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Numerical minimization of energy functionals in continuum mechanics using hp-FEM in MATLAB
Moskovka, Alexej ; Frost, Miroslav ; Valdman, Jan
Many processes in mechanics and thermodynamics can be formulated as a minimization of a particular energy functional. The finite element method can be used for an approximation of such functionals in a finite-dimensional subspace. Consequently, the numerical minimization methods (such as quasi-Newton and trust region) can be used to find a minimum of the functional. Vectorization techniques used for the evaluation of the energy together with the assembly of discrete energy gradient and Hessian sparsity are crucial for evaluation times. A particular model simulating the deformation of a Neo-Hookean solid body is solved in this contribution by minimizing the corresponding energy functional. We implement both P1 and rectangular hp-finite elements and compare their efficiency with respect to degrees of freedom and evaluation times.
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Numerical implementation of incremental minimization principle for materials with multiple rate-independent dissipative mechanisms
Frost, Miroslav ; Moskovka, Alexej ; Sedlák, Petr ; Valdman, Jan
The incremental energy minimization approach is a compact variational formulation of the evolutionary boundary value problem for constitutive models of materials with a rate-independent response. Although it can be easily applied to many conventional models, its main advantages arise when applied to models with multiple strongly coupled dissipation mechanisms, where the direct construction of the coupled yield conditions and flow rules may be challenging. However, this usually requires a more complex numerical treatment of the resulting sequence of time-incremental boundary value problems resolved via the finite element method. This contribution presents, compares and discusses two genuine minimization approaches - the staggered solution procedure relying on alternating minimization and the monolithic approach employing global minimization - for an advanced constitutive model of shape memory alloys.
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Nerovnosti Friedrichsova a Poincarého typu a jejich výpočet
MOSKOVKA, Alexej
Tato bakalářská práce se zabývá teorií Friedrichsových a Poincarého nerovností a konstant v nich vystupujících, které mají široké využití v matematické analýze, zejména ve funkcionální analýze a v teorii parciálních diferenciálních rovnic. Základní myšlenkou nerovností je omezení L-norem funkcí vzhledem k L-normám jejich gradientů. Výpočet konstant se dá provést analyticky pro jednoduché geometrické oblasti nebo přibližně numericky. Zde detailně provádíme explicitní výpočet konstant pro interval, obdélník a kvádr. Dále realizujeme numerický výpočet pro interval, obdélník a také pro mezikruží, pro které přesnou hodnotu konstant neznáme.
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