for Two-Dimensional Linear Multi-Field Media

This paper presents an efficient and accurate numerical technique, based on a scaled boundary finite element method (SBFEM), that is capable of solving two-dimensional, second-order, linear, multi-field boundary value problems. Basic governing equations are established in a general, unified context allowing the treatment of various classes of linear problems such as steady-state heat conduction problems, steady-state flow in porous media, linear elasticity, linear piezoelectricity, and linear piezomagnetic and piezoelectromagnetic problems. A scaled boundary finite element approximation is also formulated within a general framework integrating the influence of the distributed body source, general boundary conditions, contributions of the general side-face data, and the flexibility of scale boundary approximations. Standard procedures for numerical integration, search of eigenvalues and eigenvectors, determination of particular solutions, and solving a system of linear algebraic equations are adopted. After fully tested with available benchmark solutions, the proposed SBFEM is applied to solve various classes of linear problems under different scenarios to demonstrate its vast capability, computational efficiency and robustness.This paper presents an efficient and accurate numerical technique, based on a scaled boundary finite element method (SBFEM) that is capable of solving two-dimensional, second-order, linear, multi-field boundary value problems. Basic governing equations are established in a general, unified context allowing the treatment of various classes of linear problems such as steady-state heat conduction problems, steady-state flow in porous media, linear elasticity, linear piezoelectricity, and linear piezomagnetic and piezoelectromagnetic problems. A scaled boundary finite element approximation is also formulated within a general framework integrating the influence of the distributed body source, general boundary conditions, contributions of the general side-face data, and the flexibility of scaled boundary approximations. Standard procedures for numerical integration, search of eigenvalues and eigenvectors, determination of particular solutions, and solving a system of linear algebraic equations are adopted. After fully tested with available benchmark solutions, the proposed SBFEM is applied to solve various classes of linear problems under different scenarios to demonstrate its vast capability, computational efficiency and robustness