Transient free-surface seepage in three-dimensional general ...

Engineering Analysis with Boundary Elements 46 (2014) 51–66

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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound

Transient free-surface seepage in three-dimensional general anisotropic media by BEM K. Rafiezadeh a,n, B. Ataie-Ashtiani a,b a b

Department of Civil Engineering, Sharif University of Technology, P.O. Box 11155-9313, Tehran, Iran National Centre for Groundwater Research & Training, Flinders University, GPO Box 2100, Adelaide, SA 5001, Australia

art ic l e i nf o

a b s t r a c t

Article history: Received 26 December 2013 Received in revised form 30 April 2014 Accepted 30 April 2014 Available online 6 June 2014

Kinematic boundary condition is usually used when dealing with transient free-surface flow problems in isotropic media. When dealing with anisotropic problems, a transformation can transform the anisotropic media to an equivalent isotropic media for seepage analysis, but the kinematic boundary condition cannot be used directly in the transformed media. A generalization of the kinematic boundary condition along any arbitrary direction is derived for use in the transformed domain for general threedimensional anisotropic problems. A boundary element method for solving transient free-surface seepage problems is developed and the treatment of the proposed kinematic boundary condition in the boundary element method is given. Three examples have been solved to show the reliability and flexibility of the model. Examples are verified with some available experimental and numerical cases to show the accuracy of the model for predicting the phreatic surface and it is shown that anisotropy has a very important and non-neglecting effect in the behavior and the shape of phreatic surface. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Seepage Free-surface Boundary element method Anisotropic Kinematic boundary condition Dam Unconfined aquifer

1. Introduction Anisotropy is a characteristic of geo-materials when seepage problems are considered. Sedimentary deposits are often anisotropic with flow occurring more readily along the plane of deposition than across it [1]. Earth structures such as earth dams, levies, highways, etc. are usually built by depositing and compacting layers of soil. The resulted earth structure is highly anisotropic when water flow through the structure is studied. For examples in earth fill dams the ratio K h =K v (horizontal to vertical hydraulic conductivity ratio) may exceed 20 [2]. Boundary element method (BEM) is a powerful method for solving seepage problems. This method was developed in the 1960s as an efficient numerical technic [3]. A great advantage of the boundary element method is the formulation, which is written on the domain boundaries. Therefore only the boundaries of the domain should be discretized [4]. The reduction of one dimension decreases the cost of discretization and generally may decrease the memory storage and solution effort in many cases. As mesh preparation is one of the most costly parts of the analysis [5], BEM can have a significant advantage of computational cost, especially in three-dimensional problems, as there is a great reduction in the number of nodes and elements, besides mesh

n

Corresponding author. E-mail address: rafi[email protected] (K. Rafiezadeh).

http://dx.doi.org/10.1016/j.enganabound.2014.04.025 0955-7997/& 2014 Elsevier Ltd. All rights reserved.

preparation efforts. When dealing with moving boundary problems such as free-surface seepage problems, the mesh should be modified at each time-step, and clearly, using BEM would effectively enhance the efficiency of the analysis not only for preprocessing but also for mesh manipulations in the process of the transient problems solution in each time-step or steady-state problems in each trial iteration. When dealing with isotropic seepage problems, the governing equation is the Laplace equation. Solution of Laplace equation with BEM is available in the literature for both two- and threedimensions [6]. When the porous media is anisotropic, the governing equation is not Laplace equation anymore. In general anisotropy, there is a tensor of hydraulic conductivity in place of a scalar value and the governing equation is [7] kxx ∂2 φ=∂x2 þkyy ∂2 φ=∂y2 þ kzz ∂2 φ=∂z2 þ 2kxy ∂2 φ=∂x∂y þ 2kyz ∂2 φ=∂y∂z þ2kxz ∂2 φ=∂x∂z ¼ 0 when the porous media has the orthotropic hydraulic conductivity, the cross-derivatives vanish and one deals with a simpler governing equation kxx ∂2 φ=∂x2 þ kyy ∂2 φ=∂y2 þkzz ∂2 φ=∂z2 ¼ 0 [8]. For solving the general anisotropic or orthotropic seepage problems with BEM, two different approaches have been used. The first approach is the fundamental solution approach and the goal is to find a fundamental solution to the corresponding governing equation and develop the BEM formulation based on the proposed fundamental solution. A fundamental solution for the orthotropic seepage governing equation has been given by

52

K. Rafiezadeh, B. Ataie-Ashtiani / Engineering Analysis with Boundary Elements 46 (2014) 51–66

Nomenclature x, y, z coordinates in original domain xn, yn, zn coordinates in rotated domain xnn, ynn, znn coordinates in transformed domain x† ; y† ; z† coordinates along an arbitrary line in original domain x†nn ; y†nn ; z†nn coordinates along an arbitrary line in transformed domain φ potential head p seeping fluid pressure ρ seeping fluid density g magnitude of gravity of the Earth n soil porosity g gravity vector of the Earth n outward normal to boundary unit vector ∂φ=∂n normal to boundary gradient of potential head q flux vector

Brebbia and Chang [8]. Steady state BEM solution of free-surface seepage for two-dimensional cases was given by Bruch et al. [9] using the fundamental solution approach. A set of equilibrium conditions between zones was provided and the singularity when multiple zones were met in a single node or on the boundaries was studied. Bruch and Grilli [10,11] then extended their work to transient free-surface flow. Weakly singular integral equations for Darcy's flow in anisotropic media were used by Rungamornrat and Wheeler [12]. Rungamornrat [13] also used the fundamental solution approach and the governing equations were established based on a pair of weakly singular weak-form integral equations for fluid pressure and fluid flux in general anisotropic media. More complicated set of fundamental solutions for the problem were then used. As the governing equation of seepage is similar to other potential problems such as heat transfer, the fundamental solution approach has also been used for solving potential and heat transfer problems. A fundamental solution approach has been used to solve the anisotropic potential problems with indirect BEM by Zhang et al. [14]. A comparison of BEM formulations for steady state anisotropic heat conduction in two dimensions based on the fundamental solution has been given by Mera et al. [15]. Fundamental solution for the general anisotropic Laplacian operator has been given in [16] and [17]. Solution of this equation by the fundamental solution approach has been presented in [17] for applications in heat conduction problems. A second approach that has been used widely is the transformation approach. In this approach the homogenous anisotropic domain of seepage is transformed geometrically in a way that the seepage governing equation on the transformed domain is the Laplace equation. The method was used for solving the anisotropic seepage problems with graphical flow nets method in classical soil mechanics [18,19]. This approach was used to solve the potential problems with BEM in two- and three-dimensions. This approach has been used to solve the two-dimensional orthotropic seepage problems by BEM [20–23]. Shiah and Tan [24] used this approach to solve two-dimensional general anisotropic potential problems. Then they extended their work to three-dimensional general anisotropic domain solution by BEM [25]. This approach was also used for multi-domain general anisotropic three-dimensional heat transfer by BEM [26]. This approach has been used for seepage applications by Lafe et al. [27] to solve flow problems in unconfined three-dimensional aquifers. Although the three-dimensional aquifers were dealt with in [27], the effective dimension of an aquifer system was reduced to two using of the Dupuit assumptions, as in many practical cases sufficiently accurate results could

K K xx ,

hydraulic conductivity tensor K yy , K zz diagonal elements of the hydraulic conductivity tensor K xy , K xz , K yz cross-diagonal elements of the hydraulic conductivity tensor kxn ; kyn ; kzn hydraulic conductivity in principal axes directions V 1 ; V 2 ; V 3 Three eigenvectors of hydraulic conductivity tenor ∇ Nabla operator: ð∂=∂x; ∂=∂y; ∂=∂zÞ det½ determinant operator of a matrix ½ fig i-th column of a matrix (k) kth time-step I 1 , I 2 , I 3 invariants of hydraulic conductivity tensor R rotating transformation matrix S stretching transformation matrix direct transformation matrix T η†nn ðx†nn ; y†nn ; tÞ a function that defines z†nn on the free-surface D Material differentiation operator Dt

be obtained with less than a full three-dimensional analysis. Based on the fact that the horizontal dimensions of the aquifer are often much larger than the vertical dimension, they assumed a ‘nearly horizontal’ flow as a good approximation. Although the nearly horizontal assumption is a good approximation for problems like aquifers, in many other civil engineering structures like dams, sheet-piles, etc. it is not the case because the horizontal dimensions are not much larger than the vertical dimensions. Rafiezadeh and Ataie-Ashtiani [28] used this approach for three-dimensional seepage in multi-domain orthotropic confined seepage problems by BEM and provided the transformation of the Neumann boundary conditions and equilibrium equations (conservation of the flux across the interface boundaries of adjacent domains) for multidomain problems. Rafiezadeh and Ataie-Ashtiani [7] then extended their work to deal with general anisotropy and provided BEM solution to confined multi-domain general anisotropic seepage problems and provided the generalized transformations of Neumann boundary conditions and equilibrium equations as well. A transient unconfined problem is a moving-boundary problem in which the free-surface location changes as time progresses. Kinematic boundary condition as given by Bear [29] and Liggett and Liu [20] can be used to provide a boundary condition on the free-surface. Liggett and Liu [20] have used the kinematic boundary condition for two- and three-dimensional isotropic seepage analysis with BEM along the vertical line [20]. Writing the isotropic free-surface boundary condition in the finite-difference form, Liggett and Liu [20] implemented the kinematic boundary condition in the BEM solution of free-surface seepage problems. For many problems it is convenient to be able to describe the change of the potential on the free surface along an arbitrary straight line (not necessarily vertical). Liggett and Liu [20] generalized the kinematic free-surface boundary condition along an arbitrary straight line for two-dimensional case. To the best of authors' knowledge, no transient BEM solution for the free-surface seepage problems has been given in threedimensional anisotropic media yet. The main objective of this work is to provide a BEM solution for three-dimensional freesurface seepage problems in an anisotropic domain. The transformation approach given by the authors in [7] is used for BEM treatment of the general anisotropy. To extend our previous confined orthotropic [28] and general anisotropic seepage analysis studies [7] to free-surface problems, the generalized form of kinematic boundary condition is derived for an anisotropic domain along any arbitrary straight line in three-dimensions. Implementation of the derived boundary condition and its finite difference


form in the BEM is also provided in this study. Formulation for relocating the transient location of free-surface nodes are presented as well. A smoothing algorithm is proposed for soft transient behavior of the free-surface and a simple algorithm for enhancing the position of nodes in neighborhood of seepage face is used. Finally the previous 3D BEM code (SEEPBEM3D) [7] is developed to handle free-surface anisotropic problems and is used to solve three examples.

53

kzn ¼ λ3 ¼ I 1 kxn kyn

ð10Þ

v ¼ ðI 1 =3Þ2 I 2 =3

ð11Þ

s ¼ ðI 1 =3Þ3 I 1 I 2 =6 þI 3 =2

ð12Þ

s 1 θ ¼ cos 1 3=2 3 v

ð13Þ

I 1 ¼ kxx þ kyy þ kzz

ð14Þ

2. Governing equation

I 2 ¼ kxx kyy þ kyy kzz þ kzz kxx kxy kyz kxz

We can define the specific discharge q (or seepage velocity) as the volume flow rate per unit total area (pores plus particles) normal to the flow. By neglecting fluid and soil compressibility and assuming saturated flow the mass conservation equation to [20]:

I 3 ¼ kxx kyy kzz þ 2kxy kyz kxz ðkxx kyz þ kyy kxz þ kzz kxy Þ

ð16Þ

V 1x ¼ ½kxy kyz Bx kxz ½kyz kxz C x kxy

ð17Þ

∇:q ¼ 0

V 1y ¼ ½kyz kxz C x kxy ½kxz kxy Ax kyz

ð18Þ

V 1z ¼ ½kxz kxy Ax kyz ½kxy kyz Bx kxz

ð19Þ

V 2x ¼ ½kxy kyz By kxz ½kyz kxz C y kxy

ð20Þ

V 2y ¼ ½kyz kxz C y kxy ½kxz kxy Ay kyz

ð21Þ

V 2z ¼ ½kxz kxy Ay kyz ½kxy kyz By kxz

ð22Þ

V 3x ¼ ½kxy kyz Bz kxz ½kyz kxz C z kxy

ð23Þ

V 3y ¼ ½kyz kxz C z kxy ½kxz kxy Az kyz

ð24Þ

V 3z ¼ ½kxz kxy Az kyz ½kxy kyz Bz kxz

ð25Þ

Ax ¼ kxx kxn

ð26Þ

Ay ¼ kxx kyn

ð27Þ

Az ¼ kxx kzn

ð28Þ

Bx ¼ kyy kxn

ð29Þ

By ¼ kyy kyn

ð30Þ

Bz ¼ kyy kzn

ð31Þ

C x ¼ kzz kxn

ð32Þ

C y ¼ kzz kyn

ð33Þ

C z ¼ kzz kzn

ð34Þ

2

ð1Þ

Considering Darcy's law as: q ¼ K∇φ

ð2Þ

where φ is the potential head and is defined by φ ¼ p=ρg þ z

ð3Þ

where p is the fluid pressure, ρ is the density of fluid, g ¼ jgj is the gravity acceleration, z is the upward vertical coordinate (in the opposite direction of gravity vector g, i.e. vertical unit vector is k ¼ g=g) and K is hydraulic conductivity tensor defined as follows: 2 3 kxx kxy kxz 6k 7 K ¼ 4 xy kyy kyz 5 ð4Þ kxz

kyz

kzz

Combining Eqs. (1) and (2) results into: ∇:ðK∇φÞ ¼ 0

ð5Þ

Assuming homogeneity of hydraulic conductivity tensor, by expanding Eq. (5) and substituting K from Eq. (4), the governing equation for seepage in general anisotropic media is as following: 2

kxx

2

2

2

2

2

∂ φ ∂ φ ∂ φ ∂ φ ∂ φ ∂ φ þ 2kyz þ 2kxz ¼0 þ kyy 2 þkzz 2 þ 2kxy ∂x∂y ∂y∂z ∂x∂z ∂x2 ∂y ∂z

ð6Þ

If an isotropic media is assumed, the diagonal elements of the hydraulic conductivity tensor are all equal to K and all off-diagonal elements are equal to zero, thus the governing equation will be simplified to the Laplace equation.

3. Domain transformation It has been shown by Rafiezadeh and Ataie-Ashtiani [7] that the following transformation matrix T transforms the governing equation of anisotropic seepage to the Laplace equation 2 3 ðkzn =kxn Þ1=2 V 1x ðkzn =kxn Þ1=2 V 1y ðkzn =kxn Þ1=2 V 1z 6 7 1=2 1=2 1=2 7 ð7Þ T¼6 4 ðkzn =kyn Þ V 2x ðkzn =kyn Þ V 2y ðkzn =kyn Þ V 2z 5 V 3x V 3y V 3z where kxn ¼ λ1 ¼

pffiffiffi I1 þ 2 v cos θ 3

ð8Þ

kyn ¼ λ2 ¼

pffiffiffi I1 π 2 v cos θ þ 3 3

ð9Þ

2

2

2

2

ð15Þ 2

The transformation matrix T transforms the physical coordinate axes to the distorted one as shown in the following: 8 9 8 9 > > < xnn > = = ynn ¼ T y ð35Þ > > :z > ; : > ; z nn As shown in [7], transformation T is a combination of a rotating transformation R and stretching transformation S defined as following: 2 3 V 1x V 1y V 1z 6 7 R ¼ 4 V 2x V 2y V 2z 5 ð36Þ V 3x V 3y V 3z 8 9 8 9 > > < xnn > = < xn > = ynn ¼ S yn > > > :z ; :z > ; nn

n

ð37Þ

54


T¼SR

ð38Þ

8 9 8 9 > > < xn > = = yn ¼ R y > > :z > ; : > ; z n 8 9 8 9 > > < xnn > = < xn > = ynn ¼ S yn > > > :z ; :z > ; nn

ð39Þ

ð40Þ

is

the

material

differentiation

operator.

Defining †nn

†nn

vy†nn j

þ vz†nn k

†nn

, in which i

†nn

,j

†nn

,k

†nn

þ

†nn

,

are unit vectors in x

y†nn and z†nn directions and knowing that v†nn ¼ q†nn =n [18] where n is the porosity of the soil, Eq. (44) is determined as:

∂η† nn q† nn † nn † nn þ U∇ ðz η† nn Þ ¼ 0 on z† nn ¼ η† nn ∂t n

ð45Þ

T transformation is a rotating–stretching transformation where xn and yn coordinates are stretched in a way that the stretched media have the equivalent isotropic hydraulic conductivity kzn in all directions. Considering Darcy's law:

Finally, the transformed governing equation is ∂ φ ∂2 φ ∂2 φ þ þ ¼0 ∂xnn 2 ∂ynn 2 ∂znn 2

D

vx†nn ¼ dx†nn =dt, vy†nn ¼ dy†nn =dt, vz†nn ¼ dz†nn =dt and v†nn ¼ vx†nn i

n

2

where

ð41Þ

q† nn ¼ kzn ∇† nn φ

ð46Þ

Inserting Eq. (46) into (45) we have ∂η† nn kz ¼ n ½∇† nn φ U∇† nn ðz† nn η† nn Þ on z† nn ¼ η† nn ∂t n

4. Three-dimensional transient free-surface boundary condition along an arbitrary line

The unit normal vector on the free-surface can be defined as

Fig. 1 shows the schematic of free-surface seepage in an earth dam. The free-surface is taken to be the surface of constant atmospheric pressure and an unsaturated zone exists above this surface. The free-surface and is shown as blue color surface in Fig. 1. This constant atmospheric pressure p can be chosen as zero without loss of generality. We define x y z coordinate system as a reference right-handed coordinate system in which the zcoordinate is defined as an upward vertical coordinate, i.e. in the opposite direction of the gravity. The potential head defined in Eq. (3) is thus defined by φ ¼ z ðon free surfaceÞ

ð42Þ

The transient free-surface boundary condition along an arbitrary line AB shown in Fig. 1 shall be defined. Point F is the intersection of this line and the free-surface. We may define an auxiliary right-handed coordinate system x† y† z† that the z† coordinate is along the line that we need the free-surface boundary condition. Assume that the porous media of the earth dam body is a general anisotropic media. The coordinate transformation T defined in Eq. (7) is applied to reach xnn ynn znn coordinate system in which the Laplace equation is the governing equation for seepage problem. x†nn y†nn z†nn coordinate system is the transformation of x† y† z† coordinates under T. For unconfined saturated flow in the distorted media, the freesurface elevation is described by z† nn ¼ η† nn ðx† nn ; y† nn ; tÞ

ð43Þ

where η†nn is a scalar function of x†nn , y†nn and t. The material derivative of z†nn η†nn vanishes on z†nn ¼ η†nn [20], we have D † nn ðz η† nn Þ ¼ 0 Dt

Fig. 1. Arbitrary line and defined coordinates.

ð47Þ

† nn

n

" †nn 2 †nn 2 # 1=2 ∇†nn ðz†nn η† nn Þ ∂η ∂η ¼ 1þ ¼ †nn þ ∇†nn ðz† nn ηnn† Þ ∇ ðz†nn η† nn Þ ∂x†nn ∂y†nn

ð48Þ Considering the normal derivative definition, ∂φ ¼ n† nn U∇† nn φ ∂n† nn

ð49Þ

By using Eq. (49) in (48) and rearranging we have " † nn 2 † nn 2 #1=2 ∂η ∂η ∂φ † nn † nn † nn † nn þ ∇ φ U ∇ ðz η Þ ¼ 1 þ ∂x† nn ∂y† nn ∂n† nn From Eqs. (47) and (50) " † nn 2 † nn 2 #1=2 ∂η kz ∂η ∂η ∂φ ¼ n 1þ þ on z† nn ¼ η† nn ∂t n ∂x† nn ∂y† nn ∂n† nn † nn

ð50Þ

ð51Þ

A relationship between x†nn y†nn z†nn and xnn ynn z coordinate systems is required. x† y† z† coordinate system is defined as a right-handed coordinate system in which the z† axis is in direction of the line in which we need the free-surface boundary condition along it. Assume that nn

k

† nn

nn

nn

¼ ax i þ ay j þ az k

nn

ð52Þ

Therefore in the transformed domain, we need the transient †nn free-surface boundary condition along the direction of k . Coordinates of a specific point in the two coordinate systems are related to each other by the following matrix equation (see

ð44Þ

Fig. 2. Seepage-face height enhancement.


Using the chain rule and knowing that T 1 ¼ TT (See the Appendix), we have

Appendix for details) 9 8 nn 9 8 † nn > > = = ynn ¼ TT y† nn > > > ; : nn ; : † nn > z z 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ax 2 6 6 TT ¼ 6 0 4 ax

ax ay pffiffiffiffiffiffiffiffiffiffi ffi 1 ax 2

az pffiffiffiffiffiffiffiffiffiffi ffi 1 ax 2

ay

55

ð53Þ

ax az ffi pffiffiffiffiffiffiffiffiffiffi

ð55Þ ð56Þ

3

7 7 pffiffiffiffiffiffiffiffiffiffiffi 7 1 ax 2 5 az 1 ax 2 ay

∂η† nn pffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi∂η† nn ∂η† nn ¼ 1 ax þ ax nn nn † nn ∂x ∂z ∂x ax ay ∂η† nn ∂η† nn az ∂η† nn ∂η† nn p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nn þ ay nn ¼ þ nn † nn ∂z ∂y 1 ax 2 ∂x 1 ax 2 ∂y

ð54Þ

Using Eqs. (35) and (42), on the free-surface φ ¼ ðkxn =kzn Þ1=2 V 1z xnn þ ðkyn =kzn Þ1=2 V 2z ynn þ V 3z znn

Fig. 3. Flowchart of the model.

ð57Þ

56


From Eqs. (53) and (57) we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi φ ¼ ½ 1 ax 2 ðkxn =kzn Þ1=2 V 1z þ ax V 3z x† nn " # ax ay az 1=2 1=2 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkxn =kzn Þ V 1z þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkyn =kzn Þ V 2z þ ay V 3z y† nn 1 ax 2 1 ax 2 " # ay ax az þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkxn =kzn Þ1=2 V 1z þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkyn =kzn Þ1=2 V 2z þ az V 3z z† nn 1 ax 2 1 ax 2

Using Eqs. (51) and (60) we have " † nn 2 † nn 2 #1=2 ∂φ λkz ∂η ∂η ∂φ ¼ n 1þ þ on z† nn ¼ η† nn ∂t n ∂x† nn ∂y† nn ∂n† nn

ð62Þ

which is the transient free-surface boundary condition along an arbitrary line for the transformed geometry of a general anisotropic media.

ð58Þ On the free-surface z†nn ¼ η†nn ðx†nn ; y†nn Þ, thus hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i φ¼ 1 ax 2 ðkxn =kzn Þ1=2 V 1z þ ax V 3z x† nn " # ax ay az 1=2 1=2 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkxn =kzn Þ V 1z þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkyn =kzn Þ V 2z þ ay V 3z y† nn 1 ax 2 1 ax 2 " # ay ax az þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkxn =kzn Þ1=2 V 1z þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkyn =kzn Þ1=2 V 2z þ az V 3z η† nn 1 ax 2 1 ax 2 ð59Þ As the free-surface along the z†nn -direction is needed, x†nn and y are constant along that direction, thus †nn

∂φ ∂η† nn ¼λ on z† nn ¼ η† nn ∂t ∂t

ð60Þ

where " # ax az ay 1=2 1=2 λ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkxn =kzn Þ V 1z þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkyn =kzn Þ V 2z þaz V 3z 1 ax 2 1 ax 2 on z† nn ¼ η† nn

ð61Þ

Fig. 5. BEM mesh of the rectangular dam.

Table 1 Hydraulic conductivity tensors of Example I. Case name

Hydraulic conductivity tensor 2

Isotropic

Orthotropic

Anisotropic

3 1 0 0 6 7 3 4 0 1 0 5 10 ðm=sÞ 0 0 1 2 3 1 0 0 60 1 0 7 3 4 5 10 ðm=sÞ 0 0 0:1 2 3 1 0:1 0:2 6 7 3 4 0:1 1:1 0:5 5 10 ðm=sÞ 0:2 0:5 1

Table 2 Boundary conditions of rectangular dam. Face

Boundary condition

x ¼0, 5 m o yo 5 m, 0o z o 10 m x ¼5 m, 5 mo yo 5 m, 0o zo 2 m x ¼5 m, 5 mo yo 5 m, 2o zo 10 m z ¼0, 5 m o yo 5 m, 0 ox o 5 m y¼ 5 m, 0o z o 10 m, 0o x o5 m y¼ 5 m, 0 oz o 10 m, 0o x o 5 m z ¼10, 5 m oy o5 m, 0o x o5 m

φ¼ 10 m φ¼ 2 m φ¼ z ∂φ/∂n¼ 0 ∂φ/∂n¼ 0 ∂φ/∂n¼ 0 Free-surface condition

Fig. 4. Dimensions of the rectangular dam.


5. BEM treatment Using the transformation, Laplace equation is the governing equation on the transformed domain Ωnn with transformed boundary Γ nn . As it is very well established in the BEM literature for the Laplace equation, we then have [6] Z Z ∂φ ∂G G dΓ ¼ φ dΓ ð63Þ c i φi þ Γ nn ∂nnn Γ nn ∂nnn where G is named as the Green's Function and is equal to 1=4πr nn when the region is three dimensional and the governing equation is the Laplace Equation. r nn is the distance from any point i to the integration point on the transformed boundary. ci is between

57

0 and 1 and is equal to 1 if the point i is inside the boundary, 1/2 if point i is on a smooth boundary and is equal to zero when point i is outside the boundary. If point i is on a non-smooth part of the boundary (a boundary corner), ci is equal to the special angle of the corner divided by 4π. The spatial angle s defined as the surface area of the sector of the unit sphere that is centered at the corner point and lies inside the domain [4]. BEM treatment for the general problem of anisotropic seepage is given in [28] and [7]. Readers are referred to [7] and [28] for further details. Here we describe how the free-surface boundary condition is implemented into SEEPBEM3D model [7]. The free-surface boundary condition in Eq. (62) can be discretized in time using a finite difference method within a small time interval, Δt, which gives the potential at time (k)Δt as follows: " † nn 2 † nn 2 #1=2 λkzn Δt ∂η ∂η 1þ þ n ∂x† nn ∂y† nn " # ∂φ ∂φ θ þ ð1 θÞ ∂n ðk þ 1Þ ∂n ðkÞ

φðk þ 1Þ ¼ φðkÞ

ð64Þ

in which θ is a time-weighting factor that positions the derivative between time levels (k) and (kþ 1). We can re-write Eq. (64) as: ∂φ þβ ð65Þ φðk þ 1Þ ¼ α ∂n ðk þ 1Þ where α¼

λγ † θkzn Δt n

β ¼ φðkÞ "

λγ † ð1 θÞkzn Δt ∂φ n ∂n ðkÞ

∂η† nn γ ¼ 1þ ∂x† nn †

Fig. 6. Comparison of free-surface of rectangular dam.

ð66Þ

2

∂η† nn þ ∂y† nn

ð67Þ

2 #1=2 ð68Þ

Eq. (65) represents the three-dimensional free-surface kinematic boundary condition along an arbitrary line for general anisotropic media when one have used the transformation approach for dealing with anisotropy. This equation establishes a relation between φ and ∂φ=∂n at the time-level (k þ1). We emphasize that

Fig. 7. Effect of anisotropy in the free-surface transient descent of rectangular dam.

58


Fig. 8. Mesh deformations of rectangular dam.

Table 3 Hydraulic conductivity tensors of Example II. Case name


3 1 0 0 6 7 3 4 0 1 0 5 10 ðm=sÞ 0 0 1 2 3 1 0 0 6 7 3 0 1 0 4 5 10 ðm=sÞ 0 0 0:1 3 2 1 0:1 0:2 6 7 3 4 0:1 1:1 0:5 5 10 ðm=sÞ 0:2 0:5 1

Isotropic

Orthotropic

Anisotropic

this equation is only applicable on the free surface with a fixed †nn value of the x†nn and y†nn coordinate (i.e. along the k direction). When in each time-step the potential has been calculated, the free-surface is repositioned. Assume a node on the free-surface nn nn with coordinates ðxnn 0ðkÞ ; y0ðkÞ ; z0ðkÞ Þ in time-step (k). The equation of †nn line passing through this node along the k is xnn xnn 0ðkÞ ax

Fig. 9. Effect of anisotropy in the phreatic surface of rectangular dam.

¼

ynn ynn 0ðkÞ ay

¼

znn znn 0ðkÞ

ð69Þ

az

Writing Eq. (57) for time-step (k þ1) and using Eq. (69) we have 2 6 6 4

Fig. 10. Convergence of the free-surface of rectangular dam.

az

0

0

az

ðkxn =kzn Þ1=2 V 1z ðkyn =kzn Þ1=2 V 2z 9 8 a xnn ax znn > > 0ðkÞ > > = < z 0ðkÞ nn ¼ az ynn 0ðkÞ ay z0ðkÞ > > > > ; : φðk þ 1Þ

38 9 ax > xnn > 7< nn = ay 7 y 5> > V 3z : znn ;ðk þ 1Þ ð70Þ

By solving the system of three equations in Eq. (70) for xnn , ynn and znn , the new position of the node on the free-surface in time nn nn step (kþ 1)is determined, i.e. ðxnn 0ðk þ 1Þ ; y0ðk þ 1Þ ; z0ðk þ 1Þ Þ: The general boundary condition for seepage face is φ ¼ z. Cooley [30] stated that this boundary condition is necessary but


59

Fig. 11. Dimensions of trapezoidal dam.

Ashtiani [7] code that is developed for free-surface problems is given in Fig. 3.

6. Simplified special Cases Some simplified special cases of the general presented formulas are considered here. 6.1. Free-surface boundary condition along an arbitrary line for three-dimensional orthotropic media

Fig. 12. BEM mesh of trapezoidal dam.

In this case, no rotation is needed for transformation and the rotation Matrix R in Eq. (36) should be taken as a unit matrix, so V 1z ¼ V 2z ¼ 0 and V 3z ¼ 1. Eq. (61) is simplified to " # ay ax az λ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkxn =kzn Þ1=2 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkyn =kzn Þ1=2 þ az on z† nn ¼ η† nn 1 ax 2 1 ax 2 ð71Þ

Table 4 Boundary conditions of example II.

Other relations then can be used without any change. Face

Boundary condition

x ¼ 0, 5 mo yo 5 m, 0o zo 5 m x þ z¼ 7 m, 5 m o yo 5 m, 0o zo 1 m x þ z¼ 7 m, 5 m o yo 5 m, 1o zo 5 m z ¼ 0, 5 mo yo 5 m, 0o xo 7 m y ¼ 5 m, 0o zo 5 m, 0o xo 7 m, xþ zr 7 m y ¼5 m, 0o z o5 m, 0o xo 7 m, xþ z r7 m z ¼ 5, 5 mo yo 5 m, 0o xo 2 m

φ¼5 m φ¼1 m φ¼z ∂φ/∂n ¼0 ∂φ/∂n ¼0 ∂φ/∂n ¼0 Free-surface condition

not sufficient and using just this condition and the early methods of Rubin [31] and Neuman [32], the seepage-face could be located too high. This may cause recession of the water table to be retarded at the time-steps where the problem occurred. Another condition must be added; that is, the seepage face must also be allowed to attain its lowest possible elevation consistent with the seepage-face boundary conditions[30,33]. In our analyses we also found the similar condition and the seepage-face height was located above the neighboring free-surface nodes. To enhance the seepage-face height, a simple algorithm for adjustment of the seepage-face height was implemented. For this adjustment, we force that the z-coordinate of each free-surface node which is neighbor to a seepage-face node should not be greater that the average z-coordinate of neighboring free-surface nodes which are not neighbor to a seepage-face node (Fig. 2). After calculating the new position of all free-surface nodes, a smoothing procedure is run for smoothing the free-surface. A plane of regression is passed through the new position of the considered free-surface node and all of its neighbor free-surface nodes new positions. We then reset the position of each freesurface node by intersecting the line passing through the node (Eq. (69)) and the regressed plane. After repositioning the free-surface nodes, all non-free-surface nodes that their location is dependent to the phreatic surface should be repositioned. A flowchart of the general algorithm of the SEEPBEM3D Rafiezadeh and Ataie-

6.2. Free-surface boundary condition along an arbitrary line for three-dimensional isotropic media In isotropic case, kxn ¼ kyn ¼ kzn ¼ K. Eq. (71) then is simplified to

" # ay ax az λ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þaz on z† nn ¼ η† nn 1 ax 2 1 ax 2

ð72Þ

6.3. Free-surface boundary condition along a vertical line for threedimensional isotropic media When the free-surface boundary condition is prescribed along a vertical line, ax ¼ ay ¼ 0 and az ¼ 1 and thus λ ¼ 1 on z† nn ¼ η† nn and so Eq. (62) changes to " 2 2 #1=2 ∂φ K ∂η ∂η ∂φ ¼ 1þ on z† nn ¼ η† nn þ ∂t n ∂x ∂y ∂n

ð73Þ

ð74Þ

which is the form of the kinematic boundary condition presented by [20].

7. Examples Three examples are solved with the present BEM model. In the first example a rectangular dam is simulated. In the second example a trapezoidal dam with a slanted seepage-face is solved and in the third example a well in an unconfined aquifer is studied.

60


Fig. 13. Comparison of free-surface profiles in isotropic case.

Fig. 14. Effect of anisotropy in free-surface transient descent of trapezoid dam.


61

Fig. 15. Mesh deformations of trapezoidal dam.

Table 5 Hydraulic conductivity tensors of example III. Case name


Isotropic

Orthotropic

Anisotropic

Fig. 16. Effect of anisotropy in the phreatic surface of trapezoidal dam.

Fig. 17. Convergence of the free-surface of trapezoidal dam.

7.1. Example I. Rectangular dam The first example is a rectangular dam and has been solved by [34–43]. The dam height is 10 m with upstream full water depth. The downstream water depth is 2 m and the dam width is 5 m. Dam crest length is 10 m. To show the effect of anisotropy on the free-surface profile, three cases are considered here. The first case is the isotropic case. The second case is an orthotropic case with the vertical hydraulic conductivity of one-tenth of the horizontal hydraulic conductivity. The third case involves a general

3 9:57 0 0 6 0 9:57 0 7 4 5 (in./min) 0 0 9:57 2 3 9:57 0 0 6 7 0:957 0 5 (in./min) 4 0 0 0 9:57 2 3 10 1 4 6 7 4 1 11 5 5 (in./min) 4 5 10

anisotropic case with non-vanishing off-diagonal elements in the hydraulic conductivity tensor. Hydraulic conductivity tensors of the three cases are given in Table 1 and the effective porosity is equal to 0.3. A schematic of the dam is given in Fig. 4. The surface boundaries of the dam are discretized with triangular elements and a structured mesh. The discretization includes 1034 nodes and 2062 three-node linear isoparametric triangular elements. The discretized three-dimensional model at the initial condition is shown in Fig. 5. We have used 13-point Gauss quadrature for calculating the integrals in the numerical method. The defined boundary conditions at the initial step are given in Table 2. At the initial step, the free-surface is placed horizontally at the top of the model, i.e. z¼10 m. Because the downstream water depth is 2 m, the free-surface descends transiently and finally reached to equilibrium around a profile which is the steady-state free-surface profile. The time-step for the transient BEM analysis is 10 s. After 8000 s, the profile reaches to almost an equilibrium in the isotropic case. Computed location of free-surface at the 8000 s in isotropic case is compared with the steady-state results of Oden and Kikuchi [34], Lacy and Prevost [35], Borja and Kishnani [36], Bradet and Tobita [37], Ayvaz et al. [38], Herreros et al. [39], Ayvaz and Karahan [40], Darbandi et al. [41], Bazyar and Graili [42] and Kazemzadeh-Parsi and Daneshmand [43] and is depicted in Fig. 6. The results show a very good agreement of the calculated freesurface position with the results of other studies. The transient descend of the free-surface at a 2000 s intervals are shown in Fig. 7. In isotropic and orthotropic cases the freesurface is the same in any vertical plane, i.e. the behavior is twodimensional, but in the anisotropic case, the profile of free-surface is not two-dimensional and the seepage-face height is clearly greater at Y ¼ 5 m than Y ¼ þ5 m and the free-surface has a descending slope in the direction of y-coordinate. After 8000 s the free-surface descends and therefore the domain geometry and

62


Fig. 18. Dimensions of well in example III.

location of nodes and elements change. The changed geometry is shown in Fig. 8. The effect of anisotropy can be observed on the position of phreatic surfaces of the three cases after 8000 s in Fig. 9. The variations of the free-surface position at the seepage-face with time for isotropic, orthotropic and three points (left/middle/ right) of anisotropic cases are shown in Fig. 10. 7.2. Example II. Trapezoidal dam

Fig. 19. BEM mesh of the well.

Table 6 Boundary conditions of example III. Face

Boundary condition

r ¼ 76.8 in. (195.072 cm), 0 rz r 48 in. (121.92 cm) 0 rr r 48 in. (121.92 cm) z ¼ 0 r ¼ 4.8 in. (12.192 cm), 0 r z r12 in. (30.48 cm) r ¼ 4.8 in. (12.192 cm), 12 in. (30.48 cm) r z 0 rr r 48 in. (121.92 cm) z ¼ 48 in. (121.92 cm)

φ ¼48 in. (121.92 cm) ∂φ/∂n¼ 0 φ ¼12 in. (30.48 cm) φ ¼z Free-surface condition

Fig. 20. Comparison of free-surface profiles in isotropic case.

For the second example a trapezoidal dam with slanted seepage face is solved. The dam height is 5 m with upstream full water depth. The downstream water depth is 1 m and the dam width is 2 m on top and 7 m on bottom giving a 451 slope of the downstream face. Dam crest length is 10 m. To show the effect of anisotropy on the free-surface profile, two anisotropic solutions also have been given and the results will be compared with the isotropic case. The first case is the isotropic case. The second case is an orthotropic case with the vertical hydraulic conductivity of one-tenth of the horizontal hydraulic conductivity. The third case involves a general anisotropic case with non-vanishing off-diagonal elements in the hydraulic conductivity tensor. Hydraulic conductivity tensors of the three cases are given in Table 3. The effective porosity of the soil is equal to 0.3. A schematic of the dam with dimensioning is given in Fig. 11. The surface boundaries of the dam are discretized with triangular elements and a structured mesh. The discretization includes 1034 nodes and 2062 three-node linear isoparametric triangular elements. The discretized three-dimensional model at the initial condition is shown in Fig. 12. We have used 13-point Gauss quadrature for calculating the integrals in the numerical method. Boundary conditions should be defined to solve the problem. At the upstream boundary, head is constantly prescribed to be 5 m. On the downstream, in the lower part below the tail-water, the potential head is prescribed to be equal to 1 m while above the tail-water the seepage-face exists where the potential head is prescribed to be equal to the z-coordinate of the node. On the freesurface, the generalized kinematic boundary condition is considered. On bottom and left and right boundaries, a no-flux condition is imposed. The defined boundary conditions at the initial step are given in Table 4. The initial free-surface is placed horizontally at the top of the model, i.e. z ¼5 m. Because the downstream water depth is 1m, the free-surface descends gradually and finally it


reaches to a steady-state free-surface profile. The time-step for the transient BEM analysis is 10 s. The problem runs for 8000 s to reach to steady-state equilibrium for the isotropic case. Computed location of free-surface at the 8000 s in isotropic case is compared with the steady-state results of Oden and Kikuchi [34], Lacy and Prevost [35], Borja and Kishnani [36], Bradet and Tobita [37], Bazyar and Graili [42] and KazemzadehParsi and Daneshmand [43] and is depicted in Fig. 13. The results match very well with the results of other studies.

63

Positions of the free-surface at 2000 s intervals are shown in Fig. 14. The free-surface descends gradually and hence the geometry of the seepage domain changes in each time-step. The geometries of the seepage domain after 8000 s in different cases are shown in Fig. 15. Effect of anisotropy in the phreatic surfaces of the three cases after 8000 s can be observed in Fig. 16. Again like Example I, in isotropic and orthotropic cases the free-surface is the same in any vertical plane, i.e. the behavior is two-dimensional, but in the anisotropic case, the profile of free-surface is not two-

Fig. 21. Effect of anisotropy in free-surface transient descent in the well and final steady-state shape.

64


dimensional and the seepage-face height is clearly greater at Y¼ 5 m than Y¼ þ5 m and the free-surface has a descending slope in the direction of y-coordinate. The time-descend of the free-surface at the seepage-face for isotropic, orthotropic and three points (left/middle/right) of anisotropic cases is shown in Fig. 17. It can be seen that the anisotropy has considerable effect both in transient and steady-state behavior of the seepage. The rate of descend of the free surface is much faster in isotropic case than in orthotropic case. Reason for this is that hydraulic conductivity of the orthotropic material in the vertical direction is one-tenth of horizontal one. Because of vertical compaction, such anisotropy is common in earth dams and so it is an important note for the engineer that this type of anisotropy should not be neglected when the hydraulic behavior of an earth dam is studied. Third case shows that general anisotropy usually cannot be simplified to two-dimensional modeling as the effect of the anisotropy cannot be neglected. Although the general anisotropy occasionally happens in dam bodies, it often happens in dam foundations. Again if the engineer is studying the hydraulic behavior of a dam considering the effect of porous foundations, a full three-dimensional analysis is essential and again the effect of anisotropy cannot be neglected.

agreement to the experimental results than other mentioned numerical solutions. To compare the three cases, the position of the free-surface at 1, 2 and 3 min are shown in Fig. 21. The free-surface at 30 min that is almost the steady-states free-surface is also shown in this figure.

7.3. Example III – Hall's well A fully-penetrated well was experimentally tested in a sand tank by Hall [44]. This model has been considered as the third example as the problem was previously solved numerically by other researchers. The problem is solved by Taylor and Luthin [45] using finite difference method, by Cooley [30] using finite element method and by Clement et al. [46] using a different finite difference scheme. The well is a 4.8 in. (12.192 cm) radius fullypenetrating well in a sand tank of radius 76.8 in. (195.072 cm) and 48 in. (121.92 cm) height. The potential head on the exterior cylindrical boundary is 48 in. (121.92 cm) and the water depth in the well is kept constantly at 12 in. (30.38 cm) Although the problem was previously tested and solved only for isotropic soil [44,45,30,46], to show the results in non-isotropic cases, two additional orthotropic and general anisotropic cases were also solved in this research. The hydraulic conductivity tensors of first, second and third cases which are named isotropic, orthotropic and anisotropic, respectively, are given in Table 5. The effective porosity is equal to 0.3. A schematic of the well with dimensioning is given in Fig. 18. The surface boundaries of the problem are discretized with triangular elements and a structured mesh. These boundaries include two circular boundaries at the top and bottom of the tank, and two cylindrical boundaries for the outer boundary of the tank and the well. The discretization includes 1040 nodes and 2080 three-node linear isoparametric triangular elements. The discretized three-dimensional model at the initial condition is shown in Fig. 19. We have used 13-point Gauss quadrature for calculating the integrals in the numerical method. At the initial step, the free surface is set to be at the top of the model and the water depth in the well is 12 in.(30.48 cm) When seepage starts, the free-surface gradually descends until it reaches to a steady equilibrium state. The prescribed boundary conditions of the model in the first step are given in Table 6. The time-step for the transient BEM analysis is 0.1 min. After 30 min, the profile will reach to the steady state equilibrium in all cases. Computed location of free-surface at the 30 min in isotropic case is compared with the steady-state experimental results of Hall [44] and numerical results of Taylor and Luthin [45], Cooley [30] and Clement et al. [46] and is depicted in Fig. 20. The results show that the present method is in almost perfect agreement with the experimental results of Hall [44] and seems that has a better

Fig. 22. Free-surface cones in (a) isotropic, (b) orthotropic and (c) anisotropic cases.

Fig. 23. Phreatic surface close to the well in (a) isotropic, (b) orthotropic and (c) anisotropic cases.


65

considered to study the flexibility and ability of such implementation for transient free-surface problems in three-dimensional general anisotropic media. From the solved examples it can be concluded that the transformation approach can be effectively evolved for transient free-surface problems by the presented formulation for generalized kinematic boundary condition in a transformed media. Comparing the results from solved examples considering anisotropy (general or orthogonal) with isotropic ones show that the effect of anisotropy on the shape and position of the phreatic surface is not negligible. Fig. 24. Intersection of free-surface and the well in isotropic, orthotropic and anisotropic cases.

Acknowledgments The authors would like to thank the two reviewers and Editor for their comments, which have improved the quality of this paper.

Appendix We assume a linear rotating transformation matrix which † transforms k to k ¼ ax iþ ay j þ az k, i.e. †

k ¼ Tk

ðA1Þ †

We can assume that we may transform k to k by two successive transformations around x and yaxes, i.e. Fig. 25. Convergence of the free-surface in well to the steady state.

The depression cones of free-surfaces in the three cases are shown in Fig. 22 in steady state. It is seen that the seepage-face and free-surface intersection is a circle in isotropic case, a saddle in orthotropic case and an ellipse in anisotropic case. A closer view of the phreatic surface near the well is shown in Fig. 23. The intersection line of free-surface and seepage-face is shown in Fig. 24. As time progresses, the phreatic surface declines until it reaches to an equilibrium. At the beginning, because the phreatic surface is in the highest position, the rate of descend of the freesurface is large, but as the free surface declines, the rate of declining decreases. Seepage face height is plotted in time in Fig. 25 to show the convergence of the phreatic surface to a state of equilibrium.

T ¼ Rx Ry

ðA2Þ

in which Rx is 2 1 0 6 a Rx ¼ 4 0 0 b and Ry is a 2 c 6 Ry ¼ 4 0 d

a rotating transformation around xaxis 3 0 7 b5 a

rotating transformation around yaxis 3 0 d 7 1 05 0 c

Using Eqs. (A3) and (A4) in (A2) we have 2 3 c 0 d 6 7 a bc 5 T ¼ 4 bd ad b ac

ðA3Þ

ðA4Þ

ðA5Þ

Using Eq. (A5) in (A1) we have

8. Conclusions The previous works of the authors [28,7] for developing a three-dimensional multi-purpose BEM seepage analysis code was extended to deal with the transient free-surface anisotropic problems. It was shown that the ordinary kinematic boundary condition for seepage in isotropic media is no-longer valid in the transformed domain of an anisotropic media and a generalization for the kinematic boundary condition is needed for such problems. In this study a generalization to the kinematic boundary condition has been presented. The implementation of this boundary condition in the BEM analysis was also conducted in this study. Adding small pre- and post-processing to a standard BEM code allows for anisotropic analysis if the transformation approach is used. Using the generalized kinematic boundary condition in the model simply allows for solving unconfined moving boundary seepage problems, as the generalized boundary condition provides a simple closed form algebraic relation between potential and normal to boundary flux in the transformed domain. Examples have been

d ¼ ax

ðA6Þ

bc ¼ ay

ðA7Þ

ac ¼ az

ðA8Þ

From the orthogonal transformation matrix Ry , we have 2

c2 þ d ¼ 1 From Eqs. (A9) and (A6) we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ¼ 7 1 ax 2

ðA9Þ

ðA10Þ

To choose between signs we can investigate for an identical transformation. For an identical transformation the matrix Ry should be the identity matrix, i.e. c ¼ 1. Thus the positive sign is the correct answer. The minus sign results in a left-handed coordinate system while the plus sign results in a right-handed coordinate system. Thus we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ¼ 1 ax 2 ðA11Þ From Eqs. (A11) and (A7) we have

66


pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ¼ ay = 1 ax 2

ðA12Þ

From Eqs. (A11) and (A8) we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ¼ az = 1 ax 2

ðA13Þ

Substituting Eqs. (A6),(A11)–(A13) in (A4) we have the transformation in the following form: 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1 ax 2 0 ax 6 ax ay 7 az 6 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ffi ay 7 7 1 ax 2 1 ax 2 ðA14Þ T¼6 6 7 4 pffiffiffiffiffiffiffiffiffiffi ay ax az ffi pffiffiffiffiffiffiffiffiffiffiffi az 5 2 2 1 ax

1 ax

From Eqs. (A4),(A5),(A12) and (A13) we know Rx Rx T ¼ I and Ry Ry T ¼ I

ðA15Þ

Thus we have T 1 ¼ ðRx Ry Þ 1 ¼ Ry 1 Rx 1 ¼ Ry T Rx T ¼ ðRx Ry ÞT ¼ TT

ðA16Þ

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