Mathematics

Time series, signals and chaotic dynamics

Thu 19 Mar - 2020

As is well known, in many scientific studies, there are quantities measured in the form of time series, or signals, derived either from experimental measurements or from a series of consecutive observations. These time series are often the only source of information for characterizing the dynamics of the observed system and predicting its future behavior. Of course, many methods have been developed in the past for time series analysis, such as Fourier analysis, correlation functions, ARMA models, etc. But these classical methods are largely based on the assumption that time series evolve...

Category: 
Mathematics

Variation, virtual displacement and stationary value of a functional

Wed 18 Mar - 2020

The so-called extreme principle of mechanics (e.g. principle of minimum potential energy) and the principle of virtual work (e.g. principle of virtual displacements) play a decisive role in understanding FEM and deriving the basic equations. The calculus of variations is available for the mathematical treatment of this principles. The variational calculus is a higher teaching of maxima and minima. We start with the concept of the variation of a function $y(x)$. A variation of the function $y(x)$ is defined as

$$\delta y = {\bar y}(x) - y(x).$$

Here ${\bar y}$ is a...

Category: 
Mathematics

Application of Newmark-β method in Structural Dynamics

Wed 04 Mar - 2020

In the field of analysis of structures, it is often desired to calculate the dynamic response of a structure subjected to general dynamic excitation. In order to achieve the calculation of the response, usually numerical integration schemes are applied. In most cases the structure is also discretized by the finite element method resulting in a dynamic system of differential equations, in matrix form with a large number of unknowns. Special numerical integration methods have been developed for these cases like the Linear Acceleration method, the Central Difference method, the HHT-α method...

Category: 
Computing, Engineering, Mathematics, Programming

Inertia Relief Analysis

Fri 27 Sep - 2019

Inertia relief is an effective technique for the analysis of structures without supports, often combined with the finite element method. A structure without supports basically exhibits a rigid body motion. In order to evalute the elastic behaviour of the structure and calculate stresses, strains etc. the rigid body motion must be eliminated from the structure. Simply put, the structure must be "relieved" of inertial effects in order for a static equilibrium to be established. Examples of such structures are aircrafts, launch vehicles, spacecrafts, marine structures, machine components etc...

Category: 
Engineering, Mathematics

An Introduction to the Finite Element Method

Wed 26 Jun - 2019

The Finite Element Method (FEM) can be seen as a numerical computation method for the solution of problems of mathematical physics. The concept of mathematical physics must clarify the universality of the method. To the problems of mathematical physics, which nowadays to a large extend are solved with the help of FEM, belong primarily:

problems of structural mechanics (statics and dynamics), where we understand here under the term structure a structural framework in the broadest sense, not only special types of structures such as trusses and frames, surface or body structures, but...
Category: 
Engineering, Mathematics

Basic Linear Algebra in Gaea

Wed 19 Jun - 2019

The Gaea package " array " is giving access to some of the most useful functions in Gaea, i.e. basic linear algebra operations. In a previous blog post we have seen the available element-wise operations. In this post we are going to have a look to the rest of package " array ". More specifically this library provides the following:

element-wise operations of arrays, elementary operations of arrays, such as [+], [-], [*], dot, cross, other operations of arrays, such as diag, eye, linspace, ones, transpose, zeros, a user interface for taking slices of array-like objects, an extensible...
Category: 
Computing, Mathematics