Uncertainty quantification for random domains using periodic random variables

We consider uncertainty quantification for the Poisson problem subject to domain uncertainty. For the stochastic parameterization of the random domain, we use the model recently introduced by Kaarnioja, Kuo, and Sloan (SIAM J. Numer. Anal., 2020) in which a countably infinite number of independent random variables enter the random field as periodic functions. We develop lattice quasi-Monte Carlo (QMC) cubature rules for computing the expected value of the solution to the Poisson problem subject to domain uncertainty. These QMC rules can be shown to exhibit higher order cubature convergence rates permitted by the periodic setting independently of the stochastic dimension of the problem. In addition, we present a complete error analysis for the problem by taking into account the approximation errors incurred by truncating the input random field to a finite number of terms and discretizing the spatial domain using finite elements. The paper concludes with numerical experiments demonstrating the theoretical error estimates.Comment: 38 pages, 3 figure

Uncertainty quantification for random domains using periodic random variables

We consider uncertainty quantification for the Poisson problem subject to domain uncertainty. For the stochastic parameterization of the random domain, we use the model recently introduced by Kaarnioja et al. (SIAM J. Numer. Anal., 2020) in which a countably infinite number of independent random variables enter the random field as periodic functions. We develop lattice quasi-Monte Carlo (QMC) cubature rules for computing the expected value of the solution to the Poisson problem subject to domain uncertainty. These QMC rules can be shown to exhibit higher order cubature convergence rates permitted by the periodic setting independently of the stochastic dimension of the problem. In addition, we present a complete error analysis for the problem by taking into account the approximation errors incurred by truncating the input random field to a finite number of terms and discretizing the spatial domain using finite elements. The paper concludes with numerical experiments demonstrating the theoretical error estimates

On Effective Material Parameters of Thin Perforated Shells under Static Loading

One of the defining properties of thin shell problems is that the solution can be viewed as a linear combination of local features, each with its own characteristic thickness-dependent length scale. For perforated shells it is thus possible that for the given dimensionless thickness, the local features dominate, and the problem of deriving effective material parameters becomes ill-posed. In the general case, one has to account for many different aspects of the problem that directly affect the effective material parameters. Through a computational study we derive a conjecture for the admissible thickness-ranges. The effective material parameters are derived with a minimisation process over a set of feasible instances. The efficacy of the conjecture and the minimisation process is demonstrated with an extensive set of numerical experiments

Conformal moduli of symmetric circular quadrilaterals with cusps

We investigate moduli of planar circular quadrilaterals that are symmetric with respect to both coordinate axes. First we develop an analytic approach that reduces this problem to ODEs and then devise a numerical method to find out the accessory parameters. This method uses the Schwarz equation to determine a conformal mapping of the unit disk onto a given circular quadrilateral. We also give an example of a circular quadrilateral for which the value of the conformal modulus can be found in analytic form. This example is used to validate the numeric calculations. We also apply another method, the so called hpFEM, for the numerical calculation of the moduli. These two different approaches provide results agreeing with high accuracy

Asymptotic and numerical analysis of the eigenvalue problem of a clamped cylindrical shell

We are interested in the asymptotic analysis of the eigenvalue problem of clamped cylindrical shells. We analyze the lowest eigenvalues as a function of the shell thickness t, the asymptotic behavior of the respective eigenfunctions, and show how the different displacement components and parts of the energy scale in t. As a consequence, we are able to single out the numerical difficulties of the problem, which, surprisingly for a formally bending inhibited problem, include the presence of locking. Extensive numerical tests are included

On the asymptotic behaviour of shells of revolution in free vibration

We consider the free vibration problem of thin shells of revolution of constant type of geometry, focusing on the asymptotic behaviour of the lowest eigenfrequency, as the thickness tends to zero. Numerical experiments are computed using two discretization methods, collocation and finite elements, each corresponding to a different shellmodel. Our results are in agreement with theoretical results obtained using interpolation theory and cited in literature

Free vibrations for some Koiter shells of revolution

The asymptotic behaviour of the smallest eigenvalue in linear Koiter shell problems is studied, as the thickness parameter tends to zero. In particular, three types of shells of revolution are considered. A result concerning the ratio between the bending and the total elastic energy is also provided, by using the general theory detailed in [L. Beir˜ao da Veiga, C. Lovadina, An interpolation theory approach to Shell eigenvalue problems (submitted for publication); L. Beir˜ao da Veiga, C. Lovadina, Asymptotics of shell eigenvalue problems, C.R. Acad. Sci. Paris 9 (2006) 707–710]