The element-free Galerkin method for the dynamic Signorini contact problems with friction in elastic materials

doi:10.1016/j.amc.2021.126696

Applied Mathematics and Computation

Volume 415, 15 February 2022, 126696

https://doi.org/10.1016/j.amc.2021.126696 Get rights and content

Highlights

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The element-free Galerkin method is proposed for the dynamic Signorini contact problems with friction in elastic materials and its error estimates are obtained.
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The Dirichlet boundary conditions and the constrained conditions are imposed by the penalty method.
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The error estimates indicate that the convergence order depends on the spatial step, the time step, the largest degree of a complete Pascal's monomial basis in the moving least-squares approximation and the penalty factor.

Abstract

The element-free Galerkin method is proposed for the dynamic Signorini contact problems with friction in elastic materials. The Dirichlet boundary conditions and the constrained conditions are imposed by the penalty method. The error estimates of the element-free Galerkin method indicate that the convergence order depends on the spatial step, the time step, the largest degree of a complete Pascal's monomial basis in the moving least-squares approximation and the penalty factor. Numerical examples verify our theoretical results.

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 $S^{d}$ be the space of second order symmetric tensors on $R^{d}$ (d = 1, 2, 3) and i, j = 1, ⋅⋅⋅, d. Summation over repeated indices is adopted here and hereafter. Define $u \cdot v = u_{i} v_{i}, \forall u, v \in R^{d},$ $σ : τ = σ_{ij} τ_{ij}, \forall σ, τ \in S^{d} .$

Let Ω be a non-empty bounded domain with the Lipschitz boundary Γ. Define $H = {[L^{2} (Ω)]}^{d} = {u = (u_{i}) | u_{i} \in L^{2} (Ω)},$ $Q = {[L^{2} (Ω)]}^{d \times d} = {σ = (σ_{i j}) | σ_{i j} = σ_{j i} \in L^{2} (Ω)},$ $H_{1} = {u = (u_{i}) | ɛ (u) \in Q} \subset {[H^{1} (Ω)]}^{d},$

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 $β = γ = \frac{1}{2}$ 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 $v |_{Γ_{D}} = 0$ and the constraint condition $v_{ν} |_{Γ_{S}} \leq 0$ by the penalty method.

Let $l_{0} (u, v) = \int_{Γ_{D}} u \cdot v d_{Γ} = {(u, v)}_{Γ_{D}}, l_{1} (u, v) = \int_{Γ_{S}} u_{ν}^{+} \cdot v_{ν} d_{Γ} = {(u_{ν}^{+}, v_{ν})}_{Γ_{S}}$ , where $u_{ν}^{+} = max {u_{ν}, 0}$ . Set the penalty factor α > 0, then the penalty formulation of Problem PV^k is as follows.

Problem ${PV}_{α}^{k}$ : Find $u_{α}^{k} = {u_{α n}^{k}}_{n = 0}^{N} \subset H_{1}$ such that $\begin{matrix} (u_{α n}^{k}, v - u_{α n}^{k}) + \frac{k^{2}}{2 ρ} a (u_{α n}^{k}, v - u_{α n}^{k}) + α l_{0} (u_{α n}^{k}, v - u_{α n}^{k}) + α l_{1} (u_{α n}^{k}, v \end{matrix}$

Error estimates for the full discrete scheme of the EFG method by the penalty method

In the Newmark scheme, assume that ${u_{n}^{k}, {\dot{u}}_{n}^{k}, {\ddot{u}}_{n}^{k}}$ are known at t_n, we shall solve

${u_{n + 1}^{k}, {\dot{u}}_{n + 1}^{k}, {\ddot{u}}_{n + 1}^{k}}$ at t_n₊₁. So, we shall construct the quintic Hermite interpolating polynomial u^k(x,t) to approximate $u^{k} = {u_{n}^{k}}_{n = 0}^{N}$ of Problem PV^k based on ${u_{n}^{k}, {\dot{u}}_{n}^{k}, {\ddot{u}}_{n}^{k}}_{n = 0}^{N}$ . Namely, $u^{k} (x, t_{n}) = u_{n}^{k}, {\dot{u}}^{k} (x, t_{n}) = {\dot{u}}_{n}^{k}, {\ddot{u}}^{k} (x, t_{n}) = {\ddot{u}}_{n}^{k} .$

From the definition of piecewise quintic Hermite interpolating polynomials [31], it holds that $u^{k} (x, s) = u_{n - 1}^{k} (x) A_{1} + u_{n}^{k} (x) A_{2} + {\dot{u}}_{n - 1}^{k} (x) B_{1} + {\dot{u}}_{n}^{k} (x) B_{2}$ $+ {\ddot{u}}_{n - 1}^{k} (x) C_{1} + {\ddot{u}}_{n}^{k} (x) C_{2},$ where $A_{1} (s) = [6 {(s - t_{n - 1})}^{2}$

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 $v^{h} = 2 u_{α n}^{h k}$ , v^h = 0 in (23) of Problem ${PV}_{α}^{h k}$ , we have $(u_{α n}^{hk}, u_{α n}^{hk}) + \frac{k^{2}}{2 ρ} a (u_{α n}^{hk}, u_{α n}^{hk}) + α l (u_{α n}^{hk}, u_{α n}^{hk}) \geq \frac{k^{2}}{2 ρ} < L_{n}, u_{α n}^{hk} > + (u_{α n - 1}^{hk} + k {\dot{u}}_{α n - 1}^{hk}, u_{α n}^{hk}),$ $- (u_{α n}^{hk}, u_{α n}^{hk}) - \frac{k^{2}}{2 ρ} a (u_{α n}^{hk}, u_{α n}^{hk}) - α l (u_{α n}^{hk}, u_{α n}^{hk}) \geq - \frac{k^{2}}{2 ρ} < L_{n}, u_{α n}^{hk} > - (u_{α n - 1}^{hk} + k {\dot{u}}_{α n - 1}^{hk}, u_{α n}^{hk}),$ thus $\begin{matrix} (u_{α n}^{hk}, u_{α n}^{hk}) + \frac{k^{2}}{2 ρ} a (u_{α n}^{hk}, u_{α n}^{hk}) + α l (u_{α n}^{hk}, u_{α n}^{hk}) \\ = \frac{k^{2}}{2 ρ} < L_{n}, u_{α n}^{hk} > + (u_{α n - 1}^{hk} + k {\dot{u}}_{α n - 1}^{hk}, u_{α n}^{hk}) . \end{matrix}$

Here $l (v^{h}, v^{h}) = \int_{Γ_{D}} v^{h} \cdot v^{h} d a + \int_{Γ_{S}} {(v_{ν}^{h})}^{+} \cdot v_{ν}^{h} d a$ , 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).

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Citation 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.
An implicit Galerkin meshfree scheme based on the element-free Galerkin (EFG) and the backward Euler methods is analyzed for general second-order parabolic problems. The penalty method is used to impose Dirichlet boundary conditions, and a priori approximation error has been derived to confirm its validity. The existence and uniqueness of the weak solution for the penalized parabolic problems under homogeneous mixed boundary conditions are proved, offering an insight for numerical discretizations. With an orthogonal projection being defined, the continuous error bounds with the penalty factor for the semi-discrete approximation are obtained, and meanwhile, the discrete error bounds with the penalty factor for the fully discrete approximation are also achieved, which provide a theoretical explanation for the value of a reasonable penalty factor. Finally, numerical examples are conducted to verify the theoretical results.
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The element-free Galerkin method for the dynamic Signorini contact problems with friction in elastic materials

Highlights

Abstract

Introduction

Section snippets

Mathematical problems of the dynamic Signorini contact problems with friction in elastic materials and their equivalent variational inequalities

Error estimates of the semi-discrete scheme in time of the equivalent variational inequalities by the Newmark scheme

The full discrete scheme of the EFG method for the equivalent variational inequalities by the penalty method

Error estimates for the full discrete scheme of the EFG method by the penalty method

The computational framework of the EFG method with the penalty factor

Numerical examples

Conclusion

Acknowledgment

Comptes Rendus - Mathématique

Nonlinear Anal. Real World Appl.

Math. Comput. Simul.

Eng. Fract. Mech.

Comput. Methods Appl. Mech. Eng.

Comput. Math. Appl.

Appl. Math. Model.

Finite Elem. Anal. Des.

Appl. Numer. Math.

Existence of solutions of a dynamic Signorini's problem with nonlocal friction in viscoelasticity

Zeitschrift Für Angewandte Mathematik Und Physik

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.