The Finite Element Method

FEM for Two-Dimensional Solids (Finite Element Method) Part 4

Elements With Curved Edges Using high order elements, elements with curved edges can be used in the modelling. Two relatively frequently used higher order elements of curved edges are shown in Figure 7.20(a). In formulating these types of elements, the same mapping technique used for the linear quadrilateral elements (Section 7.4) can be used. In […]

FEM for Two-Dimensional Solids (Finite Element Method) Part 5

Solution Process Let us now try to relate the information we provided in the input file with what is covered in this topic. As before, the first sets of data usually defined are the nodes and their coordinates. Then, there are the element cards containing the connectivity information. The importance of this information has already […]

FEM for Plates and Shells (Finite Element Method) Part 1

Introduction In this topic, finite element equations for plates and shells are developed. The procedure is to first develop FE matrices for plate elements.Unlike the 2D solid elements in the previous topic, plate and shell elements are computationally more tedious as they involve more Degrees Of Freedom (DOFs). The constitutive equations may seem daunting to […]

FEM for Plates and Shells (Finite Element Method) Part 2

Higher Order Elements For an eight-node rectangular thick plate element, the deflection and rotations can be summed as where the shape function Ni is the same as the eight-node 2D solid element given by Eq. (7.52). The element constructed will be a conforming element, as w, θχ and θγ are continuous on the edges between […]

FEM for Plates and Shells (Finite Element Method) Part 3

Solution Process Looking at the mesh in Figure 8.6, one can see that quadrilateral shell elements are used. Therefore, the equations for a linear, quadrilateral shell element must be formulated by ABAQUS. As before, the formulation of the element matrices would require information from the nodal cards and the element connectivity cards. The element type […]

FEM for 3D Solids (Finite Element Method) Part 1

Introduction A three-dimensional (3D) solid element can be considered to be the most general of all solid finite elements because all the field variables are dependent of x, y and z. An example of a 3D solid structure under loading is shown in Figure 9.1. As can be seen, the force vectors here can be […]

FEM for 3D Solids (Finite Element Method) Part 2

Hexahedron Element Strain Matrix Consider now a 3D domain, which is divided in a proper manner into a number of hexahedron elements with eight nodes and six surfaces, as shown in Figure 9.9. Each hexahedron element has nodes numbered 1, 2, 3, 4 and 5, 6, 7, 8 in a counter-clockwise manner, as shown in […]

FEM for 3D Solids (Finite Element Method) Part 3

Higher Order Elements Tetrahedron Elements Two higher order tetrahedron elements with 10 and 20 nodes are shown in Figures 9.13(a) and (b), respectively. The 10-node tetrahedron element is a quadratic element. Compared with the linear tetrahedron element (four-nodal) developed earlier, six additional nodes are added at the middle of the edges of the element. In […]

FEM for 3D Solids (Finite Element Method) Part 4

Elements With Curved Surfaces Using high order elements, elements with curved surfaces can be used in the modelling. Two relatively frequently used higher order elements of curved edges are shown in Figure 9.18(a). In formulating these types of elements, the same mapping technique used for the linear hexadron elements (Section 9.3) can be used. In […]

Special Purpose Elements (Finite Element Method) Part 1

Crack Tip Elements In fracture mechanics, much interest for analysts is on the tip of the crack, as it is a singularity point where the stress field becomes mathematically infinite. When modelled with the conventional, polynomial-based finite elements discussed in previous topics, the finite element approximations are usually quite bad unless a very dense mesh […]

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