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Adaptive finite elements for obstacle problems

Tom Gustafsson

TL;DR

The paper addresses obstacle-type inequality constraints in PDEs by formulating them as a mixed variational problem with a Lagrange multiplier, enabling the unknown coincidence (contact) set to be resolved via $h$-adaptive mesh refinement. The main method combines bubble-enriched $P_2-P_0$ finite elements with a primal–dual active-set (semismooth Newton) linearisation and a robust a posteriori error estimator to drive refinement. The authors demonstrate this approach on three engineering-inspired applications—membrane contact, elastoplastic torsion, and cavitation in hydrodynamic bearings—showing accurate resolution of contact/cavitation zones, efficient convergence, and substantial DOF reductions compared to uniform meshes. They also discuss practical implementation aspects, including geometry handling, reuse of prior solutions across mesh refinements, and the implications of estimator design on adaptive performance. Overall, the work provides a validated, adaptable framework for solving complex obstacle problems with meaningful physical interpretations and practical computational benefits.

Abstract

We summarise three applications of the obstacle problem to membrane contact, elastoplastic torsion and cavitation modelling, and show how the resulting models can be solved using mixed finite elements. It is challenging to construct fixed computational meshes for any inequality-constrained problem because the coincidence set has an unknown shape. Consequently, we demonstrate how $h$-adaptivity can be used to resolve the unknown coincidence set. We demonstrate some practical challenges that must be overcome in the application of the adaptive method.

Adaptive finite elements for obstacle problems

TL;DR

The paper addresses obstacle-type inequality constraints in PDEs by formulating them as a mixed variational problem with a Lagrange multiplier, enabling the unknown coincidence (contact) set to be resolved via -adaptive mesh refinement. The main method combines bubble-enriched finite elements with a primal–dual active-set (semismooth Newton) linearisation and a robust a posteriori error estimator to drive refinement. The authors demonstrate this approach on three engineering-inspired applications—membrane contact, elastoplastic torsion, and cavitation in hydrodynamic bearings—showing accurate resolution of contact/cavitation zones, efficient convergence, and substantial DOF reductions compared to uniform meshes. They also discuss practical implementation aspects, including geometry handling, reuse of prior solutions across mesh refinements, and the implications of estimator design on adaptive performance. Overall, the work provides a validated, adaptable framework for solving complex obstacle problems with meaningful physical interpretations and practical computational benefits.

Abstract

We summarise three applications of the obstacle problem to membrane contact, elastoplastic torsion and cavitation modelling, and show how the resulting models can be solved using mixed finite elements. It is challenging to construct fixed computational meshes for any inequality-constrained problem because the coincidence set has an unknown shape. Consequently, we demonstrate how -adaptivity can be used to resolve the unknown coincidence set. We demonstrate some practical challenges that must be overcome in the application of the adaptive method.

Paper Structure

This paper contains 14 sections, 1 theorem, 52 equations, 26 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Applications
  3. Deformation and contact of a membrane
  4. Elastoplastic torsion problem
  5. Hydrodynamic lubrication with cavitation
  6. Variational formulation
  7. Discretisation
  8. Linearisation
  9. Implementation
  10. Computational demonstrations
  11. Membrane contact problem
  12. Plastic torsion of I beam
  13. Cavitation in a hydrodynamic bearing
  14. Discussion

Key Result

Theorem 1

There exists $C>0$ such that where

Figures (26)

  • Figure 1: A visualisation of the kinematics of the torsion problem, $\vec{u}(x,y,z) = (-\theta yz, \theta xz, 0)$, when applied to a square shaft. The coordinate $z$ is along the length of the shaft and $\theta$ refers to the magnitude of the rotation. The colouring is based on the magnitude of the displacement field with darker hue corresponding to smaller displacement magnitude.
  • Figure 2: A schematic of a hydrodynamic bearing (top left), and its unraveled planar representation in a computational domain $\Omega = (-R\pi, R\pi) \times (0, L)$ (top right). The red cylinder represents a shaft which is fitted inside the bearing represented by the gray cylinder. The space between the shaft and the bearing is filled by a thin layer of lubricant with thickness $d(x)$. The lubricant pressure is periodic over the dashed parts of the boundaries.
  • Figure 3: Adaptive schemes may overrefine if a curved domain (left) is approximated by a polygonal domain (right) without modifying the error indicator.
  • Figure 4: (Red-green-blue refinement) The mesh on the left is being refined by marking one triangle for refinement. The red triangle is marked and split into four. To avoid hanging nodes and reduction in the mesh quality, the blue triangles are split into three and the green triangle is split into two. The yellow triangles are not split. The refined mesh is depicted on the right.
  • Figure 5: (Primal-dual active set method) The iterations required for the primal-dual active set method to converge can be reduced significantly by projecting the previous solution to the refined mesh (top) versus starting from a zero intial guess (bottom).
  • ...and 21 more figures

Theorems & Definitions (1)

  • Theorem 1: e.g., Gustafsson--Stenberg--Videman gustafsson2017