1,730 research outputs found

    Kollokaatiomenetelmä stokastiselle elastisuustehtävälle epävarmassa alueessa

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    In this thesis, we formulate a method for determining how quantities such as stress in an elastic body change depending on its shape. This stochastic elasticity problem has important applications in structural analysis and design, such as determining how manufacturing flaws affect the properties of an object. We assume that the shape of the object depends on some stochastic parameters, and use a combination of multivariate interpolation and conformal mappings to solve the problem. The interpolation allows us to reduce the stochastic problem to a collection of deterministic elasticity problems, which are solved by using existing finite element analysis software, and the conformal mappings are used to accommodate the varying shape of the object. A sparse grid interpolation scheme is used to diminish the curse of dimensionality related to multivariate interpolation. We define model problems involving two stochastic parameters, for both 2D and 3D objects. The implementation of the method is described in detail, and numerical results are provided for the model problems. With as few as 29 deterministic problems, we reach the point where the interpolation accuracy cannot be improved due to the inherent inaccuracy of the finite element solutions.Tässä diplomityössä esittelemme menetelmän, jolla voidaan määrittää kuinka jännityksen tai jonkin muun suureen arvo muuttuu elastisessa kappaleessa kun sen muoto vaihtelee. Tällä stokastisella elastisuustehtävällä on tärkeitä sovelluksia rakenteiden analyysissä ja suunnittelussa; sitä käyttäen voidaan esimerkiksi määrittää valmistusvirheiden vaikutus kappaleen ominaisuuksiin. Oletamme kappaleen muodon riippuvan joistakin stokastisista parametreista, ja käytämme usean muuttujan interpolointia sekä konformikuvauksia tehtävän ratkaisemiseksi. Interpoloinnin avulla stokastinen ongelma voidaan muuntaa kokoelmaksi deterministisiä tehtäviä, jotka ratkaistaan käyttäen elementtimenetelmää; konformikuvauksien avulla käsitellään stokastisuuden aiheuttama kappaleen muodon vaihtelu. Usean muuttujan interpolointiin liittyvän dimensionaalisuuden kirouksen lievittämiseksi käytämme Smolyakin konstruktioon perustuvaa harvan hilan interpolointimenetelmää. Mallitehtävinä käytämme kahdesta stokastisesta parametrista riippuvia kappaleita, tarkastellen sekä kaksi- että kolmiulotteista tapausta. Kuvailemme menetelmän toteutuksen yksityiskohtaisesti, ja esittelemme mallitehtävien numeeriset tulokset. Menetelmä saavuttaa jo 29 deterministisen tehtävän avulla pisteen, jonka jälkeen interpolaation tarkkuutta ei enää voida parantaa elementtimenetelmälle luontaisesta epätarkkuudesta johtuen

    Interactive Topology Optimization

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    A numerical method for finite-strain mechanochemistry with localised chemical reactions treated using a Nitsche approach

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    In this paper, a novel finite-element based method for finite-strain mechanochemistry with moving reaction fronts, which separate the chemically transformed and the untransformed phases, is proposed. The reaction front cuts through the finite elements and moves independently of the finite-element mesh, thereby removing the necessity for remeshing. The proposed method solves the coupled mechanics-diffusion–reaction problem. In the mechanical part of the problem, the force equilibrium and the displacement continuity conditions at the reaction front are enforced weakly using a Nitsche-like method. The formulation is applicable to the case of large deformations and arbitrary constitutive behaviour, and is also consistent with the minimisation of the total potential energy

    Reconstruction and Simulation of Cellular Traction Forces

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    Biological cells are able to sense the stiffness, geometry and topography of their environment and sensitively respond to it. For this purpose, they actively apply contractile forces to the extracellular space, which can be determined by traction force microscopy. Thereby cells are cultured on elastically deformable substrates and cellular traction patterns are quanti- tatively reconstructed from measured substrate deformations, by solving the inverse elastic problem. In this thesis we investigate the influence of environmental topography to cellular force generation and the distribution of intracellular tension. For this purpose, we reconstruct traction forces on wavy elastic substrates, using a novel technique based on finite element methods. In order to relate forces to single cell-matrix contacts and different structures of the cytoskeleton, we then introduce another novel variant of traction force microscopy, which introduces cell contraction modeling into the process of cellular traction reconstruction. This approach is robust against experimental noise and does not need regularisation. We apply this method to experimental data to demonstrate that different types of actin fibers in the cell statistically show different contractilities. We complete our investigation by simulation studies considering cell colonies and single cells as thermoelastically contracting continuum coupled to an elastic substrate. In particular we examined the effect of geometry on cellular behavior in collective cell migration and tissue invasion during tumor metastasis

    A hybrid method for inversion of 3D DC resistivity logging measurements

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    This paper focuses on the application of hp hierarchic genetic strategy (hp-HGS) for solution of a challenging problem, the inversion of 3D direct current (DC) resistivity logging measurements. The problem under consideration has been formulated as the global optimization one, for which the objective function (misfit between computed and reference data) exhibits multiple minima. In this paper, we consider the extension of the hp-HGS strategy, namely we couple the hp-HGS algorithm with a gradient based optimization method for a local search. Forward simulations are performed with a self-adaptive hp finite element method, hp-FEM. The computational cost of misfit evaluation by hp-FEM depends strongly on the assumed accuracy. This accuracy is adapted to the tree of populations generated by the hp-HGS algorithm, which makes the global phase significantly cheaper. Moreover, tree structure of demes as well as branch reduction and conditional sprouting mechanism reduces the number of expensive local searches up to the number of minima to be recognized. The common (direct and inverse) accuracy control, crucial for the hp-HGS efficiency, has been motivated by precise mathematical considerations. Numerical results demonstrate the suitability of the proposed method for the inversion of 3D DC resistivity logging measurements
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