[**T**riangular **U**nstructured
**M**esh with **B**eziers
in a **L**agrangian **E**nvironment]

The TUMBLE project is supported by the ALADDIN and SANGRIA programs at Carnegie Mellon Univeristy.

## Introduction

## Applications

## TUMBLE Project Links: |

The TUMBLE project is supported by the ALADDIN and SANGRIA programs at Carnegie Mellon Univeristy.

**Primary Authors:**- Gary L. Miller (glmiller at cs.cmu.edu)
Todd Phillips (tp517 at andrew.cmu.edu)

Mark J. Olah (mjo at andrew.cmu.edu)

The TUMBLE package is a testbed for
next-generation meshing technology. Its primary features are:

- Fully Lagrangian framework (most meshing packages are Eulerian or Eulerian/Lagrangian hybrids)
- Explicit and dynamic boundary tracking
- Curved elements - Currently TUMBLE uses a second order Bezier basis for both geometric and functional data.
- Modern mesh manipulation algorithms with theoretical guarantee of correctness and optimality

Thus not only does TUMBLE deal with moving elements, but also with elements with a non-linear geometry. The concepts involved with moving (Lagrangian) meshes, and curved (Bezier) elements are relatively new and untested. Accordingly, the main use of the TUMBLE package is as a testbed for new algorithms and techniques relating to Lagrangian meshes, and meshes with Bezier elements.

The ultimate goals of the authors of TUMBLE are:

- Producing a fully Lagrangian three dimensional meshing package
- Providing tools for that package to be run on massively parallel machines
- Providing sound theoretical results for the correctness and complexity of the underlying algorithms

In this light, we will attempt to provide links to relevant papers, which describe the theoretical results on which TUMBLE is based. The reader will see that nearly every data-structure and algorithm employed by the TUMBLE package is extensible to three dimensions, and also to a parallel computing framework. The main reason for the existence of TUMBLE is that while many algorithms and techniques useful for Lagrangian meshing are known, their relationship within a complete and general meshing framework is not clear. TUMBLE will demonstrate the feasibility of the techniques in two-dimensions, where prototyping and computational costs are less, yet it will not loose sight of the ultimate, three dimensional goal.

For more information see the Detailed Introduction to TUMBLE.

One of the main advantages of
working in
a Lagrangian environment is that it is much more
natural to solver certain problems in such a framework. In
particular fluid simulations are very natural to solve with Lagrangian
techniques, as the elements in the mesh represent regions of
fluid. Thus, the Lagrangian techniques are able to explicitly
track fluid interfaces as they move and deform over time. In
addition the Navier-Stokes equations which govern the dynamics of
Newtonian Fluids, require solving a non-linear system, when formulated
as an Eulerian problem. But in the Lagrangian framework,
Navier-Stokes reduces to a linear problem.
The primary application for which TUMBLE is used at this time is the simulation of blood flow at a microscopic level. Traditionally this is an extraordinarily ddifficult problem, as the interfaces between fluid and cells are dynamic and not well understood. The ability of TUMBLE to formulate the problem in a Lagrangian framework greatly simplifies many of the difficulties. |