FEM: Finite Elements Method – Definition and Principles

Have ever you asked: what is finite elements method? Or what are the definitions and principles of finite elements methods? In this article, we will briefly answer these questions:

The main aim of Finite Elements Method (FEM) applied to engineering is simulates physical events of the real world in a virtual environment created by a computer. The FEM is part of Computer Aided Engineering (CAE), and with it is possible to simulate several events for design interests, like structural analysis, thermal, acoustic, electromagnetic, fluids etc. All them with linear or non-linear behavior and with the possibility to check interaction between parts or more that one physical event at once (multiphysics analysis).

The principle behind FEM is to divide the body (or event) in a finite number of elements (discretization) that will compose the total. All of the small elements have a specific behavior that can be described by maths equations.  These will form a Differential Equations Matrix that can be solved by a numeric method, which in turn can be easier worked out by a computer. As the system is made by a matrix of finite elements, the solution isn´t exact, but with an approximate solution (convergence). As much smaller and numerous are the elements, more accurate the result will be, by the other hand, more computerizing power and time will be required to solve the equation system.

Figure 1 - Discretization of the body in finite elements. Font: ESSS

Figure 2 - Discretization of a continuous function in other linear. How much smaller is the interval, more accuracy about original function.

Figure 3 - FEM isn´t an exact solution. Better quality of inputs, better quality for outputs.

The small elements can have different forms, such as triangular, quadrangular or hexahedrons. For each application, some specific form can provide better results. All elements are connected by points, called nodes or nodal points, and is on these that calculations are done, considering itself´s behavior and of immediate neighbor, through differential equations (from derivations, rate variations); and values of intermediate points are solved by interpolation deduction. This group of nodes and elements is called of mesh, characterized by its density. How bigger, more small elements there are.

Figure 4 - Mesh with its nodes and elements. Model from ESSS

Types of Analysis to be Considered:

The analysis can be static or dynamic, and linear or non-linear. Actually, all real events are non-linear, but in special conditions, some of them can be analyzed like a linear behavior, and with this approach, it´s easier to solve some problems.

1.       Structural Analysis: Used to provide information about strain, displacement and stress distribution for a body under loadings.

·         Static Analysis: The loading doesn´t change with time.

Linear: Force (F) and strain (ε) are proportional;

Nonlinear: Force (F) and strain (ε) aren´t proportional. The rigidity changes with strain. The boundary conditions changes with strain.

Some reasons to linear and non-linear behavior:

                  I.            Elastic regime and plastic deformation on Strain (ε) x Stress (σ) material curve;

Figure 5 - Linear and nonlinear behavior.

For elastic regime (σ= E ε), very low strain (ε) and very low geometry changes (Ѵ≥0.3), the analysis can saw as linear.

                  I.            Nonlinearity due changes in the body geometry:

Figure 6 - Nonlinearity due geometry modification Font: ESSS.

              III.            Non-linearity due variation of contact between bodies:

Figure 7 - Contact between bodies cause non-linearity Font: ESSS.

·        ·         Dynamic Analysis: The loading changes with time.

Modal: Estimate natural frequencies and vibration modals;

Harmonic: Determines the body response under an harmonic (sinusoidal) loading;

Transient Dynamic: Determines displacements amplitude and stress of a body under a loading that variates aleatory with time (no defined function);

Spectral: It is a sequence of Modal Analysis, but to calculate behavior due randomly variation in the frequency domain. Usually for simulates situations like earthquakes;

Explicit Dynamics: This is used for high non-linearity and contacts, in which there is high inertial forces and great deformations, capable to change the mesh form through time and material and loading proprieties. This analysis is usefully to crash-tests, falls, conformations and forging.

Figure 8 – Explicit Dynamics for crash test simulation. Font: ESSS

Fatigue Analysis: For estimate the life cycle of a body under cyclic loading. This is a kind of post-processing.

Buckling Modal Analysis: This evaluates what is the critical load to lead the structure to collapse in an unstable balance.

2. Fluids Analysis: Used to describe how would be a flow internal or external through a body and its distribution of pressure, velocity and temperature. Flow can be laminar, turbulent, compressible or incompressible. It´s possible to analyze complex phenomes like chemical reactions and multiphase flows.

                  3. Thermal Analysis: Provides information about the distribution of temperature, flow                        and thermal gradient in a body under heat transfer through conduction, convection or                            radiation.

Linear Solution: The material thermal properties don´t change with time or temperature and there is no radiation.

Nonlinear Solutions: The material thermal properties, boundary conditions and loading can change with temperature. Radiation influence is considered. This analysis is analogous with the structural nonlinearity.

Steady-state: Analysis the thermal equilibrium condition, which there isn´t influenced by time. Temperature distribution and thermal flow are stabilized. The energy balance supports to calculate the steady-state. Is usefully to aid structural analysis.

Transient Analysis: Useful when variables change with time from an initial condition well knew, to some final process, which could end in a steady-state. The most important for this analysis is to estimate the time required to run all thermal process. Is possible with it simulate the phases changing. The accuracy of the analysis depends on time steps chose, wherein the beginning process normally needs smaller time steps.

                   4. Multiphysics analysis: With multiphysics analysis is possible to simulate structural,                      fluids, thermals and others at once. In real conditions and process, the piece normally is                        affected by simultaneous physics events, as well as they are mutual influenced. This can                      save time and achieve designs of products more reliable.

Figure 9 - Example of a multiphysics analysis. Font: ESSS.