# On the effect of the contact surface definition in the Cartesian grid finite element method

- José Manuel Navarro-Jiménez
^{1}Email authorView ORCID ID profile, - Manuel Tur
^{1}, - Francisco Javier Fuenmayor
^{1}and - Juan José Ródenas
^{1}

**5**:12

https://doi.org/10.1186/s40323-018-0108-5

© The Author(s) 2018

**Received: **2 January 2018

**Accepted: **3 May 2018

**Published: **15 May 2018

## Abstract

The definition of the surface plays an important role in the solution of contact problems, as the evaluation of the contact force is based on the measure of the gap between the solids. In this work three different methods to define the surface are proposed for the solution of contact problems within the framework of the 3D Cartesian grid finite element method. A stabilized formulation is used to solve the contact problem and details of the kinematic description for each surface definition are provided. The three methods are compared solving some numerical tests involving frictionless contact with finite and small deformations.

## Keywords

## Introduction

In recent years some alternatives to standard Finite Element methods have been developed under the category of immersed boundary methods [1–3], also known as fictitious or embedded domain methods. The common idea in these methods is that the FE mesh is obtained by discretizing a simple domain (usually cuboid) which fully embeds the analysis domain, but is independent of the analysis boundaries, which may be complex. Within this category is the Cartesian grid finite element method (cgFEM) for solving elasticity problems in 2D [4] and 3D [5]. The main differentiating features of cgFEM with respect to other immersed boundary methods are that the cgFEM is able to consider the CAD geometry (represented by NURBS) for the numerical integration and the use of a stabilized Lagrange multiplier method for the imposition of Dirichlet boundary conditions (see [5] and [6] for further details).

In order to solve the contact problem with cgFEM we use a stabilized Lagrangian formulation first presented in [7]. The method has similarities with Nitsche-based formulations proposed in [8–11] with a relevant difference in the stabilizing stress field. In our case we use a smooth field calculated with the Zienkiewicz and Zhu Superconvergent patch recovery (SPR) technique [12–14]. In a first approach, the developed contact formulation was applied to cgFEM considering a linear facet discretization of the boundary, based on the intersections between the Cartesian grid with the CAD geometry.

Several attempts to enhance the definition of the contact boundaries have been developed in the framework of body-fitted meshes, usually known as surface smoothing, using diverse techniques such as Hermite, Bezier spline and NURBS interpolations [15–18], Gregory patches [19] or Nagata patches [20]. It is proven in these works that the enhancement of the contact surfaces results in more accurate solutions and increased robustness of the contact algorithm. A relevant contribution in the consideration of CAD geometries is the isogeometric analysis [21] (and its applications in contact simulation, e.g. [22, 23]), in which the basis functions for the approximation of the solution are the same used for the CAD definition. There are also NURBS-enriched formulations as in [24, 25], where isogeometric basis functions are included only in the contact elements.

As the cgFEM is able to consider the CAD geometry, it seems appropriate to use this surface definition to improve the gap measure. In [26] the deformed surface is defined as a combination of the undeformed CAD geometry and the finite element displacement field. This paper can be considered as an extension of [26], where we study the effect of the surface definition (hence the contact gap) when solving frictionless contact problems with cgFEM. In addition to the previous approaches, linear facets and a combination of FE solution and NURBS surface, in this work we propose a new method in which the deformed configuration is defined as a NURBS surface, i.e., the control points of the original CAD surface are updated such that the new configuration fits the finite element displacement field of the contact surface.

The paper is structured as follows: in “Contact kinematics” section the kinematic variables of the problem are stated. The different alternatives to define the contact surface are presented in “Discretization of contact kinematics” section. The formulation used to solve the contact problem is described in “Stabilized Lagrangian contact formulation” section. Finally the different methods are compared with some numerical tests in “Numerical examples” section.

## Contact kinematics

*slave*and

*master*bodies respectively. \(\Gamma _c^{(i)}\) is the part of body (

*i*) that can interact with the other body. Let \(\mathbf {X}\) be the initial configuration of a given material point in \(\Omega ^{(i)}, i = 1, 2\). We describe the motion of \(\Omega ^{(i)}\) with the mapping \(\varvec{\varphi }:\Omega \longrightarrow \mathbb {R}^3\). Therefore \(\mathbf {x}^{(i)}=\varvec{\varphi }\left( \mathbf {X}^{(i)},t\right) \) for a given point at time

*t*. Since we are solving quasi-static problems, we will omit the time variable and assume that the load increments are small enough. Then, the position vector for any point in \(\Omega ^{(i)}\) is given as

## Discretization of contact kinematics

The finite element (FE) approximation of these continuum variables introduces two important sources of error. One is related to the discretization of the analysis domain \(\Omega _{\textit{h}}\) which usually differs from the original \(\Omega \). The approximation of the continuum displacement with the FE variable \(\mathbf {u}^{\textit{h}}\) introduces the discretization error. We define this field from the nodal value \(\mathbf {u}\) using linear shape functions, \(\mathbf {u}^{\textit{h}}=N_k \mathbf {u}_k\), where \(\mathbf {u}_k\) is the displacement of node *k*.

### Previous considerations regarding cgFEM

#### Surface topology with the *Marching cubes* algorithm

#### Convective to local coordinates transformation

*h*the size of the element. The partial derivatives of this mapping with respect to the convective coordinates are involved in the kinematic variables definition and can be formulated as:

### Variation of kinematic variables

### Surface definition using linear facets

### Surface definition using NURBS and FE displacements

*p*and

*q*respectively, each one defined along two knot vectors with

*n*and

*m*control points. \(\mathbf {P}_{i,j}\) are the coordinates of the \(n\times m\) control points of the surface. Equation (12) can be simplified for further developments as:

*i*,

*j*):

*i*,

*j*) to the unique index

*k*, hence, we can rewrite the NURBS surface as a vector-matrix multiplication:

### Displacement of the NURBS surface matching the FE solution

Note that in both proposed alternatives the NURBS surface is implicitly considered through the numerical integration, and in the last one the nodes of the Cartesian grid are coupled with the control points of the contact NURBS through the gap definition. However, no additional degrees of freedom are included over the boundary and, in contrast with NURBS-enriched contact formulations as [24], the standard FE interpolation is kept inside the domain.

## Stabilized Lagrangian contact formulation

*g*. The substitution of \(\lambda _N\) in the numerical integration of (25) yields the following equation:

*E*is the elastic modulus,

*h*is the mesh size, \(H_g\) is the quadrature weight and, \(\left| J\right| _g\) is the Jacobian of the transformation.

### Linearization of kinematic variables

## Numerical examples

### Contact between plane surfaces

*x*is the out-of-plane direction. Both solids have common elastic material properties, \(E = 115 {\text {GPa}}\) and \(\nu = 0.3\). At the initial configuration, the contact surfaces are overlapping and vertical displacement \(d=-1.6\times 10^{-6}\, {\text {m}}\) is applied on the upper face of the upper body. Displacements along

*y*direction are constrained on the upper face of body 2 and on the lower face of body 1. We use a 2D plane strain overkill solution from [30] as a reference for the discretization error evaluation, so symmetry conditions are applied to the faces parallel to the

*yz*plane. The lateral faces of body 1 are loaded with \(p_y=4\cdot 10^{11}(0.01-z)z\ Pa\) and \(p_z=10\cdot 10^{11}(0.01-z)z\ Pa\).

Non-conforming Cartesian grids are used on both bodies. Figure 8 shows some of the uniformly refined meshes used for the analysis. Starting with the first discretization in Fig. 8 each element is subdivided into 8 new elements to build the following mesh.

The convergence of the relative error in energy norm is shown in Fig. 9 for a sequence of 4 meshes using linear elements, \(\mathscr {H}_8\). The results show that, for all the surface definitions, optimal convergence rate of the error in energy norm (represented by the triangle) is achieved. Only two meshes were solved with the fitting NURBS definition due to the high amounts of nodes coupled in the following meshes.

The original surface definitions consist in linear NURBS for both solids. The degree of the contact surfaces was modified in order to increase the flexibility of the surfaces when performing the NURBS fitting. Figure 10 shows the vertical displacements \(u^h_y\) along a line located on the top surface of the lower solid for the cases of linear facets and fitting NURBS definitions, which are very similar. The line in red represents the NURBS surface resulting from the fitting problem Eq. (18).

When the contact occurs between planar surfaces there is practically no difference in the definition of the surfaces using the three presented methods, and the gap measurement is trivial. Therefore, as expected, all methods have results with a similar precision.

### Contact between curved surfaces, finite deformations

The second example considers the contact interaction between elastic solids with a toroidal shape with major radius \(R = 2\,{\text {cm}}\) and minor radius \(r = 0.5\,{\text {cm}}\). Figure 11 shows the initial position of the bodies in contact. A positive displacement is imposed along the *y* direction over the purple surfaces in 5 incremental steps of 0.1*cm*. All the DOFs are constrained over the blue surfaces. A Neo-Hookean material is used with \(E = 116 GPa\) and \(\nu = 0.3\).

Three different discretizations have been considered in this case, using the same uniform refinement process described in the previous example. Figure 12 shows the analysis meshes for one of the solids. No results were obtained when using linear facets with the first of the meshes due to loss of convergence caused by the surface discretization being extremely coarse. However, the same coarse mesh had no convergence problems using the other two surface definitions, thanks to the consideration of the exact geometry. The last mesh was not solved using the NURBS displacement method due to the high amount of nodes coupled by each surface, which results in non-viable computational cost.

Figure 13 shows the evolution of the reaction forces over the constrained surfaces during the load for each analysis. Note that although all analyses have similar results, the reaction forces when solving mesh 2 with linear facets is clearly lower than the rest of the analyses, including the results obtained with NURBS+FE and NURBS fitting for the coarse mesh. This is mainly due to the lower precision in the gap measurement with linear facets.

The values of \(\varvec{\sigma }_y\) at the final load step for all the performed analyses is shown in Fig. 14. The results are similar for the different methods, with the maximum stress value increasing with the refinement of the mesh.

The same problem was solved using a Neo-Hookean material with \(E = 7\,{\text {MPa}}\), \(\nu = 0.45\), and 15 displacement increments of 0.1*cm* along the *y* direction. Two methods are compared in this test, first the linear facets definition with mesh number 2 (Fig. 12b) and the NURBS + FE method with the coarse mesh (Fig. 12a). The deformed configuration of the solids for the last load step is shown in Fig. 15. It can be seen that despite the use of a coarser discretization, the results with the NURBS + FE method are similar to those obtained with linear facets and a finer mesh.

### Contact between curved surfaces, small deformations

The last example consists in a small deformations contact simulation between three torus. The geometric parameters are the same as in the previous example. For this problem a linear elastic material has been considered, with \(E = 115 \,{\text {GPa}}\) and \(\nu = 0.3\), and only one increment of 0.05*cm* has been applied in the *y* direction over the purple surfaces, shown in Fig. 16. The problem was solved using linear facets and NURBS + FE definition, with the meshes in Fig. 12a, b respectively.

Figure 17 shows the resulting Von Mises stress at the central solid for both cases. A substantial difference between the maximum stress values can be appreciated in this cases. As the deformations in this problem are small, the stress is mainly due to the contact interaction, and the initial gap measure becomes crucial. With convex contact surfaces the linear facets definition estimates less penetration than the actual geometries have, thus producing lower values of stress, even with a higher number of degrees of freedom than in the case of NURBS contact surfaces.

## Conclusions

Three different alternatives have been presented to define the contact surfaces within the Cartesian grid finite element method: a linear facet representation, a combination of NURBS surface and FE displacements and the fitting of a NURBS surface to the FE displacements. The first option, is the most simple and fastest of all three in terms of procedure and implementation. The surface integration quadratures are based on linear triangles whose rules are well known. The ray-tracing algorithm becomes a linear equation, thus having an analytical solution. Therefore the gap is easily computed. In terms of implementation, the normal vector is constant along a surface subdomain (triangle) reducing the number of terms in the calculation of the kinematic variables. On the other hand, this method has lower precision in the gap measure, which can affect the robustness of the method, specially with coarse discretizations.

The use of NURBS surfaces combined with FE solution provides with better results compared with linear facets, as the actual CAD geometry is considered regardless of the used discretization. In all the tests analysed the precision of the solution computed in terms of energy error or stresses is always greater or equal than that obtained with linear facets. This is specially true for coarse discretizations, due to the enhanced gap measure. Some drawbacks of this method are related to the computational cost of the quadrature rules creation [5] and the solution of the ray-tracing algorithm (non-linear equation). In terms of implementation, more terms are involved in the evaluation of the kinematic variables and its variations. However, the total computational cost is not compromised, as the results obtained with NURBS surfaces and coarse meshes have a similar quality as those obtained with finer meshes and linear facets.

From an analytical point of view, the NURBS fitting definition has interesting features with respect to the mixed NURBS and FE definition. The evaluation of the kinematic variables and the ray-tracing solution are simpler as there is only a NURBS involved. However, the high coupling of degrees of freedom for fine discretizations should be addressed, as the computational cost grows exponentially. For these reasons, the combination of NURBS and FE solution seems to be the most versatile and robust option to define the contact surfaces in the framework of the cgFEM.

## Declarations

### Author's contributions

MT carried out the theoretical developments and supervised the implementation. JMN implemented the developments, collected data and drafted the manuscript. MT, JJR and FJF revised the manuscript. All authors read and approved the final manuscript.

### Acknowledgements

The authors wish to thank the Spanish Ministerio de Economia y Competitividad the Generalitat Valenciana and the Universitat Politècnica de València for their financial support received through the projects DPI2017-89816-R, Prometeo 2016/007 and the FPI2015 program.

### Competing interests

The authors declare that they have no competing interests.

### Availability of data and materials

Not applicable.

### Ethics approval and consent to participate

Not applicable.

### Funding

Funded by Spanish Ministerio de Economia y Competitividad (project DPI2017-89816-R), Generalitat Valenciana (project Prometeo 2016/007) and Universitat Politècnica de València (program FPI2015).

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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