# Static and Quasi-Static Simulation

Maximum Kinetic Energy Attrition Method and Force Imbalance Method for static and quasi-static simulations.

## Introduction

Static simulation is useful for finding the equilibrium configurations for models having one or more degrees of freedom. The equilibrium is defined as that configuration where all forces and moments acting on all parts in the model at rest equal zero.

Quasi-static means that the system is being driven extremely slowly so as not to engage any transient dynamics. Mathematically, quasi-static simulation is a sequence of static simulations performed over a given duration. It is useful when your model contains time-dependent forces or motions and you are interested in the sequence of equilibrium configurations your model passes through.

MotionSolve provides two distinct methods for static and/or quasi-static simulation.

## Maximum Kinetic Energy Attrition Method (MKEAM)

This method is suitable for static simulation only. It is based on the numerical integration of a special dynamic formulation of the static problem, as outlined below:
1. The model is formulated as a dynamics problem from which all damping has been removed. The result is a conservative system whose total energy, defined as the sum of kinetic and potential energies, remains invariant with time.
2. Numerical integration is started. As integration, the kinetic energy of the system is monitored. When a peak (maximum) is detected, integration stops and backtracks as necessary to locate the peak in time, within some precision. Since the system is conservative, this instant also corresponds to a valley (minimum) for potential energy.
3. At this point, all velocities and accelerations in the model are set to zero, leading to zero kinetic energy. Then, the integration is restarted.
4. Steps 1-3 constitute one iteration. If the peak is located perfectly, then the model will already be at the equilibrium configuration. However, due to the discrete nature of the integrator, this is usually not the case. Thus, it is necessary to repeat steps 1-3 until the process converges as defined by three convergence parameters:
• Maximum Kinetic Energy Tolerance (Max K.E. Tol.) - Denotes the change in maximum kinetic energy from one iteration to the next.
• Maximum Delta q Tolerance (Max Delta q Tol.) - Denotes the maximum change in generalized coordinates from one iteration to the next.
• Maximum Iterations (Max Iterations) - Denotes the maximum number of iterations before the simulation stops.
The MKEAM method has the following characteristics:
1. It only finds the stable equilibria.
2. It is suitable for problems where the equilibrium configuration is far from the model configuration. A simple example of such a model is a spring-loaded pendulum that must make several complete revolutions to reach the equilibrium configuration.
3. It works well for contact dominated models.
4. It is only applicable to static solution.
5. It is computationally expensive and provides less accurate results compared to FIM.

## Force Imbalance Method (FIM)

This method is suitable for static and quasi-static simulation. It sets all the velocity and acceleration terms in the equations of motion to zero to obtain a system of nonlinear algebraic equations. The generalized coordinates and constraint forces are the unknowns. This system is then solved using a variant of Newton-Raphson to find the equilibrium configuration.

There are two variations of FIM available in MotionSolve:
1. FIM_S is a fixed step solution where the Jacobian is strictly re-evaluated for each Newton-Raphson iteration.
2. FIM_D is a variable step solution where the Jacobian re-evaluation frequency is a variable as well. In general, this solution is faster compared to FIM_S, but comes with potential accuracy loss.
The FIM method has following benefits:
1. It is suitable for static and quasi-static simulation.
2. It is a fairly reliable method and works well for a large class of problems.
3. Its computational performance is superior compared to the MKEAM method.
4. The accuracy of the solution is higher compared to the MKEAM method.
5. The FIM method makes no distinction among equilibriums based on stability. In other words, it is equally capable of finding both stable, unstable, and neutral equilibrium configurations. It converges to the solution closest to the initial configuration.
6. The FIM method has difficulty in cases where the equilibrium configuration is far from the model configuration and for models dominated by discontinuous effects such as contact.
Note: For quasi-static simulations, the step size you provided is adhered to by FIM_S. Too large of a step size may cause convergence difficulties.

## Usage

The MKEAM, FIM_S, and FIM_D solvers have their individual benefits and drawbacks. In certain situations, it can be beneficial to use them in combination, for example, to run a static analysis using MKEAM first, followed by a static or quasi-static analysis using FIM_S or FIM_D. A summary of each solver is given below:

MKEAM FIM_S FIM_D
Static Simulation Yes Yes Yes
Quasi-Static Simulation No Yes Yes
Basin of Attraction Global Local Local
Handles Discontinuities Well Poor Poor
Performance Slowest Average Fastest
Accuracy Least Most Average