An alternative multi-region BEM technique for layered soil problems
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1 An alternative multi-region BM technique for layered soil problems D.B. Ribeiro & J.B. Paiva Structural ngineering Department, São Carlos ngineering School, University of São Paulo, Brazil. Abstract Different formulations may be found in the literature for the numerical simulation of layered soils. Among these options, the boundary element method (BM) detaches because it requires only the boundary discretisation of the infinite domains involved. The classical way of simulating infinite layered solids with the BM is by imposing equilibrium and displacements compatibility along the contacts. There is, however, an alternative technique which is more accurate because it does not impose these conditions along the interfaces. In this alternative technique, the regions displacement fundamental solutions are related and for each equation all domains are considered as a unique solid. In this chapter, it is demonstrated that this alternative formulation is suited for the simulation of layered soil problems, representing such media as an infinite solid composed of different homogeneous, isotropic and elastic linear domains in contact. In the end of the chapter, this formulation is employed in a threedimensional problem, comparing the results with the classical technique and with an analytical solution. Keywords: Boundary element method, Layered soil, Alternative multi-region technique, Static. Introduction In the literature, many works are dedicated to the simulation of layered soil problems. In such works, when a structure is considered interacting with the soil, WIT Transactions on State of the Art in Science and ngineering, Vol 43, 00 WIT Press ISSN (on-line) doi:0.495/ /
2 6 Recent Developments in Boundary lement Methods one may observe that the mathematical models employed for the structures are closer to reality than the ones used for the soil. This discrepancy is due to the type of materials involved in each case. While structural materials, such as concrete and steel, are produced with some type of technological control, the soil materials are formed due to various conditions of weathering in rocks. The result is a chaotic media where the characteristics are difficult to be numerically modelled, including discontinuities, heterogeneity and anisotropy. Due to these factors, it is necessary to assume simplifications in order to make possible a numerical simulation of layered soil problems with reasonable time processing. In this chapter, the main simplifications adopted are: Representing the soil as an infinite domain in radial directions; Considering the soil as a composition of homogeneous, isotropic, and linear elastic domains; Considering only the static case. In this case, the problem comes down to the static analysis of infinite multiregion domains. Focusing this theme, many formulations are available in the literature and each one of them implies on advantages and disadvantages. However, depending on the problem to be solved, one technique may become more attractive than the others. When it is possible, a good choice is to employ analytical solutions. When correctly programmed, they give trustful results in little processing time. In reference [], for example, an analytical expression is deduced for a circular load applied to the surface of a two-layer infinite domain. The main disadvantage of these solutions is that they are not versatile, suiting only specific situations. If analytical solutions can not be used, then a numerical approach may become attractive. Although the finite element method (FM) is popular, it has some disadvantages compared with other options such as the boundary element method (BM). The FM requires the discretisation of the infinite domain, implying on a high number of elements and leading to a large and, sometimes, impracticable processing time. In order to reduce these inconveniences, some authors use infinite elements together with finite elements, as done in reference []. The main advantage of the BM is that only the boundaries of the domains involved require discretisation. This allows reducing the problem dimension, implying on less processing time. This advantage is explored in several works, and more developments are turning the BM even more attractive to future applications. One of them is using infinite boundary elements, as performed in reference [3]. To consider two or more domains in contact with the BM, several techniques may be employed. The most popular one, which may be consulted in reference [4], is based on imposing equilibrium and compatibility conditions for all interface points between every pair of domains in contact. Using these relations, the matrices obtained for each domain applying the BM formulation WIT Transactions on State of the Art in Science and ngineering, Vol 43, 00 WIT Press ISSN (on-line)
3 Recent Developments in Boundary lement Methods 63 are coupled and all the unknown boundary values may be obtained by solving the same system of equations. In spite of its simplicity, one disadvantage of this formulation is that the final system of equations has blocks of zeros. These blocks can become numerous depending on the number of domains considered, increasing the size of the system and consequently the processing time. In addition to that, the conditions imposed on the contacts do not represent satisfactorily the media continuum, what may cause inaccuracies on the results. The multi-region method presented in reference [5] for two-dimensional elastic and potential problems eliminates the need of equilibrium and compatibility conditions among the interfaces. This approach improves the media continuity compared with the classical formulation, leading to more accurate results. This technique is modified in reference [6] for bending plate analysis with the BM, latter used for bending moments calculation in [7] and finally adapted for three-dimensional elastic problems in reference [8]. The aim of this chapter is to present this alternative multi-region BM technique, which is suitable for layered soil problems. The soil is represented as a multi-region media, similar to the Gibson soil approach. In such a way, the domain is modelled with variable elasticity module and a constant Poisson ratio, which may be considered a disadvantage in certain cases. Nevertheless, by testing this formulation in problems with different Poisson ratios, one may verify that the error introduced by an average Poisson ratio consideration may be considered of little relevance for displacement calculation. In the end, it is viable to employ this formulation in more general problems. Alternative multi-region formulation In Figure, a homogeneous, isotropic and linear elastic domain Ω is presented. The boundary is denoted by, is the elasticity module and ν is the Poisson Figure : Problem with one region. WIT Transactions on State of the Art in Science and ngineering, Vol 43, 00 WIT Press ISSN (on-line)
4 64 Recent Developments in Boundary lement Methods ratio. Point x is called field point, and represents a general point of. Point y is called source point, and can be placed in any point inside Ω, outside Ω or at the boundary. Versor η is normal to at x, and r is the distance between points x and y. The equilibrium of this solid body can be represented by a boundary integral equation called Somigliana identity, which for homogeneous, isotropic and linear-elastic domains is: () ij j ij j ij j c ( y) u ( y) + p ( x, y) u ( x)d ( x) = u ( x, y) p ( x)d ( x). This equation is written for the source point y, where the displacement is uj ( y ). The constant c ij depends on the Poisson ratio and on the position of y. The field point x goes through all boundary, where displacements are uj ( x ) and tractions are pj ( x ). The integral kernels uij ( x, y ) and pij ( x, y ) are Kelvin three-dimensional fundamental solutions for displacements and tractions, respectively, and may be written as follows: u = [(3ν 4) δ + r, r, ], 6 πµ ( ν ) r ij ij i j r pij = [( ) 3,, ] ( )( νδij + rir j + ν ηir, j η jr, i ). 8 π( ν) r η () (3) In eqns () and (3), δ ij is unitary for i = j and zero for all other cases, η is the normal versor at point x (see Figure ) and µ is given by: µ = ( + ν ). (4) The objective is to obtain an integral equation similar to eqn (), but valid for an arbitrary number of domains in contact. In such a way, a demonstration will be held for the most simple case, with only two regions as illustrated in Figure, and the resulting expression will then be extended for an arbitrary number of domains. In Figure, the regions have the same Poisson ratio ν and different elasticity modules, for region Ω and for region Ω. The boundary of region Ω is divided into two parts, and. The part of which is in contact with region Ω is denoted by and the rest of it is named. Analogously, for region Ω, the boundary is divided into for the contact and for the free surface. Consequently: = +, = +. (5) WIT Transactions on State of the Art in Science and ngineering, Vol 43, 00 WIT Press ISSN (on-line)
5 Recent Developments in Boundary lement Methods 65 Kelvin displacement fundamental solutions for regions Ω and Ω may be written as: where u u = [(3ν 4) δ + r r ], 6 πµ ( ν ) r ij ij, i, j = [(3ν 4) δ + r r ], 6 πµ ( ν ) r ij ij, i, j (6) (7) µ =, µ =. ( + ν ) ( + ν ) (8) In such a way, the fundamental solutions may be related as follows: where u = u u = u + u, (9) ij ij ij ij ij ij = i j. (0) One may observe in eqn (3) that the traction fundamental solution does not depend on the elasticity module, except when it is the same for domains Ω and Ω. Thus, Kelvin traction fundamental solution may be represented as p ij for both domains. Considering region Ω as a reference, the deduction is started by writing the Somigliana identity, represented in eqn (). In order to reduce the expressions, the variables in parenthesis will be suppressed. In such a way: () ij j + ij j = ij j c u p u d u p d. Note that if the source point is outside Ω, then the free-term c ij is zero; if it is inside, c ij = and if it is on the boundary, then c ij may be determined using standard BM techniques. Considering eqn (5), the integrals of eqn () may be divided into two parts as shown below: () ij j + ij j + ij j = ij j + ij j c u p u d p u d u p d u p d. An analogous equation may be written for region Ω : (3) ij j + ij j + ij j = ij j + ij j c u pud pud u pd u pd. WIT Transactions on State of the Art in Science and ngineering, Vol 43, 00 WIT Press ISSN (on-line)
6 66 Recent Developments in Boundary lement Methods When eqns () and (3) are added, a single equation which considers both regions is obtained for the point: ( cij + cij ) uj + pijujd + pijujd + pijujd + pijujd = uijpjd + uijpjd uijpjd uijpjd. + + (4) Some terms of eqn (4) may be related. Starting with the left side, the integral over is equal to the one over with an inverted signal. This occurs because the exactly same functions are being integrated only by inverting the integration direction. Thus: d + d = 0. (5) pu ij j pu ij j Analysing now the right side of eqn (4) and considering eqn (9), the following arrangement is made: uijpjd + uijpjd = uijpjd ( ) + + = + uij uij pjd uij pjd uij pjd + d = d. uijpj uijpj (6) The two integrals added inside the parenthesis are equal to zero for the same reason that eqn (5) is zero. One more simplification is possible on the right side of eqn (4) by substituting eqn (9) in one more term, as shown below: uijpjd = uij + uij pjd = uijpjd + d. uij pj After these deductions, eqns (5) (7) are substituted in eqn (4). The result is ( cij + cij ) uj + pijujd + pijujd = uij pjd uijpj uijpj uijpj + d + d + d. Reorganising the terms, the following equation is obtained: ( cij + cij ) uj + pijujd + pijujd = uij pjd uijpj ( uijpj uijpj ) + d + d + d. (7) (8) (9) WIT Transactions on State of the Art in Science and ngineering, Vol 43, 00 WIT Press ISSN (on-line)
7 Recent Developments in Boundary lement Methods 67 The terms inside the parenthesis are functions of the boundary tractions; however, it is necessary to transform them into functions of the boundary displacements. This transformation is possible using eqn (3), isolating one of its integrals for further substitution in eqn (9). In such a way: d d d d. (0) uijpj = cijuj + pijuj + pijuj uijpj Substituting eqn (0) into eqn (9) and combining the integrals inside the parenthesis, the subsequent equation is obtained: ( cij + cij ) uj + pijujd + pijujd = uij pjd uijpj cij pijuj pijuj ( ) + d + + d + d. ( ) c c u p u p u ij + ij j ij jd ij jd + + pijujd + pijujd uijpjd uijpjd. = + Note that two integrals on the left side can still be combined as follows: () () pu ij jd pu ij jd = pu ij jd = pujd. ij The result of eqn (3) substitution into eqn () is: c c u p u p u p u ij + ij j ij jd ij jd ij jd + + uijpj uijpj = d + d. (3) (4) quation (4) is only valid for two domains. xtending it to an arbitrary number of domains, it becomes: nd ne nc s e mn cijs uj pijujd e pijujd mn e mn s + e + c = = = ne = uij pjd e. e e= (5) In eqn (5), the second summation signal is positive because the integration direction was changed from nm to mn. The total number of domains is nd, WIT Transactions on State of the Art in Science and ngineering, Vol 43, 00 WIT Press ISSN (on-line)
8 68 Recent Developments in Boundary lement Methods Table : xamples of c ij. Point position c ij c ij c ij Outside Ω and Ω Inside Ω 0 Over boundary 0 ( ) Over boundary + the number of contact boundaries is nc and the number of external boundaries is ne. The first summation represents the coefficient cij ( y ) of eqn (). Before calculating it, all coefficients c ijs, one for each domain, must be known. Considering all boundaries smooth in Figure, Table presents some examples for this summation calculation. quation () is used as a starting point to obtain the BM system of equations which solution leads to the unknowns of the problem. If the same steps are repeated for eqn (5), valid for multi-regions, a similar system of equations is obtained. The unknowns of this new system are the non-prescribed boundary values plus the interface displacements. The total number of unknowns is reduced compared with the classic multi-region technique, once the interface tractions are not included in this case. This justifies why the alternative formulation leads to less time processing. A better interface continuity is also guaranteed, once all regions are modelled as a unique solid. 3 Results This example aims to analyse an infinite multi-region domain problem with the alternative BM technique. The domain considered is composed by two layers of different elasticity module, as illustrated in Figure 3. Layer has a 9000 kn m elasticity module, a 0.5 Poisson ratio and 5m of thickness. Layer has a 900 kn m elasticity module, a 0.5 Poisson ratio and infinite thickness. Both layers are infinite in radial directions. A vertical circular kn m uniform loading with a 7.5m diameter is applied at the top layer surface. This problem was simulated using a 53 node and 88 boundary element mesh, which may be visualised in Figure 4. It is composed by triangular elements with linear shape functions. Figure 4(a) presents a general view of the mesh and Figure 4(b) presents a detail of the central area. This mesh is employed at the surface and at the layers contact, extending to a distance from which the displacements and tractions could be considered negligible. For the nodes positioned at this WIT Transactions on State of the Art in Science and ngineering, Vol 43, 00 WIT Press ISSN (on-line)
9 Recent Developments in Boundary lement Methods 69 Figure 3: Problem to be analysed. limits, the boundary values are imposed to be zero to better simulate their far field behaviour. Hence, it was considered not necessary to close the boundary at the limits. As commented in the introduction of this chapter, in reference [], an analytical solution is deduced for a circular load applied to the surface of a two-layer infinite domain. Using the values adopted in this example and applying the solution given in reference [], a m vertical displacement is obtained for the central point of the circle. Considering this same point, a m displacement was obtained using the classical formulation with the mesh of Figure 4. mploying the alternative technique and the same mesh, a m vertical displacement was obtained. Both Figure 4: Mesh employed. WIT Transactions on State of the Art in Science and ngineering, Vol 43, 00 WIT Press ISSN (on-line)
10 70 Recent Developments in Boundary lement Methods values numerically calculated may be considered very close to the analytical one, leading to the conclusion that both formulations achieved accurate results in this example. 4 Conclusions In this chapter, an alternative BM formulation suited for layered soil problems was presented. lecting one domain as a reference and establishing relations between its displacement fundamental solution and the ones of the other regions allows integrating all domains as a unique solid. This approach eliminates the need of equilibrium and compatibility relations between the different media, guaranteeing a better continuity along the interfaces. This reduces possible inaccuracies that may occur due to traction approximations along the contacts. A numerical example was presented, in which a two-layer half space was simulated employing the classical and the alternative formulations. The values obtained were compared with an analytical expression. The numerical results were very close to the analytical one, meaning that both techniques were accurate. These low errors were expected, once the problem analysed may be considered simple. References [] Burmister, D.M., The general theory of stresses and displacements in layered systems. I. Journal of Applied Physics, 6, pp , 945. [] Seo, C.G., Yun, C.B. & Kim, J.M., Three-dimensional frequency-dependent infinite elements for soil structure interaction. ngineering Structures, 9, pp , 007. [3] Moser, W., Duenser, C. & Beer, G., Mapped infinite elements for threedimensional multi-region boundary element analysis. International Journal for Numerical Methods in ngineering, 6, pp , 004. [4] Brebbia, C.A. & Dominguez, J., Boundary lements: An Introductory Course, Computational Mechanics Publications: London, 99. [5] Venturini, W.S., Alternative formulations of the boundary element method for potential and elastic problems, ngineering Analysis with Boundary lements, 9, pp , 99. [6] Paiva, J.B. & Aliabadi, M.H., Boundary element analysis of zoned plates in bending, Computational Mechanics, 5, pp , 000. [7] Paiva, J.B. & Aliabadi, M.H., Bending moments at interfaces of thin zoned plates with discrete thickness by the boundary element method. ngineering Analysis with Boundary lements, 8, pp , 004. [8] Ribeiro, D.B. & Paiva, J.B., An alternative multi-region BM technique for three-dimensional elastic problems. ngineering Analysis with Boundary lements, 33, pp , 009. WIT Transactions on State of the Art in Science and ngineering, Vol 43, 00 WIT Press ISSN (on-line)
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