A. KONYUKHOV, K. SCHWEIZERHOF 17

3.1. Construction of Kinematics and Numerical

Algorithms for Various Contact Pairs

geom

riant form for various geo-

) procedures. Thus, fundamental

ection routines leads to “projection do-

conditions:

r independent of approxima-

ariant form for arbitrary

set of relative motions

h many publications, they can be sum-

m

nding to a

ce

n “master-slave” contact al-

go

The following open problems are stated as goals for the

etrically exact theory

Development of the unified geometrical formulation

of contact conditions in cova

metrical 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

(joint). The description will be fully based on the differ-

ential geometry of specific features forming a continuum,

because it is carried out in the local coordinate systems

attached to this feature: this is the Gaussian surface coor-

dinate system in the case of surface; the Serret-Frenet

basis in the case of a curved line; the coordinate system

standard for rigid body rotation problem (e.g. via the

Euler angles) the case of a point in. This general de-

scription is forming a geometrically exact theory for

contact interaction.

A full set of contact pairs requires various closest

point projection (CPP

problems of existence and uniqueness of closest point

projection routines corresponding to the following situa-

tions are investigated: point-to-surface, point-to-line,

line- to-line.

A solution of existence and uniqueness problems of

closest point proj

mains” as the “maximal searching domains“.

Derivation of a unified covariant description of

various applicable methods to enforce contact

Lagrange multipliers methods, penalty methods, aug-

mented Lagrange multipliers method. Consistent tangent

matrices are given in closed covariant form possessing a

clear geometrical structure.

Description of all geometrical situations in a co-

variant form which is a-prio

tions of these geometrical features leads to straightfor-

ward numerical algorithms for the implementation with

any order of approximation for finite elements including

iso-geometric finite elements.

Generalization of classical Coulomb law into a

complex interface laws in cov

geometry of the surfaces (e.g. coupled anisotropic fric-

tion and adhesion for surfaces).

Development of the curve-to-curve contact model

allowing considering the co mplete

between curves including a rotational interaction (this is

a novel in the current theory and has not been possible in

earlier theories).

Though, the specific points of the proposed theory are

developed throug

arized under the unified aim, see more detail in mono-

graph [10]. In order to construct a numerical algorithm

for a certain contact pair, first of all, it is identified that

the closest distance between contacting bodies is a natu-

ral measure of the contact interaction. The procedure is

introduced via the closest point projection procedure

(CPP), solution of which requires the differentiability of

the function represen ting the parameterization of the sur-

face of the contacting body. Analysis of the solvability

for the CPP procedure, see more in [8], allows then to

classify all types of all possible contact pairs discussed

earlier. Starting with a consideration of C2-continuous

surfaces, the concept of the projection domain is intro-

duced as a domain from which any potential contact

point can be uniquely projected, and therefore, the nu-

merical contact algorithm can be further constructed.

This domain can be constructed for utmostC1-continuous

surfaces. If the surfaces contain edges and vertex then the

CPP procedure should be generalized in order to include

the projection onto edges and onto vertexes.

The main idea for application for the contact is then

straightforward – the CPPprocedure correspo

rtain geometrical feature gives a rise to a special, in

general, curvilinear 3D coordinate system. This coordi-

nate system is attached to a geometrical feature and its

convective coordinates are directly used for further defi-

nition of the contact measures. Thus, all contact pairs

listed earlier should be described in the corresponding

local coordinate system. The requirement of the exis-

tence for the generalized CPP procedure leads to the

transformation rule between types of contact pairs ac-

cording to which the corresponding coordinate system is

taken. Thus, the all contact pairs can be uniquely de-

scribed in most situations.

A surface-to-surface contact pair, see Figure 1, is de-

scribed via the well know

rithm based on the CPP procedure onto the surface.

This projection allows defining a coordinate system as

follows:

12123 12

,, ,rx xx xxnxx

(1)

12

,rx x

Vector is a vector for the “slave” point,

12

,

x

is a parameterization of the “master” surface,

nx x

12

, is a normal vector to the surface. Equation (1)

in fact, a coordinate transformation in which

e coordinates 123

,,

describes,

convectiv

xx

are used for measure

of contact interaction: the first two 12

,

x are meas-

ures for the tangent internd the third coordinate

3action a

is a penetration – the measure of normal

on-penetrability condition. This transformation is valid

lied only if the solution of the corresponding surface

CPP procedure exists. Initially, the computational algo-

rithm is constructed for non-frictional contact interaction

of smooth surfaces. Here the description starts in the co-

ordinate system given in equation (1), however, due to

the small penetration it is mostly falling into the descrip-

app

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