Strong form meshfree collocation method for frictional contact between a rigid pile and an elastic foundation

Abstract

In this paper, a strong form meshfree collocation method is developed for two-dimensional single-body frictional contact problems. In this approach, a point-wise Taylor series approximation and generalized moving least squares approach is used to construct numerical differential operators at discrete points within the domain. The differential operators are then used to spatially discretize and solve the governing partial differential equations. Contact constraint conditions are formulated with the penalty approach. To demonstrate the efficiency of the method, benchmark problems in frictionless and frictional contact relevant to a rigid pile and an elastic foundation contact are provided. The numerical results are also compared with the finite element solutions to verify robustness and accuracy of the method.

Appendices

Appendices

Appendix

The global system of equation can be written

$$\begin{aligned} \begin{aligned} \left[ \begin{array}{lllll} \begin{array}{ll} K_{I1J1}^{\text {int}}& K_{I2J1}^{\text {int}}\\ K_{I1J2}^{\text {int}} & K_{I2J2}^{\text {int}} \end{array}& & 0 & \\ & \left[ \begin{array}{ll} K_{I1J1}^{D}& 0\\ 0& K_{I2J2}^{D} \end{array} \right]& & \\ & & \left[ \begin{array}{ll} K_{I1J1}^{N} & K_{I2J1}^{N} \\ K_{I1J2}^{N} & K_{I2J2}^{N} \end{array} \right] & \\ & 0& & \left[ \begin{array}{ll} K_{I1J1}^{c} & K_{I2J1}^{c} \\ K_{I1J2}^{c} & K_{I2J2}^{c} \end{array} \right] \end{array} \right] \left[ \begin{array}{l} u^{int}_{1I}(\mathbf{x}_I)\\ u^{int}_{2I}(\mathbf{x}_I)\\ u^{D}_{1I}(\mathbf{x}_I)\\ u^{D}_{2I}(\mathbf{x}_I)\\ u^{N}_{1I}(\mathbf{x}_I)\\ u^{N}_{2I}(\mathbf{x}_I)\\ u^{c}_{1I}(\mathbf{x}_I)\\ u^{c}_{2I}(\mathbf{x}_I) \end{array} \right]-\\ \left[ \begin{array}{l} -b_{1}(\mathbf{x}_I)\\ -b_{2}(\mathbf{x}_I)\\ u_{1I}(\mathbf{x}_I)\\ u_{2I}(\mathbf{x}_I)\\ t_{1}(\mathbf{x}_I)\\ t_{2}(\mathbf{x}_I)\\ t_{1c}(\mathbf{x}_I)\\ t_{2c}(\mathbf{x}_I) \end{array} \right]=0 \end{aligned} \end{aligned}$$

(53)

The components of $K^{int}$ matrix are

$$\begin{aligned} \begin{aligned}&K_{I1J1}^{\text {int}} =[(\lambda +2\mu ) \varvec{\phi }_{IJ}^{(2,0)} + \mu \varvec{\phi }_{IJ}^{(0,2)}], \\&K_{I2J1}^{\text {int}} =(\lambda + \mu ) \varvec{\phi }_{IJ}^{(1,1)}, \\&K_{I1J2}^{\text {int}} =(\lambda + \mu ) \varvec{\phi }_{IJ}^{(1,1)},\\&K_{I2J2}^{\text {int}} =\mu \varvec{\phi }_{IJ}^{(2,0)} + (\lambda + 2\mu ) \varvec{\phi }_{IJ}^{(0,2)}. \end{aligned} \end{aligned}$$

(54)

The components of $K^{D}$ matrix are

$$\begin{aligned} \begin{aligned} K_{I1J1}^{D} = \varvec{\phi }_{IJ}^{(0,0)}, \quad K_{I2J2}^{D} =\varvec{\phi }_{IJ}^{(0,0)}. \end{aligned} \end{aligned}$$

(55)

The components of $K^{N}$ matrix are

$$\begin{aligned} \begin{aligned}&K_{I1J1}^{N} =(\lambda +2\mu ) \varvec{\phi }_{IJ}^{(1,0)} n_1 + \mu \varvec{\phi }_{IJ}^{(0,1)} n_2, \\&K_{I2J1}^{N} =\lambda \varvec{\phi }_{IJ}^{(0,1)} n_1 + \mu \varvec{\phi }_{IJ}^{(1,0)} n_2, \\&K_{I1J2}^{N} =\mu \varvec{\phi }_{IJ}^{(0,1)} n_1 + \lambda \varvec{\phi }_{IJ}^{(1,0)} n_2, \\&K_{I2J2}^{N} =(\lambda +2\mu ) \varvec{\phi }_{IJ}^{(0,1)} n_2+ \mu \varvec{\phi }_{IJ}^{(1,0)} n_1. \end{aligned} \end{aligned}$$

(56)

The components of $K^{c}$ matrix for stick case are

$$\begin{aligned} \begin{aligned} K_{I1J1}^{c}&=K_{I1J1}^{N} +, K_{I1J1}^{stick}, \quad K_{I2J1}^{c} =K_{I2J1}^{N} +, K_{I2J1}^{stick}, \\ K_{I1J2}^{c}&=K_{I1J2}^{N} +, K_{I1J2}^{stick}, \quad K_{I2J2}^{c} =K_{I2J2}^{N} +, K_{I2J2}^{stick}, \end{aligned} \end{aligned}$$

(57)

where $K_{I1J1}^{N}$, $K_{I2J1}^{N}$, $K_{I1J2}^{N}$, and $K_{I2J2}^{N}$ can be obtained from Eq. (56). The component of stiffness matrix $K^{stick}$ in Eq. (57) defined as

$$\begin{aligned} \begin{aligned} K_{I1J1}^{stick}&=\varvec{\phi }_{IJ}^{(0,0)} H(g(\mathbf{x}_J)) + [\epsilon _{\text {N}} \nu _1 \nu _1 + \epsilon _T \tau _1 \tau _1], \\ K_{I2J1}^{stick}&=\varvec{\phi }_{IJ}^{(0,0)} H(g(\mathbf{x}_J)) + [\epsilon _{\text {N}} \nu _2 \nu _1 + \epsilon _T \tau _2 \tau _1], \\ K_{I1J2}^{stick}&=\varvec{\phi }_{IJ}^{(0,0)} H(g(\mathbf{x}_J)) + [\epsilon _{\text {N}} \nu _1 \nu _2 + \epsilon _T \tau _1 \tau _2], \\ K_{I2J2}^{stick}&=\varvec{\phi }_{IJ}^{(0,0)} H(g(\mathbf{x}_J)) + [\epsilon _{\text {N}} \nu _2 \nu _2 + \epsilon _T \tau _2 \tau _2]. \end{aligned} \end{aligned}$$

(58)

The components of $\mathbf{f}^c _{stick}$ vector are

$$\begin{aligned} \begin{aligned} t_{1} ^c (\mathbf{x}_J)&=\epsilon _{\text {N}}<g (\mathbf{x}_J)> \nu _1 - \epsilon _T H(g (\mathbf{x}_J)) (u_k \tau _k) \tau _1 \\ t_{2} ^c (\mathbf{x}_J)&=\epsilon _{\text {N}} <g (\mathbf{x}_J)> \nu _2 - \epsilon _T H(g (\mathbf{x}_J)) (u_k \tau _k) \tau _2 . \end{aligned} \end{aligned}$$

The components of $K^{c}$ matrix for stick case are

$$\begin{aligned} \begin{aligned} K_{I1J1}^{c}&=K_{I1J1}^{N} +, K_{I1J1}^\mathrm{slip}, \quad K_{I2J1}^{c} =K_{I2J1}^{N} +, K_{I2J1}^\mathrm{slip}, \\ K_{I1J2}^{c}&=K_{I1J2}^{N} +, K_{I1J2}^\mathrm{slip}, \quad K_{I2J2}^{c} =K_{I2J2}^{N} +, K_{I2J2}^\mathrm{slip}. \end{aligned} \end{aligned}$$

(59)

The component of stiffness matrix $K^\mathrm{slip}$ in Eq. (59) defined as

$$\begin{aligned} \begin{aligned} K_{I1J1}^\mathrm{slip}&=\varvec{\phi }_{IJ}^{(0,0)} H(g(\mathbf{x}_J)) + [ \epsilon _{\text {N}} \nu _1 \nu _1 - \mu \epsilon _{\text {N}} \text {sign} (u_k \tau _k) \nu _1 \tau _1], \\ K_{I2J1}^\mathrm{slip}&=\varvec{\phi }_{IJ}^{(0,0)} H(g(\mathbf{x}_J)) + [\epsilon _{\text {N}} \nu _2 \nu _1 - \mu \epsilon _{\text {N}} \text {sign} (u_k \tau _k) \nu _2 \tau _1], \\ K_{I1J2}^\mathrm{slip}&=\varvec{\phi }_{IJ}^{(0,0)} H(g(\mathbf{x}_J)) + [\epsilon _{\text {N}} \nu _1 \nu _2 -\mu \epsilon _{\text {N}} \text {sign} (u_k \tau _k) \nu _1 \tau _2], \\ K_{I2J2}^\mathrm{slip}&=\varvec{\phi }_{IJ}^{(0,0)} H(g(\mathbf{x}_J)) + [ \epsilon _{\text {N}} \nu _2 \nu _2 - \mu \epsilon _{\text {N}} \text {sign} (u_k \tau _k) \nu _2 \tau _2]. \end{aligned} \end{aligned}$$

(60)

The components of $\mathbf{f}^c _\mathrm{slip}$ vector are

$$\begin{aligned} \begin{aligned} t_{1} ^c (\mathbf{x}_J)&=\epsilon _{\text {N}}<g (\mathbf{x}_J)> \nu _1 - \mu \epsilon _{\text {N}}<g (\mathbf{x}_J)> \text {sign} (u_k \tau _k) \tau _1 \\ t_{2} ^c (\mathbf{x}_J)&=\epsilon _{\text {N}}<g (\mathbf{x}_J)> \nu _2 - \mu \epsilon _{\text {N}} <g (\mathbf{x}_J)> \text {sign} (u_k \tau _k) \tau _2. \end{aligned} \end{aligned}$$