C
DOCUMENTATION AND
USER INSTRUCTIONS OF
PROGRAM `SANOPT-F'
This appendix deals with the program SANOPT-F which is used for the
free vibration and buckling analysis and shape optimization of shells of
revolution and prismatic shells and folded plates with right and curved
planform. First an overview of the program structure is given. This is
followed by detailed input instructions. Finally, the specimen input data
les are presented.
C-1
C-2
C. Documentation and user instructions of program SANOPT-F
C. Documentation and User Instructions of
Program SANOPT-F
C.1 Program Documentation
C.1.1 Overview of the Program
Program SANOPT-F deals with the linear elastic free vibration analysis, buckling analysis and optimization of shells of revolution and also prismatic shells
of rectangular and curved planform which are supported on diaphragms at two
opposite edges with the other two edges arbitrarily restrained. The analysis is
carried out using a family of new, curved, variable thickness, C(0), MR, FEs
and FSs. The program contains: (a) a shape modelling tool using cubic splines
and an automatic mesh generation program, (b) FE and FS analysis capability
for shells of revolution and prismatic folded plates and shells respectively, and
(c) an optimization algorithm SQP.
The dierent parts of the programs are written in modular form so that
they can easily be incorporated into other programs. This means that all of
the input into and output out of subroutines is done via argument lists. The
dimensioning, however, is done centrally for all subroutines. This allows for
easy, `error free' re-dimensioning of the programs. However, some modularity
is lost.
C.1.2 Block Structure and Main Routines
The top level of the program structure for SANOPT-F is given in Figure C.1.
Only the important subroutines are shown.
FIMA
DATPRE
PRE
SANOPT-F
SENSIT
FEMFS
OPTIMA
FIMA
Fig. C.1.
Main block diagram of program SANOPT-F.
C.1 Program documentation
C-3
1. FIMA handles the le management. This subroutine open and close the
necessary data les.
2. DATPRE reads the main data for the geometry denition, analysis and
optimization such as design variables, material properties and boundary
conditions.
3. PRE is the shape denition and mesh generation routine and generates
meshes of varying sizes, types and element thickness.
4. SENSIT updates the values of the `shape' and `thickness' design variables
after every perturbation and evaluates the new values of the nodal point
coordinate and thickness using classical cubic spline interpolation.
5. FEMFS carries out the FE or FS free vibration and buckling analysis for the
structure on which the elements or strips are generated. It also calculates
the derivatives of the weight and eigenvalue for a specied number of design
variables.
6. OPTIMA updates the FE/FS results to identify the objective function and
constraints values and transfers them to the optimizer.
Next the function of the various routines in the geometry modelling routine
PRE, shape denition routine for perturb geometry SENSIT, analysis routine
FEMFS and optimization routine OPTIMA will be described.
Subroutine PRE is a pre-processor routine in which the geometry of the structure is dened and mesh generation is carried out. It is an implementation of
the procedures described in Chapter 2. Figure C.2 shows the block diagram of
subroutine PRE. In the following section brief descriptions of the functions of
various routines called by subroutine PRE are presented.
PARAM
GENODE
CORRIT
PRE
FIXIT
INTPAR
BANDWM
CONVER
Fig. C.2.
Block diagram of subroutine PRE.
1. PARAM calculates the control points for all cubic B-spline segments. The
result is stored in an array for later use in mesh generation.
2. GENODE generates nodes on a dened geometry.
C-4
C. Documentation and user instructions of program SANOPT-F
3. CORRIT is called when there is a need to carry out geometric corrections
in convergence studies.
4. FIXIT automatically assigns the xity data for the new generated nodes.
5. INTPAR interpolates the values of the thickness using classical cubic spline
interpolation.
6. BANDWM carries out the bandwidth minimization prior to the use of the
subspace iteration solver.
7. CONVER rearrange the coordinate, element connectivity, and xity data
for the analysis routine FEMFS.
Subroutine SENSIT is a implementation of the shape denition and mesh generation for sensitivity analysis and structural optimization described in Section
2.6. SENSIT calculates the new position of the coordinates (or thickness) of
the design variables and update the nodal coordinates (or thickness) after every perturbation which is required for sensitivity analysis. Figure C.3 shows
the block diagram of subroutine SENSIT.
DISTRB
GETCNT
SENSIT
DISTHK
INTPAR
Fig. C.3.
Block diagram of subroutine SENSIT.
1. DISTRB updates the new coordinates of the `shape' design variables after
each perturbation.
2. GETCNT calculates the new control points of the cubic spline for the perturbed geometry.
3. DISTHK updates the new coordinates of the `thickness' design variables
after each perturbation.
4. INTPAR interpolates the new values of the nodal coordinates or nodal
thickness using classical cubic spline interpolation.
Subroutine FEMFS is an implementation of the formulation described in Chapters 7 and 8 for structural analysis and Chapter 9 for sensitivity analysis. The
block diagram of subroutine FEMFS is shown in Figure C.4. In the following
section brief descriptions of the subroutines called by FEMFS are presented.
1. OUTPUT is an output subroutine and prints the data generated from the
routine PRE.
2. JUNCTN sets up direction cosines for the ends of each element.
C.1 Program documentation
C-5
OUTPUT
JUNCTN
ANGLES
STIFF
FEMFS
BUCKM
MASSM
DSTIFF
BSOLVE
DEGVAL
Fig. C.4.
Block diagram of subroutine FEMFS.
3. ANGLES nds the value of subtention angle at each node of each element.
4. STIFF sets up the stiness matrix for MR prismatic shell of rectangular or
curved planform strips or shell of revolution elements.
5. BUCKM sets up the geometric stiness matrix for buckling analysis.
6. MASSM sets up the mass matrix for free vibration analysis.
7. DSTIFF calculates the dierences in stiness, geometric stiness and mass
matrices terms after each perturbation of the design variables | this is
required in the semi-analytical method for the sensitivity analysis.
8. BSOLVE solves for the smallest eigenvalues and corresponding eigenvectors
using the subspace iteration method. The details of this subroutine can be
found in reference [1].
9. DEGVAL evaluates the sensitivities of the eigenvalues.
Subroutine OPTIMA is the actual interface for the communication between the
analysis program and the mathematical programs. It controls the optimization
by calling dierent optimizer interface routines. Figure C.5 shows the block
diagram of subroutine OPTIMA.
1. SQPSET is carry out sequential quadratic programming based on the algorithm developed by Powell [2].
2. SQPSTP terminate the SQP optimization program. Extracts a message
from the SQPSET writes it into the screen and output le.
3. PAROPT extracts the objective and constraints functions values and their
respective derivatives from the information computed in the previous routines.
C-6
C. Documentation and user instructions of program SANOPT-F
SQPSET
SQPSTP
PAROPT
MMASUB
OPTIMA
FIMA
DOTSET
DOT
SQPSET
SQPSTP
Fig. C.5.
Block diagram of subroutine OPTIMA.
4. MMASUB is a interface routine for MMA algorithm by Svanberg [3].
5. FIMA handles the le management. This subroutine open and close the
necessary data les.
6. DOTSET sets up some optimization parameters and initialise the optimizer
for DOT.
7. DOT is a interface routine for DOT optimization package by Vanderplaats
[4].
The interface routines for these three optimizers are implemented so far.
However, the program structure allows for simple integration of more algorithms. Only SQP algorithm is given in the current version of the program.
DOT and MMA optimization packages are a commercial package and is not
included in the program.
C.1.3 File Structures
Program SANOPT-F uses the following les for input and output:
unit 8 contains the output of the analysis and optimization results.
unit 10 reads the control parameters, material properties, boundary condi-
tion, geometry data and linking conditions.
unit 12 is used to read the optimization data concerning design variables
and constraints.
unit 15 is used in cases where geometric correction is required and contains
the coordinates of the key points.
C.2 Input instructions
C-7
unit 16 contains the stiness matrices of the original shape.
unit 17 contains the geometric stiness or mass matrices of the original
shape.
unit 18 contains the stiness matrices of the perturbed shape.
unit 19 contains the geometric stiness or mass matrices of the perturbed
shape.
unit 20 contains the results of the eigenvalue solver, that is frequencies,
eigenvectors, etc.
unit 24 contains the gradient of stiness matrix.
unit 25 contains the gradient of mass or geometric stiness matrix.
unit 40-60 are the plotting les for each optimization iteration and contain the plotting parameters such as eigenvectors, coordinates and element
connectivities.
C.1.4 Main Dimensions and Limitations of the Program
The program at present can handle 250 elements/strips, 500 points, 4 material
types, 20 geometry segments, 75 geometry key points, 20 design variables and
30 constraints.
C.2 Input Instructions
C.2.1 Main Structure of Input Data
In the following section user instructions for preparing the input data are
presented. Program SANOPT-F requires two basic data sets:
Data set 1 is read from unit 10 and contains information about the num-
ber of design variables, the parameters for shape denition, the geometry
data, linking data, xity condition information, material data and loading
conditions.
Data set 2 is read from unit 12 and contains the optimization data including;
the number of design variables, the constraints, upper and lower bounds of
the constraints, upper and lower bounds on the design variables and the
initial value of the design variables.
A le lename.cor is used in cases where geometric correction is required
and contains the coordinates of the key points.
C-8
C. Documentation and user instructions of program SANOPT-F
Data set 1.1: Title of problem | One record (a80)
cols. variable name
1-80
title
description
Title of the problem to be solved
Data set 1.2: Type of problem | One record (16i5)
cols. variable name
1-5
ndvab
description
Number of design variables
= 0, analysis
> 0, optimization
Data set 1.3: Title of output | One record (a80)
cols. variable name
1-80
title
description
Title of the program output
Data set 1.4: Output parameters | One record (16i5)
cols. variable name
description
koutp(i)
Control array for output information,
= 0, do not write
= 1, write into output le (see user hints)
1-5
koutp(1)
Analysis input data
6-10
koutp(2)
Eigenvalues
11-15
koutp(3)
Eigenvectors
16-20
koutp(4)
Plotting data
21-25
koutp(5)
Subspace solver output
26-30
koutp(6)
Gradient of eigenvalues
31-35
koutp(7)
Undened
36-40
koutp(8)
Undened
41-45
koutp(9)
Undened
46-50
koutp(10)
Optimization output
Data set 1.5: Analysis control title data | One record (a80)
cols. variable name
1-80
text
description
Title of the analysis control data
C.2 Input instructions
Data set 1.6: Parameters for analysis | One record (16i5)
cols. variable name
description
1-5
nprob
Type of analysis
= 1, buckling analysis
= 2, free vibration analysis
6-10
ntype
Type of analysis
= 1, prismatic shell with rectangular planform
= 2, prismatic shell with curved planform
= 3, shell of revolution
11-15
nnode
Number of nodes in each element or strip
16-20
ngaus
Number of Gauss point (see user hints)
21-25
nharm
Number of harmonics
26-30
nmode
Number of modes
31-35
nmats
Number of material sets
Data set 1.7: Geometry control title data | One record (a80)
cols. variable name
1-80
text
description
Title of the geometry control data
Data set 1.8: Parameters for geometry | One record (16i5)
cols. variable name
description
1-5
nsegg
Number of segments in the structure
6-10
npoig
Number of key points in the structure
16-20
ntang
Spline end condition (see user hints)
11-15
npolr
Number of key points expressed in polar coordinates
21-25
ncorr
Geometry correction switch (see user hints)
= 0, no correction required
= 1, correction required
26-30
nnbog
Number of boundary conditions
31-35
npara
Number of parameters to be interpolated (=1)
C-9
C-10
C. Documentation and user instructions of program SANOPT-F
Data set 1.9: Parameters for analysis | One record (2f10.5)
cols.
variable name
description
1-10
tleng
Span of the structure (see user hints)
= in length units for rectangular planforms
= in radians for curved planforms
11-20
coeff
Elastic foundation constant
Data set 1.10: Mesh parameters title data | One record (a80)
cols.
variable name
1-80
text
description
Title of the data to be read
Data set 1.11: Mesh parameters data
Two records for each segment.
First record (16i5) :
cols.
variable name
1-5
jsegb
6-10
nnseb(jsegb)
description
Segment number
Number of points in each segment
Second record (i5, 6f15.8) :
1-5
lnogb
Point number in each segment
6-20
coobt
h-coordinate of the segment
21-35
delta
Mesh density
Data set 1.12: Geometry denition title | One record (a80)
cols.
variable name
1-80
text
description
Title of geometry denition
Data set 1.13: Geometry denition
One record for each key point. Total of npoig records. (i5, 2f15.9, 4i5)
cols.
variable name
1-5
jpoin
description
Key point number
6-20 coorg(jpoin,1) x1 -coordinate of key point
21-35 coorg(jpoin,2) x2 -coordinate of key point
36-40 lsvab(jpoin,1) Design variable number of x1 -coordinate
41-45 lsvab(jpoin,2) Design variable number of x2 -coordinate
C.2 Input instructions
C-11
Data set 1.14: Coordinates of origin (polar coord.) points
Omit this record, if npolr = 0; one record for each point expressed in
polar coordinates. Total of npolr records. (i5, 2f15.9, 4i5)
cols.
variable name
description
1-5
npoln(ipolr ) = i
6-20
cpolr(i,1)
Value of x1 -coordinate of origin
21-35
cpolr(i,2)
Value of x2 -coordinate of origin
Key point number
Data set 1.15: Shape design variable linking parameters
Omit this record, if nlink = 0; one record for each linking. (16i5)
cols.
variable name
description
1-5
jlink
6-10
linkp(jlink,1)
Identication of segment rst key point
11-15
linkp(jlink,2)
Identication of segment last key point
16-20
linkp(jlink,3)
Key points common direction, x1 or x2
21-25
linkp(jlink,4)
Design variable number
Linking number
Data set 1.16: Segment data
Three sets of records for each segment. Total of nsegg records.
First record set (16i5) :
cols.
variable name
description
1-5
jsegg
6-10
nnseg(jsegg)
Number of key points in each segment
11-15
maseg(jsegg)
Material set number of each segment
Segment number
Second record set (2i5, 4f15.9): (repeated) for each key point of segment
cols.
variable name
1-5
lnodg
6-10
ltvab(jsegg,innse)
description
Key point number
Design variable number of key points
innse (see user hints)
11-25 paras(jsegg,innse,1) Thickness value at key point innse
C-12
C. Documentation and user instructions of program SANOPT-F
Third record set (16i5) :
cols. variable name
description
1-5
icotg(1)
Tangent indicator for cubic spline at start of
segment
6-10
icotg(2)
Tangent indicator for cubic spline at end of segment
= 0, natural spline condition,
= 1, horizontal tangent specied,
= 2, vertical tangent specied,
= 3, inclined tangent specied
Data set 1.17: Fixity condition title | One record (a80)
cols. variable name
1-80
text
description
Fixity condition title
Data set 1.18: Fixity condition data
One record for each xity condition. Total of nnbog records (16i5)
cols. variable name
1-5
description
nboug(iboun) Key point number at which boundary condition
is specied
6-10
ifixg(1)
Constraint on u-displacement ,
= 0, no constraint
= 1, constraint
11-15
ifixg(2)
Constraint on v -displacement
16-20
ifixg(3)
Constraint on w-displacement
21-25
ifixg(4)
Constraint on -rotation
26-30
ifixg(5)
Constraint on -rotation
31-35
ifixg(6)
Constraint on z -rotation
Data set 1.19: Output section title | One record (a80)
cols. variable name
1-80
text
description
Title of output section
C.2 Input instructions
C-13
Data set 1.20: Output section data | One record (i5, 6f15.8)
cols.
variable name
1-5
nsect
6-20
ysect(1)
Location of the 1st section along the span
21-35
ysect(2)
.
46-55
ysect(nsect)
Location of the 2nd section
.. along the span
..
..
Location of the nsectth section along the span
..
..
.
description
Number of output sections
Data set 1.21: Material properties title | One record (a80)
cols.
variable name
1-80
text
description
Title of the material type
Data set 1.22: Material properties data
One record for each material type. Total of nmats records. (i5, 6e15.8)
cols.
variable name
1-5
jmats
description
Material set number
6-20 props(jmats, 1) Elastic modulus E
21-35 props(jmats, 2) Poisson's ratio 36-50 props(jmats, 3) Material density C-14
C. Documentation and user instructions of program SANOPT-F
Data set 2.1: Title of problem | One record (a80)
cols. variable name
1-80
title
description
Title of the problem to be solved
Data set 2.2: Optimization parameters | One record (14i5)
cols. variable name
description
1-5
ntopt
Type of optimizer (see user hints)
= 1, MMA
= 2, DOT optimization package
= 3, SQP
6-10
ncons
Total number of constraints
11-15
kobje
Type of objective function
= 1, weight minimization
= 2, maximization of fundamental frequency or
maximization of sum of frequencies
= 3, maximization of lowest frequency
16-20
ktcon
Type of constraint
= 1, eigenvalues
= 2, weight
= 3, eigenvalues and weight
21-25
maxit
Maximum number of optimization iterations to
be performed
26-30
nlink
Number of shape design variable linking (see
user hints)
Data set 2.3: Initial values of the design variables | One record (*)
cols. variable name
xdvab(1)
.
..
..
.
xdvab(ndvab)
description
Initial value of 1st design
.. variable
..
..
Initial value of ndvabth design variable
C.2 Input instructions
C-15
Data set 2.4: Lower bounds of the design variables { One record (*)
cols. variable name
description
Lower bounds of 1st design
.. variable
..
..
th
Lower bounds of ndvab design variable
xdmin(1)
.
..
..
.
xdmax(ndvab)
Data set 2.5: Upper bounds of the design variables { One record (*)
cols. variable name
description
Upper bounds of 1st design
variable
..
...
.
Upper bounds of ndvabth design variable
xdmax(1)
.
..
..
.
xdmax(ndvab)
Data set 2.6: Bounds of the constraints { One record (*)
cols. variable name
description
Bounds of 1st constraint.
..
..
.
th
Bounds of ncons constraint
cfmax(1)
.
..
..
.
cfmax(ncons)
Geometric correction le: lename.cor
These records only exist if ncorr 6= 0. Two records for each segment.
First record set (6i5) :
cols. variable name
description
jsegg
Segment number for which correction is required
icorr
Correction switch,
= 0, no correction required
= 1, put all points on a straight line
= 2, put all points on a circular segment
Second record set (3e20.9). This record exists if icorr = 2 :
circx
Coordinate of origin: x-coordinate
circy
Coordinate of origin: y -coordinate
circr
Radius of circular segment
C-16
C. Documentation and user instructions of program SANOPT-F
C.2.2 User Hints
The purpose of user hints is to guide the user in the preparation of the data.
In the following some comments are made about various parts of the input le.
In the data sets (*) indicates that free format is used for inputting the data.
If ndvab is set to zero in data set 1.2 then optimization input le (data set
2) is omitted and the program does not carry out the optimization.
The output control ags (data set 1.4) allow to switch on/o output of
modules. For example setting koutp(3)=1 means that the output of the
eigenvectors is switched on.
ngaus is equal to 1, 2 and 3 for linear, quadratic and cubic FEs and FSs
respectively.
The variable ncorr is a ag for geometric correction. If set to 1, geometric
correction is used and the FE nodes are moved onto their exact location.
Furthermore, the subroutine PRE reads in the kind of correction to be carried out (lename.cor). The values are 0 for no correction, 1 for a straight
line and 2 for a circle. In the case of ncorr = 1, irrespective of the kind of
correction, for each segment ve real numbers must be given.
If C(0) continuity is desired at the junction of two or more segments then
ntang = 0 is specied.
npara species the number of parameters for each nodal point which would
be read in data set 1.16. The parameters will be interpolated from the background mesh to the generated FE/FS nodes. For instance, for the free vibration analysis, the parameter interpolated was the cross-sectional thickness
of structure.
tleng = 0.0 for shells of revolutions.
In data set 1.13, the shape design variable numbers are assigned to the
lsvab(ipoin,idime) array which species which key point and which coordinate of this key point is taken as the design variable. The key points
having the same lsvab(ipoin,idime) number behave in a similar manner.
We select the length or height of a segment as a design variable. The variable
nlink species the number of shape design variable linking rules, the value of
which is read in data set 2.2. In data set 1.15, linkp(ilink,1) rst key point
of segment, linkp(ilink,2) last key point of segment, linkp(ilink,3)
common coordinates of these key points (x1 or x2 ) and linkp(ilink,4)
design variable number must be provided for each link rule. Then the length
of a segment connected to that key points is a design variable.
The variable npolr species the number of polar coordinate systems, the
values of which are read in data set 1.14; the rst integer species key
C.3 Specimen input data les
C-17
point. Then centre of polar coordinate system is specied in the appropriate
cartesian coordinate system.
In data set 1.16 any segment key point for which a design variable number
is dened is taken as a design variable. The thickness of some key points can
be same value during optimization iteration. In that case, thickness design
variable number of this key points is identied with the same design variable
number.
As mentioned in Chapter 2, the thickness values along the shell segment
are interpolated using the thickness values at some or all the key points.
If any key point is to be excluded from the interpolation process then an
identication number is used which in the present case is 99999.999999999
(see section on specimen input data).
Segments are dened by a sequence of geometry or key points (data set
1.16). The end conditions of segments (icotg(1) and icotg(2)) inuence
the shape of the geometry (see Chapter 2). The values which can be specied
are 0 for natural end condition, 1 for horizontal alignment, 2 for vertical
alignment, 3 for alignment between rst two or last two key points of a
segment. The two values are for the start and the end of a segment.
As can be seen from data set 1.20 | boundary conditions are associated
with the geometric model and are later on transferred to the FE/FS model
after mesh generation has been completed.
C.3 Specimen Input Data Files
We now present some specimen input data les for the examples taken from
Chapter 9 for prismatic plates and shells.
C.3.1 Conical Shell
Problem denition: In the rst example, we consider the optimization of the
conical shell shown in Figure C.6. The optimization results for this example
are presented in Section 6.3.2. The objective function is the maximization of
the fundamental frequency with a constraint that the enclosed volume should
be kept constant 1 . The shell is clamped at its lower edge and free at its upper
edge. The following (concrete) material properties are assumed: modulus of
elasticity, E = 30 GP a, Poisson's ratio, = 0:15 and mass density, =
2410 kg=m3. The FE analysis is carried out using 3-noded quadratic elements.
1
Program SANOPT-F is modied to incorporate the enclosed volume constraint
for this problem.
C-18
C. Documentation and user instructions of program SANOPT-F
0.13 m
4
3
2.768 m
r4
r3
4.17 m
2
1
r2
r1
s1
s2
Fig. C.6.
1.25 m
=
= rr13 =
= rr24
0.13 m
Conical shell | initial geometry (thickness t = 0:0625 m).
Geometry denition: The geometry of the conical shell is dened using four
key points and three straight segments. In the rst stage of the optimization,
shape change alone is considered. In the present study, the radial coordinates
of key points 1 and 2 are linked as are key points 3 and 4. Thus, the two design
variables used for the optimization are the radial coordinate of the rst and
third segments; in other words, the design variables are the radial coordinates
of key points 1 and 3.
Mathematical denition: The mathematical description of the problem can be
written as:
Maximize !f und (s1 ; s2 )
(C:1)
subject to
s1 1:0; s2 1:0
s1 3:0; s2 3:0
Vencl 3:699 :
nonlinear constraint
Vencl 3:76
design variable bounds
(C:2)
(C:3)
Starting values: s(1)
= 1:25, s(1)
= 2:768.
1
2
A typical input les for SANOPT-F is displaced in Table C.1 and C.2.
They are for conical shell shown in Figure C.6. Data set 1, concerning the
general data and geometry data is given in Table C.1. Second data le which
includes optimization parameters for data set 2 is presented in Table C.2.
C.3 Specimen input data les
C-19
Table C.1. The data set 1 for conical shell.
conical shell
2
ganl freq emod graf berk ganl dfre undf undf opar
1
1
0
0
0
0
0
0
0
1
prob ntyp node gaus harm mode mats
2
3
3
2
5
1
1
segg poig polr tang corr nbog para
3
4
0
0
0
1
1
0.000
0.0
Backgroundmesh:
1
2
1
0.0000
0.1
2
1.0000
0.1
2
2
2
0.0000
0.1
3
1.0000
0.1
3
2
3
0.0000
0.1
4
0.0000
0.1
Definition of Geometry:
1 1.2500
0.00000
1
0
2 1.2500
0.13000
1
0
3 2.7680
4.30000
2
0
4 2.7680
4.43000
2
0
1
2
1
1
0 0.0625
2
0 0.0625
3
3
2
2
1
2
0 0.06250
3
0 0.06250
3
3
3
2
1
3
0 0.0625
4
0 0.0625
3
3
Fixity Conditions:
1
1
1
1
1
1
output section
1
1.5
Material Property:
1 30.0e9
0.1500
2410.0
0.0
0.0
data set
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13
1.16
1.17
1.18
1.19
1.20
1.21
1.22
C-20
C. Documentation and user instructions of program SANOPT-F
Table C.2. The data set 2 for conical shell.
topt ncon obje tcon maxi link
2
2
2
2 20
0
1.25
2.768
1.0
1.0
4.0
4.0
60.0
58.0
data set
2.1
2.2
2.3
2.4
2.5
2.6
conical shell opt.
C.3.2 Square Plate
Problem denition: A square plate to be optimized has side lengths a, with
an initial thickness value of t. The objective is to maximize the fundamental
frequency subject to the constraint that the volume of the plate remains constant. The results presented here are obtained using cubic FSs. The square
plate is simply supported at two ends and the other two edges are clamped.
The following material properties and dimensions are assumed: modulus of
elasticity, E = 2:6, Poisson's ratio, = 0:3, mass density, = 1:0, length of
the plate, b = 1:0 and thickness of the plate t = 0:1. The optimization results
are presented in Chapter 7 in non-dimensional form.
y
1.0
t1
x
1.0
Fig. C.7.
s1
= t1
t2
t3
s2
= t2
s4
= t3
x
Geometry of the square plate and locations of the design variables.
Geometry denition: From considerations of symmetry, only half of the plate
is analysed for boundary conditions Sh /C/Sh /C. The symmetric half of the
plate is modelled using one segment and three key points. For the quadratic
thickness representation, three design variables are considered. The geometry
of the plate and locations of the design variables are show in Figure C.7.
Mathematical denition: The shape optimization problem described above may
be summarised formally as follows:
C.3 Specimen input data les
Maximize !f und (s1 ; s2 ; s3 )
subject to
design variable bounds
C-21
(C:4)
s1 0:00; s2 0:00, s3 0:00
s1 0:20; s2 0:28, s3 0:25
(C:5)
V (s1 ; s2 ; s3 ) 0:099
V (s1 ; s2 ; s3 ) 0:1001
(C:6)
nonlinear constraint
:
Starting values: s(1)
= 0:1, s(1)
= 0:1, s(1)
= 0:1.
1
2
3
As an example of input les required to describe analysis and optimization
consider the cross-section of square plate shown in Figure C.7. The data set 1
and 2 input les for this problem are given in Tables C.3 and C.4 respectively.
Table C.3. The data set 1 for square plate.
data set
square Plates s/c/s/c
1.1
3
1.2
ganl freq emod graf berk ganl dfre undf undf opar
1.3
1
1
0
0
0
0
0
0
0
1
1.4
prob ntyp node gaus harm mode mats (anal. cont. param.) 1.5
2
1
4
3
1
1
1
1.6
segg poig polr tang corr nbog para (geom. cont. data)
1.7
1
3
0
0
0
2
1
1.8
1.000
0.0
1.9
Backgroundmesh:
1.10
1
2
1.11
1
0.0000
0.075
2
1.0000
0.075
Definition of Geometry:
1.12
1 0.0000
0.00000
0
0
1.13
2 0.2500
0.00000
0
0
3 0.5000
0.00000
0
0
1
3
1
1.16
1
1 0.1
2
2 0.1
3
3 0.1
3
3
Fixity Conditions:
1.17
1
1
1
1
1
1
1.18
3
1
1
0
0
1
output section
1.19
1
0.25
1.20
Material Property:
1.21
C-22
C. Documentation and user instructions of program SANOPT-F
1
2.600
0.3000
1.0000
0.0
0.0
1.22
Table C.4. The data set 2 for square plate.
topt ncon obje tcon maxi link (square plate opt.)
3
1
2
2 20
0
0.1
0.1
0.1
0.0001
0.0001
0.00000001
0.27
0.27
0.27
0.05
data set
2.1
2.2
2.3
2.4
2.5
2.6
C.3.3 Cylindrical Shell Segment with Curved Planform
Problem denition: The following example deals with a cylindrical shell segment with curved planform which is clamped at its lower edge and free at the
other edge as shown in Figure C.8. The following (concrete) material properties are assumed: Young's modulus E = 28 GP a, Poisson's ratio = 0:15 and
material density = 2400 kg=m3 . In this optimization problem, the objective
is to maximize the fundamental frequency with a constraint that the total
volume of the structure should remain constant.
s2
= t3
2
z4
1 rad
r2
1
s1
= t1
x1
(a)
(b)
Typical cross-section of cylindrical shell segment | position of design
variables: (a) plan view and (b) cross-sectional view.
Fig. C.8.
Geometry denition: The geometry of the shell and the denition of the design
variables are shown in Figure C.8. The shell is modelled using one curved
segment with a total of three key points. Two thickness design variables are
considered. Using variable linking, linear thickness variations are maintained
within the segment.
Mathematical denition: The shape optimization problem described above may
be expressed in mathematical terms as:
C.3 Specimen input data les
Maximize !f und (s1 ; s2 )
subject to
design variable bounds
nonlinear constraint
(C:7)
s1 0:25; s2 0:25
s1 0:25; s2 0:25
V (s1 ; s2 ) 141:4
V (s1 ; s2 ) 141:6
C-23
:
(C:8)
(C:9)
Starting values: s(1)
= 0:25, s(1)
= 0:25.
1
2
The input les listing of cylindrical shell segment problem for data set 1
and 2 are given in Table C.5 and C.6 respectively.
Table C.5. The data set 1 for cylindrical shell segment.
data set
cylindrical shell segment with curved planform
1.1
2
1.2
ganl freq emod graf berk ganl dfre undf undf opar
1.3
1
1
0
0
0
0
0
0
0
1
1.4
prob ntyp node gaus harm mode mats (ana. cont. param)
1.5
2
2
4
3
1
1
1
1.6
segg poig polr tang corr nbog para (geo. cont. data)
1.7
1
3
1
0
0
1
1
1.8
1.0
0.0
1.9
Backgroundmesh:
1.10
1
2
1.11
1
0.0000
0.5
2
1.0000
0.5
Definition of Geometry:
1.12
1 40.0000
0.00000
0
0
1.13
2 10.0000
45.00000
0
0
3 30.0000
10.00000
0
0
2 30.0
0.0
1.14
1
3
1
1.16
1
1
0.25
2
099999.999999999
3
2
0.25
3
3
Fixity Conditions:
1.17
1
1
1
1
1
1
1.18
output section
1.19
1
0.5
1.20
Material Property:
1.21
1 28.0e04 0.1500
2400.0
0.0
0.0
1.22
C-24
C. Documentation and user instructions of program SANOPT-F
Table C.6. The data set 2 for cylindrical shell segment.
data
topt ncon obje tcon maxi link (cylin. shell segment)
3
1
2
2 20
0
0.25
0.25
0.01
0.01
0.6
0.6
141.5
set
2.1
2.2
2.3
2.4
2.5
2.6
References
[1] Bathe, K.J., Finite element procedures in engineering analysis, Prentice{Hall,
Englewood Clis, New Jersey, 1982.
[2] Powell, M.J.D., Algorithms for nonlinear constraints that use Lagrangian functions, Math. Progr., 14, 224-248, 1978.
[3] Svanberg, K., The method of moving asymptotes - a new method for structural
optimization, Int. J. Num. Meth. Eng., 13, 359-373, 1987.
[4] DOT, Design Optimization Tools, Users Manuals, Version 5.0, Vanderplaats Research & Development, Inc., 1999.