Shikui Chen, Wei Chen, Sanghoon Lee

[Steve Yenisch]

Paper: www.mae.ufl.edu/nkim/ISSMO/ShikuiChenPaper.pdf
Presentation: www.mae.ufl.edu/nkim/ISSMO/ShikuiChenPresentation.pdf

Robust Shape and Topology Optimization of Compliant Mechanisms Considering Random Field Uncertainty
Shikui Chen, Wei Chen and Sanghoon Lee

Department of Mechanical Engineering, Northwestern University, Evanston, IL, USA, [e-mail]

ABSTRACT
In this paper, we attempt to address the cutting-edge problem of robust shape and topology optimization (RSTO) of compliant mechanisms with consideration of random field uncertainty, such as material property. The proposed approach is based on the state-of-the art level set methods for shape and topology optimization and the latest research development in design under uncertainty. Conventional robust design, usually posed as a continuous optimization problem in finite dimensions, is extended to an infinite-dimensional shape and topology optimization problem and uncertainty is considered as a new dimension in addition to space and time. We illustrate that a level-set-based RSTO problem can be mathematically formulated by expressing the statistical moments of a response as functionals of geometric shapes and random field. To characterize the high-dimensional random-field material uncertainty with a reduced set of random variables, the Karhunen-Loeve expansion is employed, which is essentially a spectral representation of the random field using a reduced set of random variables and the eigenfunctions of its covariance function as expansion bases. Once the reduced set of random variables is identified, the univariate dimension-reduction (UDR) quadrature rule is employed for calculating statistical moments of the design response. The combination of the above techniques not only greatly reduces the computational cost in evaluating the statistical moments but also enables a semi-analytical approach that introduces the shape sensitivity of the statistical moments using shape sensitivity analysis. The application of our approach to compliant mechanism design shows that the proposed RSTO method can lead to designs with completely different topologies and superior robustness compared to their deterministic counterparts. Although the current contents of this paper are focused on Gauss-type material uncertainties, the proposed method is generic and can be easily extended to robust topology optimization subject to other types of uncertainties, such as Gauss/Non-Gauss type loading and geometric uncertainties.

[Nam Ho Kim]

Dear Shikui:

Thank you for the interesting paper. At the last iten of the conclusion slide, you mentioned that the interaction between the boundary condition and uncertainty is important. Could you elaborate further how this interaction affects the optimum design?
Thanks,

[Shikui Chen]

Dear Professor Kim,

Thanks for your interests in our work. In my research process, I observed that the material uncertainty is not the only factor that impacts the final design. The final configuration of the robust design is a coeffect of the the boundary conditions and the uncertainty. For example, in a 2D bridge beam design problem the deterministic design and the robust design are quite different. But when it comes to a cantilever beam design problem where all the settings are the same as the cantilever beam problem except the boundary condition, the deterministic and robust designs are very similar.
This conclusion is based on my observation. I have not digged deep into the problem to make a quantitative study. I hope this can answer your question.

Thanks!
Shikui

[Saad Mukras]

Shikui
I enjoyed you paper and presentation which were well detailed. I have a question about the stress constraints. Could you please comment on how the performance and the robustness will be affected (in comparison to the deterministic design) if you included the stress constraints. Thanks

[Shikui Chen]

Saad,

A very good question. Stress constraint is indeed an important issue in CM design. But in the current contents of the paper, the focus is to study the impact of random field material uncertainties on the topology optimization result. We employ the conventional spring model in this work, where the stress is controlled by constraining the input displacement. The "robust design" here refers to a design with a more stable geometric advantage which is insensitive to the material field variations. But in the current framework, we can take stress into account by treating it as a global constraint. Hope this answers your question.

Thanks!

Shikui

[Shikui Chen]

May 21, 2009 10:47:45 GMT -5 @Shikui Chen said:

Dear Professor Kim,

Thanks for your interests in our work. In my research process, I observed that the material uncertainty is not the only factor that impacts the final design. The final configuration of the robust design is a coeffect of the the boundary conditions and the uncertainty. For example, in a 2D bridge beam design problem the deterministic design and the robust design are quite different. But when it comes to a cantilever beam design problem where all the settings are the same as the cantilever beam problem except the boundary condition, the deterministic and robust designs are very similar.
This conclusion is based on my observation. I have not digged deep into the problem to make a quantitative study. I hope this can answer your question.

Thanks!
Shikui

Dear Professor Kim,

Thanks for your interests in our work. In my research process, I
observed that the material uncertainty is not the only factor that impacts
the final design. The final configuration of the robust design is a coeffect

of the the boundary conditions and the uncertainty. For example, in a 2D
bridge beam design problem the deterministic design and the robust design
are quite different (Please see our DETC paper www.mae.ufl.edu/nkim/ISSMO/Level_Set_Based_RSTO_final.pdf ). But

when it comes to a cantilever beam design problem where all the settings are

the same as the bridge beam problem except the boundary condition, the
deterministic and robust designs are very similar. This conclusion is based
on my observation. I have not digged deep into the problem to make a
quantitative study. I hope this can answer your question.

If you have any question, please let me know.

Thanks!
Shikui

[Kogiso]

I am also interested in your conclusion statement, "The interaction between the boundary condition and the uncertainties determines the topology of the final design".
Do you consider that the uncertainty of the boundary condition can be modeled by the spectral approach ?

My second question:
How long is the computational time for 3-D example ?

[Shikui Chen]

May 22, 2009 2:24:25 GMT -5 @Kogiso said:

I am also interested in your conclusion statement, "The interaction between the boundary condition and the uncertainties determines the topology of the final design".
Do you consider that the uncertainty of the boundary condition can be modeled by the spectral approach ?

My second question:
How long is the computational time for 3-D example ?

Kogiso,
Thanks for your questions!
Let me answer your first question first. In this work, the spectral method is employed to quantify the material uncertainty in a random field form. I am not clear if the boundary condition uncertainties shares the same essence as the material field or not. So I can not simply say YES or NO to your question. But with my intuition, they are not the same problem.
As to your second problem, the computing time depends on the number of finite elements used in analysis and the number of integration points used in statistic moments calculation. The deterministic CM design in this paper cost about 20 hours. The robust design took about one and half days.

Shikui

[Kogiso]

Dear Shikui Chen:

Thank you for your reply.
> with my intuition, they are not the same problem.
I think so, too.

Computational time about 20 hours for deterministic and one and half day for robust design is within my estimation.