Domain Decomposition Algorithm for Coupling of ...

Domain Decomposition Algorithm for Coupling of Finite Element and Boundary Element Methods A. Eslami Haghighat & S. M. Binesh

Arabian Journal for Science and Engineering ISSN 1319-8025 Volume 39 Number 5 Arab J Sci Eng (2014) 39:3489-3497 DOI 10.1007/s13369-014-0995-9

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Author's personal copy Arab J Sci Eng (2014) 39:3489–3497 DOI 10.1007/s13369-014-0995-9

RESEARCH ARTICLE - CIVIL ENGINEERING

Domain Decomposition Algorithm for Coupling of Finite Element and Boundary Element Methods A. Eslami Haghighat · S. M. Binesh

Received: 31 October 2012 / Accepted: 28 February 2013 / Published online: 7 March 2014 © King Fahd University of Petroleum and Minerals 2014

Abstract In several engineering problems, especially the ones associated with the unbounded domains, the coupling of the finite element method (FEM) and the boundary element method (BEM) improves the efficiency of the numerical analysis. Due to the complexity of direct coupling techniques, iterative domain decomposition method became a popular approach. However, in the conventional domain decomposition algorithms, the boundary conditions at the interface boundary must be of one type, i.e., Neumann/Dirichlet. In this paper a new algorithm is presented for the iterative coupling of the FEM and the BEM. Both Dirichlet and Neumann boundary conditions are assumed simultaneously in different parts of the interface boundary and an iterative procedure is conducted by two relaxation parameters to solve the coupled problem. To demonstrate the accuracy of the proposed method, some numerical examples are investigated at the end of the paper.

1 Introduction

A. E. Haghighat Civil Engineering Department, Isfahan University of Technology, Isfahan, Iran

The finite element method (FEM) and the boundary element method (BEM) are powerful numerical techniques which are widely used for solving various problems in applied science and engineering. Each method has its individual merits. For certain categories of problems, neither the FEM nor the BEM is well-matched and it is rational to couple these two methods to combine their advantages and reduce their disadvantages. Unfortunately, the conventional formulations of the BEM and the FEM are not directly compatible and cannot be linked as they stand. In this regard, the discretized equations for the BEM and the FEM sub-domains are combined to couple the BEM and the FEM directly. See, e.g., References [1–5] not to mention many others. Direct coupling of the FEM and the BEM equations suffers from some deficiencies that are stated briefly as follows:

S. M. Binesh (B) Faculty of Civil and Environmental Engineering, Shiraz University of Technology, Shiraz, Iran e-mail: [email protected]

• The algorithm for direct coupling of the FEM and the BEM is highly complicated when compared with that of each single method [6].

Keywords Finite element method · Boundary element method · Iterative coupling · Interface boundary

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• The application of collocation BEM in the direct coupling process leads to an un-symmetric global stiffness matrix and requires some symmetrization techniques [7, 8]. However, utilizing the symmetric Galerkin BEM rather than the collocation BEM results in a symmetric global stiffness matrix [9]. • Direct coupling of the FEM and the indirect type of BEM is not straightforward, because some density functions are introduced in the indirect BEM formulations which may complicate the coupling process [10,11]. • When a non-linear FEM sub-domain directly combines with a linear BEM sub-domain, iterations are performed for both linear and non-linear parts of the global domain and hence, unnecessary iterations are performed for linear BEM sub-domain. • When the global domain is decomposed into more than two sub-domains, the direct combination of the equations is difficult.

In order to reduce the stated inconveniences associated with the direct coupling of the FEM and the BEM, several studies devoted to the development of iterative coupling methods have appeared [6,12–15]. These iterative coupling techniques, which are also called the domain decomposition method (DDM), can be implemented by a single type of numerical method to reduce the dimension of matrices or to capture the non-homogeneity of the problem domain [12– 14]. It may also be utilized to couple two arbitrary numerical techniques such as BEM/FEM or BEM/mesh-free techniques [16–19]. Details of different approaches of DDM are summarized in References [20,21]. One of the challenges in the DDM is the convergence of the procedure. Some studies proposed an estimation for the optimum value of the relaxation parameter which speeds up the convergence [18,21], and some others introduced an interval in which the convergence is assured [6,22]. In this paper, a new algorithm of DDM is presented. Both displacement and traction are assumed simultaneously in the different parts of the interface boundary and an iterative procedure is conducted by two relaxation parameters to solve the coupled problem. To verify the efficiency of the proposed approach, two problems including a cavity in a full space subjected to uniform pressure on its wall and an elastic halfspace under the loading of a rigid strip footing have been analyzed. The results of the analyses have been compared with the closed form solutions.

2 FEM Formulation The principle of the FE method consists in minimizing the total potential  given by [23]:

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 = 

1 T ε · σ d − 2



 uT · b d −



uT · t d

(1)

t

where σ and ε are stress and strain tensors respectively, u is the displacement field, b is the body force vector, t is the prescribed boundary traction,  is the whole domain and t is the part of boundary on which tractions are imposed. Using the stationary condition for Eq. (1), and substituting the nodal displacement vector into derived relation, the following equation can be obtained [23]: k F uF = F F

(2)

where, kF , uF , F F are stiffness matrices, the displacement vector, and the force vectors respectively.

3 BEM Formulation The BEM is a numerical technique used in the analysis of a homogenous elastic medium, especially, for problems with large or infinite domains. Neglecting the body forces, the governing boundary integral equation for an elastic, isotropic and homogenous medium can be obtained from Betti’s Reciprocal theorem as follows [8]:  ci j (η)u i (η) = − pi∗j (ξ, x)u j (x) dL(x) L



+

p j (x)u i∗j (ξ, x) dL(x)

(3)

L

where u i , p j and ci j are displacement, traction, and jump term respectively. ci j is equal to δi j for interior nodes. However, for nodes on a smooth boundary, the value of ci j is equal to 21 δi j , where δi j is the Kroneker Delta function. u i∗j and pi∗j are fundamental solutions of the displacement and traction, respectively. For two dimensional elastic medium, the kernels can be written as:     1 1 ∗ (4) (3 − 4υ) ln δi j + r,i r, j ui j = 8π μ(1 − υ) r pi∗j =

−1 {[(1 − 2υ)δi j + 2r,i r, j ]r,n 4π(1 − υ)r −(1 − 2υ)[r,i n j − r, j n i ]}

(5)

where r , is the distance between the source and the receive points. Vector n is the outward unit normal vector on the boundary, υ is the Poisson’s ratio and μ is the shear modulus of elastic material. The boundary integral equation can be solved by the discretization of the boundary to the elements and nodes. Hence, the convoluted form can be obtained as follows:

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u c 2q−1 u 2q

 +

⎧ NE ⎪ ⎨ ⎪

e=1 ⎩L

e

p∗ N dL

⎫ ⎪ ⎬ e

⎪ ⎭

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ue =

⎧ NE ⎪ ⎨ ⎪

e=1 ⎩L

e

u∗ N dL

⎫ ⎪ ⎬ e

⎪ ⎭

pe

(6) where, L e is length of the element, q denotes the degree of freedom of each node, NE is the total number of elements, and N is the shape function matrix which approximates the displacement and stress within the element as a function of nodal values. For a quadratic element with three nodes 1, 2 and 3   N2 0 N3 0 N1 0 (7) N= N2 0 N3 0 N1 0 where, Ni stands for the value of shape function at node i. The right hand side of Eq. (6) has a weak singularity for which, the closed form solution can be obtained in the case of the constant and the linear boundary elements. However, for quadratic element, numerical integration using Gauss points is required. Strong singularities in the left hand side term of Eq. (6) can be solved indirectly using the rigid body motion. Imposition of Eq. (6) at each node, results into a system of equations as follows: Hu = Gp

(8)

where, H and G are influence coefficient matrices, and u and p are displacement and traction vectors at the nodes, respectively. Transferring unknown parameters to the left hand side, leads to the following equation: Ax = B y

(9)

where, the components of A and B are obtained from H and G. x and y are the vectors of the known and the unknown parameters. The unknown displacements and the tractions can be determined by solving the above equation. It is noteworthy that the traction p can be converted to the equivalent force by  (10) f = M p, M = N T N dL Using Eqs. (10) and (8) can be re-written in terms of displacement and force as follows: Hu = Q f Q = G M −1

(11)

4 Mixed Neumann–Dirichlet Scheme As mentioned earlier, direct coupling of the FEM and the BEM is not an efficient procedure, and thus the studies in this area tend towards the coupling of the FEM and the BEM through the iterative algorithms, which are called DDM. In

these coupling methods, the coefficient matrices for the FEM and the BEM are not combined and the basic formulation of each method is preserved during the coupling process. Several algorithms such as the sequential Schwarz Dirichlet–Neumann scheme [22,24], parallel Schwarz Dirichlet–Neumann scheme [25] and parallel Schwarz Neumann–Neumann scheme [26], have been proposed for the DDM. To describe the simplest iterative coupling method (i.e., sequential Schwarz Dirichlet–Neumann scheme), consider a 2-D model in which the global domain is decomposed into two sub-domains (Fig. 1). One of the sub-domains is discretized by the FEM and the other is simulated by the BEM. Now, let us define the following vectors (Fig. 1): {u B } is the displacement in the BEM sub-domain, {u IB } is the displacement on the interface boundary approached from the BEM sub-domain, {u B B } is the displacement in the BEM sub-domain except {u IB }, {u F } is the displacement in the FEM sub-domain, {u IF } is the displacement on the interface boundary approached from the FEM sub-domain, {u FF } is the displacement in the FEM sub-domain except {u IF }. Considering the mentioned vector we have   I T (12) {u B } = u B B uB  F I T {u F } = u F u F (13) Similarly, the traction and force vectors can be defined for the BEM and the FEM sub-domains, respectively. Now, set initial values of {u I } at the interface, i.e., {u IB } = 0, and consider the BEM sub-domain to solve Eq. (10) for n ) as follows: the forces at the interface (i.e., f IB        u BB  f BB = Q (14) HB HI Q B I n u nIB f IB where, subscripts I and B are respectively, stand for the interface and the BEM sub-domain and superscript n denotes the n = − f n to the interface number of step. By imposing f IF IB boundary approached from the FEM sub-domain and solving Eq. (2) for the displacement we have      u FF f FF K FF K FI = (15) n K IF K II u nIF f IF where, subscript F stands for the FEM. The displacement at the BEM sub-domain interface can be updated as follows: n n u n+1 IB = (1 − α) u IB + α u IF

(16)

where, α is the relaxation parameter (acceleration factor). The updated displacement is substituted to the initial considered value and the process is repeated until    n+1  u IB or IF − u nIB or IF   n  ε (17) u  IB or IF

where ε is the acceptable error.

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Fig. 1 Decomposition of the global domain to the FEM and the BEM sub-domains

Fig. 2 Decomposition of the global domain in the mixed Neumann–Dirichlet scheme

As it is stated above, the interface boundary conditions can be of Dirichlet or Neumann type. However, the assigned interface boundary condition for each sub-domain must be of one type. For instance, if Neumann boundary condition is assigned to the interface boundary of the FEM sub-domain, the whole interface boundary associated with the FEM subdomain must be of Neumann type. This is also true for the BEM sub-domain. Nevertheless, it is possible for one subdomain to contain only Neumann type boundary conditions and hence, the sub-domain could not be analyzed generally. To deal with such problems, a new algorithm is presented here. Based on this method, a suitable condition is assigned to the interface boundary, which is compatible with the other boundary conditions of the sub-domain. In this regard, the boundary condition at some parts of the sub-domain interface is of the Dirichlet type and at the remaining parts is of the Neumann type. The privilege of this approach is its compatibility with any type of boundary conditions of the subdomains. Considering Fig. 2, the required steps of the proposed approach which is called “ mixed Neumann–Dirichlet scheme” are as follows:

•

n = − f n and u n n f IB IFi IB j = u IF j (compatibility and equii librium condition) ⎡ ⎤ ⎤ ⎡ u BB f BB ⎢ n ⎥  ⎢ n ⎥  • Solve HB HIi HI j ⎣ u IBi ⎦ = Q B Q Ii Q I j ⎣ f IBi ⎦ n u nIB j f IB j n for u nIBi and f IB j

n n • Update u n+1 IFi = (1 − αi ) u IFi + αi u IBi n+1 n n • Update f IF j = (1 − α j ) f IF j − α j f IB j

where, αi and α j are the relaxation parameters in order to accelerate convergence. The convergence criterion is defined as follows:        n+1 n  − u u IFi − u nIFi  u n+1 IB j  IB j     + ε (18)  n   n  u IFi  u IB j  where, ε is an acceptable error.

5 Numerical Study • Choose initial value u 0IFi for some nodes on the boundary 0 for the other nodes. of the FE sub-domain and f IF j • Do for n = 0, 1, 2, . . . until convergence ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ f FF u FF K FF K FIi K FI j ⎢ n ⎥ ⎢ n ⎥ n • Solve ⎣ K IFi K IIii K IIi j ⎦ ⎣ u IFi ⎦ = ⎣ f IFi ⎦ for f IF i n n u IF j f IF j K IF j K IIi j K II j j and u nIF j

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5.1 A Pressurized Cavity in an Elastic Full Space A pressurized cylindrical cavity in an elastic full space is analyzed by the proposed domain decomposition algorithm. As shown in Fig. 3, the inside wall of the cavity is subjected to a uniform stress σ0 . Considering the plane strain condition, the closed form solutions of the stress and displacement values are as follows [27]:

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Fig. 3 Cylindrical cavity in an elastic full space subjected to uniform pressure

u cr =

σ0 R 2 2μr

σrc = −σθc =

Fig. 4 The finite element and the boundary element sub-domains around the cavity

(19) σ0 R 2 r2

(20)

where, u cr is the radial displacement, σrc and σθc are, respectively, the radial and the hoop stresses at each point of the surrounding domain. r is the distance from the center of the circular cross section, R is the radius of the cavity, and μ is the shear modulus of elastic medium. The geometry and the loading condition of the model are axi-symmetric. Hence the response at any point depends on its distance from the center of the cavity. Figure 4 shows the near field and the far field regions modeled by the FEM and the BEM respectively. Quadratic quadrilateral elements with 8 nodes and 16 Gauss points are used to model the finite element sub-domain. The boundary elements have to be compatible with finite elements at the interface boundary. Therefore, quadratic boundary elements have been used. Note that the interior boundary of the BEM sub-domain should be discritized and numbered in clockwise manner. The Young modulus and Poisson’s ratio of the elastic material considered are 100 MPa and 0.2, respectively. The uniform pressure σ0 and R are, respectively, assumed as 10 MPa and 1 m. The non-dimensional accepted error, ε, is fixed to 0.005 and the displacement and force relaxation parameters are set to be 0.12 and 0.1, respectively. The initial traction and displacement boundary conditions at the interface are assumed to be zero. The lower semi-circular interface

Fig. 5 a Radial displacement and b radial stresses at points around the cavity

boundary is of Dirichlet boundary condition and the upper part is of Neumann type. Figure 5 shows the results obtained from the proposed DDM for the radial displacements and stresses at different distances r. The results are also compared with the closed form solution. As shown in this figure, there is an excellent agreement between the results of analyses by coupling method and the results of exact solution. Based on the selected values of error and the relaxation parameters, the mixed Neumann–Dirichlet scheme will converge after 22 iterations, as shown in Fig. 6. Note that due to the traction free and zero displacement assumptions at the

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α i =0.5

α i =0.35 α i =0.05

Number of iterations

60

40

20

Fig. 6 The trend of convergence of the coupled FEM/BEM for αi = 0.1 and α j = 0.12

0 0.00

0.04

0.08

αj

0.12

0.16

Fig. 8 The variation of the required number of iterations with the values of αi and α j

5.2 Rigid Strip Footing Resting on an Elastic Half-Space

Fig. 7 Deformation of the interface boundary at different iterations

interface boundary in the first iteration, the obtained error, using Eq. (18), is infinite in this step. Figure 7 shows the positions of the interface boundary at the first, the third and the last iterations. The computational time of the proposed DDM is a function of selected error and the relaxation parameters. However, the relaxation parameters are not independent from each other, and for a given value of αi , a limiting value of α j exists beyond which the convergence is not guaranteed. The effects of αi and α j values on the required number of iterations are shown in Fig. 8. Other factors such as mesh size and the assigned initial values at the boundary interface can also affect the time of computation.

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In this example, an elastic half-space is subjected to a prescribed settlement due to the presence of a rigid strip footing of width B. This problem is analyzed by the coupled FEM/BEM. A mesh with the extension of 1.5 B from the center of the strip footing is selected as the finite element mesh and the surrounding medium is modeled by the BEM (Fig. 9). In order to satisfy the compatibility at the interface, a quadratic element with three nodes is adopted for the boundary elements. The optimum values of the relaxation parameters for the mixed Neumann–Dirichlet scheme were determined by trial and error process as αi = 0.24 and α j = 0.29. Half of the arc length at the interface is of Neumann type and the other half is of Dirichlet type boundary condition. As a basis for comparison, the normalized stress at the center of the footing is considered. The exact solution for the normalized stress beneath the foundation can be obtained by [27]: σ Ftotal

=

1 √ π 1 − d2

(21)

where Ftotal is the total force resulted from the integration of the tractions under the footing and d is the distance from the center of the footing. The value of exact solution, obtained from Eq. (20), for the traction at the footing center is 0.3183. The coupled FFM/BEM gives 0.3034 for this quantity, having 4.68 % deviation from the exact result and confirms the accuracy of proposed algorithm.

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Fig. 9 The FEM and BEM sub-domains for strip footing problem

Fig. 10 The trend of convergence for strip footing problem

Figure 10 shows the trend of convergence to the exact solution result. The non-dimensional acceptable error is taken as 0.001 in present example. As it is obvious, after a limited number of iterations (i.e., 11), convergence is achieved. The speed of convergence and the rate of error descending in each step are attributed to the selection of relaxation parameters and also the tolerable error. The deformed shape of the problem domain is shown in Fig. 11. The closed form solution for the displacement profile of the free surface is also available as [27]: ⎞ ⎛  x2 F(1 − υ) ⎝ x − 1⎠ (22) ln + δ= πG a a2 In the above equation, F is the resultant vertical load, and the parameters a and x are half of the footing length and the distance of point on the free surface from the footing

center, respectively. The free surface displacement profile for the proposed coupled method is shown in Fig. 12. As it is obvious, there is an excellent agreement between the proposed method and the exact solution which confirms the accuracy of proposed technique.

6 Conclusion A domain decomposition algorithm for the coupling of the FEM and the BEM is introduced in this paper. Similar to other domain decomposition algorithms, in the proposed method, the governing equations of sub-domains are solved by an iterative process and the rate of convergence considerably depends on the assumed relaxation parameters. However, the latter method utilizes two relaxation parameters, which extend the degree of freedom of the technique for obtain-

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Fig. 11 Deformed shape of problem domain

Fig. 12 Free surface displacement profile for strip footing problem

ing the suitable values for the relaxation parameters, and consequently, decreasing the number of iterations. The main advantage of the proposed method is its compatibility with any type of boundary conditions of each sub-domain. The efficiency and the accuracy of the proposed method have been investigated by solving two numerical examples. Excellent agreements have been observed between the results of the numerical analyses and the results of the exact solutions.

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