Contents  Introduction  Preliminary discussion  Finite Element  Method  Adaptation of the  FEM  Appendix  Printable version  Discuss this article  in the forums The Series  Part 1  Part 2  Part 3 (coming soon)

Preliminary discussion

Some definitions

Hypothesis of continuity

The basic structure of the matter is characterized by non-uniformity and discontinuity attributable to its various subdivisions: molecules, atoms and subatomic particles. The concern in this document is to replace the actual system of particles with a continuous distribution of matter. There is the clear implication in such an approach that any small volumes which could be considered here are enough to contain a lot of particles. Random fluctuations in the properties of the material are not considered to be important.

Definition of the studied system

The studied system is a solid, which is sometimes referred to by the name of body. Some of the main abilities of a solid are:

• to maintain its shape without need of a container;
• to resist continuous shear and tension.

Mechanics of solids

In contrast with rigid body statics and dynamics, which treat the external behaviour of bodies, the mechanics of solids are concerned with the relationship of external effects (i.e. forces and moments) to internal stresses and strains.

Types of strains

External forces actions on a body may be classified as surface forces and body forces. A surface force is of the concentrated type when it acts at a point; a surface force may also be distributed uniformly or non-uniformly over a finite area. Body forces act on volumetric elements rather than surfaces and are attributable to fields such as gravity and magnetism.

Units

All the units follow the international standard (IS).

The right-handed system is used by default.

Right-handed system

Which method to use?

Aim of the project

According to the first part of this document, the first target of the Hyperion Project is to calculate the deformations of a solid with a distribution of forces. As above the main specifications impose the following conditions for the simulation:

• Real-time;
• Realistic solution
• Generation of a large variety of shapes for the body;
• Application of usual boundary conditions;
• Elastic hypothesis

Analytical vs. Numerical methods

So what are the classical means to calculate the system deformations? To succeed in this task, classic analytical methods used in the mechanics of solids are limited. Strength of Materials has resolved some problems but they are too specific. Navier relations for plates and shells are also not enough generic. In conclusion, the analytical methods seem to be less adapted than the numerical method.

Nowadays, the most powerful method to resolve physical problems is a numerical method called the Finite Element Method (FEM).

Next : The Finite Element Method