   # Non-linear finite element method The finite element .The finite element method is first developed

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• The finite element method

Traditionally finite element analysis (FEA) is used to perform strength analysis. FEA is in most cases cheaper to use, than perform a destructive test on a prototype. Moreover a finite element simulation provides a detailed insight into the behaviour of the structure. The finite element method is not meant to replace tests, completely, but as partial replacement and complementary to tests. For example some crash tests on new car models are carried out still, but dozens, or more likely, hundreds of crash simulations are performed. The crash tests serve then especially as a benchmark for crash simulations.

The finite element method is first developed in 1943, for the simulation of a vibration problem. In 1956, the method was adapted for more general application. In the 70s of the previous century the method was commercially applied, in particular in aerospace industry, nuclear industry, motorcar industry, and defense industry. In the following period FEA has become a generally accepted tool in all industries concerned with structural strength of their products.

One of the first computers

Non-linear finite element method

The non-linear finite element method is based on the linear finite element method. Increased computational power of computers has opened the opportunities for more complex simulations of structural behaviour. When a load is doubled the deflection is twice as large is true for a linear finite element analysis. That is valid for a certain type of structures and for certain loads. When loading increases to such a level that plasticity occurs, linear behaviour does not apply anymore, obviously. This type of behaviour is called non-linear.

The three most important sources of non-linearity are plasticity or partial failure, geometrically like buckling and finally contact between components of the structure. Recalling the mentioned sources on non-linearity it is clear that the non-linear FEA can be well applied for production simulation. For example deep drawing of a metal component, material non-linearity (plasticity), contact between blank and mould, and large displacements all play an important role.

• Deep drawing, simulation and test

Overview non-linear phenomena

The three already mentioned sources on non-linearity might all be combined.

non-linear type: material

non-linear type: geometry

non-linear type: boundary conditions

Plasticity large displacements contact between parts

Creep large strains follower forces

visco-elasticity and hyper-elasticity

buckling and stability

failure

Explicit simulations

Explicit finite elements analysis is used for short-term processes. Examples are crash analyses, collisions, impact, falling objects, etc. This analysis type is more suitable for analyzing large non-linearities than the standard (implicit) finite elements analysis.

Crash analysis

Technology behind the finite element method

The structure is divided into a finite number of small discrete parts (elements). For each of these elements a simplified behaviour is assumed. This behaviour is described by mathematical expressions. By assembling all these elements a (large) set of equations. Computers are extremely good in solving large sets of equations.

The assembly and solving of the finite element equations is shown schematically in the following figure.

• Schematic view of finite element analysis

Description of the individual steps

Dividing a structure into the elements is generally performed by employing a mesh generator or a pre-processor. The number of elements can easily exceed a few hundred thousands. When the model is divided into elements, stiffness must be assigned to every element. The stiffness is defined by a combination of material and specific properties like a thickness, a cross-sectional area, etc. The extra definition depends on the element type. After defining the stiffness for all elements, the boundary conditions must be defined. Boundary conditions consist of the support of the structure and the applied loads. Loads might be mechanical loads, thermal loads, magnetic loads, etc.

When the definition is finished the finite element program can do its job. When the input is correct, and describing the effects we have in mind, the finite element program guarantees equilibrium. The finite element program also takes care of the compatibility of all elements.

Most of the computational effort is then spent on solving the set of equations. The solution consists of the displacements on the nodes of the finite element model. These displacements are then used to compute the internal forces, stresses and strains in the model. This completes the solutions.

Finally the solution is visualised (post-processing) and the quality of both analysis and the structure is assessed.

Characteristics of Finite Element Method (FEM)

FEM simulated complex structural behaviour and allows therefore an early intervention in the design process.

Using FEM requires less physical prototypes.

FEM does not replace physical tests. In general FEM and tests are complementary to each other, although FEM reduces the number of physical tests.

More design alternatives can be analysed in less time.

You can assess stresses, strains, displacements and other responses everywhere in the model, even inside the structure.

FEM is a method to approximate the real physical behaviour. The approximation is only sufficiently accurate when the quality of the finite element model is good and the material properties are accurate and the representation of the loads and other boundary conditions are correct and the chose solution technique is the correct one.

Assign properties and boundary conditions

Solve equations and analyse results

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