Stabilised finite element methods for a bending moment formulation of the Reissner-Mindlin plate model

This work presents new stabilised finite element methods for a bending moments formulation of the Reissner-Mindlin plate model. The introduction of the bending moment as an extra unknown leads to a new weak formulation, where the symmetry of this variable is imposed strongly in the space. This weak problem is proved to be well-posed, and stabilised Galerkin schemes for its discretisation are presented and analysed. The finite element methods are such that the bending moment tensor is sought in a finite element space constituted of piecewise linear continuos and symmetric tensors. Optimal error estimates are proved, and these findings are illustrated by representative numerical experiments.     

Enforcing Efficient Equilibria in Network Design Games via Subsidies

The efficient design of networks has been an important engineering task that involves challenging combinatorial optimization problems. Typically, a network designer has to select among several alternatives which links to establish so that the resulting network satisfies a given set of connectivity requirements and the cost of establishing the network links is as low as possible. The Minimum Spanning Tree problem, which is well-understood, is a nice example.In this paper, we consider the natural scenario in which the connectivity requirements are posed by selfish users who have agreed to share the cost of the network to be established according to a well-defined rule. The design proposed by the network designer should now be consistent not only with the connectivity requirements but also with the selfishness of the users. Essentially, the users are players in a so-called network design game and the network designer has to propose a design that is an equilibrium for this game. As it is usually the case when selfishness comes into play, such equilibria may be suboptimal. In this paper, we consider the following question: can the network designer enforce particular designs as equilibria or guarantee that efficient designs are consistent with users’ selfishness by appropriately subsidizing some of the network links? In an attempt to understand this question, we formulate corresponding optimization problems and present positive and negative results.     

Hybrid Difference Methods for PDEs

We propose the hybrid difference methods for partial differential equations (PDEs). The hybrid difference method is composed of two types of approximations: one is the finite difference approximation of PDEs within cells (cell FD) and the other is the interface finite difference (interface FD) on edges of cells. The interface finite difference is obtained from continuity of some physical quantities. The main advantages of this new approach are that the method can applied to non-uniform grids, retaining the optimal order of convergence and stability of the numerical method for the Stokes equations is obtained without introducing staggered grids.     

Dynamic Model Adaptation for Multiscale Simulation of Hyperbolic Systems with Relaxation

In numerous industrial CFD applications, it is usual to use two (or more) different codes to solve a physical phenomenon: where the flow is a priori assumed to have a simple behavior, a code based on a coarse model is applied, while a code based on a fine model is used elsewhere. This leads to a complex coupling problem with fixed interfaces. The aim of the present work is to provide a numerical indicator to optimize to position of these coupling interfaces. In other words, thanks to this numerical indicator, one could verify if the use of the coarser model and of the resulting coupling does not introduce spurious effects. In order to validate this indicator, we use it in a dynamical multiscale method with moving coupling interfaces. The principle of this method is to use as much as possible a coarse model instead of the fine model in the computational domain, in order to obtain an accuracy which is comparable with the one provided by the fine model. We focus here on general hyperbolic systems with stiff relaxation source terms together with the corresponding hyperbolic equilibrium systems. Using a numerical Chapman–Enskog expansion and the distance to the equilibrium manifold, we construct the numerical indicator. Based on several works on the coupling of different hyperbolic models, an original numerical method of dynamic model adaptation is proposed. We prove that this multiscale method preserves invariant domains and that the entropy of the numerical solution decreases with respect to time. The reliability of the adaptation procedure is assessed on various 1D and 2D test cases coming from two-phase flow modeling.     

Towards Smooth Particle Methods Without Smoothing

In this article we present a new class of particle methods which aim at being accurate in the uniform norm with a minimal amount of smoothing. The crux of our approach is to compute local polynomial expansions of the characteristic flow to transport the particle shapes with improved accuracy. In the first order case the method consists of representing the transported density with linearly-transformed particles, the second order version transports quadratically-transformed particles, and so on. For practical purposes we provide discrete versions of the resulting LTP and QTP schemes that only involve pointwise evaluations of the forward characteristic flow, and we propose local indicators for the associated transport error. On a theoretical level we extend these particle schemes up to arbitrary polynomial orders and show by a rigorous analysis that for smooth flows the resulting methods converge in \(L^\infty \) without requiring remappings, extended overlapping or vanishing moments for the particles. Numerical tests using different passive transport problems demonstrate the accuracy of the proposed methods compared to basic particle schemes, and they establish their robustness with respect to the remapping period. In particular, it is shown that QTP particles can be transported without remappings on very long periods of time, without hampering the accuracy of the numerical solutions. Finally, a dynamic criterion is proposed to automatically select the time steps where the particles should be remapped. The strategy is a by-product of our error analysis, and it is validated by numerical experiments.     

An improved factor based approach to precedential constraint

In this article I argue for rule-based, non-monotonic theories of common law judicial reasoning and improve upon one such theory offered by Horty and Bench-Capon. The improvements reveal some of the interconnections between formal theories of judicial reasoning and traditional issues within jurisprudence regarding the notions of the ratio decidendi and obiter dicta. Though I do not purport to resolve the long-standing jurisprudential issues here, it is beneficial for theorists both of legal philosophy and formalizing legal reasoning to see where the two projects interact.     

A linear algebra approach to OLAP

Inspired by the relational algebra of data processing, this paper addresses the foundations of data analytical processing from a linear algebra perspective. The paper investigates, in particular, how aggregation operations such as cross tabulations and data cubes essential to quantitative analysis of data can be expressed solely in terms of matrix multiplication, transposition and the Khatri–Rao variant of the Kronecker product. The approach offers a basis for deriving an algebraic theory of data consolidation, handling the quantitative as well as qualitative sides of data science in a natural, elegant and typed way. It also shows potential for parallel analytical processing, as the parallelization theory of such matrix operations is well acknowledged.     

Abstraction and approximation in fuzzy temporal logics and models

Recently, by defining suitable fuzzy temporal logics, temporal properties of dynamic systems are specified during model checking process, yet a few numbers of fuzzy temporal logics along with capable corresponding models are developed and used in system design phase, moreover in case of having a suitable model, it suffers from the lack of a capable model checking approach. Having to deal with uncertainty in model checking paradigm, this paper introduces a fuzzy Kripke model (FzKripke) and then provides a verification approach using a novel logic called Fuzzy Computation Tree Logic* (FzCTL*). Not only state space explosion is handled using well-known concepts like abstraction and bisimulation, but an approximation method is also devised as a novel technique to deal with this problem. Fuzzy program graph, a generalization of program graph and FzKripke, is also introduced in this paper in consideration of higher level abstraction in model construction. Eventually modeling, and verification of a multi-valued flip-flop is studied in order to demonstrate capabilities of the proposed models.     

Model checking dynamic pushdown networks

A dynamic pushdown network (DPN) is a set of pushdown systems (PDSs) where each process can dynamically create new instances of PDSs. DPNs are a natural model of multi-threaded programs with (possibly recursive) procedure calls and thread creation. Thus, it is important to have model checking algorithms for DPNs. We consider in this work model checking DPNs against single-indexed LTL and CTL properties of the form \({\bigwedge f_i}\) such that f _i is a LTL/CTL formula over the PDS i. We consider the model checking problems w.r.t. simple valuations (i.e., whether a configuration satisfies an atomic proposition depends only on its control location) and w.r.t. regular valuations (i.e., the set of the configurations satisfying an atomic proposition is a regular set of configurations). We show that these model checking problems are decidable. We propose automata-based approaches for computing the set of configurations of a DPN that satisfy the corresponding single-indexed LTL/CTL formula.