FLAC3D Theory and Background • Theoretical Background

Theoretical Background

The theoretical background for FLAC3D is presented in this section. Much of FLAC3D is a direct extension from the two-dimensional version of FLAC. Refer to the FLAC User’s Manual for further background on the explicit finite difference method, which is the basis for both FLAC and FLAC3D.

There are, however, distinct differences between the two- and three-dimensional formulations, specifically in the 2D-to-3D extension of the mathematical model that describes the mechanics of the system. This section describes the three-dimensional formulation and its implementation in FLAC3D.

Section Outline

• Formulation of a 3D Explicit Finite Difference Model
  • Mathematical Model Description
    • Conventions
    • Stress
    • Rate of Strain and Rate of Rotation
    • Equations of Motion and Equilibrium
    • Boundary and Initial Conditions
    • Constitutive Equations
  • Numerical Formulation
    • Finite Difference Approximation to Space Derivatives
    • Nodal Formulation of the Equations of Motion
    • Explicit Finite Difference Approximation to Time Derivatives
    • Constitutive Equations in Incremental Form
    • Large- and Small-Strain Modes
    • Mechanical Timestep Determination for Numerical Stability
    • Mechanical Damping
      • Local Nonviscous Damping
      • Combined Damping
• Grid Discretization
  • Mixed Discretization for a Hexahedral Grid
  • Nodal Mixed Discretization for a Tetrahedral Grid
    • Nodal mixed discretization on strain
    • Nodal mixed discretization on stress
• Numerical Implementation
  • Body Discretization
  • Initial and Boundary Conditions
  • Main Calculation Steps
  • Strain-Rate Calculation
  • Stress Calculation
  • Nodal Mass Calculation
  • Out-of-Balance Force Calculation
  • Velocity and Displacement Calculations
  • Geometry Update Calculation
  • Energy Calculation in FLAC3D
• References