Structures with Shape and Topology Optimization
By Jennifer Krawec
Engineering Project Proposal
Advisor: Ernesto Gutierrez-Miravete
Abstract
In industry, structural optimization is a major concern when designing mechanical
systems. The aim of structural optimization is to determine the structural design variables
(geometry: shape and size) in order to minimize an objective function (cost or weight)
while satisfying given constraints. Structural optimization can be subdivided into
topology and shape optimization. Topology optimization consist of optimizing the
material layout within a given design space, and for a given set of loads and boundary
conditions, such that the resulting structure meets a prescribed set of performance targets.
While shape optimization is as the term indicates, optimizing the geometry of a specific
shape.
Background/Introduction
Traditionally, engineers performed structural optimization by trial and error. This is a
very costly and time consuming approach. Currently, the modern approach consists of
using numerical softwares that simultaneously anaylzes and optimizes the design. This
approach streamlines the optimization process by automatating the iterations, therefore
decreasing the time required by the operator and increasing the solution efficiency.
Finite element based topology, sizing and shape optimization tools are typical used as
part of a two phase design process. Firstly, a topology optimization is performed to obtain
a first view on an optimal configuration for the structure and initial design with optimal
load paths. Next, the suggested configuration is interpreted to form an engineering design
and this design is then optimized using detail sizing and shape optimization methods with
real design requirements.
There are several steps involved when solving a structural optimization problem. After
selecting the test case certain parameters of the specific test case can be define like the
volume, perimeter, boundary conditions and applied forces, etc. In parallel the
optimization code can be developed, which consist of defining the mesh, velocity field,
objective function, level-set function, density function, volume, perimeter, and etc. This
code will be executed several times until the optimal design is achieved.
Problem Description
This project will consist of optimizing a simply triangular arch with a point load in the
center of the arch. The boundary conditions will be fixed-fixed in the both the x and y
direction on either ends of the arch. The force applied will be in the negative direction
and have a value of 1,000lbs. This is shown in Figure 1 below:
F
Fx2
Fx1
Fy1
Fy2
Figure 1: Free Body Diagram of the Arch
If time permits a stress analysis will be performed using either COMSOL or ANSYS to
compare the non-optimized and optimized structures. Performing this analysis will
require material property inputs; E and v will be that of steel.
Methodology/Approach
The approach for this project is to use previously developed topology optimization code,
Scilab using methods developed by Gregorie Allaire. The below procedure will outline
the steps/methodology that were used to develop this topology optimization code.
1. Topology Optimization Section
1.1. Setup of problem

The structure is assumed to be composed of a linear elastic material and
possess Neuman and Dirichlet boundary conditions. Volume forces and
surface loads will be applied.

The displacement field in the linear elastic system will need to be
determined.

The objective functions need to be calculated, which is the total amount
of work done by the structure, also known as the compliance. The
compliance equation will also consist of geometrical contrainst, constant
volume and perimeter Lagrange multiplier.
1.2. Shape derivative

To calculate the objective function equation a gradient method is applied
using shape derivatives. Shape derivative measure the sensitivity of
boundary disturbance.
1.3. Shape representation using level-set method

The level-set function needs to be defined. The level-set function tracks
the boundary, interfaces, and the shape of the structure. Therefore, the
structures shape and curvature can be calculated.
1.4. Optimization algorithm

Initiate the level-set function

Iterate until convergence of optimal structure

Reinitialize the level-set function
2. Implementation using Scilab
2.1. 3 Files

Functions.sci is the first code that will be executed, which encompasses all
of the functions used in the main program

The second file includes all the parameters need for optimizing my
structure.

The last code to run is the main routine code, optalg.sce. This is the actual
optimization algorithm that requires the above 2 files in order to determine
the optimal structure
2.2. Parameter and settings

Define working domain and resolution

Define how many and size of holes

Set Lagrange multipliers

Define the setting of the problem, the forces acting on structure, fixed
portions (boundary conditions) of the structure, and passive parts (volume
and perimeter) of the working domain
2.3. Initialization and re-initialization

Guess an optimal topology of final structure – initialization

In order to prevent the level-set function from becoming too steep or too
flat, at every five iterations re-initialize of the mesh is required. This is
performed while solving the convection equation
2.4. Finite element analysis

Use a finite element routine for rectangular mesh in order to solve for the
velocity field
2.5. Use compliance to solve convection equation
2.6. Checking objective function

If objective function decreased move onto next iteration

If objective function didn’t decrease, solve level-set function with fewer
iterations
The second part of this approach is to develop finite element models of my nonoptimized and optimized structure and perform a stress analysis. Either COMSOL or
ANSYS will be used to perform this comparison. This analysis will be based on the
geometry constraints, applied forces, boundary conditions, and mechanical properties of
steel.
Resources Required
Scilab (similar to Matlab)
COMSOL/ASYS (potentially)
Expected Outcomes
The outcome of this project is to become proficient in the process of structural
optimization. This includes understanding the elements of the optimization codes and
modifying the codes to optimize my specific structure. If time allows I will utilize either
COMSOL or ANSYS to perform my stress analysis comparison.
Milestone/Deadline List
Task
Date
Submit proposal
9/30/2010
Download Scilab
10/02/2010
Get access to COMSOL/ANSYS
10/07/2010
Understand current Scilab codes
10/12/2010
Submit first progress report
10/21/2010
Start to adapt current Scilab code to my project
11/4/2010
Submit second progress report
11/11/2010
Run Scilab code
11/18/2010
Run stress comparison between optimized and non-optimized shapes
11/29/2010
Submit final draft
12/2/2010
Submit final report and presentation
12/16/2010
References

http://www.iitk.ac.in/directions/directsept04/Iyenger~new.pdf

http://www.cmap.polytechnique.fr/~jouve/papers/cras.pdf

http://www-math.mit.edu/phase2/UJM/vol1/CPHILL~1.PDF

http://www.cmap.polytechnique.fr/~allaire/levelset_en.html

ftp://ftp.math.ucla.edu/pub/camreport/cam06-44.pdf

Allaire’s website