This Article Appeared in a Journal Published by Elsevier. the Attached

(This is a sample cover image for this issue. The actual cover is not yet available at this time.)

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright Author's personal copy

Finite Elements in Analysis and Design 64 (2013) 24–35

Contents lists available at SciVerse ScienceDirect

Finite Elements in Analysis and Design

journal homepage: www.elsevier.com/locate/ﬁnel

Three dimensional smoothed fixed grid finite element method for the solution of unconfined seepage problems

Mohammad Javad Kazemzadeh-Parsi a,n, Farhang Daneshmand b a Mechanical Engineering Department, Shiraz Branch, Islamic Azad University, P.O. Box 71955-845, Shiraz, Iran b Department of Bioresource Engineering, McGill University, 21111 Lakeshore Road, Ste. Anne de Bellevue, QC, Canada H9X 3V9 article info abstract

Article history: A three dimensional numerical analysis for unconfined seepage problems in inhomogeneous and Received 13 October 2011 anisotropic domains with arbitrary geometry is presented in this paper. The unconfined seepage Received in revised form problems are nonlinear in its nature due to unknown location of the phreatic surface and nonlinear 2 June 2012 boundary conditions which complicates its solution. The presented method is based on the application Accepted 2 September 2012 of non-boundary-fitted meshes and is an extension of the recently proposed two dimensional smoothed fixed grid finite element method. The main objective of using this method is to facilitate Keywords: solution of variable domain problems and improve the accuracy of the formulation of the boundary Smoothed fixed grid finite element method intersecting elements. In this method, the gradient smoothing technique is used to obtain the element Unconfined seepage matrices. This technique simplifies the solution significantly by reducing the volume integrals over the Variable domain problems elements into area integrals on the faces of smoothing cells. To locate the free surface, an initial guess Non-boundary-fitted meshes Inhomogeneous for the unknown geometry is selected and modified in each iteration to eventually satisfy nonlinear Anisotropic boundary condition. The application of the proposed technique for three dimensional seepage problems is carried out for different examples including rectangular, trapezoidal and semi-cylindrical dams and the results are compared with those available in the literature. & 2012 Elsevier B.V. All rights reserved.

1. Introduction seepage problem with complicated geometry, boundary conditions and inhomogeneous and/or anisotropic material properties The unconfined seepage problems (USP) have long been a is difficult to solve analytically and must be treated by using main concern in the analysis of hydraulic structures, oil reservoirs numerical methods. and mining [1]. It has adverse influence on the stability of dams Different numerical approaches have been used to predict the and deep foundations and must be taken into account in daily seepage behavior, for example, the boundary element method [3], maintenance of earth dams. For example, a sudden loss of water the finite difference method [4], the finite volume method [5], the level can create very high gradients in the dam and cause finite element method [6,7] and meshfree methods [8]. Among structural failure. these methods, the boundary element method reduces the pro- The main purposes of free-surface seepage analysis are deter- blem dimension and simplifies the solution of variable domain mination of free-surface profile, velocity and pressure distribu- problems but it has some difficulties in inhomogeneous domains tions. In the other words, the solution of general three and nonlinear problems. The finite element method (FEM) is the dimensional unconfined seepage problems requires determina- most powerful and widely used method in many engineering tion of the fluid head within the domain as well as the position of problems. However application of the FEM in the solution of the phreatic surface. This is a difficult task because determination variable domain problems requires mesh modification in succes- of the phreatic surface, as a part of the solution, involves sive iterations. In the last decade, some meshless or meshfree intrinsically non-linear characteristics. Analytical solutions are methods have been proposed to reduce the mesh dependency of generally possible for particular cases with linear governing the FEM. Their computational time and difficulties in satisfying equations and simple boundary conditions with homogenous essential boundary conditions are two main drawbacks of mesh- material [2]. On the other hand, a three dimensional unconfined less methods. The other methods have also difficulties with boundary fitted meshes. An alternative approach for the solution of variable domain

n problems is using Non-Boundary-Fitted Meshes (NBFM) in which Corresponding author. E-mail addresses: [email protected], [email protected] the mesh modiﬁcation is completely omitted. Some applications (M.J. Kazemzadeh-Parsi), [email protected] (F. Daneshmand). of NBFM for seepage problems have been reported in Refs. [9–12].

M.J. Kazemzadeh-Parsi, F. Daneshmand / Finite Elements in Analysis and Design 64 (2013) 24–35 25

An important issue in NBFM is the formulation of Boundary 2. Unconfined seepage problems Intersecting Elements (BIE). One of the recently proposed methods in this context is the Smoothed Fixed Grid Finite Element Method The main objective of an unconfined seepage analysis is to locate (SFGFEM). This method has been developed and used by the authors the phreatic surface as shown in Fig. 1. The water elevation at for the solution of inverse geometry heat conduction problems of upstream and downstream reservoirs is designated by hu and hd, homogeneous [13,14] and functionally graded materials [15].The respectively. The domain can be divided into wet and dry regions applicability of the SFGFEM has also been examined in the solution where their interface is shown by the phreatic surface. The piezo- of unconfined seepage problems in inhomogeneous and anisotropic metric head h in the wet region O can be calculated as follows [5]: two dimensional domains [16]. Due to complexities associated with p the USPs, only two dimensional cases have been considered by h ¼ þz ð1Þ rg many researchers and to the best knowledge of the authors, the only works dealing with the 3D case of USP are Refs. [17–22]. where p, r and g are the fluid pressure, the fluid density and the In the present work, the 3D SFGFEM is introduced and used in gravitational acceleration, respectively. The fluid velocity vector v the solution of steady USPs in inhomogeneous and anisotropic can be presented in terms of the permeability tensor k(x) and domains with arbitrary geometry. In this method, the Gradient gradient of piezometric head as follows: Smoothing Technique (GST) is used to evaluate the element matrices. GST has been introduced for the first time in the Cell v ¼kðxÞrh ð2Þ based Smoothed Finite Element Method (CS-FEM) in Ref. [23]. The The media is considered anisotropic and inhomogeneous where Face-based Smoothed Finite Element Method (FS-FEM) is another the permeability tensor is assumed to vary as a function of coordi- variation of this method and has been used for nonlinear nates. The continuity equation for the problem can be written as problems [24]. Other variations of this method have also been U implemented, for example, acoustic problems [25], heat transfer r ðkðxÞrhÞ¼0 ð3Þ problems [26–28] and 3D visco-elastoplastic problems with This is the governing equation for the piezometric head within tetrahedral meshes [29]. The stability and convergence of the problem domain O. We have the following boundary conditions smoothed finite element formulation have been investigated by on the upstream surface GU and downstream surface GD using the G-space theory [30–32]. Evaluation of element matrices in NBFM requires domain inte- h ¼ hu on GU ð4Þ grals which must be computed over the internal parts of BIEs. h ¼ hd on GD However the internal parts of these elements has no predefined shape and hence the standard integration schemes such as Gauss The zero normal velocity condition on the impermeable sur- formulation cannot be used for evaluation of element matrices. This face GI can also be written as difficulty is serious in three dimensions because volume integration ðkðxÞrhÞUn ¼ 0onGI ð5Þ is more complicated than area integration [33]. The GST reduces the volume integrals over the internal parts of BIEs to the area integrals The seepage face GS is in atmospheric pressure and the over the surfaces of the smoothing cells and reduces the computa- boundary condition on this face is given by tional complexity of the problem considerably. The applicability of the proposed method in the solution of unconfined seepage h ¼ z on GS ð6Þ problems for inhomogeneous and anisotropic three dimensional The following two simultaneous (over determined) boundary domains is studied here by solving numerical examples and conditions should also be satisfied on the phreatic surface G : comparing the results with those available in the literature. P (a) zero normal velocity (impermeable boundary condition) and In the present paper, a brief summary on the mathematical (b) atmospheric pressure. description of unconfined seepage problems and boundary conditions is presented in Section 2. The formulation of the proposed ðkðxÞrhÞUn ¼ 0onGP ð7aÞ method in three dimensions is then presented in Section 3. Boundary parameterization and updating formula for the key h ¼ z on GP ð7bÞ points are presented in Section 4. Finally, numerical examples and conclusions are given in Sections 5 and 6, respectively. where n is the unit vector normal to the boundary surface.

Fig. 1. Schematic representation of a 3D unconﬁned seepage problem. Author's personal copy

26 M.J. Kazemzadeh-Parsi, F. Daneshmand / Finite Elements in Analysis and Design 64 (2013) 24–35

Our aim in the present study is to develop an iterative process based on the 3D SFGFEM to solve the USP in domains with arbitrary geometry and permeability.

3. 3D smoothed ﬁxed grid ﬁnite element method

SFGFEM uses the benefits of NBFMs. A typical NBFM consists of three types of elements: Internal Elements (IE), Boundary Inter- secting Elements (BIE) and External Elements (EE). The IEs and BIEs are considered as active elements. The nodes involved in the active elements are also defined as active nodes and therefore the total number of degree of freedoms is equal to the total number of active nodes. Eight nodded hexahedral brick elements with tri-linear shape functions are used in this work to approximate the field variable. The interpolation function of the field variable in each active element can be written in terms of the nodal values as [34]: hhðxÞ¼NT H ð8Þ where N is the shape functions vector and H is the nodal values vector. In the present work, to use the GST, each element is divided into eight smoothing cells. The smoothed gradient of the e shape functions in smoothing cell S denoted by rNS and can be found using GST as follows [16,23]: Z e rNS ¼ fSðxÞrNdO ð9Þ OS

Fig. 3. Division of a boundary intersecting element into eight smoothing cells showing the faces of the internal part of one of the cells.

Fig. 4. The centroidal coordinate system.

In the above equation, rN is the gradient of the shape functions

vector and OS is the domain of smoothing cell S. The division of IEs and BIEs into smoothing cells is schematically shown in Figs. 2 and

3, respectively. The piecewise constant smoothing kernel fS(x)is assumed to be constant within each cell and vanish elsewhere [23], ( 1=VS xAOS fSðxÞ¼ ð10Þ 0 x 2= OS

where VS is the volume of the smoothing cell S. Performing integration-by-part on the right hand side of Eq. (9) and using Fig. 2. Division of an internal element into eight smoothing cells. Eq. (10), the smoothed gradient of the shape functions within Author's personal copy

M.J. Kazemzadeh-Parsi, F. Daneshmand / Finite Elements in Analysis and Design 64 (2013) 24–35 27

smoothing cell S can be obtained as [16]: where Ni is the shape function vector evaluated at the centroid Z of ith face of the cell and Ai is its area. e 1 T rNS ¼ Nn d@O ð11Þ V S @OS where qOS is the boundary surfaces of smoothing cell S and n is the outward unit normal vector on qOS. Regarding Eq. (11), the smoothed gradient vector of the shape functions, in each cell, can be obtained via area integration over the surfaces of the cell. e h Now, the smoothed gradient vector of the ﬁeld variable, rhS , within the smoothing cell S can be represented as

e h eT rhS ¼ BS H ð12Þ e where BS is known as the smoothed gradient matrix and is defined as Z e e 1 T BS ¼ rNS ¼ Nn d@O ð13Þ V S @OS Note that since the integration in Eq. (9) is defined over the internal part of the smoothing cell, the area integrations in Eqs. (11) and (13) should be performed over the surfaces of the internal part of the smoothing cells. Fig. 3 shows a BIE and the faces of the internal part of one of its cells. A simple and accurate algorithm is proposed in this paper to approximate the area integral in Eq. (13). Depending on orientation of the smoothing cells with respect to the domain boundary, the cell surface qOS can be decomposed to 4, 5, 6, or 7 faces. For example, the cell shown in Fig. 3 is decomposed into 7 faces. Therefore the integral in Eq. (13) can be broken down to integrals on these faces. Now, if these faces were flat (zero curvature), the normal vector n remain Fig. 5. Reconstruction of the phreatic surface via a set of triangular facets. The vertices are considered as key points. constant and leave the integral. The hexahedral elements shape functions are trilinear functions and therefore the general form of the integrals in Eq. (13) can be represented as follows: Z f ¼ ða0 þa1xþa2yþa3zþa4xyþa5yzþa6zxþa7xyzÞdA ð14Þ A where A is a face of a smoothing cell. A schematic representation of such face is shown in Fig. 4. By introducing a centroidal coordinate system for each face (see Fig. 4) and using the coordinate transformation between the centroidal and global coordinate systems, the above integral reduced to Z f ¼ a0Aþa1xAþa2yAþa3zAþa4 xyAþ x0y0dA Z Z A

þa5 yzAþ y0z0dA þa6 zxAþ z0x0dA AZ Z A Z Z

þa7 xyzAþx y0z0dAþy z0x0dAþz x0y0dAþ x0y0z0dA A A A A ð15Þ where ðx,y,zÞ is the centroidal coordinate of the face. Note that the ﬁrst tree parentheses in this equation have some similarities to product of inertia of the face. For example, the second term in these parentheses will vanish due to symmetry of the face (this is the case for internal elements while regular hexagonal elements are used). It must also be noted that these terms can be ignored for elements which are located far from the axis of the coordinate system. It is also true for the integrals in the fourth parenthesis. Therefore, in the present work, we ignore the integrals in the Eq. (15) and this equation reduced to f ¼ða0 þa1xþa2yþa3zþa4xyþa5yzþa6zxþa7xyzÞA ð16Þ In the other words, the integral in Eq. (13) can be approxi- mated using the following one point integration rule: X e 1 B ¼ N nT A ð17Þ S V i i i S i Fig. 6. Flowchart for the solution of the unconﬁned seepage problems. Author's personal copy

28 M.J. Kazemzadeh-Parsi, F. Daneshmand / Finite Elements in Analysis and Design 64 (2013) 24–35

Fig. 7. Rectangular dam (Example 1). (a) Geometric dimension and (b) structured uniform non-boundary-ﬁtted mesh.

Table 1 Permeability values and number of iterations for example 1.

Case Isotropic permeability coefﬁcient No. of iterations

Case 1 k¼115 Case 2 k¼1þ3.8x 15 Case 3 k¼203.8x 17 Case 4 k¼201.9z 17 Case 5 k¼1þ1.9z 17 Case 6 k¼2019x/519y/9þ38xy/45 24

Fig. 8. Initial guess and phreatic surfaces for cases 1–6 (Example 1). Author's personal copy

M.J. Kazemzadeh-Parsi, F. Daneshmand / Finite Elements in Analysis and Design 64 (2013) 24–35 29

Now, converting the differential equation given in Eq. (3) and Eq. (19) one can approximate k(x) in each smoothing cell with a natural boundary conditions to the integral weak form and then linear function of the coordinates and then obtain the integral in Eq. introducing the interpolation equations, Eqs. (8) and (12), and (19) as follows: using the Galerkin method, the discrete form of governing X8 equations can be obtained as [34] eT e K ¼ ðV SBS kSBSÞð20Þ KH ¼ R ð18Þ S ¼ 1 where K is the element coefficient matrix and R is the element load where kS is the permeability tensor evaluated at the centroidal point vector due to natural boundary conditions. In general, calculation of of the smoothing cell S and VS is its volume. the coefficient matrix K requires integration over the internal part of the elements. Each element is also divided into eight smoothing Table 2 cells. Therefore, the integration must be performed over the internal Permeability values and number of iterations for example 2. parts of the smoothing cells as follows: Z Case Isotropic permeability coefficient No. of iterations X8 eT e K ¼ B kðxÞB d ð19Þ S S O Case 1 k¼116 OS S ¼ 1 Case 2 k¼1þ2.714x 22 Case 3 k¼202.714x 23 where OS is the internal part of smoothing cell S. Note that the e Case 4 k¼203.8z 16 smoothed gradient matrix B is constant in each smoothing cell. In S Case 5 k¼1þ3.8z 20 the present work, an inhomogeneous media with arbitrary perme- Case 6 k¼2019x/71.9yþ19xy/35 20 ability function k(x) is considered. To evaluate the integrals in

Fig. 9. Phreatic surface (Example 1). (a) Comparison with 2D solutions and (b) phreatic surface for three different cross sections for case 6.

Fig. 10. Trapezoidal dam (Example 2). (a) Geometric dimensions and (b) unstructured non-boundary-ﬁtted mesh. Author's personal copy

30 M.J. Kazemzadeh-Parsi, F. Daneshmand / Finite Elements in Analysis and Design 64 (2013) 24–35

The appealing feature of the foregoing algorithm is that the there is no need for geometric mapping in performing integra- volume integrals in the formulation of coefficient matrix K,at tions [23]. first, are reduced to area integrals on the faces of smoothing The load vector R in Eq. (18) can be evaluated by using the cells and are eventually reduced to the summation given in following integration over the natural boundaries: Eq. (20).ThisisanimportantpointintheformulationofBIEs, Z in which the internal parts of these elements do not have R ¼ Nðv nÞd@O ð21Þ a predefined shape and the use of standard integration Gnatural algorithms such as Gauss integration formula is not straight forward. Less sensitivity of the results to the geometric shape where Gnatural represent the boundaries on which natural boundary of the elements is the next advantage of the use of GST because conditions are specified (impermeable face GI and phreatic surface

Fig. 11. Initial guess and phreatic surfaces (Example 2).

Fig. 12. Comparison of the phreatic surfaces with 2D solutions (Example 2). Author's personal copy

M.J. Kazemzadeh-Parsi, F. Daneshmand / Finite Elements in Analysis and Design 64 (2013) 24–35 31

GP, in the present study). Note that the loading vector R vanishes on obtained as follows [16]: these boundaries due to zero normal velocity condition. new old old zi ¼ zi þðhizi Þa1 ð23Þ

old new where, zi and zi are the elevations of the ith key point in the 4. Solution algorithm previous and the next iterations, respectively and hi is the computed piezometric head in this key point. In this paper, boundary parameterization is adopted by defining a More attention should also be paid for the key points located set of key points which reconstruct the phreatic surface. As shown in on the downstream surface of the dam (seepage face GS). Apply- Fig. 5, the phreatic surface is formed by a set of triangular facets ing boundary condition given in Eq. (6) for key points located on which its vertices are selected as key points. As shown in this figure, the seepage face GS, leads the boundary condition given in the coordinates of the ith key point, xi, is represented using a base Eq. (7b) to be automatically satisfied. It means that Eq. (23) point B ,adirectionvectore and a distance parameter r as follows: i i i cannot be useful for updating the position of the key points on GS. To overcome this difficulty, the boundary condition given in xi ¼ Bi þriei ð22Þ Eq. (7a) is used to provide a suitable updating formula for these

For each key point, the base point Bi and direction vector ei are key points [16]. The following formula is used here to obtain the considered as given parameters but the distance ri is considered new elevation of these key points. as the only unknown shape parameter. new ¼ old þð Þ ð Þ Now, the USP is reduced to the problem of determination of zn zn v n a2 24 shape parameters ri in such a way that satisfies the over where a2 is under-relaxation parameter and v and n are fluid determined boundary conditions given in Eqs. (7a) and (7b).To velocity vector and unit vector normal to phreatic surface at these solve this problem, an iterative strategy is used which starts with key points, respectively. an initial guess for the shape parameters. The SFGFEM is then After obtaining the new elevation of the key points form used to solve the governing equation given in Eq. (3) with new Eqs. (23) or (24), the new shape parameter ri for ith key boundary conditions given in Eqs. (4–6), (7a). According to the point can be obtained using Eq. (22) as the following updating obtained field variable, the shape parameters must be modified to formula. satisfy the second boundary condition given in Eq. (7b) on phreatic surface. By considering an under-relaxation parameter new 1 new ri ¼ ðzi BziÞð25Þ a1, the new elevation of the ith key point for the next iteration is ezi

Table 3 Permeability values and number of iterations for examples 3 and 4.

Case Anisotropic No. of No. of permeability iterations for iterations for coefﬁcients example 3 example 4

Case 1 kxx¼kyy ¼kzz ¼11414

kxy ¼kyz¼kzx¼0

Case 2 kxx¼kzz ¼1, kyy ¼215 20

kxy ¼kyz¼kzx¼0

Case 3 kxx¼kzz ¼1, kyy ¼420 23

kxy ¼kyz¼kzx¼0

Case 4 kxx¼kzz ¼1, kyy ¼623 27

kxy ¼kyz¼kzx¼0 Fig. 13. Phreatic surfaces for three different cross sections in case 6 (Example 2).

Fig. 14. Rectangular dam with side seepage faces (Example 3). (a) Geometric dimensions and (b) the non-boundary-ﬁtted mesh. Author's personal copy

32 M.J. Kazemzadeh-Parsi, F. Daneshmand / Finite Elements in Analysis and Design 64 (2013) 24–35

where Bzi and ezi are the elevations of the base point and the vertical selected to check the validation of the proposed method and the component of the direction vector of the key point, respectively. program. To this goal, the results are compared with those given The foregoing iterative process is continued until the change of in the literature. In the ﬁrst two examples, the seepage problem in the shape parameters between two successive iterations falls inhomogeneous and isotropic earth dams is considered while the within the desired accuracy. A simple ﬂowchart showing this anisotropic case is investigated in the next two examples. process is presented in Fig. 6. 5.1. Example 1, rectangular cross section dam

5. Numerical examples In the ﬁrst example, a rectangular cross section dam made of inhomogeneous and isotropic media is considered. Geometric Based on the numerical approach proposed in this paper, a dimensions, upstream and downstream reservoirs are presented computer program is developed. Some typical examples are in Fig. 7(a). A structured uniform NBFM (see Fig. 7(b)) is used to

Fig. 15. Phreatic surfaces (Example 3).

Fig. 16. Location of phreatic surfaces for two cross sections (Example 3). (a) Section at y¼0 and (b) section at y¼5. Author's personal copy

M.J. Kazemzadeh-Parsi, F. Daneshmand / Finite Elements in Analysis and Design 64 (2013) 24–35 33

Fig. 17. Cylindrical dam (Example 4). (a) Geometric dimensions and (b) the non-boundary-ﬁtted mesh.

Fig. 18. Phreatic surfaces (Example 4). Author's personal copy

34 M.J. Kazemzadeh-Parsi, F. Daneshmand / Finite Elements in Analysis and Design 64 (2013) 24–35 solve USP and to determine the phreatic surface. Six different cases of the permeability functions are considered and presented in Table 1. A flat surface is selected for initial guess of the phreatic surface (see Fig. 8). The USP is then solved for each case using the proposed method and the phreatic surfaces are shown in Fig. 8. The permeability function for cases 1–5 are assumed to be independent of y coordinate and therefore the results of 3D analysis of these cases can be compared with those obtained from a 2D analysis. The 2D solution of cases 1–5 has been previously reported by the authors in Ref. [16] and Fig. 9(a) shows the results of 3D and 2D analysis. As it can be seen, the 3D results are in good agreement with 2D ones. For case 6, the permeability is assumed to be a function of x and y. Therefore it might be considered as a general 3D case. The phreatic surface for three different cross sections at y¼0, y¼4.5 and y¼10 is shown in Fig. 9(b). The number of iterations for each case is also Fig. 19. Location of phreatic surfaces for a section at y¼5.0 (Example 4). stated in Table 1. This table shows that the proposed method converged for all cases with a reasonable number of iterations. problem and is shown in Fig. 17(b). Similar to the previous example, 5.2. Example 2, trapezoidal cross section dam the media is also considered anisotropic with the same coefficients of permeability. The numbers of iterations are also presented in In this example, an earth dam with trapezoidal cross section is Table 3 and the obtained phreatic surfaces are plotted in Fig. 18 for considered to evaluate the applicability of the proposed methods in different cases. Due to symmetry, the location of phreatic surface at the irregular domains and usability of unstructured meshes. The y¼5isshowninFig. 19. As can be seen, increasing the permeability media is also considered as inhomogeneous and isotropic. The coefficient in the y direction leads to lowering the phreatic surface problem dimensions, upstream and downstream reservoirs are due to increasing the side seepage flow. shown in Fig. 10(a) and unstructured NBFM used for the solution of this example is shown in Fig. 10(b). Six permeability functions selected in this example are given in Table 2. A flat surface is selected 6. Conclusion as the initial guess and the results are shown in Fig. 11 for all cases. The results for 2D analysis for cases 1–5 are available in the The problem of unconfined flow through porous media is literature [16]. The results obtained by the current 3D analysis complex in nature and can be categorized as variable domain and previous works are reported and compared in Fig. 12. It can problem in which the application of classical numerical methods be seen that phreatic surfaces obtained by the present 3D analysis based on boundary fitted meshes is not straight forward. In this are in good agreement with 2D solutions in all cases. paper, the numerical procedure introduced in Ref. [16] by the For case 6, the permeability is assumed to be a function of x authors has been extended for 3-dimensional seepage problems and y coordinates and the phreatic surface given by the present with anisotropic material and arbitrary geometry. The proposed analysis is shown in Fig. 13 for three different sections. Table 2 method was based on the smoothed fixed grid finite element also reports the number of iterations for each case. approach with non-boundary-fitted meshes. The element matrices for the boundary intersecting elements were calculated 5.3. Example 3, rectangular cross section with side flow using the gradient smoothing technique. This method reduces the volume integrations into area integrations on the faces of An anisotropic rectangular cross section earth dam with side smoothing cells. Several numerical examples were solved includ- seepage faces is considered in this example. The main goal of ing rectangular trapezoidal and semicircular domains. To validate considering such geometry is to check the applicability of the the results, the obtained phreatic surfaces were compared with proposed method when the seepage face composed of different those available from 2D solutions. In all examples, the phreatic planes as shown in Fig. 14(a). This example also evaluates the surfaces have been obtained with reasonable number of itera- performance of the method in anisotropic materials. Problem tions. It is believed that the use of the proposed method simplifies dimensions, upstream and downstream reservoirs are shown in this the analysis of 3D variable domain problems and provides an figure. The NBFM is also shown in Fig. 14(b). Four different cases of effective tool for three dimensional unconfined seepage problems. permeability are selected in this example and are given in Table 3. After solving the problem, the obtained phreatic surfaces are shown in Fig. 15. The number of iterations is also given in Table 3.The Acknowledgments locations of phreatic surface at sections y¼0andy¼10 are identical due to symmetry and therefore the phreatic surface in sections at The results presented in this paper are from a research project y¼0andy¼5areshowninFig. 16(a) and (b) respectively for supported financially by Islamic Azad University, Shiraz Branch, different cases. The results show that by increasing the permeability Shiraz, Iran. Partial support of the first author by the Islamic Azad in the y direction, the seepage flow passed through the side faces University, Shiraz Branch is also appreciated. increased and as it can be seen from Figs. 15 and 16 the phreatic surface lowered to the level of downstream reservoir. References 5.4. Example 4, cylindrical dam [1] H.B. Wang, W.Y. Xu, R.C. Xu, Q.H. Jiang, J.H. Liu, Hazard assessment by 3D In the last example, a different problem with curved seepage stability analysis of landslides due to reservoir impounding, Landslides 4 (2007) 381–388. face is considered. The problem domain and its specifications are [2] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press, given in Fig. 17(a). An unstructured mesh prepared for this London, 1959. Author's personal copy

M.J. Kazemzadeh-Parsi, F. Daneshmand / Finite Elements in Analysis and Design 64 (2013) 24–35 35

[3] A.K. Chugh, H.T. Falvey, Seepage analysis in a zoned anisotropic medium by [19] C.S. Desai, B. Baseghi, Theory and verification of residual flow procedure for the boundary element method, Int. J. Numer. Anal. Methods Geomech. 8 3-D free surface seepage, Adv. Water Resour. 11 (1988) 192–203. (1984) 399–407. [20] N.H. Sharif, N.E. Wiberg,Adaptive, ICT procedure for non-linear seepage flows [4] Y. Jie, G. Jie, Z. Mao, G. Li, Seepage analysis based on boundary-fitted with free surface in porous media, Commun. Numer. Methods Eng. 18 (2002) coordinate transformation method, Comput. Geotech. 31 (2004) 279–283. 161–176. [5] M. Darbandi, S.O. Torabi, M. Saadat, Y. Daghighi, D. Jarrahbashi, A moving- [21] M.T. Ayvaz, H. Karahan, Modeling three-dimensional free-surface flows using mesh finite-volume method to solve free-surface seepage problem in multiple spreadsheets, Comput. Geotech. 34 (2007) 112–123. arbitrary geometries, Int. J. Numer. Anal. Methods Geomech. 31 (2007) [22] Q.H. Jiang, S.S. Deng, C.B. Zhou, W.B. Lu, Modeling unconfined seepage flow 1609–1629. using three-dimensional numerical manifold method, Journal of Hydrody- [6] O.C. Zienkiewicz, P. Mayer, Y.K. Cheung, Solution of anisotropic seepage by namics 22 (2010) 554–561. finite elements, J. Eng. Mech. 92 (1966) 111–120. [23] G.R. Liu, K.Y. Dai, T.T. Nguyen, A smoothed finite element for mechanics [7] G. Gioda, C. Gentile, A nonlinear programming analysis of unconfined steady- problems, Comput. Mech. 39 (2007) 859–877. state seepage, Int. J. Numer. Anal. Methods Geomech. 11 (1987) 283–305. [24] T. Nguyen-Thoi, G.R. Liu, K.Y. Lam, G.Y. Zhang, A face-based smoothed finite [8] G. Li, J. Ge, Y. Jie, Free surface seepage analysis based on the element-free element method (FS-FEM) for 3D linear and geometrically non-linear solid method, Mech. Res. Commun. 30 (2003) 9–19. mechanics problems using 4-node tetrahedral elements, Int. J. Numer. [9] K.J. Bathe, M.R. Khoshgoftaar, Finite element free surface seepage analysis Methods Eng. 78 (2009) 324–353. without mesh iteration, Int. J. Numer. Anal. Methods Geomech. 3 (1979) [25] Z.C. He, G.R. Liu, Z.H. Zhong, X.Y. Cui, G.Y. Zhang, A.G. Cheng, A coupled edge/ face-based smoothed finite element method for structural-acoustic pro- 13–22. blems, Appl. Acoust. 71 (2010) 955–964. [10] C.S. Desai, G.C. Li, Residual flow procedure and application for free surface [26] S.C. Wu, G.R. Liu, H.O. Zhang, X. Xu, Z.R. Li, A node-based smoothed point flow in porous media, Adv. Water Resour. 6 (1983) 27–35. interpolation method (NS-PIM) for three-dimensional heat transfer pro- [11] A. Cividini, G. Gioda, An approximate F.E. analysis of seepage with a free blems, Int. J. Therm. Sci. 48 (2009) 1367–1376. surface, Int. J. Numer. Anal. Methods Geomech. 8 (1984) 549–566. [27] S.C. Wu, G.R. Liu, H.O. Zhang, G.Y. Zhang, A node-based smoothed point [12] H. Zheng, D.F. Liu, C.F. Lee, L.G. Tham, A new formulation of Signorini’s type interpolation method (NS-PIM) for three-dimensional thermoelastic pro- for seepage problems with free surfaces, Int. J. Numer. Anal. Methods blems, Numer. Heat Transfer Part A: Appl. 54 (2008) 1121–1147. Geomech. 64 (2005) 1–16. [28] Z.B. Zhang, S.C. Wu, G.R. Liu, W.L. Chen, Nonlinear transient heat transfer [13] M.J. Kazemzadeh-Parsi, F. Daneshmand, Solution of geometric inverse heat problems using the meshfree ES-PIM, Int. J. Nonlinear Sci. Numer. Sim. 11 conduction problems by smoothed fixed grid finite element method, Finite (2010) 1077–1091. Elem. Anal. Des. 45 (2009) 599–611. [29] T. Nguyen-Thoi, G.R. Liu, H.C. Vu-Do, H. Nguyen-Xuan., A face-based [14] M.J. Kazemzadeh-Parsi, F. Daneshmand, Cavity shape identification with smoothed finite element method (FS-FEM) for visco-elastoplastic analyses convective boundary conditions using non-boundary-fitted meshes, Numer. of 3D solids using tetrahedral mesh, Comput. Methods Appl. Mech. Eng. 198 Heat Transfer, Part B 57 (2010) 283–305. (2009) 3479–3498. [15] M.J. Kazemzadeh-Parsi, F. Daneshmand, Inverse geometry heat conduction [30] G.R. Liu, On G space theory, Int. J. Comput. Methods 6 (2009) 257–289. analysis of functionally graded materials using smoothed fixed grid [31] G.R. Liu, A G space theory and a weakened weak (W2) form for a unified finite elements, Inverse Probl. Sc. Eng. (2012), http://dx.doi.org/10.1080/ formulation of compatible and incompatible methods: Part I theory, Int. J. 17415977.2012.686998. Numer. Methods Eng. 81 (2010) 1093–1126. [16] M.J. Kazemzadeh-Parsi, F. Daneshmand, Unconfined seepage analysis in earth [32] G.R. Liu, A G space theory and a weakened weak (W2) form for a unified dams using smoothed fixed grid finite element method, Int. J. Numer. Anal. formulation of compatible and incompatible methods: Part II applications, Methods Geomech. 36 (2011) 780–797. Int. J. Numer. Methods Eng. 81 (2010) 1127–1156. [17] J. Caffrey, J.C. Bruch Jr, Three-dimensional seepage through a homogeneous [33] F. Daneshmand, M.J. Kazemzadeh-Parsi, Static and dynamic analysis of 2D dam, Adv. Water Resour. 2 (1979) 167–176. and 3D elastic solids using the modified FGFEM, Finite Elem. Anal. Des. 45 [18] C.S. Desai, J.G. Lightner, S. Somasundaram, A numerical procedure for three- (2009) 755–765. dimensional transient free surface seepage, Adv. Water Resour. 6 (1983) [34] J.N. Reddy, An Introduction to the Finite Element Method, 2nd edition, 175–181. McGraw-Hill, New York, 1993.