Numerical Boundary Integral Equation Research Articles

A numerical boundary integral equation analysis for standard linear viscoelastic media made of functionally graded materials

Although structures made of functionally graded materials have been studied by many researchers, no research may be found in literature on boundary element analysis of the functionally graded viscoelastic structures. In the present paper, a 2D boundary element formulation capable of modeling time-dependent functionally graded materials (FGM) is presented. A numerical implementation of the Somigliana identity in terms of the displacements is developed to solve 2D problems of the exponentially graded viscoelasticity. The FGM concept can be applied to various materials, for structural and functional purposes. In this model, only Green functions of the nonhomogeneous elastostatic problems are needed with material properties that vary continuously along a given dimension. Results reveal that the boundary element approach can successfully be employed for the present complicated problem for arbitrary time histories of the applied loads and arbitrary boundary conditions, without the need to use relaxation functions or mathematical transformations.

Role of geomorphic and hydraulic parameters in governing pore water seepage from salt marsh sediments

A numerical boundary integral equation model has been used to study the effects of marsh geomorphology and hydraulic parameters on the seepage of pore water into tidal channels and its subsequent recharge by tidal inundation. In general, the results show that regardless of the geomorphic configuration the locus of maximum seepage flux occurs at or near the intersection of the creek bank and the channel water surface. Over a tidal cycle, typically two thirds of the total seepage discharge occurs through the creek bank with only about a third discharging from the channel bottom. In most cases the volume of seepage discharging through the creek bank exceeds by a factor of two or more that emanating from the channel bottom. Of the creek bank discharge, up to half may occur through the seepage face that develops above the tide line at tidal stages below mean tide. The ratio of creek bank to bottom seepage only approaches unity when the thickness of the aquifer beneath the bottom is similar to or greater than the width of the channel bottom and the ratio of conductivity to specific yield exceeds 0.5. For given hydraulic parameters, total seepage volume varies directly with creek bank slope and aquifer thickness, whereas these factors are inversely related to the ratio of the two seepage components. Channel bottom width, on the other hand, has no influence on total seepage volume but greatly increases the component ratio as the width approaches zero. Since most of the seepage discharge is derived from sediments within several meters of the creek bank, variations in seepage discharge driven by variations in marsh geomorphology might play a role in the productivity of creekside cord grass. Several hypotheses are offered and discussed regarding the effects of geomorphology on creekside productivity. These results also indicate that placement of seepage meters only on the channel bottom will not give samples or measures representative of the total seepage.

A modeling study of the dynamics of pore water seepage from intertidal marsh sediments

A numerical boundary integral equation model has been used to simulate tidally driven transient variations in pore water seepage from salt marsh sediments into tidal channels and its subsequent recharge by tidal inundation. In general the results show that the maximum seepage discharge occurs at or near the intersection of the creek bank and the channel water surface. Over a tidal cycle typically two-thirds of the total seepage discharge occurs through the creek bank with only about a third discharging from the channel bottom. Of the creek-bank discharge up to a third occurs through the seepage face that develops above the tide line at tidal stages below mean tide. These results indicate that placement of seepage meters only on the channel bottom will not give samples or measures representative of the total seepage. Of the total recharge only about 5% occurs through the upper part of the creek bank with the remainder infiltrating vertically through the marsh platform during early stages of tidal submergence. For the platform recharge about 80% occurs within 3 m of the creek bank. Thus, most of the water that seeps out of marsh sediments is derived from sediments that lie within several meters of the creek bank and accordingly has had a relatively short residence (one to two years) in the marsh. Compared to the distal portion of the marsh this relatively rapid flushing may enhance the productivity of Spartina alterniflora in the creek-bank environment and control the differential generation of radium quartet isotopes.

Wave forces on a large, horizontal submerged cylinder

Abstract A numerical boundary integral equation method combined with a non-linear time stepping procedure is used for the calculation of wave forces on a large, submerged, horizontal circular cylinder. As the method is based on potential theory, all computations are performed in the inertia dominated domain, that is, for small Keulegan-Carpenter numbers. Computations are carried out for the Eulerian mean current under wave trough level equal to zero. When the cylinder is moved towards the sea bed the computations show that the inertia coefficients increase significantly, which is associated with a blockage effect. Furthermore, the effect of the wave steepness is reduced when the submergence of the cylinder is increased. In the vicinity of the free water surface the vertical inertia coefficient is highly dependent upon the wave steepness, which tends to reduce it, whereas the horizontal inertia coefficient is only slightly dependent on the wave steepness. Computations are also carried out for cylinder diameters comparable with the wave length. Finally, inertia coefficients computed by the present method are compared with some analytical results by Ogilvie [(1963), First and second order forces on a cylinder submerged under a free surface. J. Fluid Mech. 16, 451–472]. As long as the assumptions leading to Ogilvie's theory are fulfilled (cylinder radius small compared to the wave length), the results are quite similar.

Effects of non‐welded interfaces on guided SH‐waves

The effects of non‐welded interfaces on guided SH‐waves are examined analytically and numerically using the displacement‐discontinuity model to describe the mechanical shear stiffnesses of the interfaces. A modal solution for guided SH‐waves is developed for a layer in non‐welded contact with underlying and overlying half‐spaces. For low interface stiffnesses, the modal solution degenerates to the SH‐plate wave solution and for high interface stiffnesses, to the Love channel wave solution. It is demonstrated that a single non‐welded interface separating two half‐spaces is incapable of supporting an SH‐interface wave. The effects of interface stiffness on the phase velocities and displacement profiles of guided SH‐waves are examined for a low velocity layer in non‐welded contact with bounding half‐spaces using the modal solution and a numerical boundary integral equation code. The results of this analysis demonstrate that the magnitude of the interface stiffness can have a first order effect on the velocities, amplitudes and displacement profiles of guided SH‐waves in a low velocity layer which should make these waves highly detectable in crosshole seismic surveys.

The asymptotic computational ansatz: Application to critical angle beam transmission boundary integral equation solution

Surface wave generation arising from critical angle beam transmission at a smooth but arbitrarily curved fluid–solid interface is studied using a numerical boundary integral equation (BIE) technique that incorporates a priori field information to significantly reduce computation. The technique substitutes a GTD-type computational ansatz into the BIE, where the phase factors of the ansatz are prescribed via asymptotic analysis, thus transforming the unknown variables to slowly varying amplitude factors. In situations where the asymptotic evaluation of these amplitude factors becomes intractable or inaccurate, common numerical integral equation methods are employed to determine the amplitude factors. The technique yields exact (in the numerically convergent sense) results for high-frequency scattering problems unsolvable by prevalent analytic or numeric techniques. Numerical implementation of the technique for critical angle beam transmission is demonstrated.

Boundary integral equation fracture mechanics analysis of plane orthotropic bodies

In this paper, a quick and efficient means of determining stress intensity factors, KI and KII, for cracks in generally orthotropic elastic bodies is presented using the numerical boundary integral equation (BIE) method. It is based on the use of quarter-point singular crack-tip elements in the quadratic isoparametric element formulation, similar to those commonly employed in the BIE fracture mechanics studies in isotropic elasticity. Analytical expressions which enable KIand KII to be obtained directly from the BIE computed crack-tip nodal traction, or from the computed nodal displacements, of these elements are derived. Numerical results for a number of test problems are compared with those established in the literature. They are accurate even when only a very modest number of boundary elements are used.

Axisymmetric boundary integral equation analysis of interface cracks between dissimilar materials

The numerical boundary integral equation (BIE) method with quadratic quarter-point crack-tip singular elements is used to analyse interface cracks between dissimilar material in axisymmetry. Such crack problems present modelling difficulties using conventional procedures for obtaining the stress intensity factors. This is because of the oscillatorily singular nature of the stresses in the vicinity of the bimaterial interface crack-tip. Analytical expressions for the direct evaluation of the fracture characterising parameters from the BIE numerical results of displacements or tractions are derived. Three different crack problems are investigated, two of which have known solutions in the literature. Excellent agreement between the BIE results and these other established solutions are obtained even with relatively coarse mesh discretisations. The present study illustrates the ease with which the BIE method may be used in the fracture analysis of both straight and curved binaterial interface cracks.

On the Numerical Boundary Integral Equation Method for Three-Dimensional Dynamic Shear Crack Problems

The numerical boundary integral equation method for the solution of dynamic shear crack problems is discussed in detail, including the questions of the numerical approximation, stability, and efficiency. In particular, the necessary conditions for stability are developed. In addition to a discussion of the original algorithm proposed by Das (1980), a new version is proposed which is more efficient for some important particular cases, due to the fact that the domain of integration for the necessary convolutions is limited only to the slipping part of the crack plane instead of the entire disturbed area of the crack plane.

Stress redistributions in thick-walled cylinders due to the introduction of a cross-bore after autofrettage

The numerical boundary integral equation method for three-dimensional elastostatics is employed for the stress analysis of a thick-walled cylinder with a radial cross-bore. The geometries considered are for cylinders with radius ratio k = 2.0 and 2.25, and with the cross-bore radius one-quarter that of the cylinder bore. Results showing the elastic stress distributions in an internally pressurized un-autofrettaged cylinder with the radial hole are presented. Using the classical superposition principle, the stress redistributions in partially autofrettaged tubes, due to the introduction of a cross-bore after autofrettage, are also obtained. The maximum percentage overstrain that can be induced in the cylinder with k = 2.0 without re-yielding occurring when the radial hole is introduced, is found to be only 35 per cent. The corresponding percentage overstrain for the cylinder with k = 2.25 is 27.5 per cent. From the computational results, it is also found that the variations of the resultant residual hoop stresses across the cylinder wall near the cross-bore and in the radial plane containing its axis, may be represented by a bi-linear form. Further, the limiting internal pressure that the vessel can sustain without re-yielding occurring, is not increased if the process of autofrettage is applied prior to the introduction of the cross-bore.

Elastic-Plastic Analysis of Thin Sheets with Pressurized Circular Holes

Using the numerical boundary integral equation (BIE) method for two-dimensional elasto-plasticity based on the quadratic isoparametric formulation, solutions for the deformations and stresses in two 2024-T81 aluminum sheets with pressurized circular holes are obtained. Results are presented showing the plastic zone developments in these structures, the variations of the stresses when subjected to the applied radial pressures at the hole peripheries, as well as the residual stress distributions when the loads are completely removed. Where appropriate, the computed solutions are compared with those obtained analytically for a similar problem.

Numerical boundary integral equation method for elastodynamics. 1 : Cole, D M; Kosloff, D D; Minster, J B Bull Seismol Soc Am, V68, N5, Oct 1978, P1331–1357

Numerical boundary integral equation method for elastodynamics. 1 : Cole, D M; Kosloff, D D; Minster, J B Bull Seismol Soc Am, V68, N5, Oct 1978, P1331–1357

A numerical boundary integral equation method for elastodynamics. I

abstract The boundary initial value problems of elastodynamics are formulated as boundary integral equations. It is shown that these integral equations may be solved by time-stepping numerical methods for the unknown boundary values. A specific numerical scheme is presented for antiplane strain problems and a numerical example is given.