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It is of great importance for the development of new products to find the best possible topology or layout for given design objectives and constraints at a very early stage of the design process (the conceptual and project definition phase). The topology optimization result using material distribution method is a density distribution of the finite elements in the design domain. It is used in an increasing rate by for example the car, machine and aerospace industries as well as in materials, mechanism and Micro Electro Mechanical Systems (MEMS) design.


More and more, the human being is increasingly aware of the necessity of saving natural

resources. This fact is the main motivation for researching optimum designs.

Structural engineers also adopted this trend in the design of new structures or modification of existent ones. In this context, a structure can be considered as an amount of distributed material over a design domain, in order to support loads (static or dynamic), absorb and distribute energy and transmit it to the supports. One of the goals of optimum design is best distribute the available material into the design domain. This problem is solved by the most universal kind of structural optimization, i.e., optimizing the topology of a structure.

Topology optimization is also referred as layout optimization or in general term as Shape optimization. Structural topology optimization has been becoming an interesting area of research in the structural optimization community. The importance of this type of optimization lies in the fact that the choice of the appropriate topology of a structure in the conceptual phase is generally the most decisive factor for the efficiency of a novel product.

2. Topology Optimization

The topology of a structure is defined as a spatial arrangement of structural members and joints or internal boundaries. Consequently, topology optimization means varying the connectivity between structural members of discrete structures or between domains of continuum structures, as can be seen in Fig. 1.1.

Figure 1.1. Variation of topology.

Figure 1.2. Conceptual Process


2.1 Classification Of Topology Optimization

Two classes of approaches, the so-called material or Micro-approaches, and the geometrical or Macro-approaches are available. There are essential conceptual differences between these two types of approaches.

Based on this topology, subsequent shape optimization is usually carried out such as to yield a design that is optimal with regard to both topology and shape.

It is of three types:-

2.11 Homogenization Method

2.12 Evolutionary structural optimization (ESO)

2.13 Density- Distribution approach

3. Structural Optimization

3.1 Basic concepts and definitions

Structural optimization aims to increase the structural performance of components and mechanical systems in a systematic way. Thus, firstly we need to identify which design variables best describes the features of some component. Then, by modifying these variables, following some criteria, we obtain the best solution, among a set of solutions.

Design variables for a typical structural optimization problem can be the elements size, structural configuration, mechanical or physical properties of materials, or other qualitative aspects for the project being analyzed. Cost function, also known as objective function is the scalar function to be minimized (or maximized) during optimization process. Constraints are conditions imposed to the physical problem, representing the limit of the admissible space.


The topology optimization result using material distribution is a “density distribution” of the finite elements in the design domain. The objective function is to minimize the compliance of the structure, or maximize the stiffness of the structure. In the literature many examples are available, given here are some of the examples based on minimum compliance.

The topology optimization has been done using the standard Matlab topology optimization code.

The main program is called from the Matlab prompt by the line

top(nelx, nely, volfrac, penal, rmin)


nelx and nely = the number of elements in the horizontal and vertical directions, respectively,

volfrac = the volume fraction,

penal = the penalization power and

rmin = the filter size (divided by element size).

Other variables as well as boundary conditions are defined in the Matlab code itself and can be edited if needed. For each iteration in the topology optimization loop, the code generates a picture of the current density distribution. In all the examples of static problems a square bilinear 4-node element has been assumed.

Example: Simply Supported beam.

A popular example of the minimum compliance is the MBB beam. Here is is the design domain and F is the unit force applied on the beam.

Figure 4.1 Design domain of the beam

The default boundary conditions correspond to half of the “MBB-beam” (Figure 4.1).

The load is applied vertically in the upper left corner and there is symmetric boundary conditions along the left edge and the structure is supported horizontally in the lower right corner. The material has Young’s modulus, E = 1N/m2 and poisson’s ratio (?) = 0.3. Figure 4.2 shows the resulting density distribution obtained by the code called with the input line


i.e., nelx = 60

nely = 20

volume fraction = 0.5

penalization = 3

rmin = 1.5.


The topology optimization results using the density distribution approach incorporated with computer program gives the optimal shape of the structure, giving a clear position of material (black) and void (white) in the final figure. It saves both time and money.

The following conclusion is derived from the present investigation of static and dynamic topology optimization problems:

· Solving dynamic problems in topology optimization is far from a trivial exercise. Numerical challenges such as

(i) localized eigenmodes in areas of low density, associated with low eigenvalue; and

(ii) different material interpolation schemes result in different optimal topologies.

· These problems are not encountered in static problems, though many, if not all, topology optimization problems suffer from numerical problems such as checkerboarding; one-node connected hinges, mesh dependency and local minima.

· The problem of localized eigenmodes is overcome successfully using the material interpolation scheme.

· The optirnality criteria appears to be more prone to this problem and further work is required to resolve this problem.

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