# Introduction

Nonlinear finite element analyses confront users with many choices. An understanding of the fundamental concepts of nonlinear finite element analysis is necessary if you do not want to use the finite element program as a black box. The purpose of this manual is to describe the numerical methods included in Radioss.

Radioss belongs to the family of hydro-codes, in which the material is considered as a non viscous fluid. These hydro-codes found their origin in the work supported by the American Department of Energy at the end of the 70's and which led to software like DYNA2D/3D, HEMP, PRONTO, STEALTH, HONDO and WHAM.

- A 3D Lagrangian formulation for mesh description
- An explicit time integration scheme, leading to small time steps
- Simplicity, under integrated finite element models
- Element by element assembly of nodal forces leading to in memory codes and low I/O requirements as compared to implicit approaches where matrix assembly and inversion is required every time step
- Non-iterative approaches
- Penalty methods based contact
- Highly vectorized implementation.

This first section introduces the notations which will be used throughout the document. An introduction to kinematics is also given.

- Material and spatial coordinates
- Mesh description
- Kinematic and kinetic descriptions
- Stress rates and stresses in solids
- Updated and total Lagrangian formulations
- Equations of equilibrium
- Principle of virtual power and the physical names of power terms.

The small strain formulation is also introduced.

The finite element formulation of the virtual power principle is introduced in Finite Element Formulation, leading to the discretized equations of equilibrium.

Dynamic Analysis deals with time discretization and the integration schemes. Stability and time step concepts are also discussed.

Different finite element models are presented in Element Library. Tetrahedral solid elements, hexahedral solid and solid-shell elements, 3 and 4-node shell elements, 2-node truss and beam elements and spring elements are successively presented.

Kinematic Constraints deals with kinematic constraints, i.e. constraints placed on nodal velocities.

Linear stability is introduced in Linear Stability.

The very important concept of interfaces is considered in Interfaces. Interfaces allow the solution of contact and impact conditions between two parts of a model. The different interface types available in Radioss are presented.

Material laws are discussed in Material Laws.

In Monitored Volume, the formulations of different kinds of monitored volumes are presented in detail. Airbag theory is also developed.

Static deals with the use of explicit algorithms to model quasi-static or static problems. Different approaches are discussed: slow dynamic computation, dynamic relaxation, viscous relaxation and energy discrete relaxation. The dynamic relaxation approach is developed. The /DYREL and /DAMP options are introduced in this chapter.

Radioss Parallelization concerns the presentation of the fundamentals in Radioss parallelization.

In the ALE, CFD and SPH Theory Manual, the ALE formulation is presented in ALE Formulation.

Finally, the last sections are respectively dedicated to the Computational Aero-Acoustic and the Smooth Particle Hydrodynamics formulations.