VNE

Two- and Three-dimensional variable-node elements have been developed using moving least square (MLS) approximation and point interpolation. For two-dimensional quadrilateral elements, the variable-node finite elements were proposed to allow an arbitrary number of nodes on element edge for four-node linear elements and nine-node quadratic elements. Moreover, based on an eight-node hexahedral element in three-dimensional domains, the variable-node element was developed to allow additional nodes on the element face as well as element edge. These elements satisfy the basic properties of finite elements, such as partition of unity, linear or quadratic completeness, Kronecker delta condition. Therefore, the variable-node elements are directly used in the framework of the conventional FEM, without any process such as projection, interpolation, and imposition of constraints on the interface between the different meshes. In addition, when the domain is constructed with the variable-node elements, the system matrix remains symmetric, and then the symmetric solver is applicable to efficiently obtain the solution. That is, the use of this element makes it possible to connect the different-level meshes in a seamless way, satisfying nodal connectivity and compatibility across the interface.

Fig. 1 A two-dimensional (4+k+l+m+n)-node element

Fig. 2 A three-dimensional (8+3p+3q+3r+pq+qr+rp)-node element

application of variable-node elements

Simulation of crack propagation with the aid of crack elements

Two-dimensional finite 'crack' elements for simulation of propagating cracks are developed using the moving least square (MLS) approximation. The mapping from the parental domain to the physical element domain is implicitly obtained from MLS approximation, with the shape functions and their derivatives calculated and saved only at the numerical integration points. The MLS-based variable-node elements are extended to construct the crack elements, which allow the discontinuity of crack faces and the crack-tip singularity.

[Related paper] Y.-S. Cho and S. Im, "MLS-based variable-node elements compatible with quadratic interpolation: Part II. Application for finite crack element", International Journal for Numerical Methods in Engineering, Vol. 65, Issue 4, pp. 517-547, 2006.

The edge cracking

Fig. 3 Contour plots of von Mises stress for edge cracking and spalling.

Double cantilever beam with kinked crack

Fig. 4 Contour plots of von Mises stress for double cantilever beam with the kinked crack.

Variable-node elements combined with stabilized conforming nodal integration

Two-dimensional (4+k+l+m+n)-node elements are proposed to handle non-matching mesh problems and microstructure modeling. It is noted that these elements, which are even highly irregular and distorted, show an good performance in terms of accuracy and stability when it is combined with stabilized conforming nodal integration. As shown in Figs. 5 and 6, they also pass the patch test without any ambiguity.

[Related paper] J.H. Lim, D. Sohn, J.H. Lee, and S. Im, "Variable-node finite elements with smoothed integration techniques and their applications for multiscale mechanics problems", Computers & Structures, Vol. 88, pp. 413-425, 2010.

Fig. 5 The patch test result of 10-node polygonal star-like element.

Fig. 6 The patch test result of 6-node polygonal honeycomb elements

Frictionless sliding contact analysis using variable-node elements

Sliding contact mechanics problems is well known as one of major difficult problems to resolve in conventional finite elements. In the case of sliding contact problems as illustrated in Fig. 7, commercial software may not pass the patch test as wells as even show undesirable numerical oscillation in terms of contact pressure. However, after treating their contact interface by the variable-node elements, we obtain a stable and accurate solution as seen in Fig. 8 and Fig. 9.

[Related paper] J.H. Kim, J.H. Lim, J.H. Lee, and S. Im, "A new computational approach to contact mechanics using variable-node finite elements", International Journal for Numerical Methods in Engineering, Vol. 73, pp. 1966-1988, 2008.

Fig. 7 Finite element model of sliding contact body

Fig. 8 Contact pressure of sliding contact body

Fig. 9 Stress distribution of sliding contact body

Finite element analysis of quasistatic crack propagation in brittle media with voids or inclusions

[Related paper] D. Sohn, J.H. Lim, Y.-S. Cho, J.H. Kim, and S. Im, "Finite element analysis of quasistatic crack propagation in brittle media with voids or inclusions", Journal of Computational Physics (2011) [doi:10.1016/j.jcp.2011.05.016

Descriptions of problem and scheme

▣ Models of Double Cantilever beam (DCB) with a kinked crack and boundary conditions

- under the concentrated loading on the left end (BC1)

- under the constraint of uniform displacements on the top and bottom faces (BC2)

▣ Type of material

- Homogeneous material

- Heterogeneous material: porous, fiber-reinforced composite material

▣ The modeling for heterogeneous DCB specimen

Consider 10 coarse-level elements for representative volume elements (RVEs), each of which has its own random distribution of various sizes of inhomogeneities (voids or fibers). The heterogeneous specimen is finally made up by randomly picking one of 10 RVEs for each coarse-element site of the specimen.

▣ Three-level modelling for DCB specimen with the initial kinked crack

- Region of coarse-level mesh Ω₁: far from the crack tip

to treat the material inhomogeneities in the sense of coarse-graining through homogenization

- Region of intermediate-level (or inhomogeneity-level) mesh Ω₂: near the crack tip

to model the presence of microstructures in detail

- Region of fine-level mesh Ω₃: very close to the crack tip

to capture a stiff elastic field due to crack tip singularity

All elements inside or touched by the circle, of which radius is R with the origin at the crack tip, are swiched to the region of intermediate-scale mesh. The interface between the different-scale meshes is straightforwardly connected by variable-node finite elements.

Numerical results

▣ Homogeneous material

▣ Porous material

▣ Fiber-reinforced composite material

▣ Comparison of crack path

Crack paths under BC1

Crack paths under BC2