ODE Solver

The ordinary differential equations (ODE) equation 5.579 and equation 5.591, governing the 1-DOF translation and rotation of the boundaries/volumes respectively, are solved numerically in Simerics-MP. Specifically, to calculate a boundary/volume movement and displacement for remeshing, three explicit time-marching schemes are adopted to integrate the ODE equations: Stiff, Euler and Runge-Kutta explicit solver.

Integration of One-DOF Translation Equation

Substituting equation 5.581 - equation 5.583 into equation 5.579 and grouping the explicit force terms into a single term, for brevity, we rewrite the 1-DOF translation equation of motion in the following form:

5.602

where the explicitly calculated force term is:

5.603

With given initial and boundary conditions, the displacement of the solid body is obtained by integrating equation 5.602 using explicit time-marching schemes. Over time-step, , we have the general formulations as follows:

5.604

5.605

where the weighting factors sum to unity:

5.606

With the choice of weighting factors, different schemes can be easily derived. As examples, the Euler and Runge-Kutta explicit schemes are given below:

Euler Explicit Solver (1st–Order)

With , and , we have the Euler explicit scheme in the following:

5.607

5.608

Runge-Kutta Explicit Solver

The often-used Runge-Kutta solvers are 2^nd-order and 4^th-order explicit schemes, which are presented in the following:

Second-Order Scheme

	5.609
	5.610

Fourth-Order Scheme

	5.611
	5.612

where

	5.613
	5.614
	5.615
	5.616

Stiff Solver (Explicit)

In addition to the standard Euler and Runge-Kutta schemes, Simerics-MP has developed its own stiff solver to integrate the translation 1-DOF ODE equation. It is the default method for dynamic motions of solid bodies.

Integration of One-DOF Rotation Equation

As for the translation, substituting equation 5.593 - equation 5.594 into equation 5.591 and grouping the explicit torque terms into a single term for brevity, we rewrite the 1-DOF rotation equation of motion, equation 5.591 in the following form:

5.617

where the explicitly calculated torque term is:

5.618

With given initial and boundary conditions, the angle of rotation is obtained by integrating equation 5.617 using explicit time-marching schemes. Over time-step, , we have the general formulations as follows:

	5.619
	5.620

where the weighting factors sum to unity:

5.621

With the choice of weighting factors, different numerical schemes can be easily derived. Again, the Euler and Runge-Kutta explicit schemes are given below: