Adaptive refinement in advection–diffusion problems by anomaly detection: A numerical study

We consider advection–diffusion–reaction problems, where the advective or the reactive term is dominating with respect to the diffusive term. The solutions of these problems are character-ized by the so-called layers, which represent localized regions where the gradients of the solutions are rather large or are subjected to abrupt changes. In order to improve the accuracy of the computed solution, it is fundamental to locally increase the number of degrees of freedom by limiting the computational costs. Thus, adaptive refinement, by a posteriori error estimators, is employed. The error estimators are then processed by an anomaly detection algorithm in order to identify those regions of the computational domain that should be marked and, hence, refined. The anomaly detection task is performed in an unsupervised fashion and the proposed strategy is tested on typical benchmarks. The present work shows a numerical study that highlights promising results obtained by bridging together standard techniques, i.e., the error estimators, and approaches typical of machine learning and artificial intelligence, such as the anomaly detection task

General features of the energy landscape in Lennard-Jones like model liquids

Features of the energy landscape sampled by supercooled liquids are numerically analyzed for several Lennard-Jones like model systems. The properties of quasisaddles (minima of the square gradient of potential energy W=|grad V|^2), are shown to have a direct relationship with the dynamical behavior, confirming that the quasisaddle order extrapolates to zero at the mode-coupling temperature T_MCT. The same result is obtained either analyzing all the minima of W or the saddles (absolute minima of W), supporting the conjectured similarity between quasisaddles and saddles, as far as the temperature dependence of the properties influencing the slow dynamics is concerned. We find evidence of universality in the shape of the landscape: plots for different systems superimpose into master curves, once energies and temperatures are scaled by T_MCT. This allows to establish a quantitative relationship between T_MCT and potential energy barriers for LJ-like systems, and suggests a possible generalization to different model liquids.Comment: 8 pages, 5 figure

Isogeometric Analysis in advection-diffusion problems: tension splines approximation

We present a novel approach, within the new paradigm of isogeometric analysis introduced by Hughes et al., to deal with advection dominated advection-diffusion problems. The key ingredient is the use of Galerkin approximating spaces of functions with high smoothness, as in IgA based on classical B-splines, but particularly well suited to describe sharp layers involving very strong gradients

Construction of planar quintic Pythagorean-hodograph curves by control-polygon constraints

In the construction and analysis of a planar Pythagorean–hodograph (PH) quintic curve r(t), t∈[0,1] using the complex representation, it is convenient to invoke a translation/rotation/scaling transformation so r(t) is in canonical form with r(0)=0, r(1)=1 and possesses just two complex degrees of freedom. By choosing two of the five control–polygon legs of a quintic PH curve as these free complex parameters, the remaining three control–polygon legs can be expressed in terms of them and the roots of a quadratic or quartic equation. Consequently, depending on the chosen two control–polygon legs, there exist either two or four distinct quintic PH curves that are consistent with them. A comprehensive analysis of all possible pairs of chosen control polygon legs is developed, and examples are provided to illustrate this control–polygon paradigm for the construction of planar quintic PH curves

Cubature rules based on bivariate spline quasi-interpolation for weakly singular integrals

In this paper we present a new class of cubature rules with the aim of accurately integrating weakly singular double integrals. In particular we focus on those integrals coming from the discretization of Boundary Integral Equations for 3D Laplace boundary value problems, using a collocation method within the Isogeometric Analysis paradigm. In such setting the regular part of the integrand can be defined as the product of a tensor product B-spline and a general function. The rules are derived by using first the spline quasi-interpolation approach to approximate such function and then the extension of a well known algorithm for spline product to the bivariate setting. In this way efficiency is ensured, since the locality of any spline quasi-interpolation scheme is combined with the capability of an ad--hoc treatment of the B-spline factor. The numerical integration is performed on the whole support of the B-spline factor by exploiting inter-element continuity of the integrand

Splines Parameterization of Planar Domains by Physics-Informed Neural Networks

The generation of structured grids on bounded domains is a crucial issue in the development of numerical models for solving differential problems. In particular, the representation of the given computational domain through a regular parameterization allows us to define a univalent mapping, which can be computed as the solution of an elliptic problem, equipped with suitable Dirichlet boundary conditions. In recent years, Physics-Informed Neural Networks (PINNs) have been proved to be a powerful tool to compute the solution of Partial Differential Equations (PDEs) replacing standard numerical models, based on Finite Element Methods and Finite Differences, with deep neural networks; PINNs can be used for predicting the values on simulation grids of different resolutions without the need to be retrained. In this work, we exploit the PINN model in order to solve the PDE associated to the differential problem of the parameterization on both convex and non-convex planar domains, for which the describing PDE is known. The final continuous model is then provided by applying a Hermite type quasi-interpolation operator, which can guarantee the desired smoothness of the sought parameterization. Finally, some numerical examples are presented, which show that the PINNs-based approach is robust. Indeed, the produced mapping does not exhibit folding or self-intersection at the interior of the domain and, also, for highly non convex shapes, despite few faulty points near the boundaries, has better shape-measures, e.g., lower values of the Winslow functional

IgA-BEM for 3D Helmholtz problems using conforming and non-conforming multi-patch discretizations and B-spline tailored numerical integration

An Isogeometric Boundary Element Method (IgA-BEM) is considered for the numerical solution of Helmholtz problems on 3D bounded or unbounded domains, admitting a smooth multi-patch representation of their finite boundary surface. The discretization spaces are formed by C0 inter-patch continuous functional spaces whose restriction to a patch simplifies to the span of tensor product B-splines composed with the given patch NURBS parameterization. Both conforming and non-conforming spaces are allowed, so that local refinement is possible at the patch level. For regular and singular integration, the proposed model utilizes a numerical procedure defined on the support of each trial B-spline function, which makes possible a function-by-function implementation of the matrix assembly phase. Spline quasi-interpolation is the common ingredient of all the considered quadrature rules; in the singular case it is combined with a B-spline recursion over the spline degree and with a singularity extraction technique, extended to the multi-patch setting for the first time. A threshold selection strategy is proposed to automatically distinguish between nearly singular and regular integrals. The non-conforming C0 joints between spline spaces on different patches are implemented as linear constraints based on knot removal conditions, and do not require a hierarchical master-slave relation between neighbouring patches. Numerical examples on relevant benchmarks show that the expected convergence orders are achieved with uniform discretization and a small number of uniformly spaced quadrature nodes

Relaxation processes in harmonic glasses?

A relaxation process, with the associated phenomenology of sound attenuation and sound velocity dispersion, is found in a simulated harmonic Lennard-Jones glass. We propose to identify this process with the so called microscopic (or instantaneous) relaxation process observed in real glasses and supercooled liquids. A model based on the memory function approach accounts for the observation, and allows to relate to each others: 1) the characteristic time and strength of this process, 2) the low frequency limit of the dynamic structure factor of the glass, and 3) the high frequency sound attenuation coefficient, with its observed quadratic dependence on the momentum transfer.Comment: 11 pages, 3 figure