Overview

The ContactStructuralMechanicsApplication (CSMA) adds computational contact mechanics to the StructuralMechanicsApplication of Kratos Multiphysics. It implements segment-to-segment (mortar) contact with dual Lagrange multipliers, enforced either with an augmented Lagrangian method (ALM), a penalty method or multipoint constraints (MPC), for frictionless and frictional (Coulomb) problems in 2D, 3D and axisymmetric settings, plus mortar mesh tying of non-conforming meshes. Everything needed to run a contact simulation is included: the contact conditions (with consistently linearised, automatically generated tangent matrices), the contact search (bounding-volume trees, oriented bounding boxes, self-contact detection), the semi-smooth Newton strategies and active-set convergence criteria, a dedicated mixed displacement/Lagrange-multiplier linear solver, and the Python processes and solvers that drive them from a standard ProjectParameters.json.

Examples from the KratosMultiphysics/Examples repository: double arch (frictionless and frictional), cylinder in a ring, hyperelastic tubes, contacting cylinders with adaptive remeshing, self-contact.

The contact problem in one paragraph

Two (or more) deformable bodies \(\Omega^1\) and \(\Omega^2\) may come into contact along a priori unknown portions of their boundaries \(\Gamma_c^1\) (the slave side) and \(\Gamma_c^2\) (the master side). Contact adds three sources of non-linearity to the structural problem: the constraints are inequalities (Hertz–Signorini–Moreau conditions: no penetration, compressive pressure, complementarity), the active set of nodes in contact is unknown, and, with friction, the stick/slip state of each active node is unknown too. The application discretises the interface with the mortar method (integrals of the constraints over the slave surface, weighted by dual shape functions so that the coupling matrix \(\mathbf{D}\) is diagonal), enforces the constraints with an augmented Lagrangian or a penalty functional, and solves the resulting non-smooth problem with a semi-smooth Newton–Raphson scheme in which the active set is updated inside the Newton loop through a non-linear complementarity (NCP) function.

Figure: Basic definition of the contact problem (thesis Fig. 4.3). The slave point $$\mathbf{x}^1$$ is projected along the normal $$\mathbf{n}$$ onto the master surface, $$\hat{\mathbf{x}}^2$$; the gap is measured along $$\mathbf{n}$$ and the tangential slip in the local frame $$(\boldsymbol{\tau}_1,\boldsymbol{\tau}_2)$$.

Formulations at a glance

Capability matrix

Where the theory comes from

The formulation, the search algorithms and the benchmarks are documented in Chapter 4 (“Contact mechanics”) and Appendices A, C, D and E of the PhD thesis of the application’s author:

V. Mataix Ferrándiz, Innovative mathematical and numerical models for studying the deformation of shells during industrial forming processes with the Finite Element Method, PhD thesis, Universitat Politècnica de Catalunya (UPC), Barcelona, 2020. UPCommons PDF.

The theory pages of this documentation reproduce the relevant derivations, figures, algorithms and tables of the thesis and map every concept to the code. The mortar/dual-Lagrange-multiplier machinery follows A. Popp’s work (TU München, 2012) and the augmented Lagrangian treatment follows Alart–Curnier and Cavalieri–Cardona; see the Bibliography.

A short history

The application was created in August 2016 as a container for a general mortar contact condition, grew through 2017–2019 with the dual-mortar ALM frictionless formulation and its automatic-differentiation code generation, the penalty and frictional formulations, frictional laws, self-contact detection, the explicit solver and contact-driven adaptive remeshing, added the MPC-based formulation in 2020 (the year the thesis was defended), refactored all convergence criteria in 2021, and has been in maintenance mode since 2022 (about 3 150 commits in total).

Architecture in one picture

A contact simulation is a normal StructuralMechanicsAnalysis: the presence of contact_settings (or mpc_contact_settings) in solver_settings makes the structural solver wrapper pick the contact solvers of this application; the contact process listed under processes builds the interface model parts, runs the contact search and creates the mortar conditions; the contact strategy, builder-and-solver, convergence criteria and (optionally) the mixed linear solver do the rest. See Architecture.

Documentation map

The source tree also contains a README.md in every folder of the application (custom_conditions, custom_strategies, custom_processes, custom_utilities, custom_frictional_laws, custom_linear_solvers, custom_master_slave_constraints, python_scripts, tests, automatic_differentiation) that summarises that folder and links back here.

Status, authors and citation

If you use the application in your work, please cite the thesis above and the Kratos reference:

@phdthesis{MataixFerrandiz2020,
  author = {Mataix Ferr{\'a}ndiz, Vicente},
  title  = {Innovative mathematical and numerical models for studying the deformation of shells during industrial forming processes with the Finite Element Method},
  school = {Universitat Polit{\`e}cnica de Catalunya},
  year   = {2020},
  url    = {https://upcommons.upc.edu/bitstream/2117/328952/1/TVMF1de1.pdf}
}
@article{Dadvand2010,
  author  = {Dadvand, Pooyan and Rossi, Riccardo and O{\~n}ate, Eugenio},
  title   = {An object-oriented environment for developing finite element codes for multi-disciplinary applications},
  journal = {Archives of Computational Methods in Engineering},
  volume  = {17}, number = {3}, pages = {253--297}, year = {2010},
  doi     = {10.1007/s11831-010-9045-2}
}

Thesis figures © Vicente Mataix Ferrándiz, PhD thesis “Innovative mathematical and numerical models for studying the deformation of shells during industrial forming processes with the Finite Element Method”, UPC 2020, reproduced by the author. Figures marked “inspired by” are the author’s redrawings of the cited sources.