Computer methods in applied

mechanics and engineering

ELSEVIER Comput. Methods Appl. Mech. Engrg. 174 (1999) 355-369

www.elsevier.com/locate/cma

Finite element approximation of Quasi-3D shallow water equations E d i e M i g l i o , A l f i o Q u a r t e r o n i , * F a u s t o Sa le r i

Politecnico di Milano, Dipartimento di Matematica "F. Brioschi" P.zza Leonardo da Vinci 32, 20133 Milano, Italy

Received 27 April 1998; revised 17 June 1998

A b s t r a c t

A new method to solve the Quasi-3D shallow water equations is proposed. This method combines a suitable mass-preserving finite element approach in the horizontal plane with a conventional conforming finite element (or finite difference) scheme along the vertical. © 1999 Elsevier Science S.A. All rights reserved.

1. Introduction

Free surface flows are encountered in various types of phenomena, e.g. in ocean and coastal engineering to simulate tidal currents and transport of pollutants, in meteorology for weather prediction, etc.

The equations modelling free surface flows are derived from the three dimensional Navier-Stokes equations for an incompressible fluid. One of the most important assumption made in the context of free surface flows is that the pressure is hydrostatic: this hypothesis is valid only for the so-called long waves when the vertical accelerations are small (i.e. when the wavelength is much greater than the height of the wave itself). This is very often the case when the horizontal dimensions of the domain is much greater than the vertical one. Under this assumption a family of models can be derived.

The 21) Shallow Water Equations (2D-SWE) are obtained integrating along the vertical the horizontal components of the momentum equation as well as the continuity equations. The associated system, which is written in terms of average horizontal velocity, is essentially two-dimensional.

The so-called Quasi-3D models are generated by integrating along the vertical only the continuity equation. In practice the domain is divided into several layers and in each layer the physical variables (velocities) vary in a prescribed manner (typically they are either constant or linear).

Three different kinds of stratification are currently used: - t h e so-called sigma transformation, a topographically conformal vertical coordinate system [9]; - t h e isopycnal system, which uses equal density surfaces; - the fixed strata system, which consists of an a priori subdivision into layers (parallel to the horizontal plane)

of fixed thickness. In this paper we follow the latter approach and introduce a model that can be called 3D-ML-SWE (which

stands for three-dimensional multi-layers shallow water equations). It is worthwhile to notice that the 3D-ML-SWE is consistent with the 2D-SWE; as a matter of fact, in the limit case of a single layer the 3D-ML-SWE model reduces to 2D-SWE one.

The approach we advocate can be described as follows. At first we divide the computational domain into layers and then we use a finite element approximation (over unstructured triangular grid) to represent the

* Corresponding author.

0045-7825/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PlI: S 0 0 4 5 - 7 8 2 5 ( 9 8 ) 0 0 3 0 4 - 1

356 If. Miglio et al. / Comput. Methods Appl. Mech. Engrg. 174 (1999) 355 369

horizontal velocity in each layer. The lowest order Raviart-Thomas finite elements are employed to describe the horizontal velocity: by combination with a special technique of numerical integration, we obtain a scheme that can be reinterpreted as a pure finite volume approach. In particular mass conservation is guaranteed. Along the vertical direction we use a finite difference approach (that can also be regarded as mono-dimensional linear finite element scheme).

Time advancing is achieved through an Eulerian-Lagrangian approach (i.e. a characteristics Galerkin method) that has two main features:

- it is an upwind scheme and is therefore well suited for convection-dominated problems; - it is stable with a mild stability criterion, allowing therefore the use of a large time-step when appropriate. An outline of this paper is as follows: in Section 1 we review the mathematical models for free surface flows,

the associated boundary conditions and we derive a weak form suitable for the Quasi-3D model. Section 2 treats the temporal discretization based on the Lagrangian derivative. In Section 3 we introduce the discretization of the physical domain and the finite element approximation. Then, we analyze the algebraic form of the problem and show that it is very suitable to be faced by a parallel strategy. Finally, we show the consistency of 3D-ML-SWE scheme with the classical 2D-SWE in the case of a single layer. In Section 4 we propose some numerical tests of various complexity with the aim of outlining the good properties of our method: in particular we verify that the scheme is mass-preserving and that the CPU time is growing sublinearly with respect to the number of the layers.

2. M a t h e m a t i c a l m o d e l s for free s u r f a c e f lows

Consider an incompressible fluid in a three-dimensional domain f2 which is normal with respect to z and is vertically bounded by the surfaces z = r/(x, y, t) and z = -[9(x, y). There, [9(x, y) denotes the depth of the point (x, y) from a reference level and q(x, y, t) the elevation of the tree surface from the same reference level, so to have H(x, y, t) = h(x, y) + rl(x, y, t) where H(x, y, t) is the height of fluid at point (x, y). Let us denote with ~(2 the

projection of ~ on the xv plane; every integral over ~ will be written as follows: f~ ( . . . ) d r 2 = t" f ~TIx, v,t ) fn , J t,~,~) (- . -) dz) d~2.

The three-dimensional governing equations for the fluid motion can be derived from the incompressible Navier-Stokes equations after turbulent averaging (see [10,18]):

DV 1

dt p ~ - ~ - + f Vp + ~zA~ v + 7zz oz ' (1)

V . V = 0 , (2)

where V = (u, v, w) v is the velocity vector (u and v are the horizontal velocity components, w is the vertical velocity component), f = (fv, - fu , g)V is the external force vector (g is the gravity acceleration, f the Coriolis

z= rl (x,y,t)

X

Fig. 1. Vertical section of the domain.

E. Miglio et al. / Comput. Methods Appl. Mech. Engrg. 174 (1999) 355-369 357

parameter), ~ and /~ are, respectively, the horizontal and vertical eddy viscosity coefficients ( typically /z < < u: we will consider # = 0 and 1, = const throughout the domain),

D 0 0 0 0 - - - + w - -

Dt Ot + u ~ x + v 03 ' Oz

is the total derivative (which is also called material, or Lagrangian derivative), V and V. are the gradient and divergence operators with respect to x, y and z and, finally,

32 02

- cgx 2 c9), °

is the Laplace operator with respect to the variables x and y. The long waves hypothesis implies that w < < u and w << v so the vertical velocity can be neglected with

respect to the horizontal ones (see [7]), hence the z-component of the momentum equation reduces to

10p p Oz - g " (3)

Since p is constant, integrating (3) along the vertical we obtain that the pressure is hydrostatic [11]:

P = Po + Pg07 - z ) , (4)

where Po is the atmospheric pressure (which we suppose to be constant over the whole free surface). Substituting Eq. (4) into the x and y components of the momentum equation we have [19, 20]:

D v 0 2v

Dt - - g V ~ 7 + u Oz 2 + f ~ (5)

V. V = 0 , (6)

with v = (u, v) v, fry = (fv, - f u ) 'r and V is the gradient operator with respect to the variables x and y. Now, it is necessary to assign the boundary conditions to close the system (5 ) - (6 ) : the quantities relative to

the bottom will be denoted by the subscript b and those relative to the free surface by the subscript s. In particular, wind stresses are given at the free surface (/[[,):

3v r ~ = K ] W I W o n e , (7)

where W is the wind velocity vector, IwI is the intensity of the same vector and K is a proportionality constant. On the free surface the fluid moves with velocity equal to that of the surface itself, therefore:

DT/_ 07/ 0~/ 0~7 W, -- Dt -- at ~ U' Oxx + V~ ofy o n ~ . (8)

At the bottom ~ we prescribe the bottom stresses using a Chezy formula:

Ou givI u az - c 2 v on K h , (9)

where C is the Chezy coefficient, together with the free-slip condition:

Dh O~ Or/ w ; , - D t - - u h o x + V h - - on 4 , . (10) 0y

In alternative to (9), 10) one could assume a no slip condition:

u = v = w = O on ~ . (11)

Finally, we must assign the condition on the ' lateral ' boundary; there are two possibilities: - on a fictitious water-water boundary (e.g. on the open sea) that we denote by ~ we assign the elevation, i.e.

H = H o where for all (x, y) E ( F o U ~ ) H 0 is a prescribed function of t;

358 E. Miglio et al. / Comput. Methods Appl. Mech. Engrg. 174 (1999) 355-369

- on a coast, denoted by Fc, we set to zero the normal component of the velocity solution, i.e. fcc v - n dy = 0 (n is the unit exterior normal to Fc).

The first assumption towards a simplification of the model is achieved upon integrating the continuity equation (6) along the vertical (i.e. between - t l and r/) and using the kinematical condition (8): this is the base of Quasi-3D models [2]. The equations on which we will develop our 3D-ML-SWE scheme are then

De O2V -67 = -uv~vr/+ " ~ +Lv, OZ

(12)

o. ( f ) 0-7 + v • v d z = 0 . (13)

REMARK 1 (2D-SWE). A further simplification would arise if we should integrate over the vertical equation (5), and introducing average velocities U = 1/H f"~ u dz and V= 1/H f ~ v dz; in this way we obtain

o g g l w l w g l g P - - 0-T + ( V - V ~ ) V = -gV, f f /+ ~ C2 H V +f,v (14)

Or/ 0-7 + V -.. ( H V ) : 0 . (15)

where V = (U,V) T and f,y = (fV, - f U ) T. This is precisely the 2D-SWE model [1].

To derive a suitable weak form of (12), (13), we introduce the following space of vector functions:

Ho.c(div; ,(2) = {7": 7" E (L2(~2)) 2, div 7 ~ L2(~(2), 7" n = 0 on Fc}. (16)

Let us multiply the momentum equation (12) by the vector test function w(x, y, z)= 7"(x, y)~(z)= (r~p, L~p) v where 7" is an arbitrary vector of Ho,c(div;/2) and ~o an arbitrary function of H l([_[?, r/]); then integrate over the three-dimensional domain. Similarly, we multiply the continuity equation by ~p @ L2(/2) and integrate over /2. Using the Green's formula and taking into account the boundary conditions (7), (9) we obtain:

find (v, r/) ~ ~ ( / ) ) × L2(/2) such that

g;" (f]r/Vxy. dz)dr/ ~,J- t, Dt = g

f . ( f , oo o, ) ) - P 0~" OZ dz d l 2 - g r / w . n d z dy

+ ] d/2,

+ f a £ , , . w d ~ 2 V w ~ 72(/)), (17)

f , ( f ) -0-~-Od/2+ V y. vdz ~ P d / 2 = 0 V O E L 2 ( / 2 ) , (18) 2 2 b

where we have set 7 / ' ( / ) )= Ho.c(div,/2) × HI( [ -~? , r/l).

REMARK 2. In Eq. (17) (and in the following sections) we deliberately neglect the vertical velocity in the Lagrangian derivative i.e. D/Dt = O/Ot + u (O/Ox) + v(OlOy). We will recover the vertical velocity field with a post-processing based on the use of the continuity equation (see Remark 3.2). An alternative approach can be found in [12].

E. Miglio et al. / Comput. Methods Appl. Mech. Engrg. 174 (1999) 355-369 359

3. Temporal discretization

To discretize the convective term a method based on a Lagrange-Galerkin (or characteristics Galerkin) approach is considered [4,5,8,14,15].

Let us consider the total derivative:

Dv Ov O Dt (t, x) = ~ - + (v- V)v = ~ v(r, X(T; t, x)) , (19)

where X(7-; t, x) is the solution of the following problem:

{ dX(7; t, x) v T. ~ T - - - = ( ' X(~'; t, x)) for 7- E (0, t) ,

X(t; t, x) = x . (20)

From a geometric point of view X ( . ) = X(.; t , x ) is the parametric representation of the streamlines: X0-; t , x ) is the position at time ~- of a particle which has been driven by the field v and that occupied the position x at time t.

The total derivative can be discretized as follows:

D v v ( t . + l , x ) - v ( t . , X ( t , , ; t . + ~ , x ) ) Dt ( t , + , , x ) ~ At (21)

Problem (20) is a system of ordinary differential equations. For its discretization it is possible to use a backward Euler scheme, or a more accurate fourth-order Runge-Kutta scheme. Since (20) is nonlinear, to compute X ( t ; t ~,x) we will use the velocity at time t .

The overall time discretization method that we use is based on the following assumption: - t h e Lagrangian derivative is discretized as in (21); - the elevation and the vertical diffusive term in the momentum equation and the velocity in the free surface

equation, are discretized implicitly. We therefore obtain the set of equations:

n + l ~ n + l v - v " ( X ) = - A t g .... ~ + u A t - -

02V ~+l

0• 2 (22)

~z+l n • D n + l ~q --r / +AtV~ . dz = 0 . (23)

4. Numerical approximation of 3D-ML-SWE

4.1. The discretization o f the domain

The physical three-dimensional domain is embedded in a parallelepipedon composed of ~/" layers; in the sequel with 5~ k we identify the layer k whose thickness 6zk is fixed. A layer is said to be active if it is wet. The number of active layers depends on the total depth; therefore it is not constant over the whole domain and can also change in time, accounting for the variation of the free surface. In particular the thickness of the lowermost active layer (denoted by the index ko) depends on the bottom shape, while the thickness of the uppermost active layer (whose index is Y{') varies in space and time according to the free surface location.

To discretize the two-dimensional domain /2 we will consider an unstructured triangular mesh J],: the mesh is placed at the mid-point of each layer. The vertical distance between the grids of the layers k and k + 1 will be denoted by ~ Z k + l / 2 = [~Z k -~- ~Zk+,]/2 (see Fig. 2). It is important to underline that the mesh is the same on each layer (see Fig. 3).

360 E. Miglio et al. / Comput. Methods Appl. Mech. Engrg. 174 (1999) 355-369

triangular meshes

Z k+l/2

Zk-I 0

i ........................ '

Vk÷ I

V k

Vk_l

6-Z k Ik

Fig. 2. Notations for the layers.

4.2. The finite element approximation

For every integer r > 0 we denote by Pr(T) the space of polynomials of degree ~< r on each triangle T E ~ , and consider the Raviar t -Thomas vector finite element space of lowest order (see [16]):

x A g 0 ( T ) = (~o(T))2 + XPo(T) = { v = ( ~ ) + C(y ) , a, b, c" ~ R } . (24)

Let us now introduce the following finite element spaces:

Qh = {q E Ho.c(div; 11): q[r E ~ql-o(T),

U h : {4t E Lz(J2): ~Plr E P(,(T)},

W~: = {q~ E H ' ( [ - i ) , r/]): ~Pl+, E P , ( g , ) } .

J-+},

To approximate the semi-implicit scheme (22), (23) we introduce the following problem (which is a fully discrete space-time approximation of (17)-(18)):

edge I~ f r e e surface ~

, I

o- - °~ ,

4:)..",,,.

bottom

(a) (b)

Fig. 3. The column of fluid associated to triangle T (a) and the generic face 5~ associated to the edge l (b).

E. Miglio et al. / Comput. Methods Appl. Mech. Engrg. 174 (1999) 355 369 361

find v E Q;, X W~ and r /C U h such that

f,f," o,,+' - f, f," ~o At " r d z d ~ 2 = g C r f ' + ~ V v. r d z d ~

f, f," oo',+' f, f," - v r" Oz d z d . ( 2 - g ~rl"+~'r'n d z d y s2 b i~ [3

+£, ,<,-(Uo,+, I d,,

f,f" + ~ f 2 ~ . . r d z d D , V r ~ © ; , , V ~ W ~ . , (25) 2 1) "

f , ,+ ; ) r/ -- 3 7 v , ,+ l At gt d~(2 + V,~.' dz g* dY2 = 0 , V g, E U;,. (26)

It is worthwhile to notice that the boundary condition on the bottom has been linearized considering the modulus of the velocity at time n.

Denoting by Ned and Nel the number of (oriented) edges e; and triangles ~ in the mesh, the approximate solution for the horizontal velocity is represented as follows:

, ; / N e d

v = ~ ~ (J;)kr~(X)~k(Z), (27) k kip / - I

with (J~)k = f~; vk .n do-, l = 1 . . . . . Ned, ~ E Q;, and ~o~ C W<. Let us consider the two triangles facing on the ith edge (see Fig. 4): the couple (T, l) is equal to (T ~ , i) or

(T~ ~, i '), m is i, j, k when T = T ~ or i ', j ' , k ' when T = T ~ ; r / = x - x l is the vector joining the vertex x; to the generic point x E T; n m is the outward normal to the edge era; supp(r,.) denotes the support of the function r;; IT I and le;I are, respectively, the area of T and the length of edge e,; 6# is the Kronecker symbol.

to the ith edge we associate the following shape function g E Qh:

r,.(x)lT - x - x ; 21TI ' ~ = supp(r,.) = T~; 'u T{j ' , (28)

1 ~lr.n, ,[e 4,,, div r , . I r - IT [ , ., : ~ . (29)

We approximate the elevation by piecewise constant functions on each triangle, so we can write:

?,,(x) = ~] ,~j~(x), (30) / - - I

k

6) i ~.~'Oi T)" -

[] --normal flux of ~ - ¢ l e v a l i o n • =vertical velocity k' horizontal velocity

Fig. 4. Notations for the R~ o shape functions. Fig. 5. Degrees of freedom.

362 E. Miglio et al. / Comput. Methods Appl. Mech. Engrg. 174 (1999) 355 369

where ~Tj = ~h,[r? J = 1 . . . . . N~ and 0 / ~ Uh. Finally, the vertical velocities are calculated on the upper and lower face of each computational cell: they are

constant on that faces and vary linearly along the vertical in each layer (see Fig. 5). To simplify the algebraic form of the problem (and hence reduce its computational cost) it is desirable to

diagonalize the mass matrix. In this context two kinds of lumping procedures are required: one for the ~p-mass

m a t r i x , the other for the ~ ' -mass m a t r i x . For the former one we can use the following quadrature formula:

f [ ~ ' { = 0 f ° r s ~ a k , q~p~dz =~z~ for s = k (31)

while for the latter we employ the circocentric quadrature formula proposed in [17]:

= 0 for i ~ r ,

f (~," 7,) dx (32) dr ~, ~ = ~ for i = r ,

where d~ is the distance between the circocenters of the two adjoining triangles T(~ ~ and T~ ~. Using the lumped terms (31), (32), the algebraic form of the momentum equation reads:

077 n ~I "('* j ~tvk g At n+l n*l

- '

[ ( l ~ l n + l __ ( l , l n + l , ( j ~n + I n + l ] v a t ,o l ,k+ 1 ~ot~ k ~ I 'k - - ( J I ) k I , k = k o . . . . . 9`{ (33)

-1- ~ ~Z' k ~ I ~ ~ k 1/2

where:

" = g~kv~(X) • ~ d x d z .

In Eq. (33) for k = 9'( the first term in square brackets becomes f .y , xlwIw, r, dx = ~ accounting for the boundary conditions on the free surface; similarly for k = k o the second term in square brackets becomes:

n 2 n+ 1 ( g l v ~ l / C ) ( J t ) k o accounting for boundary condition at the bottom.

Now, we can obtain the following system for each face w with l 1 . . . . . Ned on the vertical direction (see Fig. 3):

A j T + , G ' / g A t . ,,+, ,,+1 - +

?~¢~t

where:

A t =

-~.~r 8z.vr 1 + ~.f- + ~.vr : :

0 - - O~k/)+ 2

0 0

with 03 k = ( u A t ) l ( a z ~ i/2), and

j,/+L l ~ j ~ "+l / - - . , ! '

n + l n

(34)

~Zk~ + 1

0 0 -

-- ~.7{'-- 1

I,,nl - 4 + g A t e 2 - - ko+ 1 O~kcl+ 1

(35)

G i' = 072 n

i 072 n ~ l V ko

Let us consider the weak form of the free surface equation: substituting the integral between - t) and r /with a summation and recalling that the test function for the elevation selects a triangle, we obtain

E. Miglio et al. I Comput. Methods Appl. Mech. Engrg. 174 (1999) 3 5 5 - 3 6 9 3 6 3

f T n ~ I n f T J rL - rl, ,,- i 8~" = 0 At dx + V~," ~ v , . , ,

, ., k = k 0

using the divergence theorem and noting that the elevation is constant on each triangle, we have

IT, I ,7, - , 7 , ~ + E E "'''+~ " ,~llk 8Zk = O, (36) k=k 0 I= 1

where T, is the area of the triangle T and the index l runs over the edges of that triangle. In matrix form we have

,( .) IT,, 71.~ -~7,. ,,., 8Z.jI~j,. 2 I ~_~zT i n + l At + 8Zvl" J,r. + _ " - -~ ,3" , .3 = 0 s = 1 , . . . , N e l , (37)

where 8Z T.~j, 8Z~2,. 8Zf3 denote the transpose of the vectors 81. at time n on the edges of the triangle T..,

REMARK 3. The lumping formula (31), although not very accurate, has the nice feature that the scheme which is derived from it, enjoys the same algebraic structure obtained from a pure finite difference approach (see [6]). A more accurate quadrature formula could be based on the trapezoidal rule:

~,+~ f = 0 f o r s # k , i (38) ~R'~& dz~. 8Zk+112 + ~Zk-~/2 for s = k s~ , 2 '

The algebraic form of the equations (33), (36) in this case becomes:

[ rl,,~ , t rl.,, ) .Jr 3 ]r, II~ ' ~ + E E (J ,)~- ' ~z, = 0 , (39)

k--kll t I

n 5~lv k g At ,,+j ,,+1

( s , )7 ' - ~, + T , (,7~,,,-,~,~,,) [ ,,+, ,-, ,jr,+, ( j r ,+, ] ,., ,xt (4)~+,_ - (J,)~ , , , , g _- ,~, ,~-~

+ - - 8zk+,77 - , k = k o . . . . . K , (40) ASk Zk I/2

where

Z~Zk = ( ~ k + l / 2 JC- ~k 1/2 2

From a computational point of view it is desirable to generate a system in which the only unknown is the elevation: formal elimination of {J'/+~} in (37) by (34) yields (see Fig. 6):

,,÷l [ g At2 v ~ ).~.l + (~ZVA i ~Z), 2 r L l + [ T ~ 7 ~ - ( S Z A -~ Z " gAt2 I ,, F ~ l ) 1 ~

g At2 y 1 I/ g At2 T ) + ~ ( ~ Z A '~Z)}3 -rL',!, +' ~ ( S Z a l~Z)'[,

-w,.2 MIK.I~,2 (aZTA ' aZ)'g_ - . ~ . ~ MILI~_~ (SZ"A-' 8Z).~ 3

A t T I n T 1 n = r / ' ~ - l ~ . , [ { ( S Z a G),, +(SZTA-~G)~Iz+(SZ A G),3}, s = 1 . . . . . Nel . (41)

Matrices A.~, A2, A, 3 are positive definite hence ( s z T A - ~ 8/.), i are non-negative numbers. The system (41) is symmetric, positive definite, and has only four non-zero terms on each row. Hence it can be efficiently solved by the preconditioned conjugate gradient method.

Once the elevation is available we can obtain the fluxes {JT+~} using (34). Now, we have to compute the

364 E. Miglio et al. / Comput. Methods" Appl. Mech. Engrg. 174 (1999) 355-369

• e d ~

Fig. 6. T h e e l eme n t S and its ne ighbours .

vertical velocity using a finite volume approach. Let us integrate Eq. (2) on the generic prism with triangular base and use the divergence theorem:

f e V ' V d V = f;, (42)

On the upper and lower faces of the prism V" n is the equal to the vertical velocity component. Considering the generic prism of the k-th layer (with base T,) the previous equation reads:

3

ILl(w - , ) + (J,)k = o , ( 4 3 ) j 1

where j runs over the edges of T, and gz~t is the thickness of the prism on layer k evaluated in the mid-point of the edges of T . Starting from the bottom where w = 0 we can compute the vertical velocity on each column of fluid.

New total depth in the mid-point of the edges is defined as follows:

n + l a ~ l n + l H t = max(0, h I + r/r, ~, h t + "r/r~ ( , ) , 1 = 1 . . . . . Ned, (44)

where h / is the bathimetry evaluated in the mid-point of the edge l and r/r ~, are the elevations on the triangles facing on the same edge. At the end we must update the layers thickness.

REMARK 4. It is worthwhile to notice that if a mesh composed of rectangles is employed one obtains from (33) the finite difference scheme proposed [6].

4.3. Aspects of parallelization

In 3D hydrodynamic simulation the number of unknowns is very high, in particular when dealing with large areas characterized by complex geometry (irregular boundaries and large bathymetry variation). Hence to minimize the computational time a parallelization strategy is recommended. The numerical method that we are advocating is especially suitable to be parallelized, as pointed out in the flow diagram of Fig. 7.

In block 2." the construction of system (34) and then the calculation of the terms (SZTA ~gZ) and (gZTA IG) can be performed independently for each face ~ with 1 = 1 . . . . . Ned (and so also the computation of fluxes of the horizontal velocity when the elevation is obtained, see block 5 in the diagram). We balance the load on each processor either with faces or layers (this balancing is used also in blocks 6 and 7). To minimize the communication of data we use the approach developed in [13];

In block 4: in the solution of system (41) by the conjugate gradient method the matrix-vector products are naturally parallelizable (see [13]);

In block 6: the post-processing to reconstruct the vertical velocity field by Eq. (43) can be carried out separately for each column of fluid;

In block 7: the updating of the total depth (see Eq. (44)) and consequently the redefinition of the layers' thicknesses can be accomplished independently for each face ~w with l = 1 . . . . . Ned.

E. Miglio et al. / Comput. Methods Appl. Mech. Engrg. 174 (1999) 355-369

For each time step:

365

Compute the lagrangian derivative

[equations (20)]

Symmetric tridiagonal 2 system for each face

[equation (33)]

Post-processing: reconstruction of the

vertical velocity [formula (43)]

Calculation of the horizontal velocity from systems of

point 2

61- 1

51 Discrefizafion of the free

surface equation [formula (37)] [

Fig. 7. Flow chart of the scheme for the 3D-ML-SWE.

d e p J I

Updating the total I and the layer thickness

[formula (44)]

4.4. 2 D - S W E is the limit case o f the 3 -D-ML-SWE

It is interesting to notice that in the case in which in our 3D-ML-SWE model we choose a single layer the n n

vectors 8Z t reduce to H~, k 0 = 3{'= 1 (so the index k can be omitted), and Eqs. (33), (37) become:

Pz

. . . . , 3~v, g At At K I W ] W _ g Atlv"[ -,,+, J ' - ~'l + ~ (rl~'~'' - r[~''+') + ''--~- r214 '~ J ' , 1 = 1 . . . . . Ned,

H/@/ ~ " ' / (45)

i ( r/"* I - r/' "~ IT, + H , i J , q +H~2J~2 +H~fl.~3 = 0 , s = 1 . . . . . Nel . (46)

~ ,q + ]

where J~ is the vertical averaged flux on the edge 1. The 2D-consistency property of 3D-ML-SWE is very important, in fact in the simulation of large domains, in

which shallow areas are present, the scheme reduces automatically to 2D-SWE yielding a great computational saving.

Eqs. (45) and (46) can also be considered as a finite volume approximation of the classical 2D-SWE [3].

5. Numerical validation

.5.1. Gaussian hill

This is a very simple test case but it is interesting because is a close system for which it is possible to perform a mass balance. Let us consider a rectangular domain of 20 m x 20 m in which the mean water depth is 1 m. Initially the fluid is at rest and the elevation is described by the following equation:

" q ( x , y ) = 0 . 1 . e --°25c~x-I° 2+ y to)21

The grid used in the simulation is composed of 2634 triangles and 1398 nodes; the time step is set to 0.01 s and

366 E, Miglio et al, / Comput. Methods Appl. Mech. Engrg. 174 (1999) 355-369

the vertical viscosity is 0.01; in the present simulation 10 layers are employed. In Fig. 8 the elevation and the vertical section of the velocity field at time t = 1.75 s and t = 3 s are shown.

In Tables 1 and 2 the mass-balance and the CPU time are shown: it is possible to see that the scheme is completely mass-preserving and that the CPU time increases less than linearly with the number of layers.

I

Y

1 ::l - t

z X - A x i s

l

•

0 125 ~ ,~ I I~ Iql

r._l:~ :~"~,, %~P~'~ z X-Axis

Fig. 8. E l eva t ion and ver t ica l sect ion o f the ve loc i ty field at t = 1.75 s (upper) and t = 3 s ( lower ) for the G a u s s i a n hill.

Tab l e 1

M a s s ba lance : the er ror is the d i f f e r ence be tween the initial m a s s and the m a s s at t i m e t

T i m e Er ro r

1.0

1.5

2.0

2.5

3.0

1 .70530E - 012

5 . 1 1 5 9 0 E - 013

1 .02318E - 012 - 1 . 1 3 6 8 7 E - 0 1 3

1 .08002E - 012

E. Miglio et al. / Comput. Methods Appl. Mech. Engrg. 174 (1999) 355-369

Table 2 CPU time vs. number of layers

367

Number of layers CPU time 1 (2D-SWE) 221.9 5 314.5 10 346.7 15 490.8

5.2. The Venice Lagoon

Venice Lagoon is the largest wetland area on the Mediterranean (50 km long and 11 km wide): the area is composed of patches of land that are periodically submerged by high tides and areas of shallow water traversed

3 - 3 x 1 0 4 ~ - . ; . . , , , ; _ . . -

3"2X|04~: !',:i [; !" "~ ": i(' :i ::, J"~('i'(! ; . ( ~ ~ i { i i i ' i i ( ~ ' ! ! ' ~ ~

A 1 . 4 x 1 0 4 2 . 0 x 1 0 4 2 . 5 x 1 0 4

4 . -~ 3 . 3 × 1 0 . . . . . . . . . a-~ ,- . .L- ,-M~..~-- ,- ,~,~.~'7, ".,,. " . ~ ' ¥ l~,y~zr~. ~ ~ ? ¢~ - ~ ~ . , ~ , ~ • - . ~

3 . 0 × 1 0 ~q

2 . 9 x 1 0 ;: . i . . . . .

~ : - . . ¢. • , ;

2 7 × 1 o ~ - g ~ / f - ~ " e ~ :~--, - • ~ k ~ ~ ~ : / / • .'~, . ' .~, , ' , : - ~ . . i ~ . • . "L ' _ '~ , . : . . , . . . - . . . . . . . . • . • , , . . . , .

~ t ~ x ¢ . ~ ' - ' . : : ~ , ~ - ~ . ~ : , ; . . ' - ' . . G . .,~, 5 ~ I " " " . " . ' ' . ' , ' . . .

2 6 x 1 0 4 ~ 5 . "" {-'~ "N":~Y"'" ~'---':: " ' ) " ~ / " "" * " " " " : 1 . 4 x 1 0 4 2 . 0 x 1 0 4 2 . 5 x 1 0 4

Fig. 9. Venice Lagoon: velocity field near the Lido inlet during high tide (upper) and low tide (lower).

3 6 8 E. Miglio et al. / Comput. Methods AppI. Mech. Engrg. 174 (1999)3.~5-369

by natural and artificial canals. The maximum depth is about 50 m. The lagoon is separated from the sea by a strip of sand (Cavallino coast, the islands of the Lido and Pellestrina, and the Sottomarina coast) with only three inlets (the Lido to the north, Malamocco in the middle and Chioggia to the south).

The mesh used in the simuhttion is composed of 31 454 triangles and 15 773 nodes; the maximum number of active layers is 24 and they are placed in correspondence of the tbllowing z-coordinates: 0.1, 0.2, 0.3, 0.4, 0.5 meters above mean sea level, and 0.1, 0.2,0.3, 0.4~ 0.5, 1, 2, 4, 6, 8, 10, 12, 14, 16, 20, 27, 35, 40, 50ms below mean sea level. As of boundary conditions, on the Adriatic Sea an M 3 tide of 0.5 m amplitude and a period of 12 h has been specified. Time step is set to 60s , Chezy coefficient is 60 and the vertical viscosity is 0.01.

In Fig. 9 the velocity field near the Lido inlet is shown; in Fig. 10 the elevation during the high tide and low

rgl

I

D.424

~).398

0.371

0.345

0.318

0.292

0.265

0.239

0.2t2

0.186

0.159

0.I33

0.106

0.0796

0.053

O. 0265

0

0 . 0 x l 0 ° 1 . 0 x l 0 4 2 . 0 x 1 0 4 3 .0X10 4 4 . 0 x 1 0 4

X-Axi s

3,154

).1t2

3.0709

3.0295

.0,0118

-0.0532

~.09~.5

-0.136

[ -0.I 77

-0.26

-030I

. 0343

-0.384

-0.425

-0.467

-0_508

0 . 0 × 1 0 ° 1 . 0 x l 0 n 2 . 0 × 1 0 4 3 . 0 × 1 0 4 4 . 0 x 1 0 4

X - A x i s

Fig. 10. Elevation in the Venice Lagoon during high tide (upper) and low tide (lower).

E. Miglio et al. / Comput. Methods Appl. Mech. Engrg. 174 (1999)355-309 369

tide is shown. These results are reported with the only purpose of showing the numerical properties of our model (robustness and capability to deal with arbitrary complex geometries). The validation against real experimental data will make the object of a future report.

6. Conclusions

In th is p a p e r w e h a v e p r o p o s e d a s e m i - i m p l i c i t f ini te e l e m e n t s c h e m e for 2 D and 3D s h a l l o w w a t e r e q u a t i o n s :

the v e l o c i t y f ie ld is r e p r e s e n t e d u s i n g the s o - c a l l e d R a v i a r t - T h o m a s f in i te e l e m e n t s b a s e d on f lux c o n s e r v a t i o n

on e l e m e n t edges . Th i s pa r t i cu l a r c h o i c e fo r the f in i te e l e m e n t s p a c e l eads to a v e r y s i m p l e a l g e b r a i c

f o r m u l a t i o n : at e a c h t i m e s t ep w h a t is r e q u i r e d is s i m p l y the so l u t i on o f a set o f s y m m e t r i c and t r i d i a g o n a l

s y s t e m s for the f luxes a n d a s y m m e t r i c a n d p o s i t i v e de f in i t e s y s t e m for the e l e v a t i o n . M o r e o v e r the use o f a

c o n s e r v a t i v e f o r m o f the f r ee su r f ace e q u a t i o n a l low us to ob t a i n a m e t h o d that is m a s s - p r e s e r v i n g .

T h e s c h e m e c a n a l so be r e g a r d e d as a f ini te v o l u m e a p p r o x i m a t i o n o f the S W E and in the c a s e w e use a s ing le

l aye r w e r e o b t a i n the c l a s s i ca l 2 D - S W E .

Acknowledgment

We g ra t e fu l l y a c k n o w l e d g e P r o f e s s o r V i n c e n z o Casu l l i o f U n i v e r s i t y o f T r e n t o fo r h a v i n g p r o v i d e d us t he

b a t h y m e t r y o f the V e n i c e L a g o o n .

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