The element-free Galerkin method for the dynamic Signorini contact problems with friction in elastic materials
Introduction
Contact problems with friction between deformable bodies can be found in daily life and industrial production. The deformation of elastic bodies will be different due to different friction contact laws which are imposed on the friction contact boundary. The classical friction contact laws include the Signorini's friction law, the Tresca's friction law, the Coulomb's friction law and so on. In practice, the partial differential equations of contact problems with friction may be static or dynamic. Because contact forces are strongly related to the time in dynamic problems, these dynamic problems are challenging mathematical problems. Therefore, the dynamic Signorini contact problems with friction are very important in the research. Many researchers have done plenty of meaningful work in this field. Cao et al. [1] presented the existence and uniqueness of solutions for a dynamic Signorini's contact problem. Cocou [2,3] discussed the existence and uniqueness of solutions of a dynamic Signorini's problem with nonlocal friction and dynamic contact problems with Coulomb friction in viscoelasticity. With the development of computers, a large number of efficient numerical methods have been applied to contact problems with friction. Space adaptive finite element methods was proposed by Blum et al. [4] for dynamic Signorini problems and error estimates were also obtained. Doyen et al. [5] gave time-integration schemes for the dynamic Signorini problem by the finite element method and also analyzed this problem by the modified mass method [6,7]. In [8], Kosior et al. combined the boundary element method and the domain decomposition method to analyze the frictional contact problem. Most of the classical numerical methods above are very effective and have strict mathematical theory to ensure its convergence. But the disadvantage of these methods is that they have strong requirements for the mesh generation, which will cost a lot of computational time to deal with the large deformation. Because the deformation of the contact surface often causes mesh distortion in dynamic contact problems with friction, meshless methods have appeared.
Meshless methods [9] are new numerical methods, which have advantages in dealing with large deformations, singularities or dynamic crack growth. Meshless methods can be divided into two types: collocation methods (e.g., the radial basis functions collocation method) and Galerkin methods (e.g., the element-free Galerkin method). In terms of collocation methods for contact problems, Almasi et al. [10] used a strong form meshfree collocation method to solve the frictional contact problem on a rigid obstacle. However, the traditional continuous differentiable function space cannot be used to describe the contact boundary function in contact problems, the mathematical description of contact problems is often in the sense of weak solutions. So the meshless Galerkin method is naturally introduced to solve various contact problems.
In 1992, Nayroles et al. [11] introduced the moving least-squares (MLS) approximation [12] into the Galerkin method and presented the diffuse element method (DEM). Belytschko et al. [13] developed the element-free Galerkin (EFG) method on the basis of the DEM. Henceforth, the EFG method has a wide application in fracture and crack growth [14], crack propagation [15], dynamic fracture [16,17]. The EFG method approximates unknown functions by the MLS approximation and gives the weak form of the problem by the Galerkin method. Because the mathematical models of contact problems with friction are variational inequalities, the EFG method has already been applied to variational inequalities recently. For instance, Ding et al. [18] presented the convergence analysis of the EFG method for the second kind of elliptic variational inequalities, of which the Dirichlet boundary conditions are imposed by the penalty method. Shen et al. [19] discussed the error estimates for a contact problem with the Tresca friction or the simplified Coulomb friction in elastic materials by the EFG method. In the latest paper of Ding et al. [20], the EFG method was proposed for a quasistatic contact problem with the Tresca friction. In this paper, we shall develop the EFG method for the dynamic Signorini contact problems with friction in elastic materials and give error estimates of the EFG method.
In contact problems, especially in dynamic contact problems, the classical numerical methods based on mesh-generation, e.g., the finite element method (FEM), require the high-quality mesh around the dynamic contact boundary. However, in general, mesh-generation will cost lots of time, and the quality of the mesh is not necessarily guaranteed. The EFG method, whose trial and test functions are obtained by the MLS approximation, is one of meshless methods. It has the advantage of no mesh-generation. Moreover, its formulation is based on the Galerkin method, so it has the high accuracy and the good stability similar to the FEM. Nevertheless, unlike the FEM method, the shape function of the EFG method does not have the properties of the Kronecker delta function. Therefore, we shall choose the penalty method to apply the Dirichlet boundary condition. The penalty method will lead to difficulties in the theoretical analysis because the solution of the penalty problem does not satisfy the Kronecker delta condition on the Dirichlet boundary. Therefore, the solution of the penalty problem does not belong to the solution space of the original problem. Its error analysis is particularly important.
This paper is organized as follows. There are mathematical problems of the dynamic Signorini contact problems with friction in elastic materials and their equivalent variational inequalities in Section 2. In Section 3, we give error estimates of the semi-discrete scheme in time of the equivalent variational inequalities by the Newmark scheme. In Section 4, we introduce the penalty method to deal with the Dirichlet boundary conditions and the constrained conditions, and obtain the full discrete scheme of the equivalent variational inequalities by the EFG method. Error estimates of the full discrete scheme by the EFG method are given in Section 5. We shall construct the computational framework of the EFG method with the penalty factor in Section 6. In Section 7, there are two numerical examples to verify our theoretical results. The last is a brief conclusion.
Section snippets
Mathematical problems of the dynamic Signorini contact problems with friction in elastic materials and their equivalent variational inequalities
Some notations and preliminaries used here and hereafter are introduced briefly. Further details can be referred to [21,22,23].
Let be the space of second order symmetric tensors on (d = 1, 2, 3) and i, j = 1, ⋅⋅⋅, d. Summation over repeated indices is adopted here and hereafter. Define
Let Ω be a non-empty bounded domain with the Lipschitz boundary Γ. Define
Error estimates of the semi-discrete scheme in time of the equivalent variational inequalities by the Newmark scheme
For the semi-discrete scheme in time, the most used numerical methods in computational mechanics are back differentiation formulas (BDF) methods and generalized-α methods [26], also called Hilbert-Hughes-Taylor-α methods. There are two parameters β and γ in generalized-α methods. For simplicity, we choose the Newmark method [4,27] with in generalized-α methods, which is unconditionally stable with the second order accuracy. In this way, there will not be too many parameters in the
The full discrete scheme of the EFG method for the equivalent variational inequalities by the penalty method
Because shape functions of the MLS approximation do not have the property of the Kronecker Delta function, we have to impose the Dirichlet boundary condition and the constraint condition by the penalty method.
Let , where . Set the penalty factor α > 0, then the penalty formulation of Problem PVk is as follows.
Problem : Find such that
Error estimates for the full discrete scheme of the EFG method by the penalty method
In the Newmark scheme, assume that are known at tn, we shall solve
at tn+1. So, we shall construct the quintic Hermite interpolating polynomial uk(x,t) to approximate of Problem PVk based on . Namely,
From the definition of piecewise quintic Hermite interpolating polynomials [31], it holds thatwhere
The computational framework of the EFG method with the penalty factor
In this section, we shall give the computational procedures of the EFG method with the penalty factor. First, let , vh = 0 in (23) of Problem , we havethus
Here , which has a
Numerical examples
In this section, we shall give two numerical examples to demonstrate the convergence and theoretical results of the EFG method for the dynamic Signorini contact problems with friction. Both of numerical examples have exact solutions. The first numerical example is in the square domain, the other is in Z-shaped irregular domain. The aim of numerical examples is to test the validity and accuracy of the EFG method, and to verify the relationship between the theoretical error estimates and the
Conclusion
In this paper, we present the EFG method for the dynamic Signorini contact problems with friction in elastic materials. For the variational inequality of this problem, the domain and the time derivative are discretized by the EFG method and the Newmark scheme, respectively. The penalty method is adopted to deal with the Dirichlet boundary conditions and the constrained conditions. The error estimates of the EFG method depend not only on the spatial step and the time step but also the largest
Acknowledgment
This work was supported by the National Natural Science Foundation of China (No. 11401416 and No. 11771319).
References (31)
- et al.
Existence of solutions for a dynamic Signorini's contact problem
Comptes Rendus - Mathématique
(2006) A class of dynamic contact problems with Coulomb friction in viscoelasticity
Nonlinear Anal. Real World Appl.
(2015)- et al.
Meshless methods: a review and computer implementation aspects
Math. Comput. Simul.
(2008) - et al.
Crack propagation by element-free Galerkin methods
Eng. Fract. Mech.
(1995) - et al.
Element-free Galerkin method for wave propagation and dynamic fracture
Comput. Methods Appl. Mech. Eng.
(1995) - et al.
Convergence analysis and error estimates of the element-free Galerkin method for the second kind of elliptic variational inequalities
Comput. Math. Appl.
(2019) - et al.
Error estimates for a contact problem with the Tresca friction or the simplified Coulomb friction in elastic materials by the element-free Galerkin method
Appl. Math. Model.
(2020) - et al.
Analysis of dynamic frictional contact problems using variational inequalities
Finite Elem. Anal. Des.
(2001) Error estimates for the moving least-square approximation and the element-free Galerkin method in n-dimensional spaces
Appl. Numer. Math.
(2016)Existence of solutions of a dynamic Signorini's problem with nonlocal friction in viscoelasticity
Zeitschrift Für Angewandte Mathematik Und Physik
(2002)
Space adaptive finite element methods for dynamic Signorini problems
Comput. Mech.
Time-integration schemes for the finite element dynamic Signorini problem
SIAM J. Sci. Comput.
Analysis of the modified mass method for the dynamic Signorini problem with coulomb friction
SIAM J. Numer. Anal.
Convergence of a space semi-discrete modified mass method for the dynamic Signorini problem
Commun. Math. Sci.
Analysis of frictional contact problem using boundary element method and domain decomposition method
Int. J. Numer. Methods Eng.
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2023, Computer Methods in Applied Mechanics and EngineeringThe element-free Galerkin method for the variational–hemivariational inequality of the dynamic Signorini–Tresca contact problems with friction in elastic materials
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2022, Applied Numerical MathematicsCitation Excerpt :In addition, there are some theoretical results of the penalty-based EFG method for time-dependent problems. The error estimates of a full discrete EFG scheme for quasistatic contact problems are presented in Refs. [11,33], while the similar analysis for the dynamic Signorini contact problems is developed in Ref. [12]. We mention that the analysis shown in Refs. [11,12,33] only considers the effect of the penalty factor on errors of the full discrete scheme.