Robust Discretization of Flow in Fractured Porous Media

Flow in fractured porous media represents a challenge for discretization methods due to the disparate scales and complex geometry. Herein we propose a new discretization, based on the mixed finite element method and mortar methods. Our formulation is novel in that it employs the normal fluxes as the mortar variable within the mixed finite element framework, resulting in a formulation that couples the flow in the fractures with the surrounding domain with a strong notion of mass conservation. The proposed discretization handles complex, non-matching grids, and allows for fracture intersections and termination in a natural way, as well as spatially varying apertures. The discretization is applicable to both two and three spatial dimensions. A priori analysis shows the method to be optimally convergent with respect to the chosen mixed finite element spaces, which is sustained by numerical examples

The Construction of Verification Models for Embedded Systems

The usefulness of verification hinges on the quality of the verification model. Verification is useful if it increases our confidence that an artefact bahaves as expected. As modelling inherently contains non-formal elements, the qualityof models cannot be captured by purely formal means. Still, we argue that modelling is not an act of irrationalism and unpredictable geniality, but follows rational arguments, that often remain implicit. In this paper we try to identify the tacit rationalism in the model construction as performed by most people doing modelling for verification. By explicating the different phases, arguments, and design decisions in the model construction, we try to develop guidelines that help to improve the process of model construction and the quality of models

Functional Analysis and Exterior Calculus on Mixed-Dimensional Geometries

We are interested in differential forms on mixed-dimensional geometries, in the sense of a domain containing sets of $d$-dimensional manifolds, structured hierarchically so that each $d$-dimensional manifold is contained in the boundary of one or more $d + 1$ dimensional manifolds. On any given $d$-dimensional manifold, we then consider differential operators tangent to the manifold as well as discrete differential operators (jumps) normal to the manifold. The combined action of these operators leads to the notion of a semi-discrete differential operator coupling manifolds of different dimensions. We refer to the resulting systems of equations as mixed-dimensional, which have become a popular modeling technique for physical applications including fractured and composite materials. We establish analytical tools in the mixed-dimensional setting, including suitable inner products, differential and codifferential operators, Poincar\'e lemma, and Poincar\'e--Friedrichs inequality. The manuscript is concluded by defining the mixed-dimensional minimization problem corresponding to the Hodge-Laplacian, and we show that this minimization problem is well-posed

Nonextensive diffusion as nonlinear response

The porous media equation has been proposed as a phenomenological ``non-extensive'' generalization of classical diffusion. Here, we show that a very similar equation can be derived, in a systematic manner, for a classical fluid by assuming nonlinear response, i.e. that the diffusive flux depends on gradients of a power of the concentration. The present equation distinguishes from the porous media equation in that it describes \emph{% generalized classical} diffusion, i.e. with $r/\sqrt Dt$ scaling, but with a generalized Einstein relation, and with power-law probability distributions typical of nonextensive statistical mechanics

Visualizing Multiple Quantile Plots

Multiple quantile plots provide a powerful graphical method for comparing the distributions of two or more populations. This paper develops a method of visualizing triple quantile plots and their associated confidence tubes, thus extending the notion of a QQ plot to three dimensions. More specifically, we consider three independent one-dimensional random samples with corresponding quantile functions Q1, Q2 and Q3, respectively. The triple quantile (QQQ) plot is then defined as the three-dimensional curve Q(p) = (Q1(p);Q2(p);Q3(p)); where 0Confidence region;empirical likelihood;quantile plot;three-sample com- parison

Contingency and inevitability in science - Instruments, interfaces and the independent world

It is argued that the meaning of inevitability and contingency depends on the position someone has in the realism/constructivism debate. Furthermore, it is argued that analyzing what we mean by inevitable versus contingent knowledge adds a new dimension to the realism/constructivism debate. Scientific realist and social constructivist views of science often lead to, respectively, too high expectations and too low confidence in what science can do. This controversy is not always productive for scientific practices that work in the context of practical applications (e.g., the engineering sciences). The aim of this paper is to make a contribution to a more viable view of those scientific practices. The approach taken is to construe two philosophical stances within which the meaning of inevitability and contingency is examined. The first stance is called metaphysical realism. It is construed such that it fits Hacking’s (2000) ideas on the inevitability of scientific knowledge. The second stance is called epistemological constructivism. The two stances agree that there is an independent world which sets limits to our knowledge, but they disagree on whether we must assume the existence of a pre-given, or even, knowable structure in the world. The dissonance between the two stances is reduced such that just one significant epistemological issue remains, namely, which part(s) of science are inevitable? Next, the contingency of science is interpreted in terms of Giere’s (2006) notion of scientific perspectivism. This view emphasizes the role of (different kinds of) instruments in providing epistemic access to the world – which is why we cannot attain mirror-like knowledge of the world. Both stances agree on this view. Yet, when considering the apparent presuppositions of perspectivism an additional epistemological issue arises, namely, whether a clear distinction can be made between the object and representations (i.e., perspectives) of that object. In order to be more faithful to concrete scientific practice, it is proposed to consider (different kinds of) instruments as interfaces rather than perspectives on something. In the epistemological constructivist stance, interfaces transform material or symbolic or electronic or whatever input, which cannot be directly perceived or known by us, to output that can be perceived, experienced and/or conceived (e.g., numbers, tables, graphs). Finally, the materiality of (different kinds of) instruments is crucial for explaining which parts of science are inevitabl