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Element differential method for contact problems with non-conforming contact discretization

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Abstract

In this paper, a new strong-form numerical method, the element differential method (EDM) is employed to solve two- and three-dimensional contact problems without friction. When using EDM, one can obtain the system of equations by directly differentiating the shape functions of Lagrange isoparametric elements for characterizing physical variables and geometry without the variational principle or any integration. Non-uniform contact discretization is used to enhance contact conditions, which avoids performing identical discretization along the contact surfaces of two contact objects. Two methods for imposing contact constraints are proposed. One method imposes Neumann boundary conditions on the contact surface, whereas the other directly applies the contact constraints as collocation equations for the nodes within the contact zone. The accuracy of the two methods is similar, but the multi-point constraints method does not increase the degrees of freedom of the system equations during the iteration process. The results of four numerical examples have verified the accuracy of the proposed method.

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Data availability

The authors state that all the data necessary to replicate the results are presented in the manuscript. Relevant parts of the code can be shared by contacting the corresponding author.

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Funding

This work was supported by the National Natural Science Foundation of China (12072064,12272081) and the Natural Science Foundation of Liaoning Province, China (2022-MS-138).

Author information

Authors and Affiliations

  1. School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian, 116024, China

    Wei-Long Fan, Xiao-Wei Gao & Hai-Feng Peng

  2. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, 116024, China

    Xiao-Wei Gao

  3. School of Infrastructure Engineering, Nanchang University, Nanchang, 330031, China

    Yong-Tong Zheng

  4. Institute of Continuum Mechanics, Leibniz University Hannover, Hannover, Germany

    Bing-Bing Xu

Authors
  1. Wei-Long Fan
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  2. Xiao-Wei Gao
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  3. Yong-Tong Zheng
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  4. Bing-Bing Xu
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  5. Hai-Feng Peng
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Corresponding authors

Correspondence to Xiao-Wei Gao or Hai-Feng Peng.

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Appendix A

Appendix A

1.1 The description of symbols and abbreviations

The symbols and abbreviations used in this paper:

Γ

 = 

Boundary

Ω

 = 

Domain

xi

 = 

The ith coordinate

x, x, y, z

 = 

Coordinates

u

 = 

Displacement

t

 = 

Traction

n

 = 

Outward normal of the surface

Dijkl

 = 

Elastic constitutive tensor

y

 = 

Coordinates after contact

gn

 = 

Normal gap after contact

Gn

 = 

Normal gap before contact

σn

 = 

Normal contact stress

ξ, ξ, η, ζ

 = 

Intrinsic coordinates

Nα, Nβ

 = 

Shape function

M, S

 

Number of surfaces

A

 = 

Coefficient matrix

z

 = 

Vector of unknown

b

 = 

Vector of known

\(K_{ij}^\alpha\)

 = 

Coefficient matrix for node

G

 = 

Coefficient matrix

C

 = 

Coefficient matrix by contact equations

∆

 = 

Change of variable

Superscripts

  

A, B

 = 

Object

m, n, p

 = 

Number of interpolation points

α

 = 

Node

f

 = 

Element surface

k

 = 

kth iteration

Subscripts

  

C

 = 

Contact

N

 = 

Not in contact

T

 = 

Neumann boundary

U

 = 

Dirichlet boundary

i, j, k, l

 = 

The ith, jth, kth, lth component

α, β

 = 

Node

abbreviations

  

EDM

 = 

Element differential method

FEM

 = 

Finite element method

BEM

 = 

Boundary element method

MFM

 = 

Mesh-free method

FDM

 = 

Fnite difference method

SFEM

 = 

Strong-form finite element method

FBM

 = 

Finite block method

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Fan, WL., Gao, XW., Zheng, YT. et al. Element differential method for contact problems with non-conforming contact discretization. Engineering with Computers (2024). https://doi.org/10.1007/s00366-024-01963-7

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  • DOI: https://doi.org/10.1007/s00366-024-01963-7

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