Tarbiat Modarres University (T.M.U) In The Name of Allahvelos0.ltt.mech.ntua.gr/ERCOFTAC/PROC06-GC/resources/7_April/session_6/Ercoftac 06...Necessity &Target • Because in our country

In The Name of AllahTarbiat Modarres University (T.M.U)

Optimum Design Program for Shape of Arch Dams

(ODPSAD)

Jalal AkbariPhD Candidate of Civil Engineering, Tarbiat Modares University ,Iran

M.T.AhmadiProfessor of Civil Engineering, Tarbiat Modares University, Iran

H.MoharramiAssistant Professor of Civil Engineering, Tarbiat Modares University, Iran

Peter Jan Pahl Professor of Civil Eng., TU Berlin ,Germany

ERCOFTAC 2006 ,5-7 April, Gran Canari, Spain

Tarbiat Modarres University (T.M.U)

OutlineOutline

• Main Target

• Steps of ODPSAD Code

• Dynamic Loads & Optimization algorithm

• Design Variables Description

• Constraints & Objective Function

• Results

• Conclusion


Necessity &Target

• Because in our country (Iran) there are many new sites (150) for making new arch dams the necessity of shape optimization of arch dams is important in our country.

• Developing a practical and professional code for design optimum shape of arch dams with considering Engineering and constructionnecessities.

Inputs:

• Arbitrary shape of canyon or topography data• Height of arch dam • material properties of rock and concrete

Outputs:

• Optimum geometry of arch dam with satisfying design and construction demands


l Geometry of Daml Geometry of arch dam is made with cubic Hermit splines. In this case

characteristics such as ( thicknesses , radius ,…) are known in 4 levels Hb=0, Hm=0.40*Hdam, Hu=0.75*Hdam, Hc=Hdam . In the arbitrary heights characteristics are determined with splines interpolation.

l Mesh Generationl For 3 load cases ( dead load , hydrostatic load and Earthquake load) meshes

are generated automatically with 8 node brick elements .

l Finite element analysisl For above load cases finite element analysis is done with FEAPPv and stress

results saved for optimization stage.

l Sensitivity analysisl Gradients of objective function and constraints are carried out by finite

difference method

l Optimizationl Enhanc design variables with one of optimization algorithms

Main Steps of ODPSAD Program


Dynamic Load

There are 3 sorts of dynamic load

l Time Historyl Response Spectruml Equivalent Static Load

l in practice time history analysis is applied in form of shock, impact or harmonic loads so rarely is used in general form in optimization process.

l optimization for dynamic loads is time consuming process l design space is disjoint for dynamic loadsl The number of constraints in time history load is increased dramatically

With considering above notes Response spectrum load is used as dynamic load


Optimization Method

There are 2 kinds method in arch dam shape optimization

l Gradient Based Methods GBM :(SLP,SQP,Penalty function, MFD,SD,…)l Statistical Based Methods SBM :( GA ,NN,ES,SA,….)

Advantage and disadvantage of above methods

l Gradient computation in GBM is main drawback & time consuming processl The probability of trapping in local optimum is highl The rate of convergence in GMB is good

l In SBM gradient computation don’t requiredl The probability of trapping in local optimum is low l In SBM The constraint problem must be changed to unconstraint probleml In shape optimization and in the case that heavy finite element analysis is need

the SBM are very weak

NDV=40 NFEM=26000Time= 325 h (13 day)1.6 GHZ

NDV=40 NFEM=3000Time= 433 h (1.4 day)1.6 GHZ

SBM(GA) GBM (SQP)


cθC :Dam Coordinate system location (Cx,Cy,Cz)

: The rotation of arch dam in site

Design VariablesThere are 40 design variables in this optimization process


Tb,Tm,Tu,Tc : Thickness of crown cantilever in 4 levels

Pb,Pm,Pu,Pc: Overhang parameters of crown in 4 levels


Rb,Rm,Ru,Rc: radius of curvature for crown cantilever at 4 stations


Tbr,Tmr,Tur,Tcr: Thickness of dam body at right abutment at 4 stations

Tbl,Tml,Tul,Tcl: Thickness of dam body at left abutment at 4 stations


Sb,Sm,Su,Sc: Starting points of thickness variables zone in left side

tb,tm,tu,tc : Starting points of thickness variables zone in right side


dbr,dmr,dur,dcr : Excavation in right abutment at 4 stations

dbl,dml,dul,dcl : Excavation in left abutment at 4 stations


Gradient Computation

l Because there is not explicit formulation between design variables and objective function & constraints, these functions are approximated via Taylor expansion series.

l Nonlinear functions are replaced with sequential approximated quadratic (Objective function) and linear functions ( constrains)

l Gradient of objective function is computed with central finite difference method and gradient of constraints with forward finite difference method

l To increase the rate of convergence, design variables are modified as bellow

i

i


l The volume of dam body and excavation volume at abutments is considered as objective function

l In shape optimization of arch dams, the 3 sorts of constraints should satisfy the demands of design and construction requirements:

l Geometrical constraints ( there are 7 constraints )

l Stress constraints (there are variable constraints)

l Stability constraints ( there 8 constraints)

l In this research SQP (Sequential Quadratic Programming) algorithm is used as an optimizer

Objective function & Constraints


Geometrical constraints


Stress constraints


Location of elements & nodes for stress constraints evaluation


Stability Constraint


Structure of ODPSAD


Initial Design Variables

l Coordinate of site systeml Hdam =13l XS=0.0l YS=0.0l ZS=0.0l Coordinate of dam systeml XD= 0.0l YD=15.0l TETA=0.l Overhangs in 4 levelsl Pc=1.00l Pu=0.l Pm=0.l Pb= 0. 5l Thickness of Crownl Tc= .l Tu=1 .l Tm=1 .l Tb=2 .

l Radius of curvature .l Rc=l Ru=1 5l Rm= 5l Rb= 5.

l Thicknesses in rightl Tcr=l Tur=15l Tmr=2l Tbr=30l Thicknesses in leftl Tcl=l Tul=15l Tml=2l Tbl=30l Width Raito in rightl Sci=.5l Sui=.5l Smi= 5l Sbi=.5

l Width Raito in leftl tci=.5l tui=.5l tmi=.5l tbi=.5 l Excavation in abut.l Dci=3l Dui=5;l Dmi=6l Dbi=7l Material Propertiesl Ec=22 Gpal Wc=2400l Ww=1000l Nu=.18l Er=8Gpal Valley shape nodesl Trapezoidal shape



Geometry of Dam is made by 40 D.V


Geometry of Dam & Foundation is made by the same 40 D.V

ResultsTarbiat Modarres University (T.M.U)

Evolution of crown cantilever shape

Evolution of crown cantilever shape


Objective function convergence

350000

400000

450000

500000

550000

600000

650000

1 2 3 4 5 6 7 8 9 10 11 12 13

Iterations

Vo

lum

e(m

3)

Crown cantilever Thickness

0

20

40

60

80

100

120

140

5 10 15 20 25 30 35

Thickness(m)

Dam

Hei

ght(m

)

Initial

Optimum


Conclusion

l This code has minimum interact with user ( user must guess suitable initial shape)

l In this code dynamic allocation and pointer capability of Fortran 90 is used

l The time consuming process in this research is gradient calculations because of using finite difference method instead of analytical sensitivity analysis or semi analytical method

l Good initial shape has very impact in number of iterations and convergence rate

l With starting from any arbitrary shape ( arbitrary design variables) optimum shape don’t obtain. Suitable initial shape is required .

l The range of lower and upper limits of design variables in optimization algorithm have too effect in design variables enhancement so these limits must be select carefully.

l Using GUI and suitable post processor will develop in the future


Thanks for your attention

Documents

Tarbiat Modarres University (T.M.U) In The Name of Allahvelos0.ltt.mech.ntua.gr/ERCOFTAC/PROC06-GC/resources/7_April/session_6/Ercoftac 06...Necessity &Target • Because in our country