Coupled boundary-element scheme for eddy-current computation

The mathematical foundation of a symmetric boundary-element method for the computation of eddy currents in a linear homogeneous conductor which is exposed to an alternating magnetic field is presented. Starting from the A-based variational formulation of the eddy-current equations and a related transmission problem, the problem inside and outside the conductors is reformulated in terms of integral equations on the boundary of the conductors. Surface currents occur as new unknowns of this direct formulation. The integral equations can be coupled in a symmetric fashion using the transmission conditions for the vector potential A and the magnetic field H. The resulting variational problem is elliptic in suitable trace spaces. A conforming Galerkin boundary-element discretization is employed, which relies on surface edge elements and provides quasi-optimal discrete approximations for the tangential traces of A and H. Surface stream functions supplemented with co-homology vector fields ensure the vital zero divergence of the discrete equivalent surface currents. Simple expressions allow the computation of approximate total Ohmic losses and surface forces from the discrete boundary data.     

Antisymmetric bending analysis of symmetric laminated plates including transverse shear and normal effects

Antisymmetric bending analysis of symmetric laminated plates is presented here. The transverse shear and normal strain and stress effects on bending of such laminates are considered. The displacement fields and the transverse shear and normal stress fields are assumed to preserve the displacement and traction continuity conditions at the interface between layers. A set of twelfth-order governing equations and consistent boundary conditions are given from a mixed variational theorem. Solutions for simply-supported cross-ply plates and a strip are discussed. The numerical results are compared with elasticity solutions and results given from other theories. The present theory is found to agree closely with three-dimensional elasticity solutions.     

Interface waves along an anisotropic imperfect interface between anisotropic solids

First and second order asymptotic boundary conditions are introduced to model a thin anisotropic layer between two generally anisotropic solids. Such boundary conditions can be used to describe wave interaction with a solid-solid imperfect anisotropic interface. The wave solutions for the second order boundary conditions satisfy energy balance and give zero scattering from a homogeneous substrate/layer/substrate system. They couple the in-plane and out-of-plane stresses and displacements on the interface even for isotropic substrates. Interface imperfections are modeled by an interfacial multiphase orthotropic layer with effective elastic properties. This model determines the transfer matrix which includes interfacial stiffness and inertial and coupling terms. The present results are a generalization of previous work valid for either an isotropic viscoelastic layer or an orthotropic layer with a plane of symmetry coinciding with the wave incident plane. The problem of localization of interface waves is considered. It is shown that the conditions for the existence of such interface waves are less restrictive than those for Stoneley waves. The results are illustrated by calculation of the interface wave velocity as a function of normalized layer thickness and angle of propagation. The applicability of the asymptotic boundary conditions is analyzed by comparison with an exact solution for an interfacial anisotropic layer. It is shown that the asymptotic boundary conditions are applicable not only for small thickness-to-wavelength ratios, but for much broader frequency ranges than one might expect. The existence of symmetric and SH-type interface waves is also discussed.     

Boundary element analysis of thermal stress intensity factors for interface griffith and cusp cracks

The thermal stress intensity factors for interface cracks of Griffith and symmetric lip cusp types under vertical uniform heat flow in a finite body are calculated by the boundary element method. The boundary conditions on the crack surfaces are insulated or fixed to constant temperature. The relationship between the stress intensity factors and the displacements on the nodal point of a crack-tip element is derived. The numerical values of the thermal stress intensity factors for an interface Griffith crack in an infinite body are compared with the previous solutions. The thermal stress intensity factors for a symmetric lip cusp interface crack in a finite body are calculated with respect to various effective crack lengths, configuration parameters, material property ratios and the thermal boundary conditions on the crack surfaces. Under the same outer boundary conditions, there are no appreciable differences in the distribution of thermal stress intensity factors with respect to each material property. However, the effect of crack surface thermal boundary conditions on the thermal stress intensity factors is considerable.     

Symmetric Galerkin boundary integral formulation for interface and multi-zone problems

Domains containing an ‘internal boundary’, such as a bi-material interface, arise in many applications, e.g. composite materials and geophysical simulations. This paper presents a symmetric Galerkin boundary integral method for this important class of problems. In this situation, the physical quantities are known to satisfy continuity conditions across the interface, but no boundary conditions are specified. The algorithm described herein achieves a symmetric matrix of reduced size. Moreover, the symmetry can also be invoked to lessen the numerical work involved in constructing the system of equations, and thus the method is computationally very efficient. A prototype numerical example, with several variations in the boundary conditions and material properties, is employed to validate the formulation and corresponding numerical procedure. The boundary element results are compared with analytical solutions and with numerical results obtained with the finite element method. © 1997 John Wiley & Sons, Ltd.     

Preconditioning for boundary element methods in domain decomposition

This paper is concerned with asymptotically almost optimal preconditioning techniques for the solution of coupled elliptic problems with piecewise continuous coefficients by domain decomposition methods. Spectrally equivalent, two- and multilevel interface preconditioners are proposed and analyzed. They are applied to two basic formulations: strongly elliptic skew-symmetric problems and symmetric, positive definite variational problems; the former involves the classical boundary potentials from the Calderon projections and the latter is based on the Steklov–Poincaré operators associated with subdomains of the decomposition. The preconditioners considered are shown to be robust with respect to both mesh-parameters and jumps in the coefficients.     

A Galerkin boundary node method for biharmonic problems

A Galerkin boundary node method (GBNM) is developed in this paper for solving biharmonic problems. The GBNM combines an equivalent variational form of boundary integral formulations for governing equations with the moving least-squares approximations for construction of the trial and test functions. In this approach, only a nodal data structure on the boundary of a domain is required. In addition, boundary conditions can be implemented accurately and the system matrices are symmetric. The convergence of this method and numerical examples are given to show the efficiency.     

Added mass computations by integral equation methods

This paper presents boundary integral equation procedures for calculating the added mass matrix required to determine hydroelastic vibrational modes of tanks or immersed structures. The liquid is assumed inviscid and incompressible and no surface waves are admitted. Two symmetric variational formulations of the boundary integral equation are derived, including the free surface condition and possible conditions of symmetry. These are approximated by a curved finite element method and numerical results are presented for an axisymmetric tank, immersed cantilever plates and a three-dimensional fuel storage container.     

A symmetric and positive definite variational BEM for 2-D free vibration analysis

A general BEM model for structural dynamics is derived by using a symmetric and positive definite variational formulation. The functional employed involves domain displacements and boundary tractions and displacements. These variables are taken to be independent of one another. The boundary variables are expressed in terms of their nodal values while the domain displacement field is approximated by a linear combination of static fundamental solutions. The source point of the latter is located outside the domain. The resolving system is a linear system and for free vibration a classic linear algebraic eigenvalue problem is inferred. The stiffness and mass matrices are symmetric and positive definite and the domain integral, when associated with the inertial term, can be transformed into a boundary integral. Numerical results are presented to prove the efficiency of the method.     

BIEM-based variational principles for elastoplasticity with unilateral contact boundary conditions

The structural step problem for elastic-plastic internal-variable materials is addressed in the presence of frictionless unilateral contact conditions. Basing on the BIEM (boundary integral equation method) and making use of deformation-theory plasticity (through the backward-difference method of computational plasticity), two variational principles are shown to characterize the solution to the step problem: one is a stationarity principle having as unknowns all the problem variables, the other is a saddle-point principle having as unknowns the increments of the boundary tractions and displacements, along with the plastic strain increments in the domain. The discretization by boundary and interior elements transforms the above principles into well-posed mathematical programming formulations belonging to the symmetric Galerkin BEM formulations (with features such as a symmetric sign-definite coefficient matrix, double integrations, and hypersingular integrals).     

Finite element iterative techniques for determining the interface boundary between Laplace and Poisson domains—characteristic analysis of a field effect transistor

Two kinds of techniques for solving a shape determination problem are proposed. The determination of the interface boundary between two domains governed by Poisson and Laplace equations under the compatible and constraint condition is considered. Influence coefficient and inverse variational approaches are examined by using the iterative finite element procedure. A two-dimensional model of a junction-type field effect transistor is a test example. The determination of its interface boundary and the prediction of the potential distribution and static characteristic are demonstrated.     

Transient dynamic crack analysis in piecewise homogeneous, anisotropic and linear elastic composites by a spatial symmetric Galerkin-BEM

Transient elastodynamic analysis of two-dimensional, piecewise homogeneous, anisotropic and linear elastic solids containing interior and interface cracks is presented in this paper. To solve the initial boundary value problem, a spatial symmetric time-domain boundary element method is developed. Stationary cracks subjected to impact loading conditions are considered. Elastodynamic fundamental solutions for homogenous, anisotropic and linear elastic solids are implemented. The piecewise homogeneous, anisotropic and linear elastic solids are modeled by the multi-domain technique. The spatial discretization is performed by a symmetric Galerkin-method, while a collocation method is utilized for the temporal discretization. An explicit time-stepping scheme is obtained for computing the unknown boundary data. Numerical examples are presented and discussed to show the effects of the interface cracks, the material anisotropy, the material combination and the dynamic loading on the dynamic stress intensity factors.     

A coupled symmetric BE–FE method for acoustic fluid–structure interaction

A coupled symmetric BE–FE method for the calculation of linear acoustic fluid–structure interaction in time and frequency domain is presented. In the coupling formulation a newly developed hybrid boundary element method (HBEM) will be used to describe the behaviour of the compressible fluid. The HBEM is based on Hamilton's principle formulated with the velocity potential. The state variables are separated into boundary variables which are approximated by piecewise polynomial functions and domain variables which are approximated by a superposition of static fundamental solutions. The domain integrals are eliminated, respectively, replaced by boundary integrals and a boundary element formulation with a symmetric mass and stiffness matrix is obtained as result. The structure is discretized by FEM. The coupling conditions fulfil C¹-continuity on the interface. The coupled formulation can also be used for eigenfrequency analyses by transforming it from time domain into frequency domain.     

Stress analysis of axisymmetric shear deformable cross-ply laminated circular cylindrical shells

A generalized mixed theory for bending analysis of axisymmetric shear deformable laminated circular cylindrical shells is presented. The classical, first-order and higher-order shell theories have been used in the analysis. The Maupertuis–Lagrange (M–L) mixed variational formula is utilized to formulate the governing equations of circular cylindrical shells laminated by orthotropic layers. Analytical solutions are presented for symmetric and antisymmetric laminated circular cylindrical shells under sinusoidal loads and subjected to arbitrary boundary conditions. Numerical results of the higher-order theory for deflections and stresses of cross-ply laminated circular cylindrical shells are compared with those obtained by means of the classical and first-order shell theories. The effects, due to shear deformation, lamination schemes, loadings ratio, boundary conditions and orthotropy ratio on the deflections and stresses are investigated.     

Comparison between a finite element and a composite method for a three-dimensional potential problem

This paper is concerned with the solution of the three-dimensional potential problem for electromagnetic river gauging. It extends previous ideas of joining finite elements in an interior region to one infinite external element treated by the boundary integral method^1,2 to this case where there are two external infinite elements representing the river and the ground. The boundary continuity conditions on the infinite river–ground interface, as well as the internal–external interfaces, are dealt with by introducing a variational principle with relaxed continuity requirement³.     

Approximate imposition of boundary conditions in immersed boundary methods

We analyze several possibilities to prescribe boundary conditions in the context of immersed boundary methods. As basic approximation technique we consider the finite element method with a mesh that does not match the boundary of the computational domain, and therefore Dirichlet boundary conditions need to be prescribed in an approximate way. As starting variational approach we consider Nitsche's methods, and we then move to two options that yield non‐symmetric problems but that turned out to be robust and efficient. The essential idea is to use the degrees of freedom of certain nodes of the finite element mesh to minimize the difference between the exact and the approximated boundary condition. Copyright © 2009 John Wiley & Sons, Ltd.     

A regular variational boundary model for free vibrations of magneto-electro-elastic structures

In this paper a regular variational boundary element formulation for dynamic analysis of two-dimensional magneto-electro-elastic domains is presented. The method is based on a hybrid variational principle expressed in terms of generalized magneto-electro-elastic variables. The domain variables are approximated by using a superposition of weighted regular fundamental solutions of the static magneto-electro-elastic problem, whereas the boundary variables are expressed in terms of nodal values. The variational principle coupled with the proposed discretization scheme leads to the calculation of frequency-independent and symmetric generalized stiffness and mass matrices. The generalized stiffness matrix is computed in terms of boundary integrals of regular kernels only. On the other hand, to achieve meaningful computational advantages, the domain integral defining the generalized mass matrix is reduced to the boundary through the use of the dual reciprocity method, although this implies the loss of symmetry. A purely boundary model is then obtained for the computation of the structural operators. The model can be directly used into standard assembly procedures for the analysis of non-homogeneous and layered structures. Additionally, the proposed approach presents some features that place it in the framework of the weak form meshless methods. Indeed, only a set of scattered points is actually needed for the variable interpolation, while a global background boundary mesh is only used for the integration of the influence coefficients. The results obtained show good agreement with those available in the literature proving the effectiveness of the proposed approach.     

An augmented lagrangian approach to material discontinuities in meshless methods

The purely nodal discretization, typical of meshless methods, turns out in the necessity of properly defining both the external boundary and the material interfaces which may be present in the analysis of mechanical problems. Each material domain is dicretized independently, and interface conditions are imposed into the variational formulation by the augmented Lagrangian method. This allows for a numerically efficient formulation where the number of approximation variables is unchanged by the presence of any number of interface constraints.     

A hybrid flux model for heat conduction problems

A hybrid method of solution for the linear problem of heat conduction in a body is presented. The variational support is a two‐field functional whose arguments are heat flux, which meets a priori inner thermal equilibrium, and temperature on the boundary of the body. The stationary conditions of the functional are the Fourier's law and the prescribed boundary conditions. This variational framework allows to develop a finite element model that exhibits good accuracy, especially in the presence of geometry irregularities in a mesh. Copyright © 2000 John Wiley & Sons, Ltd.     

Implementation of a symmetric boundary element method in transient heat conduction with semi‐analytical integrations

A time‐convolutive variational hypersingular integral formulation of transient heat conduction over a 2‐D homogeneous domain is considered. The adopted discretization leads to a linear equation system, whose coefficient matrix is symmetric, and is generated by double integrations in space and time. Assuming polynomial shape functions for the boundary unknowns, a set of compact formulae for the analytical time integrations is established. The spatial integrations are performed numerically using very efficient formulae just recently proposed. The competitiveness from the computational point of view of the symmetric boundary integral equation approach proposed herein is investigated on the basis of an original computer implementation. Copyright © 1999 John Wiley & Sons, Ltd.