### 6.1.1. Calculation of Particle Variables

The particle displacement is calculated using forward Euler integration of the particle velocity over timestep, .

As , the particle displacement is given as:

| (6–1)

where the superscripts o and n refer to old and new values respectively, and is the initial particle velocity. In forward integration, the particle velocity calculated at the start of the timestep is assumed to prevail over the entire step. At the end of the timestep, the new particle velocity is calculated using the analytical solution to the particle momentum equation:

| (6–2)

is the sum of all forces acting on a particle. This equation is an example of the generic transport equation:

| (6–3)

where is a generic transported variable, subscript indicates the value of the variable in the surrounding fluid, is a linearization coefficient and is a general nonlinear source. The analytical solution of the generic transport equation above can be written as:

| (6–4)

The fluid properties are taken from the start of the timestep.

In the calculation of forces and values for and , many fluid variables, such as density, viscosity and velocity are needed at the position of the particle. These variables are always obtained accurately by calculating the element in which the particle is traveling, calculating the computational position within the element, and using the underlying shape functions of the discretization algorithm to interpolate from the vertices to the particle position.

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