# Structural Application

## General Description

The KRATOS Structural Application contains a number of elements, conditions, strategies and constitutive laws that deal with the Finite Element Analysis of problems in structural mechanics. It covers 2D and 3D calculations in linear and nonlinear static and dynamic structural mechanics. Additional utilities are capable of considering contact problems, the deactivation and reactivation of elements and a large number of boundary conditions.

An arc:

A metal dome:

A simple Beam:

### Theory

Ones of the aim of this application is to solve the well known set of Poissson equations. The Poisson equations express in mathematical form the behavior of many physical problems. The simplest form of the Poisson equations is as follows:

$\quad \text{In 1D} \quad k \frac{\partial^{2} u}{\partial x^{2}} + Q(x) = 0 \quad \text{in} \quad \Omega$

$\quad \text{In 2D} \quad k \frac{\partial^{2} u}{\partial x^{2}} + k \frac{\partial^{2} u}{\partial y^{2}} + Q(x,y) = 0 \quad \text{in} \quad \Omega$

$\quad \text{In 3D} \quad k \frac{\partial^{2} u}{\partial x^{2}} + k \frac{\partial^{2} u}{\partial y^{2}} + k \frac{\partial^{2} u}{\partial z^{2}} + Q(x,y,z) = 0 \quad \text{in} \quad \Omega$

$\quad \quad \quad \quad \quad \nabla\cdot\mathbf{u} = 0 \quad \text{in} \quad \Omega$

$\mathbf{u} = \mathbf{u_{0}} \quad \text{in} \quad \Omega, t=0$

$\mathbf{u} = \mathbf{0} \qquad \text{in} \Gamma, t\in ]0,T[$

where u is the displacement fields of the problem and Q is the source term. The form of this equations assumes that k is constant. In fact k can be a function of the position and even of the problem unknown u and its derivatives, as it happens in non-linear problem.