Spectral collocation methods in solid mechanics
| Thesis title in Czech: | Spektrální metody v úlohách mechaniky pevných látek |
|---|---|
| Thesis title in English: | Spectral collocation methods in solid mechanics |
| Key words: | spektrální metody|Čebyševovy body|resampling|linearizovaná elasticita |
| English key words: | spectral collocation method|Chebyshev points|resampling|linearised elasticity |
| Academic year of topic announcement: | 2021/2022 |
| Thesis type: | Bachelor's thesis |
| Thesis language: | angličtina |
| Department: | Mathematical Institute of Charles University (32-MUUK) |
| Supervisor: | doc. Mgr. Vít Průša, Ph.D. |
| Author: | hidden - assigned and confirmed by the Study Dept. |
| Date of registration: | 11.11.2021 |
| Date of assignment: | 11.11.2021 |
| Confirmed by Study dept. on: | 02.12.2021 |
| Date and time of defence: | 16.06.2022 09:00 |
| Date of electronic submission: | 09.05.2022 |
| Date of submission of printed version: | 09.05.2022 |
| Date of proceeded defence: | 16.06.2022 |
| Opponents: | prof. doc. Ing. Jan Zeman, Ph.D. |
| Advisors: | RNDr. Karel Tůma, Ph.D. |
| Guidelines |
| The thesis objective is to investigate the applicability of spectral collocation methods in numerical solution of boundary value problems in solid mechanics. The thesis will start with the solution of simple classical problems in linearised elasticity theory that have an analytical solution, such as the semi-infinite elastic medium subject to a surface load. This will subsequently allow one to easily asses the performance of the given numerical method via the direct comparison of the numerical solution with the exact analytical solution. Once the methodology is tested in the linearised setting, the thesis can proceed either with the focus on subtleties of the numerical method (Fourier transformed version of the spectral collocation method, preconditioning of spectral differentiation matrices) or with the treatment of more advanced physical models (phase field models for the response of shape memory alloys). Numerical methods will be implemented in Matlab. |
| References |
| Aurentz, J. L. and L. N. Trefethen (2017). Block operators and spectral discretizations. SIAM Review 59 (2), 423–446.
Brubeck, P. D. and L. N. Trefethen (2021). Lightning Stokes solver. Miller, K. S. (1981). On the inverse of the sum of matrices. Mathematics Magazine 54 (2), 67–72. Rezaee-Hajidehi, M. and S. Stupkiewicz (2020). Phase-field modeling of multivariant martensitic microstructures and size effects in nano-indentation. Mech. Mater. 141, 103267. Saada, A. S. (1974). Elasticity: theory and applications. New York: Pergamon Press Inc. Pergamon Unified Engineering Series, Vol. 16. Sneddon, I. N. (1950). Fourier transforms. New York: McGraw–Hill. Trefethen, L. N. (2000). Spectral methods in MATLAB, Volume 10 of Software, Environments, and Tools. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). Tůma, K., M. Rezaee-Hajidehi, J. Hron, P. Farrell, and S. Stupkiewicz (2021). Phase-field modeling of multivariant martensitic transformation at finite-strain: computational aspects and large-scale finite-element simulations. Comput. Methods Appl. Mech. Eng. 377, 113705. Weideman, J. A. and S. C. Reddy (2000). A MATLAB differentiation matrix suite. ACM Trans. Math. Softw. 26 (4), 465–519. |