Enzo Tonti

Finite formulation of electromagnetic field

We show that the equations of electromagnetism can be directly obtained in a finite form, i.e., discrete, thus avoiding the traditional discretization methods of Maxwells differential equations. The finite formulation can be used with unstructured meshes in two and three dimensions and easily permits to obtain fourth-order convergence.

Finite Formulation of the Electromagnetic Field

This paper shows that the equations of electromagnetism can be directly obtained in a finite (=discrete) form, i.e. without going throught the differential formulation. This finite formulation is a natural extension of the network theory to electromagnetic field and it is suitable for computational electromagnetics.

The Mathematical Structure of Classical and Relativistic Physics

1 Introduction.- Part I Analysis of variables and equations.- 2 Terminology revisited.- 3 Space and time elements and their orientation.- 4 Cell complexes.- 5 Analysis of physical variables.- 6 Analysis of physical equations.- 7 Algebraic topology.- 8 The birth of the classification diagrams.- Part II Analysis of physical theories.- 9 Particle dynamics.- 10 Electromagnetism.- 11 Mechanics of deformable solids.- 12 Mechanics of fluids.- 13 Other physical theories.- Part III Advanced analysis.- 14 General structure of the diagrams.- 15 The mathematical structure.- Part IV Appendices.- A Affine vector fields.- B Tensorial notation.- C On observable quantities.- D History of the diagram.- D.1 Historical remarks.- E List of physical variables.- F List of symbols used in this book.- G List of diagrams.- References.

Finite formulation and domain-integrated field relations in electromagnetics - a synthesis

Complementary formulations of the integral type have established themselves as the most adequate approach to computational electromagnetics. This paper proposes a computational strategy that benefits from the advantages offered by the finite formulation of the electromagnetic (EM) field, employing integral field quantities and dual meshes, and by the domain-integrated field relations approach to EM field computation.

Why starting from differential equations for computational physics

The computational methods currently used in physics are based on the discretization of differential equations. This is because the computer can only perform algebraic operations. The purpose of this paper is to critically review this practice, showing how to obtain a purely algebraic formulation of physical laws starting directly from experimental measurements. In other words, we can get an algebraic formulation avoiding any arbitrary process of discretization of differential equations. This formulation has the great merit of maintaining close contact between the mathematical description and the physical phenomenon described.

A new meshless approach for subject-specific strain prediction in long bones: Evaluation of accuracy.

BACKGROUND The Finite Element Method is at present the method of choice for strain prediction in bones from Computed Tomography data. However, accurate methods rely on the correct topological representation of the bone surface, which requires a massive operator effort, thus restricting their applicability to clinical practice. Meshless methods, which do not rely on a pre-defined topological discretisation of the domain, might greatly improve the numerical process automation, but currently their application to biomechanics is negligible. METHODS A meshless implementation of an innovative numerical approach based on a direct discrete formulation of physical laws, the Cell Method, was developed to predict strains in a cadaver femur from Computed Tomography data. The model accuracy was estimated by comparing the predicted strains with those experimentally measured on the same specimen in a previous study. As a reference, the results were compared to those obtained with a state-of-the-art finite element model. FINDINGS The Cell Method meshless model predicted strains highly correlated with the experimental measurements (R2=0.85) with a good global accuracy (RMSE=15.6%). The model performed slightly worse than the finite element one, but this was probably due to the need to sub-sample the original data, and the lower order of the interpolation used (linear vs parabolic). INTERPRETATION Although there is surely room for improvement, the accuracy already obtained with this meshless implementation of the Cell Method makes it a good candidate for some clinical applications, especially considering the full automation of the method, which does not require any data pre-processing.

Sulla struttura formale delle teorie fisiche

Starting from the observation that among physical quantities of every physical theory there are many ones that are naturally referred to the basic geometrical and chronometrical objects, it is shown how one may obtain a classification scheme for the physical quantities of every physical theory. This permits to assign a definite behaviour of physical quantities under time reversal and space inversion. The association of physical quantities with chrono-geometrical objects gives a motivation for the introduction in physical theories of the notions of multivectors, fiber bundles, exterior differential forms and connection theory. Typical equations of physics are shown to be described by a single mathematical process, the coboundary process (exterior differential). The basic equations of physics are then divided into two classes: metric independent equations, i.e. purely topological equations, and metric dependent equations, the phenomenological equations. Classification schemes for the main physical theories close the paper.

The Origin of Analogies in Physics

The paper gives the reasons for analogies in physics showing that they arise from the natural link between global physical variables and the space elements, i.e. points, lines, surfaces and volumes.

Analysis of Physical Equations

While physical variables describe the quantitative attributes of physical systems, the equations linking them describe the quantitative behaviour of phenomena, i.e. the physical laws. We will distinguish four kinds of equations used in physics, and for each kind we put into evidence their mathematical structure. The main classification is that of defining equations, topological equations, equations of behaviour and phenomenological equations.

General Structure of the Diagrams

We start with diagram [GEN1], which shows the relation of exterior differential forms with the classification diagram. The physical variables associated with space elements endowed with an inner orientation are described by multicovectors and, hence, by even differential forms. In contrast, the physical variables associated with space elements endowed with an outer orientation are described by pseudo multicovectors and, hence, by odd differential forms.