Predictive Real Unified Continuum modeling of multi-physics in the Digital Math framework
Johan Jansson (firstname.lastname@example.org)
Patricia Lopez Sanchez
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 .
This work is developed as part of the Digital Math framework  - as the
foundation of modern science based on constructive digital
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
The Galerkin part of the method is formulated as below in FEniCS notation:
First order conservation law approach to solid mechanics
Bonet et. al. in  and  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.
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.
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
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
 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