Towards discrete octonionic analysis

In recent years, there is a growing interest in the studying octonions, which are 8-dimensional hypercomplex numbers forming the biggest normed division algebras over the real numbers. In particular, various tools of the classical complex function theory have been extended to the octonionic setting in recent years. However not so many results related to a discrete octonionic analysis, which is relevant for various applications in quantum mechanics, have been presented so far. Therefore, in this paper, we present first ideas towards discrete octonionic analysis. In particular, we discuss several approaches to a discretisation of octonionic analysis and present several discrete octonionic Stokes' formulae

Abstract modelling: towards a typed declarative language for the conceptual modelling phase

Modelling languages have become an indispensable aid to practising engineers. They offer modelling at a high level of abstraction backed by features such as automatic simulation and even derivation of production code. However, partly because of the offered automation, modelling languages are limited to specific application areas: to our knowledge, no modelling language supports mathematical physics modelling in its full generality. Yet, when developing large, coupled, multiphysics models, there is a clear need for such an overarching language to ensure the coherence of the model as a whole, even if submodels ultimately are realised in modelling languages targeting specific domains or are pre-existing. In prior work, it was demonstrated how treating models as abstract objects in category theory offers one way to ensure coherence of key aspects for composite models. Type theory offers complementary approaches. This paper presents a first step towards a language supporting abstract modelling in mathematical physics with the aim of ensuring coherence of coupled multiphysics models early in the design process. To that end, following the approach of Functional Hybrid Modelling (FHM), we discuss how a language supporting quite general modelling equations can be realised as an embedding in Haskell. The appeal of the proposed approach is that only very few core concepts are needed, which greatly simplifies the semantics. The appeal of an embedded realisation as such is that much of the language infrastructure comes for free

Finite element exterior calculus with script geometry

Finite element method is probably the most popular numerical method used in different fields of applications nowadays. While approximation properties of the classical finite element method, as well as its various modifications, are well understood, stability of the method is still a crucial problem in practice. Therefore, alternative approaches based not on an approximation of continuous differential equations, but working directly with discrete structures associated with these equations, have gained an increasing interest in recent years. Finite element exterior calculus is one of such approaches. The finite element exterior calculus utilises tools of algebraic topology, such as de Rham cohomology and Hodge theory, to address the stability of the continuous problem. By its construction, the finite element exterior calculus is limited to triangulation based on simplicial complexes. However, practical applications often require triangulations containing elements of more general shapes. Therefore, it is necessary to extend the finite element exterior calculus to overcome the restriction to simplicial complexes. In this paper, the script geometry, a recently introduced new kind of discrete geometry and calculus, is used as a basis for the further extension of the finite element exterior calculus.publishe

Operator calculus approach to comparison of elasticity models for modelling of masonry structures

The solution of any engineering problem starts with a modelling process aimed at formulating a mathematical model, which must describe the problem under consideration with sufficient precision. Because of heterogeneity of modern engineering applications, mathematical modelling scatters nowadays from incredibly precise micro- and even nano-modelling of materials to macro-modelling, which is more appropriate for practical engineering computations. In the field of masonry structures, a macro-model of the material can be constructed based on various elasticity theories, such as classical elasticity, micropolar elasticity and Cosserat elasticity. Evidently, a different macro-behaviour is expected depending on the specific theory used in the background. Although there have been several theoretical studies of different elasticity theories in recent years, there is still a lack of understanding of how modelling assumptions of different elasticity theories influence the modelling results of masonry structures. Therefore, a rigorous approach to comparison of different three-dimensional elasticity models based on quaternionic operator calculus is proposed in this paper. In this way, three elasticity models are described and spatial boundary value problems for these models are discussed. In particular, explicit representation formulae for their solutions are constructed. After that, by using these representation formulae, explicit estimates for the solutions obtained by different elasticity theories are obtained. Finally, several numerical examples are presented, which indicate a practical difference in the solutions

THE PROBLEM OF COUPLING BETWEEN ANALYTICAL SOLUTION AND FINITE ELEMENT METHOD

This paper is focused on the first numerical tests for coupling between analytical solution and finite element method on the example of one problem of fracture mechanics. The calculations were done according to ideas proposed in [1]. The analytical solutions are constructed by using an orthogonal basis of holomorphic and anti-holomorphic functions. For coupling with finite element method the special elements are constructed by using the trigonometric interpolation theorem

The characteristics of 2D and 3D modelling approach in calibration of reinforced concrete frames cyclic behaviour

A 3D FE micromodel of a bare RC frame was developed. The model is based on validated 2D micromodel. The 3D model obtained higher response when compared to its 2D counterpart. Consequently, a calibration of the frame was initiated. Calibration involved modifying parameters that govern the plastic behaviour of the computational model, such as fracture energy, plastic displacement and direction of plastic flow. It was shown that the greatest effect in lowering the response had the direction of plastic flow. Plastic flow direction was selected as -0.1 as it has greatest correlation with the experimental data. Negative value denotes that material volume will decrease due to crushing.publishe

Error Estimates for the Coupling of Analytical and Numerical Solutions

In this paper we present error estimates for a continuous coupling of an analytical and a numerical solution for a boundary value problem with a singularity. A solution of the Lamé–Navier equation with a singularity caused by a crack is considered as an example. The analytical solution near a singularity is constructed by using complex function theory and coupled continuously with the finite element solution. The objective of this paper is to estimate the coupling error, which cannot be covered by the classical theory of the finite element method

Mathematical Modelling by Help of Category Theory: Models and Relations between Them

The growing complexity of modern practical problems puts high demand on mathematical modelling. Given that various models can be used for modelling one physical phenomenon, the role of model comparison and model choice is becoming particularly important. Methods for model comparison and model choice typically used in practical applications nowadays are computation-based, and thus time consuming and computationally costly. Therefore, it is necessary to develop other approaches to working abstractly, i.e., without computations, with mathematical models. An abstract description of mathematical models can be achieved by the help of abstract mathematics, implying formalisation of models and relations between them. In this paper, a category theory-based approach to mathematical modelling is proposed. In this way, mathematical models are formalised in the language of categories, relations between the models are formally defined and several practically relevant properties are introduced on the level of categories. Finally, an illustrative example is presented, underlying how the category-theory based approach can be used in practice. Further, all constructions presented in this paper are also discussed from a modelling point of view by making explicit the link to concrete modelling scenarios

Boundary values of discrete monogenic functions over bounded domains in R3

In this paper we are going to study boundary values for discrete monogenic functions over bounded spatial domains. After establishing the discrete Stokes formula and the Borel–Pompeiu formula we are going to construct discrete Plemelj–Sokhotzki formulae, discrete Plemelj projections and discrete Hardy spaces. A further extension to the n-dimensional case can be done in a straightforward way based on the results presented in this paper.publishe

Discrete Hardy spaces for bounded domains in Rn{\mathbb {R}}^{n}

Discrete function theory in higher-dimensional setting has been in active development since many years. However, available results focus on studying discrete setting for such canonical domains as half-space, while the case of bounded domains generally remained unconsidered. Therefore, this paper presents the extension of the higher-dimensional function theory to the case of arbitrary bounded domains in R^n. On this way, discrete Stokes’ formula, discrete Borel–Pompeiu formula, as well as discrete Hardy spaces for general bounded domains are constructed. Finally, several discrete Hilbert problems are considered.publishe