Comput Mech (2005) 36: 245–254 DOI 10.1007/s00466-004-0661-2
ORIGINAL PAPER
George S. A. Fam Æ Youssef F. Rashed
The Method of Fundamental Solutions applied to 3D structures with body forces using particular solutions
Received: 1 July 2004 / Accepted: 18 January 2005 / Published online: 24 May 2005 Springer-Verlag 2005
Abstract This paper presents the Method of Fundamental Solutions for three-dimensional elastostatics with body forces. The gravitational body loading is considered as an example for the treated body forces. A new set of particular solutions corresponding to such loading is derived using Ho¨rmander operator-decoupling technique, and the relevant particular solution expressions for displacements, tractions and stresses are derived and given explicitly. Several examples are tested and the results confirm the validity and efficiency of the presented method. Keywords The method of fundamental solutions Æ body forces Æ particular solutions Æ Hormonder method Æ three-dimensional elastostatics
1 Introduction Nowadays, the main purpose of researchers in computational mechanics is to improve numerical techniques for solving problems governed by differential equations. The more the proposed numerical technique is powerful and reliable, the more it appeals to engineers to be used in practical applications. The term ‘‘powerful and reliable’’ is measured by the computing speed, storage requirements and the pre-processing and post-processing requirements. It is also dependent on how suitable the method is when dealing with large applications. The finite element method (FEM) [1] is considered to be the most suitable method as it fits many (not all) of the formerly mentioned requirements. It has a strong theoretical basis, and is well-tested in various engineering applications. Despite the huge advances in the FEM G. S. A. Fam Wessex Institute of Technology, Southampton, SO40 7AA, UK Y. F. Rashed (&) Dept. of Structural Engineering, Cairo University, Giza, Egypt E-mail:
[email protected] Tel.: 0020-10-511-2949
developments, still a lot of research is carried out seeking more improvements. Such improvements can be done within the FEM itself, or using alternative methods, such as the boundary element methods (BEM) [2]. The BEM added many advantages to the numerical solution procedures such as improving the accuracy and reducing the pre-processing requirements; however, disadvantages had also appeared such as the nature of the fully populated matrices involved, which deteriorates the numerical stability especially for large problems. Many researches have been carried out to overcome such disadvantage, e.g., the use of multi region technique [3], or the multi pole expansion method [4]. However, to the authors’ opinion, still a lot of additional research has to be done in seeking for more improvements. The popular BEM formulation presented in the literature is well known as the direct BEM formulation [2], which is mainly a form of application of the Betti reciprocal theory [2]. Other researchers, such as Crouch and Starfield [5] used an alternative BEM formulation named the indirect formulation, which is based on superposition of the influence of sources at field or observation points. Butterfield and Brebbia [6] proved that both the direct and the indirect BEM are equivalent. The indirect BEM has an advantage over the direct method, i.e., it can be used via a discrete version. In other words, the formulation can be set up by using a simple summation of the influence of the sources. According to the previous experience of the authors [7], the obtained accuracy of the solution in the indirect formulation is superb compared to other numerical methods. The discrete (meshless) version of the indirect BEM, where the sources are placed away from the body is called in the literature by various names, such as the Charge Simulation method [8], the Modified Trefftz method [9], the Fundamental Collocation Method [10], the Indirect Discrete Boundary Method [11, 12] or, more frequently, the Method of Fundamental Solutions (MFS) (e.g., [7, 13, 14, 15]). The MFS appeared in the work of several researchers in different fields of applications. For elasticity, previous work was reported by Patterson and Sheikh [16], Re-
246
dekop [17] and Murashima et al. [8] for planar problems and Karageorghis and Fairweather [18] and Redekop and Thompson [19] for axisymetric problems and by Patterson and Sheikh et al. [9, 12], Wearing and Sheikh [11], Redekop and Cheung [20] and Poullikas et al. [21] for three-dimensional problems. Patterson and Sheikh [9] studied stress concentration in a perforated plate under tension and bending of a rectangular bar. The sources were located along the normal to the boundaries at the collocation points at a distance equal to double of the inter-nodal spacing. Patterson, Sheikh and Scholfield [12] studied the application of the MFS on a 3-D circular bar with a spherical cavity. The sources were placed along the normal too but at a distance of five times the minimum inter-nodal spacing. Wearing and Sheikh [11] presented the stress distribution around a spherical hole in a cylindrical bar subjected to tensile loading. The sources were placed in a similar way to that of references [9] and [12]. Redekop and Cheung [20] studied the cases of an infinite domain with a spherical cavity, a solid cube and a cube with a spherical cavity, each under uniaxial tension. For the infinite domain the sources were placed on a spiral fictitious surface inside the cavity. For the two other cases, the sources were placed on planes parallel to the cube boundaries with an offset distance of 0.1–5 times the dimensions of the cube. Poullikas et al. [21] presented the kernels for displacement in cartesian form and used non-linear least squares algorithms for minimizing the error, using 1000–5000 iterations. They presented in their work simple cases of a cube and a cylinder subjected to dirichlet conditions. Other studies for the application of the MFS to different problems were carried out by many researchers such as Fam and Rashed [7] for potential problems, Chen [13] for non linear thermal explosions, Koopman et al. [22] for acoustics, Karageorghis and Fairweather [14] for biharmonic equation, Burguess and Mahajereen [10] and also Alves et al. [23] for Poisson equation. Partridge and Sensale [15] used the MFS in combination with the Dual Reciprocity Method (DRM) for diffusion and diffusion convection. Golberg et al. [24] used the method for time dependant problems. Other meshless techniques appeared in the literature such as the Element-free Galerkin methods [25], the Kernel methods, the Moving Least Squares method, the partition of unity methods [26], the Radial Basis Function collocation method [27], the Direct Trefftz method [28], The Boundary Particle method [29], Meshless Local Petrov–Galerkin, and Local Boundary Integral Equation [30]. One of the topics, which represented a real challenge for boundary element researchers, has been the treatment of domain integrals, such as the body force term, as it requires domain discretization which spoils the major advantage of boundary methods. Several researchers (see, e.g., [31]) summarized methods of transforming such integrals to boundary ones. These techniques are generally useful for the boundary type methods such as the BEM. However, for meshless
methods such as the MFS, these techniques are not efficient, as they spoil the advantage of having no mesh. Recently, Medeiros and Partridge [32] applied DRM to treat body forces in two-dimensional elasticity problems using the MFS. This could be advantageous when the body force term cannot be described by a known function. However, in case of gravitational loading, the DRM is inefficient as it approximate particular solution, which can be obtained analytically. Pape and Banerjee [33] computed analytical particular solutions for the treatment of body loading in the direct BEM for twodimensional elasticity problems. In this paper, the MFS is reviewed for three dimensional elasticity problems. A new set of particular solutions, corresponding to displacements, strains, stresses and tractions, is derived to treat gravitational loading. Ho¨rmander method is used as a systematic technique to derive particular solution expressions. The implemented formulation is however general and can be easily extended to other types of body loads or domain integrals. The main advantage of this method is that the effect of body forces is added directly to the equation as a mathematical expression, i.e., no integration is involved. This formulation is then implemented into computer code. The presented examples cover the application of the presented formulation in problems with body loads and the results demonstrate the validity and accuracy of the presented formulation. It has to be noted that equations in the following sections are written using the indicial notation.
2 The method of fundamental solutions The Method of Fundamental Solutions (MFS) is based on calculating the solution (stress, strain, traction or displacement) at the target point as a superposition of the effect of fictitious sources located at arbitrary chosen positions and it can be represented as follows: N X uti ðXm Þ ¼ UT ij ðXm ; nn Þ uj ðnn Þ þ utib ðXm Þ ð1Þ n¼1
where, UTij*(Xm, nn)is the fundamental solution kernel (for traction or displacement) calculated between point Xm belonging to the solution domain X and the source nn, uj ðnn Þ is the intensity of the fictitious source nn uti(Xm)is the component of the solution (traction or displacement) at point Xm, in the direction of i. utib ðX m Þ is the expression presenting body loads. It has to be noted that theoretically, the problem domain is considered to be infinite, and the solution domain is represented by boundary conditions along the solution domain boundaries (Fig. 1). In order to obtain the expressions for the fundamental solution, some fundamental concepts in elasticity are reviewed in brief [34]. The governing differential equation for 3-D elasticity (Navier equation) for a homogeneous isotropic linearly elastic material (with
247
Fig. 1 The solution domain W embedded in an infinite space and the placement of sources
Young’s modulus E and Poisson’s ratio m) in terms of displacements uj can be written in the following form: Lij ðuj Þ þ bi ¼ 0
ð2Þ
where bi is the body forces term, Lij is the Navier differential operator, and it can be defined as follows: G Lij ¼ Gr2 dij þ oi oj ð3Þ 1 2m or G G o1 o1 o1 o2 Gr2 þ 12m 12m ; L¼ G G Gr2 þ 12m o2 o2 12m o2 o1 E in which, G ¼ 2ð1þmÞ is the shear modulus. The solution of this differential equation is expressed as the summation of a complementary solution uci and a particular solution upi
ui ¼
uci
þ
upi
ð4Þ
The complementary solution satisfies the following homogeneous partial differential equation (recall equation (2)) Lij ðucj Þ ¼ 0
ð5Þ
Equation (5) can be solved using the well-known MFS in equation (1), where Uij ðX ; nn Þ ¼
1 16pGð1 mÞrðX ; nn Þ ð3 4mÞdij þ r;i ðX ; nn Þr;j ðX ; nn Þ
Fig. 2 Tension in a bar under its own weight
1 Uij;k ðX ; nn Þ þ Ukj;i ðX ; nn Þ 2 From equations (6) and (7), we get 1 dik r;j ðX ; nn Þ eijk ðX ; nn Þ ¼ 16pGð1 mÞr2 ðX ; nn Þ eijk ðX ; nn Þ ¼
3r;i ðX ; nn Þr;j ðX ; nn Þr;k ðX ; nn Þ ð1 2mÞ½dij r;k ðX ; nn Þ þ dkj r;i ðX ; nn Þ
ð7Þ
ð8Þ
From Hook’s law [2], E m dik ehjh ðX ; nn Þ þ eijk ðX ; nn Þ rijk ðX ; nn Þ ¼ ð1 þ mÞ ð1 2mÞ ð9Þ Therefore, the stress kernels can be obtained as follows: E rijk ðX ; nn Þ ¼ 16pGð1 mÞð1 þ mÞr2 ðX ; nn Þ ð1 2mÞ½dik r;j ðX ; nn Þ dik r;j ðX ; nn Þ dkj r;i ðX ; nn Þ 3r;i ðX ; nn Þr;j ðX ; nn Þr;k ðX ; nn Þ
ð6Þ
in which r(X, nn)is the Euclidean distance between X and nn , ri ðX ; nn Þ ¼ xi ðnn Þ xi ðX Þ is the component of r in the direction of xi , dij is the Kronecker delta symbol and i ðX ;nn Þ r;i ðX ; nn Þ ¼ orðXox;ni n Þ ¼ rrðX ;nn Þ is the derivative of r w.r.t. the coordinate xi, represents the displacement value at any point X due to concentrated unit point load acting at the source nn (Kelvin’s fundamental solution of elastostatics [2]). The strain kernels can be obtained as follows:
ð10Þ The tractions kernel can be obtained using the following stress-traction relationships Tik ðX; nn Þ ¼ rijk ðX; nn Þnk ðX Þ
ð11Þ
Substituting from the stresses kernel (10), it gives 1 Tih ðX ; nn Þ ¼ 8pð1 mÞr2 ðX ; nn Þ r;n ðX ; nn Þ½ð1 2mÞdih þ 3r;i ðX ; nn Þr;h ðX ; nn Þ þð1 2mÞ½r;i ðX ; nn Þnh ðX Þ r;h ðX ; nn Þni ðX Þ
ð12Þ
248 Fig. 3 Deflection along the bar centerline loaded by its selfweight
T Wi ðX Þ ¼ qgdi3 det co Ladj ij
3 Particular solution derivation
ð19Þ
In order to obtain the solution of the complete problem, the expressions for the particular solution has to be obtained first. The particular solution for equation (2) is any solution that satisfies
which can be written as follows (recall equation (18))
Lij ðupj ðX ÞÞ þ bi ðX Þ ¼ 0
A suitable particular solution for (20) can be obtained in the following form
ð13Þ
r6 Wi ðX Þ ¼
with no boundary conditions. The considered body force here is the self weight and is expressed as
Wi ðX Þ ¼
bi ðX Þ ¼ qgdi3
where:
ð14Þ ð15Þ
The steps to derive such expression for the particular solution are briefly shown. Further explanation of the method can be found in [35, 36, 37]. Considering the differential equation (15), and following Ho¨rmander operator decoupling technique, the cofactor matrix of the adjoint operator can be obtained as follows G2 co adj 2ð1 mÞdij r4 r2 oi oj Lij ¼ ð16Þ 1 2m For 3D elasticity equations, the cofactor matrix is selfadjoint, therefore T co adj Lij ¼ co Ladj ð17Þ ij and its determinant can be computed as follows: T 2G3 ð1 mÞ ¼ r6 det co Ladj ij 1 2m
ð1 2mÞqg ðxh ðX Þxh ðX ÞÞ3 di3 2G3 ð1 mÞ 5040
ð20Þ
ð21Þ
xh ðX Þxh ðX Þ ¼ x1 ðX Þ2 þ x2 ðX Þ2 þ x3 ðX Þ2
Substituting from (14) into (13) Lij ðupj ðX ÞÞ ¼ qgdi3
qgð1 2mÞ di3 2G3 ð1 mÞ
ð18Þ
According to Ho¨rmander [37], a scalar potential Yi is needed to be computed from the following relationship:
Finally, the particular solution of the original Navier equation can be obtained as follows [37] upi ðX Þ ¼co Ladj ij Wj ðX Þ
ð22Þ
It has to be noted that the relevant derivatives of Yj used to compute the above expression (22) are r2 oij Wj ðX Þ ¼
ð1 2mÞqg di3 60G3 ð1 mÞ 2xi ðX Þxj ðX Þ þ dij xh ðX Þxh ðX Þ
ð23Þ
and r4 Wj ðX Þ ¼
qgð1 2mÞ di3 xh ðX Þxh ðX Þ 12G3 ð1 mÞ
ð24Þ
This leads to the following expression for the particular solution upi qg upi ðX Þ ¼ 60Gð1 mÞ ð25Þ ½ð9 10mÞdi3 xh ðX Þxh ðX Þ 2xi ðX Þx3 ðX Þ
249 Fig. 4 Normal stress distribution along the bar centerline
and upj;i ðX Þ ¼
qg 2ð9 10mÞdj3 xi ðX Þ 2di3 xj ðX Þ 60Gð1 mÞ 2dij x3 ðX Þ ð27Þ
Fig. 5 Bending of cantilever under its own weight
Differentiating the displacement expression given in (25) it gives qg upi;j ðX Þ ¼ 2ð9 10mÞdi3 xj ðX Þ 60Gð1 mÞ ð26Þ 2dj3 xi ðX Þ 2dij x3 ðX Þ
Fig. 6 Deflection along the beam neutral axis
Substituting in the relationship (7), the particular solution expression for the strains can be written as follows qg ð8 10mÞ dj3 xi ðX Þ þ di3 xj ðX Þ epij ðX Þ ¼ 60Gð1 mÞ 2dij x3 ðX Þ ð28Þ
250 Fig. 7 Absolute error in deflection along the beam neutral axis
Fig. 8 Normal Stresses over the cross section at the middle of the beam
Fig. 9 Absolute error in normal stresses over the cross section at the middle of the beam
and ephh ðX Þ ¼
qg ½10ð1 2mÞx3 ðX Þ 60Gð1 mÞ
rpij ðXÞ ¼ ð29Þ
From equation (9), the particular solution for stresses can be written as follows
qg ð4 5mÞ dj3 xi ðX Þ þ di3 xj ðX Þ 15ð1 mÞ ð1 5mÞdij x3 ðX Þ
ð30Þ
251
u=0
greater than the number of sources), at which the boundary conditions are known, are chosen. The following system of equations is obtained through collocation at these points Which can be written in the indicial notation format as follows: UTij ðXm ; nn Þ:uj ðnn Þ ¼ uti ðXm Þ utip ðXm Þ
t=0
ð33Þ
where utip ðXm Þ is the particular solution expression (for displacements or tractions) at point Xm. The only unknown in the above linear system of equations, is the {u} vector, which can be easily obtained by solving the linear system of equations (32). Hence, the displacement ui ðX m Þ and traction ti ðX m Þ at any point can be obtained as follows N X ui ðX m Þ ¼ Uij ðXm ; nn Þ uj ðnn Þ þ upi ðX m Þ ð34Þ n¼1
Fig. 10 Fixed Slab under its own weight
ti ðX m Þ ¼
N X
Tij ðXm ; nn Þ uj ðnn Þ þ tip ðX m Þ
ð35Þ
n¼1
and from (11), the tractions particular solution is Similarly, for stresses rik ðXm Þ and strains eik ðXm Þ, at any point can be computed as follows obtained as follows N X qg ð4 5mÞ xi ðX Þn3 ðX Þ þ di3 xj ðX Þnj ðX Þ rik ðXm Þ ¼ tip ðX Þ ¼ rijk ðXm ; nn Þ:uj ðnn Þ þ rpik ðX m Þ ð36Þ 15ð1 mÞ n¼1 N ð1 5mÞni ðX Þx3 ðX Þ ð31Þ X eik ðXm Þ ¼ eijk ðXm ; nn Þ:uj ðnn Þ þ epik ðX m Þ ð37Þ n¼1 It has to be noted that expressions (25), (28), (30) and (31) can be used to obtain the particular solution term of equation (4). 1 0 u1 ðn1 Þ 1B 0 C UT11 ðX1 ; n1 Þ UT12 ðX1 ; n1 Þ UT13 ðX1 ; n1 Þ UT11 ðX1 ; n2 Þ UT13 ðX1 ; nN Þ B u2 ðn1 Þ C C B CB B UT ðX ; n Þ UT ðX ; n Þ UT ðX ; n Þ UT32 ðX1 ; n1 Þ UT33 ðX1 ; n1 Þ CB u3 ðn1 Þ C B 1 1 1 1 1 1 21 22 23 C CB B CB u1 ðn2 Þ C B UT11 ðX ; n Þ C 2 1 CB B CB . C B . CB . C B .. AB . C @ C C B A @ UT33 ðXM ; nN Þ UT31 ðXM ; n1 Þ u3 ðnN Þ 1 1 0 p 0 ut1 ðX1 Þ ut1 ðX1 Þ B ut2 ðX2 Þ C B utp ðX1 Þ C C C B 2 B C C B B B ut3 ðX1 Þ C B ut3p ðX1 Þ C C C B p B ð32Þ ¼ B ut ðX Þ C B ut ðX Þ C B 1 2 C B 1 2 C C C B B .. .. C C B B A A @ @ . . p ut3 ðXM Þ ut3 ðXM Þ
4 Numerical implementation
5 Numerical examples
In order to implement equation (1) numerically, the intensity of the sources uj ðnn Þ needs to be determined first. Hence, a number of boundary points (equal or
In this section, several examples are presented to demonstrate the validity of the present formulation. The results for displacements and stresses are compared
252 Fig. 11 Deflection along the slab horizontal centerline
Fig. 12 3-D deformed shape of the slab
Fig. 13 Moment distribution along the slab horizontal centerline
to their analytical values or to values obtained from other numerical methods. A parametric study is presented to demonstrate the effect of the number of sources and boundary points on the numerical accuracy. The location of sources can be chosen arbitrary [7], however, in the following examples sources are placed around the problem domain with a chosen offset to form a similar pattern to that of the boundary points.
Example 1 – Bar under its own weight This example represents a 3-D bar (10 · 1 · 1 m) fixed from the top and loaded by its own weight (see Fig. 2). The analytical solution can be obtained as follows: qg 2 d¼ z L2 ð38Þ 2E where z is the distance measured from the free end of the bar, d is the deflection of the bar at distance z, q is the
253
density of the bar material (=104 kg/m3), g is the gravitational acceleration (=10 m/s2), L is the total length of the bar. Results at points chosen along the bar centerline for displacement and stresses are demonstrated in Figs. 3 and 4, respectively. A distribution of (2 · 2) boundary points on the fixed end, and (1 · 1) on the other sides gave reasonable accuracy (within 5% error). By increasing the number of points to (2 · 2) on all sides, the obtained numerical results were exactly the same as the analytical ones for both displacements and stresses (almost 0% error).
Example 2 – Bending of cantilever under its own weight A prismatic cantilever beam (5 · 1 · 0.5 m) subjected to bending under its own weight, as shown in Fig. 5, is considered in this example. The analytical value of deflection at the beam centerline can be computed from the following expression: t¼
q:g:b:h:x2 2 x þ 6L2 4Lx 24EIu
ð39Þ
where, q is the density of the bar material (q ¼ 5 · 103 kg/m3), g is the gravitational acceleration (= 10 m/s2), b, h and L are the width (= 1 m), height (= 0.5 m) and length (= 5 m) of the beam respectively, x is the distance measured from the fixed end of the beam, Iu is the area moment of inertia for the beam cross section (Iu ¼ bh3/12), Results for displacement and stresses along the beam and their corresponding absolute error are shown in Figs. 6, 7, 8, 9. A distribution of (10 · 2) boundary points on the sides and (2 · 2) on the end faces gave acceptable results (within 9% error at the free end for displacement values, and about 9% in normal stress calculations). However, a (10 · 3), (6 · 3) and (4 · 3) pattern of boundary points on the sides, together with a (3 · 3) on both ends, gave highly accurate results (within 0.8% error for displacements, and 0.35% error for normal stresses). A lower number of boundary points (3 · 3) gave almost the same accuracy for displacement but with lower accuracy for normal stresses values (error up to 2.5%).
Example 3 – Fixed slab under its own weight A slab of (5 · 5 · 0.2 m) subjected to bending load resulting from its own weight (Fig. 10) is considered in this example. The slab is fixed from the 4 sides, whereas the upper and lower (5 · 5 m) faces are free (ti ¼ 0, i ¼ 1, 2 and 3). The deflection of the slab is computed using the presented MFS formulation, and is compared to results obtained from direct BEM formulation for thick plates (Reissner’s plates). The deflection along the horizontal
centerline in the direction of X is demonstrated in Fig. 11, whereas the 3-D deflected shape is demonstrated in Fig. 12. The moment distribution along the same horizontal axis is demonstrated also in Fig. 13. The results shown for the presented MFS were obtained using 90 collocation points on the boundaries (5 · 5 on upper and lower free side, and 5 · 2 on each fixed side), and are compared to direct BEM mesh of 20 quadratic elements. From Figs. 11 and 13, it can be seen that the results for both MFS and direct BEM are almost identical (difference within 4%).
6 Conclusions The implementation of the MFS for 3D elastostatics applications with body forces was discussed in this paper. A new set of particular solutions for displacements, tractions and stresses were derived and given explicitly using Ho¨rmander operator decoupling technique. Several examples for the application of the presented method were given showing highly accurate results with relatively low number of boundary points. It has be noted that the formulation presented here for self weight body forces can be extended using the same technique to represent other types of body forces. Acknowledgement This research is supported by the EPSRC grant number GR/R45505.
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