Researchers in the Materials and Thermal Structures
Branch at the NASA Langley Research Center in Hampton, VA, are exploring
advanced metal manufacturing methods for rapid prototyping, fabrication and
repair of aerospace structures. Electron beam freeform fabrication (EBF^{3})
is a layer addition technique that produces complex, unitized, structural
metallic parts with strengths comparable to that of wrought product, directly
from computer-aided design (CAD) data. Since the EBF^{3} process is
digitally controlled, EBF^{3} can be used to perform repairs in remote
locations, and it offers the potential for reductions in part manufacturing
costs and weight, and increased material processing performance to enhance
mission success for aircraft, launch vehicles, and spacecraft.

However, before this vision can be realized a number of technical issues related to the computational modeling and optimization of the fabrication process, practical applications, and the relationship between an application and the selection of parameters that impact the quality of the process must be addressed. The proposed work here is a step in that direction. We propose starting a joint VT/IALR/Langley integrated analysis/design/prototyping research program of complex structures. As a first step, the team will focus on following technical challenge: (I) integrated stiffened aluminum and titanium plates for aircraft structural panels subjected to complex in-plane and out-of-plane loads, and (II) implement these designs using some of the RPT capabilities available at IALR and Danville Community College.

The team at Virginia Tech will consist of __ Dr. Rakesh K. Kapania__,
and two graduate students. Dr. Don Moffitt of the

**1. ****Functionally
Graded, Optimally Stiffened Panels**

Throughout this research, efforts will be made to develop scaling parameters that will enhance the value of the present research for panels that have arbitrary dimensions, curvatures and loads.

The optimal
designs arrived at in the previous two applications will be evaluated for their
ability to be fabricated using the processes available at the Danville
Community College (DCC)/Regional Center for Applied Technology and Training
(RCATT). The two graduate students
who would work on the two application problems will spend at least three months
at Danville to ensure that the CAD (Computer-Aided Design) files work
adequately on the processes available at DCC before they are sent to the NASA
Langley Research Center for EBF^{3} processing.

**Interaction
with Langley**:

We plan to have a continuous interaction with Karen Taminger and other researchers at Langley by using the communication power of the Internet. We will have a monthly Telecon and will make a presentation over the internet. The slides will be placed on a secured web site and will be available to NASA Engineers before the meetings.

Key techniques
and progress

Here is a summary
for the first quarterly achievements and progress:

·
NURBS
representation for both the stiffeners and plate of a panel

·
Triangulation
mesh generation using DistMesh

·
Finite
element analysis and structural optimization via MATLAB/NASTRAN interface

·
Size
(cross-sectional dimensions) optimization for fixed orientations and curvatures
of stiffeners

·
** p**-method via

NURBS
(Non-uniform rational *B*-Splines) has already been a standard for CAGD (computer
aided geometry design). Our
main reference
in this topic is one of the most used textbooks, *Curves and Surfaces for Computer Aided Geometric Design: A Practical
Guide*, by G. Farin [1] .

·
**Smart bed
example**

We have used
the NURBS to represent the placement of the fiber optic sensor, the deformed
and un-deformed mattress surfaces as well as body pressure distribution for a
smart bed (Fig. 2) [2]. The locations of the control points of the NURBS curve
for the fiber optic sensor are design variables. The objective is to find a
placement of the fiber optic sensor so as to maximize the integrated norm-2
curvature change of the fiber curve due to a patient’s movement on the
bed mattress. The constraint is
placed on the length and initial curvature of the fiber optic curve.

Figure 3 shows an example of the placement optimization
of the fiber optic sensor using GA, implemented as the Java Design Tool for the
Fiber Optic Sensors for the Integrated Smart Bed. The cyan curve is the initial
fiber curve while the white dots of the polygon are the control points of the
NURBS for the initial fiber curved. The yellow curve is the deformed fiber
curve, which is determined by the bilinear interpolation from the mattress
surface grid points.

·
**Blade
stiffened panel of EBF ^{3}**

While we can borrow certain ideas from the smart
bed work, we decided to give up using Java language to program in current work.
Instead, we are using __ MATLAB__,
which has many useful MATLAB tool boxes (e.g. Spline
Toolbox and Optimization Toolbox), as the main programming language to develop
the interfaces between a user and computer and between a user and other
application software, such as MSC.NASTRAN or ANSYS.

Though the Spline Toolbox
in MATLAB has the capability to represent NURBS, it is not in the convenient
form as used by most CAGD packages. Therefore, we choose the * NURBS
Toolbox* by D.M. Spink
[3].

As shown in Fig. 4 and 5, the axes of stiffeners (blue curves) are represented by NURBS curves, where the control points (red circles) determine the placements of the stiffeners.

While the ends of stiffeners can be arbitrary positions within and on the plate boundary, we choose to place the ends of the stiffeners on the boundary edges (Fig. 5) to avoid the undesirable stress concentration problem due to a sudden stiffness change. The green circles in the Fig. 5 are the prescribed boundary points. Both end control points and inner control points of the stiffener curves are generated randomly and automatically to demonstrate the freedom of the stiffened panel representation using NURBS.

In
addition, the curvilinear mid-surface of the panel can also be represented by NURBS,
as shown in Fig. 6. The surface is determined by the red control points.

Finally,
the width and height distributions along the axes of the curved stiffeners are
also described using NURBS, see Fig. 7.

**Automatic** mesh generation during optimization has been
one of our most important concern for this stiffened panel optimization
problem, where both cross-sectional sizes and placements or orientations of the
stiffeners as design variables.

For size
optimization, automation in mesh generation is not a necessity because one time
satisfactory meshing can be used in the whole process of the optimization.

For
placement optimization of stiffeners, however, the mesh grid points keep
changing, at least in grid point positions. Sometimes, re-meshing is
dispensable or advisable.

Our previous work on the smart bed includes dealing
with a similar problem, where an optimal placement of the fiber optic curve
needed to be found. As the stiffness contribution of the fiber optic cable to
the mattress model is too small, the fiber is treated as absolutely flexible.
This allows us to use a very coarse rectangular mesh for the mattress, and the
curve points of the fiber are not required to coincide with those of the
mattress (Fig. 8a). The deformed fiber curve is determined from the deformed
mattress surface by an approach similar to the __ bilinear
interpolation __from the grid points of the mattress surface.

For the stiffened
panel work, however, the stiffeners are attached mainly to provide the
structural support for thin and easily deformed plate so that both the load
carrying capability and natural vibration frequency of the panel are increased;
thus, the stiffness of the stiffeners cannot be ignored.

First, we
may generate **dense** mesh for plate and align the grid points of
stiffeners to coincide with those of the plate (Fig. 8b). This approach may not
apply to the problem at hands when the eigen-problems
for buckling and vibration frequency are to be solved.

Second, we
may choose to align the gridlines of the plate as shown in Fig. 8c so that all
of the stiffener grid points coincide with the grid points of the plate for a given
stiffener meshing. This will lead to an extremely non-uniform mesh. We can
release this problem by first aligning the grid points to some prescribed mesh grids such as in
Fig. 8b but with a much less density, and then move back to previous locations
after generating the gridlines that pass all the grid points for stiffeners.
Moving mesh grid points is a necessity to get a more uniform mesh. Some further
work, including literature review, needs to be done for moving rectangular
mesh.

Third, we
may choose to use coarse mesh for both plate and stiffeners. A grid point of a
stiffener can be either on the gridline(s) of plate mesh or within a
rectangular element, as also seen in Fig. 8a. Then, special rectangular plate elements
with extra mid-side grid points and inner grid points are needed. The most
popular rectangular plate elements, such as 8-noded or 9-noded, will work when
the grid point of the stiffeners are apart enough. There is a limitation to use this kind
of element, such as 8-noded element, because it is suggested that the mid-side
node location on an edge be on middle 1/3 edge length. To remove this
limitation, moving mesh is also a necessity.

Higher
order or ** p**-version of elements
is one of our concerned via

A
comprehensive literature review on automatic mesh generation has
not been made yet as we
focus on the availability of the mesh generation software. __ Reference
4__ gives the link to a summary of different mesh generation
software, their capability, and source code availability, of which the

·
Source
code in MATLAB is available as free software

·
The code
is short and simple and easy to be used and modified

·
The mesh
quality is mostly acceptable

DistMesh generates
unstructured triangular and tetrahedral meshes! It uses a special distance
function to describe the relationship between a point and the boundary of the
domain, Delaunay function in MATLAB to generate
triangular mesh, and move the mesh by solving a truss equilibrium problem while some prescribed grid point
locations can be fixed. This gives us the freedom first to generate arbitrary
grid points of stiffeners’ axes, and then generate the triangular mesh
for plate so that the grid points for stiffeners are part of grid points for
the plate.

Figure 9a
shows an example of triangulation meshing using DistMesh,
modified to curry our current requirements. The mesh does not look very nice as
it is a very coarse mesh with unevenly distributed grid points of the four
curved stiffeners for mesh distortion testing. The mesh with high quality can be
obtained when the mesh density is increased and relatively uniformly
distributed grid points for stiffeners are given.

For
convenience, the consistence or continuity between stiffeners and plate are
enforced only by the connection of grid points. Therefore, we allow two
consecutive or adjacent grid points of a stiffener to jump over several plate
elements. For the same reason, the intersections among stiffeners or a
stiffener itself are not calculated or connected by grid points. That is,
stiffeners can overlap or penetrate with each other.

The finite
element mesh for a curvilinear stiffened panel, as seen from Fig. 1, can be
simply mapped or ** deformed** from
a reference flat stiffened panel with similar geometric mid-surface dimensions.
This is easily implemented using NURBS representation for both flat and curved
plate.

It is
noticeable that Figs. 1, 6 and 7 display both plate and stiffeners with
thickness. MATLAB provides a good graphical interface and rendering
implementation. Its **patch** function provides a means for solid body
display. We implement the thickness display using **mid-surface**, its **boundary**
and **normal**, and **thickness **information to build up **big patches**
for top surface, bottom surface, and side surface (s) of the solid body, either
plate or stiffeners. Each of the big patches is composed of many small
triangular or rectangular patches. We wrote a small package of about a dozen of
MATLAB functions to implement the 3-D thickness display. It will be very
helpful and pleasant to visualize the stiffener placements and cross-sectional size
variations.

MSC.NASTRAN
and/or ANSYS are used for response analysis, design sensitivity analysis, and optimization,
in the first stage. The Optimization Toolbox in MATLAB will be used for
placement optimization while response and design sensitivity analyses are
performed using MSC.NASTRAN and/or ANSYS. Working on a finite element software
in MATLAB is also an option, which can speed-up the over all procedure.
Repeatedly calling external software, such as NASTRAN, may be time consuming.

A
MATLAB/MSC.NASTRAN interface is being developed for the stiffened panel
optimization. We decided to create this interface for the following reasons:

·
MATLAB
is not only an easy to learn programming language, but also has included most
useful tool boxes for numerical simulation and graphical displaying

·
Application
software (including NASTRAN, ANSYS, etc.) in other popular languages, such as,
FORTRAN, C, JAVA, etc., can be executed as external functions in MATLAB environment

·
The NURBS
tool is freely available in MATLAB version and easily implemented

·
Automatic
mesh generator, such as DistMesh, is freely available
in MATLAB

·
NASTRAN
and/or ANSYS have been standards for structural analysis and optimization, but
lacking or weak on NURBS representation and flexible programming

With the MATLAB/MSC.NASTRAN
interface, a user can easily define the design problems and create Bulk Data
file for MSC.NASTRAN to perform response analysis, and optimization. It also
takes NASTRAN itself as external function, invokes it, and read the results
from NASTRAN output to display and further process in the MATLAB environment.

The
elements considered for NASTRAN will include CTRIA, CQUAD, for 2-D component
modeling (e.g. plate and thin-walled stiffeners), CBAR and CBEAM for 1-D
component modeling (thick and thin-walled stiffeners when warping is not an
issue).

Similarly,
MATLAB/ANSYS interface will be developed, when it is deemed necessary.

The
following are some selective examples we have run through to illustrate and
test our current capability and understand the problems at hands.

**3.1 ****Response
analysis examples**

**3.1.1
****Elastic stability of a plate**

This example
is adopted from __ http://www.mscsoftware.com/support/online_ex/Nastran/Nas101/AppendixH.pdf__,
used for

1) Testing
of NASTRAN data created

2) Testing of mesh distortion
as shown in Fig. 9a, due to introduction of the four stiffeners.

The model is described as follows:

1) Find
the unit critical compressive stress of a flat rectangular plate. (The elements
of stiffeners are not included.)

2) Equal uniform
compression of 6.89E+5 Pa on two opposite (short) edges.

3) All edges are simply supported.

4) Model properties are given in Table
1.

The result
in Fig. 9b and many other testing results show that for buckling load, the mesh
distortion due to the introduction of the stiffeners is trivial. The mesh
generator DistMesh works fine.

**3.1.2
****Natural
frequency of a plate**

**3.1.3
****Stress analysis of a plate**

**3.2 ****Structural
optimization examples**

In the
examples that follow, the typical material properties will be refereed from
Table 2.

The
geometrical dimensions, boundary loading and support conditions are denoted in
Fig. 10 for a planar panel.

** a** and

*N*_{x} and *N*_{y}
are the in-plane normal load densities in *X*
and *Y* -directions, respectively; *Q*_{xy} is the shear load density. The
in-plane load density is in the term of load per unit length. The out-plane load
can also be added though it is not denoted in the figure.

Though it
is not very noticeable in Fig. 10, along boundary of the mid-surface of the
plate are prescribed boundary fixed points. The number of boundary fixed grid
points can be prescribed for each of the four *numbered* edges. The ends of every stiffener are connected to those
boundary points. Therefore, the meshes for both plate and stiffeners are
generated based on those prescribed boundary information. By default, the grid
points are set to be uniformly distributed on each edge, but it is not a
necessity.

Internally,
the boundary fixed points are numbered along edge 1 from the corner between
edges 4 and 1 to the corner between edges 1 and 2, then along edge 2 to the
corner between edges 2 and 3, then along edge 3 to the corner between edges 3
and 4, finally along edge 4 to the corner between edges 3 and 1. That is, the
counting order follows the right-hand screw rule, with the thumb pointing in
the direction of the normal of the mid surface of the plate, but starting from
the corner between edges 4 and 1.

Externally,
however, it is convenient to give a location of a boundary fixed point by a
number pair for each edge. For edge
1, count the fixed points from edge 2; for edge 2, count from edge 1; for edge
3, count from edge 2; and for edge 4, and count from edge 1. Note that does not
follow the right-hand screw rule this time but in positive X or Y direction!

For
documentation purpose and also for better understanding the examples that
follow, we take Fig. 9a as an example to illustrate an approach how we can
handle placement/orientation optimization for stiffened panels.

Let Nbfix1,
Nbfix2, Nbfix3, and Nbfix4 be numbers of fixed points on edge 1, 2, 3, and 4,
respectively. Let Nbfix be the total number of
boundary fixed points.

·
Nbfix1 =
10; Nbfix2 = 15; Nbfix3 = 10; Nbfix4 = 15;

·
Nbfix = Nbfix1 +
Nbfix2 + Nbfix3 + Nbfix4 - 4;

Let EID be
edge ID and *i*
be *i*^{th}
fixed point on edge EID. Then, the connection between (EID, *i*) and internal ID *I* is,

·
For EID
= 1, *i* = 1
to Nbfix1, *I*(EID, *i*) = Nbfix1 – *i* +1* *;

·
For EID
= 2, *i* = 1
to Nbfix2, *I*(EID, *i*) = Nbfix1 + *i** *– 1;

·
For EID
= 3, *i* = 1
to Nbfix3, *I*(EID, *i*) = Nbfix1 + Nbfix2 – 1 + *i* - 1;

·
For EID
= 4, *i* = 2
to Nbfix4, *I*(EID, *i*) = Nbfix1 + Nbfix2 – 1 + Nbfix3
– 1 + Nbfix4 -2 - *i** *+ 2;

·
For EID
= 4, *i* = 1,
*I*(EID, *i*) =1.

Now, we
illustrate how to represent the stiffeners’ boundary layout by its end
points’ locations on the boundary edges.

Let (EID_{1}
*i*_{1}; EID_{2} *i*_{2};)_{st}
denote the edge location of the first (EID_{1} , *i*_{1}) and second (EID_{2} , *i*_{2}) end points, respectively on the boundary edges, for
stiffener *st*,
and put all edge locations of the stiffeners (four of them, for example,) in a
column, we get a stiffener layout representation like

*Edgelocation* = [(EID_{1} *i*_{1}; EID_{2} *i*_{2};)_{1}

(EID_{1} *i*_{1}; EID_{2} *i*_{2};)_{2}

(EID_{1} *i*_{1}; EID_{2} *i*_{2};)_{3}

(EID_{1} *i*_{1}; EID_{2} *i*_{2};)_{4}]

Therefore,
the stiffeners’ boundary layout for Fig. 9a is

*Edgelocation** *= [ 1 5; 4 12;

1 7; 3 5;_{}

2 6; 2 12;

2 13; 3 10;]

To
summarize, the boundary layout for any stiffened panel as described in Fig. 10
can be determined by the numbers of fixed points on four edges [Nbfix1 Nbfix2
Nbfix3 Nbfix4] and the stiffeners’ boundary edge location *Edgelocation*.
Note that when the boundary fixed points are dense enough, we have got a layout
representation for any placement/orientation of (straight) stiffeners for any
stiffened panel. When mid control points are added for stiffeners, curved
stiffeners with any placement/orientation are also represented.

**3.2.1
****Size optimization of a blade stiffened plate**

**A general design description** for a given stiffened panel layout, planar
dimensions of the plate, **a** and **b**, and
the materials of the plate and stiffeners:

1.
**Find** the uniform thickness *t* of the
plate, and the width and height distributions of a stiffener *st*, (*w*** h**)

2.
to **minimize**
the weight of the blade stiffened panel,

Objective = *weight*

3.
**subjected to** the minimum buckling load factor constraint,

*λ*** **≥

and von-mises stress
constraint on plate,

*σ*_{von-mises} *≤
σ_{a}*

and side
constraints on size design variables,

*t*_{min} ≤
** t** ≤

*w*^{st}_{min} ≤ *w*^{st} ≤ *w*^{st}_{max},

*h*^{st}_{min} ≤ *h*^{st} ≤ *h*^{st}_{max}

At this time, the size optimization with variable cross-sections is implemented element-wisely by taking the width and height of the cross-section independent from those of rest of the elements for stiffeners while assuming a uniform cross-section within an element. The optimization results at grid points will be smoothed by taking the average of values of the adjacent elements. The variable linking is not considered. The number of design variables will be very large and the convergence may be hard and slowed down. This limitation may be removed using NURBS representation for the cross-sections and using CBEAM or CQUAD element, instead of CBAR. It will be implemented when the optimization is performed within MATLAB environment.

**Example 1**. ** Regular grid fashion
– larger spacing of stiffeners**. Uniform via variable
cross-section of a stiffener is studied.

Planar dimensions and materials of a stiffened panel are given in Table 3. Layout (201) and edge locations of stiffeners' ends are given in Table 4. Initial sizes and bounds of cross-sectional dimensions are given in Table 5.

For **a**/**b**=1, simply supported on four edges,
the material properties in Table 2, the critical stress due to buckling, with **the compression on edges 1 and 3**, is
[6]

*σ*_{cr}* = k
π^{2}E*(

* = *4* π^{2}
*70 GPa

=1.0014 MPa;

The unit pressure load density is chosen

*P*_{0}
= [*N*_{x0};
*N*_{y0};
*Q*_{xy0}];
*N*_{x0}
= *σ*_{cr}_{ }**t**** **= 5.0072e+003 N/m; *N*_{y0} = 0; *Q*_{xy0}
= 0;

The actual load is obtained by multiplication of a
load factor *λ*

** P** =

The allowable minimum buckling loads considered are included in the following array

*λ*_{a} = [0.5 1. 5. 10. 15. 20. 25. 30. 35. 40. 45.
50.];

The design
optimization parameters used in NASTRAN is normally

$---1---|---2---|---3---|---4---|---5---|---6---|---7---|---8---|---9---|--10---|

$ Design Optimization Parameters

$DOPTPRM PARAM1 VAL1 PARAM2 VAL2 PARAM3 VAL3 PARAM4 VAL4

$

DOPTPRM P2 15 DESMAX 25 DELP 0.5 GMAX 0.005

CONVDV 0.0005 CONVPR 0.0005 P1 1

If there is
a convergence is not achieved, DELP is set to 0.5. The maximum allowed design
cycles DESMAX is extended to 50 in some cases.

** This set
of parameters is used thru all the examples that follow**.

The scale of the design problem is as follows:

NUMBER OF GRID POINTS = 375

NUMBER OF CBAR ELEMENTS = 64

NUMBER OF CTRIA3 ELEMENTS = 688

NUMBER OF DESIGN VARIABLES = 129 (9 for uniform case)

RUN TERMINATED DUE TO HARD CONVERGENCE TO AN OPTIMUM AT CYCLE NUMBER = 21.

The 3-D display with thickness in Fig. 11b gives much better vision than in Fig. 11c.

By examination of NASTRAN f06 file, we find that only buckling load constraint is active in all buckling constraint levels. Also see Fig. 11d.

A little benefit can be observed for the larger spacing of stiffeners from Fig. 11e using variable cross-sections of stiffeners. The minimum weight under buckling load constraint level 10 seems much lower than the expected from other part of the results. It might be due to the hard convergence of the problem itself. Different initial values of design variables should be used for a stable result.

**Example 2**. ** Regular grid fashion
– smaller spacing of stiffeners**. Uniform via variable cross-section
of a stiffener is studied.

Planar dimensions and materials of a stiffened
panel are given in Table 3.** Layout (207)**
and edge locations of stiffeners' ends are given in **Table 6**. Initial sizes and bounds of cross-sectional dimensions are
given in Table
5.

The loading is chosen as the same as in Example 1. It is a longitudinal load.

The allowable minimum buckling loads considered are included in the following array

*λ*_{a} = [1.
10. 20. 30. 40. 50.];

The scale of the design problem is as follows:

NUMBER OF GRID POINTS = 425

NUMBER OF CBAR ELEMENTS = 80

NUMBER OF CTRIA3 ELEMENTS = 786

NUMBER OF DESIGN VARIABLES = 161 (11 for uniform case)

RUN TERMINATED DUE TO HARD CONVERGENCE TO AN OPTIMUM AT CYCLE NUMBER = 17.

The 3-D display with thickness in Fig. 12b gives much better vision than in Fig. 12c.

By examination of NASTRAN f06 file, we find that only buckling load constraint is active in all buckling constraint levels. Also see Fig. 12d.

A little more benefit can be observed for smaller spacing of stiffeners from Fig. 12e using variable cross-sections of stiffeners. The minimum weights under buckling load constraint level 40 and 50 seem much lower than the expected from other part of the results. It might be due to the hard convergence of the problem itself. Different initial values of design variables should be used for a stable result.

**Example 3**. ** Geodesic fashion – larger spacing of stiffeners**.
Uniform cross-section of a stiffener is considered.

Planar dimensions and materials of a stiffened
panel are given in Table 3.** Layout (212)**
and edge locations
of stiffeners' ends are given in **Table 7**.
Initial sizes and bounds of cross-sectional dimensions are given in Table 5.

The loading is chosen as the same as in Example 1. It is a longitudinal load.

The allowable minimum buckling loads considered are included in the following array

*λ*_{a} = [1.
10. 20. 30. 40. 50.];

The scale of the design problem is as follows:

NUMBER OF GRID POINTS = 379

NUMBER OF CBAR ELEMENTS = 68

NUMBER OF CTRIA3 ELEMENTS = 696

NUMBER OF DESIGN VARIABLES = 9

RUN TERMINATED DUE TO HARD CONVERGENCE TO AN OPTIMUM AT CYCLE NUMBER = 24.

The 3-D display with thickness in Fig. 13b gives much better vision than in Fig. 13c.

By examination of NASTRAN f06 file, we find that only buckling load constraint is active in all buckling constraint levels. Also see Fig. 13d.

The minimum weight under buckling load constraint level 10 seems much lower than the expected from other part of the results. It might be due to the hard convergence of the problem itself. Different initial values of design variables should be used for a stable result. See Fig. 13e.

A little higher weight can be observed for geodesic stiffeners than regular grid ones as in Fig. 11e of example 1, due to the longitudinal loading condition.

**Example 4**. ** Geodesic fashion –
smaller spacing of stiffeners**. Uniform cross-section of a
stiffener is considered.

Planar dimensions and materials of a stiffened
panel are given in Table 3.** Layout (211)**
and edge locations of stiffeners' ends are given in **Table 8**. Initial sizes and bounds of cross-sectional dimensions are
given in Table
5.

The loading is chosen as the same as in Example 1. It is a longitudinal load.

The allowable minimum buckling loads considered are included in the following array

*λ*_{a} = [1.
10. 20. 30. 40. 50.];

The scale of the design problem is as follows:

NUMBER OF GRID POINTS = 436

NUMBER OF CBAR ELEMENTS = 90

NUMBER OF CTRIA3 ELEMENTS = 806

NUMBER OF DESIGN VARIABLES = 13

RUN TERMINATED DUE TO HARD CONVERGENCE TO AN OPTIMUM AT CYCLE NUMBER = 33.

The 3-D display with thickness in Fig. 14b gives much better vision than in Fig. 14c.

By examination of NASTRAN f06 file, we find that only buckling load constraint is active in all buckling constraint levels. Also see Fig. 14d.

The minimum weights under different buckling load constraint levels seem much smoother than previous examples. See Fig. 14e. A little lower weight can be observed for smaller spacing of stiffeners than in Fig. 13e of example 3.

**Example 5**. ** Geodesic fashion – large spacing of curved stiffeners**. Uniform cross-section of a
stiffener is considered.

Planar dimensions and materials of a stiffened
panel are given in Table 3.** Layout
(2120)** and edge locations
of stiffeners' ends are given in **Table 9**
the same as (**Table 7)**. There is one middle control point for
each NURBS curve of the stiffeners’. The four middle points are generated
randomly. Initial sizes and bounds of cross-sectional dimensions are given in
Table 5.

The loading is chosen as the same as in Example 1. It is a longitudinal load.

The allowable minimum buckling loads considered are included in the following array

*λ*_{a} = [1.
10. 20. 30. 40. 50.];

The scale of the design problem is as follows:

NUMBER OF GRID POINTS = 379

NUMBER OF CBAR ELEMENTS = 68

NUMBER OF CTRIA3 ELEMENTS = 696

NUMBER OF DESIGN VARIABLES = 9

RUN TERMINATED DUE TO HARD CONVERGENCE TO AN OPTIMUM AT CYCLE NUMBER = 12.

The 3-D display with thickness in Fig. 15b gives much better vision than in Fig. 15c.

By examination of NASTRAN f06 file, we find that only buckling load constraint is active in all buckling constraint levels. Also see Fig. 15d.

The minimum weights under different buckling load constraint levels seem smooth. See Fig. 15e. A little higher weight can be observed for curved stiffeners than straight ones as in Fig. 13e of example 3, possibly due to the uniform load density along boundary edges.

We have ** initially** achieved the following at
this time:

· Successfully represent the planar and curvilinear stiffened panel using NURBS.

·
Finite element mesh generation is initially
investigated and a mesh generator __ DistMesh
- A Simple Mesh Generator in MATLAB__
is obtained and modified for triangulation meshing that can include arbitrarily
curved stiffeners.

· A MATLAB/NASTRAN interface, along with a thickness display package, is developed and still under development for the panel definitions, automatic mesh generation, NASTRAN data generation, NASTRAN invoking, and for reading and displaying response analysis and optimization results from NASTRAN output files.

·
**Sizing
optimization for a planar stiffened panel is tested for different given layouts (orientations and curvatures) of stiffeners
under longitudinal compression load.**

·
** p** – method
(via

Our current success in above is only initial one in every aspect, and subjected to testing and modification, and improvement so that it can be user-friendly, efficient, and robust.

Specifically, we will address the following in the next quarters.

· Further modify the DistMesh so that it can both generate fine mesh and be robust.

·
Further review literature about finite element
meshing, especially for quadrilateral elements for ** p**-methods.

·
Customize the thickness display package so that
is can be used for ** p**-methods and implanted to other applications.

·
Increase the capability of the MATLAB/NASTRAN
interface so that it can handle more elements (CQUAD, CBEAM, etc.), responses
(displacements, stresses, and design sensitivities) and user-friendly (easily
to be used by anyone rather than programmer himself). Especially, it needs to
be able to handle multiple cases, and read the results from any design cycles.
It will also be extended to ** p**-version elements.

· Testing examples will include different boundary conditions, and loading cases, as well as materials. Size optimization for curvilinear panel will be performed. Optimization considering natural frequency will also be tested.

· The placement/orientation optimization methods and implementation will be further investigated.

·
The ** p**-method, relevant meshing,
implementation environment will be further tested, and investigated.

· The necessity of creating a MATLAB/ANSYS interface will be studied.

· The necessity and possibility to implement the structural analysis and optimization uniquely in MATLAB without using other external software will be considered.

References

1. G.
Farin: “Curves and Surfaces for Computer Aided
Geometric Design: A Practical Guide”. 2^{nd} Ed. Academic Press,
Inc., 1990.

2. J. Li, R.K. Kapania,
W.B. Spillman, Jr.: “Placement Optimization of
Fiber Optic Sensors for a Smart Bed Using Genetic Algorithms”. *Paper
AIAA-2004-4334, 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization
Conference, August 30–-September 1, 2004/Albany, NY USA.*

3.
__http://www.aria.uklinux.net/nurbs.php3__

4.
__http://www-users.informatik.rwth-aachen.de/~roberts/software.html__

5.
MSC.NASTRAN 2004 Reference Manual

6. __http://www-math.mit.edu/~persson/mesh/__

7. N.W. Murray: “Introduction to the theory of thin-walled structures”. Clarendon Press – Oxford, 1984.

8. MSC.NASTRAN 2004** **Quick
Reference Guide

9. __http://www.aoe.vt.edu/~jing/EBF3/__

10. __http://www.mscsoftware.com/support/online_ex/Nastran/Nas101/AppendixH.pdf__