FEA Basics

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Glossary

 

3D surface representation: A model of a 3 dimensional surface made up (usually) of plate elements that share vertices and thus approximate the complete shape of the physical surface they represent. The larger the number of plate elements (and hence smaller in individual size), the closer the model can approximate the actual shape of the physical surface.

 

Brick finite element: A brick element is a 3-D six-sided, hexahedral element.  A linear brick element is defined by 8 nodes- one at each corner of the hexahedral. This means that the field variable (i.e. displacement field) is interpolated from nodal values using linear interpolating polynomials, and the shape of the hexahedral is defined by straight lines and flat surfaces (i.e. linear functions). A quadratic brick element is defined by 20 nodes- one at each of the corners of the 8 corners of the hexahedral, and one node located along each of the 12 edges of the hexahedral element. The field variable is interpolated from nodal values using quadratic interpolating polynomials, and the shape of the hexahedral may by defined by quadratic functions. Thus, a side or face of a quadratic brick element may match a portion of the geometry of a cylinder.  All other things being equal, brick elements are most accurate when interior angles are 90 degrees and the aspect ratio is 1, i.e. the brick element is a cube. 

The Linear Brick Element. The element need not be aligned with the coordinate axis or have square corners.

 

A brick element has three degrees of freedom per node corresponding to displacement components in the three coordinate system directions at each node. For a given number of nodal degrees of freedom, brick elements are more accurate than tetrahedral elements and should be used over tetrahedrals. However, automatic mesh generation of complex 3-D geometry with brick elements is problematic with current tools, whereas most meshing tools today have fairly reliable automatic tetrahedral mesh generators that will mesh relatively complex geometry.  Thus, due to limitations in automatic mesh generation capability, the user may be forced to use tetrahedral elements. Note that biological systems typically are much more irregular, i.e. geometrically complex, than engineered products or manufacturing processes that finite element tools have been developed for.

 

Finite element analysis: Analysis of the physical behavior of a finite element model. In this analysis a given physical "treatment" is applied (such as force loading on some elements), followed by computation of the effects of this treatment on other elements of the FEM. More specifically, the analysis phase involves solving a set of simultaneous algebraic equations in which the unknown variables that are solved for in this system of equations are the unknown nodal degrees of freedom. Once the values of the unknown nodal degrees of freedom are found, the unknown reaction forces are determined. Next, the spatial variation of the primary field variables within each element is computed using the element's predefined interpolating polynomials and the element’s nodal values. Thus, for solid elements this results in mathematical functions that completely define the displacement field within every finite element. These functions are then differentiated to obtain the complete strain field within each element which, when combined with known material properties of the element, yields the element stress field. In summary, the finite element solution will yield 1) the reaction forces necessary to maintain static equilibrium of the system, 2) the displacement field ( i.e. displacements of the material through out the 3-D domain), the complete strain tensor field, and the complete stress tensor field.

 

Finite element model: A computer-based model of a physical object or system composed of contiguous set of finite elements and associated nodes that model the volume of the object. The finite elements are assigned material properties such as elasticity, thermal conductivity, etc. to account for the material behavior associated with the physical phenomenon being modeled. A finite element model (FEM) is the dataset used for finite element analysis (FEA). Individual finite elements may have different material properties, loading conditions, size, shape, etc., enabling the most complicated physical systems to be modeled.

 

Nodes: The points that define the geometry of finite elements. For the simplest shaped finite elements, these points are located at the vertices of the polygon (triangle or quadrilateral shape) or polyhedron (tetrahedral or hexahedral shape) that represent the element geometry. Nodes can also be located midway between element corners, resulting in curved edges and curved faces.

 

Nodes also have degrees of freedom associated with them. Degrees of freedom are the primary variables that are directly solved for in a finite element analysis. For structural problems nodal degrees of freedom are the values of the displacements or translations of the material at each node in each coordinate direction for solid elements. For plate finite elements, nodal degrees of freedom include, in addition to translations,  rotations of the material about each coordinate axis.  Thus, a plate finite element will have 6 degrees of freedom per node (3 translations and 3 rotations) and a solid finite element will have 3 degrees of freedom per node. Before a finite element analysis is conducted, most nodal degrees of freedom are unknown, except at locations where the model is being constrained against motion. Constrained degrees of freedom are known; the values of these degrees of freedom are specified before the analysis is conducted. However, prescribed degrees of freedom have unknown forces corresponding to these prescribed values. These forces are called reaction forces. Conversely, all unknown degrees of freedom have known forces prescribed.

 

Plate finite element: A plate finite element is similar in shape to the plate geometry element except the element has thickness associated with it, material properties, and an underlying structural behavioral model that accounts for the stiffness of the physical structure represented by the element. The points that define the geometry of finite elements are called nodes. Each node of a plate finite element has 6 degrees of freedom corresponding to displacements of the material at the node in the three coordinate directions and rotation of the material at the nodal location about each of the three coordinate axis.  Thus, a three-noded triangle plate finite element has 18 degrees of freedom.

 

Plate geometry element: A geometry model element that has area but not volume. The simplest example is a triangular element defined by three vertices, which is by definition planar. Non-planar plate elements defined by 4 or more vertices are possible.

 

Polygon model: A model composed of polygonal elements or plates. The model can be "watertight" and thus define a volume, or it can be open and only define a surface. Watertight polygon models are the forms used to build the polyhedron models used in finite element analysis.

 

Tet-4: This is the simplest possible 3-D finite element, consisting of only 4 nodes with 12 total degrees of freedom- three degrees of freedom at each of the element's four nodes corresponding to displacements in the three principle coordinate directions. The Tet-4 element admits linear displacement fields, which means the displacements within the element in the three coordinate directions are interporalated from the element's nodal displacements using known linear interpolating polynomial functions.  This also means that the strain tensor field, which is given by derivatives of the displacement field, is constant over the element volume. For this reason, this element is also called a constant strain tetrehedral (CST). If the material properties are also assumed to be constant over the element volume, the element is also a constant stress element.

Four- Noded Tetrahedral Element

 

Tet-10: This is a quadratic tetrahedral element defined by four vertix nodes and six nodes placed along the six edges of a tetrahedral. Ideally, these midside nodes should be placed at the center point between the two vertices of the edge. However, the element can be quadratic in shape with curved edges. The element has 30 total degrees of freedom corresponding to displacements in the three principle coordinate directions at each of the element's ten nodes. The displacement field within the element is interpolated from the element's nodal displacements using quadratic interpolating polynomial functions. This means that the strain tensor field can vary linearly over the element volume. If the material properties are also assumed to be constant over the element volume, then the stress tensor field can also vary linearly over the element volume. This element performs much better under bending type loads than the Tet-4 element.

Ten- Noded Tetrahedral Element (note that element edges may be curved)

 

Tetrahedral element: Tetrahedral elements are 3-D elements that have the shape of a tetrahedral. There are two common types of tetrahedral elements: linear, 4 noded tetrahedrals (tet-4) and quadratic 10 noded tetrahedrals (tet-10). Linear tetrahedrals are not well suited for problems that involve bending type loads, particular of thin structures. If they are to be employed for such applications, then it is recommended that the element size be such that at least 3 elements lie in the thickness direction of the structure.

 

Copyright - biomesh.org, 2007

 

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