COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Analysis of Concentrated Boundary Loads in the Scaled Boundary
Finite Element Method
T. H. Vu *, A. J. Deeks
School of Civil and Resource Engineering, University of Western Australia, Perth, WA 6009 Australia
Email: [email protected] , [email protected]
Abstract This paper introduces a technique for solving concentrated boundary load problems using the scaled
boundary finite element method. By employing fundamental solutions for the displacements and the stresses, the
solution is computed as summation of a fundamental solution part and a regular part. The singularity of the solution is
modelled exactly by the fundamental solution, and only the regular part, which enforces the boundary conditions of the
domain onto the fundamental solution, needs to be approximated in the solution space of the basic scaled boundary
finite element method. A solution with high accuracy is obtained when solving problems which have not previously
been solved effectively, such as problems with a concentrated load on the boundary, and problems which have not
previously been solved with the scaled boundary finite element method, such as the problem with a concentrated load
acting on one of the side faces (the straight boundary sections which intersect at the scaling center of the domain).
Examples provided illustrate that the new approach is simple to implement and accurate for solving concentrated
boundary load problems using the scaled boundary finite element method.
Key words: scaled boundary finite element method, dipole image technique, fundamental solution, Flamant’s
solution, Cerruti’s solution.
INTRODUCTION
The scaled boundary finite element method [1] models linear elastostatics problems excellently and out-performs the
finite element method (FEM) when solving problems involving unbounded domains or stress singularities. Currently
concentrated loads on the boundary are included in a finite element manner by arranging the nodes at the loading
locations so that the input concentrated loads can be modelled as nodal forces. In 2D problems the displacements and
stresses in all directions vary in proportion to r −1 , with r representing the distance between the source point (where
the concentrated load is located) and the point of interest. The solutions are singular at the source point and no longer
remain in the solution space of the scaled boundary finite element method. Consequently, this approach is not efficient
as an extremely fine mesh is required near the concentrated load location to approximate the singularity. But no matter
how far the mesh is refined, clearly the numerical solution can never model exactly the singularity. In many cases the
domain is modelled with side faces (the straight boundary sections which intersect at the scaling center of the domain)
to utilize the advantage of the scaling center in producing an analytical solution at a point of singularity, such as at a
sharp corner point or a stress discontinuity point. However, until now there has been no attempt made to solve
problems subjected to concentrated loads on the side faces. This paper introduces a technique to solve problems
involving concentrated loads on the boundary and on the side faces efficiently with the scaled boundary finite element
method by using fundamental solutions.
The technique presented in this paper was inspired by the way that the Green’s function can be computed with high
accuracy by splitting it into a fundamental solution and a regular part, and the finite element method used only to solve
for the regular part of the function [2-6]. For two dimensions, the fundamental solution is the exact solution at an
observation point in an infinite or semi-infinite plane when a unit concentrated force acts at a distant source point. The
regular part matches the boundary values of the fundamental solution with the enforced boundary conditions. Since the
regular part does not need to represent singularities or step variations in the applied loading function, this approach is
particularly efficient. Here the same technique is used to produce the displacement field and the stress field due to
concentrated loads in the case of linear elastostatics problems, but with the regular part of the solution being obtained
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with the scaled boundary finite element method. The proposed procedure includes following steps: The domain of the
unbounded or bounded problem is discretised normally using the scaled boundary finite element method. The
fundamental solutions for the displacements and the stresses are weighted by the value of the concentrated load. A
boundary value problem (BVP) is formulated to solve for the regular part. The boundary values for this problem are
those boundary values necessary to restore the actual boundary conditions of the primal problem in the presence of the
weighted fundamental solutions. The regular part of the solution is a smooth function if the problem is not polluted by
any adverse effect. This is computed very efficiently in the solution space of the basic scaled boundary finite element
model of the problem. The complete final solution is recovered as the summation of the weighted fundamental solution
and the computed approximate regular part. In this paper the proposed technique is applied to 2D problems, but an
extension to any linear elastostatics problem in 3D with concentrated boundary/ side face loads of arbitrary direction is
straightforward as long as there are suitable fundamental solutions available. Examples are provided for
demonstration. The results show that the proposed technique can solve a domain subjected to the concentrated loads to
a high level of accuracy using a comparatively coarse mesh.
FUNDAMENTAL SOLUTION
When loading an infinite plane with a unit concentrated load located at a particular source point x = ( x1 , x 2 ) , the
displacement field caused by the load is defined by the fundamental solution for displacement g 0 [x ] . The associated
stress field is the fundamental solution for stress g1 [x ] . In the notation for location coordinates, the subscripts 1 and 2
denote the x axis and the y axis respectively. In the fundamental solution, the subscripts 0 and 1 denote the appropriate
derivative of the primary solution variable, the displacement. These solution functions are associated with the position
of the source x , and they vary according to position of observation y . Hence the point x is a parameter, while the
observation point y is the true variable for the functions. For example, g 0 [x ] is a function
g 0 [x ] = g 0 ( y , x )
(1)
Basically fundamental solutions specify the exact solution for the unit force case P = (1, 1) . However this study deals
with any concentrated load P = (P1 , P2 ) . For simplicity it is convenient to use the name fundamental solution to
describe the basic fundamental solution weighted by the actual applied concentrated load P = (P1 , P2 ) from this point
forward. For the load P = (P1 , P2 ) in 2D, the fundamental solution matrix with respect to the displacements is
U ( y , x ) = [P1
P2 ][ g 01 ( y , x )
U 12 ⎤
⎡U
g 02 ( y , x )] = ⎢ 11
⎥
⎣U 21 U 22 ⎦
(2)
The three associated fundamental solutions for stresses form the matrix D where
D ( y , x ) = [P1
P2 ][ g111 ( y , x )
g122 ( y, x )
⎡D
g112 ( y, x )] = ⎢ 111
⎣ D211
D122
D222
D112 ⎤
D212 ⎥⎦
(3)
The traction vectors at the point y are computed by forming the scalar product between the stress tensor and the unit
vector normal to the surface. The traction matrix is
T ( y , x ) = [P1
T12 ⎤
⎡T
P2 ][t01 ( y , x ) t02 ( y, x )] = ⎢ 11
⎥
⎣T21 T22 ⎦
(4)
In the matrix component U ij , the index i shows the direction of the concentrated load and the index j represents the
displacement component of interest. In the matrix component Dkij , the index k indicates the direction of the
concentrated load and the indices ij represent the corresponding stress indices.
There are several different formulae available to describe the fundamental solution, depending on the location of the
concentrated load and the nature of the domain.
The most widely used case is that of a full plane loaded in its interior, in which case the displacements are provided by
the so-called Kelvin solution. The fundamental solution consists of a symmetric matrix U, called the Somigliana
matrix [2, 3].
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In the case of a half-plane where the source point is still in the interior but close to the boundary, a dipole image
solution technique can be employed to obtain the fundamental solution [7, 8]. In this technique an extra image source,
which is an image of the main source mirrored on the opposite side of the boundary, is included to cancel the boundary
condition violations caused by the load singularity. The half plane fundamental solution is consists of the summation
of the Kevin solution with a complementary part, which is the contribution of the image source to the main solution.
When the source point is on the half-plane boundary, the fundamental solution is the called the Flamant solution when
the point load is normal to the boundary, and the Cerruti solution when the point load is tangential to the boundary.
These solutions can be found in many textbooks and other related literature. This study uses the formulae detailed by
Kachanov et al [9]. For the sake of simplicity, only the case of a straight boundary is considered in this study. For the
reader who is interested in finding more information on this topic, a study to extend the dipole image technique to
cover for case when the source point is near or on a curve boundary can be found in Wang, Sloan et al [10], and the
references contained therein.
REGULAR PART
The solution of a particular problem of domain Ω subjected to a concentrated load and certain boundary conditions
can be computed as the superposition of an appropriate fundamental solution and a regular part. Consequently, the
displacement field of the problem is computed as
u[x ] = P g 0 [x ] + u R
(5a)
and in similar manner, the stress field is computed as
σ [x ] = P g1 [x ] + σ R
(5b)
The relation between the fundamental solution and the regular part can be described with reference to Fig. 1.
P
P
(a)
(b)
Figure 1: Infinite plate versus the boundary value problem
Fig. 1(a) shows an infinite plate loaded under the concentrated load P . For this situation the displacement field and
the stress field are defined by P g 0 [x ] and P g1 [x ] subsequently. Fig. 1(b) shows the domain of a BVP with Dirichlet
boundary conditions on the left hand side and bottom edges. Although the load is still the same as for the infinite plate,
the displacement field and the stress field of the 2D BVP are different from P g 0 [x ] and P g1 [x ] due to the effects of
the enforced boundary conditions. The regular functions u R and σ R need to be added to P g 0 [x ] and P g1 [x ]
respectively to match the boundary values of these fundamental solutions with the appropriate values prescribed at the
supports.
If the square plate is cut out of the infinite plate along the dashed line, to maintain the equilibrium along the imaginary
cuts the external tractions need to be put in to balance the internal tractions which are caused by the stress field
P g1 [x ] evaluated on the internal boundary. For Neumann boundaries in the actual problem, the boundary condition
for the regular part should be the prescribed boundary tractions subtract these external tractions. For the Dirichlet
boundaries in the actual problem, the prescribed displacements need to balance P g 0 [x ] along the restrained surfaces.
In case the Dirichlet boundary conditions of the main problem have non-zero prescribed deformation, the prescribed
displacement for the regular part is the difference between the actual prescribed displacement and P g 0 [x ] . This
discussion indicates the basic principles from which the BVP for finding the regular part can be formulated.
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THE BOUNDARY VALUE PROBLEM FOR COMPUTING THE REGULAR PART
From the previous discussion about the relationship between the fundamental solution and the regular part, the
boundary conditions for the BVP are defined by the following two equations, assuming that only the concentrated load
presents and there are no prescribed tractions on the Neumann boundaries:
uRi + Uij = ui
on ΓD ;
t Ri + Tij = 0
on ΓN .
(6)
The variational form of regular part of the problem is
a (u Ri , v ) = − ∫ Tij . v ds
∀v ∈ V
(7)
S
which is formulated over the domain Ω ⊂ R 2 . As usual the solution space V is the Sobolev space
V = H 1 (Ω) × H 1 (Ω) .
The boundary conditions of the domain for the regular part of the solution are
uRi = −Uij + ui
on ΓD ;
t Ri = −Tij
on ΓN .
(8)
The exact solution of this BVP is usually not possible to find. Instead, an approximation u Ri ; h , p must be found in the
sub space V h , p ⊂ V , where h refers to the size of the element and p refers to the polynomial order of the shape
function. In this work the BVP is solved using the scaled boundary finite element method in the usual way. Although
the fundamental solutions are evaluated by different sets of formulae depending on the location of the concentrated
load in the domain, this basic procedure for formulating and solving the BVP for the regular part remains unchanged.
EXAMPLES
In the scaled boundary finite element method, concentrated loads acting on the side faces have not been treated before.
In the current approach [11], the solution can be obtained only for distributions of side face load that can be modelled
as a power function of the radial coordinate. Since any continuous function can be represented by a Taylor’s series, this
limits the loads to be continuous functions of the radial coordinate. The new approach can handle the case when the
side face load consists of one or more concentrated loads with a high level of accuracy. In the following examples, the
Flamant solution and the Cerruti solution are subsequently used to solve the domain subjected to the normal to side
face point load and the tangential to side face point load respectively. For verification purposes, the problems are
solved again using the dipole image technique.
The convergence of the new technique is examined by analysing the problem of a plane strain square plate. The plate
is fixed along the left hand side and the bottom edges. In the scaled boundary model the scaling centre is located at the
top corner on the right. This arrangement forms a domain with two free side faces and a fixed discretised boundary.
The discretised boundary is formulated with four spectral elements. A single concentrated load of unit value is
subsequently applied at various locations to examine the performance of the new technique for each individual set of
fundamental solutions. Data is specified in N and mm. The material constants used are: Young’s
modulus E = 250000 N mm −2 and Poisson’s ratio ν = 0.3 . The problem is illustrated in Fig. 2.
The quality of the solution field needs to be examined to judge the performance of the new technique. In each example,
the domain deformation and a contour plot of the stress component which is relevant to the direction of the point load
are presented. The relative error in the approximation is described in terms of the scalar energy norm formed using the
stress field. The discrete relative error is computed on a series of meshes with uniformly increased polynomial order to
examine the convergence rate of the approximation. Since the Lobatto polynomials are anti-symmetric for odd order
and symmetric for even order and the problem is anti-symmetric, the odd order shape functions will contribute more to
the solution than the even ones. This suggests that if the mesh starts with order p = 1 , an increment of 2 be adopted and
the mesh refined subsequently to p = 3 , p = 5 , p = 7 and so on. For convenience, the reference solution, which
“better” approximates the exact solution, is defined by the next uniform refinement in this series.
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B
1000
C
B = (500, 0)
C = (500, -1)
(a)
(b)
Figure 2: A plane strain plate subjected to a single concentrated load subsequently at various locations
1. Point load on the boundary – Solution of a domain subjected to a horizontal point load acting on the side face
In this example, the horizontal concentrated load is applied at point B (Fig. 2). Rather than employing the fundamental
solution for the full plane, a half-plane solution is used, where the side face to which the load is applied becomes the
boundary of the half-plane. Since the load acts tangential to this half-plane, the Cerruti solution [9] is employed as the
fundamental solution. The deformation is plotted in Fig. 3(a) while the normal stress σ xx is contoured in Fig. 3(b).
The accuracy of the solution is demonstrated by examining the convergence of the error in energy norm. The
convergence achieved when increasing uniformly the polynomial order of the Lobatto shape functions is plotted in
Fig. 5.
(a) Exaggerated displacement field
(b) σ xx contour plot
Figure 3: Solution for the plane strain square plate subjected to a side face
concentrated load acting tangential at source point (500, 0)
2. Point load close to the boundary – Horizontal direction When the point load is located comparatively close to
the boundary of the domain, the Kelvin solution generates rapidly varying Neumann and Dirichlet boundary
conditions for the regular part of the solution in the vicinity of the load. This means that a higher degree of refinement
of the mesh used to solve the regular part is required. However, this can be avoided by using the fundamental solution
for a point load applied near the surface of a half-plane. Such a fundamental solution can be generated using the dipole
image technique [7], which introduces some extra complementary parts to eliminate the disturbance appearing in the
Kelvin solution due to the singularity at the source point. The problem is solved in a similar way to the case when the
point load is in the interior of the domain, but with the Kelvin solution replaced by the summation of the Kelvin
solution and the complementary part to form the fundamental solution. For comparison purposes, the problem solved
here is formed by moving the tangential point load on the boundary at point B vertically downward 1mm into the
domain interior. The new location of the source point is indicated by point C in Fig. 2. The computed solution can be
verified by comparison with the solution produced above using the Cerruti solution, since point C is extremely close to
point B, with a difference of only 1mm in vertical coordinate. It is expected that the solutions obtained with the source
point in these two locations should be almost coincident.
The calculated displacement field is shown in Fig. 4(a) and contours of the normal stress σ xx are plotted in Fig. 4(b).
The convergence of the solution is presented in Fig. 5, which also shows the convergence rate obtained when the load
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is applied tangential to the boundary. Since the regular part of both problems is practically identical, there is no
noticeable difference between the convergence rates, as expected.
(b) σxx contour plot
(a) Exaggerated displacement field
Figure 4: Solution for the plane strain square plate subjected to a horizontal concentrated
load extremely close to the boundary at source point (500, -1)
Figure 5: Convergence for the solution of the square plane strain subjected to horizontal concentrated load,
including comparison of the case with the point load acting tangential to the boundary and the case
with the point load extremely close to the boundary
As expected, the solutions from the two examples vary from each other by very small values when comparing
corresponding quantities. When graphed, the difference is hardly noticeable when the results for coincident pairs of
points are compared. In terms of theory, it is expected that the solution for a point load applied at an interior point
( x1 ,− d ) of a half-plane approaches the solution for a point load applied on the half-plane boundary at point ( x1 , 0)
when d → 0 . The purpose of verifying the solution implementation for these two examples is that the similarity in
these results implies that the new technique works well for each individual set of half-plane fundamental solutions, and
it describes the trend of the matching solutions accurately. A similar solution verification process is now attempted for
the case of the point load acting in the vertical direction. Details of this study are presented in following two sections.
3. Point load on the boundary – Solution of a domain subjected to a vertical point load normal to the side face
When the concentrated load is applied in the vertical direction to the top face of the block at point B, the Flamant
solution [9] is employed to describe the fundamental solution. To be compatible with the conventional direction of the
point load in the Flamant problem, the vertical load is applied downwards in the negative y direction.
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The final calculated solution is presented as follows: the displacement field is plotted in Fig. 6(a), the normal stress
σ yy is contoured in Fig. 6(b) and the convergence of the solution is graphed in Fig. 8. The solution converges rapidly,
satisfying the boundary conditions and showing the expected characteristics.
(b) σyy contour plot
(a) Exaggerated displacement field
Figure 6: Solution for the plane strain square plate subjected to a side face concentrated
load acting vertically downward at source point (500, 0)
4. Point load close to the boundary – Vertical direction As in the situation when the horizontal load is close to the
boundary, when the vertical load is applied in the domain but close to the boundary the Kelvin solution leads to rapidly
varying boundary conditions near the source point in the BVP for the regular part. Consequently it is better to use a
half-plane fundamental solution. Here the problem with a concentrated load normal to the side face is modified by
moving the source point 1mm downward to the position point C (Fig. 2). The dipole image technique is employed in a
similar manner to that described for the horizontal load in the last example. The disturbance near the boundary due to
the singularity at source point is cancelled out by adding the complementary parts into the Kelvin solution.
The solution is presented in a similar way to the previous examples: the displacement field is plotted in Fig. 7(a) and
the relevant stress component σ yy is contoured in Fig. 7(b). The convergence rate of the solution is plotted in Fig. 8
together with that for the case with the point load normal to the side face to allow comparison. As in the problems
containing the horizontal load, close agreement between the two solutions is confirmed.
(b) σyy contour plot
(a) Exaggerated displacement field
Figure 7: Solution for the plane strain square plate subjected to a downward vertical concentrated
load extremely close to the boundary at source point (500, -1)
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When looking at various convergence curves of the computed solutions for this square plane strain plate, it is observed
that the solution converges at a rapid rate, and an error of around 3% is obtained when the mesh is refined to 122
degrees of freedom, which corresponds to a uniform polynomial order of 15 for 4 elements in the mesh.
Figure 8: Convergence for the solution of the square plane strain subjected to a vertical concentrated
load, including comparison of the case with the point load acting normal to the boundary
and the case with the point load extremely close to the boundary
CONCLUSION
This paper presents a new technique for solving linear elastostatic problems subjected to concentrated boundary loads
using the concept of the fundamental solution. The solution for the problem is broken into a fundamental solution and
a regular part which is approximated by the usual scaled boundary method. The BVP for solving the regular part is
formulated from the fundamental solution. Results from the examples provided illustrate the high accuracy and
efficiency of the proposed technique. The new technique enhances the solution of domains subjected to concentrated
loads on the boundary and solves problems which have not previously been solved with the scaled boundary finite
element method, namely domains subjected to concentrated loads on the side faces.
The wide range of applications which involve boundary concentrated loads indicates the advantages of implementing
this new technique in the scaled boundary finite element method. Theoretically it is possible to extend the new
technique to treat the problems other than the concentrated load case by employing integration and the principle of
superposition to recover the corresponding fundamental solution for a certain load distribution. In addition, although
this paper covers the 2D case only, the new technique can be extended straightforwardly to problems in three
dimensions where appropriate fundamental solutions exist.
REFERENCES
1. Wolf JP, The Scaled Boundary Finite Element Method. Wiley Chichester, 2003.
2. Hartmann F. The mathematical foundation of structural mechanics. Springer, Berlin, Germany, 1985.
3. Hartmann F. Introduction to boundary elements. Springer, Berlin, Germany, 1989.
4. Grätsch T, Hartmann F. Finite element recovery techniques for local quantities of linear problems using
fundamental solutions. Computational Mechanics, 2003; 33: 15-21.
5. Grätsch T, Hartmann F. Pointwise error estimation and adaptivity for the finite element method using
fundamental solutions. Computational Mechanics, published online(9/12/2005), 2005.
6. Grätsch T, Bathe K. A posteriori error estimation techniques in practical finite element analysis. Computers &
Structures, 2005; 83: 235-265.
7. Telles JCF, Brebbia CA. Boundary element solution for half-plane problems. International Journal of Solids and
Structures, 1981; 17(12): 1149-1158.
8. Yang JD, Kelly DW. Isles JD. A posteriori pointwise upper bound error estimates in the finite element method.
International Journal for Numerical Methods in Engineering, 1993; 36(8): 1279-1298.
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9. Kachanov M, Shafiro B, Tsukrov I. Handbook of elasticity solutions. Kluwer Academic Publishers, Boston,
USA, 2003.
10. Wang, S, Sloan I, Kelly D. Computable error bounds for pointwise derivatives of a Neumann problem. IMA
Journal of Numerical Analysis, 1998; 18: 251-271.
11. Deeks AJ, Wolf JP. A virtual work derivation of the scaled boundary finite-element method for elastostatics.
Computational Mechanics, 2002; 28(6): 489-504.
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