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# Doing Topology Optimization Explicitly and Geometrically - A New Moving Morphable Components Based Framework

Structural topology optimization, which aims at placing available material within a prescribed design domain appropriately in order to achieve optimized structural performances, has received considerable research attention since the pioneering work of Bendsoe and Kikuchi. Many approaches have been proposed for structural topology optimization and it now has been extended to a wide range of physical disciplines such as acoustics, electromagnetics and optics.

Most of the topology optimization approaches such as element-based SIMP method and node-based level set method are geometrically implicit in nature and therefore make them difficult to embed geometrical features into topological designs. This is quite inconsistent with the current CAD environment where geometry objects are represented explicitly. Besides, since the design model and the finite element analysis model are tightly coupled in these methods, huge number of design variables and significant computational efforts are involved when designs with high resolutions are required. This constitutes a big bottleneck to enhance the efficiency of topology optimization especially for three-dimensional and mutli-physics problems.

In the present work, we intend to demonstrate ** how to do topology optimization in an explicit and geometrical way**. To this end, a new computational framework for structural topology optimization based on the concept of moving morphable components is proposed. Compared with the traditional pixel or node point-based solution framework, the proposed solution paradigm can incorporate more geometry and feature information into topology optimization directly and therefore render the solution process more flexibility. It also has the great potential to reduce the computational burden associated with topology optimization substantially.

This work has been accepted by ASME Journal of Applied Mechanics (*J.App.Mech.*, 2014: doi: 10.1115/1.4027609) and

and has been published online.

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## Comments

## Very nice work

Dear Prof. Guo,

It's a very nice work. I think this method can be viewed as an appraoch to do the optimization of a discrete truss (or frame) like structure with a continuum modelling. Of course, this method can provide much more information than the pure truss structure optimization and significantly reduce the optimization freedom compared with conventional topological methods. I have a few questions about this method:

1. Is there theoretical proof of the convergence of current method? Since the current method reduces the optimization parameters, the result of which may be viewed as a subset of those based on the conventional method, will the optimization finally stop at some local minimal configuration (i.e. not the best results can be obtained with the conventional approaches)?

2. In the current method, the structure topology can be changed by hiding some of the components. This indicates the total material volume will be reduced too, am I right? My guess is that the optimum design for structure stiffness should use as much as possible materials.

3. Related to 2, I have another question that how to define the thickness for the overlapped region (joint or fully overlapped part) ?

Best,

Teng

## Dear Teng, the following are my thoughts on your comments

Dear Teng,

Thank you very much for your kind words and comments. The following are some of my thoughts:

(1) I think the main contribution of this work is to provide a way to do topology optimization in an explicit and geometrical way, which cannot

be achieved easily in the current toology optimization framework. As mentioned in the manuscript, although superellipse curve is used

in the present work to represent the shape of a component explicitly, in fact, well-established NURBS and iso-parameteric modeling

techniques can also be employed to descibe the geometry of the component. In this sense, our approach establish a link between the

conventional shape optimization and topology optimization, which is currently under intensive investigation. Within the proposed

computational framework, in principle all mehthods developed for shape optimization can be employed topology optimization problems.

Futhermore, since our approach is geometrically explicit in nature, it provides a natural application yard for modern FEM analysis techniques,

such as Isogeometry Analysis. We intend to report the corresponding results in subsequent work.

(2) As for the issue of local minimum, first I would like to point out that since the objective functionals in topology optimization are usually not

lower-semi-continuous with respoct to the weak* topology of L^(infinity) sapce, usually regularization treatments are required to guarantee the

well-posedness. Under this circumstance, the topology of a optimal design is always not too complicated. As long as the final topology does

can be represeneted by the initial set of components, our approach can guarantee to include the global optimum in the design space and

therefore has the opportunity to find it. Since topology optimization problem is non-convex in nature, usually global optimum cannnot be

expected for both conventional and the proposed numerical solution approaches. As shown in our paper, standard benchmark examples,

our appoach can obtain the same results as those obtianed by conventional methods. Futhermore, since our optimization problem

is a finite dimensional one, convergence to local optimum can be guaranteed if appropriate optimization algorithms (e.g., SQP, SNOPT) are

adopted.

(3) You are right the optimum design for structural stiffness should use all available materials. Actually, in all our test examples, the volume

contsraints are active for optimal sloutions. The total material volume will not be reduced through “hiding” or "overlapping" of components

since the volume of "useful" components will expand smartly during the course of optimization.

(4) In the proposed framework, the overlapping region between two components can be easily identified by an "AND" operation on their respective

level set functions (i.e.,

xin the overlapping regioniffmin (\Phi_1(x), Phi_2(x)). Once the overlapping region is identified, we can constructa inscribed circle of this region to estimate its "thickness".

Best regards，

Xu Guo