Infinite boundary elements for the dynamic analysis of machine foundations

This paper describes an infinite boundary element technique for the dynamic analysis of three-dimensional machine foundations. The core region is modelled with quadratic elements while infinite boundary elements are used to discretize the unbounded far-field outside the immediate vicinity of the loaded area. The dynamic decay function assumed in the analysis is based on Rayleigh wave attenuation away from the centre of the foundation. Effective integration schemes for oscillatory integrals have been developed. Illustrative examples are presented in order to demonstrate the applicability and accuracy of the proposed infinite boundary element. © 1997 John Wiley & Sons, Ltd.     

Infinite boundary elements

Special types of boundary elements are discussed which can be used for the modelling of surfaces which extend to infinity. The theoretical background and details of implementation are discussed. On test examples it is shown that the elements perform extremely well even for cases in which they are located close to the area of interest. A practical application of the use of the elements for the modelling of mining excavations is given.     

Infinite boundary elements for electromagnetics

Consider a two‐dimensional plane wave transverse magnetic mode scattering from a perfectly electric conducting ground plane. Let the ground plane be of infinite extent and comprise two regions, a near field and a far field. In the far field, let the ground plane be flat and let us choose the co‐ordinates (x, y) such that it lies on the axis y=0. Over the interior region, let the profile of the ground plane change such that it can lie partially above and also partially below the axis y=0. Finally, let us assume that the source of the excitation lies above the ground plane. To model this general class of problems, a method of moments electric field integral equation formulation is proposed which uses infinite boundary elements to model the far field and boundary elements to model the near field. In the far field, the field variable is approximated by the highest order terms in the far‐field asymptotic expansion. The integrals over the infinite boundary elements are infinite in extent and contain oscillatory terms and hence require special integration rules. The formulation is tested for the specific problem of a semi‐circular cylindrical protrusion of radius a lying above an infinite flat ground plane, such that ka=1 where k is the wave number. This problem is chosen because it has an analytic solution in the form of a Bessel function expansion; hence, the accuracy of the formulation can be tested. In particular, the radar cross section results for various angles of incidence of the plane wave source are calculated and compared with analytic results. Copyright © 2007 John Wiley & Sons, Ltd.     

Finite elements in space and time for dynamic contact problems

Dynamic obstacle and Signorini problems are discretized by continuous and discontinuous finite elements in space and time. The resulting discrete problems are attributed to a standard implicit time‐stepping scheme through relaxation of impact phenomena and suitable numerical integration. The method can cope with dynamic contact problems, which is shown by an analysis of a model problem. Moreover, numerical examples demonstrate that it is actually able to approximate the solution of dynamic contact problems, which are not fully covered by the theory. Copyright © 2008 John Wiley & Sons, Ltd.     

A boundary elements formulation for 3D fretting-wear problems

Computational wear modeling is an extremely time-consuming problem, especially the 3D cases. In this work, a 3D boundary element method (BEM) formulation for wear modeling is proposed and applied to simulate 3D fretting-wear problems under gross sliding and partial slip conditions. The present formulation applies the BEM to approximate the elastic response of solids, and an augmented Lagrangian formulation to solve the contact problem. Contact restrictions fulfilment is established by a set of projection functions, and wear on contact surfaces is computed using the Archard wear law. The BEM proves to be a very suitable numerical method for this kind of mechanical interaction problems, considering only the boundary degrees of freedom involved in the problem and obtaining a very good approximation of contact tractions with a low number of elements. This is very interesting in terms of computational cost reductions of wear modeling, specially in 3D problems. In that regard, an acceleration strategy is applied to the proposed algorithm. It allows to obtain very important reductions on wear simulation times. The proposed methodology is therefore an efficient numerical tool for 3D fretting-wear problems modeling.     

Mapped infinite elements for 3-D vector potential magnetic problems

Numerous engineering problems, especially those in electromagnetics, often require the treatment of the unbounded continua. Mapped infinite elements have been developed for the solution of 3-D magnetic vector potential equations in infinite domain that may be used in conjunction with the standard finite elements. The electromagnetic field equations are written in terms of the magnetic vector potential for the infinite domain, and 3-D mapped infinite eiement formulation based on these equations is presented in detail. A series of magnetostatics and eddy current problems are solved to demonstrate the validity and efficiency of the procedure. These numerical results indicate that the combined finite–infinite element procedure is computationally much more economical for the solution of unbounded electromagnetic problems, especially when using the vector potential formulation, as the number of system equations decreases substantially compared to the finite element only procedure. The present procedure shows promise for the treatment of large practical industrial 3-D eddy current problems with manageable computer resources.     

Expansion of periodic boundary condition for 3-D FEM analysis usingedge elements

The application of periodic boundary conditions in the analysis of three-dimensional magnetic fields by finite element methods (FEMs) leads to a substantial reduction of computation labor and storage. The expansion of the condition for magnetostatic curl-curl formulation with the magnetic vector potential employing edge tetrahedral elements is discussed. Differences between the definitions of the condition for nodal and edge elements are examined. The vectorial nature of edge elements is emphasized and associated difficulties in the formulation and application of the condition are carefully analyzed and overcome. Details for computer implementation are given and a simple test problem to verify the validity of the software is proposed. Advantages gained when the condition is used for TEAM Workshop problem 13 as an example are shown     

Hypersingular quarter-point boundary elements for crack problems

The present paper deals with the study and effective implementation for Stress Intensity Factor computation of a mixed boundary element approach based on the standard displacement integral equation and the hypersingular traction integral equation. Expressions for the evaluation of the hypersingular integrals along general curved quadratic line elements are presented. The integration is carried out by transformation of the hypersingular integrals into regular integrals, which are evaluated by standard quadratures, and simple singular integrals, which are integrated analytically. The generality of the method allows for the modelling of curved cracks and the use of straight line quarter-point elements. The Stress Intensity Factors can be computed very accurately from the Crack Opening Displacement at collocation points extremely close to the crack tip. Several examples with different crack geometries are analyzed. The computed results show that the proposed approach for Stress Intensity Factors evaluation is simple, produces very accurate solutions and has little dependence on the size of the elements near the crack tip.     

Three-dimensional infinite boundary elements for contact problems

A three-dimensional infinite boundary element method for the modelling of half space contact problems is presented. The infinite surface of the half space is discretized into a finite number of elements with the elements extending to infinity mapped onto finite elements. A special mapping scheme that handles the singular integrals over those infinite boundary elements is introduced. The infinite boundary element scheme treats the effect of infinity correctly, leading to excellent accuracy which is verified for a half space subjected to uniform pressure in a circular region and a square region on the surface. The utility of the program is then illustrated through the analysis of the stress concentration at the surface of an oblate spheroidal cavity, serving as a model for a naturally occurring void, embedded in the half space subjected to normal and tangential surface tractions.     

Singular boundary elements for three-dimensional elasticity problems

The focus of this paper is a set of semi-discontinuous, traction-singular surface elements introduced to help the rigorous boundary integral analysis of problems in three-dimensional solid mechanics. In contrast to the singular boundary elements developed for linear fracture mechanics where the square-root singularity is of primary interest, traction shape functions featuring the proposed four- and eight-node boundary elements can be used to represent power-type singularities of arbitrary order, such as those arising at non-smooth material boundaries and interfaces. Apart from being capable of rigorously handling traction singularities and discontinuities across the domain boundaries and interfaces, these elements also permit a smooth transition to adjacent regular elements. Complemented with a family of suitable displacement and geometry shape functions, the singular surface elements are incorporated into a regularized boundary integral equation method and shown, through a set of benchmark results, to perform well for both static and dynamic problems.     

The 3-D thermoelastic boundary element method: semi-analytical integration for subparametric triangular elements

In this paper a semi-analytical integration scheme is described which is designed to reduce the errors that can result with the numerical evaluation of integrals with singular integrands. The semi-analytical scheme can be applied to quadratic subparametric triangular elements for use in thermoelastic problems. Established analytical solutions for linear isoparametric triangular elements are combined with standard quadrature techniques to provide an accurate integration scheme for quadratic subparametric triangular elements. The use of subparametric elements provides an efficient means for coupling thermal and elastostatic analyses. It is possible for the same mesh to be employed, with linear isoparametric elements used for thermal analysis and quadratic subparametric elements used for deformation analysis. Numerical tests are performed on simple test problems to demonstrate the advantages of the semi-analytical approach which is shown to be orders of magnitude more accurate than standard quadrature techniques. Moreover, the expected increased accuracy with subparametric elements is also demonstrated. © 1998 John Wiley & Sons, Ltd.     

Uniform bicubic B-splines applied to boundary element formulation for 3-D scalar problems

In this work, uniform bicubic B-spline functions are used to represent the surface geometry and interpolation functions in the formulation of boundary-element method (BEM) for three-dimensional problems. This is done as a natural generalization of cubic B-spline curves, introduced by Cabral et al, for two-dimensional problems. Three-dimensional scalar problems, with particular applications to Laplace and Helmholtz equations, are considered.     

Shape design sensitivity analysis via material derivative-adjoint variable technique for 3-D and 2-D curved boundary elements

A general approach to shape design sensitivity analysis of three- and two-dimensional elastic solid objects is developed using the material derivative-adjoint variable technique and boundary element method. The formulation of the problem is general and first-order sensitivities in the form of boundary integrals for the effect of boundary shape variations are derived for an arbitrary performance functional. Second-order quadrilateral surface elements (for 3-D problems) and quadratic boundary elements (for 2-D problems) are employed in the solution of primary and adjoint systems and discretization of the boundary integral expressions for sensitivities. The accuracy of sensitivity information is studied for selected global performance functionals and also for boundary state fields at discrete points. Numerical results are presented to demonstrate the accuracy and efficiency of this approach.     

Hermitian cubic boundary elements for two-dimensional potential problems

A hermite interpolation based formulation is presented for the boundary element analysis of two-dimensional potential problems. Two three-noded Hermitian Cubic Elements (HCE) are introduced for the modelling of corners or points with non-unique tangents on the boundary. These elements, along with the usual two-noded HCE, are used in numerical examples. The results obtained show that faster convergence can be achieved using HCE compared with using Lagrange interpolation type Quadratic Elements (QE), for about the same amount of computing resources.     

Blended infinite elements for paraboblic boundary value problems

A common method for numerically approximating two-point parabolic boundary value problems of the form u_t = L[u]+f(u) defined of the semi-infinite strip S = [0, 1]×[0, ∞] is to first discretize the spatial operator in the differential equation and then solve for the time evolution. Such an approach typically involves solving a system of algebriaic equations at a sequence of time steps. In this paper we take a different approach and subdivide S into a collection of semi-infinite substrips S_i = [x_i, x_i+1]×[0, ∞], and use blending function techniques to derive finite parameter functions e_i(x, t) defined on S_i. Spectral matching methods are used in deriving e_i to ensure that (u ? e_i) can be made small on S_i. Galerkin's method, with associated integration sover the entire space-time domain S, is then used to generate approximations to u(x, t) based upon the so defined infinite element (e_i, S_i). Approximations are hence found for all (x, t) in S by solving one well structed system of algebraic equations. We apply the method to several linear and non-linear problms.     

Curved composite boundary elements for Kirchhoff plate-bending problems

Composite boundary elements are formed for plate-bending problems by attaching a circular sector to a triangle. Elements utilizing cubic and quintic interpolation are developed for convex and concave boundaries. Smaller error and less numerical effort result when triangular elements are replaced by the composite boundary elements on clamped and most simply supported convex, boundaries. For domains with concave boundaries, greater accuracy is obtained when the quintic composite boundary element replaces the triangular element in fine mesh configurations.     

Curved composite boundary elements for second-order two-dimensional problems

Finite elements with a curved edge often require relatively large numerical effort to form.^1–4 Relatively simple triangular elements with a single curved edge are developed in this paper for second-order, two-dimensional problems. A convex boundary element is formed as a composite of two straight-edged triangles and a circular sector. Application of the convex composite boundary element results in less numerical effort for a comparable error in circular and elliptical domain test problems than the application of straight-edged elements in all cases shown, and in most cases when compared to the curved isoparametric elements. For domains with concave boundaries, the application of straight-edged, concave composite boundary and curved isoparametric elements give comparable accuracies and numerical efforts because of a fortuitous cancellation of error that occurs with straight-edged elements in this case.     

Application of boundary element method to 3-D problems of coupled thermoelasticity

This paper deals with a new boundary element method for analysis of the quasistatic problems in coupled thermoelasticity. Through some mathematical manipulation of the Navier equation in elasticity, the heat conduction equation is transformed into a simpler form, similar to the uncoupled-type equation with the modified thermal conductivity which shows the coupling effects. This procedure enables us to treat the coupled thermoelastic problems as an uncoupled one, A few examples are computed by the proposed BEM, and the results obtained are compared with the analytical ones available in the literature, whereby the accuracy and versatility of the proposed method are demonstrated.     

A study of boundary layers in plates using Mindlin-Reissner and 3-D elements

Finite elements based on Mindlin–Reissner theory and three-dimensional theory are used to study the distribution of shear forces and twisting moments in plates with various simple support conditions. Differences between the results obtained using these two theories are highlighted. A crude adaptive mesh refinement procedure is applied to improve the accuracy of the finite-element analysis.     

Parallel processing of quadratic boundary elements for axisymmetric elasto-static problems

The concern of this paper is on improving the computational efficiency of boundary element methods (BEM) through the development of parallel algorithms for use on massively parallel machines. The application is on the axisymmetric elasto-static problems with quadratic boundary elements. Different ways of parallel approaches are discussed and a parallel approach suited to the BEM numerical process is developed. Numerical results from both the parallel algorithm and a serial algorithm are given in the paper to illustrate the efficiency of the parallel approach.