Liquid crystal elastomers (LCEs) are rubbers that are comprised of weakly cross-linked liquid crystal polymers with orientationally ordered side-chain and main-chain mesogenic rods. The remarkable property of LCEs is the coupling between orientation order and mechanical deformation, which makes these rubbers very sensitive to external stimuli, such as illumination and other applied fields, leading to large and fast shape deformations. A great deal of the experimental and theoretical results on LCEs has been obtained during the last decade. However, a full characterization of LCEs still remains elusive, especially for the dynamic responses of LCEs. Recently, the investigator and his collaborators proposed a non-local continuum model to understand the dynamics of nematic LCEs. The simulation of the model demonstrated that the proposed model can successfully capture shape changing phenomena and some other features of LCEs that were observed from real experiments. The model thus provides a solid basis for further exploration of the dynamics of LCEs. However, due to the intrinsic complexity of the physical processes underlying the dramatic responses of LCEs, the numerical treatment of the model is very challenging. In this project, the investigator focuses on developing efficient and reliable numerical methods for solving the derived equations originating from the proposed model on nematic LCEs. The success of this project will provide an important tool to enhance the understanding of LCEs. Further, the deep understanding of the dynamic responses of LCEs is crucial to their technological applications including sensors, actuators, deformable adaptive optical elements, micro-fluidic pumps, etc.
Liquid crystal elastomers (LCEs) are soft complex materials. The salient feature of LCEs is that relatively small external effects, such as changes in temperature or onset of illumination, can result in large and fast shape deformations. Due to this remarkable property, LCEs have the potential for real technological applications including sensors, actuators, deformable adaptive optical elements, micro-fluidic pumps, etc. To fully exploit these materials, the investigator and his collaborators have already proposed a mathematical model that can successfully capture many dynamic features of LCEs, such as shape changing phenomena. However, due to the intrinsic complexity of the physical processes underlying the dramatic responses of LCEs, both theoretical study and simulation of the proposed model are very challenging. In this project, the investigator seeks to develop efficient and reliable numerical methods for the simulation. The success of the research will provide a powerful tool to improve the understanding of LCEs, and the deep understanding is essential to the real technological applications of these materials. Moreover, the methods developed in the research will be useful for the modeling of other related complex soft matter systems, and will therefore have a lasting value in the computational materials science community.
for NSF DMS-1016504 Wei Zhu University of Alabama, Tuscaloosa During the last four academic years, supported by the NSF grant DMS-1016504, and with my collaborators, I have conducted research in two different fields, with one for the simulation of liquid crystal elastomers (LCEs) and the other for the modeling and the development of numerical methods for problems in mathematical imaging. In this period, seven papers on image processing have been accepted (six of them have been published), one paper on the simulation of LCEs has been submitted and is under the third round of review, and another paper on the study of the preconditioner developed in the simulation of LCEs is in preparation. The detailed information of these papers is listed as follows: 1. Wei Zhu, On the study of a novel preconditioner for Chebyshev spectral collocation method, in preparation. 2. Wei Zhu, Simulation on Liquid crystal elastomers using spectral method with a new preconditioner, under the third round of review (submitted on May 2013). 3. Sung Ha Kang, Wei Zhu, and Jackie Shen, Illusory shapes via corner fusion, Accepted by SIAM J. Imaging Sci., May 2014. 4. Wei Zhu, Xue-Cheng Tai, and Tony F. Chan, A fast algorithm for a mean curvature based image denoising using augmented Lagrangian method, A. Bruhn et al. (Eds.): Global Optimization Methods, LNCS 8293, pp. 104-118, 2014. 5. Wei Zhu, Xue-Cheng Tai, and Tony F. Chan, Image segmentation using Euler's elastica as the regularization, Journal of Scientific Computing, 57, pp. 414-438, 2013. 6. Andy M. Yip and Wei Zhu, A fast modified Newton method for curvature based denoising of 1D signals, Inverse Problems and Imaging, 7(3), pp. 1075-1097, 2013. 7. Wei Zhu, Xue-Cheng Tai, and Tony F. Chan, Augmented Lagrangian method for a mean curvature based image denoising model, Inverse Problems and Imaging, 7(4), pp. 1409-1432, 2013. 8. Wei Zhu, Sung Ha Kang, and George Biros, A geodesic-active-contour-based variational model for short axis cardiac-MR image segmentation, International Journal of Computer Mathematics, 90(1), pp. 124-139, 2013. 9. Wei Zhu and Tony Chan, Image denoising using mean curvature of image surface, SIAM J. Imaging Sci., 5, pp. 1-32, 2012. Brief summary: In [2], we proposed a novel preconditioner for spectral collocation method, which has proved to be very efficient in the simulation of LCEs and thus provides a powerful tool for further exploration of the dynamic behaviors of liquid crystal elastomers (LCEs), an active material that has the potential to assume great technological applications. Moreover, this novel preconditioner can also be applied for solving a wide range of elliptic equations. In [1], its efficiency has been tested and compared with the well-known Orszag’s finite difference based preconditioning, and the results demonstrate that for some elliptic equations, the new preconditioner is more efficient. For those works in image processing [3~9], with my collaborators, I developed a novel variational model for image denoising. This model is able to achieve some new goals in the denoising process that cannot be obtained by those existing denoising models, such as the preservation of corners and image contrasts. We also developed efficient numerical methods for the associated optimization problem. The model has received lots of attention in the community and its regularizer has been utilized to accomplish other imaging tasks.
