Large deformation frictional contact analysis with immersed boundary method

Abstract

This paper proposes a method of solving 3D large deformation frictional contact problems with the Cartesian Grid Finite Element Method. A stabilized augmented Lagrangian contact formulation is developed using a smooth stress field as stabilizing term, calculated by Zienckiewicz and Zhu Superconvergent Patch Recovery. The parametric definition of the CAD surfaces (usually NURBS) is considered in the definition of the contact kinematics in order to obtain an enhanced measure of the contact gap. The numerical examples show the performance of the method.

Appendices

Variation of normal and tangent vectors

We recall here (15) for the calculation of $\delta \mathbf n ^{(1)}$.

$$\begin{aligned} \mathbf{n ^{(1)}}&=\frac{{\hat{\mathbf{n }}^{(1)}}}{||\hat{\mathbf{n }}^{(1)}||}; \qquad {\hat{\mathbf{n }}^{(1)}} = \mathbf s ^{(1)}_{\xi }\times \mathbf s ^{(1)}_{\eta }\end{aligned}$$

(51)

$$\begin{aligned} \delta \mathbf n ^{(1)}&=\frac{\delta \mathbf s ^{(1)}_{\xi }\times \mathbf s ^{(1)}_{\eta }+\mathbf s ^{(1)}_{\xi }\times \delta \mathbf s ^{(1)}_{\eta }}{\left\| \hat{\mathbf{n }}^{(1)}\right\| }\nonumber \\&\quad -\,\frac{\mathbf{n ^{(1)}}}{\left\| {\hat{\mathbf{n }}^{(1)}}\right\| } \left[ \mathbf n ^{(1)}\cdot (\delta \mathbf s ^{(1)}_{\xi }\times \mathbf s ^{(1)}_{\eta }+\mathbf s ^{(1)}_{\xi }\times \delta \mathbf s ^{(1)}_{\eta })\right] \end{aligned}$$

(52)

For the calculation of the variation of the tangent vectors $\mathbf s ^{(1)}_{\xi }$ and $\mathbf s ^{(1)}_{\eta }$ we start from (16). We will only describe the calculation of $\delta \mathbf s ^{(1)}_{\xi }$ as the other term, $\delta \mathbf s ^{(1)}_{\eta }$, has an identical procedure:

$$\begin{aligned} \mathbf s ^{(1)}_{\xi }&=\,\frac{\partial \mathbf x ^{(1)}}{\partial \xi }=\frac{\partial S(\xi ,\eta )}{\partial \xi }\nonumber \\&\quad +\,\sum _j\left( \frac{\partial N_j}{\partial \zeta ^e_1} \frac{\partial \zeta ^e_1}{\partial \xi }+\frac{\partial N_j}{\partial \zeta ^e_2}\frac{\partial \zeta ^e_2}{\partial \xi }+\frac{\partial N_j}{\partial \zeta ^e_3} \frac{\partial \zeta ^e_3}{\partial \xi }\right) \mathbf u ^{(1)}_j \end{aligned}$$

(53)

$$\begin{aligned} \delta \mathbf s ^{(1)}_{\xi }&=\delta \left( \frac{\partial \mathbf x ^{(1)}}{\partial \xi }\right) =\frac{\partial \delta \mathbf u ^{(1)}}{\partial \xi }\nonumber \\&=\sum _j \left( \frac{\partial N_j}{\partial \zeta ^e_1}\frac{\partial \zeta ^e_1}{\partial \xi }+\frac{\partial N_j}{\partial \zeta ^e_2}\frac{\partial \zeta ^e_2}{\partial \xi } +\frac{\partial N_j}{\partial \zeta ^e_3}\frac{\partial \zeta ^e_3}{\partial \xi }\right) \delta \mathbf u ^{(1)}_j \end{aligned}$$

(54)

The linearization of all these variables has the same structure as the variation, so the variations $\delta \mathbf n ^{(1)}$, $\delta \mathbf s ^{(1)}_{\xi }$ and $\delta \mathbf s ^{(1)}_{\eta }$ can be directly substituted for the increments $\varDelta \mathbf n ^{(1)}$ , $\varDelta \mathbf s ^{(1)}_{\xi }$ and $\varDelta \mathbf s ^{(1)}_{\eta }$.

Linearization of $\varDelta ^t\mathbf g _t$

We recall the definition of $\varDelta ^t\mathbf g _t$ here:

$$\begin{aligned} \varDelta ^t\mathbf g _t= \frac{\mathbf{p }_T-\frac{\kappa E}{h}\mathbf{T }_n\left( \mathbf x ^{(1)}-\mathbf{x }^{(2)}\left( \varvec{\xi }_t\right) \right) }{\left\| \mathbf{p }_T-\frac{\kappa E}{h}\mathbf{T }_n\left( \mathbf x ^{(1)}-\mathbf{x }^{(2)}\left( \varvec{\xi }_t\right) \right) \right\| } \end{aligned}$$

(55)

If we use the simplification of (56), the linearization of $\varDelta ^t\mathbf g _t$ can be expressed as in (57)

$$\begin{aligned} \varDelta ^t\mathbf g _t= & {} \,\frac{\hat{\mathbf{d }}}{||\hat{\mathbf{d }}||}; \qquad \hat{\mathbf{d }}= \mathbf p _T+\frac{\kappa E}{h}\mathbf T _n\left( \mathbf x ^{(2)}\left( \varvec{\xi }_t\right) -\mathbf x ^{(1)}\right) \nonumber \\\end{aligned}$$

(56)

$$\begin{aligned} \varDelta \varDelta ^t\mathbf g _t= & {} \,\frac{\varDelta \hat{\mathbf{d }}}{\left\| \hat{\mathbf{d }}\right\| }-\frac{\varDelta ^t\mathbf g _t}{\left\| \hat{\mathbf{d }}\right\| }\left[ \varDelta ^t\mathbf g _t\cdot \varDelta \hat{\mathbf{d }}\right] \end{aligned}$$

(57)

Finally, for the linearization of $\hat{\mathbf{d }}$ we can rearrange Eq. (56) as:

$$\begin{aligned} \hat{\mathbf{d }}=\mathbf p _T+\frac{\kappa E}{h}\left\{ \left( \mathbf x ^{(2)}-\mathbf x ^{(1)}\right) -\left[ \left( \mathbf x ^{(2)}-\mathbf x ^{(1)}\right) \cdot \mathbf n ^{(1)}\right] \mathbf n ^{(1)}\right\} \end{aligned}$$

(58)

With this definition we have a clearer linearization term, which is the following:

$$\begin{aligned} \varDelta \hat{\mathbf{d }}= & {} \,\frac{\kappa E}{h}\left\{ \varDelta \mathbf u - \left[ \varDelta \mathbf u \cdot \mathbf n ^{(1)}+ \left( \mathbf x ^{(2)}-\mathbf x ^{(1)}\right) \cdot \varDelta \mathbf n ^{(1)}\right] \mathbf n ^{(1)}\right. \nonumber \\&\left. +\,\left[ \left( \mathbf x ^{(2)}-\mathbf x ^{(1)}\right) \cdot \mathbf n ^{(1)}\right] \varDelta \mathbf n ^{(1)}\right\} \end{aligned}$$

(59)

where $\varDelta \mathbf u = \varDelta \mathbf u ^{(2)}\left( \varvec{\xi }_t\right) -\varDelta \mathbf u ^{(1)}$. Notice that the local coordinates of the master body are not unknowns, but the coordinates from the last converged step.