A higher order polytopal method for contact mechanics with Tresca friction

TL;DR

This work develops and analyzes a second-order Discrete De Rham (DDR) discretization for contact mechanics with Tresca friction on fracture networks modeled as co-dimension-1 interfaces. The scheme uses a mixed displacement–Lagrange multiplier formulation with displacement DOFs attached to vertices, edges, faces, and elements, and a face-wise constant multiplier on fractures, enabling piecewise quadratic reconstruction and robust performance in the quasi-incompressible limit. A discrete Korn inequality and a nonstandard $H^{-1/2}$-type inf-sup condition yield well-posedness, while a detailed error analysis shows $\mathcal O(h^{\tfrac{1}{2}+\mathfrak{r}})$ convergence under $H^{\tfrac{3}{2}+\mathfrak{r}}$ regularity, with locking-free behavior for the primal variable as $\nu \to \tfrac{1}{2}$. Numerical experiments across frictionless and Tresca cases, including a 3D discrete fracture–matrix model, validate the theoretical results and demonstrate improved accuracy and efficiency over lower-order methods, highlighting the method’s suitability for complex fractured geometries in subsurface applications.

Abstract

In this work, we design and analyze a Discrete de Rham (DDR) scheme for a contact mechanics problem involving fractures along which a model of Tresca friction is considered. Our approach is based on a mixed formulation involving a displacement field and a Lagrange multiplier, enforcing the contact conditions, representing tractions at fractures. The approximation space for the displacement is made of vectors values attached to each vertex, edge, face, and element, while the Lagrange multiplier space is approximated by piecewise constant vectors on each fracture face. The displacement degrees of freedom allow reconstruct piecewise quadratic approximations of this field. We prove a discrete Korn inequality that account for the fractures, as well as an inf-sup condition (in a non-standard $H^{-1/2}$-norm) between the discrete Lagrange multiplier space and the discrete displacement space. We provide an in-depth error analysis of the scheme and show that, contrary to usual low-order nodal-based schemes, our method is robust in the quasi-incompressible limit for the primal variable~(displacement). An extensive set of numerical experiments confirms the theoretical analysis and demonstrate the practical accuracy and robustness of the scheme.

Figures (8)

• Figure 6.1: (Test case from Section \ref{['subsec.6.1']}). Relative $L^2$-norm of errors $\mathbf{u}-\Upsilon^2_{h}(\underline{\mathbf{u}}_h)$, $[ [\mathbf{u}] ]-[ [\underline{\mathbf{u}}_h] ]_h$, $\nabla\mathbf{u}-\nabla\Upsilon^2_{h}({\underline{\mathbf{u}}_h})$, and $\lambda_{\mathbf{n}}-\lambda_{h,\mathbf{n}}$ versus the cubic root of the number of cells, for (a) Cartesian, (b) tetrahedral, and (c) Hexa-cut mesh families.
• Figure 6.2: (Test case from Section \ref{['subsec.6.2']}). Relative $L^2$-norm errors of $\mathbf{u}-\Upsilon^2_{h}(\underline{\mathbf{u}}_h)$, $[ [\mathbf{u}] ]-[ [\underline{\mathbf{u}}_h] ]_h$, $\nabla\mathbf{u}-\nabla\Upsilon^2_{h}({\underline{\mathbf{u}}_h})$, and $\lambda_{\mathbf{n}}-\lambda_{h,\mathbf{n}}$ versus the cubic root of the number of cells, for (a) Cartesian, (b) tetrahedral, and (c) Hexa-cut mesh families.
• Figure 6.3: ((a) Test case from Section \ref{['subsec.6.1']} and (b) Test case from Section \ref{['subsec.6.2']}). Comparison of $L^2$-norm of $\nabla\mathbf{u}-\nabla\Upsilon^2_{h}(\underline{\mathbf{u}}_h)$ of the lowest-order method jhr with the current higher-order scheme for a tetrahedral mesh.
• Figure 6.4: Fracture network for the test case in Section \ref{['subsec.6.4']}
• Figure 6.5: Results for the test case in Section \ref{['subsec.6.4']}: (a) contact state classification, where the values indicate: 0 if $\left|\boldsymbol{\lambda}_{h,\tau}\right| < g$ and $[ [\underline{\mathbf{u}}_h] ]_{h,\mathbf{n}} <0$; 1 if $\left|\boldsymbol{\lambda}_{h,\tau}\right| < g$ and $[ [\underline{\mathbf{u}}_h] ]_{h,\mathbf{n}} =0$; 2 if $\left|\boldsymbol{\lambda}_{h,\tau}\right| = g$ and $[ [\underline{\mathbf{u}}_h] ]_{h,\mathbf{n}} <0$; and 3 if $\left|\boldsymbol{\lambda}_{h,\tau}\right| =g$ and $[ [\underline{\mathbf{u}}_h] ]_{h,\mathbf{n}} =0$; and (b) normal displacement jump obtained using the DDR discretisation with 123k cells and 8.8k fracture faces.

Theorems & Definitions (38)

• Theorem 3.1: Discrete Korn Inequality
• proof
• Definition 3.1: ${H}^{-1/2}$-like norm on $L^2(\Gamma)^3$
• Remark 3.1: Comparison with the norm in jhr
• Theorem 3.2: Discrete inf-sup condition
• proof
• Proposition 3.3: Existence and uniqueness result
• proof
• Remark 4.1: Definition of the interpolator
• Lemma 4.1: Properties of the reconstruction operators