Predictive Real Unified Continuum modeling of multi-physics in the Digital Math framework


  • Johan Jansson (
  • Massimiliano Leoni
  • Måns Andersson
  • Patricia Lopez Sanchez
  • Mats Stading
  • Ridgway Scott
  • Claes Johnson


In this paper we present the predictive Real Unified Continuum (RUC) cG(1)cG(1) Direct FEM Simulation methodology, allowing general phenomena such as multiphase fluid-structure interaction (FSI), general interface motion, implicit contact, fracture, plasticity, etc., The methodology also allows application of Direct duality-based adaptive error control, which we will investigate in future work. RUC is a generalization of the Unified Continuum framework which we developed previously in [1].

This work is developed as part of the Digital Math framework [2] - as the foundation of modern science based on constructive digital mathematical computation.

The formulation allows mesh motion, similar to an Arbitrary-Lagrange-Eulerian (ALE) setting, and mesh modification, and importantly allows part or the entire mesh to be fixed, representing zero mesh velocity. The demonstration cases below have been computed with zero mesh velocity to investigate the limits of this setting.

Possible performance improvements can be gained by general mesh modification primitives (in e.g. the Omega_h framework), and/or improved particle-based quadrature.

Real Unified Continuum modeling

The methodology is a Direct FEM simulation of the first principle equations, here in multi-phase incompressible form, where we include constitutive laws for a Newtonian fluid and Neo-Hookean solid.

These first principle equations are discretized by the Direct FEM approach, meaning Galerkin-Least-Squares (GLS) stabilization with shock-capturing.

The Galerkin part of the method is formulated as below in FEniCS notation:

F_G = z*inner(u, grad(rho))*nu*dx
F_G += z*(inner(rho*grad(u)*u + grad(p),v) - theta*inner(sigma, grad(v)) + nnu*inner(grad(u),grad(v)) - rho*dot(f,v))*dx
F_G += z*(inner(dot(u, grad(sigma)) + \
                    theta*(2*rho*mumu*epsilon(u) + grad(u)*sigma + sigma*grad(u).T), y))*dx

and in corresponding strong form in Latex notation:

$$ \begin{equation*} %\label{eq:nseinc2} \begin{array}{rcll} \rho (\partial_t {\bf u} + ({\bf u} \cdot \nabla) {\bf u}) - \nabla \cdot \sigma &=& 0 \,\, &\mbox{in } Q,\\ \nabla \cdot {\bf u} &=& 0 \,\, &\mbox{in }Q,\\ \partial_t \rho + ({\bf u} \cdot \nabla) \rho &=& 0 \,\, &\mbox{in }Q,\\ \partial_t \theta + ({\bf u} \cdot \nabla) \theta &=& 0 \,\, &\mbox{in }Q,\\ \sigma &\equiv& \bar{\sigma} - pI\\ \bar{\sigma}_f &\equiv& 2\mu_f \epsilon\\ \bar{\sigma} &\equiv& \theta \bar{\sigma}_f + (1 - \theta) \bar{\sigma}_s\\ \partial_t \bar{\sigma}_s - 2\mu_s \epsilon - \nabla {\bf u} \bar{\sigma}_s - \bar{\sigma}_s \nabla {\bf u}^{\top} &=& 0 \end{array} \end{equation*} $$

First order conservation law approach to solid mechanics

Bonet et. al. in [1] and [2] have developed a mathematical framework for solid mechanics with the promise of translating the success and systematic approach of the success of stabilized FEM for fluid mechanics, which they denote "first order conservation law approach to solid mechanics". One important aspect is that the methods are "locking-free" allowing efficient solution of e.g. thin structures.

We see that our RUC methodology as formulated in this paper automatically satisfies the Bonet criteria, unlocking great potential for efficiency and generality.

Application: Human heart

We here investigate a fully FSI canonical heart model, with elastic heart walls and valves, driven by a force compressing the heart walls. We observe a pumping motion with opening and closing of the valves, and implicit contact in the valves.

Valve configuration 1.

Valve configuration 2.

Application: Swallow

We here investigate a canonical swallowing configuration, as part of the Swallow project. A "bolus" represening a food parcel, passes through a tube representing the esophagus.

High stiffness

Low stiffness

Application: Turning in machining in the ENABLE project

We here investigate a canonical turning case in metal machining, defined from our participation in the ENABLE project. A tool cuts through a moving solid material, representing the metal being machined.


We here introduce the Real Unified Continuum (RUC) modeling methodology. We present canonical cases representing several of the outstanding challenges in the field. The simulations qualitatively capture the salient phenomena of the challenges, demonstrating the power of the methodology.

In future work we will focus on quantitative verification and validation, and in integrating the Direct duality-based adaptive error control method in the framework.

Digital Math: Reproducible Results and Open Source Software

The software for reproducing the results in the paper is available as part of the distribution for the Digital Math: Human workshop.

Contact Johan Jansson ( for instrutions how to run the software in a Google Cloud virtual machine in an easy web interface (free credits).


[1] Hoffman, Johan, Johan Jansson, and Michael Stöckli. "Unified continuum modeling of fluid-structure interaction." Mathematical Models and Methods in Applied Sciences 21.03 (2011): 491-513.

[2] Digital Math

[3] Bonet, Javier and Gil, Antonio J and Lee, Chun Hean and Aguirre, Miquel and Ortigosa, Rogelio, "A first order hyperbolic framework for large strain computational solid dynamics. Part I: Total Lagrangian isothermal elasticity", Computer Methods in Applied Mechanics and Engineering, 2015

[4] Gil, Antonio J and Lee, Chun Hean and Bonet, Javier and Ortigosa, Rogelio, "A first order hyperbolic framework for large strain computational solid dynamics. Part II: Total Lagrangian compressible, nearly incompressible and truly incompressible elasticity", Computer Methods in Applied Mechanics and Engineering, 2016

[5] Bonet, Javier,, WCCM 2014