Reduced Lagrange Multipliers

GitHub CI Documentation Indent

This repository contains C++ implementations of reduced Lagrange multiplier methods for mixed-dimensional coupling problems, built on top of deal.II.

Overview

The codebase is centered on:

  • bulk elasticity and Poisson solvers with immersed coupling;

  • reduced-order coupling operators and tensor-product spaces;

  • benchmark parameter sets and coupled workflows built around deal.II.

Quick Start

mkdir -p build
cd build
cmake -DDEAL_II_DIR=/path/to/deal.II ..
cmake --build . -j
./elasticity path/to/input.prm
cd ..
python3 -m pip install -r doc/requirements.txt
./scripts/build_doc.sh

Repository Layout

  • include/, source/: library headers and implementations.

  • apps/: executable entry points derived from app_*.cc.

  • tests/, gtests/: regression and GoogleTest-based test suites.

  • doc/: Doxygen and Sphinx source for the published documentation site.

  • scripts/: helper scripts, including scripts/build_doc.sh.

  • prms/, benchmarks/: parameter files, meshes, and benchmark assets.

  • blood/: auxiliary 1D hemodynamics code and data used by coupled workflows.

Notes

  • This repository may contain large benchmark/data files and local experiment artifacts.

  • Some coupled-elasticity paths depend on an external lib1dsolver library and related inputs under blood/.

Coupled 3D/1D workflow

For setup/build/run notes specific to the coupled elasticity problem, see COUPLED_PROBLEM.md.

References

  • Giovanni Alzetta and Luca Heltai, Multiscale modeling of fiber reinforced materials via non-matching immersed methods, Computers & Structures, 239 (2020), 106334. DOI: https://doi.org/10.1016/j.compstruc.2020.106334

  • Camilla Belponer, Alfonso Caiazzo, and Luca Heltai, Mixed-dimensional modeling of vascular tissues with reduced Lagrange multipliers (2025). Local PDF: doc/papers/2309.06797v2.pdf

  • Luca Heltai and Alfonso Caiazzo, Multiscale modeling of vascularized tissues via nonmatching immersed methods, International Journal for Numerical Methods in Biomedical Engineering, 35(12) (2019), e3264. DOI: https://doi.org/10.1002/cnm.3264

  • Luca Heltai, Alfonso Caiazzo, and Lucas O. Muller, Multiscale Coupling of One-dimensional Vascular Models and Elastic Tissues, Annals of Biomedical Engineering, 49 (2021), 3243-3254. DOI: https://doi.org/10.1007/s10439-021-02804-0

  • Luca Heltai and Paolo Zunino, Reduced Lagrange multiplier approach for non-matching coupling of mixed-dimensional domains, Mathematical Models and Methods in Applied Sciences, 33(12) (2023), 2425-2462. DOI: https://doi.org/10.1142/S0218202523500525

  • Yashasvi Verma, Jakob Schattenfroh, Ingolf Sack, Silvia Budday, Paul Steinmann, and Luca Heltai, Simulation Platform to Evaluate Inversion Techniques for Magnetic Resonance Elastography Data (2026).

License

See LICENSE.md.