Spatial integration

The basic idea of the finite element method is to replace the original continuous problem (infinitely dimensional) by a similar finite dimensional problem. This process is called discretization, where the entity discretized is the variational form (2.57). The first step comprises the geometrical approximation of the bodies’ domain, on which the boundary value problem is defined. Thus, the continuum domain is discretized by Ne finite elements, such as:

h h 1 Ω Ω Ω , Ne i i   (3.2)

where Ω represents the discrete domain of h Ω and Ωh_i denotes the domain of a generic finite element. The operator stands for the addition operation between all elements. Figure 3.2 presents the spatial discretization of a continuous deformable body, which is defined by a set of finite elements, each one composed by a set of nodes, indicated by dots. All degrees of freedom in the discrete system are associated with these nodes. The shape of each element is completely characterized by the coordinates of the nodes attached to it and the associated shape functions. Note that for a finite number of nodes, some points of the continuous body have no counterpart in the discretized geometry and vice versa, as shown in Figure 3.2. This difference occurs only in the boundary, which is a very important location for accurately imposing boundary conditions, particularly during the contact treatment.

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Figure 3.2. Continuous body and its discretized representation using a finite element mesh composed by finite elements and nodes.

For the spatial discretization of the deformable body, standard three-dimensional isoparametric finite elements are employed. Based on the isoparametric concept, the same shape function is applied to interpolate both the geometry and the independent field variables (deformations or stresses), such as:

h h h h 1 1 ( ) , and ( ) , Nn Nn i i i i i i N N    





where Nn defines the number of nodes of the finite element, N_i represent the element shape functions (Lagrange polynomials), while h

i

x and h

i

u denote the unknown nodal coordinates and displacements, respectively. The shape functions N ξ_i( ) are defined with respect to the reference element geometry or parameter space, commonly denoted as natural coordinates ξ( ,ξ ξ ξ_{1 2}, )₃ for three-dimensional problems (see Figure 3.3). For general shape functions, see [Zienkiewicz 00a]. The use of shape functions (3.3) in the finite element method introduces restrictions on the solution and weighting spaces defined in (2.22). In the discrete setting, these spaces only contains a finite number of solution and weighting functions, respectively, which is expressed mathematically in terms of finite dimensional subspaces h_{ and} h _{ .}

After the division of the body domain (volume of integration) into a finite number of solid elements, the integration over each element domain is performed approximated via numerical integration procedures. The Gauss quadrature method is used to integrate all field variables over the solid element domain. This method is easily employed in the reference configuration (canonical domain) of the finite element, depicted in Figure 3.3

(left). For every finite element, the coordinates of each point defined in the Euclidean space are related with the local coordinates of the element through the shape functions (3.3), as shown schematically in Figure 3.3. Then, the elemental contributions are sorted into global vectors based on the assembly operator, which manages the position of each local vector quantity into the global vector, such as:

Ω Ω

General Finite Element Framework 65 h h 1 Ω Ω ( ) Ω ( ) Ω , i Nel i i d d    



_A

where A is the standard finite element assembly operator.

Currently, the finite element library of DD3IMP is composed by tetrahedral and hexahedral solid elements, namely: (i) 4-node linear tetrahedral; (ii) 8-node tri-linear hexahedral; (iii) 10-node quadratic tetrahedral; (iii) 20-node serendipity hexahedral; (iv) 10- node quadratic tetrahedral and (v) 27-node tri-quadratic hexahedral. Tetrahedral elements are geometrically more adaptable and easier to handle in automatic meshing of complex shapes than hexahedral elements [Tekkaya 09]. Nevertheless, standard linear tetrahedral elements are overly stiff, very much sensitive to mesh distortion and the plastic incompressibility constraint results in volumetric locking. In fact, since a tetrahedral element is geometrically a degenerated hexahedral element, more tetrahedral elements are required to achieve the same level of accuracy as in hexahedral elements [Benzley 95]. The tri-linear hexahedral isoparametric elements, when associated with a full integration scheme, present a deficient behaviour in elastoplastic problems [Nagtegaal 74]. The element stiffness increases causing the occurrence of artificial hydrostatic stresses, which leads to a complete deterioration of the solution. This effect can be eliminated using a selective reduced integration method, where a reduced integration is used only in particular terms of the stiffness matrix. Regarding the method currently implemented in DD3IMP, the hydrostatic component of the velocity field gradient is considered constant in the whole element (calculated at its central point), replacing its evaluation in each integration point [Hughes 80]. On the other hand, the uniform reduced integration applied to all terms of the stiffness matrix prevents volumetric locking in nearly incompressible cases, nevertheless introduces spurious zero-energy deformation modes that lead to hourglassing. Presently three integration methods are available in DD3IMP code: (i) Full Integration (FI); (ii) Uniform Reduced Integration (URI) and (iii) Selective Reduced Integration (SRI). The elemental stiffness matrix and the nodal force vector definition that results from the discretization of the linearized principle of virtual velocities can be found in [Oliveira 08] for each integration method.

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Figure 3.3. Isoparametric 8-node tri-linear hexahedral solid element.