Probabilistic Godunov-type hydrodynamic modelling under multiple uncertainties: robust wavelet-based formulations

Intrusive stochastic Galerkin methods propagate uncertainties in a single model run, eliminating repeated sampling required by conventional Monte Carlo methods. However, an intrusive formulation has yet to be developed for probabilistic hydrodynamic modelling incorporating robust wetting-and-drying and stable friction integration under joint uncertainties in topography, roughness, and inflow. Robustness measures are well-developed in deterministic models, but rely on local, nonlinear operations that can introduce additional stochastic errors that destabilise an intrusive model. This paper formulates an intrusive hydrodynamic model using a multidimensional tensor product of Haar wavelets to capture fine-scale variations in joint probability distributions and extend the validity of robustness measures from the underlying deterministic discretisation. Probabilistic numerical tests are designed to verify intrusive model robustness, and compare accuracy and efficiency against a conventional Monte Carlo approach and two other alternatives: a nonintrusive stochastic collocation formulation sharing the same tensor product wavelet basis, and an intrusive formulation that truncates the basis to gain efficiency under multiple uncertainties. Tests reveal that: (i) a full tensor product basis is required to preserve intrusive model robustness, while the nonintrusive counterpart achieves identically accurate results at a reduced computational cost; and, (ii) Haar wavelets basis requires at least three levels of refinements per uncertainty dimension to reliably capture complex probability distributions. Accompanying model software and simulation data are openly available online

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Reduced order modeling of convection-dominated flows, dimensionality reduction and stabilization

We present methodologies for reduced order modeling of convection dominated flows. Accordingly, three main problems are addressed. Firstly, an optimal manifold is realized to enhance reducibility of convection dominated flows. We design a low-rank auto-encoder to specifically reduce the dimensionality of solution arising from convection-dominated nonlinear physical systems. Although existing nonlinear manifold learning methods seem to be compelling tools to reduce the dimensionality of data characterized by large Kolmogorov n-width, they typically lack a straightforward mapping from the latent space to the high-dimensional physical space. Also, considering that the latent variables are often hard to interpret, many of these methods are dismissed in the reduced order modeling of dynamical systems governed by partial differential equations (PDEs). This deficiency is of importance to the extent that linear methods, such as principle component analysis (PCA) and Koopman operators, are still prevalent. Accordingly, we propose an interpretable nonlinear dimensionality reduction algorithm. An unsupervised learning problem is constructed that learns a diffeomorphic spatio-temporal grid which registers the output sequence of the PDEs on a non-uniform time-varying grid. The Kolmogorov n-width of the mapped data on the learned grid is minimized. Secondly, the reduced order models are constructed on the realized manifolds. We project the high fidelity models on the learned manifold, leading to a time-varying system of equations. Moreover, as a data-driven model free architecture, recurrent neural networks on the learned manifold are trained, showing versatility of the proposed framework. Finally, a stabilization method is developed to maintain stability and accuracy of the projection based ROMs on the learned manifold a posteriori. We extend the eigenvalue reassignment method of stabilization of linear time-invariant ROMs, to the more general case of linear time-varying systems. Through a post-processing step, the ROMs are controlled using a constrained nonlinear lease-square minimization problem. The controller and the input signals are defined at the algebraic level, using left and right singular vectors of the reduced system matrices. The proposed stabilization method is general and applicable to a large variety of linear time-varying ROMs

Structure preserving schemes and kinetic models for approximating measure valued solutions of hyperbolic equations

In this thesis we consider approximate schemes and models for hyperbolic conservation laws. Systems of conservation laws are fundamental mathematical models and have received a lot of attention from the point of view of analysis, modelling and computations. They include the wave equations in elastic media and fundamental equations in fluid mechanics. We consider structure preserving schemes and kinetic models for approximating measure valued solutions of hyperbolic equations. Such solutions are of interest given their application to problems in uncertainty quantification and in statistical inference. This thesis contains new results on (i) the design of new schemes for the computation of entropy consistent approximations, with particular emphasis on the consistency of the computational algorithms to entropic measure valued solutions for HCL, (ii) the introduction of discrete and generalised kinetic models designed to directly approximate measure valued solutions by using a combination of approximate Young measures and the kinetic formulation of the conservation law and (iii) stability analysis of generalised viscus kinetic models. We obtain uniqueness within a particular class of vanishing viscosity limits of these models and of their corresponding measure valued solutions