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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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Elastic properties of blood veins with a scaffold
Frost, Miroslav ; Maršík, František (advisor) ; Kružík, Martin (referee)
Presented master's thesis deals with modeling of a NiTiNOL wire under thermal and uniaxial mechanical loading. NiTiNOL can undergo reversible martensitic phase transformation and thus belongs among shape memory alloys. In the form of a thin wire it is used in many applications (e.g. as a reinforcement for veins). MT is studied with respect to the extended non-equilibrium thermomechanics of mixtures and the Clusius-Clapeyron equation is derived for it. A new phenomenological model iRLOOP, developed at AS CR, simulating thermomechanical behavior of a NiTiNOL wire is mathematically formulated. Restrictions on tting functions in proposed hysteresis mechanism are derived from the second law of thermodynamics. The existence and uniqueness of the solution of an initial problem are proven for the superelasticity model. Experiments are compared with results modeled by numerical implementation of iRLOOP.
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Modeling of phase transformations in shape memory materials
Frost, Miroslav ; Maršík, František (advisor) ; Rohan, Eduard (referee) ; Seiner, Hanuš (referee)
Title: Modeling of phase transformations in shape memory materials Author: Miroslav Frost Department: Mathematical Institute of Charles University Supervisor: Prof. Ing. František Maršík, DrSc., Mathematical Institute of Charles University Abstract: This thesis presents a new thermomechanical three-dimensional con- stitutive model of NiTi-based shape memory alloys. The model was formulated within the framework of generalised standard models and it features a novel form of the dissipation function, which combines contributions stemming from the phase transformation between austenite and martensite and from the reorienta- tion of martensite. The change in the material response associated with the phase transformation between austenite and R-phase as well as material anisotropy and tension-compression asymmetry are also covered. The time-evolutionary problem of a quasistatic mechanical loading of a NiTi body with prescribed temperature evolution was formulated and analyzed within the framework of energetic so- lutions. The corresponding time-incremental minimization problem provided a conceptual algorithm utilized in the numerical treatment. The constitutive mod- el was implemented into the finite element package Abaqus. Several numerical simulations were performed and compared with experiments. Keywords:...
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Elastic properties of blood veins with a scaffold
Frost, Miroslav ; Kružík, Martin (referee) ; Maršík, František (advisor)
Presented master's thesis deals with modeling of a NiTiNOL wire under thermal and uniaxial mechanical loading. NiTiNOL can undergo reversible martensitic phase transformation and thus belongs among shape memory alloys. In the form of a thin wire it is used in many applications (e.g. as a reinforcement for veins). MT is studied with respect to the extended non-equilibrium thermomechanics of mixtures and the Clusius-Clapeyron equation is derived for it. A new phenomenological model iRLOOP, developed at AS CR, simulating thermomechanical behavior of a NiTiNOL wire is mathematically formulated. Restrictions on tting functions in proposed hysteresis mechanism are derived from the second law of thermodynamics. The existence and uniqueness of the solution of an initial problem are proven for the superelasticity model. Experiments are compared with results modeled by numerical implementation of iRLOOP.
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