This repository implements the Reduced Lagrange Multiplier method for non-matching coupling of mixed-dimensional domains, as described in the paper:

Reduced Lagrange multiplier approach for non-matching coupling of mixed-dimensional domains
Authors: Luca Heltai, Paolo Zunino
Published in Mathematical Models and Methods in Applied Sciences (2023)
DOI: 10.1142/S0218202523500525

Overview

In many physical problems, especially those involving heterogeneous spatial scales, we encounter coupled partial differential equations (PDEs) defined on domains of different dimensions embedded into each other. Examples include:

Flow through fractured porous media
Fiber-reinforced materials
Modeling small circulation in biological tissues

This repository provides a computational framework for coupling PDEs across dimensions using a reduced Lagrange multiplier approach. The method ensures stability and robustness, with particular attention to the smallest characteristic length of the embedded domain.

Features

General Mathematical Framework: This framework provides tools for analyzing and approximating coupled PDEs across different dimensions.
Non-Matching Coupling: The method applies to non-matching interfaces, where coupling constraints are enforced via Lagrange multipliers.
Dimensionality Reduction: Supports model reduction techniques that simplify complex 3D problems into more tractable ones by working in lower-dimensional spaces, and employing a reduced Lagrange multiplier space.
Inf-Sup Stability: Ensures stable and robust numerical solutions across a range of configurations and mesh sizes.

Installation

To use this code, you need a working deal.II version (at least version 9.5). Clone the repository and run the following commands:

git clone https://github.com/luca-heltai/reduced_lagrange_multiplier.git
cd reduced_lagrange_multiplier
mkdir build
cd build
cmake -DDEAL_II_DIR=/path/to/deal.II ..
make

Online documentation

The documentation is built and deployed at each merge to master. You can find the latest documentation here:

https://luca-heltai.github.io/reduced_lagrange_multipliers/

Licence

See the file LICENSE.md for details

## USE OF COUPLED ELASTICITY

Install FVM(1D)

Clone the fvm repo
run docker: docker run -ti -v ./:/fvm --platform linux/amd64 dealii/dealii:v9.6.0-jammy
cd /fvm/
make -f MakefileGCCDesktop clean all

Compile the coupled code

create the build directory with cmake cmake -DCMAKE_CXX_FLAGS=-fopenmp . (this includes OMP, necessary for the 1D)
cd build
compile with ninja -jX (X = number of proc)

run in parallel as

export OMP_NUM_THREADS=1

mpirun -np n ./build/coupled_elasticity_debug <path_to_input_3d> <path_to_input_1d> <couplingSampling> <couplingStart> 0 if we only want the the 1D simulation then set coupling Start to 100 if we only want the 3D Simulation then only give <path_to_input_3d>

random error "invalid template argument" solved by changing the order od #include in app_*