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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 condi- tions 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]. 0168-874X/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.finel.2012.09.001 Author's personal copy 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