ANNÉE ACADÉMIQUE 2014-2015

BACHELOR PROJECT

13th Benchmark Workshop on the Numerical Analysis of Dams

Damien Scantamburlo (student in Civil Engineering Bachelor 6 EPFL)

Sacha Laffely (student in Civil Engineering Bachelor 6 EPFL)

02 Juin 2015

(M. Molinari, M. Corrado, M. Anciaux)

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Table of contents

1 Table of illustrations ........................................................................................................ - 3 -

2 Acknowledgement .......................................................................................................... - 5 -

3 Introduction .................................................................................................................... - 5 -

4 Presentation of the case ................................................................................................ - 5 -

4.1 Location .................................................................................................................. - 6 -

4.2 Geometry and behaviour ........................................................................................ - 6 -

4.3 Material properties .................................................................................................. - 7 -

5 Data provided ................................................................................................................. - 7 -

5.1 Mesh of the dam ..................................................................................................... - 7 -

5.2 Acceleration time histories ...................................................................................... - 8 -

6 Implementation of new elements in Akantu .................................................................... - 9 -

6.1 Implementation ....................................................................................................... - 9 -

6.2 Testing the convergence of the hexahedron element ........................................... - 10 -

7 Load computation ......................................................................................................... - 13 -

7.1 Taking account of the damping ............................................................................. - 13 -

7.2 Taking account of the seismic load ....................................................................... - 15 -

7.3 Taking account of the fluid-structure interaction .................................................... - 16 -

7.3.1 Practical application in our study case .......................................................... - 17 -

7.3.2 Surface recognition and projection difficulties in Akantu ................................ - 18 -

7.4 Taking account of the static load ........................................................................... - 18 -

7.5 Taking account of the thermal gradient ................................................................. - 19 -

8 Combine the loading case ............................................................................................ - 20 -

9 Summary of the solving step in Akantu ........................................................................ - 21 -

10 Modal shape analysis ............................................................................................... - 22 -

10.1.1 Theoretical aspect ......................................................................................... - 22 -

10.1.2 Results........................................................................................................... - 23 -

11 Result ....................................................................................................................... - 26 -

11.1 Static simulation .................................................................................................... - 26 -

11.2 Dynamic simulation ............................................................................................... - 28 -

12 Remark ..................................................................................................................... - 30 -

13 Conclusion ................................................................................................................ - 30 -

14 Appendix .................................................................................................................. - 31 -

14.1 Appendix I: Heat transfer model ........................................................................... - 31 -

14.2 Appendix II: detection of the interface fluid-structure ............................................ - 33 -

14.3 Appendix III: Modal analysis algorithm (details) .................................................... - 34 -

14.3.1 Problematic on the study case ...................................................................... - 34 -

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14.3.2 How the generalized eigenvalue problem can be solved with this consideration? - 35 -

14.3.3 How we apply this in our Matlab code ........................................................... - 35 -

14.3.4 Displacement computation for the different frequency mode ........................ - 36 -

14.3.5 Physical meaning of λ .................................................................................... - 36 -

14.4 Appendix IV: Preprocessing and modification on the original mesh ..................... - 37 -

15 References ............................................................................................................... - 55 -

1 Table of illustrations

Figure 1 : Map localisation of the dam ................................................................................... - 6 - Figure 2 : Luzzone atificial lake ............................................................................................. - 6 - Figure 3 : Luzzone dam (view 1) ........................................................................................... - 6 - Figure 4 : Luzzone dam (view 2) ........................................................................................... - 6 - Figure 5 : Material properties ................................................................................................. - 7 - Figure 6 : Mesh and the reality .............................................................................................. - 7 - Figure 7 : View of the mesh though the Diana software ........................................................ - 8 - Figure 8 : Example of an acceleration time histories ............................................................. - 8 - Figure 9 : Hexahedron element with 20 nodes ...................................................................... - 9 - Figure 10 : Pentahedron element with 15 nodes ................................................................... - 9 - Figure 11 : GMSH output of hexahedron 20 ........................................................................ - 10 - Figure 12 : GMSH output of pentahedron 15 ...................................................................... - 10 - Figure 13 : Load applied ...................................................................................................... - 10 - Figure 14 : Blocked degrees of freedom ............................................................................. - 10 - Figure 15 : Deflection of the beam ...................................................................................... - 12 - Figure 16 : Graphical view of the damping ratio relative to α and β .................................... - 14 - Figure 17 : Plot of the acceleration over time on Matlab ..................................................... - 15 - Figure 18 : displacement of the boundary though time ....................................................... - 16 - Figure 19 : the downstream surface is not fully in contact with water .................................. - 18 - Figure 20 : View of the dam without the soil ........................................................................ - 18 - Figure 21 : Upstream face of the dam ................................................................................. - 18 - Figure 22 : 3D view of the temperature load ....................................................................... - 19 - Figure 23 : slice of the dam (temperature fields) ................................................................. - 20 - Figure 24 : Modal shapes for the case of full reservoir (with Westergaard) ........................ - 24 - Figure 25 : Modal shapes for the case of empty reservoir (without Westergaard) .............. - 24 - Figure 26 : Mesh refinement ................................................................................................ - 26 - Figure 27 : Blocked DOFs ................................................................................................... - 26 - Figure 28 : Silt pressure nodal forces .................................................................................. - 26 - Figure 29 : Water pressure + self-weight (dam) .................................................................. - 26 - Figure 30 : Added mass....................................................................................................... - 26 - Figure 31 : Temperature field ............................................................................................... - 26 - Figure 32 : Stress magnitude representation ...................................................................... - 27 - Figure 33 : Displacement magnitude representation ........................................................... - 27 - Figure 34 : Stress in direction x, middle section .................................................................. - 27 - Figure 35 : Z displacement, middle cross section ............................................................... - 27 - Figure 36 : Shear Stress in direction z at the foundation ..................................................... - 28 - Figure 37 : Point of interest ................................................................................................. - 28 - Figure 38 : Displacement shape at time step 1349 ............................................................. - 29 - Figure 39 : Stress in direction x at time step 1349............................................................... - 29 -

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Figure 40 : Sub-section considered to compute the Westergaard added mass .................. - 33 - Figure 41 : "Ideal" mass Matrix ............................................................................................ - 34 - Figure 42 : "Spy" of the stiffness matrix ............................................................................... - 34 - Figure 43 : "Spy" of the mass matrix ................................................................................... - 34 - Figure 44 : "Spy" of the reduced mass matrix (without blocked DOFs) ............................... - 35 -

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2 Acknowledgement

We wish to express our sincere thanks to Professor Molinari that gave us the possibility to work on this workshop. Not a lot of student had the chance to experiment the finite element method by the inside like this, on a serious project: we are a lot grateful for this reason.

We owe special thanks to Mauro Corrado for his help and support, for providing us with all the necessary facilities for the project. We express sincerely our gratitude for his patience and the time he spends to help us during long and hard “debugging session”.

We also place on record, our sense of gratitude to one and all in the LSMS, who directly or indirectly, have lent their support in this venture.

3 Introduction

The finite element method (FEM) is a well know numerical technique in the engineering domain to approximate solutions of boundary problems for partial differential equations. There exist several tools on the market that allow the user to do very complex calculation in a simple environment. In this growing market, an increasing number of software are used and seen as a black-box. It’s in this problematic that we had the chance to work on this workshop: the 13th Benchmark Workshop on the Numerical Analysis of Dams by the International Commission on Large Dam (ICOLD). This study case on the Luzzone dam give us the possibility to discover finite element code in details. The purpose is to make us aware of the importance of knowing how FEM works, what type of problems this method can encounter or create, which hypothesis and simplification FEM are doing.

This report explains at first the new thing we learned from a theoretical point of view, and in a second step the application and the corresponding result in the real study case.

4 Presentation of the case

This benchmark proposed by Dr. Russell Michael Gunn & Dr. Anton Doytchinov Tzenkov is on the Luzzone Dam, which is a Class I concrete arch dam. The purpose is to evaluate the safety under static and dynamic seismic condition. This workshop is a chance for us to experiment some complex tests on a real structure with the Akantu code.

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4.1 Location

The hydroelectric dam of Luzzone is located in the “Tessin canton” in Switzerland. It has been constructed by Alfred Stucky and inaugurated in 1962. In 1996, the upper part of the dam has been heightened with a 17 meter high wall, to increase the capacity of the reservoir by 25 %.

4.2 Geometry and behaviour

Figure 3 : Luzzone dam (view 1)

Figure 4 : Luzzone dam (view 2)

It’s a double-curvature arch dam. It has a crest length of 510 [m].

Maximum height of 225 [m].

The thickness of the crown section varies from 4.55 m at the crest to 36 m at the base.

The behaviour of the structure sounds normal, but presents some interesting aspects for the dam engineering community.

Figure 2 : Luzzone atificial lake Figure 1 : Map localisation of the dam

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4.3 Material properties

The properties of the material is given in the documentation of the workshop:

Figure 5 : Material properties

5 Data provided

5.1 Mesh of the dam

Figure 6 : Mesh and the reality

The mesh given by ICOLD presents this type of characteristics:

The mesh has been generated by the Diana software.

It has 3’102 elements and 12’419 nodes. The volume element are composed by quadratic pentahedron and quadratic hexahedron.

The water is also meshed in the files (to give the possibility to the participant to take in account the dynamical aspect of the reservoir acceleration).

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As we can see on the figure, the soil is also present in the mesh:

Figure 7 : View of the mesh though the Diana software

5.2 Acceleration time histories

Figure 8 : Example of an acceleration time histories

Three set of stochastically independent acceleration time-histories has been provided to us. Here is the basics properties of those data:

Duration of a set : 30.71[s]

The list of acceleration has been divided with a 0.01 second time step.

Total number of step to apply: 3072

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6 Implementation of new elements in Akantu

6.1 Implementation

We had to implement two new element in Akantu in order to read the mesh given by ICOLD. Actually, the mesh was essentially made up of quadratic hexahedron and pentahedron. The process used to achieve that goal was:

1. Code the quadratic shape functions for each element.

We based ourselves on documentation given online by the Aster Code.

2. Then, we had to derivate all those shape functions with respect to all axis.

We computed those derivatives using Mathematica and Matlab.

3. Calculate all the Gauss points for each element. 4. Update all the dependency through the Akantu code

The difficulty lied with implementing the connectivity for each element and find the right switch for adapting the connectivity through the different convention. It was relevant because we had to work with two different types of mesh/convention. There was the one given by ICOLD, which was written in the Diana format. For greater simplicity, we test our new element in GMSH, which has another convention for the facet connectivity.

There were also several other dependencies that we had to adjust such as the type of integration (Serendip for the hexahedron and Lagrange for the pentahedron), the dumper relative to the software Paraview etc.

After that we performed some basic patch test (compression, traction, bending) in order to verify our elements.

Figure 9 : Hexahedron element with 20 nodes Figure 10 : Pentahedron element with 15 nodes

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6.2 Testing the convergence of the hexahedron element

Here is a convergence test that we performed with the hexahedron 20 and the hexahedron 8.

The test consists of a beam of 10 m long with a section of 2 x 2 = 4 m2. One end of the beam is fixed. Then we mesh this beam either with hexahedron with 20 nodes or either with hexahedron with 8 nodes with different levels of refinement. We applied a load of 12 kN/m2 uniformly on the beam.

Then we take the y displacement at the free end of the beam in the middle of the section and we compare the different solutions with the analytical one.

The analytical deflection was calculated as follows:

𝑣 =𝑞𝐿4

8𝐸𝐼+

𝑞𝐿2

3𝐺𝐵= −1.409 ∙ 10−4 𝑚

Figure 11 : GMSH output of hexahedron 20 Figure 12 : GMSH output of pentahedron 15

Figure 13 : Load applied Figure 14 : Blocked degrees of freedom

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The self-weight wasn’t taken into account in both the analytical and the Akantu computation. We obtained those results:

First Order Hexahedron 8 Number of nodes Degrees of freedom Deflection

Refine1 99 297 -9.252E-05

Refine2 525 1575 -1.040E-04

Refine3 3321 9963 -1.079E-04

Refine4 23409 70227 -1.091E-04

Refine5 175329 525987 -1.095E-04

Second Order Hexahedron 20 Number of nodes Degrees of freedom Deflection

Refine1 321 963 -1.084E-04

Refine2 1865 5595 -1.092E-04

Refine3 12465 37395 -1.095E-04

Refine4 90593 271779 -1.096E-04

-0,00016

-0,00014

-0,00012

-0,0001

-0,00008

-0,00006

-0,00004

-0,00002

0

50 500 5000 50000 500000

dis

pla

cem

ent

at t

he

free

en

d o

f th

e

bea

m [

m]

log(DoFs)

Convergence of the displacement

Analytical solution First Order hexahedron Second Order hexahedron

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We can see that both the hexahedron 8 and the hexahedron 20 converge to the same value of displacement (around -1.096E-4). However, the mesh containing the hexahedron with 20 nodes converges faster than the other one. Let’s compare the “Refine2” case of the first order element and the “Refine1” case of the second order element. Despite the number of DoFs (degrees of freedom) of the hexahedron 8 element is greater than the hexahedron 20, the solution with the hexahedron 20 is closer to the convergence than the former. This is simply due to the possibility of “bending, curving” of the element (second order calculation).

As for the analytical solution, since we have applied only a load on the beam, by the principle of minimal potential energy, we expect to obtain a displacement inferior to the analytical solution. This is what we obtain here, our analytical solution is around 20% greater than the one we obtain in Akantu. We can see in Figure 15 the deflection of the beam amplified by 300.

Figure 15 : Deflection of the beam

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7 Load computation

7.1 Taking account of the damping

In the dynamic analysis of the structure and the foundation, the damping plays an important role. However, our knowledge of the damping is very limited, and cannot be find precisely before the construction of the dam. This is due to the fact that it’s very hard to model realistically the relation between the soil and the foundation.

It is why we will use the equivalent Rayleigh Damping:

[𝐶] = 𝛼 ∗ [𝑀] + 𝛽 ∗ [𝐾] (1)

In which [C] is the damping matrix of the physical system, [M] the mass matrix of the physical system, [K] the stiffness matrix of the system, α and β are some pre-defined constants.

The major advantage of using this equivalent form is that since we know that both matrices [K] and [M] is diagonalizable, the matrix [C] can also be diagonalized because it is a linear combination of the two other matrices.

The equation of motion is given by:

[𝑀]{�̈�} + [𝐶]{�̇�} + [𝐾]{𝑢} = {𝐹} (2)

By orthogonal transformation, the equation reduces to:

{𝜑}𝑇[𝑀]{𝜑}{�̈�} + {𝜑}𝑇[𝐶]{𝜑}{�̇�} + {𝜑}𝑇[𝐾]{𝜑}{𝜉} = {𝜑𝑇}{𝐹} (3)

We can reduce split this equation into n-uncoupled equations in the form of:

{𝜉�̈�} + 2𝜁𝑖𝜔𝑖{𝜉�̇�} + 𝜔𝑖2{ 𝜉𝑖} = {𝐹(𝑡)} (4)

In which { 𝜉𝑖} is the displacement of the structure in the transformed coordinate, ζ is the damping

ratio, ω is the natural frequency of the system, {F(t)} is the modified force vector in the transformed coordinate, {φ} is the normalized eigenvector of the system (cf. Section 10).

This above transformation is valid only when the damping matrix [C] is a function of the mass and stiffness matrix [M] and [K]. This is why it is desirable to have our matrix in the form of equation (1), because the orthogonal transformation of the damping term becomes:

{𝜑}𝑇[𝐶]{𝜑} = [𝛼 + 𝛽𝜔1 ⋯ 0

⋮ ⋱ ⋮0 ⋯ 𝛼 + 𝛽𝜔𝑛

] (5)

From equations (4) and (5), we can say:

2𝜁𝑖𝜔𝑖 = 𝛼 + 𝛽𝜔𝑖 (6)

Which we can simplify as:

𝜁𝑖 =𝛼

2𝜔𝑖+

𝛽𝜔𝑖

2 (7)

We can see that the damping ratio is proportional to the natural frequency of the system. In fact, we can see on the following graph that the damping ratio is nonlinear in the first portion (the part proportional to the inverse of the natural frequency) and beyond the variation becomes linear.

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Generally, almost 100% of the mass participation occurs in the first 10 to 20 modes (hypothesis), it is then not needed to measure all the 𝜁𝑖 (because the results aren’t important and of no practical consequence) and due to our very great number of degrees of freedom it will reduce the calculation a lot.

Instead, as we have calculated the first 12 modes in section 10, we chose to fix the damping of

𝜁1 and 𝜁6 to 5%. So we can now resolved the system of equation (7) with i = 1 and i = 6,

and find the following values of α and β:

𝛼 = 0.44511966, β = 0.00458196

Figure 16 : Graphical view of the damping ratio relative to α and β

0

0,5

1

1,5

2

2,5

3

3,5

4

0 2 4 6 8 10 12 14 16

Dam

pin

g r

atio

ζ

Natural frequency ω

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7.2 Taking account of the seismic load

Since we can’t apply directly an acceleration to the dam, we decided to integrate this acceleration to have the corresponding displacement. We will apply this new “information” through a Dirichlet boundary in Akantu.

We convert the Excel data in Matlab matrix, and we plot the different histories:

Figure 17 : Plot of the acceleration over time on Matlab

We use the composite trapezoidal numerical method to integrate two times the acceleration and get the total relative displacement with respect to the origin at the time t:

𝑢𝑟𝑒𝑙(𝑡) = ∫ �̈�𝑡

𝑡𝑜(𝑡) 𝑑𝑡 ≅

𝑡−𝑡𝑜

2𝑁∑ (�̈�𝑛

𝑁𝑛=1 + �̈�𝑛+1) (8)

For example, at time t = 28’’, urel = 1.2[cm] means that since the beginning of the earthquake, a generic point in the ground has been translated of 1.2 [cm].

If we want to know the increment displacement (∆u) at time t = ti, we have to compute:

∆𝑢𝑡𝑖 = 𝑢𝑟𝑒𝑙(𝑡𝑖) − 𝑢𝑟𝑒𝑙(𝑡𝑖−1) (9)

After this computation we have this type of vector:

∆𝑢 = {∆𝑢𝑡0 , ∆𝑢𝑡1 , … , ∆𝑢𝑡𝑛 } (10)

We dumped 𝑢𝑟𝑒𝑙(𝑡) in a .txt file, to give to Akantu the possibility to store this “array” and to apply this displacement when it will be necessary.

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Here is the plot of the 𝑢𝑟𝑒𝑙 :

Figure 18 : displacement of the boundary though time

7.3 Taking account of the fluid-structure interaction

In a static linear case, the finite element method is presented as the resolution of this system:

[𝐾] 𝒖 = 𝒇 (11)

[𝐾] = ∫[𝐵]𝑇𝐷[𝐵]𝑑𝛺 which is the rigidity matrix (12)

u represents the nodal displacement

𝑓 = ∭[𝑁]𝑇𝒃 𝑑𝛺 + ∮[𝑁]𝑇𝒕 𝑑𝑙 which is the nodal forces (13)

In the dynamic analysis, the finite element method take another form (in summary):

[𝑀]{�̈�} + [𝐾] {𝑢} = {𝑓} (14)

With [M] representing the mass matrix.

In fact, the term [𝑀] �̈� emerge as the discret operator that converts nodal accelerations to

intertial nodal forces. Pratically (in finite element code) the mass matrix can be computed in a lot of way: the discretization process is generically called “mass lumping” or simply “lumping”.

The simpler method is the Direct Mass Lumping. The process is as follow: the total mass of the element is distributed to the nodes so that a diagonally lumped mass matrix is produced.

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Akantu uses another standard procedure that is based on a variational formulation. By definition the kinetic energy (T) is defined in this general way:

[𝑇] =1

2 ∭𝜌 𝒗𝑇 𝒗 𝑑𝛺 (15)

It’s the basic generalization of the well-known formula:

𝐸𝑘𝑖𝑛𝑒𝑡𝑖𝑐𝑎𝑙 = 1

2𝑚𝑣2 (16)

According to the “FEM philosophy”, we can write this type of relation:

{𝑣} = [𝑁] {𝑢}̇ (17)

If we put (17) in (15) and take the node velocities out of the integral, it yields:

𝑇 =1

2 {𝑢}̇ 𝑇 ∭𝜌 [𝑁]𝑇[𝑁] {𝑢}̇ 𝑑𝛺 ≝

1

2 {𝑢}̇ 𝑇 [𝑀]{𝑢}̇ (18)

It is interesting to observe that [M] represent in fact the Hessian matrix of [T]:

[𝑀] = 𝜕2 [𝑇]

𝜕�̇� 𝜕�̇� = ∭𝜌 [𝑁]𝑇[𝑁] 𝑑𝛺 (19)

In Akantu the shape functions are the same of the one used in the derivation of the stiffness matrix. In that case, [M] is called the consistent mass matrix, and is denoted by [Mc].

Of course this is another subject, but mass matrices are often developed in a local element, and (as for the rigidity matrix) are globalize through the process called “matrix assembly”.

7.3.1 Practical application in our study case

In 1931, Westergaard defines the interaction between the fluid hydrodynamic and a dam in considering an added mass on the structure surface (in contact with the water). The fluid is replaced by an inertia contribution of the structure.

This approach is presented as a very good method if there is not too much outflow motion (and it’s corresponding to our case).

The Zangar formula also exists: it is more complex to apply, but takes into account the curvature of the dam. The Westergaard formula is presented like this:

𝜌𝑤 = 7

8 𝜌𝑒𝑎𝑢 √𝐻 (𝑦0 − 𝑦) (20)

with:

ρw = surface density to add to the wall of the dam at the corresponding height y

H = height of the dam

yo = related to the free water surface height

Through this surface density, we added at each node for the mass matrix [M] the mass that corresponds to the contribution of the water (in its dynamical mode).

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7.3.2 Surface recognition and projection difficulties in Akantu

The application of this practical method wasn’t very easy in our case. A method already implemented should have facilitate our work, but things turn a little more differently. The success of the debugging come to Mauro and Nicolas, thanks to their patience.

rocks.

The Appendix II give more detailed about the algorithm that has been coded to do this recognition.

7.4 Taking account of the static load

Normally, we should have only provided the surface of the upstream face (in contact with water), and the script would have done its job without problem. In fact the geometry of the dam is a little bit complex in our case.

As we can see on the Figure 19, a part of the geometry is on the upstream face, but hidden in the rocks, without having a contact with the water. The method in Akantu has been modified to use all the points of the upstream face we provided (in GMSH, cf. Figure 21) to compute the hydrostatic load and the mass matrix with Westergaard method, with the exception about the zone in the rocks.

Figure 19 : the downstream surface is not fully in contact with water

Figure 20 : View of the dam without the soil

The hydrostatic pressure has been taken into account as follow

Normal water level: 1606 [m s. m.]

The silt pressure has been applied on the wet surface of the upstream face of the dam:

Silt level (estimation): 1440 [m s. m.]

The altitude of the lowest point of the dam is 1385.2 [m].

Figure 21 : Upstream face of the dam

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7.5 Taking account of the thermal gradient

The thermal gradient is also a load case to consider: some temperature variation can end with high stresses in the structure. The documentation of the workshop give the initial temperature of the dam. They also give the variation of the annual temperatures between the upstream and the downstream face.

With this type of information, we created a heat transfer mode, where we put the temperature boundary on the two faces of the dam (see Fig. 22). After the solving, we get back the “exact” (without interpolation) internal distribution of temperature and we apply this to the structure.

You can find in Appendix I more explanation about how the heat transfer model and how FEM can solve this type of problem. In this way, the stress and the deformation will be computed in the static and dynamic simulation with the consideration of the thermal gradient. Reminder: the temperature difference in finite element method is considered during the solve step as a fictitious forces (to simulate the thermal expansion):

{ℎ} = ∭ [𝐵]𝑇[𝐸] {𝜀}𝑇 𝑑𝛺 (21)

with,

{𝜀}𝑇 = {∝ 𝑇 ∝ 𝑇 ∝ 𝑇 0 0 0} (22)

Figure 22 : 3D view of the temperature load

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Figure 23 : slice of the dam (temperature fields)

8 Combine the loading case

This table summarize the different case we have to test:

To be sure everything works together, we will in priority simulate the first load combination “DE2”, that is taking into account the self-weight, the hydrostatic pressure, the silt pressure, the temperature gradient and the serie 1 of the earthquake time histories.

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9 Summary of the solving step in Akantu

To understand completely the solving step, here is a recall:

Apply the

hydrostatic pressure,

the silt pressure and

the thermal «load»

on each node

Compute and

assembly the mass

matrix (with the

added component due

to the Westergaard

method)

Apply the

displacement (time ti)

to the boundary of the

model (external rocks

boundary)

Solve the system:

[𝐾] 𝑢 = {𝑓] + 𝐶 �̇� + [𝑀] �̈� with the dynamic explicit

method

Compute the new : �̇� �̈�

ti = t

i + 1

ti = t

0

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10 Modal shape analysis

10.1.1 Theoretical aspect

Through the assembly of the stiffness matrix, with the consideration of the mass, the finite element code has to solve this type of system:

[𝐾] {𝑢} + [𝑀] {�̈�} = {0} (23)

The researched solution can be expressed in two different functions, one is depending with the time, the other with the position:

{𝑢}(𝑡) = {𝑎}(𝑧) 𝑓(𝑡) (24)

Putting (24) in (23) gives:

[𝐾] {𝑎}(𝑧) 𝑓(𝑡) + [𝑀] {𝑎}(𝑧) 𝑓′′(𝑡) = {0} (25)

We don’t consider here the nodal force because we are searching some values that are depending of the structure itself. Some algebraic manipulation gives:

−�̈�(𝑡)

𝑓(𝑡)=

[𝐾]

[𝑀] {𝑎}(𝑧)

{𝑎}(𝑧) (26)

Since the left term is just depending with the time, the division can only give a constant (that we will call 𝜆). Here the system that we can construct:

𝑓̈(𝑡) + 𝜆 𝑓(𝑡) = 0 (27)

And,

([𝐾] − 𝜆 [𝑀]) {𝑎} = {0} (28)

The equation can be solved (28) for 𝜆, since the trivial solution {𝑎} = {0} is not relevant.

We have erased the lines and the columns in [K] and [M] that correspond to the blocked degrees of freedom, to avoid some useless computation (otherwise the 6 first value of 𝜆 that would be computed will be zero, since they are corresponding to the rigid modes).

A common generalization of the simple eigenvalue problem is called “Generalized Eigenvalue Problem” in the literature and it involves 2 matrices:

[𝐴] {𝑥} = 𝜆 [𝐵] {𝑥} ⇔ ([𝐴] − 𝜆 [𝐵]) {𝑥} = {0} (29)

The equation (28) that we want to solve is of the same form of (29).

The Appendix III give more detailed about how we compute the solution of this generalized

eigenvalue problem.

- 23 -

10.1.2 Results

Mode λi ω f [Hz]

1 38.801 6.229 0.991

2 49.353 7.025 1.118

3 115.226 10.734 1.708

4 139.996 11.832 1.883

5 199.274 14.116 2.247

6 243.226 15.596 2.482

7 314.745 17.741 2.824

8 329.919 18.164 2.891

9 447.200 21.147 3.366

10 502.219 22.410 3.567

11 504.645 22.464 3.575

12 553.808 23.533 3.745 Table 1 Natural frequencies for the case of full reservoir (with Westergaard)

Mode λi ω f [Hz] 1 245.382 15.665 2.493

2 405.656 20.141 3.206

3 727.294 26.968 4.292

4 866.291 29.433 4.684

5 1129.236 33.604 5.348

6 1544.398 39.299 6.255

7 1661.266 40.759 6.487

8 1696.892 41.193 6.556

9 1952.407 44.186 7.032

10 2415.557 49.148 7.822

11 2580.493 50.799 8.085

12 2851.004 53.395 8.498 Table 2 Natural frequencies for the case of empty reservoir (without Westergaard)

- 24 -

Figure 24 : Modal shapes for the case of full reservoir (with Westergaard)

Figure 25 : Modal shapes for the case of empty reservoir (without Westergaard)

- 25 -

If we consider only the dam (without the soil/rocks), with the water, we end with this

eigenvalue:

Mode λi ω f [Hz] 1 41,792 6,465 1,029

2 52,185 7,224 1,150

3 119,946 10,952 1,743

4 153,475 12,389 1,972

5 205,786 14,345 2,283

6 264,310 16,258 2,587

7 337,484 18,371 2,924

8 350,971 18,734 2,982

9 488,785 22,108 3,519

10 516,128 22,718 3,616

11 583,794 24,162 3,845

12 652,181 25,538 4,064 Table 3 Natural frequency for the dam without the soil

- 26 -

Figure 26 : Mesh refinement Figure 27 : Blocked DOFs

Figure 28 : Silt pressure nodal forces Figure 29 : Water pressure + self-weight (dam)

Figure 30 : Added mass Figure 31 : Temperature field

11 Result

11.1 Static simulation

We present in this section some results that we compute on our computer (time: 24 hours), with a refine mesh (one time). The coefficient of the Rayleigh damping has been updated after the consideration of the modal shape result, with the help of the Stucky engineers.

Here is mesh and the information about the blocked degrees of freedom :

On the next Figures, we can observe the silt and the water pressure. There is also a view of the added mass corresponding to the Westergaard theory, and the temperature field considered:

- 27 -

Figure 32 : Stress magnitude representation Figure 33 : Displacement magnitude representation

Figure 35 : Z displacement, middle cross section

The static resolution gives:

We are not supposed to make some stability or safety analysis (this part has been planned, and it will be done by the Stucky engineers) but we can observe the compression in the arch for example in the middle section of the dam:

Figure 34 : Stress in direction x, middle section

The maximum compressive stress σxx value (acting perpendicular in this cross section) is here around -5,87E6 [N/m2]. We can convert this in a more appropriate unit:

𝜎𝑥𝑥 = −5.87 [ 𝑁𝑚𝑚2⁄ ]

The uniaxial static compressive strength in the old concrete is 38 [N/mm2]. The compression in the concrete is very well supported.

The maximum tensile stress σxx value (acting perpendicular in this cross section) is here around 3,99E5 [N/m2]. We can convert this in a more appropriate unit:

𝜎𝑥𝑥 = 0.4 [ 𝑁𝑚𝑚2⁄ ]

Since the uniaxial static tensile strength in the old concrete is 3 [N/mm2], we can conclude the concrete supports the traction that is occurring in this section part.

z x

- 28 -

As we observe in the “Remark” section, it would be interesting (and necessary) to model the interaction between the foundation and the dam with cohesive element to check the sliding of the dam.

Here we can see the shearing stress value that there is in the z direction. The foundation have a high shear component in this axis, meaning that the dam could (maybe) slide.

11.2 Dynamic simulation

The dynamic simulation has been performed. There is a lot of analysis to make about this type of loading. In this sub section, we focus ourselves on a particular point (to show the type of information we can get from this type of resolution).

The point of interest is located here (purple marker):

Figure 37 : Point of interest

z

Figure 36 : Shear Stress in direction z at the foundation

-0,18

-0,16

-0,14

-0,12

-0,1

-0,08

-0,06

-0,04

-0,02

0

0,02

0,04

0 500 1000 1500 2000 2500 3000 3500

radia

l dis

pla

cem

ent

[m]

timestep

Radial displacement

- 29 -

Through the time we can observe on the previous graph the curve (for this point of interest) for the radial displacement (z direction):

The maximum displacement is occurring at the time step 1349 (13.49 [sec]) : 15.5 [cm]. The corresponding displacement shape at this time step (amplify by 1000 to see clearly):

Figure 38 : Displacement shape at time step 1349

The stress in the middle section appears like this (compressive stress):

Figure 39 : Stress in direction x at time step 1349

We can see clearly in this example that the tensile stress is high (1.43 [N/mm2]). The concrete doesn’t crack with this solicitation. In other time step, the stress exceeds slightly the maximum tensile resistance of the concrete. Again, it is necessary in this case to insert in this over-solicited area some cohesive elements, to model the crack in the concrete, and the joint opening.

z

- 30 -

12 Remark

The simulation of the dam under seismic condition has been completed and some results have

been computed. Through this study, we have made some basic assumptions that need to be

reminded of:

For greater simplicity, we don’t consider the cohesive element between the foundations

and the dam. This is a thing that could be improved, to simulate more precisely the

shearing interface between the soil and the concrete.

The model here is a linear material comportment. For sure, we could improve the model

with some “damage model”. This type of law of comportment are the subject of actual

work in the LSMS.

The construction step of the dam require some joints: the dam isn’t constructed in one

block. The earthquake creates some traction component in the concrete In the upper

section of the dam. In reality, this behavior could end with the opening of a joint.

An “easy” way to confirm the “rock block stability” is to add some cohesive element

between the concrete blocks to simulate more properly their sliding and their interaction.

The parameter α and β for the dumping can be adjusted with some experiment on the

fieldwork.

Since we are using Akantu trough a virtual box, we cannot launch big simulation.

Nonetheless, it could be clever to refine (with Gmsh) the mesh.

13 Conclusion

We didn’t have enough time to take into account all the complex aspect of the dam, for instance the cohesive element. But the simulation under various loading (temperature, dynamical, static) has been performed: we observe the displacement and the stress. The implementation of two new elements has taken us a lot of time: we haven’t got the time after the simulation computation to fully analyze all the result. Having the possibility to work on a real workshop through our Bachelor project has been a great chance. We discover a finite element code that is very flexible. Since everything has to be written manually, the possibility are nearly infinite. The purpose was to use Akantu through this workshop, but also compute some interesting results.

For us, the modal shape analysis was something new. We work on big matrix in Matlab, and this have made us aware of the problematic of refinement and accuracy in the finite element method.

- 31 -

14 Appendix

14.1 Appendix I: Heat transfer model

Some tools that can be useful in the next proof:

The formula of decomposition of the divergence give this relation:

𝑑𝑖𝑣({𝑢})𝑓 = 𝑑𝑖𝑣(𝑓{𝑢}) − {𝑢} 𝑔𝑟𝑎𝑑⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ (𝑓) (1)

𝑤𝑖𝑡ℎ 𝑢 𝑎 𝑣𝑒𝑐𝑡𝑜𝑟 𝑓𝑖𝑒𝑙𝑑 𝑎𝑛𝑑 𝑓 𝑎 𝑠𝑐𝑎𝑙𝑎𝑟 𝑓𝑖𝑒𝑙𝑑

If we integrate (1) on the volume, it gives:

∭ 𝑑𝑖𝑣({𝑢})𝑓 𝑑𝛺 = ∭𝑑𝑖𝑣(𝑓{𝑢}) 𝑑𝛺 − ∭{𝑢} 𝑔𝑟𝑎𝑑⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ (𝑓) 𝑑𝛺 (2)

𝑎 = 𝑏 − 𝑐 (3)

The term “b” can be rewritten with the divergence theorem on a surface integral:

𝑏 ≡ ∭𝑑𝑖𝑣(𝑓{𝑢})𝑑𝛺 = ∯(𝑓{𝑢})𝑑𝑆⃗⃗⃗⃗ = ∯(𝑓{𝑢})𝑇{𝑛}𝑑𝑆 = ∯({𝑢})𝑇{𝑛}𝑓 𝑑𝑆 (4)

The term “c” can be rewritten more clearly if we are assuming that we are in 3D:

𝑐 ≡ ∭{𝑢} 𝑔𝑟𝑎𝑑⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ (𝑓)𝑑𝛺 = ∭ [ 𝛿𝑓

𝛿𝑥 𝛿𝑓

𝛿𝑦 𝛿𝑓

𝛿𝑧] {𝑢}𝑑𝛺 (5)

We can rewrite (2) in this way with (4) and (5):

∭ 𝑑𝑖𝑣({𝑢})𝑓 𝑑𝛺 = ∯({𝑢})𝑇 {𝑛} 𝑓 𝑑𝑆 − ∭ [ 𝛿𝑓

𝛿𝑥 𝛿𝑓

𝛿𝑦 𝛿𝑓

𝛿𝑧] {𝑢} 𝑑𝛺 (6)

Problem Statement:

The most basic heat transfer equation can be written like this:

−(𝛿𝑞𝑥

𝛿𝑥+

𝛿𝑞𝑦

𝛿𝑦+

𝛿𝑞𝑧

𝛿𝑧) + 𝑄 = 𝜌 𝑐

𝛿 𝑇

𝛿𝑡 (7)

With qx, qy and qz the heat flow through the unit area Q = Q(x,y,z,t) the inner heat-generation rate per unit volume c the heat capacity

If we assume that the temperature field inside each element can be interpolate with the shape function like this:

𝑇 = [𝑁]{𝑇𝑛𝑜𝑑𝑎𝑙} (8)

- 32 -

We can apply the principle philosophy of differential discretization of FEM:

𝛿 𝑇

𝛿𝑥

𝛿 𝑇

𝛿𝑦

𝛿 𝑇

𝛿𝑧

=

𝛿 𝑁1

𝛿𝑥

𝛿 𝑁2

𝛿𝑥…

𝛿 𝑁1

𝛿𝑦

𝛿 𝑁2

𝛿𝑥…

𝛿 𝑁1

𝛿𝑧

𝛿 𝑁2

𝛿𝑥…

[𝑇] = [𝐵][𝑇] (9)

Using the Galerkin method we can rewrite (6) under integral form:

∭( (𝛿𝑞𝑥

𝛿𝑥+

𝛿𝑞𝑦

𝛿𝑦+

𝛿𝑞𝑧

𝛿𝑧) − 𝑄 + 𝜌 𝑐

𝛿 𝑇

𝛿𝑡 ) 𝑁𝑖 𝑑𝛺 = 0 (10)

⇔

∭( (𝛿𝑞𝑥

𝛿𝑥+

𝛿𝑞𝑦

𝛿𝑦+

𝛿𝑞𝑧

𝛿𝑧) + 𝜌 𝑐

𝛿 𝑇

𝛿𝑡 ) 𝑁𝑖 𝑑𝛺 = ∭𝑄 𝑁𝑖 𝑑𝛺 (11)

With:

∭ (𝛿𝑞𝑥

𝛿𝑥+

𝛿𝑞𝑦

𝛿𝑦+

𝛿𝑞𝑧

𝛿𝑧)𝑁𝑖 = ∭𝑑𝑖𝑣(𝑞)𝑁𝑖 (12)

The equation (10) is under the exact same form of the equation (6), so:

∭ (𝛿𝑞𝑥

𝛿𝑥+

𝛿𝑞𝑦

𝛿𝑦+

𝛿𝑞𝑧

𝛿𝑧)𝑁𝑖 = ∯({𝑞})𝑇 {𝑛}𝑁𝑖 𝑑𝑆 − ∭ [

𝛿𝑁𝑖

𝛿𝑥 𝛿𝑁𝑖

𝛿𝑦 𝛿𝑁𝑖

𝛿𝑧] {𝑞} 𝑑𝛺 (13)

The equation (11) can be rewritten with (13), and we end with:

∭ 𝜌 𝑐 𝛿 𝑇

𝛿𝑡𝑁𝑖 𝑑𝛺 − ∭ [

𝛿𝑁𝑖

𝛿𝑥 𝛿𝑁𝑖

𝛿𝑦 𝛿𝑁𝑖

𝛿𝑧] 𝑞 𝑑𝛺 = ∭𝑄 𝑁𝑖 𝑑𝛺 − ∯({𝑞})𝑇 {𝑛} 𝑁𝑖 𝑑𝑆

(14)

It is possible to discretize ∭𝜌 𝑐 𝛿 𝑇

𝛿𝑡𝑁𝑖 𝑑𝛺 like this for example:

(∭𝜌 𝑐 [𝑁]𝑇[𝑁]𝑑𝛺 ) {𝑇}̇ (15)

The discrete finite element equations for heat transfer have this form:

[C] {𝑇}̇ + ([𝐾𝑐] + [𝐾ℎ] + [𝐾𝑟]){𝑇} = 𝑠𝑜𝑚𝑒 𝑡𝑒𝑟𝑚 𝑟𝑒𝑙𝑎𝑡𝑒𝑑 𝑡𝑜 𝑡ℎ𝑒 𝐵. 𝐶. (16)

It is possible to write this like this with the help of the Fourier law for the x axis for instance):

𝑞𝑥 = −𝑘 𝛿𝑇

𝛿𝑥 → {𝑞} = −𝑘 [𝐵]{𝑇}

With “k” as the conductivity

Akantu solves (16) in order to compute the heat transfer model on our dam. It is very important to know in this type of approach in which way FEM works (in fact the principle is quiet the same as the deformation-stress one we learn during class).

- 33 -

14.2 Appendix II: detection of the interface fluid-structure

The algorithm that allow to detect the concrete surface that is in contact with the water works like this:

It has been necessary to add an exception to handle the upper section of the dam because in fact there is two type of concretes! Otherwise the method would have selected also the zone where there is this change of concrete.

Thanks to this detection of the boundary of the dam, the method is able to compute the relative height for each point (cf. Figure 26) and integrate with the gauss points all the local added mass component.

This graphical view allows to understand why it’s necessary to detect the “underwater dam boundary”. The height of the dam varies with the position (and this is a parameter in the Westergaard method), so the knowing of the exact position of the boundary is absolutely inevitable.

Detect all the nodes

on the upstream

face (declare in our

mesh)

For each node,

check how many

element are

connected to this

node

Iterate through this

maps of element

(for a given node)

to know what is the

material

For a given node, if

there is more than

one material for the

element connected to

it, this node is

exactly on the

boundary!

Figure 40 : Sub-section considered to compute the Westergaard added mass

- 34 -

14.3 Appendix III: Modal analysis algorithm (details)

14.3.1 Problematic on the study case

The dimension of [M] and [K] are very big: approximately 7500 x 7500 for each matrix.

There is absolutely no possibility to inverse this matrix. During this type of complete solving method, the determinant is computed: since the value in [K] for instance are big, the determinant would explode to infinite (overpass the 64 bits maximum storage).

In this study, only the dam has a mass (the soil is here to models the wave due to the earthquake, but no stress is computed inside this rock group). If the node ordering was perfect (all the node belonging to the soil are for example in the same range 1 -> 1400), we would expect a mass matrix [M] like this:

Figure 41 : "Ideal" mass Matrix

The reduction (“disassembly”) of the stiffness matrix and the mass matrix could be a very good solution (all the zeros entry will be deleted, and the dimension of the system to solve will really decrease).

Our case is not so ideal unfortunately for us: the node ordering in the structure is not fully optimized. The function “spy” in Matlab shows that fact (it shows where are the non-zeros entry in the stiffness matrix that we import from Akantu for example):

Figure 42 : "Spy" of the stiffness matrix Figure 43 : "Spy" of the mass matrix

- 35 -

Figure 44 : "Spy" of the reduced mass matrix (without blocked DOFs)

14.3.2 How the generalized eigenvalue problem can be solved with this

consideration?

[𝐴] {𝑥} = 𝜆 [𝐵] {𝑥} ⇔ ([𝐴] − 𝜆 [𝐵]) {𝑥} = {0} (1)

The system (1) can be easily transformed into a simple eigenvalue problem if the inverse of either [A] or [B] can be computed. To avoid this big calculation (and losing some important symmetric aspect of our matrix), it is possible to use the Cholesky factorization (a LU factorization with only positive definite matrix):

[𝐵] = [𝐿][𝐿]𝑇 with [𝐿]𝑇 very easily invertible (2)

Using this, the system can be transformed in:

([𝐿−1]𝐴[𝐿]𝑇−1

)[𝐿]𝑇{𝑥} = 𝜆 [𝐿]𝑇{𝑥} (3)

The relation (3) is equivalent to (simple eigenvalue problem):

[𝐴]′{𝑦} = {𝑦} (4)

This problem can be solved either by inversing or iteration (through the matrix).

14.3.3 How we apply this in our Matlab code

We import in our case three algorithm coded by some scientists, based on the theory we presented:

The EIGIFP script (G. Golub and Q. Ye, University of Kentucky, Department of

Mathematics, “An Inverse Free Preconditioned Krylov Subspace Method for Symmetric

Generalized Eigenvalue Problems”).

The BLEIGIFP script (slightly modified generalized version from the EIGIFP with a

different preconditioned algorithm) that is faster than the other, according to our

computation.

Some other code exists, for example the SPEIGIFP script, but it’s an old script (it needs some change because the new version of Matlab is not really fitted for him). The EIGS function exist also in Matlab, but it crashes several times with a system of this dimension.

This two method has a big power: the user can specify how many eigenvalues he want to compute, which one (minima, maxima), since the script is implemented in an iterative way.

- 36 -

We computed the first 12. According to the Guidelines, we don’t consider the dumping due to the soil.

14.3.4 Displacement computation for the different frequency mode

We have now found the first 12 λ value of this system:

([𝐾] − 𝜆 [𝑀]){𝑎} = {0} (5)

For λ1 = 4.13, we have for instance an equation under this form:

[𝐴] {𝑎} = {0} (6)

The physical meaning of {a} is quiet simple: the first three components are the x, y and z nodal displacements of the first point. To avoid the trivial solution {𝑎} = 0, we have to set:

{𝑎}(1) = 1

If we erase this unknown of {a} and make the corresponding change in the right and left term, we end with this type of equation:

[𝐴′]{𝑎′} = {𝑏′} ≠ {0} (7)

This equation is solved through a [LU] factorization and a backward substitution method in Matlab.

It’s not a mistake to set the three first component of 𝑎 completely arbitrarily. It’s important to

remember we are searching only a “shape” of deformation for each modal eigenvalue, not an exact displacement value.

Intuitively we force a displacement value for the first points and the system computes the proportional displacements for the other degrees of freedom. To erase the “arbitrarily” component of the solution solved, we have to normalize the displacement:

{𝑢𝑘} = {𝑎}

√{𝑎}𝑇 [𝑀] {𝑎} (8)

To have the full displacement vector for a modal eigenvalue, we have to add {𝑢𝑘} to the other blocked degrees of freedom (set to zeros because those nodes are on a fixed boundary) we

forget in the previous computation. In this way, we construct the final {𝑢𝑑} displacement vector.

14.3.5 Physical meaning of λ

𝑓̈(𝑡) + 𝜆 𝑓(𝑡) = 0 (9)

Since λ is the solution of a general oscillating equation (9), we can link this value to the natural mode of vibration of the system through those equations:

λ𝑘 = w𝑘2 (10)

𝑓𝑘 = 𝑤𝑘

2 𝜋 (11)

𝑇𝑘 = 1

𝑓𝑘 (18)

- 37 -

14.4 Appendix IV: Preprocessing and modification on the original mesh

This part could be interesting for someone that want to use “our” Gmsh mesh, to understand how it has been computed. The mesh is in Diana software like this:

The idea to convert this Diana file come from different advantages GMSH will give us :

Possibility to change order of the mesh without changing the geometry,

automatically!

Refine the mesh by splitting

Optimise the mesh with the already-implemented algorithm of GMSH

Use the reader gmsh of Akantu that is very well coded, very good support of

the “physical group”, possibility to assign material on elements groups

automatically and very easily in Akantu

Facilitate issue with the application of the loads : the Westergaard code already

implemented by Nicolas need the declaration of a surface : easy to create this

surface in GMSH

Etc.

- 38 -

We need to identify which points are on the upstream face (in contact with the water):

The list provided by the documentation (given by the Stucky’s engineers) to identify the boundary surface is presented like this:

- 39 -

Badly for us, this list is useless for us, since the point are not in the same referential as the mesh files. This point are related to the geometry files of the dam, used only if you have the Diana software…

Hopefully for us, there is some interesting groups that are declared in the mesh files:

We thought at the beginning it will be easy since there is a group called “FLUSTR” that represents the interface between the structure and the fluids! But in fact it was again useless: this group declared all the fluids element in contact with the dam, no way to extract the “surface of contact”.

We find a very interesting group in the mesh files that was not described in the documentation:

- 40 -

We directly plot in gmsh (with a script we code to switch that in the correct format) this group with the elements that correspond to each index found in the previous list

This group is absolutely useful. At first we identify which face was the upstream face: it was the “FACE1”.

A problem occurred: some point declared in the Diana files corresponding to the face1 was in fact in the face2. We see that in gmsh with the activation in the “visibility panel” of the physical name coloration (we link in our gmsh each surface element with a physical name):

- 41 -

Some lines of code later, we have the correct surface facets:

Now as we can see there is no mistake in the face surface:

- 42 -

We have isolated the face that we need (again with our home-made Matlab converter):

Assembly of the surface and the volume element in the same GMSH files

We read all the Diana files with our script, we convert it to the gmsh format, add the surface elements we pre-compute in the past section of this report, and we were able to display the mesh:

- 43 -

Add of the boundary (4 sided box)

In order to apply the boundary condition (blocked dof), we conclude that it will be easier for us to use directly the method in Akantu that allow to put some Dirichlet and Newman BC on surface element (declare in the gmsh file).

We find in the Diana mesh this node group :

Graphically we can see that there are all on the boundary we are interested in :

Unfortunately for us, there wasn’t a surface meshed on the boundary of the external ground box… We only know that this surface contains those points.

- 44 -

At first we try to create a script that re-create surface element with that list of points:

It didn’t work very well, so we decide to think a little bit more about the volume elements that contains those nodes.

We write a routine that detect all the elements that contains one point (or more) of this list. A sub-routine erase of this map all the element that have no face on the boundary. For example a quadratic hexahedron with 2 node in common with the list of the boundary points is not interesting for us, since he has no facet on the boundary. If he has 8 nodes, he would be stored.

We write a switch that check on which facet this common nodes are, and using the connectivity of the whole element, we reconstruct the facet.

- 45 -

Even with a simple script, we can see a lot of surface elements have been created. In fact we had to modify a little bit our code to take to account that an element can have 2 or more facets on the boundary as show in the next picture:

- 46 -

We add some exceptions to the code to handle the special case (multiple facets on the same boundary for the same element etc.) and we end up with that:

- 47 -

Before adding the boundary surface, the mesh is presented like this:

As we can see the surface fit “perfectly” to the volume:

- 48 -

With another representation (shaded volume):

- 49 -

As explain before, GMSH can refine the volume and he surface without problems:

Very useful thing: it could make possible to quantify the convergence around the exact solution, in relation with the mesh refinement.

- 50 -

Assembly of all surface and volume element in the same GMSH

As we can see here, the surface fit perfectly to the volume. We also add the downstream face, maybe for some result computation it could be interesting.

- 51 -

As we can see here, the surface elements are completely integrated around the volume elements :

- 52 -

Sub volume in the dam

In order to compute the initial state of the dam (stress and deformation), we need to simulate the construction of the dam part by part (see the Figure below with the construction stage numbering). The Diana mesh provides the information about the different part:

For example, on the figure that is below, we have all the nodes corresponding to the third construction stage:

We have written in our Matlab a routine that allow to detect all the volume elements that contains those nodes, and dump them in separate “msh” files (to see if the recognition procedure is working or not) :

- 53 -

The Matlab code allows to group those physical parts in the same GMSH format file:

As a reminder, now the mesh groups are like this in our mesh:

Surface of external boundary (4 sided-box)

Physical tag we choose: 2 Name you can use in Akantu (linked in gmsh files): boundary

Surface of loading (upstream face)

Physical tag we choose: 3 Name you can use in Akantu (linked in gmsh files): SURFUP

Other face of the dam (downstream face)

Physical tag we choose: 4 Name you can use in Akantu (linked in gmsh files): SURFDOWN

Major concret structure (material “5” in the Diana files) :

Physical tag in the gmsh files: 5 Name you can use in Akantu (linked in gmsh files): OLDDAM_1

- 54 -

Major concrete structure (material “5” in the Diana files) :

Physical tag in the gmsh files: 5 Name you can use in Akantu (linked in gmsh files): OLDDAM_2

Major concrete structure (material “5” in the Diana files) :

Physical tag in the gmsh files: 5 Name you can use in Akantu (linked in gmsh files): OLDDAM_3

Major concrete structure (material “5” in the Diana files) :

Physical tag in the gmsh files: 5 Name you can use in Akantu (linked in gmsh files): OLDDAM_4

Major concrete structure (material “5” in the Diana files) :

Physical tag in the gmsh files: 5 Name you can use in Akantu (linked in gmsh files): OLDDAM_5

New part (extra height) of concrete (blue part) (material 6 in the Diana files)

Physical tag in the gmsh files: 6 Name you can use in Akantu (linked in gmsh files): NEWDAM

Rock block (material 7 in the Diana files)

Physical tag in the gmsh files: 7 Name you can use in Akantu (linked in gmsh files): FOUND

- 55 -

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