Nitsche’s method for finite deformation thermomechanical contact problems

Abstract

This paper presents an extension of Nitsche’s method to finite deformation thermomechanical contact problems including friction. The mechanical contact constraints, i.e. non-penetration and Coulomb’s law of friction, are introduced into the weak form using a stabilizing consistent penalty term. The required penalty parameter is estimated with local generalized eigenvalue problems, based on which an additional harmonic weighting of the boundary traction is introduced. A special focus is put on the enforcement of the thermal constraints at the contact interface, namely heat conduction and frictional heating. Two numerical methods to introduce these effects are presented, a substitution method as well as a Nitsche-type approach. Numerical experiments range from two-dimensional frictionless thermo-elastic problems demonstrating optimal convergence rates to three-dimensional thermo-elasto-plastic problems including friction.

Appendix: A Consistency of the thermal weak form using Nitsche’s method

To show the consistency of (39), i.e. equivalence of (39) with (30) in combination with (24) and (25), we first observe, that $\bar{\beta }_c$ and ${\mathcal {P}}_\tau$ are, due to (31) and (33), consistent substitutes for $\beta _c\vert p_n \vert$ and $\varvec{{t}}_\tau \cdot \varvec{{v}}_\tau$ in (24) and (25). Next, (24) and (25) provide the following equivalences for weighted heat fluxes:

$$\begin{aligned}&\left\{ q_c(T)\right\} _{\omega _\vartheta } = \left\{ q_c(T)\right\} _{(1-\delta _c)} +(1-\delta _c-\omega _\vartheta ) \varvec{{t}}_\tau \cdot \varvec{{v}}_\tau \end{aligned}$$

(53a)

$$\begin{aligned}&\left\{ q_c(T)\right\} _{(1-\delta _c)} = \beta _c\vert p_n \vert [\![T]\!] \;\;, \end{aligned}$$

(53b)

with which (39) can be re-written as

$$\begin{aligned} \begin{aligned}&\int _{\varOmega }\delta T C_v \dot{T} \,{\mathrm {d}}\varOmega +\int _{\varOmega }(\nabla _{\varvec{{X}}} T)^{{\mathrm {T}}} k_0 \varvec{{C}}^{-1}\nabla _{\varvec{{X}}}\delta T \,{\mathrm {d}}\varOmega \\&\quad -\int _{\varOmega } \left( \frac{1}{2} T\frac{\partial \varvec{{S}}:\dot{\varvec{{C}}}}{\partial T} + \hat{R}_0\right) \delta T \,{\mathrm {d}}\varOmega -\int _{\varGamma _q} \hat{Q}_0 \delta T \,{\mathrm {d}}\varGamma \\&\quad +\int _{\gamma _c^{(1)}} \frac{\beta _c\vert p_n\vert }{ \beta _c\vert p_n\vert +\gamma _\vartheta } \left\{ q_c(T)\right\} _{(1-\delta _c)} [\![\delta T]\!] \,{\mathrm {d}}\gamma \\&\quad +\int _{\gamma _c^{(1)}} \frac{\gamma _\vartheta \beta _c\vert p_n\vert }{\beta _c\vert p_n\vert +\gamma _\vartheta } [\![T]\!] [\![\delta T]\!] \,{\mathrm {d}}\gamma \\&\quad -\theta _\vartheta \int _{\gamma _c^{(1)}} \frac{1}{\beta _c\vert p_n\vert +\gamma _\vartheta } \left\{ q_c(T)\right\} _{(1-\delta _c)} \left\{ q_c(\delta T)\right\} _{\omega _\vartheta }\,{\mathrm {d}}\gamma \\&\quad +\theta _\vartheta \int _{\gamma _c^{(1)}} \frac{\beta _c\vert p_n\vert }{\beta _c\vert p_n\vert +\gamma _\vartheta } [\![T]\!] \left\{ q_c(\delta T)\right\} _{\omega _\vartheta } \,{\mathrm {d}}\gamma \\&\quad -\int _{\gamma _c^{(1)}}\varvec{{t}}_\tau \cdot \varvec{{v}}_\tau \left( \delta _c\delta T^{(1)} + (1-\delta _c)\delta T^{(2)} \circ \chi _t \right) \,{\mathrm {d}}\gamma \\&= 0 \forall \delta T \in {\mathcal {V}}_T \;\;, \end{aligned} \end{aligned}$$

(54)

Next, let us take a closer look at the fifth and sixth line containing all terms tested with $\left\{ q_c(\delta T)\right\} _{\omega _\vartheta }$:

$$\begin{aligned} \begin{aligned}&-\theta _\vartheta \int _{\gamma _c^{(1)}} \frac{1}{\beta _c\vert p_n\vert +\gamma _\vartheta } \left\{ q_c(T)\right\} _{(1-\delta _c)} \left\{ q_c(\delta T)\right\} _{\omega _\vartheta }\,{\mathrm {d}}\gamma \\&\quad +\theta _\vartheta \int _{\gamma _c^{(1)}} \frac{\beta _c\vert p_n\vert }{\beta _c\vert p_n\vert +\gamma _\vartheta } [\![T]\!] \left\{ q_c(\delta T)\right\} _{\omega _\vartheta } \,{\mathrm {d}}\gamma \\&\quad = -\theta _\vartheta \int _{\gamma _c^{(1)}} \frac{1}{\beta _c\vert p_n\vert +\gamma _\vartheta } \left( \underbrace{\left\{ q_c(T)\right\} _{(1-\delta _c)} -\beta _c\vert p_n \vert [\![T]\!] }_{{\mathop {=}\limits ^{\text {eq.~(53b)}}}0} \right) \,{\mathrm {d}}\gamma \\&\quad = 0 \;\;. \end{aligned} \end{aligned}$$

(55)

What remains from (54) is

$$\begin{aligned} \begin{aligned}&\int _{\varOmega }\delta T C_v \dot{T} \,{\mathrm {d}}\varOmega +\int _{\varOmega }(\nabla _{\varvec{{X}}} T)^{{\mathrm {T}}} k_0 \varvec{{C}}^{-1}\nabla _{\varvec{{X}}}\delta T \,{\mathrm {d}}\varOmega \\&-\int _{\varOmega } \left( \frac{1}{2} T\frac{\partial \varvec{{S}}:\dot{\varvec{{C}}}}{\partial T} + \hat{R}_0\right) \delta T \,{\mathrm {d}}\varOmega -\int _{\varGamma _q} \hat{Q}_0 \delta T \,{\mathrm {d}}\varGamma \\&+\int _{\gamma _c^{(1)}} \frac{\beta _c\vert p_n\vert }{ \beta _c\vert p_n\vert +\gamma _\vartheta } \left\{ q_c(T)\right\} _{(1-\delta _c)} [\![\delta T]\!] \,{\mathrm {d}}\gamma \\&+\int _{\gamma _c^{(1)}} \frac{\gamma _\vartheta \beta _c\vert p_n\vert }{\beta _c\vert p_n\vert +\gamma _\vartheta } [\![T]\!] [\![\delta T]\!] \,{\mathrm {d}}\gamma \\&-\int _{\gamma _c^{(1)}}\varvec{{t}}_\tau \cdot \varvec{{v}}_\tau \left( \delta _c\delta T^{(1)} + (1-\delta _c)\delta T^{(2)} \circ \chi _t \right) \,{\mathrm {d}}\gamma \\ =&\int _{\varOmega }\delta T C_v \dot{T} \,{\mathrm {d}}\varOmega +\int _{\varOmega }(\nabla _{\varvec{{X}}} T)^{{\mathrm {T}}} k_0 \varvec{{C}}^{-1}\nabla _{\varvec{{X}}}\delta T \,{\mathrm {d}}\varOmega \\&-\int _{\varOmega } \left( \frac{1}{2} T\frac{\partial \varvec{{S}}:\dot{\varvec{{C}}}}{\partial T} + \hat{R}_0\right) \delta T \,{\mathrm {d}}\varOmega -\int _{\varGamma _q} \hat{Q}_0 \delta T \,{\mathrm {d}}\varGamma \\&+\int _{\gamma _c^{(1)}} \left\{ q_c(T)\right\} _{(1-\delta _c)} [\![\delta T]\!] \,{\mathrm {d}}\gamma \\&-\int _{\gamma _c^{(1)}} \frac{\gamma _\vartheta }{\beta _c\vert p_n\vert + \gamma _\vartheta } \left( \underbrace{ \left\{ q_c(T)\right\} _{(1-\delta _c)} - \beta _c\vert p_n\vert [\![T]\!] }_{{\mathop {=}\limits ^{\text {eq.~{(53)}}}}0} \right) [\![\delta T]\!] \,{\mathrm {d}}\gamma \\&-\int _{\gamma _c^{(1)}}\varvec{{t}}_\tau \cdot \varvec{{v}}_\tau \left( \delta _c\delta T^{(1)} + (1-\delta _c)\delta T^{(2)} \circ \chi _t \right) \,{\mathrm {d}}\gamma \\ =&\int _{\varOmega }\delta T C_v \dot{T} \,{\mathrm {d}}\varOmega +\int _{\varOmega }(\nabla _{\varvec{{X}}} T)^{{\mathrm {T}}} k_0 \varvec{{C}}^{-1}\nabla _{\varvec{{X}}}\delta T \,{\mathrm {d}}\varOmega \\&-\int _{\varOmega } \left( \frac{1}{2} T\frac{\partial \varvec{{S}}:\dot{\varvec{{C}}}}{\partial T} + \hat{R}_0\right) \delta T \,{\mathrm {d}}\varOmega -\int _{\varGamma _q} \hat{Q}_0 \delta T \,{\mathrm {d}}\varGamma \\&+\int _{\gamma _c^{(1)}} (\underbrace{\beta _c\vert p_n\vert [\![T]\!] - \delta _c \varvec{{t}}_\tau \cdot \varvec{{v}}_\tau }_{{\mathop {=}\limits ^{\text {eq.~(24)}}}q_c^{(1)}} ) \delta T^{(1)} \\&\qquad +(\underbrace{-\beta _c\vert p_n\vert [\![T]\!] -(1-\delta _c)\varvec{{t}}_\tau \cdot \varvec{{v}}_\tau }_{{\mathop {=}\limits ^{\text {eq.~25}}}q_c^{(2)}}) (\delta T^{(2)}\circ \chi _t) \,{\mathrm {d}}\gamma \\&= 0 \forall \delta T \in {\mathcal {V}}_T \;\;, \end{aligned} \end{aligned}$$

(56)

which is indeed equivalent to (30).