continuum simulation monday, 9/30/2002. class progress visualization: abstract concept (stress,2d,...

Continuum Simulation

Monday, 9/30/2002

Class ProgressVisualization: abstract concept (stress,2D, 3D), mechanical fieldStochastic simulations: random walk, Brownian movementMonte Carlo method (MC) EnsembleMolecular Dynamics (MD) Trajectory

Multiscale simulationAdaptive Mesh Refinement/Coarsening; Renormalization

Continuum simulationFinite Element Method (FEM); Material Point Method (MPM)

Continuum vs. Atomistic

Continuum Atomistic

Matter Indefinitely divisible

Atom aggregate

Interaction Stress; strain Interatomic force

Model Constitutive model

Potential model

Continuum Problems

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σij, j +bi =ρai

Equilibrium equations:

Material constitutive equations (linear):

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σij =Eijklεkl

Boundary conditions

Discretization inFinite Element Method (FEM)

The basic concept is that a body or structure may be divided into smaller elements.

Discretization in Material Point Method (MPM)

Solid line is an outline of the body analyzed.Black dots are the material points.Dashed lines show a regular, background grid for calculation.

Lagrangian & Eulerian Description

Lagrangian: quantities like position, velocity, stress, are given for a particular moving particle at varying times.

Eulerian:the velocities are given at fixed points in space as time varies. Imagine a case where both the measuring device and the frame of reference are fixed.

Approximation to Region

MPM FEM

FEM Simulation ofCrack Tip Process

MPM Simulation ofCrack Tip Process

Adaptive Mesh Refinement/Coarsening

MPM Simulation ofFragmentation