Propagating Geometry Information to Finite Element Computations

The traditional workflow in continuum mechanics simulations is that a geometry description —for example obtained using Constructive Solid Geometry (CSG) or Computer Aided Design (CAD) tools—forms the input for a mesh generator. The mesh is then used as the sole input for the finite element, finite volume, and finite difference solver, which at this point no longer has access to the original, “underlying” geometry. However, many modern techniques—for example, adaptive mesh refinement and the use of higher order geometry approximation methods—really do need information about the underlying geometry to realize their full potential. We have undertaken an exhaustive study of where typical finite element codes use geometry information, with the goal of determining what information geometry tools would have to provide. Our study shows that nearly all geometry-related needs inside the simulators can be satisfied by just two “primitives”: elementary queries posed by the simulation software to the geometry description. We then show that it is possible to provide these primitives in all of the frequently used ways in which geometries are described in common industrial workflows, and illustrate our solutions using a number of examples.

Clifford Spinors and Root System Induction: $$H_4$$ and the Grand Antiprism

Recent work has shown that every 3D root system allows the construction of a corresponding 4D root system via an ‘induction theorem’. In this paper, we look at the icosahedral case of $$H_3\rightarrow H_4$$ H 3 → H 4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan–Dieudonné theorem, giving a simple construction of the $${\mathrm {Pin}}$$ Pin and $${\mathrm {Spin}}$$ Spin covers. Using this connection with $$H_3$$ H 3 via the induction theorem sheds light on geometric aspects of the $$H_4$$ H 4 root system (the 600-cell) as well as other related polytopes and their symmetries, such as the famous Grand Antiprism and the snub 24-cell. The uniform construction of root systems from 3D and the uniform procedure of splitting root systems with respect to subrootsystems into separate invariant sets allows further systematic insight into the underlying geometry. All calculations are performed in the even subalgebra of $${\mathrm {Cl}}(3)$$ Cl ( 3 ) , including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes, and are shared as supplementary computational work sheets. This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework.

Random Walks with Invariant Loop Probabilities: Stereographic Random Walks

Random walks with invariant loop probabilities comprise a wide family of Markov processes with site-dependent, one-step transition probabilities. The whole family, which includes the simple random walk, emerges from geometric considerations related to the stereographic projection of an underlying geometry into a line. After a general introduction, we focus our attention on the elliptic case: random walks on a circle with built-in reflexing boundaries.

Four Spacetime Dimensions from Multifractal Geometry

As paradigm of complex behavior, multifractals describe the underlying geometry of self-similar objects or processes. Building on the connection between entropy and multifractals, we show here that the generalized dimension of geodesic trajectories in General Relativity coincides with the four-dimensionality of classical spacetime.

First principles of consistent physics

For a consistent picture of fundamental physics and cosmology, three first principles are proposed as the foundations. That is, quantum variational principle that provides the formalism, consistent observation principle that set physical constraints and symmetries, and spacetime inflation principle that determines physical contents (particle fields and interactions). Under these three principles, a series of supersymmetric mirror models are constructed to study various phases of the universe at different spacetime dimensions and the dynamics between the phases. In particular, mirror symmetry, as the orientation symmetry of the underlying geometry, plays a critical role in the new framework.

A simple sorting algorithm for compositions

We provide a particular measure for the degree to which an arbitrary composition deviates from increasing sorted order. The application of such a measure to the transport industry is given in the introduction. In order to obtain this measure, we define a statistic called the number of pushes in an arbitrary composition (which is required to produce sorted order) and obtain a generating function for this. The concept of a push is a geometrical one and leads naturally to several dependant concepts which are investigated. These are the number of cells which do not move in the pushing process and the number of cells that coincide before and after the pushing process (a number not less than those that do not move). The concept of a push leads to combining certain single pushes in a natural way which we define as a frictionless push. A generating function for these is also developed. The underlying geometry of the process also leads naturally to counting the largest first component of arbitrary compositions that are already in a sorted order. We provide a generating function for this.

Inhomogeneous spacetimes in Weyl integrable geometry with matter source

We investigate the existence of inhomogeneous exact solutions in Weyl Integrable theory with a matter source. In particular, we consider the existence of a dust fluid source while for the underlying geometry we assume a line element which belongs to the family of silent universes. We solve explicitly the field equations and we find the Szekeres spacetimes in Weyl Integrable theory. We show that only the isotropic family can describe inhomogeneous solutions where the LTB spacetimes are included. A detailed analysis of the dynamics of the field equations is given where the past and future attractors are determined. It is interesting that the Kasner spacetimes can be seen as past attractors for the gravitation models, while the unique future attractor describes the Milne universe similar with the behaviour of the gravitational model in the case of General Relativity.

Feasibility Achievement without the Hassle of Artificial Variables: A Computational Study

The purpose of this article is to encourage students and teachers to use a simple technique for finding feasible solution of an LP. This technique is very simple but unfortunately not much practiced in the textbook literature yet. This article discusses an overview, advantages, computational experience of the method. This method provides some pronounced benefits over Dantzig’s simplex method phase 1. For instance, it does not require any kind of artificial variables or artificial constraints; it could directly start with any infeasible basis of an LP. Throughout the procedure it works in original variables space hence revealing the true underlying geometry of the problem. Last but not the least; it is a handy tool for students to quickly solve a linear programming problem without indulging with artificial variables. It is also beneficial for the teachers who want to teach feasibility achievement as a separate topic before teaching optimality achievement. Our primary result shows that this method is much better than simplex phase 1 for practical Net-lib problems as well as for general random LPs.

Space–time geometry and the velocity of light

In this paper, we investigate the possibility of superluminal velocities and the appertaining underlying geometry. We consider three different approaches and arrive at the same status quo.