Rate-type models for metamaterials

Abstract

This thesis investigates various discretization methods for the second derivative operator under different boundary conditions. The compared methods include classical Finite Differences, Numerov's method, Discrete Fourier Transform (DFT), Discrete Sine Transform (DST), and several Chebyshev-based spectral approaches. Performance is evaluated on two Sturm-Liouville eigenvalue problems (Paine and Coffey-Evans equations) and on wave propagation simulations, including a rate-type constitutive model of a mass-in-mass metamaterial. Special attention is given to the limiting behaviour of the metamaterial model as the internal mass parameter approaches zero. The numerical results demonstrate that spectral methods based on DFT and DST consistently provide the highest accuracy and efficiency especially regarding high-index eigenvalues computation and subsequently wave propagation computation. 1

Tato diplomova ́ pra ́ce zkouma ́ ru ̊zne ́ metody diskretizace opera ́toru druhe ́ derivace pr ̌i periodicky ́ch a Dirichletovy ́ch okrajovy ́ch podmı ́nka ́ch. Srovna ́vane ́ metody jsou klasicke ́ konec ̌ne ́ diference, Numerova metoda, diskre ́tnı ́Fourierova transformace (DFT), diskre ́tnı ́ transformace pomocı ́ sinu ̊ (DST) a ne ̌kolik spektra ́lnı ́ch metod zaloz ̌eny ́ch na C ̌ebys ̌evovy ́ch polynomech. Pr ̌esnost metod je vyhodnocena na dvou Sturm-Liouvilleovy ́ch proble ́mech vlastnı ́ch c ̌ı 'sel (Paineova a Coffey-Evansova rovnice) a na simulacı ́ch s ̌ı ́r ̌enı ́ vln, zahrnujı ́cı ́ model rychlostnı ́ho typu pro jednodimenziona ́lnı ́ metamateria ́l. Pozornost je take ́ ve ̌nova ́na limitnı ́mu chova ́nı ́ tohoto metamateria ́love ́ho modelu v pr ̌ı ́pade ̌, kdy se parametr vnitr ̌nı ́ hmotnosti blı ́z ̌ı ́ k nule. Numericke ́ vy 'sledky ukazujı ́, z ̌e spektra ́lnı ́metody zaloz ̌ene ́ na DFT a DST poskytujı ́konzistentne ̌ nejvys ̌s ̌ı ́pr ̌esnost, a to zejme ́na pr ̌i vy ́poc ̌tu vlastnı ́ch c ̌ı 'sel s vysoky ́m indexem a take ́ pr ̌i simulacı ́ch s ̌ı ́r ̌enı ́ vln. 1