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10.2 Finite elements

The mathematical model is subdivided by finite elements which are connected to each other with nodes. Forces act at the nodes. The finite element is not a rigid body, the model assumes stresses and strains exist inside the finite elements. There are a few commonly-used finite elements: beam A, truss B, thin shell C, 2D or 3D continuum (D). It is possible to use several types of elements in one model. Reliability of the FEA predictions depends on the number of finite elements. If the inner stresses do not vary greatly then the number of elements does not have a significant effect on the accuracy. Solid elements may be linear (first-order elements) or parabolic (second-order elements). Linear elements have corner nodes only and their edges are straight. The minimum number of nodes for 3D elements is 4. Parabolic elements can have a node placed centrally along each edge and therefore the edges are parabolic. Given the same number of elements, the higher order elements are more accurate because they have a more complex mathematical formulation to describe the element shape (shape function). Also, they represent curved geometry more accurately. An analysis involving higher order elements requires more computational resources. There are 3 degrees of freedom in a node and 8 nodes for a brick (hexahedral) element. This means that 24 nodal displacements and 24 nodal forces must be considered. The size of the stiffness matrix that relates the nodal displacement vector with the nodal forces vector is [24*24]. The stiffness matrix components are inversely proportion to the modulus of elasticity. Zero modulus of elasticity means that there is no finite elements. Division by zero modulus of elasticity leads to numerical errors in the FEM procedures. Infinity modulus of elasticity means that a part of structure is absolutely rigid. Although in the theory of elasticity the tensile stress in the crack tip is equal to infinity, all stresses are finite in FEM. Long elements can be used if there is not a large gradient of displacements, strains and stresses. Such elements can be used in fields of uniform stress but not for stress concentrations. The stress gradient is high but still finite in zones of stress concentration. If a structure and its loads are symmetrical about the axis the problem can be solved using axisymmetric 2D finite elements. Five tetrahedral elements are enough to form a cube. Parabolic pyramid elements provide results that are at least as accurate as linear brick elements.

10.3 Meshing

A key step in finite element analysis procedure is to mesh the model. Meshing is process of breaking the model into small pieces (finite elements). The network of nodes and elements is called a mesh. There are two broad types of mesh-generation methods: structured and unstructured meshes. A structured mesh B is formed by grid-based subdividing of the geometry. Unstructured mesh are formed automatically. The size of neighboring elements can be significantly different for unstructured mesh. There are no "rows and columns" for such mesh, A. There are more nodes than elements for unstructured mesh with a large number of elements. The ratio between elements and nodes is approximately 2:1 for 2D unstructured mesh and 6:1 for 3D unstructured mesh with tetrahedral elements. Smaller mesh size h corresponds to a larger number of finite elements in the model. The calculation time increases exponentially as size decreases. The errors decrease for finer mesh but never fall to zero since FEM is always an numerical approximation. A linear element requires a finer mesh than a parabolic (quadratic) or a cubic element. Structured mesh B is preferable over an unstructured mesh, A. Rectangular 4 node elements, C are more preferable than triangular elements, B. Quadratic (second order) triangular elements, D have at least the same accuracy as first order 4 node elements, C. Rectangular 8 node elements, E are preferable over triangular second order elements, D despite their larger size. Cubic displacement approximation F does not need fine meshing. The FEM is an approximate method. The accuracy of the predictions depends on the assumptions made within the element types and the mesh. A fine mesh is required where there are stress and strain gradients (rates of change). A coarse mesh can be used in areas of reasonably constant stress or regions that are not of user's interest. Users must be able to identify regions of stress concentration. Points of interest may consist of fracture points of previously tested structure, holes, fillets, corners, contact zones, complex details, and high stress areas. The accuracy decreases if the sizes of the neighboring elements near stress concentrators are significantly different. The shape of finite elements affects the accuracy. It is preferable not to have sharp corners in finite elements. Elements with similar sides produce smaller errors. The FE mesh is built without gaps between elements. Both triangular and rectangular elements can be used in the same FE model. The nodes are numbered sequentially for manual meshing. It is forbidden to build four node elements with an obtuse (> 180o) inner corner.

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