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J. Zhang et al.
Conservation of energy is ensured by the equation
ρ
DE
Dt
= =
σ ·
d ε
dt
+ ρs
(3)
Where E is the internal energy per unit mass in the current configuration. ε = ε( x, t)
is the strain and d ε/dt is the corresponding strain rate. s is the inner heat sources.
2.2 Discretization
A series of material points are discretized in MPM to store information (shown in Fig. 1).
Fig. 1. Discrete material points
The density of a continuum can be expressed approximately by the equation
ρ(x i ) =
n p
p=1
m p δ(x i − x ip )
(4)
Where n p is the number of the discrete material points. m p is the mass and x ip is the
position of the material point P. δ is the Dirichlet function.
To formulate the governing equation of MPM, Eqs. (1) to (3) must be given in a
discrete form either.
2.3 The Steps of MPM Algorithm
The general algorithm of MPM can be given as follows.
1. A background mesh is generated.
2. The information is transferred from the material points to the grid nodes.
3. The forces of grid nodes are solved and the boundary conditions are imposed on the
grid nodes.