caneMultiphysics

caneMultiphysics is a MATLAB-based research code for finite element and isogeometric multiphysics analysis developed at the Chair of Structural Analysis, Technical University of Munich. The repository contains classical FEM solvers, NURBS-based isogeometric solvers, fluid and structural dynamics coupling, contact mechanics, thermal analysis, uncertainty quantification, and shape optimization examples.

The executable examples live under main/. The implementation modules live beside it, for example FEMThermalConductionAnalysis/, FEMComputationalFluidDynamicsAnalysis/, isogeometricThinStructureAnalysis/, and MonteCarloSimulationAnalysis/.

Quick Start

Start MATLAB in the cane repository root, then add the project to the path:

addpath(genpath(pwd))

Run a single application script:

run('main/main_FEMThermalConductionAnalysis/main_steadyStateThermalConductionAnalysis.m')

The main/ folder is the application catalog. Each subfolder groups runnable scripts by analysis area. To run another application, keep the repository on the MATLAB path and replace the path in run(...) with the desired script under main/.

Unit Tests

The unit-test driver main_runUnitTests.m covers quadrature, utility functions, isogeometric beam and membrane analysis, Kirchhoff-Love shells, FEM thermal conduction, plate in membrane action, contact, IGA/FEM CFD, FSI, and shape optimization.

run('main/main_unitTests/main_runUnitTests.m')

Benchmark Summary

Application area	Representative script	Output
FEM thermal conduction	main_thermalConductionBenchmarks.m	2 figures
FEM plane stress	main_convergenceStudyInfinitePlateWithHole.m	4 figures
FEM contact mechanics	main_HertzConvergenceStudy.m	4 figures
FEM incompressible CFD	main_steadyStateIncompressibleNavierStokesFlow.m	VTK output
FEM fluid-structure interaction	main_plotFluidStructureAndInterfaceDiscretizations.m	setup and interface figures
FEM shape optimization	main_unconstrainedCFDShapeOptimizationDrag.m	2 figures
IGA membrane with embedded cables	main_steadyStateDDMFourPointSail.m	benchmark driver
IGA Kirchhoff-Love shell	main_scordelisLoRoof.m	3 figures
IGA incompressible CFD	main_steadyStateStokesFlowInUnitSquareDomain.m	velocity field
Monte Carlo simulation	main_monteCarloSimpleBenchmark.m	1 figure

MATLAB Version

A practical minimum target is MATLAB R2015b or newer for the core examples, with the relevant domain toolboxes installed when a script asks for them. The Monte Carlo benchmark uses parfor and norminv, so it benefits from Parallel Computing Toolbox and requires Statistics and Machine Learning Toolbox for that example.

Repository Layout

Path	Role
main/	Runner scripts and benchmark applications. Start here.
inputGiD/ and gid_cases/	GiD model input data for FEM examples.
outputVTK/	VTK output written by CFD and other postprocessors.
basisFunctions/	Classical finite element basis functions and quadrature utilities.
CAGDKernel/	B-spline and NURBS geometry, refinement, base-vector, and graphics utilities.
equationSystemSolvers/	Linear-system solvers and iterative solver wrappers.
efficientComputation/	Pagewise and vectorized kernels used by larger analyses.
parsers/	GiD and model parsers.
unitTest/	MATLAB unit-test classes used by main/main_unitTests.
documentation/	Public README images and project documentation.

Main application folders:

Path	Application area
main/main_FEMThermalConductionAnalysis/	FEM thermal conduction
main/main_FEMPlateInMembraneActionAnalysis/	FEM plane stress and membrane action
main/main_FEMContactMechanicsAnalysis/	FEM contact mechanics
main/main_FEMComputationalFluidDynamicsAnalysis/	FEM computational fluid dynamics
main/main_FEMComputationalFluidStructureInteractionAnalysis/	FEM fluid-structure interaction
main/main_shapeOptimization/	CFD shape optimization
main/main_isogeometricBeamAnalysis/	IGA beam analysis
main/main_isogeometricMembraneAnalysis/	IGA membranes and embedded cables
main/main_isogeometricKirchhoffLoveShellAnalysis/	IGA Kirchhoff-Love shells
main/main_isogeometricComputationalFluidDynamicsAnalysis/	IGA computational fluid dynamics
main/main_MonteCarloSimulationAnalysis/	Monte Carlo simulation
main/main_unitTests/	Unit-test runner

Applications of the Finite Element Methods

Thermal Conduction

Representative scripts:

• main_steadyStateThermalConductionAnalysis.m
• main_transientThermalConductionAnalysis.m
• main_thermalConductionBenchmarks.m

The thermal module solves steady and transient heat-transfer problems,

$$ \rho c \frac{\partial T}{\partial t} - \nabla \cdot (k \nabla T) = Q. $$

Benchmark script: main_thermalConductionBenchmarks.m. See [1] for the finite element formulation background.

Plane Stress: Infinite Plate With a Hole

Representative scripts:

• main_convergenceStudyInfinitePlateWithHole.m
• main_steadyStateGeometricallyLinearPlateInMembraneAction.m
• main_modalAnalysisPlateInMembraneAction.m

For linear plane stress,

$$ \nabla \cdot \boldsymbol{\sigma} + \mathbf{b} = \mathbf{0}, \qquad \boldsymbol{\sigma} = \mathbf{C} : \boldsymbol{\varepsilon}(\mathbf{u}). $$

Benchmark script: main_convergenceStudyInfinitePlateWithHole.m. The plotted convergence lines connect only the actual computed GiD refinement cases. See [1] for the finite element formulation background.

Contact Mechanics: Hertz Benchmark

Representative scripts:

• main_HertzConvergenceStudy.m
• main_FEMContactLinearPlateInMembraneAction.m

The contact module handles frictionless Signorini contact,

$$ g_n \ge 0, \qquad p_n \le 0, \qquad g_n p_n = 0. $$

Benchmark script: main_HertzConvergenceStudy.m. The circular markers are the actual GiD refinement cases and MATLAB connects them with straight line segments.

Computational Fluid Dynamics

Representative scripts:

• main_steadyStateIncompressibleNavierStokesFlow.m
• main_transientIncompressibleNavierStokesFlow.m
• main_transientNavierStokesFlowTaylorGreenVortices.m

The FEM CFD module solves incompressible Navier-Stokes problems using a residual-based Variational Multiscale Method (VMS) [2-4]. In strong form,

$$\rho \left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot\nabla\mathbf{u}\right) - \nabla \cdot \boldsymbol{\sigma} = \mathbf{f}, \qquad \nabla \cdot \mathbf{u} = 0 .$$

With test functions $\mathbf{w}$ and $q$, the resolved-scale variational problem can be written at a high level as

$$A_\mathrm{NS}((\mathbf{w},q),(\bar{\mathbf{u}}_h,\bar{p}_h)) + A_\mathrm{VMS}((\mathbf{w},q),(\bar{\mathbf{u}}_h,\bar{p}_h)) = F(\mathbf{w}) .$$

where

$$A_\mathrm{NS} = \left(\mathbf{w}, \rho\left(\frac{\partial \bar{\mathbf{u}}_h}{\partial t} + \bar{\mathbf{u}}_h\cdot\nabla\bar{\mathbf{u}}_h\right)\right)_\Omega + \left(\nabla\mathbf{w},2\mu\boldsymbol{\varepsilon}(\bar{\mathbf{u}}_h)\right)_\Omega - \left(\nabla\cdot\mathbf{w},\bar{p}_h\right)_\Omega + \left(q,\nabla\cdot\bar{\mathbf{u}}_h\right)_\Omega .$$

and a residual-based VMS stabilization can be expressed as

$$A_\mathrm{VMS} = \sum_K \left(\rho\bar{\mathbf{u}}_h\cdot\nabla\mathbf{w}+\nabla q,\tau_M\mathbf{r}_M\right)_K + \sum_K \left(\nabla\cdot\mathbf{w},\tau_C r_C\right)_K .$$

The VMS split separates resolved and unresolved velocity and pressure scales,

$$ \mathbf{u} = \bar{\mathbf{u}} + \mathbf{u}', \qquad p = \bar{p} + p', $$

and models the unresolved scales from the momentum and continuity residuals,

$$ \mathbf{u}' \approx -\tau_M \mathbf{r}_M, \qquad p' \approx -\tau_C r_C. $$

Here, $\mathbf{r}_M$ and $r_C$ denote the momentum and continuity residuals on each element $K$.

Benchmark script: main_steadyStateIncompressibleNavierStokesFlow.m. The script writes VTK output under outputVTK/FEMComputationalFluidDynamicsAnalysis/.

Fluid-Structure Interaction

Representative scripts:

• main_plotFluidStructureAndInterfaceDiscretizations.m
• main_FSIFlexibleElasticStructure.m
• main_FSIRigidCylinderOnSpring.m

The FSI examples combine a Navier-Stokes fluid domain, a structural domain, and an interface Gamma_FSI where kinematic and traction compatibility are enforced:

$$ \mathbf{u}_f = \dot{\mathbf{d}}_s, \qquad \boldsymbol{\sigma}_f\mathbf{n}_f + \boldsymbol{\sigma}_s\mathbf{n}_s = \mathbf{0}. $$

Benchmark script: main_plotFluidStructureAndInterfaceDiscretizations.m. See [5-7] for the Turek FSI benchmark and related FSI/interface studies produced with this code base.

Gradient-Based CFD Shape Optimization

Representative script:

• main_unconstrainedCFDShapeOptimizationDrag.m

The shape optimization example minimizes drag for a 2D cylinder-flow problem by perturbing the cylinder radius,

$$ \frac{dJ}{dp} \approx \frac{J(p+\epsilon)-J(p)}{\epsilon}, \qquad p_{k+1} = p_k - \alpha \frac{dJ}{dp}. $$

Benchmark script: main_unconstrainedCFDShapeOptimizationDrag.m. See [8] for risk-aware CFD shape optimization work produced with this code base.

Applications of Isogeometric Analysis

Plane Stress: Curved Beam Under Tip Shear

Representative script:

• main_curvedBeamTipShear.m
• main_IGACurvedBeamTipShearConvergence.m

This benchmark uses exact NURBS geometry for a curved beam modeled as a plate in membrane action. It compares the stress field against a closed-form reference solution. The convergence plots repeat the same analytical error computation over several h-refinement levels. Related isogeometric coupling and domain-decomposition formulations are given in [9,10].

Benchmark script: main_curvedBeamTipShear.m.

3D Membranes With Embedded Cables: Four-Point Sail

Representative scripts:

• main_steadyStateDDMFourPointSail.m
• main_FoFiFourPointSail.m
• main_modalAnalysisFourPointSail.m

The membrane module covers single-patch and multipatch NURBS membranes, weak Dirichlet boundary conditions, domain decomposition, form-finding, modal analysis, transient response, and cable-coupled membrane models [9,11].

Kirchhoff-Love Shells: Scordelis-Lo Roof

Representative scripts:

• main_scordelisLoRoof.m
• main_cantileverPlate.m

The shell examples use Kirchhoff-Love kinematics, where the mid-surface is represented by a NURBS surface and rotations are implied by the surface normal:

$$ \delta W_\mathrm{int} = \int_\Omega \left( \delta \boldsymbol{\varepsilon}^\mathsf{T} \mathbf{n} {}+ \delta \boldsymbol{\kappa}^\mathsf{T} \mathbf{m} \right) , d\Omega. $$

Benchmark script: main_scordelisLoRoof.m. The contour shows the first principal membrane stress resultant n^1 on the geometrically nonlinear solution; the deformation is scaled for visualization. See [10,12,13] for nonlinear isogeometric shell analysis, multipatch shell coupling, and penalty-parameter studies.

Isogeometric Computational Fluid Dynamics

Representative scripts:

• main_steadyStateStokesFlowInUnitSquareDomain.m
• main_steadyStateStokesFlowShearCavity.m
• main_transientNavierStokesFlowTaylorGreenVortices.m

IGA CFD uses the same residual-based VMS idea [2-4] together with smooth NURBS spaces for velocity and pressure fields:

$$ \mathbf{x}(\boldsymbol{\xi}) = \sum_A R_A(\boldsymbol{\xi}) \mathbf{P}_A, \qquad R_A = \frac{N_A w_A}{\sum_B N_B w_B}. $$

The discrete trial fields are expanded directly in the NURBS basis,

$$ \bar{\mathbf{u}}_h(\boldsymbol{\xi}) = \sum_A R_A(\boldsymbol{\xi}) \mathbf{u}_A, \qquad \bar{p}_h(\boldsymbol{\xi}) = \sum_A R_A(\boldsymbol{\xi}) p_A, $$

while unresolved scales are again introduced through residual-based VMS stabilization.

Benchmark script: main_steadyStateStokesFlowInUnitSquareDomain.m.

Other Applications

Monte Carlo Simulation

Representative scripts:

• main_monteCarloSimpleBenchmark.m
• main_monteCarloSteadyStateFEMPlateInMembraneAction.m
• main_monteCarloSteadyStateIncompressibleNavierStokesFlow.m

The simple benchmark estimates statistics of

$$ Y = \exp(U), \qquad U \sim \mathcal{N}(0,1), $$

for which the exact mean is sqrt(e).

Benchmark script: main_monteCarloSimpleBenchmark.m. See [8] for uncertainty-aware design work connected to the stochastic-analysis functionality.

Notes for developers

• Some scripts write VTK files under outputVTK/. These outputs can be inspected in ParaView.
• Several examples use GiD input models from inputGiD/. If a parser error appears first, check that the expected case directory exists there.

License

caneMultiphysics is distributed under the BSD-style license in license.txt.

References

[1] T. J. R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Dover Publications, 2000.

[2] T. J. R. Hughes, "Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods," Computer Methods in Applied Mechanics and Engineering 127, 387-401, 1995.

[3] Y. Bazilevs, V. M. Calo, J. A. Cottrell, T. J. R. Hughes and A. Reali, "Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows," Computer Methods in Applied Mechanics and Engineering 197, 173-201, 2007.

[4] E. Oñate, "Derivation of stabilized equations for numerical solution of advective-diffusive transport and fluid flow problems," Computer Methods in Applied Mechanics and Engineering 151, 233-265, 1998.

[5] S. Turek and J. Hron, "Proposal for numerical benchmarking of fluid-structure interaction between an elastic object and laminar incompressible flow," in Fluid-Structure Interaction, Lecture Notes in Computational Science and Engineering 53, 371-385, 2006.

[6] G. De Nayer, A. Apostolatos, J. N. Wood, K.-U. Bletzinger, R. Wüchner et al., "Numerical studies on the instantaneous fluid-structure interaction of an air-inflated flexible membrane in turbulent flow," Journal of Fluids and Structures 82, 577-609.

[7] A. Apostolatos, G. De Nayer, K.-U. Bletzinger, M. Breuer and R. Wüchner, "Systematic evaluation of the interface description for fluid-structure interaction simulations using the isogeometric mortar-based mapping," Journal of Fluids and Structures 86, 368-399.

[8] A. Kodakkal, B. Keith, U. Khristenko, A. Apostolatos, K.-U. Bletzinger et al., "Risk-averse design of tall buildings for uncertain wind conditions," Computer Methods in Applied Mechanics and Engineering 402, 115371.

[9] A. Apostolatos, R. Schmidt, R. Wüchner and K.-U. Bletzinger, "A Nitsche-type formulation and comparison of the most common domain decomposition methods in isogeometric analysis," International Journal for Numerical Methods in Engineering 97 (7), 473-504.

[10] A. Apostolatos, M. Breitenberger, R. Wüchner and K.-U. Bletzinger, "Domain decomposition methods and Kirchhoff-Love shell multipatch coupling in isogeometric analysis," Isogeometric Analysis and Applications 2014, 73-101.

[11] A. Apostolatos, K.-U. Bletzinger and R. Wüchner, "Weak imposition of constraints for structural membranes in transient geometrically nonlinear isogeometric analysis on multipatch surfaces," Computer Methods in Applied Mechanics and Engineering 350, 938-994.

[12] M. Breitenberger, A. Apostolatos, B. Philipp, R. Wüchner and K.-U. Bletzinger, "Analysis in computer aided design: Nonlinear isogeometric B-Rep analysis of shell structures," Computer Methods in Applied Mechanics and Engineering 284, 401-457.

[13] T. Pasch, L. F. Leidinger, A. Apostolatos, R. Wüchner, K.-U. Bletzinger et al., "A priori penalty factor determination for (trimmed) NURBS-based shells with Dirichlet and coupling constraints in isogeometric analysis," Computer Methods in Applied Mechanics and Engineering 377, 113688.