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Geometrically exact theory of contact interactions is aiming on the development of the unified geometrical formulation of computational contact algorithms for various geometrical situations of contacting bodies leading to contact pairs: surface-to-surface, curve-to-surface, point-to-surface, curve-to-curve, point-to-curve, point-to-point. The construction of the corresponding computational contact algorithms is considered in accordance with the geometry of contacting bodies in covariant and closed forms. These forms can be easily discretized within various methods such as the finite element method (FEM), the finite discrete method (FDM) independently of the order of approximation and, therefore, the result is straightforwardly applied within any further method: high order finite element methods, iso-geometric finite element methods etc. As particular new development it is shown also the possibility to easy combine with the Finite Cell Method.

Computational contact mechanics has become a separated branch of computational mechanics during the last decades. Modeling of contact interactions became standard in numerous finite element software packages available for engineers. Various aspects of the numerical solution such as various enforcements of contact conditions, or the possibility to apply high order and iso-geometric type of approximation have been considered. One of the important aspects, even though being obvious for everyone,―the consistent geometrical treatment of contact― is often remaining hidden inside the computational algorithm. Contact interaction from a geometrical point of view can be seen as the interaction between deformable surfaces possessing various geometrical features such as surfaces, edges and vertexes, therefore, geometrical approaches can be exploited. Three major geometrical situations can be identified: surface-to-surface, curve-to-curve and curve-to-surface contact. Numerous publications are devoted to development of various numerical aspects of the first two contact pairs: we refer here to the major monographs only. Thus, Laursen [

The geometrically exact theory is straightforwardly applicable together with the isogeometric finite elements applications in computational contact mechanics―a comprehensive review of the state of the art is presented in De Lorenzis, Wriggers and Hughes [

It is easy to construct a model contact problem with two bodies possessing smooth surfaces as well as various geometrical features such as edges and vertexes―an example of this is an impact of two swords shown in

1. Point-To-Point contact pair;

2. Point-To-Curve contact pair;

3. Point-To-Surface contact pair;

4. Curve-To-Curve contact pair;

5. Curve-To-Surface contact pair;

6. Surface-To-Surface contact pair.

In order to construct a numerical algorithm for a certain contact pair, first of all, it is identified that the distance between contacting bodies is a natural measure of the normal contact interaction. The procedure is introduced via the Closest Point Projection (CPP) procedure, requiring the minimization of the distance between a selected point

Solution of the CPP procedure requires the differentiability of the function representing the parameterization of a surface of the contacting body. Namely, analysis of the existence and uniqueness for the CPP procedure allows then to classify all types of all possible contact pairs, see details for surfaces in [

A Surface-to-Surface contact pair is described via the well known ”master-slave” contact algorithm based on the CPP procedure onto the surface in Equation (1). This projection allows to define a coordinate system, see in

The vector

in fact, a coordinate transformation in which convective coordinates are used to measure the contact interaction:

The Curve-To-Curve (CTC) contact pair requires the projection on both curves, therefore, there is no classical “master” and “slave” and both curves are equivalent. For the description one of the two coordinate systems (Serret-Frenet coordinate system) can be assigned to the i-th curve, see in

Here, the vector

Remark 1. Representation for Curve-To-Curve contact in Equation (4) is possible until tangent

A special development is necessary to describe a Curve-To-Surface (CTS) contact pair precisely from the kinematic point of view. Contact between a curve (geometrically representing either a rope, or a beam, or an edge of the solid) and a surface can be described by various approaches:

a) a special combination of Curve-To-Curve (CTC) and Segment-To-Analytical-Surface (STAS) Algorithm;

b) a special consideration in the Darboux basis.

This type of contact, in case of rigid surfaces, is the combination of the Surface-To-Analytical-Surface and the Curve-To-Curve contact kinematics, see in [

The contact forces are represented dually in both surface and curve coordinate systems as:

where

In this approach, see details in [

on a surface are formulated via constraint equations in the Darboux basis. These conditions can be formulated as Karush-Kuhn-Tucker (KKT) conditions, see details in [

The weak formulation for the Surface-To-Surface contact pair is constructed in the surface coordinate system, Equation (2). First, the contact force/stress vector for the slave part is expressed in the local coordinate system on the master surface and is split into a normal and into a tangential part

The weak form is formulated as

The linearized expression which is necessary for the tangent matrix is derived via the covariant derivation in the surface coordinate system, Equation (2), see details in [

Since in this type of contact master and slave parts are equivalent, the virtual work can be equivalently written as an integral either along the first curve, or along the second curve and, therefore, for symmetry reasons the following expression is taken:

with corresponding normal

Considering the developed kinematics for the Curve-To-Surface interaction, we are formulating the weak form in the curve Serret-Frenet coordinate system as an integral along the curve l. It is formulated via parameters expressed either in the curve Serret-Frenet coordinate system; i.e. or via parameters defined in the Gaussian surface coordinate system. Starting with the weak form expressed at the curve, we have:

(or via surface parameters)

The linearization results will combine the linearization for both Surface-To-Surface and Curve-To-Curve contact pairs, discussed in detail in.

In this special case the Curve-To-Surface contact pair can be regarded as the equilibrium of ropes on orthotropic rough. The weak form is written as, see derivation details in [

Computationally, values of pulling

The advantages of the description in the Darboux basis are:

§ contact kinematics is precisely described;

§ an anisotropic friction law for the rope-surface interaction can be easily incorporated;

§ new analytical solutions such as a generalized Euler-Eytelwein formula can be obtained.

For the last statement, we represent here result of the theorem for the equilibrium of a rope, see details in [

Theorem. If a rope loaded by tangential forces is laying in equilibrium on a rough orthotropic surface then three following conditions (all of them) are satisfied:

1) No separation―normal reaction $N$ is positive for all points of the curve:

2) Dragging coefficient of friction

3) Limit values of the tangential forces:

The forces at both ends

The Finite Cell Method (FCM) provides a method for the computation of structures which can be described as a mixture of high-order FEM and a special integration technique, see [

Alternative contact approaches for FCM based on the application of the geometrically exact contact theory.

The following alternative contact algorithms are developed and tested:

CSTAS contact element (Cell-Surface-To-Analytical-Surface) for contact with rigid bodies;

DCTC contact element (Discrete-Cell-To-Cell)-based on the representation of the integration point as a discrete finite element for both deformable bodies.

In this approach a special contact element cell-wisely, namely, the contact element is developed based on the exact boundary inside the sub-domain/sub-cell, see more detail in [

In order to increase the performance of the cell-wise application the adaptive cell refinement with CSTAS method is applied. Fairly high dense mesh with low order of approximation: 20 × 20 finite elements of 2th order with additional 10 × 10 cell subdivision of the bounding area (more dense area in the

The simplest implementation for contact between deformable bodies can be constructed based on the nature of FCM―a set of Gauss points is represented as discrete elements and then a contact algorithm used for the Finite Discrete Method (FDM) is employed. The idea of the DCTC method is presented in

§ Convergence of the solution is better if the radius is larger;

§ The radius

§ The radius

The DCTC contact approach allows to describe efficiently the self-contact in a pore, see in

The geometrically exact theory for contact interaction allows to describe and to construct computational algorithm for all available geometrical contact pairs Surface-To-Surface, Curve-To-Curve, Curve-To-Surface etc. as well as allows to construct computational contact algorithms not only for the finite element method, but also for other methods e.g. Finite Cell Method.

Alexander Konyukhov, (2015) Geometrically Exact Theory of Contact Interactions—Applications with Various Methods FEM and FCM. Journal of Applied Mathematics and Physics,03,1022-1031. doi: 10.4236/jamp.2015.38126